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Evaluation of parameters in a two slab reactor by random noise measurements

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Evaluation of parameters in a two slab reactor by random noise measurements
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Random noise measurements
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Boynton, Allen Ross, 1936-
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vi, 126 leaves : illus. ; 28 cm.

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Neutrons ( jstor )
Noise spectra ( jstor )
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Thesis - University of Florida.
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Bibliography: leaves 123-125.
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Manuscript copy.
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Vita.

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EVALUATION OF PARAMETERS IN A TWO

SLAB REACTOR BY RANDOM

NOISE MEASUREMENTS










By
ALLEN ROSS BOYNTON


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
December, 1962














ACKNOWLEDGMENTS


The author wishes to acknowledge his gratitude for the advice and encouragement of his advisory committee. In particular, grateful acknowledgment is made to the chairman, Dr. Robert E. Uhrig, for his continued guidance and encouragement during the course of this work.

The author wishes to express his gratitude to the staff and technicians of the Department of Nuclear Engineering. J. Mueller and K. L. Fawcett constructed the early equipment, J. Wildt constructed the final equipment, and L. D. Butterfield operated the reactor for most of the runs.

The author wishes also to express his

appreciation to Dr. Herbert Kouts and V. Rajagopal of the Brookhaven National Laboratory for their loan of the random reactivity input device.












TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . . . . .

LIST OF TABLES . . . . . . .

LIST OF FIGURES . . . . . . .


Chapter
I.

II.





III.


INTRODUCTION . . . . THEORY . . . . . .

Cross-Power Spectra ......
Two Region Transfer Functions Reduction of Equations . MEASURING SYSTEM . . . .


Measuring Theory . . .
The University of Florida
Training Reactor . . .
Cross-Power Spectrum Analyzer Random Reactivity Input Device

IV. EXPERIMENTS AND RESULTS . .

Calibration . . . . .
Reactor Data . . . .

V. DISCUSSION AND CONCLUSIONS -


Page
. . ii . . iv

. . V


1 8 9
. . 8

. . 9
. . 20 . . 31 . . 38 . . 38 . . 44 . . 47
. 60 . . 65 . . 65 . . 82 . . 103


Appendix
A. CALCULATION OF TRANSIT TIME
BETWEEN SLABS . . . .

B. PHASE SHIFT AND HIGH PASS
FILTER DETAILS . . . .

C. OPERATION OF CROSS-POWER
SPECTRUM ANALYZER . . . LIST OF REFERENCES . . ...... BIOGRAPHICAL SKETCH . . . . .


S . . 108


* . . 113 S . . 119 . . 126


iii
















LIST OF TABLES


Results of Run L . . . Results of Run M . . . Results of Run P . . . Results of Run R . . . Results of Run CC . . . Results of Run Z . . . Results of Run Q. . . . Value of Transit Time
Between Slabs . . . Set Points for Phase Shifters Correction Factors for Filters


Page

. . 69

. . 70

* . 79

* . 83 . 90

* . 95

* . 101


. . 111i

* . 116

* . 118


Table


1.

2.

3.

4.

5.

6.

7.

8.


9.

10.















LIST OF FIGURES


Figure
1. Two Region System Diagram . . .

2. Normalized Amplitude Component of
R12(Lj) for Several Values of B . .

3. Phase Component of R12(LJ) for
Several Values of B and . .

4. Sequence of Operations Necessary for
Experimental Determination of Power
Spectrum . . . . . . .

5. Sequence of Operations Necessary for
Experimental Determination of CrossPower Spectrum . . . . . .

6. Location of Ion Chambers in UFTR Core 7. Data Transcribing System . . .

8. Modulator Schematic . . . . .

9. Cross-Power Spectrum Analyzer . .

10. Demodulator Schematic . . . .

11. Random Reactivity Input Device . .

12. Power Amplifier Schematic . . .

13. Normalized Amplitude Component of
G(LJ) from Runs L and M . . . .

14. Phase Component of G(LJ) from
Runs L and M . . . . . .

15. Normalized Power Spectrum of Random
Step from Run L . . . . . .


Page 10 35 36



. 42



. 42 46 48 50 52 53 61 63 71 72 73











LIST OF FIGURES (Cont'd)


Figure Page

16. Normalized Amplitude Component of
G(Lj) from Run P . . . . . 80

17. Phase Component of G(W) from Run P . 81

18. Normalized Amplitude Component of
R12(Lj) from Run R . . . . . 85

19. Phase Component of RI2(WJ) from Run R 86

20. Normalized Power Spectrum of Region 1
Output from Run R . . . . . 87

21. Normalized Amplitude Component of
RI2(GJ) from Run CC . . . . . 91
22. Phase Component of R12(W) from Run CC 92

23. Normalized Power Spectrum of Region 1
Output from Run CC . . . . . 93

24. Amplitude Component of Normalized
Cross-Power Spectrum from Run Z . . 96

25. Phase Component of Normalized CrossPower Spectrum from Run Z . . . 97

26. Normalized Power Spectrum of Region 1
Output from Run Z . . . . . 98

27. Normalized Power Spectrum of Region 1
Output from Run Q . . . . . 102












CHAPTER I


INTRODUCTION


This study is a development of an experimental method to measure the parameters peculiar to a two region reactor system. Of particular interest is the method of representing the coupling or interaction between the two regions. A transfer function approach is used.

The dynamic behavior of reactors has been

successfully analyzed from the transfer function point of view. Transfer functions have been instrumental in determining system stability and the design of control systems. The measurement of reactor transfer functions by sinusoidal oscillations, step, and ramp inputs of reactivity are standard techniques and have allowed the determination of parameters associated with the reactor kinetics equations (j) (2).*

However, the statistical nature of the fission process has led many investigators to measure dynamic reactor parameters by purely statistical means. As

Underlined numbers in parentheses refer to the list of references.











early as 1946, deHoffman (3) developed an expression for the intensity fluctuations of a chain reactor as measured by a counter with statistical response. The formulation included the effects of delayed neutrons and involved the dispersion of the number of neutrons emitted per fission. Later workers in this area were Luckow (4) who used the variance to mean ratio of the number of neutrons counted by a detector during a fixed counting time to measure the prompt neutron lifetime; Albrecht (5) who extended Luckow's analysis to include delayed neutron parameters; and Velez (6) who used the autocorrelation function of the counting rate from a reactor to measure the dynamic parameters. These investigations, while giving insight into the fundamental nature of multiplying systems and information on the influence of the statistical fluctuations in the reactor power level on such things as the precision of neutron measurements and the design of reactor control equipment, cannot furnish as much information about the complete reactor system as can the transfer function.

Fortunately, statistical considerations are related to the transfer function approach, since the power spectral density functions (or power spectra) and correlation functions form a Fourier transform pair. Correlation functions have been known to statisticians









3

for many years, yet were relatively unknown to engineers, who were analyzing frequencies and power spectra. Since the power spectrum is related to the transfer function of a linear system, engineers have been quick to use this new measuring technique.

Moore (7) (a) was among the first to relate the statistical nature of chain reactors to their transfer functions when he expressed the power spectrum of the reactor noise as a function of the square of the modulus of the transfer function. Cohn (9) used a band pass filter to measure the high frequency portion of the reactor noise spectrum which is dependent on the ratio of the effective delayed neutron fraction, to the prompt neutron lifetime, 2. This technique has become almost standard (10). The method has, however, several difficulties. First, the exact input to the system is not definable. This difficulty is attacked by Cohn (11) who assumes the reactor noise to arise from a random "noise equivalent" neutron source driving the reactor. The characteristics of this source are calculated from conventional random noise theory, and the resulting pile noise obtained through the use of the source transfer function. Bennett (12) develops the spectrum and variance of pile noise according to the formulation









of Rice (13). Recent measurements (.14) have indicated that the inherent reactor driving function spectrum deviates from white noise at low frequencies. Second, the effects of external disturbances on the final result are uncertain since the measured noise is actually a combination of reactor fluctuations and instrument noise. Third, the autocorrelation or pile noise spectrum method does not yield any direct information about the phase of the transfer function. A fourth objection to this method is that a large (greater than 5x10-4) detector efficiency is required. It may be physically impossible to place a detector in a position of high efficiency, and in the case of a counting experiment, the power level may need to be limited to avoid pile-up in the detector.

These objections are removed by crosscorrelation techniques. Balcomb (1) and Rajagopal (a) investigated the crosscorrelation of reactor outputs with random reactivity inputs. Both these investigators measured unit impulse response functions and then found the transfer function by taking the Fourier transform. It was found that the technique is-very effective in eliminating the effects of external noise and gave good results for low-level crosscorrelation signal inputs whose power was comparable to the noise power in the system.











This paper will apply this latest technique to a more complex reactor model, that of a two region reactor.

Baldwin (17) has examined the so-called "twoslab" loading of the Argonaut reactor and found that, experimentally, a single exponential stable period is observed when the system is supercritical, but that there is a tendency for independent behavior of the two slabs, resulting in so-called "flux tilting." Rod calibration experiments,indicated that the ratio of fluxes in the two fuel regions changes during the calibration procedure. It was also found that the single transfer function of a simple reactor system was inadequate to treat the two slab system.

The two (or more) region reactor has also been

of interest in connection with the phenomenon of coupled Xenon oscillations in large reactors (18).

The University of Florida Training Reactor (UFTR) is a two region coupled reactor. Each fuel region is subcritical when considered separately, but coupling between the two regions allows the system to become critical. If the multiplication factor of one region is disturbed, the other region will be disturbed only through its coupling to the first. A study of the interdependence of the outputs of the two regions, When







6

an external disturbance is given to only one of them, will reveal something about the nature of the coupling between the two regions. A representation of this interdependence is given by either the crosscorrelation function of the outputs of the two regions, or in the frequency domain, by the cross-power spectrum between

the two outputs.

The relations for the cross-power spectra between the outputs of each region are derived and are found to involve transfer functions of the individual regions. When these transfer functions are derived, they are found to depart from the conventional one region transfer functions because of the term that couples the two regions. The cross-power spectrum between the outputs of the two regions is a complex quantity and its phase part is found to be sensitive to the transit time of a disturbance between the regions. Its amplitude part as well as the phase part is found to be dependent on the value of the multiplication factor of the undisturbed region.

The cross-power spectrum between the outputs of both regions of the UFTR was measured under two conditions (a) a random stationary external input to region 1, and (b) no external input to either region.

Continuous data from two ion chambers were

amplitude modulated and stored on magnetic tape. The









7
cross-power spectra were then measured directly using band pass filters and an analog computer.

These data from the UFTR indicate that the propagation of a disturbance in one region of the reactor to the other region may adequately be described in terms of neutron waves. Also shown is the feasibility of measuring the multiplication factor of each side and the magnitude of the coupling between the two regions.













CHAPTER II


THEORY


The experiment to be analyzed consists of

measuring the cross-power spectrum of the outputs of each region of a two region reactor. These outputs are examined under two operational conditions: (a) a random stationary external input to region 1 and,

(b) no external input to either region. These crosspower spectra are then related to the transfer functions of the individual regions.

In the first section of this chapter, the

relations that involve the two coupled linear systems and their inputs and outputs are developed. Although these relations are general in nature and not restricted to reactor systems, they are developed so as to be directly applicable to the experiment. This is evidenced by the transformation of the relations from the time domain into the frequency domain so that the linear systems involved are represented by their transfer functions.









9

In the second section, these transfer functions are developed from the reactor kinetics equations that are derived for a two slab coupled reactor system.

The final section brings the cross-power spectra and transfer functions together. After certain simplifying assumptions are made, the equations are examined for the measurement of the parameters particular to the two slab reactor system.


Cross-Power Spectra

Consider a two region coupled system as shown

in Figure 1. Each region is a linear system having its own unit impulse response function hl(t) and h2(t), and outputs given by f1(t) and f2(t). The regions are coupled to one another because a certain fraction, A1, of the output of region 1 is received as an input to region 2 after a transit time, 0, and vice versa. Region 1 is also subjected to a random stationary external input, i(t).

The crosscorrelation function of the outputs

of region 1 and region 2, 012(T), is defined as (19),

T
lim 1
~12() = T-oo fl(t)f2(t+T)dt (1)


where T is a continuous displacement time independent of t. The two systems have been specified to be linear,





















Output Region 2
2(t) h 2(A)

Input H2((J) Al f1(t-)



Input
'A2f2(t'-)


Region 1 output

OuExternal Inputput hy(A)f1(t) 81

External Input


Figure 1. Two Region System Diagram









11

so the convolution integral between the input, output, and unit impulse response function is given as (19),


fa(t)
fl~t)O=


hl(A) Input(t-A)dA .


(2)


Reference to Figure 1 shows that the input to region 1 is the sum of the external input, i(t), and the fraction, A2, of the output of region 2, at a previous time corresponding to the transit time, 0, between the regions. Thus,


Input(t) = i(t) + A2f2(t- )


Equation (2) is now written as


fl(t) hl(A) i(t- ) + A2f2(t--8) dA,
(4)

which may be substituted into the definition of the crosscorrelation function between the outputs of region 1 and region 2, equation (1), to give


lim 1 T f D

--T -o


hl( ) i(t-x )


+ A2f2(t-A- ) dif2(t+T)dt.


12(T ) =









12

By reversing the order of integration, equation (5) is put into a more desirable form,


12(T)
12 (T) =fC h


T ()T-4 T


(t- )f2(t+T)dtdA


(6)


Gor T + A hl( A)T-i Tf
a _nT


E2 (t-I-O)f2(t+T)dtdA .


With this grouping, the inner integral in the first term is recognized as the crosecorrelation function between the external input and the output of region 2, while the inner integral in the second term is the autocorrelation function of the region 2 output. Equation (6) may now be written in the compact form,


OD
12(T) =


+ A2f


h1(A)i2(T+A)dA



h1()422 (T +A+O)dA.
00
h1 A)(22(T+A+ G)d A.


From equation (1), it is noticed that f2(t+) may also be expressed in terms of the convolution integral between the input, output, and unit impulse response function by,


(7)












h2(A) Input(t-A)dA.


In region 2 the only input is that fraction, Al, of the output of region 1 which reaches region 2 after the transit time, between the regions. Thus,


Input(t) = Alf 1 (t-9).


Equation (8) is written as


GD 2(t) = Al-oo


h2(A)f1(t-A-O)dA ,


(10)


which, when substituted into equation (1), gives,


12 T D 1(t)Al
r a OD


h2( 1) f(t+T-A-0 )d)dt.

(11)


Once again, inversion of the order of integration will yield a more convenient grouping,


q12(fa h 12(T)" 1 h
-oo


O2Tffl(t)fl(t+)dtd
2 Tl ooypb 01 fltfl(t+T-)(-O)dtd ,
T


(12)


OD
f2 (t) =f -0


(8)









14

in which the inner integral is recognized as the autocorrelation function of the output of region 1. Equation (12) can now be written in a more compact form, and with equation (7) relates the crosscorrelation of the outputs of region 1 and region 2 with their unit impulse response functions, their autocorrelation functions, and the crosecorrelation of the external input and the output of region 2.

CD

912(T) = A1f h2C)11(r-A-O)d (13) "OD


An obvious simplification is the case of no

external input to the two region system. Equation (13) is unchanged, but equation (7) becomes,



S12(T) = A2 hl(A)122(T++0 )dA. (14)
_O


These relations may now be transformed into the frequency domain by taking the Fourier transform of both sides of both equations. As previously noted this is done to relate the theory directly with the experiment. Many investigators (1&) (17) choose to measure crosscorrelation and autocorrelation functions (rather than


h I









15
the cross-power and power spectra) of the outputs and obtain unit impulse response functions. The Fourier transformation of the unit impulse response function, which gives the transfer function, is performed on a digital computer. If the cross-power and power spectra can be measured directly, a computer operation may be eliminated. The equipment available to the author also suggested the direct measurement of the cross-power and power spectra. Equations (7) and (13) must then be examined in the frequency domain.

The Fourier transform of equation (7) is taken by multiplying both sides of the equation by 1 ,juj where j = V-1 LJ] is the angular frequency (radians per second), and integrating with respect to T Equation (7) then becomes,

GD

-oo l( j Lid d

OD D
-a 1-oo



(15)
A2 aD Co
27T JU~?612







16
The crosscorrelation function and the cross-power spectrum are a Fourier transform pair defined by (20),


() 12 ( T ) e-JWT dT


and (16)


(17)


12 (U/)eJWT .


The term on the left side of equation (15) is then by definition the cross-power spectrum between the outputs of region
1 and region 2, 12 (J). In the first double integral on the right side of equation (15), the change of variable,


x= T + ,


(18)


is made so that the integral becomes


100
2 7 an


(19)


In this double integral, a separation of variables puts all terms involving x under one integral and all terms involving A under another. Equation (19) becomes a product of two single integrals,


[f00 O
.-co


eW hl(A)dAl D 22(x) e-JWxdx .

(20)


12L(W) =


12-O


aD '-JWl(x-))dxf
-O


hl 1)O2(x)d) .









17

The unit impulse response function and the transfer function are also a Fourier transform pair given by (19),

OD
h(t) = i-f H() jt dLJ and (21)


Soo-jt
H(L)) = h(t)e d t (22)




where H(LJ) is the system transfer function. The first factor in equation (20) is then recognized as the complex
*
conjugate of the transfer function of region 1, H1(). From equation (16), the second factor is identified as the cross-power spectrum between the external input and the output of region 2. In the second and last term on,

the right of equation (15) the change of variable y T + A + 0 (23)


allows a separation of variables as in the first integral which will give a product of two single integrals,



A2eJO hl(A) dL) 1 22(y) e-jYdy
-OD 0D
(24)

Here, the first factor in brackets is the complex comjugate








of the transfer function in region 1, and the second factor in brackets is the power spectrum of the region 2 output. Thus equation (15) may finally be written,


12(1) = (LJ) 12(LJ) + A26i 1 (L) 22


(25)


Equation (13) may be manipulated in the same manner as was equation (7). Taking the Fourier transform of both sides of equation (13) gives,


O12 dT

2io 012 7( T) 63-JUT7 dr


A 00 -jUT oa-.
aA -an~d
OQD
(26)

A change of variable on the right-hand side of equation
(26) to,


z = T -A -0 will give, after a separation of variables,


J1
27rf


(27)


S12(T)eJL T dT


= Al1io IOD


h2(A)e-j i d ( JL .
2 11() (2)dz
(28)









The left-hand side of equation (28) is recognized as the cross-power spectrum between the outputs of region 1 and region 2, 12 (); the first factor in brackets on the right-hand side as the transfer function of region 2, H2(}); and the second factor in brackets on the righthand side as the power spectrum of the region 1 output,

II(L). Equation (28) may now be written with equation
(25) to give the relation of the cross-power spectrum between the outputs of region I and region 2 to their transfer functions, their power spectra, and the crosspower spectrum between the external input and the output of region 2.


S12(L) = A H2 (W)11(w). (29) If there is no external input, equations (25) and (29) become,


S12(L) = A2e O1 *()f22(L)) and (30)


C12(W)) = Ale Y2(L) 11()) (31)


It is noted that different symbols are used for the transfer functions in the set of equations (25) and (29) than in the set of equations (30) and (31). The reasons for so symbolizing these transfer functions are given in the next section.









20

Relations may also be established for the

cross-power spectra between the external input and the outputs of the two regions. The relations are derived in the same manner as were those for 12 (), and are,



Si(L)= H(L) 1ii(J) + A2eJO H( ) 12(W) (32)


and


t12() = AJ-jL18 a2(L1) (33)


Two Region Transfer Functions

The two transfer functions, Hl1(W), and H2(L)), are now derived from the reactor kinetics equations for a coupled reactor system. Each region is regarded as a subcritical reactor with a neutron leakage interaction from the other slab. Within each region, the theory is space independent and a one group bare reactor model is assumed. Cohn (21) has commented that the one group bare reactor model is still used for practically all kinetics work, even though it has been discarded as impossibly crude for most statics calculations. His work shows that for most reflected reactors, the kinetic behavior corresponds to that of a bare reactor with the' same 1/v lifetime.











The kinetics equations for region 1 may then be written (18),


dn1 [kl(l-)-l] n1
+
dt ,


icil + S1 + E (34)
i


dcil dt


/3iklnl )icil
.l


(35)


where, n1 is the neutron density in region 1

kI is the effective multiplication factor
for region 1

9 is the overall prompt neutron lifetime
in the system

i is the fraction of fission neutrons in the
i-th group of delayed neutrons and


i

Ai is the decay constant of precursor of
i-th group of delayed neutrons

cil is the concentration of precursors of
i-th group of delayed neutrons in region 1
S1 is the neutron density from external source
neutrons in region 1.

The term E is the interaction term caused by the leakage of neutrons out from region 2. This term may be represented by some fraction, A2, of the neutron density in region 2 at some previous time corresponding to the


and











transit time, between regions. Thus


S= A2n2(t -9) (36)



Expanding this expression in a Taylor's series about t,


A2n2(t) A20 dn2(t) A202 d2n2(t)
+ = - (37) R dt 2 dt2


Baldwin (Q) estimates 0 by assuming it to be

determined by the mean thermal neutron velocity. In the UFTR, which has a 30 centimeter region of graphite between the fuel regions, this assumption would indicate a value of approximately 2 x 10-4 seconds for Such a small value would make 6 relatively unimportant in kinetics except for very short periods and unimportant in the transfer function except at very high frequencies. A more realistic assumption is that a disturbance in the neutron density in one fuel slab travels across to the opposite region with the velocity of a neutron wave. The velocity of a neutron wave is frequency dependent

but at frequencies from 1 to 100 radians per second is approximately constant at 1.4 x 104 centimeters per second in graphite (2_2). This gives a lag time, 6 in the UFTR of about 2 x 10-3 seconds, a value that makes it









23

desirable to retain at least the first few terms of the expansion of equation (37).

The kinetics equations for region 1 are then,



dnl [kl(l- )-l 1 n1+
+ Aicil + 1
dt i

(38)
A2n2 A2 dn A2e2 d2n2 dt 2 dt2


and

dcil ik n1
d = cil. (39)
dtii


A similar set of equations exists for region 2. In the experiment, a random reactivity input was given to region 1 only, while k2 remained constant. This fluctuation in the multiplication factor in region 1 will give rise to fluctuations in the neutron density and delayed neutron precursor concentrations in both regions. There may also be fluctuations in the external neutron source. It is convenient then to linearize these quantities as follows:


n1 = nl0 + 5nI n2 = n20 + n2

kI = kl0 + 5 kI c 2 = Ci20 + 65c12 (40)

ci2 = cil0+ 5cil Sl = s10 6Sl









24
When these definitions are substituted into the equations of region 1, (38) and (39),


d-(nl0+nl) =
dt


[(kO+6kl)(1-,/)-1] (n10+6nl)


+ i(c110+ 6 Ci1) +S1o+ (41)
i
A2 F[~~j (5d2 d21
S(n20 2)- (n20+6n2)~ t(n20+6n2)+


and (42)


d (c ) = dt ilO l1


- Ai(Cjlo+6c1).


The sums of the steady state portions of these equations are equal to zero, and the equations are linearized by setting the product,6 k6n, equal to zero since both quantities will be small. The equations for region 1 are then


d(ni (1-fi)nlO6k1
S 2-- + dt ,]


[kl(1 )-1 ]n1 c+ i
+1


+6sl + [ 5n2 -


2 0 2d 26n2 d6n2 2 d262
d + 2 d-


- .. and


d63cil $inl0k +1 $ikl0( n1 A 6c 5 dt 9 c


(43)


(44)


;i (klo+ 6 kl) (no+ 6 n,)









When the Laplace transform, using zero initial conditions, is taken of both sides of both equations,

A (1-J9)n1AK [kl(-J)-1 AN S + + + A iACil
(45)
+A j1 + [AN2 OAN2 + 2 2

and

SAC1 iinA klAN iACil, (46)


where s is the Laplace transform variable, the capital letters denote the Laplace transform of the 6 quantities (i.e., AN1 is the Laplace transform of 6 nl), and the small letters are now understood to represent steady state values. Equation (46) may be solved for A Cil*. When this is done and A Cil substituted into equation
(45) and terms collected,




AxI N-l + 2 AN2
=<1 (1 )nd1 + -$1+ 2 2











If there is no disturbance of the multiplication factor in region 1, then equation (47) becomes,




(48)

= 1- a + 2 ...BN22.
9 2


Since k2 is assumed to remain constant, the kinetics equations for region 2, after manipulations similar to the above, reduce to,


k 2(1- 9)-l k2 A

s +1 2

(49)
Al [1_-s + 02 AN1+A,2.



The transfer function is defined as the ratio of the system output to the system input when both are expressed in Laplace notation. The inputs and outputs used to derive the transfer functions must correspond to the physical situation. It is not uncommon to speak of two different transfer functions in connection with subcritical reactors; a source transfer function and a









27

reactivity transfer function. In the former, the input is associated with fluctuations in the external neutron source, while in the latter, fluctuations in the multiplication factor or reactivity are considered the input. In the two slab reactor system, each slab is considered as a subcritical system, so the same logic should apply. The transfer function when an external reactivity input is applied to a slab is a reactivity transfer function, while the transfer function when there is no external input is a source transfer function; the leakage of neutrons from the opposite slab is considered as an external neutron source.

This source transfer function is easily recognized in the case of an external random input to slab 1 while the reactivity of slab 2 is held constant. The fluctuations in the multiplication factor of slab 1 cause fluctuations in the neutron population in that slab. A certain fraction, AI, of this fluctuating population leaks out of region 1 and may be considered an external neutron source for region 2. This fluctuating neutron source then causes the neutron population of region 2 to fluctuate. A source transfer function for region 2 may then be formed by the ratio of the output of slab 2 to the output of slab 1 when both are in Laplace notation.









28

When there is no external reactivity input to

region 1, fluctuations still exist in the neutron population of that slab. These are the so-called selffluctuations or reactor noise. The statistical nature of the fission process gives rise to these fluctuations in reactor power levels even when the reactor is operated at steady state. It has been shown (1) that the power spectrum of reactor (single region) self-fluctuations is related to the transfer function, H(LJ), through,


11(/) = 0 + O' H(Lj) 2, (50)


where 0 and 0' are constants. The expression has two terms, the first being white noise of the detector and the second being related to the kinetics parameters. The transfer function used is the one based on reactivity, although the exact input to the system is not definable

(16)

Both slabs will exhibit these self-fluctuations, but crosscorrelation should measure the effect of the self-fluctuations of one region on the neutron density in the other region. The transfer functions to describe this situation would be source transfer functions, the self-fluctuations in one side acting as the input to the other side.







29
When a random reactivity input is given to region 1, the desired transfer function of that region is given by

dN1/n1
H1(L/) ANK/n, (51)



where A N1/n1 is the normalized output, and A K1 is the input. This ratio is obtained from equation (47) after AN2 is eliminated by substitution from equation (49). With the assumption of no external neutron source, solving equation (49) for AN2 gives,


A1 + 2s2 AN
A [1 s + 2 ...
N = (52) k2(l- )-1 k2 A 1s+ 1i


Substitution of this expression for AN2 into equation
(47) gives, after some algebraic manipulations and setting s = JL/ ande = 1 j9j$ L )202
2

H1(LJ) =
(53)

(A1A2 -2-JL kl1- )1 + k 1 9 1S ii l
JIl_ +A_ l + 2)i

T 2 L+A







30
The desired transfer function for region 2 when a random variation of reactivity is inserted in region 1 is a source-based transfer function and is given by

AN2
H2(LJ) = A N(54) A NlA1 e -JLIJO


where A N2 is the output of region 2 and A NA1 e -JLJO is its input. H2(L/) may be found from equation (49) directly. Thus from equation (49),

1
H2(W/) = (55) k2(1-, )-1 k2 A d. L)- e2 2 JL+A


When there is no external reactivity input to the two slab system, both transfer functions are thought of as source transfer functions in which the leakage from one side is considered a source to the other side. The transfer functions must then be formed by


AN1
Y1 (L/) = ___ __AN2A2e(56)
N2
ANAle-LJO








31

Y1(LJ) may be formed directly from equation (48),


1/2


kl(1i i-1 k


(57)


while Y2(LJ) is formed directly from equation (49), as was 2 (LJ)'


Y2 (L/) =


(58)


1i

k2(1-)8)-1 k2 __ i j -9 + k---I


Reduction of Equations
If only the higher frequencies ( L > 1) are
considered so that the effects of delayed neutrons need not be considered, and the parameter B is defined as


1 kl(1 -)
B1 =


then the four transfer functions are reduced to the following:


Y1 (L j) =


(59)










H1 (u) =


(1 -J ),


B + j12 11 &, -2j)JG
B2 + JCJ


1/4
H2(&/) = Y2(LA) = ,/_ ,
B2 + jLJ


Y1 (J) = *
B1 + JLJ


These expressions
(30), (31), (32),
input is given to


S12(W) =


are now used in equations (25), (29), and (33). When an external reactivity region 1,


<.-[ <,12 2 + A2 e L ( 22 }

AA2 2 2J (63) B1 J A
B2 )


4 12() =[A e ()
B2 + jL1


=(1-)/8 i( ,) + A 2e

B + J [A1A/2 e -2JLJ
+ l- (6s) B2 + J


and


, (60o)






(61)


(62)


(64)










and


i2(W) =[A (66) B2 + jL/


The transit time between the regions, 0 has been considered independent of direction. Further simplification is effected if the slabs are assumed identical except for their multiplication factors which may be different. This assumption results in the condition A1 = A2 = A, (67)


and equation (64) becomes,


[A/] ~i 11w
12() = (68) B2 + jLj

The expression for 12(LJ) when there is no external reactivity input to region 1 is identical to equation (68). The power spectrum, (11(Lj), is a real quantity but the cross-power spectrum is a complex quantity which may be expressed in terms of its phase and amplitude components. If the complex quantity R12(LJ) is defined as

) 12 ( I6
R12( j) (69) 11 ())









34

then it may be expressed in terms of its amplitude,/R12/' and its phase, p(R12), parts, as


A 2B22 + 2WJ2
-2 = (70)
B22 2

and,

F-LJcosOLJ sysinOLJ)
p(R12) = tan-1 L (71) B2cosOL) L)sin IJ

Equation (70) shows that the amplitude portion of RI2 is independent of e and that the shape of /R12 /depends only on the value of the multiplication constant of the region not disturbed by the random reactivity input. The phase portion of R12('), however, is strongly dependent on the value of 0 particularly as the product OW becomes large. Figures 2 and 3 show plots of the amplitude and phase portions of RI2(LJ) versus frequency as calculated from equations (70) and (71) for a reactor of the geometrical arrangement of the UFTR. The normalized amplitude plot (Figure 2) is made for several values of B: 100, 200, 500, and 1000. These values correspond to multiplication factors of 0.97, 0.94, 0.85, and 0.70, respectively, for a neutron lifetime of 3x10-4 seconds. As noted before, this plot is independent of














100





S90a,


9B=100 200 500 1000
'44
0

-4
I,
41i
P4 80

'4


N
-4 70
70




I I, I I I ...I .. I .... ......... I
2



2.0 5.0 10 20 50 100 200 500 Frequency, cycles per* second Figure 2. Normalized Amplitude Component of R12 (j) for Several Values of B















-60 1 100 Neutron Wave w 2 200 Neutron Wave
3 500 Neutron Wave 6
-90- 4 1000 Neutr n Wave
5 500 2x10- Seconds
S6 500 Zero 5
-120 1 2 3 4 C4J
S-150

0
w -180
P-4

-210


- -240


-270
1 1 I ... .. I I I I
1.0 2.0 5.0 10 20 50 100 200 Frequency, cycles per second

Figure 3. Phase Component of R12() for Several Values of B and









the transit time between slabs. The phase plot (Figure 3) indicates the dependence of the complex quantity R12(W) on this transit time, e For B=500, the phase plots for three different values of & are shown: first, for e equal to zero; second, for 0 corresponding to the thermal neutron velocity; and third, for e corresponding to the frequency dependent thermal neutron wave velocity. The value of the transit time between slabs in the UFTR when a disturbance on one side is assumed to travel to the other side with the velocity of a thermal neutron wave is calculated in Appendix A. The phase part of R12(LJ) is also given in Figure 3 for B values of 100, 200, and 1000, with 0 corresponding to the thermal neutron wave velocity. It is seen that for a particular value of B, such as 500, the phase portion of the quantity RI2(LJ) provides a rather sensitive test for the validity of assumptions about the nature of 0 The measurement of R12(LJ) would also allow the determination of the parameter, B, which yields information about the multiplication factor of the non-disturbed region. In a balanced reactor system such as the UFTR, in which the multiplication factors of the two regions 47
are equal, Baldwin (__) has shown that A = B (72) Thus a determination of B yields information about the magnitude of the coupling of the two regions.














CHAPTER III


MEASURING SYSTEM

In this chapter are discussed methods and

equipment used to perform the direct measurement of the cross-power spectrum of the outputs of the two regions of the UFTR. The first section contains a development of the relations necessary for the evaluation of the real and imaginary parts of the cross-power spectrum using band pass filters. These relations indicate the sequence of operations that must be executed. The second section contains a description of the University of Florida Training Reactor (UFTR). Since the measurements were made on this reactor, its pertinent features and dimensions are given. The third section examines in detail each component used to collect, store, and process data. The final section describes the random reactivity input device.


Measuring Theory

Most investigators have used autocorrelation and crosscorrelation measurements to obtain the transfer functions of single region reactor systems. This method









39

lends itself well to digital processing of data. Results of auto or crosscorrelation measurements are in terms of the unit impulse response function, while the quantity of interest is the transfer function. The power and cross-power spectra, however, may be measured directly and, as seen from the previous chapter, relate directly to the system transfer functions.

The theory of the measurement of power and

cross-power spectra is outlined below. The relation of the crosscorrelation function to the cross-power spectrum is,



(12(T) = 12(Li) e JL J d LJ (17)
-an)


The definition of the crosacorrelation function,


T
(T lm) f fl (t)f2 (t+T)dt, (1)
T-12 T-oo 2 1 2
-T


may be substituted into (17) to give,


T co
lim 1 f(t) f(t+T)dt 12 (j) eJ dlJ T-* oo2T 2
-T (73)









40
Here, fl(t) and f2(t) are voltages representing the neutron density in each slab. The integration over t cannot, of course, be performed over infinite time or over negative times, so a finite average is used, resulting in the approximation,


f(t)f2(t+TCo)dt fl1M 2 (t+r)dt =


dLJ .


(74)


The first operation in the data processing is to pass both signals through identical band pass filters, both of which are centered at the same frequency, J.1' with the same frequency band width, AL1. Equation (74) then becomes,


T
o fl (t) f2 (t+ 7) 1


dt = 12(u1)e 1 A1L

(75)


The power spectrum measurement, when fl (t)=f2(t), is considered first; then,


T
O11
TIJ fl(t) f f (t+T) I1dt = 1lwl)eil Awl
(76)


If 7' is set equal to zero, equation (76) is then written,


1


12(u) JLT











T 2

1 I


dt = t11(j)Awi1 I


(77)


where the left-hand side is the mean square value of


* This expression is solved for 11I(1)


I f1(t)


11 (L/1) =


(78)


TALJ.


Figure 4 indicates the sequence of operations necessary to solve equation (78) experimentally.
The cross-power spectrum is a complex quantity and may be expressed in terms of its real parts, Rel2, and its imaginary parts, Iml2. Use is also made of Euler's relation,


e j LIT


= coaLJ1T + jsinLJT


(79)


to reduce equation (75) to


dt =


r (80)
Re 2cos WlT-Iml2sinLJlT+j(ImcosLWT+Re sinLJ ) Aw1.


f1 L(t)]


1
fo


1 (t) I i 2 (t*F-) I












(11 (() TA(1J


fl (t)


set at
GU1


Figure 4.


Sequence of Operations Necessary for Experimental Determination of Power Spectrum


Re z2(LJ1) TALJ1


set at
Iml2 (l) T AL
121

f(t) Band pass 90 Multiplier Integratorf2 filter Phase Shifter

set at
Li


Figure 5. Sequence of Operations Necessary for Experimental
Determination of Cross-Power Spectrum









When T is set equal to zero, equation (80) reduces to,

T
f1 fl(t) f2(t) dt = Rel2 + jIml2 A l.
011
(81)

The integral on the left side of this equation is a real quantity so that when the real parts are equated,
T
Sfl(t) U1f2(t) 1 dt

Rel2 (J1) = (82) TA /I

T 6T1
Returning to equation (80), one sets T equal to 7 2W1
Since both fl(t) and f2(t) have been put through identical band pass filters and are both essentially at the same frequency, /Jl'



2t + ) = f2(t + 90) (83)
1 1

Thus equation (80) becomes,


1 fl(t) f2(t + 900) dt = -Iml2+jRel2 AJl"
0 1 i1
(84)












Once again, the integral on the left side of equation

(84) is recognized to be a real quantity, so that when the real parts are equated,
T
-f (t) f2(t + 900) 1 dt

Iml2 ((1) (85) TAW1


Equations (82) and (85) may be solved experimentally as shown in Figure 5.


The University of Florida Training Reactor

The UFTR is a 10kw, heterogeneous, light water and graphite moderated, graphite reflected, thermal reactor of the Argonaut type (23). It may be fueled with either 20 or 93 per cent enriched uranium-aluminum fuel elements. The fuel is contained in thin flat plates which are assembled in bundles of eleven plates. The reactor core contains 24 bundles of fuel plates placed in six water-filled aluminum boxes surrounded by reactor grade graphite. Four cadmium control blades, protected by magnesium shrouds, move between the fuel boxes. Twelve inches of graphite separate the two rows of fuel boxes. Aluminum headers connect to the bottom of the fuel boxes and supply the light water used for moderating and cooling. The tops of the boxes are











connected by aluminum overflow and vent pipes. This water, overflowing from the fuel boxes, flows by gravity through a cooling coil and then into a storage tank. The water is pumped from the storage tank to the fuel boxes. The storage tank prevents any sudden temperature changes in the coolant, its capacity being about six times that of the reactor system.

Air spaces 8 inches wide exist between the graphite reflector and the inside of the biological shield on the north and south sides of the reactor. The ion chambers used to detect the output of each region of the reactor were positioned in these spaces against the graphite, as shown in Figure 6. A graphite layer of 18 inches separated the chambers from the fuel boxes.

The UFTR is provided with six horizontal beam ports. These ports have a 4 inch diameter next to the core and increase to a 6 inch diameter through the rest of the shield. The core graphite stacking adjacent to the south beam port is fitted with a removable graphite plug 18 inches long by 2 inches diameter. When this plug is removed, the south beam port extends to the side of the south fuel box, at about the horizontal center plane of the core. The random reactivity device was inserted into the reactor through this facility.











ce 0 ,Biological Shields




North Chamber



.A
p. c


Figure 6. Location of Ion Chambers in UFTR Core











Cross-Power Spectrum Analyzer

The measurement of the cross-power spectrum between the outputs of the two regions of the UFTR involved both transcribing output signals on magnetic tape and subsequently processing these signals in the manner indicated in Figure 5. Figure 7 is a flow diagram of the equipment used to collect and store output signals of each region of the reactor on magnetic tape.

Fluctuations in the neutron intensity of each region were converted into current fluctuations by the two Westinghouse 6377 compensated ion chambers placed in the core. Two battery packs supplied 600 volts to each chamber. No compensating voltage was used. Each chamber was positioned outside one of the fuel slabs as shown in Figure 6. Attempts to avoid ground loops in the system included wrapping the ion chambers in transformer cloth and covering all connectors with electrical tape, since the chambers were in contact with the aluminum and graphite in the core. These precautions were found to eliminate from the measurements large amounts of 60 cycle interference.

About 15 feet of coaxial cable were used to connect the ion chambers with the Keithley Model 410 Micro-microammeters located immediately outside the UFTR














Ion
Chambers


Gain of
50


0.1 to 1000 cps


Gain of
50


Tape


Recorder


Figure 7. Data Transcribing System











shield. The micro-microammeters converted the signals from current fluctuations to voltage fluctuations which were then amplified on an Applied Dynamics Analog Computer, located in the reactor control room, so that the ac portion of each signal was about 10 volts peak to peak. The analog computer was also used to buck-out the dc produced by the steady state power level of the reactor. Two Krohn-Hite Ultra-low frequency band pass filters were used as high and low pass filters. The low cut-off frequencies were set at 0.1 cycle per second to remove all dc and any slow drifts in reactor power. Frequencies above 1000 cycles per second were of no interest in this experiment and were removed by setting the high cut-off frequencies at 1000 cycles per second. The signals were then amplitude modulated on an 80,000 cycles per second carrier. Amplitude modulation recording and playback is subject to considerable fluctuation in over-all gain (24). Because only approximate spectra were required in order to determine the mechanism of the lag between regions and to test the feasibility of the measurement of the multiplication factors of the regions, AM recording was employed. AM recording and detection equipment was also available. A schematic of the modulator is shown in Figure 8.




















To Tape Recorder


PT5 0.0015 f


0.008
I-"


Coil is #28 Wire Wound in a Double Layer on 1" Dia. 3" Long Form Tuned to Approx.
80 kc


Figure 8. Modulator Schematic


Noise Signal









51

The modulated signals were then recorded on two channels of an Ampex 500 magnetic tape recorder. A tape speed of 60 inches per second was used for both recording and playback. Modulation was necessary because of the frequency response of the tape recorder.

Figure 9 presents a flow diagram of the equipment used to measure the real and imaginary parts of the cross-power spectrum between the outputs of the two regions of the UFTR. The signals from the tape recorder were first demodulated, using the circuit shown in Figure 10. The demodulated signals were then amplified by a factor of about 25 on the analog computer in order to have signals of about 10 volts peak to peak to the band pass filters. The band pass filters were set for the same frequency and band width. A "zero" band width was used in which the upper and lower cut-off frequencies were set at the same value. The band pass filters allow frequencies outside the band width to pass through

even though these "outside" frequencies are uniformly and greatly attenuated. When the upper and lower cutoff frequencies are equal to fl, the half-power, or three db down, points are at 0.77f1 and 1.30fl. Thus, if a constant percentage or a "zero" band width is used, any error due to the "outside" frequencies is normalized. The larger the percentage band width used, the less the

















Phase Changers


set at fl


Gain
of
-- M


R Gain
SMof


High Pass
Filters


Real Part


Ln










Imaginary Part
-0
Gain of
P


lo0M1


10MI


Cross-Power Sectrum Analyzer


I


Figure 9.























From Tape Recorder 0.l f


-- R.


6AL5


To
Analyzer


0.1/ f


Demod #1 Demod #2


R1 6.8KQ R1 5.6K


R2 8.2K R2 10K


Figure 10. Demodulator Schematic









54

insertion losses in the filters become; however, the resolution also becomes poorer.

The two signals, fl(t)] and f2(t) were
1 1

then given a phase lag in a low pass filter and f2(t) I was also given a phase lead in a high pass

1

filter such that there was a +90 degree shift between it and the lagging fl(t)] and f2(t)J The phase

1 1

shifting networks were more complex than the simple filters indicated in Figure 9 because each filter fed into either a one megohm or a one-tenth megohm input resistor of an operational amplifier on the analog computer. Since the amplifier summing junctions are essentially at ground potential, either a one megohm or a one-tenth megohm resistor was in parallel with the capacitor, C, in the low pass filters and in parallel with the resistor, R, in the high pass filter. It is easily shown (25) that the amplitude, /A/, and the phase, p. portions of the transfer functions for these filters are given by, for the low pass filters,

K
fAIL + (86) (R + K)2 + 2y2(j2C2











and



PL = tan-1 [ RKUC (87) [-R + K


and for the high pass filter,


A/ RKL)C (88) (R + K)2 + R2K2W2C2


and


= tan-1 R + (89) 1RK ~C


where K is the value of the input resistor to the operational amplifier. Since the signals were passed through the band pass filters and were at essentially a single frequency, a convenient R and C were selected to give the desired 90 degree phase shift at each frequency set point. It was also found to be convenient to select a RC combination such that the amplitude, /A/, of the high pass and the low pass filters were equal at a given frequency. This was accomplished when


1 1
R = 2 c (90) WC 27Tfc









56

Precision 0.1% polystyrene capacitors were

available in values of 0.01, 0.1, and 1.0 microfarads and were used with precision 1% resistors and ten turn t3% 100KQ potentiometers to set the C and R values of the filters. The capacitor and resistor values used are given in Appendix B.

It is noted that when an R and C are selected such that equation (90) is satisfied, the values of PL and PH are such that there is a +90 degree shift between them and the attenuation is the same through both filters. This attenuation is, however, frequency dependent. The calculation of correction factors for this attenuation is straightforward and also given in Appendix B.

The three signals f(t, f 2(t) and f2(t+90o)
1 1 were then amplified by a factor of from about 20 to about 400 on the analog computer. This amplification was necessary in order to have signals of about 200 volts peak to peak for inputs into the Model 160 electronic multipliers on the analog computer. Large input signals are necessary to minimize the error that these diode network multipliers introduce for low input voltage.

*These symbols are now taken to represent the
signals after they have passed through the phase shifting networks.









Before multiplication, however, the signals were passed through high pass filters to remove any dc components and low frequency drifts introduced because of the high gain of the system. The nominal 1.0 microfarad capacitors and 10 megohm resistors used were actually 1.09 microfarad and 9.9 megohms, but were matched to within

1 per cent. The outputs of these filters were fed into the one megohm input resistors of operational amplifiers. Here again, attenuation and phase shift were introduced, especially at the lower frequencies. The phase shift was identical for all three signals and was ignored. The amplitude attenuation is calculated in Appendix B.

Since both the phase changer and high pass filter attenuations were the same for fl(t) 1 f2(t) and


f2(t+900)] the same correction was valid for all
J
U1

three. These corrections were squared, however, because the attenuations occurred before the multiplications. The separate and combined correction factors that were applied to the real and imaginary parts of the crosspower spectrum are given in Appendix B.

Multiplication was performed after the signals were passed through the high pass filters and the











products f1(t)] f 2 (t)] and fl (t)] 2(t+900)]



obtained. These products were then integrated and the integrals related to the real and imaginary parts of the cross-power spectrum. If the gain of the operational amplifiers between the band pass filters and the multipliers was M, the gain of the integrators (1/RC) was P, the voltages on the integrators were VR for the real and V for the imaginary parts for an integration time of T seconds, and the set frequency and frequency band width were f, and A fl' respectively, then the real and imaginary parts of (12 (L) were given by,


VR(CF)
Rel2 (f1) = T A (91)



and
V (CF)

Iml2(fl) = (M)2 (92) T AfpM .2P



where CF was the total correction factor for the amplitude attenuation of the phase shifters and high pass filters.

The measurement of the quantity R12(LJ) as defined by equation (69) requires that both the









59

cross-power spectrum between both outputs and the power spectrum of the output of the externally excited region be measured. The real and imaginary parts of the crosspower spectrum were measured by use of equations (91) and (92). The power spectrum of region 1 was measured with the same computer setup as was the cross-power spectrum except that fl(t) only was used instead of f1(t) and f2(t). Since the power spectrum was a real quantity, there was only a voltage, V, on the integrator giving the real parts. For the same computer setup as the cross-power spectrum, the power spectrum was given by,


v(cF)
11 (f) = (93) T A fl g)2p



Thus the real and imaginary parts of R12(fl1) were given by,



Re [ R12(fl)] VR (94) R R21) = (94)
V

and

Im R12 (f1) (95)
V









60

Thus the amplitude correction factors for the

low and high pass filters were unnecessary in the calculation of R12(f). They were employed, however, when either the cross-power or the power spectra were measured.


Random Reactivity Input Device

The reactivity input system consisted of (a) a

drive assembly of a coil in an electromagnet with spring return, (b) an absorber in the form of a piston moving in a cylinder, (c) a power amplifier for the moving coil, and (d) a random input signal.

The drive and piston assembly was constructed by V. Rajagopal and is described in his doctoral dissertation (16). Figure.11 illustrates this assembly. The drive assembly was a coil moving in the air gap of an electromagnet. The flux in the air gap was maintained by a steady current in the field coil. A 28 volt 3 amp power supply was used for the electromagnet. The force exerted by the moving coil was controlled by the current supplied from the power transistor in the power amplifier. Mechanical coupling from the moving coil to the cadmium covered piston was through a steel wire encased in a square slot between two blocks of graphite. The opposite end of the piston was connected














electromagnet tension return
cadmium rn steel heath outside spring
transmission wire cylinder






/41'


cadmium covered moving 5' long graphite movable piston coil casing for wire


Figure 11. Random Reactivity Input Device










to a tension spring mounted inside a graphite block. The graphite blocks and the cylinder of the absorber element were rigidly connected to the electromagnet.

A schematic diagram of the amplifier that drove the moving coil is given in Figure 12. The oscillator system was found to follow sinusoidal inputs in the frequency range 0.1 to 50 cycles per second at the maximum length of stroke (about 3/8 inch) while for higher frequencies, up to 100 cycles per second, about half stroke could be obtained. When the device was mounted in the south beam port, full stroke held about 0.015 per cent reactivity.

Two types of random noise generators were used as inputs to the power amplifier. The first was a commercial noise generator. The Elgenco, Inc., Low Frequency Noise Generator gave a 12 volt rms noise signal whose power spectrum was uniform from 0 to 35 cycles per second. The dc level of this signal was less than 50 millivolts.

A second random signal was constructed with a radioactive sample, GM counter, and a count rate meter. The signal was taken from the output of the flip-flop section of the count rate meter. This output was a randomly switched square wave whose mean period varied as the reciprocal of the count rate. This signal is












-25v


Power Amplifier Schematic


Figure 12.











also called a "random telegraph signal" and its power spectrum is well known as (20)


i4a2g ] 1 (96)

7ir7 U 2 + 4g2I


where g is the average number of sign changes per second of a square wave that varies between +a and -a. The power spectrum of this signal was then made white out to the desired frequency by a corresponding increase in the count rate seen by the GM tube. The signal was given the proper amplitude and dc level on the analog computer.

The reactivity device was sufficiently long to permit the piston end to be fully inserted into the south beam port extension, with the drive assembly outside the south reactor shield.














CHAPTER IV


EXPERIMENTS AND RESULTS

This chapter contains a description of the

experiments performed both for check-out and calibration and for the measurement of the parameters of interest of the UFTR. This description includes data that were taken, data reduction, and the confidence limits assigned to the results. A detailed description of the procedure used to process the data is given in Appendix C.

The first section deals with the calibration

runs in which the transfer function of a low pass filter was measured by obtaining the cross-power spectrum between a random input and the output of the filter from this input. The error analysis associated with this calibration is used with the data obtained from the reactor.

The second section presents data obtained from the reactor and data reduction. Data obtained both with and without an external input are presented.


Calibration

Performance of the cross-power spectrum analyzer and the data transcribing system were checked with the









66

measurement of the transfer function of a low pass filter.

When a linear system is excited by a random

input, the cross-power spectrum between the input and the output, [io}(J), is equal to the product of the
top

power spectrum of the input, tii(L), and the transfer function of the system, G(LJ) (19). The real and imaginary parts of the transfer function may be measured by,



Re [G(L) = (97) SiillJ)



and
lIm [ioL~
Im [G(J), = (98)
L J t iilW)


The low pass filter, the transfer function of

which was measured, was formed by a one megohm resistor followed by a 0.1 microfarad capacitor to ground. These were precision X1% components. The output of the filter fed a 0.1 megohm input resistor of an operational amplifier on the analog computer. The amplitude and phase parts of the transfer function, G(G,), were calculated from,












K
/A/G (86) )(R + K)2 + R2K2LJ2C2



and


p = tan-- I (87)
G(tJ) [R + K


where K = 0.1 M R = 1.0 MQ

C = 0.1 /f.


These two functions are plotted as solid lines versus frequency in Figures 13 and 14.

Performance of the cross-power spectrum analyzer independent of the data storage system was first examined by taking the cross-power and power spectra measurements "on line." A randomly switched step function was given as an input to the filter system. The step function was obtained from the flip-flop section of a count rate meter the input of which was supplied by a GM counter setup. The power spectrum of the input was measured as indicated by equation (93) and the real and imaginary parts of the cross-power spectrum as indicated by equations (91) and (92).









68

Two such runs were made, Run L and Run M. The

only difference between these was that the GM counter setup was adjusted to give a count rate of 1700 counts per minute in Run L and 13,500 counts per minute in Run M. Results of these runs are given in Tables 1 and 2 and plotted in Figures 13 and 14. The power spectrum of the random step input of Run L is also obtained from the measurement and given in Table 1. Note that the amplitude correction factors (Appendix B) must be used for the power spectrum. This measured spectrum is seen in Figure 15 with the theoretical spectrum as calculated for a g of 28.3 counts per second from equation (96).

The "flags" on the data points plotted in the

above figures indicate one standard deviation. No "flags" are shown to indicate the accuracy of the frequency readings, which were determined by the accuracy of the settings on the band pass filters. The manufacturer gives t5% as the accuracy of the frequency settings on the Krohn-Hite ultra low frequency band pass filters. It was necessary to calibrate the band pass filters against a frequency standard to match the filters to within one per cent of each other and the standard.

The non-statistical errors associated with the cross-power spectrum analyzer were small and are















TABLE 1

RESULTS OF RUN L


f Normalized Phase Angle Normalized Power
Amplitude of G(.) Spectrum of cycles per of G(LJ) Input
second decibels degrees decibels


0.15 99.2 -0.6 99.1 0.2 97.6 -0.4 100.1 0.3 98.4 -4.6 99.0 0.4 99.0 -2.2 93.9 0.5 99.4 -1.1 98.3 0.6 101.4 -3.0 98.2 0.8 101.1 -4.2 97.9 1.0 100.0 -4.0 99.8 1.5 100.6 -6.7 98.7 2.0 100.6 -9.1 100.1 3.0 99.6 -12.4 98.4 4.0 99.4 -15.5 99.2 5.0 100.6 -18.0 97.7 6.0 100.0 -20.4 97.3 8.0 100.0 -26.1 94.6 10.0 99.4 -30.2 92.7 15.0 97.2 -39.0 88.3 20.0 96.2 -48.1 83.5 30.0 94.2 -57.3 77.3 40.0 93.6 -65.0 74.6 50.0 91.0 -66 70.3 60.0 90.2 -74 65.9 80.0 88.6 -79 61.5 100.0 88.2 -85 56.7
150.0 87.6 -86 48.4

















TABLE 2

RESULTS OF RUN M


f Normalized Amplitude Phase Angle of G((J) of G(J) cycles per
second decibels degrees



1.0 101.8 -5.3 1.5 100.2 -5.6 2.0 100.6 -9.9 3.0 100.6 -14.0
4.0 99.2 -15 5.0 100.1 -19 6.0 99.8 -15 8.0 99.6 -27 10.0 99.4 -32 15.0 98.4 -42 20.0 97.6 -54 30.0 94.6 -63 40.0 92.4 -68 50.0 92.0 -71 60.0 89.2 -75 80.0 87.2 -78
100.0 85.8 -82 150.0 85.4 -85









110




U
9-4

o 100


0'
.
5



S 90
44 (D Run L G~
0 Run L

4)
804 Theoretical
r4 80
Curve


N
9-4
S 70
0
Z



0.2 0.5 1.0' 2.0 5.0 10 20 50 100 200 500 Frequency, cycles per second


Figure 13. Normalized Amplitude Component of G((J) from Runs L and M












-10


-20 Theoretical Curve

-30


-40
0 Run L o Run M S-50
44 % o0
0
a -60


S-70


S-80


-90
I II I I I Il I 0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second


Figure 14. Phase Component of G(W) from Runs L and M






































0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 Frequency, cycles per second


Figure 15.


Normalized Power Spectrum of Random Step from Run L


500







74

estimated to be within 2 per cent. These errors are those associated with the components of the phase shifting network, potentiometer settings, computer amplifiers, multipliers, and the integrator amplifier readout. To hold these errors to within a few per cent, all potentiometer setting and amplifier readout was executed with the null potentiometer, all equipment allowed to warm up for an hour before data processing, and the amplifiers on the computer were balanced at frequent intervals.

The accuracy of the cross-power spectrum analyzer was limited primarily by statistical considerations. In the development of the theory of the measurement of power and cross-power spectra, it was necessary to form equation (74) from equation (73) by replacing an average over all time with a finite average time, T. The standard deviation expected from the effect of the finite averaging time has been derived by Bennett and Fulton

(26). They give for the fractional standard deviation (O/N) of a measurement of the power spectrum of normally distributed noise,



O/N = 1, (99)



where T is the integration time and Af is the frequency difference between half-power point in the band pass









75

filter. The expression may be used as a first order approximation to cases involving other distributions and spectral shapes as well as to cross-power spectra.

Three voltages were measured, VR, VI and V, corresponding to the real and imaginary parts of the cross-power spectrum between the outputs of the two regions of the UFTR and to the power spectrum of the output of the externally disturbed region. Since the same integration times and frequency band widths were used for the measurement of all three voltages, all three have the same fractional standard deviation, namely,


R 1_. (100) V VR VI Tf



In most cases, a three minute integration time was used. This value was taken as a compromise between the drift in the analog computer and a reasonable statistical accuracy.

Given the standard deviation of the voltage

readings, the standard deviation of the amplitude and phase parts can be calculated from the propagation of errors formula,










.2 -2 I7 f ( 2 + fy2 (101)


which gives the standard deviation, O'f, of a function f(x,y) of the two quantities x and y which have standard deviations, Ix and 6y, respectively.

The above expression gives the standard deviation of the phase part (given by the quotient VI/vR) as,


V 2
(f = (102)
P VR V7T A (f02)


and the standard deviation of the amplitude part (given

[V 2 +[-]2
by + ) as,

V V'4 V1

[; ]4 -[~ _4
V V


V V
8
=. (103) [-]- + --.T~



As seen in Figure 13, the standard deviation of the amplitude part at the lower frequencies is rather large. The standard deviation of the phase part, however, is of a magnitude comparable with the size of the data points.








77

The data of these figures do, however, indicate the satisfactory performance of the cross-power spectrum analyzer. The data of Figure 15 demonstrate that with the correction factors of Appendix B, the analyzer may also be used successfully for the measurement of power spectra.

Referring to Figure 13, one observes that the data of Run L depart from the theoretical amplitude at frequencies beyond 50 cycles per second. This departure is a leveling out of the amplitude plot to the noise level of the analyzer. At 50 cycles per second Figure 15 shows that the random input power spectrum is down more than 30 decibels from the value at the lower frequencies. In Run M the power spectrum of the input carries more of the high frequency components and because of this, the cross-power spectrum data extend out to about 100 cycles per second. The phase data points followed the theoretical curve out to 125 cycles per second for both runs.

Run P was a run of the same type as L and M, but instead of processing the data "on line," the two signals, the random step input and the filter output, were AM modulated and stored on magnetic tape. The random step was formed from a count rate of 7100 counts per minute from the GM counter. The data from Run P as









78

processed from the tape are given in Table 3. The data points of the amplitude and phase parts of the transfer function of the filter, G(J), are shown plotted versus frequency in Figures 16 and 17. Once again, the theoretical curves are given also. The data from this run seem more scattered than the data points of Runs L and M, especially those of the amplitude part. This additional scatter must be attributed to the AM modulation, recording, and demodulation. The magnitude of the amplitude scatter is about t2 decibels, while little difference is noted in the phase part. In all other respects the results of Run P were identical to those of Runs L and M. The amplitude portion of the data departs from the theoretical curve between 60 and 100 cycles per second, a higher frequency than that at which the departure occurred in Run L and a lower frequency than that at which the departure occurred in Run M. This was to be expected since the random input was driven by a count rate higher than Run L and lower than Run M.

Run P demonstrates that the techniques and

equipment used to collect, record, and analyze the data were satisfactory.
















TABLE 3

RESULTS OF RUN P


f Normalized Amplitude Phase Angle of G(U) of G((J) cycles per
second decibels degrees


0.6 95.8 -6.1 0.8 101.2 -1.2 1.0 99.0 -6.7 1.5 98.4 -4.1 2.0 96.6 -6.5
3.0 98.6 -10 4.0 100.8 -18 5.0 98.2 -18 6.0 99.4 -25 8.0 97.0 -33 10.0 98.4 -30 15.0 100.0 -37 20.0 96.8 -51 30.0 94.2 -60 40.0 93.6 -62 50.0 90.8 -70 60.0 89.8 -72 80.0 89.2 -77 100.0 86.4 -80 150.0 86.6 -90








110


'-4 ,)
100 0





S 90
0
O Theoretical o o
-4
Curve OD
0 4J
-4
80

4

N
-,4

e 70
O
70
z

I II II II I
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500 Frequency, cycles per second


Normalized Amplitude Component of G(W)


Figure 16.


from Run P








0


-10 0


-20


-30
$4

-40


-50 Theoretical Curve o -60


-70


-80


-90
I I I I I I I I II I

0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500 Frequency, cycles per second


Figure 17. Phase Component of G(LJ) from Run P









82

Reactor Data

Some of the first reactor data were taken "on line." The external reactivity input device used for these runs was driven by a random step function at an average count rate of 200 counts per minute. Because of the low count rate and inadequate frequency response of the reactivity input, the data obtained were not reliable at frequencies beyond about two cycles per second. Although the data looked promising out to this point, little could be determined, as seen from Figures

2 and 3.

The random input device designed and constructed by Rajagopal and described in the previous chapter was used in succeeding runs. These runs were recorded and processed "off line."

The reactor was fueled with 3100 grams of

uranium-235 in the form of 93 per cent enriched uraniumaluminum fuel plates for Run R. The commercial low frequency noise generator was used as an input to the driving amplifier of the input device and the data were recorded with the reactor operating at a power level of 10 watts. The two Keithley micro-microammeters were operated on the 3x10-6 ampere scale. The results of this run are given in Table 4. Not only the phase and amplitude parts of the function R12 (WJ) but also the














TABLE 4

RESULTS OF RUN R


f Normalized Amplitude Phase Angle Normalized Power cycles of R12(L) of R12(J) Spectrum of Outper put of Region 1 second decibels degrees decibels


0.6 0.8
1.0 1.25
1.5
2.0 2.5 3.0
4.0 5.0
6.0 7.0 8.0
9.0
10.0 12.5 15.0 20.0 25.0 30.0
40.0
50.0 60.0
70.0 80.0 100.0


98.8
100.5
99.7
99.3 101.7
99.5
102.7 99.3 98.3 100.1
99.3 103.1
99.9 99.7 98.9 98.7 101.3 97.8 96.5 95.6
91.5 88.7 89.3 86.5 85.3 79.1


-6.2
0
-7.0
-5.0
-0.7
-14
-7.0
-11
-13
-12
-15
-17
-16
-23
-26
-20
-28
-39
-45
-40
-34
-19
-3.2
0
-13
-44


78.9 88.3 89.2 99.6
98.5
101.6 99.4
101.3
99.0 98.1 95.0 91.6 93.1 89.8 90.6 85.0 79.2
76.4 71.4 70.2
68.7 70.4 71.1
70.2 66.9 64.6









84

power spectrum of the output of the south region (the externally disturbed region) are given. The amplitude part of RI2(LJ), the phase part of R12 (J), and the power spectrum of the output of the south region are plotted versus frequency in Figures 18, 19, and 20, respectively.

The phase data points, up to about 25 cycles

per second, although badly scattered, are seen to follow the set of theoretical curves calculated for a lag time between the two reactor regions corresponding to the velocity of a thermal neutron wave. A B value of about 400 is indicated. Beyond a frequency of 25 cycles per second, little can be determined because of the excessive scatter of the data. The amplitude data points depart from the theoretical curve for a B of 400 beyond 15 cycles per second. It is difficult to determine whether this break in the amplitude data actually reflects a low value of B or if it signifies the end of correlation between the two reactor output signals because of low signal to noise ratio. The phase data would suggest the latter condition. Consideration of the power spectrum of the output of the south fuel slab (Figure 20) would also indicate that correlation would end much beyond 20 cycles per second. This power spectrum, which is representative of the






































0.5 1.0 2.0 5.0 10 20
Frequency, cycles per second


50 100 200


Figure 18. Normalized Amplitude Component of R12((J) from Run R


110


100





90 80 70


0.2


500








+20


+10


a 0 e

2 4
" -10-20
Curve B
-30 1 100 Neutron Wave
2 200 Neutron Wave
3 500 Neutron Wave
-40 4 1000 Neutrn Wave
5 500 2x10- seconds
6 500 Zero
-50 \
w I1 2 3 4 5 6
-60


-70


0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500 Frequency, cycles per second


Figure 19. Phase Component of R12(LJ) from Run R








120 110


100


90 80 70 60


50 40 30


ee 0


00 0 0o
0


I I I I I I I


0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 Frequency, cycles per second

Figure 20. Normalized Power Spectrum of Region 1 Output from Run R


500


0 0 ee 0









88

input to the north slab, is down about 20 decibels at 15 cycles per second from the amplitude value at the lower frequencies. Except for a hump at 60 cycles, the power spectrum levels out to the reactor and instrument noise level at frequencies beyond 25 cycles per second. Since the random reactivity input device had been found to respond satisfactorily to sinusoidal inputs from 0.1 to 25 cycles per second, the cause of the drop-off of the power spectrum at the lower frequencies was unknown.

Full stroke of the device held about 0.007 per cent reactivity. When the noise generator was used to drive the reactivity input device, its effect could not be seen by the reactor operators on the console instrumentation. Observation of the signal from either of the chambers in the reactor revealed that the input disturbance was about one-half the magnitude of the inherent reactor noise.

To obtain a larger input reactivity disturbance the absorber piston assembly was reconstructed so as to be about double the exponsed cadmium area. The tension spring was changed and the power amplifier was also modified to obtain better frequency response. With these modifications, the device was found to respond faithfully to sinusoidal inputs up to 50 cycles per second. Full stroke was found to produce more than

0.015 per cent reactivity change in the reactor.









89

For Run CC the reactor was fueled with 3500 grams of uranium-235 in the form of 20 per cent enriched uranium-aluminum alloy fuel plates. The reactor was operated at a power level of 50 watts and the two Keithley micro-microammeters were operated on the 3x10-6 ampere scale. The random reactivity input device was found to produce disturbances of almost twice the magnitude of the normal reactor noise. The power amplifier was driven by a randomly switched square wave. This random square wave was constructed with a GM counter and the flip-flop section of a count rate meter. A count rate of 20,000 counts per minute was used.

The results of Run CC are presented in Table 5. Once again the phase part of R12(LJ), the amplitude part of R12(j), and the power spectrum of the output of the south fuel region are listed in Table 5 and also plotted versus frequency in Figures 21, 22, and 23, respectively.

The phase data points, up to a frequency of 40 cycles per second, follow the theoretical curve corresponding to a B of about 450 and a lag time, 0 determined from the velocity of a thermal neutron wave. Beyond 40 cycles per second, the data points become excessively scattered. The amplitude data points break from the theoretical curve for a B of 450 at frequencies beyond 25 cycles per second. Again, it is difficult to














TABLE 5

RESULTS OF RUN CC


f Normalized Amplitude Phase Angle Normalized Power cycles of R12(L)J) of R12((J) Spectrum of Outper put of Region 1 second decibels degrees decibels


0.4 0.5 0.6 0.8 1.0
1.25
1.5
2.0 2.5 3.0
4.0 5.0 6.0 7.0
8.0 9.0
10.0 12.5 15.0
20.0 25.0 30.0
40.0
50.0 60.0
70.0 80.0
90.0 100.0
125.0 150.0


101.0 99.5 100.7 100.0
102.0 98.5 101.1
102.2 99.2 100.0 102.0 99.2 101.4 101.6 99.0 100.7
98.7 100.2 100.2 99.0 100.0 95.6 93.9 99.4 90.5
93.4 83.7 83.5 80.9 81.9
80.4


0
-2.0
-5.3
0
-4.4
-2.0
-2.1
-5.1
-5.2
-7.0
-8.5
-10
-11
-14
-17
-18
-25
-22
-25
-39
-43
-49
-64
-65
-34
-62
-84
-56
-62
-70
-116


101.4 102.1 100.7
99.3 99.5
102.6
99.2
98.8 100.1
97.0 95.8 9'4.7
90.9 89.7 88.3 85.7
84.9
78.0 71.5 66.3 59.3 58.0 52.8 52.3 53.0
48.7 48.9 48.6
47.7 46.3
43.3






































0.5 1.0


2.0 5.0 10 20 50 Frequency, cycles per second


100 200


Figure 21. Normalized Amplitude Component of R12 (J) from Run CC


110 100 100





90 80 70


0.2


500









+10


0

S-10
00
S -20


-30
Curve B e o
W -40 1 100 Neutron Wave
2 200 Neutron Wave 0 3 500 Neutron Wave 0 -50 4 1000 Neutron Wave V45 6
5 500 2x10-4 seconds
6 500 Zero 1 2 3 4
-60
0 \ Od 0 S-70 0


-80
I I ,,, I, -- IIII

0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second


Figure 22. Phase Component of R12(J) from Run CC








120 110 100 90 80 70 60 50


40 30


00


0 ~0
00G~ 0


0.5 1.0 2.0


5.0 10 20


50 100 200


Frequency, cycles per second


Figure 23. Normalized Power Spectrum of Region 1 Output from Run CC


0 0 0 0 e0 0


000
eg0


0E


0.2


500








94

determine if the break represents a lower value of B or if it represents the end of correlation. The power spectrum of the region 1 output indicates that correlation would end much beyond 25 or 30 cycles per second. At 30 cycles per second the power spectrum is down 40 decibels from the amplitude at the lower frequencies. From the calibration runs it would be expected that correlation would not continue when the input (which is the output of the south fuel region in this case) power spectrum drops off 40 decibels. There was no drop-off of the spectrum at the lower frequencies as in Run R, indicating a more satisfactory operation of the reactivity input device. The magnitude of the input signal was also greater than before resulting in the lower frequency portion of the spectrum being 50 decibels above the reactor and instrument noise level.

A check was made on the frequency at which

correlation between the two outputs of the reactor ended with a complementary run to Run CC. Run Z was taken under the same operational conditions as Run CC except that both ion chambers were located side by side outside the south fuel region. The results of this run are presented in Table 6. The amplitude and phase parts of the cross-power spectrum and the power spectrum of the output of the south fuel region are plotted versus frequency in Figures 24, 25, and 26, respectively.




Full Text
Normalized Amplitude of R,~(LJ), decibels
110
100
4
_J I I I I I I 1 I I
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 21. Normalized Amplitude Component of Ri2(lc/) from Run CC
500


74
estimated to be within 2 per cent. These errors are
those associated with the components of the phase shift
ing network, potentiometer settings, computer amplifiers,
multipliers, and the integrator amplifier readout. To
hold these errors to within a few per cent, all potenti
ometer setting and amplifier readout was executed with
the null potentiometer, all equipment allowed to warm
up for an hour before data processing, and the amplifiers
on the computer were balanced at frequent intervals.
The accuracy of the cross-power spectrum analyzer
was limited primarily by statistical considerations. In
the development of the theory of the measurement of
power and cross-power spectra, it was necessary to form
equation (74) from equation (73) by replacing an average
over all time with a finite average time, T. The stand
ard deviation expected from the effect of the finite
averaging time has been derived by Bennett and Fulton
(2,6) They give for the fractional standard deviation
(0/N) of a measurement of the power spectrum of nor
mally distributed noise,
07 N
1
(99)
where T is the integration time and Af is the frequency
difference between half-power points in the band pass


70
TABLE 2
RESULTS OF RUN M
f
cycles per
second
Normalized Amplitude
of G(U)
decibels
Phase Angle
of G(U)
degrees
1.0
101.8
-5.3
1.5
100.2
-5.6
2.0
100.6
-9.9
3.0
100.6
-14.0
4.0
99.2
-15
5.0
100.1
-19
6.0
99.8
-15
8.0
99.6
-27
10.0
99.4
-32
15.0
98.4
-42
20.0
97.6
-54
30.0
94.6
-63
40.0
92.4
-68
50.0
92.0
-71
60.0
89.2
-75
80.0
87.2
-78
100.0
85.8
-82
150.0
85.4
-85


BIOGRAPHICAL SKETCH
Allen Ross Boynton was born in Lexington,
Kentucky, on June 16, 1936. In June, 1953, he was gradu
ated from Owensboro Senior High School, Owensboro,
Kentucky. In June, 1957, he received the degree of
Bachelor of Science in Mechanical Engineering, with dis
tinction, from Purdue University. He was awarded an
Atomic Energy Commission Fellowship in September, 1957,
and received the degree of Master of Science in Engineer
ing from Purdue University in August, 1958. In September,
1958, he was admitted to the University's Graduate School.
While pursuing his graduate studies, he held positions
as Graduate Assistant and as the Chief Operator of the
University of Florida Training Reactor.
Allen Ross Boynton is married to the former
Carol Sue Baringer of Gainesville, Florida. He is a
member of Tau Beta Pi, Pi Tau Sigma, and the American
Nuclear Society.
126


This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.
December, 1962
Dean, Graduate School
Supervisory Committee


51
The modulated signals were then recorded on two
channels of an Ampex 500 magnetic tape recorder. A tape
speed of 60 inches per second was used for both record
ing and playback. Modulation was necessary because of
%
the frequency response of the tape recorder.
Figure 9 presents a flow diagram of the equipment
used to measure the real and imaginary parts of the
cross-power spectrum between the outputs of the two
regions of the UFTR. The signals from the tape recorder
. were first demodulated, using the circuit shown in
Figure 10. The demodulated signals were then amplified
by a factor of about 25 on the analog computer in order
to have signals of about 10 volts peak to peak to the
band pass filters. The band pass filters were set for
the same frequency and band width. A "zero" band width
was used in which the upper and lower cut-off frequen
cies were set at the same value. The band pass filters
allow frequencies outside the band width to pass through
even though these "outside" frequencies are uniformly
and greatly attenuated. When the upper and lower cut
off frequencies are equal to f., the half-power, or
three db down, points are at 0.77f^ and 1.30^. Thus,
if a constant percentage or a "zero" band width is used,
any error due to the "outside" frequencies is normalized.
The larger the percentage band width used, the less the


12
By reversing the order of integration, equation (5) is
put into a more desirable form,
$12 =
hl(A>T i(t- A>f2 -T
(6)
hi (A)
(t-X~ 0 ) f2 ^t+ )dtd^ .
With this grouping, the inner integral in the first term
is recognized as the crosscorrelation function between
the external input and the output of region 2, while the
inner integral in the second term is the autocorrelation
function of the region 2 output. Equation (6) may now
be written in the compact form,
oo
$i2 vA>$i2aA
'-CD
( T + ,\ + 0 )dX
(7)
From equation (1), it is noticed that f2(t +T)
may also be expressed in terms of the convolution
integral between the input, output, and unit impulse
response function by,


10
Output
f2<*>
Input
Aifi(t>0)
Input
A2f2 (t _@ )
Output
{> £i(t)
External Input
i(t)
Figure 1. Two Region System Diagram


M
Ul
to
Figure 9. Cross-Power Sectrum Analyzer


6
an external disturbance is given to only one of them,
will reveal something about the nature of the coupling
between the two regions. A representation of this inter
dependence is given by either the crosscorrelation
function of the outputs of the two regions, or in the
frequency domain, by the cross-power spectrum between
the two outputs.
The relations for the cross-power spectra between
the outputs of each region are derived and are found to
involve transfer functions of the individual regions.
When these transfer functions are derived, they are
found to depart from the conventional one region trans
fer functions because of the term that couples the two
regions. The cross-power spectrum between the outputs
of the two regions is a complex quantity and its phase
part is found to be sensitive to the transit time of a
disturbance between the regions. Its amplitude part
as well as the phase part is found to be dependent on
the value of the multiplication factor of the undis
turbed region.
The cross-power spectrum between the outputs
of both regions of the UFTR was measured under two
conditions: (a) a random stationary external input to
region 1, and (b) no external input to either region.
Continuous data from two ion chambers were
amplitude modulated and stored on magnetic tape. The


22
transit time,$, between regions. Thus
E
A2n2(t -Q )
2
(36)
Expanding this expression in a Taylor's series about t,
A2n2(t) A20 dn2(t) h29 2 d2n2(t)
(37)
E
Baldwin (17) estimates 9 by assuming it to be
determined by the mean thermal neutron velocity. In the
UFTR, which has a 30 centimeter region of graphite
between the fuel regions, this assumption would indicate
a value of approximately 2 x 10^ seconds for 9 Such
a small value would make 9 relatively unimportant in
kinetics except for very short periods and unimportant
in the transfer function except at very high frequencies.
A more realistic assumption is that a disturbance in the
neutron density in one fuel slab travels across to the
opposite region with the velocity of a neutron wave.
The velocity of a neutron wave is frequency dependent
but at frequencies from 1 to 100 radians per second is
approximately constant at 1.4 x 10^ centimeters per
second in graphite (22) This gives a lag time, 9 in
the UFTR of about 2 x 10 3 seconds, a value that makes it


LIST OF FIGURES (Cont'd)
Figure Page
16. Normalized Amplitude Component of
G(LJ) from Run P 80
17. Phase Component of G (iJ) from Run P 81
18. Normalized Amplitude Component of
Rl2(L/) from Run R 85
19. Phase Component of from Run R 86
20. Normalized Power Spectrum of Region 1
Output from Run R 87
21. Normalized Amplitude Component of
Ri2 (U) from Run CC 91
22. Phase Component of R12(L/) from Run CC 92
23. Normalized Power Spectrum of Region 1
Output from Run CC 93
24. Amplitude Component of Normalized
Cross-Power Spectrum from Run Z ... 96
25. Phase Component of Normalized Cross-
Power Spectrum from Run Z 97
26. Normalized Power Spectrum of Region 1
Output from Run Z 98
27. Normalized Power Spectrum of Region 1
Output from Run Q 102
vi


57
Before multiplication, however, the signals were passed
through high pass filters to remove any dc components
and low frequency drifts introduced because of the high
gain of the system. The nominal 1.0 microfarad capaci
tors and 10 megohm resistors used were actually 1.09
microfarad and 9.9 megohms, but were matched to within
1 per cent. The outputs of these filters were fed
into the one megohm input resistors of operational
amplifiers. Here again, attenuation and phase shift
were introduced, especially at the lower frequencies.
The phase shift was identical for all three signals
and was ignored. The amplitude attenuation is calculated
in Appendix B.
Since both the phase changer and high pass filter
, and
attenuations were the same for f^(t)
LA
,
LA
fo(t+90)
, the same correction was valid for all
LA
three. These corrections were squared, however, because
the attenuations occurred before the multiplications.
The separate and combined correction factors that were
applied to the real and imaginary parts of the cross
power spectrum are given in Appendix B.
Multiplication was performed after the signals
were passed through the high pass filters and the


4
of Rice (13). Recent measurements (14) have indicated
that the inherent reactor driving function spectrum
deviates from white noise at low frequencies. Second,
the effects of external disturbances on the final
result are uncertain since the measured noise is actu
ally a combination of reactor fluctuations and instru
ment noise. Third, the autocorrelation or pile noise
spectrum method does not yield any direct information
about the phase of the transfer function. A fourth
objection to this method is that a large (greater than
5xl0-4) detector efficiency is required. It may be
physically impossible to place a detector in a position
of high efficiency, and in the case of a counting
experiment, the power level may need to be limited to
avoid pile-up in the detector.
These objections are removed by crosscorrelation
techniques. Balcomb (15) and Rajagopal (16) investi
gated the crosscorrelation of reactor outputs with
random reactivity inputs. Both these investigators
measured unit impulse response functions and then found
the transfer function by taking the Fourier transform.
It was found that the technique is very effective in
eliminating the effects of external noise and gave
good results for low-level crosscorrelation signal
inputs whose power was comparable to the noise power
in the system.


130
120
110
100
90
80
70
60
50
40
O O
O
2
O 0o
o
o
o
000
I £ I I II! | I I
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 27. Normalized Power Spectrum of Region 1 Output from Run Q
500
102


107
Of these several equations, only one was developed for
experimental use. In making measurements on a more
complex reactor than the UFTR, other of the equations
may prove useful for experimental purposes.
This thesis was devoted to the treatment of a
two region reactor system. The general method and
experimental approach could certainly be extended to
a three or more region system. Several large reactors
have either been designed or built that have cores
composed of several rather independent regions. An
extension of the method developed in this thesis would
allow the measurement of the multiplication factor of
each region by crosscorrelating the outputs of two
regions when a random disturbance is given to one
region only.
The physical arrangement of the UFTR made it
impossible to position the two ion chambers used to
collect data any closer than 18 inches from each fuel
region without extensive modifications of the graphite
reflector surrounding the core. This fact meant that
the detectors were located in positions of low effici
ency (less than 10 4). While this had no effect on
the data taken when there was an external reactivity
input to one region, it is felt that these low chamber
efficiencies were responsible for the lack of correla
tion between the two region outputs where there was no
external input.


114
PL
tan-1
RK C
R + K
(37)
and for the high pass filter,
rkL/c
~}J (R+K)2 + R2K2L/2C2
(88)
and
PH
tan-1
R+K
rkUc
(89)
As long as the same value resistors and
capacitors are used there is a 90 degree shift between
the outputs of the low and high pass filters. It was
convenient then to select R and C values such that at
a particular frequency, the amplitude attenuation was
the same in each filter. This was accomplished when,
1 1
R = = (90)
U C 27TfC
The amplitude parts of both the low and high pass fil
ters under the condition of equation (90) are then,


43
When T is set equal to zero, equation (80) reduces to,
_1_
T
rT i
r -
fi(t)
Jo J
f2(t)
U
i
dt =
ui
Re12 + jlm12
A(Jl.
(81)
The integral on the left side of this equation is a
real quantity so that when the real parts are equated,
Rei2 ((jJ])
T 1
fl(t>
Jo J
^£2 la
dt
tA u1
(82)
Returning to equation (80), one sets T equal to .
2 U1
Since both f^(t) and f2(t) have been put through
identical band pass filters and are both essentially at
the same frequency, U
f2 (+
ir
2 LA
la
= f2(t + 90)
LA
(83)
Thus equation (80) becomes,
(t + 90)
^-Im12+j Re12
(84)


88
input to the north slab, is down about 20 decibels at 15
cycles per second from the amplitude value at the lower
frequencies. Except for a hump at 60 cycles, the power
spectrum levels out to the reactor and instrument noise
level at frequencies beyond 25 cycles per second. Since
the random reactivity input device had been found to
respond satisfactorily to sinusoidal inputs from 0.1 to
25 cycles per second, the cause of the drop-off of the
power spectrum at the lower frequencies was unknown.
Full stroke of the device held about 0.007 per
cent reactivity. When the noise generator was used to
drive the reactivity input device, its effect could not
be seen by the reactor operators on the console instru
mentation. Observation of the signal from either of
the chambers in the reactor revealed that the input
disturbance was about one-half the magnitude of the
inherent reactor noise.
To obtain a larger input reactivity disturbance
the absorber piston assembly was reconstructed so as to
be about double the exponsed cadmium area. The tension
spring was changed and the power amplifier was also
modified to obtain better frequency response. With
these modifications, the device was found to respond
faithfully to sinusoidal inputs up to 50 cycles per
second. Full stroke was found to produce more them
0.015 per cent reactivity change in the reactor.


101
TABLE 7
RESULTS OF RUN Q
Normalized Power
f Spectrum of Out
put of Region 1
cycles per second decibels
0.4
97.6
0.5
95.6
0.6
102.1
0.8
96.0
1.0
102.9
1.25
99.7
1.5
105.2
2.0
100.9
2.5
103.1
3.0
103.7
4.0
101.8
5.0
97.6
6.0
96.7
7.0
98.2
8.0
97.7
9.0
92.8
10.0
93.9
12.5
92.4
15.0
91.9
20.0
86.3
25.0
88.9
30.0
87.8
40.0
90.2
50.0
92.4
60.0
91.6
70.0
90.7
80.0
91.6
90.0
93.2
100.0
91.6
125.0
91.2
150.0
93.6
200.0
96.9


APPENDIX B
PHASE SHIFT AND HIGH PASS FILTER DETAILS
In the measurement of the cross-power spectrum
between two signals it was necessary to pass the sig
nals through band pass filters, set around the same
frequency, and then to introduce a 90 degree phase
shift between them. This was accomplished by giving
one signal a phase lag in a low pass filter and the
other a phase lead in a high pass filter such that
there was a 90 degree shift between them. The low pass
filter was formed by a resistor, R, followed by a
capacitor, C, and another resistor, K, in parallel to
ground while the high pass filter was composed of a
capacitor, C, followed by two resistors, R and K, in
parallel to ground. The amplitude, /&/> and phase, p,
parts of the transfer functions for these filters are
given by, for the low pass filter,
K
~Y (R + K)2 + r2k2U2c2
(86)
and,
113


EVALUATION OF PARAMETERS IN A TWO
SLAB REACTOR BY RANDOM
NOISE MEASUREMENTS
By
ALLEN ROSS BOYNTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1962

ACKNOWLEDGMENTS
The author wishes to acknowledge his gratitude
for the advice and encouragement of his advisory
committee. In particular, grateful acknowledgment is
made to the chairman, Dr. Robert E. Uhrig, for his
continued guidance and encouragement during the course
of this work.
The author wishes to express his gratitude to
the staff and technicians of the Department of Nuclear
Engineering. J. Mueller and K. L. Fawcett constructed
the early equipment, J. Wildt constructed the final
equipment, and L. D. Butterfield operated the reactor
for most of the runs.
The author wishes also to express his
appreciation to Dr. Herbert Kouts and V. Rajagopal of
the Brookhaven National Laboratory for their loan of
the random reactivity input device.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
LIST OF TABLES iv
LIST OF FIGURES V
Chapter
I. INTRODUCTION 1
II. THEORY 8
Cross-Power Spectra 9
Two Region Transfer Functions .... 20
Reduction of Equations 31
III. MEASURING SYSTEM 38
Measuring Theory 38
The University of Florida
Training Reactor 44
Cross-Power Spectrum Analyzer .... 47
Random Reactivity Input Device ... 60
IV. EXPERIMENTS AND RESULTS 65
Calibration 65
Reactor Data 82
V. DISCUSSION AND CONCLUSIONS 103
Appendix
A. CALCULATION OF TRANSIT TIME
BETWEEN SLABS 108
B. PHASE SHIFT AND HIGH PASS
FILTER DETAILS 113
C. OPERATION OF CROSS-POWER
SPECTRUM ANALYZER 119
LIST OF REFERENCES 1^3
BIOGRAPHICAL SKETCH 126
iii

LIST OF TABLES
Table Page
1. Results of Run L 69
2. Results of Run M 70
3. Results of Run P 79
4. Results of Run R 83
5. Results of Run CC 90
6. Results of Run Z 95
7. Results of Run Q 101
8. Value of Transit Time
Between Slabs Ill
9. Set Points for Phase Shifters 116
10. Correction Factors for Filters 118
iv

LIST OF FIGURES
Figure Page
1. Two Region System Diagram 10
2. Normalized Amplitude Component of
Ri2 (U) for Several Values of B . 35
3. Phase Component of R-^ Vj fr
Several Values of B and U 36
4. Sequence of Operations Necessary for
Experimental Determination of Power
Spectrum 42
5. Sequence of Operations Necessary for
Experimental Determination of Cross-
Power Spectrum 42
6. Location of Ion Chambers in UFTR Core 46
7. Data Transcribing System 48
8. Modulator Schematic 50
9. Cross-Power Spectrum Analyzer 52
10. Demodulator Schematic 53
11. Random Reactivity Input Device .... 61
12. Power Amplifier Schematic 63
13 Normalized Amplitude Component of
G (LJ) from Runs L and M 71
14. Phase Component of G (iJ) from
Runs L and M 72
15. Normalized Power Spectrum of Random
Step from Run L 73
v

LIST OF FIGURES (Cont'd)
Figure Page
16. Normalized Amplitude Component of
G(LJ) from Run P 80
17. Phase Component of G (iJ) from Run P 81
18. Normalized Amplitude Component of
Rl2(L/) from Run R 85
19. Phase Component of from Run R 86
20. Normalized Power Spectrum of Region 1
Output from Run R 87
21. Normalized Amplitude Component of
Ri2 (U) from Run CC 91
22. Phase Component of R12(L/) from Run CC 92
23. Normalized Power Spectrum of Region 1
Output from Run CC 93
24. Amplitude Component of Normalized
Cross-Power Spectrum from Run Z ... 96
25. Phase Component of Normalized Cross-
Power Spectrum from Run Z 97
26. Normalized Power Spectrum of Region 1
Output from Run Z 98
27. Normalized Power Spectrum of Region 1
Output from Run Q 102
vi

CHAPTER I
INTRODUCTION
This study is a development of an experimental
method to measure the parameters peculiar to a two
region reactor system. Of particular interest is the
method of representing the coupling or interaction
between the two regions. A transfer function approach
is used.
The dynamic behavior of reactors has been
successfully analyzed from the transfer function point
of view. Transfer functions have been instrumental in
determining system stability and the design of control
systems. The measurement of reactor transfer functions
by sinusoidal oscillations, step, and ramp inputs of
reactivity are standard techniques and have allowed the
determination of parameters associated with the reactor
kinetics equations (jL) (2J.*
However, the statistical nature of the fission
process has led many investigators to measure dynamic
reactor parameters by purely statistical means. As
Underlined numbers in parentheses refer to
the list of references.
1

2
early as 1946, deHoffman (3.) developed an expression
for the intensity fluctuations of a chain reactor as
measured by a counter with statistical response. The
formulation included the effects of delayed neutrons
and involved the dispersion of the number of neutrons
emitted per fission. Later workers in this area were
Luckow (4) who used the variance to mean ratio of the
number of neutrons counted by a detector during a fixed
counting time to measure the prompt neutron lifetime;
Albrecht (5.) who extended Luckow's analysis to include
delayed neutron parameters; and Velez (6) who used the
autocorrelation function of the counting rate from a
reactor to measure the dynamic parameters. These inves
tigations, while giving insight into the fundamental
nature of multiplying systems and information on the
influence of the statistical fluctuations in the
reactor power level on such things as the precision of
neutron measurements and the design of reactor control
equipment, cannot furnish as much information about the
complete reactor system as can the transfer function.
Fortunately, statistical considerations are
related to the transfer function approach, since the
power spectral density functions (or power spectra) and
correlation functions form a Fourier transform pair.
Correlation functions have been known to statisticians

3
for many years, yet were relatively unknown to engineers,
who were analyzing frequencies and power spectra. Since
the power spectrum is related to the transfer function
of a linear system, engineers have been quick to use
this new measuring technique.
Moore (7.) (Q) was among the first to relate the
statistical nature of chain reactors to their transfer
functions when he expressed the power spectrum of the
reactor noise as a function of the square of the modulus
of the transfer function. Cohn (9.) used a band pass
filter to measure the high frequency portion of the
reactor noise spectrum which is dependent on the ratio
of the effective delayed neutron fraction,^, to the
prompt neutron lifetime, £ This technique has become
almost standard (10). The method has, however, several
difficulties. First, the exact input to the system is
not definable. This difficulty is attacked by Cohn (11)
who assumes the reactor noise to arise from a random
"noise equivalent" neutron source driving the reactor.
The characteristics of this source are calculated from
conventional random noise theory, and the resulting
pile noise obtained through the use of the source
transfer function. Bennett (12) develops the spectrum
and variance of pile noise according to the formulation

4
of Rice (13). Recent measurements (14) have indicated
that the inherent reactor driving function spectrum
deviates from white noise at low frequencies. Second,
the effects of external disturbances on the final
result are uncertain since the measured noise is actu
ally a combination of reactor fluctuations and instru
ment noise. Third, the autocorrelation or pile noise
spectrum method does not yield any direct information
about the phase of the transfer function. A fourth
objection to this method is that a large (greater than
5xl0-4) detector efficiency is required. It may be
physically impossible to place a detector in a position
of high efficiency, and in the case of a counting
experiment, the power level may need to be limited to
avoid pile-up in the detector.
These objections are removed by crosscorrelation
techniques. Balcomb (15) and Rajagopal (16) investi
gated the crosscorrelation of reactor outputs with
random reactivity inputs. Both these investigators
measured unit impulse response functions and then found
the transfer function by taking the Fourier transform.
It was found that the technique is very effective in
eliminating the effects of external noise and gave
good results for low-level crosscorrelation signal
inputs whose power was comparable to the noise power
in the system.

5
This paper will apply this latest technique to
a more complex reactor model, that of a two region
reactor.
Baldwin (JL7) has examined the so-called "two-
slab" loading of the Argonaut reactor and found that,
experimentally, a single exponential stable period is
observed when the system is supercritical, but that
there is a tendency for independent behavior of the
two slabs, resulting in so-called "flux tilting."
Rod calibration experiments,indicated that the ratio
of fluxes in the two fuel regions changes during the
calibration procedure. It was also found that the
single transfer function of a simple reactor system
was inadequate to treat the two slab system.
The two (or more) region reactor has also been
of interest in connection with the phenomenon of coupled
Xenon oscillations in large reactors (18).
The University of Florida Training Reactor (UFTR)
is a two region coupled reactor. Each fuel region is
subcritical when considered separately, but coupling
between the two regions allows the system to become
critical. If the multiplication factor of one region
is disturbed, the other region will be disturbed only
through its coupling to the first. A study of the
interdependence of the outputs of the two regions, when

6
an external disturbance is given to only one of them,
will reveal something about the nature of the coupling
between the two regions. A representation of this inter
dependence is given by either the crosscorrelation
function of the outputs of the two regions, or in the
frequency domain, by the cross-power spectrum between
the two outputs.
The relations for the cross-power spectra between
the outputs of each region are derived and are found to
involve transfer functions of the individual regions.
When these transfer functions are derived, they are
found to depart from the conventional one region trans
fer functions because of the term that couples the two
regions. The cross-power spectrum between the outputs
of the two regions is a complex quantity and its phase
part is found to be sensitive to the transit time of a
disturbance between the regions. Its amplitude part
as well as the phase part is found to be dependent on
the value of the multiplication factor of the undis
turbed region.
The cross-power spectrum between the outputs
of both regions of the UFTR was measured under two
conditions: (a) a random stationary external input to
region 1, and (b) no external input to either region.
Continuous data from two ion chambers were
amplitude modulated and stored on magnetic tape. The

7
cross-power spectra were then measured directly using
band pass filters and an analog computer.
These data from the UFTR indicate that the
\
propagation of a disturbance in one region of the
reactor to the other region may adequately be described
in terras of neutron waves. Also shown is the feasi
bility of measuring the multiplication factor of each
side and the magnitude of the coupling between the two
regions.

CHAPTER II
THEORY
The experiment to be analyzed consists of
measuring the cross-power spectrum of the outputs of
each region of a two region reactor. These outputs
are examined under two operational conditions: (a)
a random stationary external input to region 1 and,
(b) no external input to either region. These cross
power spectra are then related to the transfer
functions of the individual regions.
In the first section of this chapter, the
relations that involve the two coupled linear systems
and their inputs and outputs are developed. Although
these relations are general in nature and not
restricted to reactor systems, they are developed so
as to be directly applicable to the experiment. This
is evidenced by the transformation of the relations
from the time domain into the frequency domain so
that the linear systems involved are represented by
their transfer functions.
8

9
In the second section, these transfer functions
are developed from the reactor kinetics equations that
are derived for a two slab coupled reactor system.
The final section brings the cross-power spectra
and transfer functions together. After certain simpli
fying assumptions are made, the equations are examined
for the measurement of the parameters particular to the
two slab reactor system.
Cross-Power Spectra
Consider a two region coupled system as shown
in Figure 1. Each region is a linear system having its
own unit impulse response function h^t) and h2(t), and
outputs given by f^(t) and f2(t). The regions are
coupled to one another because a certain fraction, A^,
of the output of region 1 is received as an input to
region 2 after a transit time, 9 and vice versa.
Region 1 is also subjected to a random stationary
external input, i(t).
The crosscorrelation function of the outputs
of region 1 and region 2, 012(T), is defined as (),
T
where T is a continuous displacement time independent
of t. The two systems have been specified to be linear,

10
Output
f2<*>
Input
Aifi(t>0)
Input
A2f2 (t _@ )
Output
{> £i(t)
External Input
i(t)
Figure 1. Two Region System Diagram

11
so the convolution integral between the input, output,
and unit impulse response function is given as (19),
f].(t) = j h^A) Input(t-A )<*A (2)
J-oo
Reference to Figure 1 shows that the input to region 1
is the sum of the external input, i(t), and the frac
tion, A2, of the output of region 2, at a previous time
corresponding to the transit time, 0, between the
regions. Thus,
Input(t) = i(t) + A2f2(t 0 ) (3)
Equation (2) is now written as
fl(t) =/ **1 J- ao
Mt-A) + A2f2 which may be substituted into the definition of the
crosscorrelation function between the outputs of
region 1 and region 2, equation (1), to give
lim
1 r T r oo r
12 T->od2t/ I hl^ A ) | Kt- A)
012IT> -
+ A2f2(t-A 0)
dAf2(t+r )dt. (5)

12
By reversing the order of integration, equation (5) is
put into a more desirable form,
$12 =
hl(A>T i(t- A>f2 -T
(6)
hi (A)
(t-X~ 0 ) f2 ^t+ )dtd^ .
With this grouping, the inner integral in the first term
is recognized as the crosscorrelation function between
the external input and the output of region 2, while the
inner integral in the second term is the autocorrelation
function of the region 2 output. Equation (6) may now
be written in the compact form,
oo
$i2 vA>$i2aA
'-CD
( T + ,\ + 0 )dX
(7)
From equation (1), it is noticed that f2(t +T)
may also be expressed in terms of the convolution
integral between the input, output, and unit impulse
response function by,

13
r 00
f2(t) =/ h2(A) Input(t- A)^A (8)
J-co
In region 2 the only input is that fraction,
of the output of region 1 which reaches region 2
after the transit time,0 between the regions. Thus,
Input (t) = ) (9)
Equation (8) is written as
f2 (t)
OD
h2 ( A ) fj. (t-A $ )d A ,
-oo
(10)
which, when substituted into equation (1), gives,
0
'-T
00
h2( A)fi(t+T -A0 )d Adt.
(ID
Once again, inversion of the order of integration will
yield a more convenient grouping,
0 12^> A1
tx(t)f1(t+T-X-0)dtdX.
-oo
(12)

14
in which the inner integral is recognized as the
autocorrelation function of the output of region 1. Equa
tion (12) can now be written in a more compact form, and
with equation (7) relates the crosscorrelation of the
outputs of region 1 and region 2 with their unit impulse
response functions, their autocorrelation functions, and
the crosscorrelation of the external input and the output
of region 2.
0i 2
h20uaA
(13)
An obvious simplification is the case of no
external input to the two region system. Equation (13) is
unchanged, but equation (7) becomes,
012(T) =
CD
( A )022 (7" + A + $ )dX (14)
-oo
These relations may now be transformed into the
frequency domain by taking the Fourier transform of both
sides of both equations. As previously noted this is
done to relate the theory directly with the experiment.
Many investigators (16) (17) choose to measure cross
correlation and autocorrelation functions (rather than

15
the cross-power and power spectra) of the outputs and
obtain unit impulse response functions. The Fourier
transformation of the unit impulse response function,
which gives the transfer function, is performed on a
digital computer. If the cross-power and power spectra
can be measured directly, a computer operation may be
eliminated. The equipment available to the author also
suggested the direct measurement of the cross-power and
power spectra. Equations (7) and (13) must then be
%
examined in the frequency domain.
The Fourier transform of equation (7) is taken
i / j
by multiplying both sids of the equation by p ,
2TT
where J = PT. LJ is the angular frequency (radians
per second), and integrating with respect to T Equa
tion (7) then becomes,
CD
27T
0i2 (r )0JLyrdr
-oo
1
2 7T
ao
OD r 00
Q~jUTdT vAx^jCr + Aial
J-oo
+ wj e'iUT^r
J-CO J-CD
(15)
hj_ ( A )0 22 ^ +A+@)dA*

16
The crosscorrelation function and the cross-power spectrum
are a Fourier transform pair defined by (20)/
§12(U)
(>12 (T)e~jur T
and (16)
§12UJ)QUT U .
(17)
The term on the left side of equation (15) is then by defini
tion the cross-power spectrum between the outputs of region
1 and region 2, (^.^(CV)- In the ^^rst double integral on
the right side of equation (15), the change of variable,
x = T + A (18)
is made so that the integral becomes
-JOAx-A)^ f
CD
hl( A )0i2 (x)dA
-oo
(19)
In this double integral, a separation of variables puts all
terms involving x under one integral and all terms involving
A under another. Equation (19) becomes a product of two
single integrals,
0 hx( A)dA
CD
(20)

17
The unit impulse response function and the transfer
function are also a Fourier transform pair given by (19),
h(t)
dU
and
(21)
H (U)
h(t)0jU;t dt ,
(22)
where R(U) is the system transfer function. The first
factor in equation (20) is then recognized as the complex
*
conjugate of the transfer function of region 1, ^(U)
From equation (16), the second factor is identified as
the cross-power spectrum between the external input and
the output of region 2. In the second and last term on'
the right of equation (15) the change of variable
y T + A + 6 (23)
allows a separation of variables as in the first integral
which will give a product of two single integrals,
A? 0
j u6
L -'-QO
22S"jL/Ydy
(24)
Here, the first factor in brackets is the complex comjugate

18
of the transfer function in region 1, and the second factor
in brackets is the power spectrum of the region 2 output.
Thus equation (15) may finally be written,
$12(Cu/) = H*(L/)(|i2(CJ) + A2ejLy^H*(Ly) Equation (13) may be manipulated in the same manner
as was equation (7). Taking the Fourier transform of both
sides of equation (13) gives,
jfff 0i2e'JL/T J-OO
ai r -jut c
7W e iT h2 J-GD J -0D
(26)
A change of variable on the right-hand side of equation
(26) to,
-r A -9
(27)
will give, after a separation of variables,
^2 (A ) 0
-iuX
(28)

19
The left-hand side of equation (28) is recognized as the
cross-power spectrum between the outputs of region 1 and
region 2, (j}12(L/); the first factor in brackets on the
right-hand side as the transfer function of region 2,
H2(L/)i and the second factor in brackets on the right-
hand side as the power spectrum of the region 1 output,
(U) Equation (28) may now be written with equation
(25) to give the relation of the cross-power spectrum
between the outputs of region 1 and region 2 to their
transfer functions, their power spectra, and the cross
power spectrum between the external input and the output
of region 2.
(29)
If there is no external input, equations (25) and (29)
become,
(30)
(31)
It is noted that different symbols are used for the trans
fer functions in the set of equations (25) and (29) than
in the set of equations (30) and (31). The reasons for so
symbolizing these transfer functions are given in the next
section.

20
Relations may also be established for the
cross-power spectra between the external input and the
outputs of the two regions. The relations are derived
in the same manner as were those for anc* are'
and
Two Region Transfer Functions
The two transfer functions, and H2(L>/),
are now derived from the reactor kinetics equations for
a coupled reactor system. Each region is regarded as a
subcritical reactor with a neutron leakage interaction
from the other slab. Within each region, the theory is
space independent and a one group bare reactor model is
assumed. Cohn (21) has commented that the one group
bare reactor model is still used for practically all
kinetics work, even though it has been discarded as
impossibly crude for most statics calculations. His work
shows that for most reflected reactors, the kinetic
behavior corresponds to that of a bare reactor with the'
same 1/v lifetime.

21
The kinetics equations for region 1 may then be
written (18),
n
£
+
I A
cil + Sx + E
(34)
and
dcll /?lklnl
dt i
Aicii
(35)
where, n is the neutron density in region 1
%
A
K
ii
is the effective multiplication factor
for region 1
is the overall prompt neutron lifetime
in the system
is the fraction of fission neutrons in the
i-th group of delayed neutrons and
T.
i
is the decay constant of precursor of
i-th group of delayed neutrons
is the concentration of precursors of
i-th group of delayed neutrons in region 1
is the neutron density from external source
neutrons in region 1.
The term E is the interaction terra caused by the leakage
of neutrons out from region 2. This term may be repre
sented by some fraction, A2, of the neutron density in
region 2 at some previous time corresponding to the

22
transit time,$, between regions. Thus
E
A2n2(t -Q )
2
(36)
Expanding this expression in a Taylor's series about t,
A2n2(t) A20 dn2(t) h29 2 d2n2(t)
(37)
E
Baldwin (17) estimates 9 by assuming it to be
determined by the mean thermal neutron velocity. In the
UFTR, which has a 30 centimeter region of graphite
between the fuel regions, this assumption would indicate
a value of approximately 2 x 10^ seconds for 9 Such
a small value would make 9 relatively unimportant in
kinetics except for very short periods and unimportant
in the transfer function except at very high frequencies.
A more realistic assumption is that a disturbance in the
neutron density in one fuel slab travels across to the
opposite region with the velocity of a neutron wave.
The velocity of a neutron wave is frequency dependent
but at frequencies from 1 to 100 radians per second is
approximately constant at 1.4 x 10^ centimeters per
second in graphite (22) This gives a lag time, 9 in
the UFTR of about 2 x 10 3 seconds, a value that makes it

23
desirable to retain at least the first few terras of the
expansion of equation (37).
The kinetics equations for region 1 are then,
dni
dt
[Vl-£)-i] ni V 1 ,
1 V Ai 11
i2n2 A2 0 dn A2 Q2 d2n2
2 dt + 2 2
dt
and
dc
11 = Aklnl
dt £
-
il'
(38)
(39)
A similar set of equations exists for region 2. In the
experiment, a random reactivity input was given to region
1 only, while k2 remained constant. This fluctuation in
the multiplication factor in region 1 will give rise to
fluctuations in the neutron density and delayed neutron
precursor concentrations in both regions. There may also
be fluctuations in the external neutron source. It is
convenient then to linearize these quantities as follows:
+ (3 n0
nl
+
o
H
a
ii
U n1
n2
= n20
kl
- k10 + (
J k1
r*
Ci2
= ci20
Ci2
cuo + 1
5cn
S1
= S10
<5
12
(40)

24
When these definitions are substituted into the equations
of region 1, (38) and (39),
d A 4 [(fcio+kx) (l-/)-l ] (n^n^
-(niO^On,)
X A(c4 m+ ) +S,rt+(5s.
(41)
'ilO il' 10 w "1
£
(n20+n2)-e^(n20+(5n2)+-^- ji2(n20+n2)+ ...
2
0 £
2' 2 dt
and
a, .,s A(kio+(5ki) dt CilO+ ^ cil
(42)
- A(cio+(5c1)
The sums of the steady state portions of these equations
are equal to zero, and the equations are linearized by
setting the product, k (5 n, equal to zero since both
quantities will be small. The equations for region 1 are
then
d(5ni (1-AnxoH [k10(l-^)-l] <5! Yx <5,
at z + i + T 1
+ S-L +
(43)
and
d^cn = Anioki +
at
C
1 '
(44)

25
When the Laplace transform, using zero initial
conditions, is taken of both sides of both equations,
s
A i o + o + 4AlA'
il
A e
(45)
An2 0sAn2 + 62An2 ...
and
ft. .
'il
2
i
(46)
where s is the Laplace transform variable, the capital
letters denote the Laplace transform of the (5 quanti
ties (i.e., An^ is the Laplace transform of n^) and
the small letters are now understood to represent steady
state values. Equation (46) may be solved for A C^.
When this is done and A substituted into equation
(45) and terms collected,

26
If there is no disturbance of the multiplication
factor in region 1, then equation (47) becomes,
Since k2 is assumed to remain constant, the kinetics
equations for region 2, after manipulations similar to the
above, reduce to,
s
I
A
N
2
l-@s +
e
2c2
(49)
An,+A
The transfer function is defined as the ratio of
the system output to the system input when both are
expressed in Laplace notation. The inputs and outputs
used to derive the transfer functions must correspond to
the physical situation. It is not uncommon to speak of
two different transfer functions in connection with sub-
critical reactors; a source transfer function and a

27
reactivity transfer function. In the former, the input is
associated with fluctuations in the external neutron source,
while in the latter, fluctuations in the multiplication
factor or reactivity are considered the input. In the two
slab reactor system, each slab is considered as a sub-
critical system, so the same logic should apply. The
transfer function when an external reactivity input is
applied to a slab is a reactivity transfer function, while
the transfer function when there is no external input is
a source transfer function; the leakage of neutrons from
the opposite slab is considered as an external neutron
source.
This source transfer function is easily recognized
in the case of an external random input to slab 1 while
the reactivity of slab 2 is held constant. The fluctua-
*
tions in the multiplication factor of slab 1 cause
fluctuations in the neutron population in that slab. A
certain fraction, A-^'of this fluctuating population
leaks out of region 1 and may be considered an external
neutron source for region 2. This fluctuating neutron
source then causes the neutron population of region 2
to fluctuate. A source transfer function for region 2
may then be formed by the ratio of the output of slab 2
to the output of slab 1 when both are in Laplace notation.

28
When there is no external reactivity input to
region 1, fluctuations still exist in the neutron popula
tion of that slab. These are the so-called self
fluctuations or reactor noise. The statistical nature
of the fission process gives rise to these fluctuations
in reactor power levels even when the reactor is operated
at steady state. It has been shown (_7) that the power
spectrum of reactor (single region) self-fluctuations is
related to the transfer function, E[LJ), through,
Q + Q'
H (LJ)
(50)
where Q and Q' are constants. The expression has two
terms, the first being white noise of the detector and
the second being related to the kinetics parameters. The
transfer function used is the one based on reactivity,
although the exact input to the system is not definable
(16) .
Both slabs will exhibit these self-fluctuations,
but crosscorrelation should measure the effect of the
self-fluctuations of one region on the neutron density
in the other region. The transfer functions to describe
this situation would be source transfer functions, the
self-fluctuations in one side acting as the input to the
other side.

29
When a random reactivity input is given to region
1, the desired transfer function of that region is given
by
Alj/n-,
H (U) = -T (51)
Zak-l
where A N^/n^ is the normalized output, and A is the
input. This ratio is obtained from equation (47) after
An2 is eliminated by substitution from equation (49).
With the assumption of no external neutron source, solv
ing equation (49) for A N2 gives,
A N~ =
1 @s +
ev
A Nx
s -
k2(
l
y
a i s + aa
(52)
Substitution of this expression for A n2 into equation
(47) gives, after some algebraic manipulations and
setting s = jU and 0 j(JQ {3 Q. 2 ..,
H^L/) =
(1 /?>/£
(53)
r\ A1A2 ^-2jL/0
,, ,kiy ^lAl i e
l
£ i u+Xl V AAl
£ £ lu+Xl

30
The desired transfer function for region 2 when
a random variation of reactivity is inserted in region 1
is a source-based transfer function and is given by
h2 iU) =
An^
AnXAX 0
-j uB '
(54)
where
An2 is the output of region 2 and A N^A^ 0
is its input. H2(U) may be found from equation (49)
directly. Thus from equation (49),
H 20J) =
1
¡L
k2 (1 /? )-1 + y. Ai
~o
(55)
j u -
i
£ i j lj + A ^
When there is no external reactivity input to the
two slab system, both transfer functions are thought of
as source transfer functions in which the leakage from
one side is considered a source to the other side. The
transfer functions must then be formed by
Y X(U) =
,Ni
y2(L/) =
An2 *2e~iUd
An2
Anxax 6
(56)

31
Y1(L/) may be formed directly from equation (48),
YX(L/)
j u -
2
i
I
ju+A1
(57)
while Y2(U) Is formed directly from equation (49), as
was H2 (U)
Y2(U)
jU
i'e
*2 AAl
£ £ jlz+A
(58)
Reduction of Equations
If only the higher frequencies ( U > 1) are
considered so that the effects of delayed neutrons need
not be considered, and the parameter B is defined as
B
1
1 kx(l J3 )
l
(59)
then the four transfer functions are reduced to the
following:

32
Hj^fU) =
a
Bt + jU -
[aia2/£,] e"2jL^
B0 + iU
, (60)
H2((J) = Y 2iU) =
B0 + jU
(61)
and
y]L (U) =
B, + j(J
(62)
These expressions are now used in equations (25), (29),
(30), (31), (32), and (33). When an external reactivity
input is given to region 1,
^>12 (CJ) =
(1>8)/ §12^> + A2ejiJ^ §22
Bn jU -
e
2ju0
B, j U
[Ai $12(L/) =
b9 + JL/
-
ii
(U) + A~ 0
-JU0
(63)
(64)
$i2(L/)
B, + iU -
[aia2/£2] e ~2jU^
(65)
B2 + jU

33
and
[Ve] e'iue
<|>i2(L/) =
B2 + iU
(66)
The transit time between the regions, 0 has been con
sidered independent of direction. Further simplification
is effected if the slabs are assumed identical except
for their multiplication factors which may be different.
This assumption results in the condition
= A2 = A ,
(67)
and equation (64) becomes,
$12(U)=MLe
-iud $
ll
(U)
(68)
b2 + j U
The expression for when there is no external
reactivity input to region 1 is identical to equation (68).
The power spectrum, (^^(L/), is a real quantity but the
cross-power spectrum is a complex quantity which may be
expressed in terms of its phase and amplitude components.
If the complex quantity R12((-^) is defined as
^ 12
r12(l;) =
nOJ)
(69)

34
then it may be expressed in terms of its amplitude,/r^2 />
and its phase, p(R^2), parts, as
/R12/
2B22 + 2U2
bS + U
(70)
and,
p(R12) = tan
-1
-Ucos0 U B2Sin0L7 1
. B2COS 9U Usin 0 U J
(71)
Equation (70) shows that the amplitude portion of R^2
independent of 0 and that the shape of / R12 ¡ depends
only on the value of the multiplication constant of the
region not disturbed by the random reactivity input.
The phase portion of R12(U), however, is strongly de
pendent on the value of 0 particularly as the product
Qu becomes large. Figures 2 and 3 show plots of the
amplitude and phase portions of R^2(Cu/) versus frequency
as calculated from equations (70) and (71) for a reactor
of the geometrical arrangement of the UFTR. The nor
malized amplitude plot (Figure 2) is made for several
values of B: 100, 200, 500, and 1000. These values
correspond to multiplication factors of 0.97, 0.94, 0.85,
and 0.70, respectively, for a neutron lifetime of 3xl0-4
seconds. As noted before, this plot is independent of

Normalized Amplitude of Ri ?((*/), decibels
y
b
Figure 2. Normalized Amplitude Component of for Several Values of B

o
-30
-60
-90
120
150
180
210
240
270
1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 3. Phase Component of R12(L/) for Several Values of B and

37
the transit time between slabs. The phase plot (Figure 3)
indicates the dependence of the complex quantity R^2(L/0
on this transit time, 0 For B=500, the phase plots
for three different values of 0 are shown: first, for
0 equal to zero; second, for 0 corresponding to the
thermal neutron velocity; and third, for 0 correspond
ing to the frequency dependent thermal neutron wave
velocity. The value of the transit time between slabs
in the UFTR when a disturbance on one side is assumed to
travel to the other side with the velocity of a thermal
neutron wave is calculated in Appendix A. The phase
part of R^OJ) is also given in Figure 3 for B values
of 100, 200, and 1000, with 0 corresponding to the ther
mal neutron wave velocity. It is seen that for a par
ticular value of B, such as 500, the phase portion of
the quantity R12 (U) provides a rather sensitive test
for the validity of assumptions about the nature of 0 .
The measurement of R12(L/) would also allow the deter
mination of the parameter, B, which yields information
about the multiplication factor of the non-disturbed
region. In a balanced reactor system such as the UFTR,
in which the multiplication factors of the two reqions
are equal, Baldwin (26.) has shown that
A = B £
(72)
Thus a determination of B yields information about the
magnitude of the coupling of the two regions.

CHAPTER III
MEASURING SYSTEM
In this chapter are discussed methods and
equipment used to perform the direct measurement of the
cross-power spectrum of the outputs of the two regions
of the UFTR. The first section contains a development
of the relations necessary for the evaluation of the
real and imaginary parts of the cross-power spectrum
using band pass filters. These relations indicate the
sequence of operations that must be executed. The
second section contains a description of the University
of Florida Training Reactor (UFTR). Since the measure
ments were made on this reactor, its pertinent features
and dimensions are given. The third section examines
in detail each component used to collect, store, and
process data. The final section describes the random
reactivity input device.
i
Measuring Theory
Most investigators have used autocorrelation and
crosscorrelation measurements to obtain the transfer
functions of single region reactor systems. This method
38

39
lends itself well to digital processing of data. Results
of auto or crosscorrelation measurements are in terms of
the unit impulse response function, while the quantity
of interest is the transfer function. The power and
cross-power spectra, however, may be measured directly
and, as seen from the previous chapter, relate directly
to the system transfer functions.
The theory of the measurement of power and
cross-power spectra is outlined below. The relation
of the crosscorrelation function to the cross-power
spectrum is,
f
00
The definition of the crosscorrelation function,
T
may be substituted into (17) to give,
T
lim i
T oo 2T
f1(t)f2(t+T )dt =
>12iu)eiUT -T
OO
(73)

40
Here, f^(t) and f2(t) are voltages representing the neutron
density in each slab. The integration over t cannot, of
course, be performed over infinite time or over negative
times, so a finite average is used, resulting in the
approximation,
oo
fx(t)
f2(t+r )dt =
§12(u)ejUr du
'-CO
(74)
The first operation in the data processing is to pass both
signals through identical band pass filters, both of which
are centered at the same frequency, U 1, with the same
frequency band width, Equation (74) then becomes,
_1_
T
/ fl
'0 J
u, J
u
dt = $12(L>/1)^AUl
L (75)
The power spectrum measurement, when f1(t)=f2(t),
is considered first; then,
fx(t)
u£i(t+r)
1
u
dt -
L (76)
If T is set equal to zero, equation (76) is then written,
I

41
u
dt = §11(U1)hU1 ,
(77)
where the left-hand side is the mean square value of
f^t)
This expression is solved for
<£>11,
(78)
Figure 4 indicates the sequence of operations necessary
to solve equation (78) experimentally.
The cross-power spectrum is a complex quantity
and may be expressed in terms of its real parts, Re12,
and its imaginary parts, Im12. Use is also made of
Euler's relation,
p jLJ}T
^ = cos + jsinl^T (79)
to reduce equation (75) to
T
-rl fi(t>
o
u.
f2(t+r)
dt =
U.
(80)
J^e12cos^l^Ira12sin^l'T+j (Im12cos Cu/jT" +Re12sin(A>/1T)J A

42
fl(t)
Figure
fx(t)
f2(t)
<|)ii((J1)tAu1
set at
U1
Sequence of Operations Necessary for Experimental
Determination of Power Spectrum
Re 12{U1)T^U1
set at
ux
Figure 5. Sequence of Operations Necessary for Experimental
Determination of Cross-Power Spectrum

43
When T is set equal to zero, equation (80) reduces to,
_1_
T
rT i
r -
fi(t)
Jo J
f2(t)
U
i
dt =
ui
Re12 + jlm12
A(Jl.
(81)
The integral on the left side of this equation is a
real quantity so that when the real parts are equated,
Rei2 ((jJ])
T 1
fl(t>
Jo J
^£2 la
dt
tA u1
(82)
Returning to equation (80), one sets T equal to .
2 U1
Since both f^(t) and f2(t) have been put through
identical band pass filters and are both essentially at
the same frequency, U
f2 (+
ir
2 LA
la
= f2(t + 90)
LA
(83)
Thus equation (80) becomes,
(t + 90)
^-Im12+j Re12
(84)

44
Once again, the integral on the left side of equation
(84) is recognized to be a real quantity, so that when
the real parts are equated,
fx(t)
U.
f2(t + 90)
u.
dt
lm12 ((JL) =
(85)
Equations (82) and (85) may be solved
experimentally as shown in Figure 5.
The University of Florida Training Reactor
The UFTR is a lOkw, heterogeneous, light water
and graphite moderated, graphite reflected, thermal
reactor of the Argonaut type (23>) It may be fueled
with either 20 or 93 per cent enriched uranium-aluminum
fuel elements. The fuel is contained in thin flat
plates which are assembled in bundles of eleven plates.
The reactor core contains 24 bundles of fuel plates
placed in six water-filled aluminum boxes surrounded by
reactor grade graphite. Four cadmium control blades,
protected by magnesium shrouds, move between the fuel
boxes. Twelve inches of graphite separate the two rows
of fuel boxes. Aluminum headers connect to the bottom
of the fuel boxes and supply the light water used for
moderating and cooling. The tops of the boxes are

45
connected by aluminum overflow and vent pipes. This
water, overflowing from the fuel boxes, flows by gravity
through a cooling coil and then into a storage tank.
The water is pumped from the storage tank to the fuel
boxes. The storage tank prevents any sudden temperature
changes in the coolant, its capacity being about six
times that of the reactor system.
Air spaces 8 inches wide exist between the
graphite reflector and the inside of the biological
shield on the north and south sides of the reactor. The
ion chambers used to detect the output of each region of
the reactor were positioned in these spaces against the
graphite, as shown in Figure 6. A graphite layer of 18
inches separated the chambers from the fuel boxes.
The UFTR is provided with six horizontal beam
ports. These ports have a 4 inch diameter next to the
core and increase to a 6 inch diameter through the rest
of the shield. The core graphite stacking adjacent to
the south beam port is fitted with a removable graphite
plug 18 inches long by 2 inches diameter. When this
plug is removed, the south beam port extends to the
side of the south fuel box, at about the horizontal
center plane of the core. The random reactivity device
was inserted into the reactor through this facility.

Figure 6. Location of Ion Chambers in UFTR Core

47
Cross-Power Spectrum Analyzer
The measurement of the cross-power spectrum
between the outputs of the two regions of the UFTR
involved both transcribing output signals on magnetic
tape and subsequently processing these signals in the
manner indicated in Figure 5. Figure 7 is a flow
diagram of the equipment used to collect and store out
put signals of each region of the reactor on magnetic
tape.
Fluctuations in the neutron intensity of each
region were converted into current fluctuations by the
two Westinghouse 6377 compensated ion chambers placed
in the core. Two battery packs supplied 600 volts to
each chamber. No compensating voltage was used. Each
chamber was positioned outside one of the fuel slabs as
shown in Figure 6. Attempts to avoid ground loops in
the system included wrapping the ion chambers in trans
former cloth and covering all connectors with electrical
%
tape, since the chambers were in contact with the alumi
num and graphite in the core. These precautions were
found to eliminate from the measurements large amounts
of 60 cycle interference.
About 15 feet of coaxial cable were used to
connect the ion chambers with the Keithley Model 410
Micro-microammeters located immediately outside the UFTR

50
oo
Figure 7. Data Transcribing System

49
shield. The micro-microamraeters converted the signals
from current fluctuations to voltage fluctuations which
were then amplified on an Applied Dynamics Analog Com
puter, located in the reactor control room, so that the
ac portion of each signal was about 10 volts peak to
peak. The analog computer was also used to buck-out
the dc produced by the steady state power level of the
reactor. Two Krohn-Hite Ultra-low frequency band pass
filters were used as high and low pass filters. The
low cut-off frequencies were set at 0.1 cycle per
second to remove all dc and any slow drifts in reactor
power. Frequencies above 1000 cycles per second were
of no interest in this experiment and were removed by
setting the high cut-off frequencies at 1000 cycles per
second. The signals were then amplitude modulated on an
80,000 cycles per second carrier. Amplitude modulation
recording and playback is subject to considerable fluctu
ation in over-all gain (24j. Because only approximate
spectra were required in order to determine the mechanism
of the lag between regions and to test the feasibility
of the measurement of the multiplication factors of the
regions, AM recording was employed. AM recording and
detection equipment was also available. A schematic of
the modulator is shown in Figure 8.

50
Noise
Signal
PT5 0.0015//f
To
Tape
Recorder
Figure 8.
Modulator Schematic

51
The modulated signals were then recorded on two
channels of an Ampex 500 magnetic tape recorder. A tape
speed of 60 inches per second was used for both record
ing and playback. Modulation was necessary because of
%
the frequency response of the tape recorder.
Figure 9 presents a flow diagram of the equipment
used to measure the real and imaginary parts of the
cross-power spectrum between the outputs of the two
regions of the UFTR. The signals from the tape recorder
. were first demodulated, using the circuit shown in
Figure 10. The demodulated signals were then amplified
by a factor of about 25 on the analog computer in order
to have signals of about 10 volts peak to peak to the
band pass filters. The band pass filters were set for
the same frequency and band width. A "zero" band width
was used in which the upper and lower cut-off frequen
cies were set at the same value. The band pass filters
allow frequencies outside the band width to pass through
even though these "outside" frequencies are uniformly
and greatly attenuated. When the upper and lower cut
off frequencies are equal to f., the half-power, or
three db down, points are at 0.77f^ and 1.30^. Thus,
if a constant percentage or a "zero" band width is used,
any error due to the "outside" frequencies is normalized.
The larger the percentage band width used, the less the

M
Ul
to
Figure 9. Cross-Power Sectrum Analyzer

53
From Tape
Recorder 0
|
Demod
Demod
6AL5
To
#1
#2
Rx 6.8kQ
R-L 5.6kQ
8.2kQ
iokQ
Figure 10. Demodulator Schematic

54
insertion losses in the filters become; however, the
resolution also becomes poorer.
and f2(t)
, were
J U,
The two signals, (t)
V
then given a phase lag in a low pass filter and
was also given a phase lead in a high pass
f2(t)
filter such that there was a +90 degree shift between
it and the lagging f-^(t)
u.
and f2(t)
. The phase
U,
shifting networks were more complex than the simple
filters indicated in Figure 9 because each filter fed
into either a one megohm or a one-tenth megohm input
resistor of an operational amplifier on the analog
computer. Since the amplifier summing junctions are
essentially at ground potential, either a one megohm
or a one-tenth megohm resistor was in parallel with the
capacitor, C, in the low pass filters and in parallel
with the resistor, R, in the high pass filter. It is
easily shown (25) that the amplitude, /af, and the
phase, p, portions of the transfer functions for these
filters are given by, for the low pass filters,
K
(R + K)2 + R2K2U2C2
(86)

55
and
PL
tan-1
rkUc
R + K
(87)
and for the high pass filter,
rkUc
*V (R + K)2 + r2k2U2c2
(88)
and
tan1
R + K
rkUc
(89)
where K is the value of the input resistor to the
operational amplifier. Since the signals were passed
through the band pass filters and were at essentially a
single frequency, a convenient R and C were selected to
give the desired 90 degree phase shift at each frequency
set point. It was also found to be convenient to select
a RC combination such that the amplitude, / a/, of the
high pass and the low pass filters were equal at a given
frequency. This was accomplished when
R
Uc
2 7Tfc
(90)

56
Precision O.1% polystyrene capacitors were
available in values of 0.01, 0.1, and 1.0 microfarads
and were used with precision tl% resistors and ten turn
t3% IOOkQ potentiometers to set the C and R values of
the filters. The capacitor and resistor values used
are given in Appendix B.
It is noted that when an R and C are selected
such that equation (90) is satisfied, the values of pL
and pH are such that there is a +90 degree shift be
tween them and the attenuation is the same through both
filters. This attenuation is, however, frequency
dependent. The calculation of correction factors for
this attenuation is straightforward and also given in
Appendix B.

The three signals,f^(t)
, f2(t)
, and f (t+90O)

^ J
u, J
were then amplified by a factor of from about 20 to about
400 on the analog computer. This amplification was
necessary in order to have signals of about 200 volts
' peak to peak for inputs into the Model 160 electronic
multipliers on the analog computer. Large input signals
are necessary to minimize the error that these diode
network multipliers introduce for low input voltage.
*These symbols are now taken to represent the
signals after they have passed through the phase shift
ing networks.

57
Before multiplication, however, the signals were passed
through high pass filters to remove any dc components
and low frequency drifts introduced because of the high
gain of the system. The nominal 1.0 microfarad capaci
tors and 10 megohm resistors used were actually 1.09
microfarad and 9.9 megohms, but were matched to within
1 per cent. The outputs of these filters were fed
into the one megohm input resistors of operational
amplifiers. Here again, attenuation and phase shift
were introduced, especially at the lower frequencies.
The phase shift was identical for all three signals
and was ignored. The amplitude attenuation is calculated
in Appendix B.
Since both the phase changer and high pass filter
, and
attenuations were the same for f^(t)
LA
,
LA
fo(t+90)
, the same correction was valid for all
LA
three. These corrections were squared, however, because
the attenuations occurred before the multiplications.
The separate and combined correction factors that were
applied to the real and imaginary parts of the cross
power spectrum are given in Appendix B.
Multiplication was performed after the signals
were passed through the high pass filters and the

58
products f^(t)
f2(t)
and f^(t)
f2 (t+90)
obtained. These products were then integrated and the
integrals related to the real and imaginary parts of
the cross-power spectrum. If the gain of the opera
tional amplifiers between the band pass filters and the
multipliers was M, the gain of the integrators (1/RC)
was P, the voltages on the integrators were VR for the
real and V^. for the imaginary parts for an integration
time of T seconds, and the set frequency and frequency
band width were f-^ and Af^, respectively, then the
real and imaginary parts of (^)12(L/) were given by,
Vr(CF)
Rei2(fl} ~ A
T Af. (MPP
and
Vj(CF)
In*12 (f )
Afx(M)
(91)
(92)
where CF was the total correction factor for the ampli
tude attenuation of the phase shifters and high pass
filters.
The measurement of the quantity R12(CJ) as
defined by equation (69) requires that both the

59
cross-power spectrum between both outputs and the power
spectrum of the output of the externally excited region
be measured. The real and imaginary parts of the cross
power spectrum were measured by use of equations (91)
and (92). The power spectrum of region 1 was measured
with the same computer setup as was the cross-power
spectrum except that f^(t) only was used instead of
f^(t) and 2(t). Since the power spectrum was a real
quantity, there was only a voltage, V, on the integra
tor giving the real parts. For the same computer set
up as the cross-power spectrum, the power spectrum was
given by,
V(CF)
(93)
Thus the real and imaginary parts of R^2^fl^ were
given by,
Re [r12<*1>]
(94)
V
and
V
I
(95)
V

60
Thus the amplitude correction factors for the
low and high pass filters were unnecessary in the calcu
lation of R12(f). They were employed, however, when
either the cross-power or the power spectra were
measured.
Random Reactivity Input Device
The reactivity input system consisted of (a) a
drive assembly of a coil in an electromagnet with spring
return, (b) an absorber in the forro of a piston moving
in a cylinder, (c) a power amplifier for the moving coil,
and (d) a random input signal.
The drive and piston assembly was constructed
by V. Rajagopal and is described in his doctoral dis
sertation (16). Figure. 11 illustrates this assembly.
The drive assembly was a coil moving in the air gap of
an electromagnet. The flux in the air gap was main
tained by a steady current in the field coil. A 28
volt 3 amp power supply was used for the electromagnet.
The force exerted by the moving coil was controlled by
the current supplied from the power transistor in the
power amplifier. Mechanical coupling from the moving
coil to the cadmium covered piston was through a steel
wire encased in a square slot between two blocks of
graphite. The opposite end of the piston was connected

Figure 11. Random Reactivity Input Device

62
to a tension spring mounted inside a graphite block.
The graphite blocks and the cylinder of the absorber
element were rigidly connected to the electromagnet.
A schematic diagram of the amplifier that drove
the moving coil is given in Figure 12. The oscillator
system was found to follow sinusoidal inputs in the
frequency range 0.1 to 50 cycles per second at the max
imum length of stroke (about 3/8 inch) while for higher
frequencies, up to 100 cycles per second, about half
stroke could be obtained. When the device was mounted
in the south beam port, full stroke held about 0.015
per cent reactivity.
Two types of random noise generators were used
as inputs to the power amplifier. The first was a
commercial noise generator. The Elgenco, Inc., Low
Frequency Noise Generator gave a 12 volt rms noise sig
nal whose power spectrum was uniform from 0 to 35 cycles
per second. The dc level of this signal was less than
50 millivolts.
A second random signal was constructed with a
radioactive sample, GM counter, and a count rate meter.
The signal was taken from the output of the flip-flop
section of the count rate meter. This output was a
randomly switched square wave whose mean period varied
as the reciprocal of the count rate. This signal is

-25v
Figure 12. Power Amplifier Schematic

64
also called a "random telegraph signal" and its power
spectrum is well known as (20)
$11 tu
4a2g
1
+ 4g2
(96)
where g is the average number of sign changes per
second of a square wave that varies between +a and -a.
The power spectrum of this signal was then made white
out to the desired frequency by a corresponding increase
in the count rate seen by the GM tube. The signal was
given the proper amplitude and dc level on the analog
computer.
The reactivity device was sufficiently long to
permit the piston end to be fully inserted into the
south beam port extension, with the drive assembly out
side the south reactor shield.

CHAPTER IV
EXPERIMENTS AND RESULTS
This chapter contains a description of the
experiments performed both for check-out and calibration
and for the measurement of the parameters of interest of
the UFTR. This description includes data that were
taken, data reduction, and the confidence limits assigned
to the results. A detailed description of the procedure
used to process the data is given in Appendix C.
The first section deals with the calibration
runs in which the transfer function of a low pass filter
was measured by obtaining the cross-power spectrum
between a random input and the output of the filter from
this input. The error analysis associated with this
calibration is used with the data obtained from the
reactor.
The second section presents data obtained from
the reactor and data reduction. Data obtained both
with and without an external input are presented.
Calibration
Performance of the cross-power spectrum analyzer
and the data transcribing system were checked with the
65

66
measurement of the transfer function of a low pass
filter.
When a linear system is excited by a random
input, the cross-power spectrum between the input and
the output, CD (iJ), is equal to the product of the
-1 io
power spectrum of the input, (^^(L/), and the trans
fer function of the system, G(L/) (19) The real and
imaginary parts of the transfer function may be
measured by,
Re [g(U)]
and
Im[G((J)]
Re
<£>io(u>
#<£^)
(97)
Im
[*
io
(U)
§iiJ>
(98)
The low pass filter, the transfer function of
which was measured, was formed by a one megohm resistor
followed by a 0.1 microfarad capacitor to ground.
These were precision tl% components. The output of
the filter fed a 0.1 megohm input resistor of an
operational amplifier on the analog computer. The
amplitude and phase parts of the transfer function,
G(U) were calculated from,

67
K
(86)
and
PG(Ly)
tan
-1 f rkUc
(87)
R + K
where
K = 0.1 mQ
R = 1.0 mQ
C = 0.1 /f.
These two functions are plotted as solid lines versus
frequency in Figures 13 and 14.
Performance of the cross-power spectrum analyzer
independent of the data storage system was first examined
by taking the cross-power and power spectra measurements
"on line." A randomly switched step function was given
as an input to the filter system. The step function was
%
obtained from the flip-flop section of a count rate
meter the input of which was supplied by a GM counter
setup. The power spectrum of the input was measured as
indicated by equation (93) and the real and imaginary
parts of the cross-power spectrum as indicated by equa
tions (91) and (92).

68
Two such runs were made, Run L and Run M. The
only difference between these was that the GM counter set
up was adjusted to give a count rate of 1700 counts per
minute in Run L and 13,500 counts per minute in Run M.
Results of these runs are given in Tables 1 and 2 and plot
ted in Figures 13 and 14. The power spectrum of the random
step input of Run L is also obtained from the measurement
and given in Table 1. Note that the amplitude correction
factors (Appendix B) must be used for the power spectrum.
This measured spectrum is seen in Figure 15 with the the
oretical spectrum as calculated for a g of 28.3 counts per
second from equation (96).
The "flags" on the data points plotted in the
above figures indicate one standard deviation. No "flags"
are shown to indicate the accuracy of the frequency
readings, which were determined by the accuracy of the
settings on the band pass filters. The manufacturer gives
5% as the accuracy of the frequency settings on the
Krohn-Hite ultra low frequency band pass filters. It was
necessary to calibrate the band pass filters against a
frequency standard to match the filters to within one
per cent of each other and the standard.
The non-statistical errors associated with the
cross-power spectrum analyzer were small and are

69
TABLE 1
RESULTS
OF RUN L
f
Normalized
Phase Angle
Normalized Power
Amplitude
of G(U)
Spectrum of
cycles per
of G(U)
Input
second
decibels
degrees
decibels
0.15
99.2
-0.6
99.1
0.2
97.6
-0.4
100.1
0.3
98.4
-4.6
99.0
0.4
99.0
-2.2
93.9
0.5
99.4
-1.1
98.3
0.6
101.4
-3.0
98.2
0.8
101.1
-4.2
97.9
1.0
100.0
-4.0
99.8
1.5
100.6
-6.7
98.7
2.0
100.6
-9.1
100.1
3.0
99.6
-12.4
98.4
4.0
99.4
-15.5
99.2
5.0
100.6
-18.0
97.7
6.0
100.0
-20.4
97.3
8.0
100.0
-26.1
94.6
10.0
99.4
-30.2
92.7
15.0
97.2
-39.0
88.3
20.0
96.2
-48.1
83.5
30.0
94.2
-57.3
77.3
40.0
93.6
-65.0
74.6
50.0
91.0
-66
70.3
60.0
90.2
-74
65.9
80.0
88.6
-79
61.5
100.0
88.2
-85
56.7
150.0
87.6
-86
48.4

70
TABLE 2
RESULTS OF RUN M
f
cycles per
second
Normalized Amplitude
of G(U)
decibels
Phase Angle
of G(U)
degrees
1.0
101.8
-5.3
1.5
100.2
-5.6
2.0
100.6
-9.9
3.0
100.6
-14.0
4.0
99.2
-15
5.0
100.1
-19
6.0
99.8
-15
8.0
99.6
-27
10.0
99.4
-32
15.0
98.4
-42
20.0
97.6
-54
30.0
94.6
-63
40.0
92.4
-68
50.0
92.0
-71
60.0
89.2
-75
80.0
87.2
-78
100.0
85.8
-82
150.0
85.4
-85

Normalized Amplitude of G(ij), decibels
110
J ill 1 i i I i i i
0.2 0.5 1.0' 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 13. Normalized Amplitude Component of G(CJ) from Runs L and M

o
10
20
30
40
50
60
70
80
90
-oo
Frequency, cycles per second
Figure 14. Phase Component of G(CJ) from Runs L and M

120
110
100
90
80
70
60
50
40
30
*
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 15.
Normalized Power Spectrum of Random Step from Run L

74
estimated to be within 2 per cent. These errors are
those associated with the components of the phase shift
ing network, potentiometer settings, computer amplifiers,
multipliers, and the integrator amplifier readout. To
hold these errors to within a few per cent, all potenti
ometer setting and amplifier readout was executed with
the null potentiometer, all equipment allowed to warm
up for an hour before data processing, and the amplifiers
on the computer were balanced at frequent intervals.
The accuracy of the cross-power spectrum analyzer
was limited primarily by statistical considerations. In
the development of the theory of the measurement of
power and cross-power spectra, it was necessary to form
equation (74) from equation (73) by replacing an average
over all time with a finite average time, T. The stand
ard deviation expected from the effect of the finite
averaging time has been derived by Bennett and Fulton
(2,6) They give for the fractional standard deviation
(0/N) of a measurement of the power spectrum of nor
mally distributed noise,
07 N
1
(99)
where T is the integration time and Af is the frequency
difference between half-power points in the band pass

75
filter. The expression may be used as a first order
approximation to cases involving other distributions
and spectral shapes as well as to cross-power spectra.
Three voltages were measured, V V and V,
R X
corresponding to the real and imaginary parts of the
cross-power spectrum between the outputs of the two
regions of the UFTR and to the power spectrum of the
output of the externally disturbed region. Since the
same integration times and frequency band widths were
used for the measurement of all three voltages, all
three have the same fractional standard deviation,
namely,
In most cases, a three minute integration time
was used. This value was taken as a compromise between
the drift in the analog computer and a reasonable sta
tistical accuracy.
Given the standard deviation of the voltage
readings, the standard deviation of the amplitude and
phase parts can be calculated from the propagation of
errors formula,
o'
77tAi
(100)

(101)
£
x
which gives the standard deviation, (J f, of a function
f(x,y) of the two quantities x and y which have standard
deviations, (J and 0*v, respectively,
x y
The above expression gives the standard deviation
of the phase part (given by the quotient Vl/vR) as,
(102)
and the standard deviation of the amplitude part
) as,
(given
(103)
As seen in Figure 13, the standard deviation of
the amplitude part at the lower frequencies is rather
large. The standard deviation of the phase part, how
ever, is of a magnitude comparable with the size of the
data points.

77
The data of these figures do, however, indicate
the satisfactory performance of the cross-power spectrum
analyzer. The data of Figure 15 demonstrate that with
the correction factors of Appendix B, the analyzer may
also be used successfully for the measurement of power
spectra.
Referring to Figure 13, one observes that the
data of Run L depart from the theoretical amplitude at
frequencies beyond 50 cycles per second. This departure
is a leveling out of the amplitude plot to the noise
level of the analyzer. At 50 cycles per second Figure
15 shows that the random input power spectrum is down
more than 30 decibels from the value at the lower fre
quencies. In Run M the power spectrum of the input
carries more of the high frequency components and because
of this, the cross-power spectrum data extend out to
about 100 cycles per second. The phase data points fol
lowed the theoretical curve out to 125 cycles per second
for both runs.
Run P was a run of the same type as L and M, but
instead of processing the data "on line," the two sig
nals, the random step input and the filter output, were
AM modulated and stored on magnetic tape. The random
step was formed from a count rate of 7100 counts per
minute from the GM counter. The data from Run P as

78
processed from the tape are given in Table 3. The data
points of the amplitude and phase parts of the transfer
function of the filter, G(U), are shown plotted versus
frequency in Figures 16 and 17. Once again, the the
oretical curves are given also. The data from this run
seem more scattered than the data points of Runs L and
M, especially those of the amplitude part. This addi
tional scatter must be attributed to the AM modulation,
recording, and demodulation. The magnitude of the
amplitude scatter is about -2 decibels, while little
difference is noted in the phase part. In all other
respects the results of Run P were identical to those
of Runs L and M. The amplitude portion of the data
departs from the theoretical curve between 60 and 100
cycles per second, a higher frequency than that at which
the departure occurred in Run L and a lower frequency
than that at which the departure occurred in Run M.
This was to be expected since the random input was
driven by a count rate higher than Run L and lower than
Run M.
Run P demonstrates that the techniques and
equipment used to collect, record, and analyze the data
were satisfactory.

79
TABLE 3
RESULTS OF RUN P
f
cycles per
second
Normalized Amplitude
of G(U)
decibels
Phase Angle
of G (U)
degrees
0.6
95.8
-6.1
0.8
101.2
-1.2
1.0
99.0
-6.7
1.5
98.4
-4.1
2.0
96.6
-6.5
3.0
98.6
-10
4.0
100.8
-18
5.0
98.2
-18
6.0
99.4
-25
8.0
97.0
-33
10.0
98.4
-30
15.0
100.0
-37
20.0
96.8
-51
30.0
94.2
-60
40.0
93.6
-62
50.0
90.8
-70
60.0
89.8
-72
80.0
89.2
-77
100.0
86.4
-80
150.0
86.6
-90

Normalized Amplitude of G(L/), decibels
Frequency, cycles per second
Figure 16. Normalized Amplitude Component of G(LJ) from Run P

Phase Angle of G(U) degrees
Figure 17. Phase Component of G(iJ) from Run P

82
Reactor Data
Some of the first reactor data were taken "on
line." The external reactivity input device used for
these runs was driven by a random step function at an
average count rate of 200 counts per minute. Because
of the low count rate and inadequate frequency response
of the reactivity input, the data obtained were not
reliable at frequencies beyond about two cycles per
second. Although the data looked promising out to this
point, little could be determined, as seen from Figures
2 and 3.
The random input device designed and constructed
by Rajagopal and described in the previous chapter was
used in succeeding runs. These runs were recorded and
processed "off line."
The reactor was fueled with 3100 grams of
uranium-235 in the form of 93 per cent enriched uranium-
aluminum fuel plates for Run R. The commercial low
frequency noise generator was used as an input to the
driving amplifier of the input device and the data were
recorded with the reactor operating at a power level of
10 watts. The two Keithley micro-microammeters were
operated on the 3xl0-6 ampere scale. The results of
this run are given in Table 4. Not only the phase and
amplitude parts of the function R12(C^) but also the

83
TABLE 4
RESULTS OF
RUN R
f
Normalized Amplitude
Phase Angle
Normalized Power
cycles
of r12 (U)
of Rj_2 ((jJ)
Spectrum of Out-
per
put of Region 1
second
decibels
degrees
decibels
0.6
98.8
-6.2
78.9
0.8
100.5
0
88.3
1.0
99.7
-7.0
89.2
1.25
99.3
-5.0
99.6
1.5
101.7
-0.7
98.5
2.0
99.5
-14
101.6
2.5
102.7
-7.0
99.4
3.0
99.3
-11
101.3
4.0
98.3
-13
99.0
5.0
100.1
-12
98.1
6.0
99.3
-15
95.0
7.0
103.1
-17
91.6
8.0
99.9
-16
93.1
9.0
99.7
-23
89.8
10.0
98.9
-26
90.6
12.5
98.7
-20
85.0
15.0
101.3
-28
79.2
20.0
97.8
-39
76.4
25.0
96.5
-45
71.4
30.0
95.6
-40
70.2
40.0
91.5
-34
68.7
50.0
88.7
-19
70.4
60.0
89.3
-3.2
71.1
70.0
86.5
0
70.2
80.0
85.3
-13
66.9
100.0
79.1
-44
64.6

84
power spectrum of the output of the south region (the
externally disturbed region) are given. The amplitude
part of R12 (CJ) the phase part of and the
power spectrum of the output of the south region are
plotted versus frequency in Figures 18, 19, and 20,
respectively.
The phase data points, up to about 25 cycles
per second, although badly scattered, are seen to fol
low the set of theoretical curves calculated for a lag
time between the two reactor regions corresponding to
the velocity of a thermal neutron wave. A B value of
about 400 is indicated. Beyond a frequency of 25
cycles per second, little can be determined because of
the excessive scatter of the data. The amplitude data
points depart from the theoretical curve for a B of
400 beyond 15 cycles per second. It is difficult to
determine whether this break in the amplitude data
actually reflects a low value of B or if it signifies
the end of correlation between the two reactor output
signals because of low signal to noise ratio. The
phase data would suggest the latter condition. Con
sideration of the power spectrum of the output of the
south fuel slab (Figure 20) would also indicate that
correlation would end much beyond 20 cycles per second.
This power spectrum, which is representative of the

Normalized Amplitude of R10(L/), decibels
110
100
90
80
70
0D
Ln
_J I I I I i I I I I u
0.2 0.5 1.0 2.0 5-0 10 20 500 100 200 500
Frequency, cycles per second
Figure 18. Normalized Amplitude Component of from Run R

Phase Angle of R,9(L/) degrees
+20
1
+ 10
-10
-20
-30
-40
-50
-60
-70
Curve
1
2
3
4
5
6
100
200
500
1000
500
500
Neutron Wave
Neutron Wave
Neutron Wave
Neutron Wave
2x10~4 seconds
Zero
0.2
0.5 1.0 2.0 5.0 10 20 50 100
Frequency, cycles per second
200
00
O'
500
Figure 19. Phase Component of from Run R

120
110
100
90
80
70
60
50
40
30
O o
O Q

G
Go
s
o
o
o
O
o
o o


00
_J I I I I L 1 1 1 I Ll
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
/
Frequency, cycles per second
Figure 20. Normalized Power Spectrum of Region 1 Output from Run R

88
input to the north slab, is down about 20 decibels at 15
cycles per second from the amplitude value at the lower
frequencies. Except for a hump at 60 cycles, the power
spectrum levels out to the reactor and instrument noise
level at frequencies beyond 25 cycles per second. Since
the random reactivity input device had been found to
respond satisfactorily to sinusoidal inputs from 0.1 to
25 cycles per second, the cause of the drop-off of the
power spectrum at the lower frequencies was unknown.
Full stroke of the device held about 0.007 per
cent reactivity. When the noise generator was used to
drive the reactivity input device, its effect could not
be seen by the reactor operators on the console instru
mentation. Observation of the signal from either of
the chambers in the reactor revealed that the input
disturbance was about one-half the magnitude of the
inherent reactor noise.
To obtain a larger input reactivity disturbance
the absorber piston assembly was reconstructed so as to
be about double the exponsed cadmium area. The tension
spring was changed and the power amplifier was also
modified to obtain better frequency response. With
these modifications, the device was found to respond
faithfully to sinusoidal inputs up to 50 cycles per
second. Full stroke was found to produce more them
0.015 per cent reactivity change in the reactor.

89
For Run CC the reactor was fueled with 3500 grams
of uranium-235 in the form of 20 per cent enriched
uranium-aluminum alloy fuel plates. The reactor was
operated at a power level of 50 watts and the two Keith-
ley micro-microarameters were operated on the 3x10^
ampere scale. The random reactivity input device was
found to produce disturbances of almost twice the magni
tude of the normal reactor noise. The power amplifier
was driven by a randomly switched square wave. This
random square wave was constructed with a GM counter
and the flip-flop section of a count rate meter. A
count rate of 20,000 counts per minute was used.
The results of Run CC are presented in Table 5.
Once again the phase part of R12(CJ)/ the amplitude part
of R12(L/) and the power spectrum of the output of the
south fuel region are listed in Table 5 and also plotted
versus frequency in Figures 21, 22, and 23, respectively.
The phase data points, up to a frequency of 40
cycles per second, follow the theoretical curve corres
ponding to a B of about 450 and a lag time, 6 deter
mined from the velocity of a thermal neutron wave.
Beyond 40 cycles per second, the data points become
excessively scattered. The amplitude data points break
from the theoretical curve for a B of 450 at frequencies
beyond 25 cycles per second. Again, it is difficult to

90
TABLE 5
RESULTS OF RUN CC
f
Normalized Amplitude
Phase Angle
Normalized Power
cycles
of R12
of ri2(l;)
Spectrum of Out-
per
put of Region 1
second
decibels
degrees
decibels
0.4
101.0
0
101.4
0.5
99.5
-2.0
102.1
0.6
100.7
-5.3
100.7
0.8
100.0
0
99.3
1.0
102.0
-4.4
99.5
1.25
98.5
-2.0
102.6
1.5
101.1
-2.1
99.2
2.0
102.2
-5.1
98.8
2.5
99.2
-5.2
100.1
3.0
100.0
-7.0
97.0
4.0
102.0
-8.5
95.8
5.0
99.2
-10
9'4.7
6.0
101.4
-11
90.9
7.0
101.6
-14
89.7
8.0
99.0
-17
88.3
9.0
100.7
-18
85.7
10.0
98.7
-25
84.9
12.5
100.2
-22
78.0
15.0
100.2
-25
71.5
20.0
99.0
-39
66.3
25.0
100.0
-43
59.3
30.0
95.6
-49
58.0
40.0
93.9
-64
52.8
50.0
99.4
-65
52.3
60.0
90.5
-34
53.0
70.0
93.4
-62
48.7
80.0
83.7
-84
48.9
90.0
83.5
-56
48.6
100.0
80.9
-62
47.7
125.0
81.9
-70
46.3
150.0
80.4
-116
43.3

Normalized Amplitude of R,~(LJ), decibels
110
100
4
_J I I I I I I 1 I I
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 21. Normalized Amplitude Component of Ri2(lc/) from Run CC
500

Phase Angle of R12 (CuO # degrees
+10
0
-10
-20
-30
-40
-50
-60
-70
-80
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
vD
to
Figure 22. Phase Component of R^2from Run CC

120
110
100
90
80
70
60
50
40
30
§ § a
Q O O
O o
O
O
O
O
o O o
000 0
o
o
u>
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 23. Normalized Power Spectrum of Region 1 Output from Run CC

94
determine if the break represents a lower value of B or
if it represents the end of correlation. The power
spectrum of the region 1 output indicates that corre
lation would end much beyond 25 or 30 cycles per second.
At 30 cycles per second the power spectrum is down 40
decibels from the amplitude at the lower frequencies.
From the calibration runs it would be expected that
correlation would not continue when the input (which is
the output of the south fuel region in this case) power
spectrum drops off 40 decibels. There was no drop-off
of the spectrum at the lower frequencies as in Run R,
indicating a more satisfactory operation of the reactiv
ity input device. The magnitude of the input signal was
also greater than before resulting in the lower fre
quency portion of the spectrum being 50 decibels above
the reactor and instrument noise level.
A check was made on the frequency at which
correlation between the two outputs of the reactor ended
with a complementary run to Run CC. Run Z was taken
under the same operational conditions as Run CC except
that both ion chambers were located side by side outside
the south fuel region. The results of this run are
presented in Table 6. The amplitude and phase parts of
the cross-power spectrum and the power spectrum of the
output of the south fuel region are plotted versus fre
quency in Figures 24, 25, and 26, respectively.

95
TABLE 6
RESULTS OF RUN Z
f
Normalized Amplitude
Phase Angle
Normalized Power
of Calibration
of
Spectrum of Out-
cycles
Spectrum
Calibration
put of Region 1
per
Spectrum
second
decibels
degrees
decibels
0.4
99.9
+3.0
100.1
0.5
99.3
-0.8
103.1
0.6
101.7
-0.9
99.5
0.8
98.1
-1.5
101.3
1.0
101.0
+ 2.0
101.2
1.25
101.2
+ 0.5
99.3
1.5
98.6
-0.5
100.2
2.0
101.0
0
100.7
2.5
100.5
0
99.6
3.0
99.9
-0.6
97.9
4.0
101.7
0
97.4
5.0
101.0
0
93.8
6.0
97.2
-3.0
94.6
7.0
101.7
-1.3
89.9
8.0
100.1
-0.5
87.1
9.0
96.8
-0.7
87.7
10.0
101.4
-7.7
83.6
12.5
99.9
+ 3.6
77.9
15-0
97.8
-4.5
75.4
20.0
98.9
0
67.3
25.0
96.9
-2.0
63.0
30.0
94.4
-7.5
60.0
40.0
93.8
-10.0
55.6
50.0
87.0
-5.6
55.7
60.0
80.7
-11.0
56.0
70.0
84.6
+ 8.4
54.4
80.0
80.7
+ 13.0
53.0
90.0
79.6
-13.0
52.5
100.0
85.5
-44
50.5
125.0
80.2
-70
48.4
150.0
80.1
+98
47.4
200.0
79.9
-132
46.3

Amplitude of Normalized Spectrum, decibels
vo
Figure 24. Amplitude Component of Normalized Cross-Power Spectrum from Run Z

+ 30-
co
d)
u
O'
<1>
T3
0)
f
O'
<1)
(0
J
I
+20-
+10-
-30
-40
-50
-60 -
Curve
o
0.2 0.5 1.0 2.0 5.0 10 20
Frequency, cycles per second
50 100
200
y£>
500
Figure 25. Phase Component of Normalized Cross-Power Spectrum from Run Z

120
110
100
90
80
70
60
50
40
30
CD
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 26. Normalized Power Spectrum of Region 1 Output from Run Z

99
The amplitude data points clearly indicate a
loss of correlation beyond frequencies of 20 cycles per
second. The cross-power spectrum between the ion cham
bers should be identical to the output power spectrum
of the region if there is sufficient signal above the
instrumentation and detection noise level. When the
amplitude part of this cross-power spectrum is nor
malized by the output power spectrum of the region as
determined by one of the chambers, a constant amplitude
plot should result. "This constant amplitude is the
theoretical curve given in Figure 24. The drop-off
beyond 20 cycles per second indicates that there is no
correlation between the two ion chambers beyond this
frequency. This end of correlation is also indicated
by the phase part data points. There would be no phase
shifting between the two chambers if perfect correla
tion existed. The zero phase angle line is plotted in
Figure 25. Beyond 20 cycles per second, the data
scatter indicates a loss of correlation. The output
power spectrum of the south region as determined by
one of the chambers is identical to that measured in
Run CC.
When there was no external reactivity input to
the reactor, no correlation could be detected between
the outputs of the two fuel regions. The data for Run Q

100
were collected while the reactor was fueled with the 93
per cent enriched fuel and operating at a power level
of one watt. The Keithley micro-microanuneters were
operated on the 10xl0_a ampere scale. No external re
activity input was used. Table 7 gives the data,
plotted in Figure 27, from Run Q, of the power spectrum
of the output of the south region of the reactor. This
spectrum is almost white. The detection chambers were
*
located about 18 inches from the fuel slab in positions
of low efficiency. It has been found (12.) (16) that in
such chambers the uncorrelated chamber noise dominates
the signal, and the spectra found from them are almost
flat.

101
TABLE 7
RESULTS OF RUN Q
Normalized Power
f Spectrum of Out
put of Region 1
cycles per second decibels
0.4
97.6
0.5
95.6
0.6
102.1
0.8
96.0
1.0
102.9
1.25
99.7
1.5
105.2
2.0
100.9
2.5
103.1
3.0
103.7
4.0
101.8
5.0
97.6
6.0
96.7
7.0
98.2
8.0
97.7
9.0
92.8
10.0
93.9
12.5
92.4
15.0
91.9
20.0
86.3
25.0
88.9
30.0
87.8
40.0
90.2
50.0
92.4
60.0
91.6
70.0
90.7
80.0
91.6
90.0
93.2
100.0
91.6
125.0
91.2
150.0
93.6
200.0
96.9

130
120
110
100
90
80
70
60
50
40
O O
O
2
O 0o
o
o
o
000
I £ I I II! | I I
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 27. Normalized Power Spectrum of Region 1 Output from Run Q
500
102

CHAPTER V
DISCUSSION AND CONCLUSIONS
Of particular interest in this study was the
representation of the coupling or interaction between
the two fuel regions of a two slab reactor. The
measurement of the complex quantity in the
University of Florida Training Reactor indicates that
the relatively simple two point one group bare reactor
model used in this thesis will give an adequate
description of the kinetics of a two region reactor.
The success of this representation does, however,
depend on the value taken for the transit time of a
disturbance between the two regions. The data that
have been taken from the UFTR (in particular the phase
part of Run CC, Figure 22) indicate that an adequate
representation of the transit time, 0 is one in
which a disturbance in the neutron density in one fuel
slab travels to the opposite region with the velocity
of a neutron wave.
These data have also indicated the feasibility
of determining the multiplication factor of each of
the regions by the measurement of the cross-power
103

104
spectrum between the outputs. From the measurement of
the quantity a B (defined as 1 k^l ] /J)
value of 450 is indicated for the multiplication factor
of one region independent of the other. This value was
taken from the phase data points only. It was not
possible to obtain a value from the amplitude data
points since the correlation between the two chambers
ended at a frequency lower than the break frequency
corresponding to a B value of 450. The phase data are
sufficient, however, to establish a value of B. The
calibration data indicated that the phase data points
were reliable out to higher frequencies than were the
amplitude data points.
This value of B is in good agreement with recent
measurements by Badgley (27) Badgley measured sub-
critical multiplication constants for the UFTR for both
one and two slab fuel loadings. Extrapolation of his
results indicates a B value of 48050 for a one slab
loading of 1610 grams of uranium-235 in the 93 per cent
enriched fuel plates. Chastain (28) gives an value of
2.8xl04 seconds,* measured by a pile oscillator. For
*Chastain actually reports a prompt neutron life
time of 2.4xl0~4 seconds based on a delayed neutron frac
tion of 0.0065. In a compact core such as that of the
UFTR, an effective delayed neutron fraction of about
0.0075 should be used. His data then give a neutron
lifetime of 2.8xl0~4 seconds.

105
a neutron lifetime of 2.8x10^ seconds, a B value of 450
corresponds to a multiplication factor of 0.874. Criti
cal loading data with the 93 per cent enriched fuel
indicate that the reactor would be critical with a one
slab loading of 2005 grams of uranium-235. With a normal
two slab loading of the fully enriched fuel, a total load
ing of 3225 grams of uranium-235, or 1612.5 grams per slab,
produces a critical mass. The two regions are therefore
very strongly coupled/ the interaction between them is
worth about 390 grams of uranium-235. As much as 1950
grams of uranium-235 can be loaded into one region of the
reactor. These attempts to load the reactor to critical
in a one slab configuration gave not only an estimate
of the critical mass but also an estimate of the multipli
cation factor for a given mass of fuel. For a uranium-
235 mass of 1612.5 grams, the following estimates of the
effective multiplication factor were obtained: loading
of May 25, 1961, 0.870.08; loading of March 21, 1962,
0.790.08; loading of April 3, 1962, 0.820.08; and
loading of June 15, 1962, 0.76^0.08. These results may
be summarized to give a multiplication factor of
0.81^0.04 for one region independent of the other when
the two slab system is critical. This value is in fair
agreement with the B value of 450 obtained by the cross-
power spectrum measurement.

106
The UFTR is certainly not the ideal reactor on
which to conduct a study of loosely coupled two region
reactors, since its two fuel regions are separated by
only 12 inches of graphite and are rather strongly
coupled. This strong coupling results in a large
value of B which moves the break frequency of the
amplitude part of toward the higher frequencies.
In order to measure this break frequency, the input to
the second region should contain sufficient power in
all frequencies out to and slightly beyond the break
frequency. Although the random reactivity input device
may deliver a signal whose power spectrum is flat
beyond the break frequency of interest, the fuel slab
that receives this signal acts very much as a low pass
filter and will attenuate the higher frequencies. Thus
the input to the second fuel slab will not necessarily
have a flat power spectrum beyond the break frequency.
It is felt that the UFTR is probably as strongly
coupled a reactor for which the cross-power spectrum
between the outputs of the two regions will give
information.
Several equations were developed in Chapter II
involving the cross-power spectra between the two out
puts of the fuel regions and between the random
reactivity input and each of the fuel region outputs.

107
Of these several equations, only one was developed for
experimental use. In making measurements on a more
complex reactor than the UFTR, other of the equations
may prove useful for experimental purposes.
This thesis was devoted to the treatment of a
two region reactor system. The general method and
experimental approach could certainly be extended to
a three or more region system. Several large reactors
have either been designed or built that have cores
composed of several rather independent regions. An
extension of the method developed in this thesis would
allow the measurement of the multiplication factor of
each region by crosscorrelating the outputs of two
regions when a random disturbance is given to one
region only.
The physical arrangement of the UFTR made it
impossible to position the two ion chambers used to
collect data any closer than 18 inches from each fuel
region without extensive modifications of the graphite
reflector surrounding the core. This fact meant that
the detectors were located in positions of low effici
ency (less than 10 4). While this had no effect on
the data taken when there was an external reactivity
input to one region, it is felt that these low chamber
efficiencies were responsible for the lack of correla
tion between the two region outputs where there was no
external input.

APPENDIX A
CALCULATION OF TRANSIT TIME BETWEEN SLABS
In this section a theoretical determination of
the lag time, 0 of a neutron wave traveling between
the two regions of the UFTR is made. It is assumed
that the neutron wave originates in the center of one
fuel slab and travels to the center of the opposite
slab.
The velocity, V, of a neutron wave, corrected
for leakage, is given by (2),
where
(jJ = frequency, radians per second
D = diffusion coefficient
(52 = (7T/a)2 + (7T/b)2 + (1/L)2
a,b = transverse dimensions of the wave
L diffusion length
v = neutron velocity.
108

109
Thompson (2_9) gives the following values at
thermal energies for the fuel and graphite regions of
the UFTR:
D
L
Fuel
0.25 cm.
0.08112 cm.-^
1.76 cm.
Graphite
0.917 cm.
0.00C342 cm.
51.8 cm.
These results are given for a fuel slab consisting of a
homogeneous mixture of water, aluminum, and uranium.
It is also assumed that the reactor is loaded with 3500
grams of the 20 per cent enriched fuel.
For the reactor loaded with 3100 grams of the
fully enriched fuel, £ for the fuel section is 0.0723
a
centimeters-^. The diffusion coefficient and diffusion
length of the fuel region will also change from the
values given for the 20 per cent enriched fuel. These
changes are small enough, however, and the lag time in
the fuel is a small enough fraction of the total lag
time, that the values for the 20 per cent enriched fuel
may be used for the fully enriched fuel also.
The dimensions a and b are taken as 264 and 142
centimeters. These dimensions correspond to the com
plete diffusion interface and include some of the graph
ite surrounding the fuel slab. The neutron velocity is
taken as 2200 meters per second.

110
In the fuel section then,

2
' Tf'
2
+
7T"
2
1
264
142
+
1.76
2 0.3234.
(105)
Since the wave must travel through halves of two
fuel sections, the lag time through one full fuel sec
tion is calculated. This lag time may be found by
dividing the thickness of the fuel region, 13.34 centi
meters, by the wave velocity as determined by equation
(104). Substitution of the fuel region value gives,
9
fuel
4.02x10
~U
-2
[(1.78xl04)2+U2] 2 1.78X104
(106)
The expansion
(1 + x)^ = 1 + x x2 + x^ .. (107)
2 8 16
is useful in the evaluation of 0 for small values
fuel
of LJ The value of the lag time in the fuel at various
frequencies is given in Table 8.

Ill
TABLE 8
VALUE OF
TRANSIT
TIME BETWEEN
SLABS
Frequency
cycles/second
Lag
Time in Milliseconds
Fuel
Graphite
Total
0.5
0.213
2.39
2.60
1
0.213
2.39
2.60
2
0.213
2.39
2.60
5
0.213
2.39
2.60
10
0.213
2.35
2.56
20
0.213
2.26
2.47
50
0.213
1.99
2.20
100
0.213
1.63
1.84
200
0.213
1.25
1.46
500
0.213
0.830
1.04
1000
0.209
0.594
0.803

112
to be
The value of
in the graphite region is found

2
T('
2
JL
' TX
2
1
_264_
142
51.8
(52 = 0.001012 .
(108)
The thickness of the graphite section separating the two
fuel regions is 30.48 centimeters. When this distance
is divided by the velocity of a neutron wave in graphite
as determined by equation (104), the quotient is the
lag time in the graphite.
2.04x102)2+CJ2] -2.04X102
(109)
Values of the lag time in the graphite and the
total lag time are also given in Table 8.
e
graphite
4.80x10
U
-2
['

APPENDIX B
PHASE SHIFT AND HIGH PASS FILTER DETAILS
In the measurement of the cross-power spectrum
between two signals it was necessary to pass the sig
nals through band pass filters, set around the same
frequency, and then to introduce a 90 degree phase
shift between them. This was accomplished by giving
one signal a phase lag in a low pass filter and the
other a phase lead in a high pass filter such that
there was a 90 degree shift between them. The low pass
filter was formed by a resistor, R, followed by a
capacitor, C, and another resistor, K, in parallel to
ground while the high pass filter was composed of a
capacitor, C, followed by two resistors, R and K, in
parallel to ground. The amplitude, /&/> and phase, p,
parts of the transfer functions for these filters are
given by, for the low pass filter,
K
~Y (R + K)2 + r2k2U2c2
(86)
and,
113

114
PL
tan-1
RK C
R + K
(37)
and for the high pass filter,
rkL/c
~}J (R+K)2 + R2K2L/2C2
(88)
and
PH
tan-1
R+K
rkUc
(89)
As long as the same value resistors and
capacitors are used there is a 90 degree shift between
the outputs of the low and high pass filters. It was
convenient then to select R and C values such that at
a particular frequency, the amplitude attenuation was
the same in each filter. This was accomplished when,
1 1
R = = (90)
U C 27TfC
The amplitude parts of both the low and high pass fil
ters under the condition of equation (90) are then,

115
K
V (R + K)2 + K2
(110)
while the phase parts are
and
PL = tan
-1
K
R + K
(111)
p = tan
H
-1
R + K
K
(112)
The R and C values at each frequency set point are given
in Table 9.
Polystyrene capacitors were used in values of 1,
0.1, and 0.01 microfarads. The R values were set on 10
turn 0.1 megohm potentiometers. Any additional resist
ance was provided by adding precision resistors in
series with the potentiometers. These t1% resistors
were available in values of 0.1, 0.2, 0.5, and 1.0
megohms. The potentiometers were used as variable re
sistors and their set points are also given in Table 9.
The attenuation of the signals through the phase
changing filcers was calculated from equation (110) for
the two values of input resistors used, 0.1 and 1.0
megohms. When this attenuation was normalized to one

116
TABLE 9
SET POINTS FOR PHASE SHIFTERS
External
0.1 Megohm
f
C
R
Resistors
Potentiometers
cycles per
micro-
second
farads
megohms
0.1 megohms
set point
0.1
1.0
1.5916
10-5
084
0.15
1.0
1.0610
10-0
390
0.2
1.0
0.7958
5-2
042
0.3
1.0
0.5305
5-0
695
0.4
1.0
0.3979
2-1
021
0.5
1.0
0.3183
2-1
817
0.6
1.0
0.2653
2-0
347
0.7
1.0
0.2274
2-0
726
0.8
1.0
0.1989
1-0
011
0.9
1.0
0.1768
1-0
232
1.0
1.0
0.1592
1-0
408
1.25
1.0
0.1273
1-0
727
1.5
1.0
0.1061
1-0
939
2.0
1.0
0.0796
0-0
2 04
2.5
1.0
0.0637
0-0
363
3.0
1.0
0.0530
0-0
470
4.0
1.0
0.0398
0-0
602
5.0
1.0
0.0318
0-0
682
6.0
1.0
0.0265
0-0
735
7.0
1.0
0.0227
0-0
773
8.0
1.0
0.0199
0-0
801
9.0
1.0
0.0177
0-0
823
10.0
1.0
0.0159
0-0
841
12.5
1.0
0.0127
0-0
87 3
15.0
1.0
0.0106
0-0
894
20.0
0.1
0.0796
0-0
204
25.0
0.1
0.0637
0-0
363
30.0
0.1
0.0530
0-0
470
40.0
0.1
0.0398
0-0
602
50.0
0.1
0.0318
0-0
682
60.0
0.1
0.0265
0-0
735
70.0
0.1
0.0227
0-0
773
80.0
0.1
0.0199
0-0
801
90.0
0.1
0.0177
0-0
823
100.0
0.1
0.0159
0-0
841
125.0
0.1
0.0127
0-0
873
150.0
0.1
0.0106
0-0
894
200.0
0.01
0.0796
0-0
204

117
for the complete isolation of the filter, i.e. for K
very large, the reciprocal squared of the attenuation
was the correction factor to be applied for the phase
change filters. These correction factors are given in
Table 10.
The attenuation due to the high pass filter
before multiplication was calculated from equation (88).
The outputs of these filters were always fed into 1.0
megohm input resistors. Once again, the square of the
reciprocal of this attenuation was the correction factor
for the high pass filters. This correction factor is
also given in Table 10 as well as the combined correc
tion factor, (CF), which takes both phase shifter
attenuation and high pass filter attenuation into account.
Once again it is noted that these corrections
were unnecessary in the measurement of the quantity
^12^^^' ^ut were used when cross-power or power spectra
were measured with the analyzer.

118
TABLE 10
CORRECTION FACTORS FOR FILTERS
f Correction
Factor for
cycles Phase Changer
per K=1.0 K=0.1
second megohm megohm
Correction
Factor for
High Pass
Filter
Combined System
Correction Factor (CF)
K=1.0 K-0.1
megohm megohm
0.1
3.86
144.0
3.59
13.8
517.0
0.15
2 .63
68.2
2.08
5.46
142.0
0.2
2.10
61.9
1.64
3.55
102.0
0.3
1.66
20.3
1.30
2.16
26.4
0.4
1.49
13.0
1.15
1.71
14.9
0.5
1.36
9.20
1.10
1.50
10.1
0.6
1.29
7.03
1.07
1.38
7.52
0.7
1.25
5.86
1.05
1.31
6.15
0.8
1.22
4.94
1.04
1.27
5.12
0.9
1.19
4.37
1.03
1.23
4.50
1.0
1.17
4.16
1.02
1.19
4.25
1.25
1.14
3.10
1.02
1.16
3.16
1.5
1.11
2.62
1.01
1.12
2.65
2.0
1.08
2.10
1.01
1.09
2.12
2.5
1.05
1.82
1.00
1.05
1.82
3.0
1.05
1.66
1.00
1.05
1.66
4.0
1.04
1.46
1.00
1.04
1.46
5.0
1.04
1.37
1.00
1.04
1.37
6.0
1.02
1.30
1.00
1.02
1.30
7.0
1.02
1.26
1.00
1.02
1.26
8.0
1.02
1.21
1.00
1.02
1.21
9.0
1.01
1.19
1.00
1.01
1.19
10.0
1.01
1.17
1.00
1.01
1.17
12.5
1.01
1.12
1.00
1.01
1.12
15.0
1.01
1.10
1.00
1.01
1.10
20.0
1.08
2.10
1.00
1.08
2.10
25.0
1.05
1.82
1.00
1.05
1.82
30.0
1.05
1.66
1.00
1.05
1.66
40.0
1.04
1.46
1.00
1.04
1.46
50.0
1.04
1.37
1.00
1.04
1.37
60.0
1.02
1.30
1.00
1.02
1.30
70.0
1.02
1.26
1.00
1.02
1.26
80.0
1.02
1.21
1.00
1.02
.1.21
90.0
1.01
1.19
1.00
1.01
1.19
100.0
1.01
1.17
1.00
1.01
1.17
125.0
1.01
1.12
1.00
1.01
1.12
150.0
1.01
1.10
1.00
1.01
1.10
200.0
1.08
2.10
1.00
1.08
2.10

APPENDIX C
OPERATION OF CROSS-POWER
SPECTRUM ANALYZER
The measurement of the quantity R12(i*y)/ defined
by equation (69), requires that three voltages be mea
sured on the cross-power spectrum analyzer; V, related
to the power spectrum of the output of the externally
disturbed region of the reactor through equation (93),
VR, related to the real part of the cross-power spectrum
between the outputs of the two regions of the reactor by
equation (91), and Vj, related to the imaginary part of
the cross-power spectrum between the outputs of the two
regions of the reactor by equation (92).
Figure 9 is a schematic of the cross-power
spectrum analyzer. The tape recorder and demodulators
are included in the figure but were not needed, of
course, for "on line" data processing.
The first step in the processing of each data
run was the amplification of the ac portions of the two
signals from the demodulators (or the two signals
directly in the case of "on line" processing) to about
10 volts peak to peak. A gain of about 25 was found
119

120
to be sufficient for most runs. Once this gain was
established it was used throughout the run.
The first step taken at each frequency point was
to set the band pass filters and the R and C values of
the phase shifters. The frequency set points on the
band pass filters were determined by a previous calibra
tion with a signal generator. The R and C values used
were those as listed in Table 9 of Appendix B.
The most convenient way to measure the power
spectrum of the output of the externally excited region
of the reactor was to feed its signal into both chan
nels of the analyzer. This procedure also afforded a
check on the frequency settings of the band pass filters.
The output of one band pass filter was fed to the verti
cal axis of a dual beam oscilloscope and the output of
the other band pass filter to the horizontal axis of
the oscilloscope. Since the input signals to the band
pass filters were identical, the Lissajous figure formed
on the oscilloscope was a straight line. Any departure
from a straight line presentation was corrected by a
fine adjustment of the band pass filter settings.
X
The gain, M, of the amplifiers between the phase
shifters and the high pass filters was then adjusted so
that the signals had a strength from 100 to 200 volts

121
peak to peak. Two potentiometers could be adjusted in
each of the three signal paths. All potentiometers
were set with the null potentiometer. It was also pos
sible to set the gain of the amplifiers at the output
of the phase changers at either 1 or 10. M values of
from 40 to 2500 were employed.
The time constant of the integrating amplifiers
was set from 0.0001 to 0.0003. Once again, the potenti
ometers used for these settings were set with the null
potentiometer. It was desirable to select a time
constant such that about 50 volts would be collected
across the integrator during the three minute integra
tion time.
The measurement of the power spectrum of the
externally disturbed region of the reactor was performed
by feeding its signal, fj^t), into both channels of the
analyzer, making the setting described above, turning
the analog computer to operate for three minutes then
switching it to hold, and reading the voltage on the
integrating amplifier with the null potentiometer. The
voltage, V, related to the power spectrum appeared
across the integrator ordinarily giving the real part
voltage during cross-power spectrum analysis.
After V was measured, the signal representing
the output of the other region of the reactor, f2(t)r

122
was used in the second channel of the analyzer instead
of f^(t). Then at each frequency set point, the same
settings and procedures were used to measure the
voltages VR and Vj as were used to measure the voltage
V.

LIST OF REFERENCES
1. E. P. Wigner, Use of the Pile Oscillator for the
Measurement of Pile Constants, CP-3066, June 23, 1945.
2. A. F. Henry, The Application of Reactor Kinetics to
the Analysis of Experiments, Nuclear Sci. and Eng., _3,
52-70 (1959).
3. F. deHoffman, Intensity Fluctuations of a Neutron
Chain Reactor, MDDC-382, LADA-256 (declassified
October 7, 1946).
4. W. K. Luckow, "The Evaluation of Nuclear Reactor
Parameters from Measurements of Neutron Statistics,"
Ph.D. Dissertation, University of Michigan, 1958.
5. R. W. Albrecht, The Measurement of Dynamic Nuclear
Reactor Parameters by Methods of Stochastic Processes,
Trans. Am. Nuclear Soc., 4, 2 (1961) pp. 311-312.
6. C. Velez, "Autocorrelation Functions of Counting Rate
in Nuclear Reactors and Their Application to the
Design of Reactor Control Instrumentation," Ph.D.
Dissertation, University of Michigan, 1959.
A less complete description of this work is found in
C. Velez, Autocorrelation Functions of Counting Rate
in Nuclear Reactors, Nuclear Sci. and Eng., 6, 414-
419 (1959).
7. M. N. Moore, The Determination of Reactor Transfer
Functions from Measurements at Steady Operation,
Nuclear Sci. and Eng.. 3, 387-394 (1958).
8. M. N. Moore, The Power Noise Transfer Function of a
Reactor, Nuclear Sci. and Eng., 6, 448-452 (1959).
9. C. E. Cohn, Determination of Reactor Kinetic Para
meters by Pile Noise Analysis, Nuclear Sci. and Eng.,
5, 331-335 (1959).
123

124
10. C. W. Griffin and J. G. Lundholm, Measurement of the
SRE and KEWB Prompt Neutron Lifetime Using Random
Noise and Reactor Oscillation Techniques, NAA-SR-
3765, Oct. 1959.
11. C. E. Cohn, A Simplified Theory of Pile Noise,
Nuclear Sci. and Eng., 1_, 472-475 (1960) .
12. E. F. Bennett, The Rice Formulation of Pile Noise,
Nuclear Sci. and Eng., 8, 53-61 (1960).
13. S. O. Rice, Mathematical Analysis of Random Noise,
from N. Wax, "Selected Papers on Noise and Stochastic
Processes," pp. 133-294, Dover, New York, 1954.
14. C. W. Griffin and R. L. Randall, Reactor Power
Spectrum Measurements, Trans. Am. Nuclear Soc., _5,
1 (1962) pp. 174-175.
15. J. D. Balcomb, "A Crosscorrelation Method for Measur
ing the Impulse Response of Reactor Systems," Ph.D.
Dissertation, Massachusetts Institute of Technology,
June 1961.
A less complete description of this work is given in
J. D. Balcomb, _et aJL., A Crosscorrelation Method for
Measuring the Impulse Response of Reactor Systems,
Nuclear Sci. and Eng., 11, 159-166 (1961).
16. V. Rajagopal, "Experimental Study of Nuclear Reactor
Internal Noise and Transfer Functions Using Random
Reactivity Variations and Correlation Analysis," Ph.D.
Dissertation, Rensselaer Polytechnic Institute, Troy,
N. Y., 1960.
A less complete description of this work is given in
V. Rajagopal, Determination of Reactor Transfer
Functions by Statistical Correlation Methods, Nuclear
Sci. and Eng.. 12, 218-224 (1962).
17. G. C. Baldwin, Kinetics of a Reactor Composed of Two
Loosely Coupled Cores, Nuclear Sci. and Eng., 6,
320-327 (1959).
18. M. A. Schultz, "Control of Nuclear Reactors and Power
Plants," 2nd ed., McGraw-Hill, New York, 1961.

125
19. Y. W. Lee, "Statistical Theory of Communication,"
John Wiley and Sons, New York, 1960.
20. J. S. Bendat, "Principles and Applications of Random
Noise Theory," John Wiley and Sons, New York, 1958.
21. C. E. Cohn, Reflected-Reactor Kinetics, Nuclear Sci.
and Eng., 13, 12-17 (1962).
22. A. M. Weinberg and E. P. Wigner, "The Physical
Theory of Neutron Chain Reactors," University of
Chicago Press, Chicago, 1958.
23. J. M. Duncan, "University of Florida Training Reactor
Hazards Summary Report," Florida Engineering and
Industrial Experiment Station, Gainesville, October,
1958.
24. R. B. Blackman and J. W. Tukey, "The Measurement of
Power Spectra," Dover, New York, 1958, p. 58.
25. J. Millman and H. Taub, "Pulse and Digital Circuits,"
McGraw-Hill, New York, 1956, pp. 28-46.
26. Bennett and Fulton, The Generation and Measurement of
Low Frequency Random Noise, Journal of Applied Physics,
22, 1187-1191, 1951.
27. R. W. Badgley, "Power Spectral Density of the Uni
versity of Florida Training Reactor Operating in the
Subcritical Region," M.S. Thesis, University of
Florida, 1962.
28. R. H. Chastain, "Transfer Function of the University
of Florida Training Reactor," M.S. Thesis, University
of Florida, Gainesville, Florida, 1961.
29. C. A. Thompson, "Neutron Flux Calculations for a
Graphite Moderated Twenty Per Cent Enriched Reactor,"
M.S. Thesis, University of Florida, Gainesville,
Florida, 1961.

BIOGRAPHICAL SKETCH
Allen Ross Boynton was born in Lexington,
Kentucky, on June 16, 1936. In June, 1953, he was gradu
ated from Owensboro Senior High School, Owensboro,
Kentucky. In June, 1957, he received the degree of
Bachelor of Science in Mechanical Engineering, with dis
tinction, from Purdue University. He was awarded an
Atomic Energy Commission Fellowship in September, 1957,
and received the degree of Master of Science in Engineer
ing from Purdue University in August, 1958. In September,
1958, he was admitted to the University's Graduate School.
While pursuing his graduate studies, he held positions
as Graduate Assistant and as the Chief Operator of the
University of Florida Training Reactor.
Allen Ross Boynton is married to the former
Carol Sue Baringer of Gainesville, Florida. He is a
member of Tau Beta Pi, Pi Tau Sigma, and the American
Nuclear Society.
126

This dissertation was prepared under the direction
of the chairman of the candidate's supervisory committee
and has been approved by all members of that committee.
It was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.
December, 1962
Dean, Graduate School
Supervisory Committee



116
TABLE 9
SET POINTS FOR PHASE SHIFTERS
External
0.1 Megohm
f
C
R
Resistors
Potentiometers
cycles per
micro-
second
farads
megohms
0.1 megohms
set point
0.1
1.0
1.5916
10-5
084
0.15
1.0
1.0610
10-0
390
0.2
1.0
0.7958
5-2
042
0.3
1.0
0.5305
5-0
695
0.4
1.0
0.3979
2-1
021
0.5
1.0
0.3183
2-1
817
0.6
1.0
0.2653
2-0
347
0.7
1.0
0.2274
2-0
726
0.8
1.0
0.1989
1-0
011
0.9
1.0
0.1768
1-0
232
1.0
1.0
0.1592
1-0
408
1.25
1.0
0.1273
1-0
727
1.5
1.0
0.1061
1-0
939
2.0
1.0
0.0796
0-0
2 04
2.5
1.0
0.0637
0-0
363
3.0
1.0
0.0530
0-0
470
4.0
1.0
0.0398
0-0
602
5.0
1.0
0.0318
0-0
682
6.0
1.0
0.0265
0-0
735
7.0
1.0
0.0227
0-0
773
8.0
1.0
0.0199
0-0
801
9.0
1.0
0.0177
0-0
823
10.0
1.0
0.0159
0-0
841
12.5
1.0
0.0127
0-0
87 3
15.0
1.0
0.0106
0-0
894
20.0
0.1
0.0796
0-0
204
25.0
0.1
0.0637
0-0
363
30.0
0.1
0.0530
0-0
470
40.0
0.1
0.0398
0-0
602
50.0
0.1
0.0318
0-0
682
60.0
0.1
0.0265
0-0
735
70.0
0.1
0.0227
0-0
773
80.0
0.1
0.0199
0-0
801
90.0
0.1
0.0177
0-0
823
100.0
0.1
0.0159
0-0
841
125.0
0.1
0.0127
0-0
873
150.0
0.1
0.0106
0-0
894
200.0
0.01
0.0796
0-0
204


50
oo
Figure 7. Data Transcribing System


CHAPTER IV
EXPERIMENTS AND RESULTS
This chapter contains a description of the
experiments performed both for check-out and calibration
and for the measurement of the parameters of interest of
the UFTR. This description includes data that were
taken, data reduction, and the confidence limits assigned
to the results. A detailed description of the procedure
used to process the data is given in Appendix C.
The first section deals with the calibration
runs in which the transfer function of a low pass filter
was measured by obtaining the cross-power spectrum
between a random input and the output of the filter from
this input. The error analysis associated with this
calibration is used with the data obtained from the
reactor.
The second section presents data obtained from
the reactor and data reduction. Data obtained both
with and without an external input are presented.
Calibration
Performance of the cross-power spectrum analyzer
and the data transcribing system were checked with the
65


14
in which the inner integral is recognized as the
autocorrelation function of the output of region 1. Equa
tion (12) can now be written in a more compact form, and
with equation (7) relates the crosscorrelation of the
outputs of region 1 and region 2 with their unit impulse
response functions, their autocorrelation functions, and
the crosscorrelation of the external input and the output
of region 2.
0i 2
h2
0uaA
(13)
An obvious simplification is the case of no
external input to the two region system. Equation (13) is
unchanged, but equation (7) becomes,
012(T) =
CD
( A )022 (7" + A + $ )dX (14)
-oo
These relations may now be transformed into the
frequency domain by taking the Fourier transform of both
sides of both equations. As previously noted this is
done to relate the theory directly with the experiment.
Many investigators (16) (17) choose to measure cross
correlation and autocorrelation functions (rather than


+ 30-
co
d)
u
O'
<1>
T3
0)
f
O'
<1)
(0
J
I
+20-
+10-
-30
-40
-50
-60 -
Curve
o
0.2 0.5 1.0 2.0 5.0 10 20
Frequency, cycles per second
50 100
200
y£>
500
Figure 25. Phase Component of Normalized Cross-Power Spectrum from Run Z


99
The amplitude data points clearly indicate a
loss of correlation beyond frequencies of 20 cycles per
second. The cross-power spectrum between the ion cham
bers should be identical to the output power spectrum
of the region if there is sufficient signal above the
instrumentation and detection noise level. When the
amplitude part of this cross-power spectrum is nor
malized by the output power spectrum of the region as
determined by one of the chambers, a constant amplitude
plot should result. "This constant amplitude is the
theoretical curve given in Figure 24. The drop-off
beyond 20 cycles per second indicates that there is no
correlation between the two ion chambers beyond this
frequency. This end of correlation is also indicated
by the phase part data points. There would be no phase
shifting between the two chambers if perfect correla
tion existed. The zero phase angle line is plotted in
Figure 25. Beyond 20 cycles per second, the data
scatter indicates a loss of correlation. The output
power spectrum of the south region as determined by
one of the chambers is identical to that measured in
Run CC.
When there was no external reactivity input to
the reactor, no correlation could be detected between
the outputs of the two fuel regions. The data for Run Q


59
cross-power spectrum between both outputs and the power
spectrum of the output of the externally excited region
be measured. The real and imaginary parts of the cross
power spectrum were measured by use of equations (91)
and (92). The power spectrum of region 1 was measured
with the same computer setup as was the cross-power
spectrum except that f^(t) only was used instead of
f^(t) and 2(t). Since the power spectrum was a real
quantity, there was only a voltage, V, on the integra
tor giving the real parts. For the same computer set
up as the cross-power spectrum, the power spectrum was
given by,
V(CF)
(93)
Thus the real and imaginary parts of R^2^fl^ were
given by,
Re [r12<*1>]
(94)
V
and
V
I
(95)
V


120
110
100
90
80
70
60
50
40
30
§ § a
Q O O
O o
O
O
O
O
o O o
000 0
o
o
u>
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 23. Normalized Power Spectrum of Region 1 Output from Run CC


5
This paper will apply this latest technique to
a more complex reactor model, that of a two region
reactor.
Baldwin (JL7) has examined the so-called "two-
slab" loading of the Argonaut reactor and found that,
experimentally, a single exponential stable period is
observed when the system is supercritical, but that
there is a tendency for independent behavior of the
two slabs, resulting in so-called "flux tilting."
Rod calibration experiments,indicated that the ratio
of fluxes in the two fuel regions changes during the
calibration procedure. It was also found that the
single transfer function of a simple reactor system
was inadequate to treat the two slab system.
The two (or more) region reactor has also been
of interest in connection with the phenomenon of coupled
Xenon oscillations in large reactors (18).
The University of Florida Training Reactor (UFTR)
is a two region coupled reactor. Each fuel region is
subcritical when considered separately, but coupling
between the two regions allows the system to become
critical. If the multiplication factor of one region
is disturbed, the other region will be disturbed only
through its coupling to the first. A study of the
interdependence of the outputs of the two regions, when


Amplitude of Normalized Spectrum, decibels
vo
Figure 24. Amplitude Component of Normalized Cross-Power Spectrum from Run Z


47
Cross-Power Spectrum Analyzer
The measurement of the cross-power spectrum
between the outputs of the two regions of the UFTR
involved both transcribing output signals on magnetic
tape and subsequently processing these signals in the
manner indicated in Figure 5. Figure 7 is a flow
diagram of the equipment used to collect and store out
put signals of each region of the reactor on magnetic
tape.
Fluctuations in the neutron intensity of each
region were converted into current fluctuations by the
two Westinghouse 6377 compensated ion chambers placed
in the core. Two battery packs supplied 600 volts to
each chamber. No compensating voltage was used. Each
chamber was positioned outside one of the fuel slabs as
shown in Figure 6. Attempts to avoid ground loops in
the system included wrapping the ion chambers in trans
former cloth and covering all connectors with electrical
%
tape, since the chambers were in contact with the alumi
num and graphite in the core. These precautions were
found to eliminate from the measurements large amounts
of 60 cycle interference.
About 15 feet of coaxial cable were used to
connect the ion chambers with the Keithley Model 410
Micro-microammeters located immediately outside the UFTR


50
Noise
Signal
PT5 0.0015//f
To
Tape
Recorder
Figure 8.
Modulator Schematic


49
shield. The micro-microamraeters converted the signals
from current fluctuations to voltage fluctuations which
were then amplified on an Applied Dynamics Analog Com
puter, located in the reactor control room, so that the
ac portion of each signal was about 10 volts peak to
peak. The analog computer was also used to buck-out
the dc produced by the steady state power level of the
reactor. Two Krohn-Hite Ultra-low frequency band pass
filters were used as high and low pass filters. The
low cut-off frequencies were set at 0.1 cycle per
second to remove all dc and any slow drifts in reactor
power. Frequencies above 1000 cycles per second were
of no interest in this experiment and were removed by
setting the high cut-off frequencies at 1000 cycles per
second. The signals were then amplitude modulated on an
80,000 cycles per second carrier. Amplitude modulation
recording and playback is subject to considerable fluctu
ation in over-all gain (24j. Because only approximate
spectra were required in order to determine the mechanism
of the lag between regions and to test the feasibility
of the measurement of the multiplication factors of the
regions, AM recording was employed. AM recording and
detection equipment was also available. A schematic of
the modulator is shown in Figure 8.


62
to a tension spring mounted inside a graphite block.
The graphite blocks and the cylinder of the absorber
element were rigidly connected to the electromagnet.
A schematic diagram of the amplifier that drove
the moving coil is given in Figure 12. The oscillator
system was found to follow sinusoidal inputs in the
frequency range 0.1 to 50 cycles per second at the max
imum length of stroke (about 3/8 inch) while for higher
frequencies, up to 100 cycles per second, about half
stroke could be obtained. When the device was mounted
in the south beam port, full stroke held about 0.015
per cent reactivity.
Two types of random noise generators were used
as inputs to the power amplifier. The first was a
commercial noise generator. The Elgenco, Inc., Low
Frequency Noise Generator gave a 12 volt rms noise sig
nal whose power spectrum was uniform from 0 to 35 cycles
per second. The dc level of this signal was less than
50 millivolts.
A second random signal was constructed with a
radioactive sample, GM counter, and a count rate meter.
The signal was taken from the output of the flip-flop
section of the count rate meter. This output was a
randomly switched square wave whose mean period varied
as the reciprocal of the count rate. This signal is


77
The data of these figures do, however, indicate
the satisfactory performance of the cross-power spectrum
analyzer. The data of Figure 15 demonstrate that with
the correction factors of Appendix B, the analyzer may
also be used successfully for the measurement of power
spectra.
Referring to Figure 13, one observes that the
data of Run L depart from the theoretical amplitude at
frequencies beyond 50 cycles per second. This departure
is a leveling out of the amplitude plot to the noise
level of the analyzer. At 50 cycles per second Figure
15 shows that the random input power spectrum is down
more than 30 decibels from the value at the lower fre
quencies. In Run M the power spectrum of the input
carries more of the high frequency components and because
of this, the cross-power spectrum data extend out to
about 100 cycles per second. The phase data points fol
lowed the theoretical curve out to 125 cycles per second
for both runs.
Run P was a run of the same type as L and M, but
instead of processing the data "on line," the two sig
nals, the random step input and the filter output, were
AM modulated and stored on magnetic tape. The random
step was formed from a count rate of 7100 counts per
minute from the GM counter. The data from Run P as


LIST OF TABLES
Table Page
1. Results of Run L 69
2. Results of Run M 70
3. Results of Run P 79
4. Results of Run R 83
5. Results of Run CC 90
6. Results of Run Z 95
7. Results of Run Q 101
8. Value of Transit Time
Between Slabs Ill
9. Set Points for Phase Shifters 116
10. Correction Factors for Filters 118
iv


124
10. C. W. Griffin and J. G. Lundholm, Measurement of the
SRE and KEWB Prompt Neutron Lifetime Using Random
Noise and Reactor Oscillation Techniques, NAA-SR-
3765, Oct. 1959.
11. C. E. Cohn, A Simplified Theory of Pile Noise,
Nuclear Sci. and Eng., 1_, 472-475 (1960) .
12. E. F. Bennett, The Rice Formulation of Pile Noise,
Nuclear Sci. and Eng., 8, 53-61 (1960).
13. S. O. Rice, Mathematical Analysis of Random Noise,
from N. Wax, "Selected Papers on Noise and Stochastic
Processes," pp. 133-294, Dover, New York, 1954.
14. C. W. Griffin and R. L. Randall, Reactor Power
Spectrum Measurements, Trans. Am. Nuclear Soc., _5,
1 (1962) pp. 174-175.
15. J. D. Balcomb, "A Crosscorrelation Method for Measur
ing the Impulse Response of Reactor Systems," Ph.D.
Dissertation, Massachusetts Institute of Technology,
June 1961.
A less complete description of this work is given in
J. D. Balcomb, _et aJL., A Crosscorrelation Method for
Measuring the Impulse Response of Reactor Systems,
Nuclear Sci. and Eng., 11, 159-166 (1961).
16. V. Rajagopal, "Experimental Study of Nuclear Reactor
Internal Noise and Transfer Functions Using Random
Reactivity Variations and Correlation Analysis," Ph.D.
Dissertation, Rensselaer Polytechnic Institute, Troy,
N. Y., 1960.
A less complete description of this work is given in
V. Rajagopal, Determination of Reactor Transfer
Functions by Statistical Correlation Methods, Nuclear
Sci. and Eng.. 12, 218-224 (1962).
17. G. C. Baldwin, Kinetics of a Reactor Composed of Two
Loosely Coupled Cores, Nuclear Sci. and Eng., 6,
320-327 (1959).
18. M. A. Schultz, "Control of Nuclear Reactors and Power
Plants," 2nd ed., McGraw-Hill, New York, 1961.


41
u
dt = §11(U1)hU1 ,
(77)
where the left-hand side is the mean square value of
f^t)
This expression is solved for
<£>11,
(78)
Figure 4 indicates the sequence of operations necessary
to solve equation (78) experimentally.
The cross-power spectrum is a complex quantity
and may be expressed in terms of its real parts, Re12,
and its imaginary parts, Im12. Use is also made of
Euler's relation,
p jLJ}T
^ = cos + jsinl^T (79)
to reduce equation (75) to
T
-rl fi(t>
o
u.
f2(t+r)
dt =
U.
(80)
J^e12cos^l^Ira12sin^l'T+j (Im12cos Cu/jT" +Re12sin(A>/1T)J A


67
K
(86)
and
PG(Ly)
tan
-1 f rkUc
(87)
R + K
where
K = 0.1 mQ
R = 1.0 mQ
C = 0.1 /f.
These two functions are plotted as solid lines versus
frequency in Figures 13 and 14.
Performance of the cross-power spectrum analyzer
independent of the data storage system was first examined
by taking the cross-power and power spectra measurements
"on line." A randomly switched step function was given
as an input to the filter system. The step function was
%
obtained from the flip-flop section of a count rate
meter the input of which was supplied by a GM counter
setup. The power spectrum of the input was measured as
indicated by equation (93) and the real and imaginary
parts of the cross-power spectrum as indicated by equa
tions (91) and (92).


37
the transit time between slabs. The phase plot (Figure 3)
indicates the dependence of the complex quantity R^2(L/0
on this transit time, 0 For B=500, the phase plots
for three different values of 0 are shown: first, for
0 equal to zero; second, for 0 corresponding to the
thermal neutron velocity; and third, for 0 correspond
ing to the frequency dependent thermal neutron wave
velocity. The value of the transit time between slabs
in the UFTR when a disturbance on one side is assumed to
travel to the other side with the velocity of a thermal
neutron wave is calculated in Appendix A. The phase
part of R^OJ) is also given in Figure 3 for B values
of 100, 200, and 1000, with 0 corresponding to the ther
mal neutron wave velocity. It is seen that for a par
ticular value of B, such as 500, the phase portion of
the quantity R12 (U) provides a rather sensitive test
for the validity of assumptions about the nature of 0 .
The measurement of R12(L/) would also allow the deter
mination of the parameter, B, which yields information
about the multiplication factor of the non-disturbed
region. In a balanced reactor system such as the UFTR,
in which the multiplication factors of the two reqions
are equal, Baldwin (26.) has shown that
A = B £
(72)
Thus a determination of B yields information about the
magnitude of the coupling of the two regions.


APPENDIX A
CALCULATION OF TRANSIT TIME BETWEEN SLABS
In this section a theoretical determination of
the lag time, 0 of a neutron wave traveling between
the two regions of the UFTR is made. It is assumed
that the neutron wave originates in the center of one
fuel slab and travels to the center of the opposite
slab.
The velocity, V, of a neutron wave, corrected
for leakage, is given by (2),
where
(jJ = frequency, radians per second
D = diffusion coefficient
(52 = (7T/a)2 + (7T/b)2 + (1/L)2
a,b = transverse dimensions of the wave
L diffusion length
v = neutron velocity.
108


7
cross-power spectra were then measured directly using
band pass filters and an analog computer.
These data from the UFTR indicate that the
\
propagation of a disturbance in one region of the
reactor to the other region may adequately be described
in terras of neutron waves. Also shown is the feasi
bility of measuring the multiplication factor of each
side and the magnitude of the coupling between the two
regions.


23
desirable to retain at least the first few terras of the
expansion of equation (37).
The kinetics equations for region 1 are then,
dni
dt
[Vl-£)-i] ni V 1 ,
1 V Ai 11
i2n2 A2 0 dn A2 Q2 d2n2
2 dt + 2 2
dt
and
dc
11 = Aklnl
dt £
-
il'
(38)
(39)
A similar set of equations exists for region 2. In the
experiment, a random reactivity input was given to region
1 only, while k2 remained constant. This fluctuation in
the multiplication factor in region 1 will give rise to
fluctuations in the neutron density and delayed neutron
precursor concentrations in both regions. There may also
be fluctuations in the external neutron source. It is
convenient then to linearize these quantities as follows:
+ (3 n0
nl
+
o
H
a
ii
U n1
n2
= n20
kl
- k10 + (
J k1
r*
Ci2
= ci20
Ci2
cuo + 1
5cn
S1
= S10
<5
12
(40)


29
When a random reactivity input is given to region
1, the desired transfer function of that region is given
by
Alj/n-,
H (U) = -T (51)
Zak-l
where A N^/n^ is the normalized output, and A is the
input. This ratio is obtained from equation (47) after
An2 is eliminated by substitution from equation (49).
With the assumption of no external neutron source, solv
ing equation (49) for A N2 gives,
A N~ =
1 @s +
ev
A Nx
s -
k2(
l
y
a i s + aa
(52)
Substitution of this expression for A n2 into equation
(47) gives, after some algebraic manipulations and
setting s = jU and 0 j(JQ {3 Q. 2 ..,
H^L/) =
(1 /?>/£
(53)
r\ A1A2 ^-2jL/0
,, ,kiy ^lAl i e
l
£ i u+Xl V AAl
£ £ lu+Xl


104
spectrum between the outputs. From the measurement of
the quantity a B (defined as 1 k^l ] /J)
value of 450 is indicated for the multiplication factor
of one region independent of the other. This value was
taken from the phase data points only. It was not
possible to obtain a value from the amplitude data
points since the correlation between the two chambers
ended at a frequency lower than the break frequency
corresponding to a B value of 450. The phase data are
sufficient, however, to establish a value of B. The
calibration data indicated that the phase data points
were reliable out to higher frequencies than were the
amplitude data points.
This value of B is in good agreement with recent
measurements by Badgley (27) Badgley measured sub-
critical multiplication constants for the UFTR for both
one and two slab fuel loadings. Extrapolation of his
results indicates a B value of 48050 for a one slab
loading of 1610 grams of uranium-235 in the 93 per cent
enriched fuel plates. Chastain (28) gives an value of
2.8xl04 seconds,* measured by a pile oscillator. For
*Chastain actually reports a prompt neutron life
time of 2.4xl0~4 seconds based on a delayed neutron frac
tion of 0.0065. In a compact core such as that of the
UFTR, an effective delayed neutron fraction of about
0.0075 should be used. His data then give a neutron
lifetime of 2.8xl0~4 seconds.


27
reactivity transfer function. In the former, the input is
associated with fluctuations in the external neutron source,
while in the latter, fluctuations in the multiplication
factor or reactivity are considered the input. In the two
slab reactor system, each slab is considered as a sub-
critical system, so the same logic should apply. The
transfer function when an external reactivity input is
applied to a slab is a reactivity transfer function, while
the transfer function when there is no external input is
a source transfer function; the leakage of neutrons from
the opposite slab is considered as an external neutron
source.
This source transfer function is easily recognized
in the case of an external random input to slab 1 while
the reactivity of slab 2 is held constant. The fluctua-
*
tions in the multiplication factor of slab 1 cause
fluctuations in the neutron population in that slab. A
certain fraction, A-^'of this fluctuating population
leaks out of region 1 and may be considered an external
neutron source for region 2. This fluctuating neutron
source then causes the neutron population of region 2
to fluctuate. A source transfer function for region 2
may then be formed by the ratio of the output of slab 2
to the output of slab 1 when both are in Laplace notation.


90
TABLE 5
RESULTS OF RUN CC
f
Normalized Amplitude
Phase Angle
Normalized Power
cycles
of R12
of ri2(l;)
Spectrum of Out-
per
put of Region 1
second
decibels
degrees
decibels
0.4
101.0
0
101.4
0.5
99.5
-2.0
102.1
0.6
100.7
-5.3
100.7
0.8
100.0
0
99.3
1.0
102.0
-4.4
99.5
1.25
98.5
-2.0
102.6
1.5
101.1
-2.1
99.2
2.0
102.2
-5.1
98.8
2.5
99.2
-5.2
100.1
3.0
100.0
-7.0
97.0
4.0
102.0
-8.5
95.8
5.0
99.2
-10
9'4.7
6.0
101.4
-11
90.9
7.0
101.6
-14
89.7
8.0
99.0
-17
88.3
9.0
100.7
-18
85.7
10.0
98.7
-25
84.9
12.5
100.2
-22
78.0
15.0
100.2
-25
71.5
20.0
99.0
-39
66.3
25.0
100.0
-43
59.3
30.0
95.6
-49
58.0
40.0
93.9
-64
52.8
50.0
99.4
-65
52.3
60.0
90.5
-34
53.0
70.0
93.4
-62
48.7
80.0
83.7
-84
48.9
90.0
83.5
-56
48.6
100.0
80.9
-62
47.7
125.0
81.9
-70
46.3
150.0
80.4
-116
43.3


26
If there is no disturbance of the multiplication
factor in region 1, then equation (47) becomes,
Since k2 is assumed to remain constant, the kinetics
equations for region 2, after manipulations similar to the
above, reduce to,
s
I
A
N
2
l-@s +
e
2c2
(49)
An,+A
The transfer function is defined as the ratio of
the system output to the system input when both are
expressed in Laplace notation. The inputs and outputs
used to derive the transfer functions must correspond to
the physical situation. It is not uncommon to speak of
two different transfer functions in connection with sub-
critical reactors; a source transfer function and a


40
Here, f^(t) and f2(t) are voltages representing the neutron
density in each slab. The integration over t cannot, of
course, be performed over infinite time or over negative
times, so a finite average is used, resulting in the
approximation,
oo
fx(t)
f2(t+r )dt =
§12(u)ejUr du
'-CO
(74)
The first operation in the data processing is to pass both
signals through identical band pass filters, both of which
are centered at the same frequency, U 1, with the same
frequency band width, Equation (74) then becomes,
_1_
T
/ fl
'0 J
u, J
u
dt = $12(L>/1)^AUl
L (75)
The power spectrum measurement, when f1(t)=f2(t),
is considered first; then,
fx(t)
u£i(t+r)
1
u
dt -
L (76)
If T is set equal to zero, equation (76) is then written,
I


75
filter. The expression may be used as a first order
approximation to cases involving other distributions
and spectral shapes as well as to cross-power spectra.
Three voltages were measured, V V and V,
R X
corresponding to the real and imaginary parts of the
cross-power spectrum between the outputs of the two
regions of the UFTR and to the power spectrum of the
output of the externally disturbed region. Since the
same integration times and frequency band widths were
used for the measurement of all three voltages, all
three have the same fractional standard deviation,
namely,
In most cases, a three minute integration time
was used. This value was taken as a compromise between
the drift in the analog computer and a reasonable sta
tistical accuracy.
Given the standard deviation of the voltage
readings, the standard deviation of the amplitude and
phase parts can be calculated from the propagation of
errors formula,
o'
77tAi
(100)


25
When the Laplace transform, using zero initial
conditions, is taken of both sides of both equations,
s
A i o + o + 4AlA'
il
A e
(45)
An2 0sAn2 + 62An2 ...
and
ft. .
'il
2
i
(46)
where s is the Laplace transform variable, the capital
letters denote the Laplace transform of the (5 quanti
ties (i.e., An^ is the Laplace transform of n^) and
the small letters are now understood to represent steady
state values. Equation (46) may be solved for A C^.
When this is done and A substituted into equation
(45) and terms collected,


69
TABLE 1
RESULTS
OF RUN L
f
Normalized
Phase Angle
Normalized Power
Amplitude
of G(U)
Spectrum of
cycles per
of G(U)
Input
second
decibels
degrees
decibels
0.15
99.2
-0.6
99.1
0.2
97.6
-0.4
100.1
0.3
98.4
-4.6
99.0
0.4
99.0
-2.2
93.9
0.5
99.4
-1.1
98.3
0.6
101.4
-3.0
98.2
0.8
101.1
-4.2
97.9
1.0
100.0
-4.0
99.8
1.5
100.6
-6.7
98.7
2.0
100.6
-9.1
100.1
3.0
99.6
-12.4
98.4
4.0
99.4
-15.5
99.2
5.0
100.6
-18.0
97.7
6.0
100.0
-20.4
97.3
8.0
100.0
-26.1
94.6
10.0
99.4
-30.2
92.7
15.0
97.2
-39.0
88.3
20.0
96.2
-48.1
83.5
30.0
94.2
-57.3
77.3
40.0
93.6
-65.0
74.6
50.0
91.0
-66
70.3
60.0
90.2
-74
65.9
80.0
88.6
-79
61.5
100.0
88.2
-85
56.7
150.0
87.6
-86
48.4


120
to be sufficient for most runs. Once this gain was
established it was used throughout the run.
The first step taken at each frequency point was
to set the band pass filters and the R and C values of
the phase shifters. The frequency set points on the
band pass filters were determined by a previous calibra
tion with a signal generator. The R and C values used
were those as listed in Table 9 of Appendix B.
The most convenient way to measure the power
spectrum of the output of the externally excited region
of the reactor was to feed its signal into both chan
nels of the analyzer. This procedure also afforded a
check on the frequency settings of the band pass filters.
The output of one band pass filter was fed to the verti
cal axis of a dual beam oscilloscope and the output of
the other band pass filter to the horizontal axis of
the oscilloscope. Since the input signals to the band
pass filters were identical, the Lissajous figure formed
on the oscilloscope was a straight line. Any departure
from a straight line presentation was corrected by a
fine adjustment of the band pass filter settings.
X
The gain, M, of the amplifiers between the phase
shifters and the high pass filters was then adjusted so
that the signals had a strength from 100 to 200 volts


110
In the fuel section then,

2
' Tf'
2
+
7T"
2
1
264
142
+
1.76
2 0.3234.
(105)
Since the wave must travel through halves of two
fuel sections, the lag time through one full fuel sec
tion is calculated. This lag time may be found by
dividing the thickness of the fuel region, 13.34 centi
meters, by the wave velocity as determined by equation
(104). Substitution of the fuel region value gives,
9
fuel
4.02x10
~U
-2
[(1.78xl04)2+U2] 2 1.78X104
(106)
The expansion
(1 + x)^ = 1 + x x2 + x^ .. (107)
2 8 16
is useful in the evaluation of 0 for small values
fuel
of LJ The value of the lag time in the fuel at various
frequencies is given in Table 8.


31
Y1(L/) may be formed directly from equation (48),
YX(L/)
j u -
2
i
I
ju+A1
(57)
while Y2(U) Is formed directly from equation (49), as
was H2 (U)
Y2(U)
jU
i'e
*2 AAl
£ £ jlz+A
(58)
Reduction of Equations
If only the higher frequencies ( U > 1) are
considered so that the effects of delayed neutrons need
not be considered, and the parameter B is defined as
B
1
1 kx(l J3 )
l
(59)
then the four transfer functions are reduced to the
following:


CHAPTER V
DISCUSSION AND CONCLUSIONS
Of particular interest in this study was the
representation of the coupling or interaction between
the two fuel regions of a two slab reactor. The
measurement of the complex quantity in the
University of Florida Training Reactor indicates that
the relatively simple two point one group bare reactor
model used in this thesis will give an adequate
description of the kinetics of a two region reactor.
The success of this representation does, however,
depend on the value taken for the transit time of a
disturbance between the two regions. The data that
have been taken from the UFTR (in particular the phase
part of Run CC, Figure 22) indicate that an adequate
representation of the transit time, 0 is one in
which a disturbance in the neutron density in one fuel
slab travels to the opposite region with the velocity
of a neutron wave.
These data have also indicated the feasibility
of determining the multiplication factor of each of
the regions by the measurement of the cross-power
103


APPENDIX C
OPERATION OF CROSS-POWER
SPECTRUM ANALYZER
The measurement of the quantity R12(i*y)/ defined
by equation (69), requires that three voltages be mea
sured on the cross-power spectrum analyzer; V, related
to the power spectrum of the output of the externally
disturbed region of the reactor through equation (93),
VR, related to the real part of the cross-power spectrum
between the outputs of the two regions of the reactor by
equation (91), and Vj, related to the imaginary part of
the cross-power spectrum between the outputs of the two
regions of the reactor by equation (92).
Figure 9 is a schematic of the cross-power
spectrum analyzer. The tape recorder and demodulators
are included in the figure but were not needed, of
course, for "on line" data processing.
The first step in the processing of each data
run was the amplification of the ac portions of the two
signals from the demodulators (or the two signals
directly in the case of "on line" processing) to about
10 volts peak to peak. A gain of about 25 was found
119


100
were collected while the reactor was fueled with the 93
per cent enriched fuel and operating at a power level
of one watt. The Keithley micro-microanuneters were
operated on the 10xl0_a ampere scale. No external re
activity input was used. Table 7 gives the data,
plotted in Figure 27, from Run Q, of the power spectrum
of the output of the south region of the reactor. This
spectrum is almost white. The detection chambers were
*
located about 18 inches from the fuel slab in positions
of low efficiency. It has been found (12.) (16) that in
such chambers the uncorrelated chamber noise dominates
the signal, and the spectra found from them are almost
flat.


CHAPTER I
INTRODUCTION
This study is a development of an experimental
method to measure the parameters peculiar to a two
region reactor system. Of particular interest is the
method of representing the coupling or interaction
between the two regions. A transfer function approach
is used.
The dynamic behavior of reactors has been
successfully analyzed from the transfer function point
of view. Transfer functions have been instrumental in
determining system stability and the design of control
systems. The measurement of reactor transfer functions
by sinusoidal oscillations, step, and ramp inputs of
reactivity are standard techniques and have allowed the
determination of parameters associated with the reactor
kinetics equations (jL) (2J.*
However, the statistical nature of the fission
process has led many investigators to measure dynamic
reactor parameters by purely statistical means. As
Underlined numbers in parentheses refer to
the list of references.
1


120
110
100
90
80
70
60
50
40
30
O o
O Q

G
Go
s
o
o
o
O
o
o o


00
_J I I I I L 1 1 1 I Ll
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
/
Frequency, cycles per second
Figure 20. Normalized Power Spectrum of Region 1 Output from Run R


120
110
100
90
80
70
60
50
40
30
CD
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 26. Normalized Power Spectrum of Region 1 Output from Run Z


LIST OF FIGURES
Figure Page
1. Two Region System Diagram 10
2. Normalized Amplitude Component of
Ri2 (U) for Several Values of B . 35
3. Phase Component of R-^ Vj fr
Several Values of B and U 36
4. Sequence of Operations Necessary for
Experimental Determination of Power
Spectrum 42
5. Sequence of Operations Necessary for
Experimental Determination of Cross-
Power Spectrum 42
6. Location of Ion Chambers in UFTR Core 46
7. Data Transcribing System 48
8. Modulator Schematic 50
9. Cross-Power Spectrum Analyzer 52
10. Demodulator Schematic 53
11. Random Reactivity Input Device .... 61
12. Power Amplifier Schematic 63
13 Normalized Amplitude Component of
G (LJ) from Runs L and M 71
14. Phase Component of G (iJ) from
Runs L and M 72
15. Normalized Power Spectrum of Random
Step from Run L 73
v


53
From Tape
Recorder 0
|
Demod
Demod
6AL5
To
#1
#2
Rx 6.8kQ
R-L 5.6kQ
8.2kQ
iokQ
Figure 10. Demodulator Schematic


Normalized Amplitude of G(ij), decibels
110
J ill 1 i i I i i i
0.2 0.5 1.0' 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 13. Normalized Amplitude Component of G(CJ) from Runs L and M


20
Relations may also be established for the
cross-power spectra between the external input and the
outputs of the two regions. The relations are derived
in the same manner as were those for anc* are'
and
Two Region Transfer Functions
The two transfer functions, and H2(L>/),
are now derived from the reactor kinetics equations for
a coupled reactor system. Each region is regarded as a
subcritical reactor with a neutron leakage interaction
from the other slab. Within each region, the theory is
space independent and a one group bare reactor model is
assumed. Cohn (21) has commented that the one group
bare reactor model is still used for practically all
kinetics work, even though it has been discarded as
impossibly crude for most statics calculations. His work
shows that for most reflected reactors, the kinetic
behavior corresponds to that of a bare reactor with the'
same 1/v lifetime.


109
Thompson (2_9) gives the following values at
thermal energies for the fuel and graphite regions of
the UFTR:
D
L
Fuel
0.25 cm.
0.08112 cm.-^
1.76 cm.
Graphite
0.917 cm.
0.00C342 cm.
51.8 cm.
These results are given for a fuel slab consisting of a
homogeneous mixture of water, aluminum, and uranium.
It is also assumed that the reactor is loaded with 3500
grams of the 20 per cent enriched fuel.
For the reactor loaded with 3100 grams of the
fully enriched fuel, £ for the fuel section is 0.0723
a
centimeters-^. The diffusion coefficient and diffusion
length of the fuel region will also change from the
values given for the 20 per cent enriched fuel. These
changes are small enough, however, and the lag time in
the fuel is a small enough fraction of the total lag
time, that the values for the 20 per cent enriched fuel
may be used for the fully enriched fuel also.
The dimensions a and b are taken as 264 and 142
centimeters. These dimensions correspond to the com
plete diffusion interface and include some of the graph
ite surrounding the fuel slab. The neutron velocity is
taken as 2200 meters per second.


122
was used in the second channel of the analyzer instead
of f^(t). Then at each frequency set point, the same
settings and procedures were used to measure the
voltages VR and Vj as were used to measure the voltage
V.


68
Two such runs were made, Run L and Run M. The
only difference between these was that the GM counter set
up was adjusted to give a count rate of 1700 counts per
minute in Run L and 13,500 counts per minute in Run M.
Results of these runs are given in Tables 1 and 2 and plot
ted in Figures 13 and 14. The power spectrum of the random
step input of Run L is also obtained from the measurement
and given in Table 1. Note that the amplitude correction
factors (Appendix B) must be used for the power spectrum.
This measured spectrum is seen in Figure 15 with the the
oretical spectrum as calculated for a g of 28.3 counts per
second from equation (96).
The "flags" on the data points plotted in the
above figures indicate one standard deviation. No "flags"
are shown to indicate the accuracy of the frequency
readings, which were determined by the accuracy of the
settings on the band pass filters. The manufacturer gives
5% as the accuracy of the frequency settings on the
Krohn-Hite ultra low frequency band pass filters. It was
necessary to calibrate the band pass filters against a
frequency standard to match the filters to within one
per cent of each other and the standard.
The non-statistical errors associated with the
cross-power spectrum analyzer were small and are


16
The crosscorrelation function and the cross-power spectrum
are a Fourier transform pair defined by (20)/
§12(U)
(>12 (T)e~jur T
and (16)
§12UJ)QUT U .
(17)
The term on the left side of equation (15) is then by defini
tion the cross-power spectrum between the outputs of region
1 and region 2, (^.^(CV)- In the ^^rst double integral on
the right side of equation (15), the change of variable,
x = T + A (18)
is made so that the integral becomes
-JOAx-A)^ f
CD
hl( A )0i2 (x)dA
-oo
(19)
In this double integral, a separation of variables puts all
terms involving x under one integral and all terms involving
A under another. Equation (19) becomes a product of two
single integrals,
0 hx( A)dA
CD
(20)


(101)
£
x
which gives the standard deviation, (J f, of a function
f(x,y) of the two quantities x and y which have standard
deviations, (J and 0*v, respectively,
x y
The above expression gives the standard deviation
of the phase part (given by the quotient Vl/vR) as,
(102)
and the standard deviation of the amplitude part
) as,
(given
(103)
As seen in Figure 13, the standard deviation of
the amplitude part at the lower frequencies is rather
large. The standard deviation of the phase part, how
ever, is of a magnitude comparable with the size of the
data points.


28
When there is no external reactivity input to
region 1, fluctuations still exist in the neutron popula
tion of that slab. These are the so-called self
fluctuations or reactor noise. The statistical nature
of the fission process gives rise to these fluctuations
in reactor power levels even when the reactor is operated
at steady state. It has been shown (_7) that the power
spectrum of reactor (single region) self-fluctuations is
related to the transfer function, E[LJ), through,
Q + Q'
H (LJ)
(50)
where Q and Q' are constants. The expression has two
terms, the first being white noise of the detector and
the second being related to the kinetics parameters. The
transfer function used is the one based on reactivity,
although the exact input to the system is not definable
(16) .
Both slabs will exhibit these self-fluctuations,
but crosscorrelation should measure the effect of the
self-fluctuations of one region on the neutron density
in the other region. The transfer functions to describe
this situation would be source transfer functions, the
self-fluctuations in one side acting as the input to the
other side.


64
also called a "random telegraph signal" and its power
spectrum is well known as (20)
$11 tu
4a2g
1
+ 4g2
(96)
where g is the average number of sign changes per
second of a square wave that varies between +a and -a.
The power spectrum of this signal was then made white
out to the desired frequency by a corresponding increase
in the count rate seen by the GM tube. The signal was
given the proper amplitude and dc level on the analog
computer.
The reactivity device was sufficiently long to
permit the piston end to be fully inserted into the
south beam port extension, with the drive assembly out
side the south reactor shield.


17
The unit impulse response function and the transfer
function are also a Fourier transform pair given by (19),
h(t)
dU
and
(21)
H (U)
h(t)0jU;t dt ,
(22)
where R(U) is the system transfer function. The first
factor in equation (20) is then recognized as the complex
*
conjugate of the transfer function of region 1, ^(U)
From equation (16), the second factor is identified as
the cross-power spectrum between the external input and
the output of region 2. In the second and last term on'
the right of equation (15) the change of variable
y T + A + 6 (23)
allows a separation of variables as in the first integral
which will give a product of two single integrals,
A? 0
j u6
L -'-QO
22S"jL/Ydy
(24)
Here, the first factor in brackets is the complex comjugate


121
peak to peak. Two potentiometers could be adjusted in
each of the three signal paths. All potentiometers
were set with the null potentiometer. It was also pos
sible to set the gain of the amplifiers at the output
of the phase changers at either 1 or 10. M values of
from 40 to 2500 were employed.
The time constant of the integrating amplifiers
was set from 0.0001 to 0.0003. Once again, the potenti
ometers used for these settings were set with the null
potentiometer. It was desirable to select a time
constant such that about 50 volts would be collected
across the integrator during the three minute integra
tion time.
The measurement of the power spectrum of the
externally disturbed region of the reactor was performed
by feeding its signal, fj^t), into both channels of the
analyzer, making the setting described above, turning
the analog computer to operate for three minutes then
switching it to hold, and reading the voltage on the
integrating amplifier with the null potentiometer. The
voltage, V, related to the power spectrum appeared
across the integrator ordinarily giving the real part
voltage during cross-power spectrum analysis.
After V was measured, the signal representing
the output of the other region of the reactor, f2(t)r


LIST OF REFERENCES
1. E. P. Wigner, Use of the Pile Oscillator for the
Measurement of Pile Constants, CP-3066, June 23, 1945.
2. A. F. Henry, The Application of Reactor Kinetics to
the Analysis of Experiments, Nuclear Sci. and Eng., _3,
52-70 (1959).
3. F. deHoffman, Intensity Fluctuations of a Neutron
Chain Reactor, MDDC-382, LADA-256 (declassified
October 7, 1946).
4. W. K. Luckow, "The Evaluation of Nuclear Reactor
Parameters from Measurements of Neutron Statistics,"
Ph.D. Dissertation, University of Michigan, 1958.
5. R. W. Albrecht, The Measurement of Dynamic Nuclear
Reactor Parameters by Methods of Stochastic Processes,
Trans. Am. Nuclear Soc., 4, 2 (1961) pp. 311-312.
6. C. Velez, "Autocorrelation Functions of Counting Rate
in Nuclear Reactors and Their Application to the
Design of Reactor Control Instrumentation," Ph.D.
Dissertation, University of Michigan, 1959.
A less complete description of this work is found in
C. Velez, Autocorrelation Functions of Counting Rate
in Nuclear Reactors, Nuclear Sci. and Eng., 6, 414-
419 (1959).
7. M. N. Moore, The Determination of Reactor Transfer
Functions from Measurements at Steady Operation,
Nuclear Sci. and Eng.. 3, 387-394 (1958).
8. M. N. Moore, The Power Noise Transfer Function of a
Reactor, Nuclear Sci. and Eng., 6, 448-452 (1959).
9. C. E. Cohn, Determination of Reactor Kinetic Para
meters by Pile Noise Analysis, Nuclear Sci. and Eng.,
5, 331-335 (1959).
123


-25v
Figure 12. Power Amplifier Schematic


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82
Reactor Data
Some of the first reactor data were taken "on
line." The external reactivity input device used for
these runs was driven by a random step function at an
average count rate of 200 counts per minute. Because
of the low count rate and inadequate frequency response
of the reactivity input, the data obtained were not
reliable at frequencies beyond about two cycles per
second. Although the data looked promising out to this
point, little could be determined, as seen from Figures
2 and 3.
The random input device designed and constructed
by Rajagopal and described in the previous chapter was
used in succeeding runs. These runs were recorded and
processed "off line."
The reactor was fueled with 3100 grams of
uranium-235 in the form of 93 per cent enriched uranium-
aluminum fuel plates for Run R. The commercial low
frequency noise generator was used as an input to the
driving amplifier of the input device and the data were
recorded with the reactor operating at a power level of
10 watts. The two Keithley micro-microammeters were
operated on the 3xl0-6 ampere scale. The results of
this run are given in Table 4. Not only the phase and
amplitude parts of the function R12(C^) but also the


Phase Angle of G(U) degrees
Figure 17. Phase Component of G(iJ) from Run P


30
The desired transfer function for region 2 when
a random variation of reactivity is inserted in region 1
is a source-based transfer function and is given by
h2 iU) =
An^
AnXAX 0
-j uB '
(54)
where
An2 is the output of region 2 and A N^A^ 0
is its input. H2(U) may be found from equation (49)
directly. Thus from equation (49),
H 20J) =
1
¡L
k2 (1 /? )-1 + y. Ai
~o
(55)
j u -
i
£ i j lj + A ^
When there is no external reactivity input to the
two slab system, both transfer functions are thought of
as source transfer functions in which the leakage from
one side is considered a source to the other side. The
transfer functions must then be formed by
Y X(U) =
,Ni
y2(L/) =
An2 *2e~iUd
An2
Anxax 6
(56)


125
19. Y. W. Lee, "Statistical Theory of Communication,"
John Wiley and Sons, New York, 1960.
20. J. S. Bendat, "Principles and Applications of Random
Noise Theory," John Wiley and Sons, New York, 1958.
21. C. E. Cohn, Reflected-Reactor Kinetics, Nuclear Sci.
and Eng., 13, 12-17 (1962).
22. A. M. Weinberg and E. P. Wigner, "The Physical
Theory of Neutron Chain Reactors," University of
Chicago Press, Chicago, 1958.
23. J. M. Duncan, "University of Florida Training Reactor
Hazards Summary Report," Florida Engineering and
Industrial Experiment Station, Gainesville, October,
1958.
24. R. B. Blackman and J. W. Tukey, "The Measurement of
Power Spectra," Dover, New York, 1958, p. 58.
25. J. Millman and H. Taub, "Pulse and Digital Circuits,"
McGraw-Hill, New York, 1956, pp. 28-46.
26. Bennett and Fulton, The Generation and Measurement of
Low Frequency Random Noise, Journal of Applied Physics,
22, 1187-1191, 1951.
27. R. W. Badgley, "Power Spectral Density of the Uni
versity of Florida Training Reactor Operating in the
Subcritical Region," M.S. Thesis, University of
Florida, 1962.
28. R. H. Chastain, "Transfer Function of the University
of Florida Training Reactor," M.S. Thesis, University
of Florida, Gainesville, Florida, 1961.
29. C. A. Thompson, "Neutron Flux Calculations for a
Graphite Moderated Twenty Per Cent Enriched Reactor,"
M.S. Thesis, University of Florida, Gainesville,
Florida, 1961.


94
determine if the break represents a lower value of B or
if it represents the end of correlation. The power
spectrum of the region 1 output indicates that corre
lation would end much beyond 25 or 30 cycles per second.
At 30 cycles per second the power spectrum is down 40
decibels from the amplitude at the lower frequencies.
From the calibration runs it would be expected that
correlation would not continue when the input (which is
the output of the south fuel region in this case) power
spectrum drops off 40 decibels. There was no drop-off
of the spectrum at the lower frequencies as in Run R,
indicating a more satisfactory operation of the reactiv
ity input device. The magnitude of the input signal was
also greater than before resulting in the lower fre
quency portion of the spectrum being 50 decibels above
the reactor and instrument noise level.
A check was made on the frequency at which
correlation between the two outputs of the reactor ended
with a complementary run to Run CC. Run Z was taken
under the same operational conditions as Run CC except
that both ion chambers were located side by side outside
the south fuel region. The results of this run are
presented in Table 6. The amplitude and phase parts of
the cross-power spectrum and the power spectrum of the
output of the south fuel region are plotted versus fre
quency in Figures 24, 25, and 26, respectively.


Phase Angle of R12 (CuO # degrees
+10
0
-10
-20
-30
-40
-50
-60
-70
-80
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
vD
to
Figure 22. Phase Component of R^2from Run CC


Normalized Amplitude of Ri ?((*/), decibels
y
b
Figure 2. Normalized Amplitude Component of for Several Values of B


45
connected by aluminum overflow and vent pipes. This
water, overflowing from the fuel boxes, flows by gravity
through a cooling coil and then into a storage tank.
The water is pumped from the storage tank to the fuel
boxes. The storage tank prevents any sudden temperature
changes in the coolant, its capacity being about six
times that of the reactor system.
Air spaces 8 inches wide exist between the
graphite reflector and the inside of the biological
shield on the north and south sides of the reactor. The
ion chambers used to detect the output of each region of
the reactor were positioned in these spaces against the
graphite, as shown in Figure 6. A graphite layer of 18
inches separated the chambers from the fuel boxes.
The UFTR is provided with six horizontal beam
ports. These ports have a 4 inch diameter next to the
core and increase to a 6 inch diameter through the rest
of the shield. The core graphite stacking adjacent to
the south beam port is fitted with a removable graphite
plug 18 inches long by 2 inches diameter. When this
plug is removed, the south beam port extends to the
side of the south fuel box, at about the horizontal
center plane of the core. The random reactivity device
was inserted into the reactor through this facility.


EVALUATION OF PARAMETERS IN A TWO
SLAB REACTOR BY RANDOM
NOISE MEASUREMENTS
By
ALLEN ROSS BOYNTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1962


ACKNOWLEDGMENTS
The author wishes to acknowledge his gratitude
for the advice and encouragement of his advisory
committee. In particular, grateful acknowledgment is
made to the chairman, Dr. Robert E. Uhrig, for his
continued guidance and encouragement during the course
of this work.
The author wishes to express his gratitude to
the staff and technicians of the Department of Nuclear
Engineering. J. Mueller and K. L. Fawcett constructed
the early equipment, J. Wildt constructed the final
equipment, and L. D. Butterfield operated the reactor
for most of the runs.
The author wishes also to express his
appreciation to Dr. Herbert Kouts and V. Rajagopal of
the Brookhaven National Laboratory for their loan of
the random reactivity input device.
ii


115
K
V (R + K)2 + K2
(110)
while the phase parts are
and
PL = tan
-1
K
R + K
(111)
p = tan
H
-1
R + K
K
(112)
The R and C values at each frequency set point are given
in Table 9.
Polystyrene capacitors were used in values of 1,
0.1, and 0.01 microfarads. The R values were set on 10
turn 0.1 megohm potentiometers. Any additional resist
ance was provided by adding precision resistors in
series with the potentiometers. These t1% resistors
were available in values of 0.1, 0.2, 0.5, and 1.0
megohms. The potentiometers were used as variable re
sistors and their set points are also given in Table 9.
The attenuation of the signals through the phase
changing filcers was calculated from equation (110) for
the two values of input resistors used, 0.1 and 1.0
megohms. When this attenuation was normalized to one


Figure 11. Random Reactivity Input Device


105
a neutron lifetime of 2.8x10^ seconds, a B value of 450
corresponds to a multiplication factor of 0.874. Criti
cal loading data with the 93 per cent enriched fuel
indicate that the reactor would be critical with a one
slab loading of 2005 grams of uranium-235. With a normal
two slab loading of the fully enriched fuel, a total load
ing of 3225 grams of uranium-235, or 1612.5 grams per slab,
produces a critical mass. The two regions are therefore
very strongly coupled/ the interaction between them is
worth about 390 grams of uranium-235. As much as 1950
grams of uranium-235 can be loaded into one region of the
reactor. These attempts to load the reactor to critical
in a one slab configuration gave not only an estimate
of the critical mass but also an estimate of the multipli
cation factor for a given mass of fuel. For a uranium-
235 mass of 1612.5 grams, the following estimates of the
effective multiplication factor were obtained: loading
of May 25, 1961, 0.870.08; loading of March 21, 1962,
0.790.08; loading of April 3, 1962, 0.820.08; and
loading of June 15, 1962, 0.76^0.08. These results may
be summarized to give a multiplication factor of
0.81^0.04 for one region independent of the other when
the two slab system is critical. This value is in fair
agreement with the B value of 450 obtained by the cross-
power spectrum measurement.


120
110
100
90
80
70
60
50
40
30
*
0.2 0.5 1.0 2.0 5.0 10 20 50 100 200 500
Frequency, cycles per second
Figure 15.
Normalized Power Spectrum of Random Step from Run L


Ill
TABLE 8
VALUE OF
TRANSIT
TIME BETWEEN
SLABS
Frequency
cycles/second
Lag
Time in Milliseconds
Fuel
Graphite
Total
0.5
0.213
2.39
2.60
1
0.213
2.39
2.60
2
0.213
2.39
2.60
5
0.213
2.39
2.60
10
0.213
2.35
2.56
20
0.213
2.26
2.47
50
0.213
1.99
2.20
100
0.213
1.63
1.84
200
0.213
1.25
1.46
500
0.213
0.830
1.04
1000
0.209
0.594
0.803


56
Precision O.1% polystyrene capacitors were
available in values of 0.01, 0.1, and 1.0 microfarads
and were used with precision tl% resistors and ten turn
t3% IOOkQ potentiometers to set the C and R values of
the filters. The capacitor and resistor values used
are given in Appendix B.
It is noted that when an R and C are selected
such that equation (90) is satisfied, the values of pL
and pH are such that there is a +90 degree shift be
tween them and the attenuation is the same through both
filters. This attenuation is, however, frequency
dependent. The calculation of correction factors for
this attenuation is straightforward and also given in
Appendix B.

The three signals,f^(t)
, f2(t)
, and f (t+90O)

^ J
u, J
were then amplified by a factor of from about 20 to about
400 on the analog computer. This amplification was
necessary in order to have signals of about 200 volts
' peak to peak for inputs into the Model 160 electronic
multipliers on the analog computer. Large input signals
are necessary to minimize the error that these diode
network multipliers introduce for low input voltage.
*These symbols are now taken to represent the
signals after they have passed through the phase shift
ing networks.


Normalized Amplitude of R10(L/), decibels
110
100
90
80
70
0D
Ln
_J I I I I i I I I I u
0.2 0.5 1.0 2.0 5-0 10 20 500 100 200 500
Frequency, cycles per second
Figure 18. Normalized Amplitude Component of from Run R


55
and
PL
tan-1
rkUc
R + K
(87)
and for the high pass filter,
rkUc
*V (R + K)2 + r2k2U2c2
(88)
and
tan1
R + K
rkUc
(89)
where K is the value of the input resistor to the
operational amplifier. Since the signals were passed
through the band pass filters and were at essentially a
single frequency, a convenient R and C were selected to
give the desired 90 degree phase shift at each frequency
set point. It was also found to be convenient to select
a RC combination such that the amplitude, / a/, of the
high pass and the low pass filters were equal at a given
frequency. This was accomplished when
R
Uc
2 7Tfc
(90)


117
for the complete isolation of the filter, i.e. for K
very large, the reciprocal squared of the attenuation
was the correction factor to be applied for the phase
change filters. These correction factors are given in
Table 10.
The attenuation due to the high pass filter
before multiplication was calculated from equation (88).
The outputs of these filters were always fed into 1.0
megohm input resistors. Once again, the square of the
reciprocal of this attenuation was the correction factor
for the high pass filters. This correction factor is
also given in Table 10 as well as the combined correc
tion factor, (CF), which takes both phase shifter
attenuation and high pass filter attenuation into account.
Once again it is noted that these corrections
were unnecessary in the measurement of the quantity
^12^^^' ^ut were used when cross-power or power spectra
were measured with the analyzer.


2
early as 1946, deHoffman (3.) developed an expression
for the intensity fluctuations of a chain reactor as
measured by a counter with statistical response. The
formulation included the effects of delayed neutrons
and involved the dispersion of the number of neutrons
emitted per fission. Later workers in this area were
Luckow (4) who used the variance to mean ratio of the
number of neutrons counted by a detector during a fixed
counting time to measure the prompt neutron lifetime;
Albrecht (5.) who extended Luckow's analysis to include
delayed neutron parameters; and Velez (6) who used the
autocorrelation function of the counting rate from a
reactor to measure the dynamic parameters. These inves
tigations, while giving insight into the fundamental
nature of multiplying systems and information on the
influence of the statistical fluctuations in the
reactor power level on such things as the precision of
neutron measurements and the design of reactor control
equipment, cannot furnish as much information about the
complete reactor system as can the transfer function.
Fortunately, statistical considerations are
related to the transfer function approach, since the
power spectral density functions (or power spectra) and
correlation functions form a Fourier transform pair.
Correlation functions have been known to statisticians


CHAPTER III
MEASURING SYSTEM
In this chapter are discussed methods and
equipment used to perform the direct measurement of the
cross-power spectrum of the outputs of the two regions
of the UFTR. The first section contains a development
of the relations necessary for the evaluation of the
real and imaginary parts of the cross-power spectrum
using band pass filters. These relations indicate the
sequence of operations that must be executed. The
second section contains a description of the University
of Florida Training Reactor (UFTR). Since the measure
ments were made on this reactor, its pertinent features
and dimensions are given. The third section examines
in detail each component used to collect, store, and
process data. The final section describes the random
reactivity input device.
i
Measuring Theory
Most investigators have used autocorrelation and
crosscorrelation measurements to obtain the transfer
functions of single region reactor systems. This method
38


84
power spectrum of the output of the south region (the
externally disturbed region) are given. The amplitude
part of R12 (CJ) the phase part of and the
power spectrum of the output of the south region are
plotted versus frequency in Figures 18, 19, and 20,
respectively.
The phase data points, up to about 25 cycles
per second, although badly scattered, are seen to fol
low the set of theoretical curves calculated for a lag
time between the two reactor regions corresponding to
the velocity of a thermal neutron wave. A B value of
about 400 is indicated. Beyond a frequency of 25
cycles per second, little can be determined because of
the excessive scatter of the data. The amplitude data
points depart from the theoretical curve for a B of
400 beyond 15 cycles per second. It is difficult to
determine whether this break in the amplitude data
actually reflects a low value of B or if it signifies
the end of correlation between the two reactor output
signals because of low signal to noise ratio. The
phase data would suggest the latter condition. Con
sideration of the power spectrum of the output of the
south fuel slab (Figure 20) would also indicate that
correlation would end much beyond 20 cycles per second.
This power spectrum, which is representative of the


Normalized Amplitude of G(L/), decibels
Frequency, cycles per second
Figure 16. Normalized Amplitude Component of G(LJ) from Run P


34
then it may be expressed in terms of its amplitude,/r^2 />
and its phase, p(R^2), parts, as
/R12/
2B22 + 2U2
bS + U
(70)
and,
p(R12) = tan
-1
-Ucos0 U B2Sin0L7 1
. B2COS 9U Usin 0 U J
(71)
Equation (70) shows that the amplitude portion of R^2
independent of 0 and that the shape of / R12 ¡ depends
only on the value of the multiplication constant of the
region not disturbed by the random reactivity input.
The phase portion of R12(U), however, is strongly de
pendent on the value of 0 particularly as the product
Qu becomes large. Figures 2 and 3 show plots of the
amplitude and phase portions of R^2(Cu/) versus frequency
as calculated from equations (70) and (71) for a reactor
of the geometrical arrangement of the UFTR. The nor
malized amplitude plot (Figure 2) is made for several
values of B: 100, 200, 500, and 1000. These values
correspond to multiplication factors of 0.97, 0.94, 0.85,
and 0.70, respectively, for a neutron lifetime of 3xl0-4
seconds. As noted before, this plot is independent of


33
and
[Ve] e'iue
<|>i2(L/) =
B2 + iU
(66)
The transit time between the regions, 0 has been con
sidered independent of direction. Further simplification
is effected if the slabs are assumed identical except
for their multiplication factors which may be different.
This assumption results in the condition
= A2 = A ,
(67)
and equation (64) becomes,
$12(U)=MLe
-iud $
ll
(U)
(68)
b2 + j U
The expression for when there is no external
reactivity input to region 1 is identical to equation (68).
The power spectrum, (^^(L/), is a real quantity but the
cross-power spectrum is a complex quantity which may be
expressed in terms of its phase and amplitude components.
If the complex quantity R12((-^) is defined as
^ 12
r12(l;) =
nOJ)
(69)


58
products f^(t)
f2(t)
and f^(t)
f2 (t+90)
obtained. These products were then integrated and the
integrals related to the real and imaginary parts of
the cross-power spectrum. If the gain of the opera
tional amplifiers between the band pass filters and the
multipliers was M, the gain of the integrators (1/RC)
was P, the voltages on the integrators were VR for the
real and V^. for the imaginary parts for an integration
time of T seconds, and the set frequency and frequency
band width were f-^ and Af^, respectively, then the
real and imaginary parts of (^)12(L/) were given by,
Vr(CF)
Rei2(fl} ~ A
T Af. (MPP
and
Vj(CF)
In*12 (f )
Afx(M)
(91)
(92)
where CF was the total correction factor for the ampli
tude attenuation of the phase shifters and high pass
filters.
The measurement of the quantity R12(CJ) as
defined by equation (69) requires that both the


44
Once again, the integral on the left side of equation
(84) is recognized to be a real quantity, so that when
the real parts are equated,
fx(t)
U.
f2(t + 90)
u.
dt
lm12 ((JL) =
(85)
Equations (82) and (85) may be solved
experimentally as shown in Figure 5.
The University of Florida Training Reactor
The UFTR is a lOkw, heterogeneous, light water
and graphite moderated, graphite reflected, thermal
reactor of the Argonaut type (23>) It may be fueled
with either 20 or 93 per cent enriched uranium-aluminum
fuel elements. The fuel is contained in thin flat
plates which are assembled in bundles of eleven plates.
The reactor core contains 24 bundles of fuel plates
placed in six water-filled aluminum boxes surrounded by
reactor grade graphite. Four cadmium control blades,
protected by magnesium shrouds, move between the fuel
boxes. Twelve inches of graphite separate the two rows
of fuel boxes. Aluminum headers connect to the bottom
of the fuel boxes and supply the light water used for
moderating and cooling. The tops of the boxes are


9
In the second section, these transfer functions
are developed from the reactor kinetics equations that
are derived for a two slab coupled reactor system.
The final section brings the cross-power spectra
and transfer functions together. After certain simpli
fying assumptions are made, the equations are examined
for the measurement of the parameters particular to the
two slab reactor system.
Cross-Power Spectra
Consider a two region coupled system as shown
in Figure 1. Each region is a linear system having its
own unit impulse response function h^t) and h2(t), and
outputs given by f^(t) and f2(t). The regions are
coupled to one another because a certain fraction, A^,
of the output of region 1 is received as an input to
region 2 after a transit time, 9 and vice versa.
Region 1 is also subjected to a random stationary
external input, i(t).
The crosscorrelation function of the outputs
of region 1 and region 2, 012(T), is defined as (),
T
where T is a continuous displacement time independent
of t. The two systems have been specified to be linear,


o
10
20
30
40
50
60
70
80
90
-oo
Frequency, cycles per second
Figure 14. Phase Component of G(CJ) from Runs L and M


118
TABLE 10
CORRECTION FACTORS FOR FILTERS
f Correction
Factor for
cycles Phase Changer
per K=1.0 K=0.1
second megohm megohm
Correction
Factor for
High Pass
Filter
Combined System
Correction Factor (CF)
K=1.0 K-0.1
megohm megohm
0.1
3.86
144.0
3.59
13.8
517.0
0.15
2 .63
68.2
2.08
5.46
142.0
0.2
2.10
61.9
1.64
3.55
102.0
0.3
1.66
20.3
1.30
2.16
26.4
0.4
1.49
13.0
1.15
1.71
14.9
0.5
1.36
9.20
1.10
1.50
10.1
0.6
1.29
7.03
1.07
1.38
7.52
0.7
1.25
5.86
1.05
1.31
6.15
0.8
1.22
4.94
1.04
1.27
5.12
0.9
1.19
4.37
1.03
1.23
4.50
1.0
1.17
4.16
1.02
1.19
4.25
1.25
1.14
3.10
1.02
1.16
3.16
1.5
1.11
2.62
1.01
1.12
2.65
2.0
1.08
2.10
1.01
1.09
2.12
2.5
1.05
1.82
1.00
1.05
1.82
3.0
1.05
1.66
1.00
1.05
1.66
4.0
1.04
1.46
1.00
1.04
1.46
5.0
1.04
1.37
1.00
1.04
1.37
6.0
1.02
1.30
1.00
1.02
1.30
7.0
1.02
1.26
1.00
1.02
1.26
8.0
1.02
1.21
1.00
1.02
1.21
9.0
1.01
1.19
1.00
1.01
1.19
10.0
1.01
1.17
1.00
1.01
1.17
12.5
1.01
1.12
1.00
1.01
1.12
15.0
1.01
1.10
1.00
1.01
1.10
20.0
1.08
2.10
1.00
1.08
2.10
25.0
1.05
1.82
1.00
1.05
1.82
30.0
1.05
1.66
1.00
1.05
1.66
40.0
1.04
1.46
1.00
1.04
1.46
50.0
1.04
1.37
1.00
1.04
1.37
60.0
1.02
1.30
1.00
1.02
1.30
70.0
1.02
1.26
1.00
1.02
1.26
80.0
1.02
1.21
1.00
1.02
.1.21
90.0
1.01
1.19
1.00
1.01
1.19
100.0
1.01
1.17
1.00
1.01
1.17
125.0
1.01
1.12
1.00
1.01
1.12
150.0
1.01
1.10
1.00
1.01
1.10
200.0
1.08
2.10
1.00
1.08
2.10


95
TABLE 6
RESULTS OF RUN Z
f
Normalized Amplitude
Phase Angle
Normalized Power
of Calibration
of
Spectrum of Out-
cycles
Spectrum
Calibration
put of Region 1
per
Spectrum
second
decibels
degrees
decibels
0.4
99.9
+3.0
100.1
0.5
99.3
-0.8
103.1
0.6
101.7
-0.9
99.5
0.8
98.1
-1.5
101.3
1.0
101.0
+ 2.0
101.2
1.25
101.2
+ 0.5
99.3
1.5
98.6
-0.5
100.2
2.0
101.0
0
100.7
2.5
100.5
0
99.6
3.0
99.9
-0.6
97.9
4.0
101.7
0
97.4
5.0
101.0
0
93.8
6.0
97.2
-3.0
94.6
7.0
101.7
-1.3
89.9
8.0
100.1
-0.5
87.1
9.0
96.8
-0.7
87.7
10.0
101.4
-7.7
83.6
12.5
99.9
+ 3.6
77.9
15-0
97.8
-4.5
75.4
20.0
98.9
0
67.3
25.0
96.9
-2.0
63.0
30.0
94.4
-7.5
60.0
40.0
93.8
-10.0
55.6
50.0
87.0
-5.6
55.7
60.0
80.7
-11.0
56.0
70.0
84.6
+ 8.4
54.4
80.0
80.7
+ 13.0
53.0
90.0
79.6
-13.0
52.5
100.0
85.5
-44
50.5
125.0
80.2
-70
48.4
150.0
80.1
+98
47.4
200.0
79.9
-132
46.3


89
For Run CC the reactor was fueled with 3500 grams
of uranium-235 in the form of 20 per cent enriched
uranium-aluminum alloy fuel plates. The reactor was
operated at a power level of 50 watts and the two Keith-
ley micro-microarameters were operated on the 3x10^
ampere scale. The random reactivity input device was
found to produce disturbances of almost twice the magni
tude of the normal reactor noise. The power amplifier
was driven by a randomly switched square wave. This
random square wave was constructed with a GM counter
and the flip-flop section of a count rate meter. A
count rate of 20,000 counts per minute was used.
The results of Run CC are presented in Table 5.
Once again the phase part of R12(CJ)/ the amplitude part
of R12(L/) and the power spectrum of the output of the
south fuel region are listed in Table 5 and also plotted
versus frequency in Figures 21, 22, and 23, respectively.
The phase data points, up to a frequency of 40
cycles per second, follow the theoretical curve corres
ponding to a B of about 450 and a lag time, 6 deter
mined from the velocity of a thermal neutron wave.
Beyond 40 cycles per second, the data points become
excessively scattered. The amplitude data points break
from the theoretical curve for a B of 450 at frequencies
beyond 25 cycles per second. Again, it is difficult to


3
for many years, yet were relatively unknown to engineers,
who were analyzing frequencies and power spectra. Since
the power spectrum is related to the transfer function
of a linear system, engineers have been quick to use
this new measuring technique.
Moore (7.) (Q) was among the first to relate the
statistical nature of chain reactors to their transfer
functions when he expressed the power spectrum of the
reactor noise as a function of the square of the modulus
of the transfer function. Cohn (9.) used a band pass
filter to measure the high frequency portion of the
reactor noise spectrum which is dependent on the ratio
of the effective delayed neutron fraction,^, to the
prompt neutron lifetime, £ This technique has become
almost standard (10). The method has, however, several
difficulties. First, the exact input to the system is
not definable. This difficulty is attacked by Cohn (11)
who assumes the reactor noise to arise from a random
"noise equivalent" neutron source driving the reactor.
The characteristics of this source are calculated from
conventional random noise theory, and the resulting
pile noise obtained through the use of the source
transfer function. Bennett (12) develops the spectrum
and variance of pile noise according to the formulation


79
TABLE 3
RESULTS OF RUN P
f
cycles per
second
Normalized Amplitude
of G(U)
decibels
Phase Angle
of G (U)
degrees
0.6
95.8
-6.1
0.8
101.2
-1.2
1.0
99.0
-6.7
1.5
98.4
-4.1
2.0
96.6
-6.5
3.0
98.6
-10
4.0
100.8
-18
5.0
98.2
-18
6.0
99.4
-25
8.0
97.0
-33
10.0
98.4
-30
15.0
100.0
-37
20.0
96.8
-51
30.0
94.2
-60
40.0
93.6
-62
50.0
90.8
-70
60.0
89.8
-72
80.0
89.2
-77
100.0
86.4
-80
150.0
86.6
-90


24
When these definitions are substituted into the equations
of region 1, (38) and (39),
d A 4 [(fcio+kx) (l-/)-l ] (n^n^
-(niO^On,)
X A(c4 m+ ) +S,rt+(5s.
(41)
'ilO il' 10 w "1
£
(n20+n2)-e^(n20+(5n2)+-^- ji2(n20+n2)+ ...
2
0 £
2' 2 dt
and
a, .,s A(kio+(5ki) dt CilO+ ^ cil
(42)
- A(cio+(5c1)
The sums of the steady state portions of these equations
are equal to zero, and the equations are linearized by
setting the product, k (5 n, equal to zero since both
quantities will be small. The equations for region 1 are
then
d(5ni (1-AnxoH [k10(l-^)-l] <5! Yx <5,
at z + i + T 1
+ S-L +
(43)
and
d^cn = Anioki +
at
C
1 '
(44)


39
lends itself well to digital processing of data. Results
of auto or crosscorrelation measurements are in terms of
the unit impulse response function, while the quantity
of interest is the transfer function. The power and
cross-power spectra, however, may be measured directly
and, as seen from the previous chapter, relate directly
to the system transfer functions.
The theory of the measurement of power and
cross-power spectra is outlined below. The relation
of the crosscorrelation function to the cross-power
spectrum is,
f
00
The definition of the crosscorrelation function,
T
may be substituted into (17) to give,
T
lim i
T oo 2T
f1(t)f2(t+T )dt =
>12iu)eiUT -T
OO
(73)


66
measurement of the transfer function of a low pass
filter.
When a linear system is excited by a random
input, the cross-power spectrum between the input and
the output, CD (iJ), is equal to the product of the
-1 io
power spectrum of the input, (^^(L/), and the trans
fer function of the system, G(L/) (19) The real and
imaginary parts of the transfer function may be
measured by,
Re [g(U)]
and
Im[G((J)]
Re
<£>io(u>
#<£^)
(97)
Im
[*
io
(U)
§iiJ>
(98)
The low pass filter, the transfer function of
which was measured, was formed by a one megohm resistor
followed by a 0.1 microfarad capacitor to ground.
These were precision tl% components. The output of
the filter fed a 0.1 megohm input resistor of an
operational amplifier on the analog computer. The
amplitude and phase parts of the transfer function,
G(U) were calculated from,


42
fl(t)
Figure
fx(t)
f2(t)
<|)ii((J1)tAu1
set at
U1
Sequence of Operations Necessary for Experimental
Determination of Power Spectrum
Re 12{U1)T^U1
set at
ux
Figure 5. Sequence of Operations Necessary for Experimental
Determination of Cross-Power Spectrum


83
TABLE 4
RESULTS OF
RUN R
f
Normalized Amplitude
Phase Angle
Normalized Power
cycles
of r12 (U)
of Rj_2 ((jJ)
Spectrum of Out-
per
put of Region 1
second
decibels
degrees
decibels
0.6
98.8
-6.2
78.9
0.8
100.5
0
88.3
1.0
99.7
-7.0
89.2
1.25
99.3
-5.0
99.6
1.5
101.7
-0.7
98.5
2.0
99.5
-14
101.6
2.5
102.7
-7.0
99.4
3.0
99.3
-11
101.3
4.0
98.3
-13
99.0
5.0
100.1
-12
98.1
6.0
99.3
-15
95.0
7.0
103.1
-17
91.6
8.0
99.9
-16
93.1
9.0
99.7
-23
89.8
10.0
98.9
-26
90.6
12.5
98.7
-20
85.0
15.0
101.3
-28
79.2
20.0
97.8
-39
76.4
25.0
96.5
-45
71.4
30.0
95.6
-40
70.2
40.0
91.5
-34
68.7
50.0
88.7
-19
70.4
60.0
89.3
-3.2
71.1
70.0
86.5
0
70.2
80.0
85.3
-13
66.9
100.0
79.1
-44
64.6


Phase Angle of R,9(L/) degrees
+20
1
+ 10
-10
-20
-30
-40
-50
-60
-70
Curve
1
2
3
4
5
6
100
200
500
1000
500
500
Neutron Wave
Neutron Wave
Neutron Wave
Neutron Wave
2x10~4 seconds
Zero
0.2
0.5 1.0 2.0 5.0 10 20 50 100
Frequency, cycles per second
200
00
O'
500
Figure 19. Phase Component of from Run R


11
so the convolution integral between the input, output,
and unit impulse response function is given as (19),
f].(t) = j h^A) Input(t-A )<*A (2)
J-oo
Reference to Figure 1 shows that the input to region 1
is the sum of the external input, i(t), and the frac
tion, A2, of the output of region 2, at a previous time
corresponding to the transit time, 0, between the
regions. Thus,
Input(t) = i(t) + A2f2(t 0 ) (3)
Equation (2) is now written as
fl(t) =/ **1 J- ao
Mt-A) + A2f2 which may be substituted into the definition of the
crosscorrelation function between the outputs of
region 1 and region 2, equation (1), to give
lim
1 r T r oo r
12 T->od2t/ I hl^ A ) | Kt- A)
012IT> -
+ A2f2(t-A 0)
dAf2(t+r )dt. (5)


21
The kinetics equations for region 1 may then be
written (18),
n
£
+
I A
cil + Sx + E
(34)
and
dcll /?lklnl
dt i
Aicii
(35)
where, n is the neutron density in region 1
%
A
K
ii
is the effective multiplication factor
for region 1
is the overall prompt neutron lifetime
in the system
is the fraction of fission neutrons in the
i-th group of delayed neutrons and
T.
i
is the decay constant of precursor of
i-th group of delayed neutrons
is the concentration of precursors of
i-th group of delayed neutrons in region 1
is the neutron density from external source
neutrons in region 1.
The term E is the interaction terra caused by the leakage
of neutrons out from region 2. This term may be repre
sented by some fraction, A2, of the neutron density in
region 2 at some previous time corresponding to the


Figure 6. Location of Ion Chambers in UFTR Core


32
Hj^fU) =
a
Bt + jU -
[aia2/£,] e"2jL^
B0 + iU
, (60)
H2((J) = Y 2iU) =
B0 + jU
(61)
and
y]L (U) =
B, + j(J
(62)
These expressions are now used in equations (25), (29),
(30), (31), (32), and (33). When an external reactivity
input is given to region 1,
^>12 (CJ) =
(1>8)/ §12^> + A2ejiJ^ §22
Bn jU -
e
2ju0
B, j U
[Ai $12(L/) =
b9 + JL/
-
ii
(U) + A~ 0
-JU0
(63)
(64)
$i2(L/)
B, + iU -
[aia2/£2] e ~2jU^
(65)
B2 + jU


112
to be
The value of
in the graphite region is found

2
T('
2
JL
' TX
2
1
_264_
142
51.8
(52 = 0.001012 .
(108)
The thickness of the graphite section separating the two
fuel regions is 30.48 centimeters. When this distance
is divided by the velocity of a neutron wave in graphite
as determined by equation (104), the quotient is the
lag time in the graphite.
2.04x102)2+CJ2] -2.04X102
(109)
Values of the lag time in the graphite and the
total lag time are also given in Table 8.
e
graphite
4.80x10
U
-2
['


CHAPTER II
THEORY
The experiment to be analyzed consists of
measuring the cross-power spectrum of the outputs of
each region of a two region reactor. These outputs
are examined under two operational conditions: (a)
a random stationary external input to region 1 and,
(b) no external input to either region. These cross
power spectra are then related to the transfer
functions of the individual regions.
In the first section of this chapter, the
relations that involve the two coupled linear systems
and their inputs and outputs are developed. Although
these relations are general in nature and not
restricted to reactor systems, they are developed so
as to be directly applicable to the experiment. This
is evidenced by the transformation of the relations
from the time domain into the frequency domain so
that the linear systems involved are represented by
their transfer functions.
8


o
-30
-60
-90
120
150
180
210
240
270
1.0 2.0 5.0 10 20 50 100 200
Frequency, cycles per second
Figure 3. Phase Component of R12(L/) for Several Values of B and


54
insertion losses in the filters become; however, the
resolution also becomes poorer.
and f2(t)
, were
J U,
The two signals, (t)
V
then given a phase lag in a low pass filter and
was also given a phase lead in a high pass
f2(t)
filter such that there was a +90 degree shift between
it and the lagging f-^(t)
u.
and f2(t)
. The phase
U,
shifting networks were more complex than the simple
filters indicated in Figure 9 because each filter fed
into either a one megohm or a one-tenth megohm input
resistor of an operational amplifier on the analog
computer. Since the amplifier summing junctions are
essentially at ground potential, either a one megohm
or a one-tenth megohm resistor was in parallel with the
capacitor, C, in the low pass filters and in parallel
with the resistor, R, in the high pass filter. It is
easily shown (25) that the amplitude, /af, and the
phase, p, portions of the transfer functions for these
filters are given by, for the low pass filters,
K
(R + K)2 + R2K2U2C2
(86)


60
Thus the amplitude correction factors for the
low and high pass filters were unnecessary in the calcu
lation of R12(f). They were employed, however, when
either the cross-power or the power spectra were
measured.
Random Reactivity Input Device
The reactivity input system consisted of (a) a
drive assembly of a coil in an electromagnet with spring
return, (b) an absorber in the forro of a piston moving
in a cylinder, (c) a power amplifier for the moving coil,
and (d) a random input signal.
The drive and piston assembly was constructed
by V. Rajagopal and is described in his doctoral dis
sertation (16). Figure. 11 illustrates this assembly.
The drive assembly was a coil moving in the air gap of
an electromagnet. The flux in the air gap was main
tained by a steady current in the field coil. A 28
volt 3 amp power supply was used for the electromagnet.
The force exerted by the moving coil was controlled by
the current supplied from the power transistor in the
power amplifier. Mechanical coupling from the moving
coil to the cadmium covered piston was through a steel
wire encased in a square slot between two blocks of
graphite. The opposite end of the piston was connected


19
The left-hand side of equation (28) is recognized as the
cross-power spectrum between the outputs of region 1 and
region 2, (j}12(L/); the first factor in brackets on the
right-hand side as the transfer function of region 2,
H2(L/)i and the second factor in brackets on the right-
hand side as the power spectrum of the region 1 output,
(U) Equation (28) may now be written with equation
(25) to give the relation of the cross-power spectrum
between the outputs of region 1 and region 2 to their
transfer functions, their power spectra, and the cross
power spectrum between the external input and the output
of region 2.
(29)
If there is no external input, equations (25) and (29)
become,
(30)
(31)
It is noted that different symbols are used for the trans
fer functions in the set of equations (25) and (29) than
in the set of equations (30) and (31). The reasons for so
symbolizing these transfer functions are given in the next
section.


13
r 00
f2(t) =/ h2(A) Input(t- A)^A (8)
J-co
In region 2 the only input is that fraction,
of the output of region 1 which reaches region 2
after the transit time,0 between the regions. Thus,
Input (t) = ) (9)
Equation (8) is written as
f2 (t)
OD
h2 ( A ) fj. (t-A $ )d A ,
-oo
(10)
which, when substituted into equation (1), gives,
0
'-T
00
h2( A)fi(t+T -A0 )d Adt.
(ID
Once again, inversion of the order of integration will
yield a more convenient grouping,
0 12^> A1
tx(t)f1(t+T-X-0)dtdX.
-oo
(12)


18
of the transfer function in region 1, and the second factor
in brackets is the power spectrum of the region 2 output.
Thus equation (15) may finally be written,
$12(Cu/) = H*(L/)(|i2(CJ) + A2ejLy^H*(Ly) Equation (13) may be manipulated in the same manner
as was equation (7). Taking the Fourier transform of both
sides of equation (13) gives,
jfff 0i2e'JL/T J-OO
ai r -jut c
7W e iT h2 J-GD J -0D
(26)
A change of variable on the right-hand side of equation
(26) to,
-r A -9
(27)
will give, after a separation of variables,
^2 (A ) 0
-iuX
(28)


15
the cross-power and power spectra) of the outputs and
obtain unit impulse response functions. The Fourier
transformation of the unit impulse response function,
which gives the transfer function, is performed on a
digital computer. If the cross-power and power spectra
can be measured directly, a computer operation may be
eliminated. The equipment available to the author also
suggested the direct measurement of the cross-power and
power spectra. Equations (7) and (13) must then be
%
examined in the frequency domain.
The Fourier transform of equation (7) is taken
i / j
by multiplying both sids of the equation by p ,
2TT
where J = PT. LJ is the angular frequency (radians
per second), and integrating with respect to T Equa
tion (7) then becomes,
CD
27T
0i2 (r )0JLyrdr
-oo
1
2 7T
ao
OD r 00
Q~jUTdT vAx^jCr + Aial
J-oo
+ wj e'iUT^r
J-CO J-CD
(15)
hj_ ( A )0 22 ^ +A+@)dA*


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
LIST OF TABLES iv
LIST OF FIGURES V
Chapter
I. INTRODUCTION 1
II. THEORY 8
Cross-Power Spectra 9
Two Region Transfer Functions .... 20
Reduction of Equations 31
III. MEASURING SYSTEM 38
Measuring Theory 38
The University of Florida
Training Reactor 44
Cross-Power Spectrum Analyzer .... 47
Random Reactivity Input Device ... 60
IV. EXPERIMENTS AND RESULTS 65
Calibration 65
Reactor Data 82
V. DISCUSSION AND CONCLUSIONS 103
Appendix
A. CALCULATION OF TRANSIT TIME
BETWEEN SLABS 108
B. PHASE SHIFT AND HIGH PASS
FILTER DETAILS 113
C. OPERATION OF CROSS-POWER
SPECTRUM ANALYZER 119
LIST OF REFERENCES 1^3
BIOGRAPHICAL SKETCH 126
iii


106
The UFTR is certainly not the ideal reactor on
which to conduct a study of loosely coupled two region
reactors, since its two fuel regions are separated by
only 12 inches of graphite and are rather strongly
coupled. This strong coupling results in a large
value of B which moves the break frequency of the
amplitude part of toward the higher frequencies.
In order to measure this break frequency, the input to
the second region should contain sufficient power in
all frequencies out to and slightly beyond the break
frequency. Although the random reactivity input device
may deliver a signal whose power spectrum is flat
beyond the break frequency of interest, the fuel slab
that receives this signal acts very much as a low pass
filter and will attenuate the higher frequencies. Thus
the input to the second fuel slab will not necessarily
have a flat power spectrum beyond the break frequency.
It is felt that the UFTR is probably as strongly
coupled a reactor for which the cross-power spectrum
between the outputs of the two regions will give
information.
Several equations were developed in Chapter II
involving the cross-power spectra between the two out
puts of the fuel regions and between the random
reactivity input and each of the fuel region outputs.


78
processed from the tape are given in Table 3. The data
points of the amplitude and phase parts of the transfer
function of the filter, G(U), are shown plotted versus
frequency in Figures 16 and 17. Once again, the the
oretical curves are given also. The data from this run
seem more scattered than the data points of Runs L and
M, especially those of the amplitude part. This addi
tional scatter must be attributed to the AM modulation,
recording, and demodulation. The magnitude of the
amplitude scatter is about -2 decibels, while little
difference is noted in the phase part. In all other
respects the results of Run P were identical to those
of Runs L and M. The amplitude portion of the data
departs from the theoretical curve between 60 and 100
cycles per second, a higher frequency than that at which
the departure occurred in Run L and a lower frequency
than that at which the departure occurred in Run M.
This was to be expected since the random input was
driven by a count rate higher than Run L and lower than
Run M.
Run P demonstrates that the techniques and
equipment used to collect, record, and analyze the data
were satisfactory.