EVALUATION OF PARAMETERS IN A TWO
SLAB REACTOR BY RANDOM
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
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 . . . . . . .
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 -
. . ii . . iv
. . V
1 8 9
. . 8
. . 9
. . 20 . . 31 . . 38 . . 38 . . 44 . . 47
. 60 . . 65 . . 65 . . 82 . . 103
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
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
. . 69
. . 70
* . 79
* . 83 . 90
* . 95
* . 101
. . 111i
* . 116
* . 118
LIST OF FIGURES
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 46 48 50 52 53 61 63 71 72 73
LIST OF FIGURES (Cont'd)
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
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
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
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
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.
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.
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.
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),
~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-)
Region 1 output
OuExternal Inputput hy(A)f1(t) 81
Figure 1. Two Region System Diagram
so the convolution integral between the input, output, and unit impulse response function is given as (19),
hl(A) Input(t-A)dA .
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,
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
hl( ) i(t-x )
+ A2f2(t-A- ) dif2(t+T)dt.
12(T ) =
By reversing the order of integration, equation (5) is put into a more desirable form,
12 (T) =fC h
T ()T-4 T
Gor T + A hl( A)T-i Tf
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,
h1()422 (T +A+O)dA.
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,
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
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.
Once again, inversion of the order of integration will yield a more convenient grouping,
q12(fa h 12(T)" 1 h
2 Tl ooypb 01 fltfl(t+T-)(-O)dtd ,
f2 (t) =f -0
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.
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)
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
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,
-oo l( j Lid d
A2 aD Co
The crosscorrelation function and the cross-power spectrum are a Fourier transform pair defined by (20),
() 12 ( T ) e-JWT dT
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 + ,
is made so that the integral becomes
2 7 an
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,
eW hl(A)dAl D 22(x) e-JWxdx .
hl 1)O2(x)d) .
The unit impulse response function and the transfer function are also a Fourier transform pair given by (19),
h(t) = i-f H() jt dLJ and (21)
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
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
Equation (13) may be manipulated in the same manner as was equation (7). Taking the Fourier transform of both sides of equation (13) gives,
2io 012 7( T) 63-JUT7 dr
A 00 -jUT oa-.
A change of variable on the right-hand side of equation
z = T -A -0 will give, after a separation of variables,
S12(T)eJL T dT
= Al1io IOD
h2(A)e-j i d ( JL .
2 11() (2)dz
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.
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)
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
icil + S1 + E (34)
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
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
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
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
A2n2 A2 dn A2e2 d2n2 dt 2 dt2
dcil ik n1
d = cil. (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:
n1 = nl0 + 5nI n2 = n20 + n2
kI = kl0 + 5 kI c 2 = Ci20 + 65c12 (40)
ci2 = cil0+ 5cil Sl = s10 6Sl
When these definitions are substituted into the equations of region 1, (38) and (39),
+ i(c110+ 6 Ci1) +S1o+ (41)
A2 F[~~j (5d2 d21
S(n20 2)- (n20+6n2)~ t(n20+6n2)+
d (c ) = dt ilO l1
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
S 2-- + dt ,]
[kl(1 )-1 ]n1 c+ i
+6sl + [ 5n2 -
2 0 2d 26n2 d6n2 2 d262
d + 2 d-
- .. and
d63cil $inl0k +1 $ikl0( n1 A 6c 5 dt 9 c
;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
+A j1 + [AN2 OAN2 + 2 2
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,
= 1- a + 2 ...BN22.
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
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
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.
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
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.
When a random reactivity input is given to region 1, the desired transfer function of that region is given by
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
(A1A2 -2-JL kl1- )1 + k 1 9 1S ii l
JIl_ +A_ l + 2)i
T 2 L+A
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(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),
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
Y1 (L/) = ___ __AN2A2e(56)
Y1(LJ) may be formed directly from equation (48),
kl(1i i-1 k
while Y2(LJ) is formed directly from equation (49), as was 2 (LJ)'
Y2 (L/) =
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 -)
then the four transfer functions are reduced to the following:
Y1 (L j) =
H1 (u) =
(1 -J ),
B + j12 11 &, -2j)JG
B2 + JCJ
H2(&/) = Y2(LA) = ,/_ ,
B2 + jLJ
Y1 (J) = *
B1 + JLJ
(30), (31), (32),
input is given to
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
4 12() =[A e ()
B2 + jL1
=(1-)/8 i( ,) + A 2e
B + J [A1A/2 e -2JLJ
+ l- (6s) B2 + J
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 ())
then it may be expressed in terms of its amplitude,/R12/' and its phase, p(R12), parts, as
A 2B22 + 2WJ2
-2 = (70)
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
9B=100 200 500 1000
I I, I I I ...I .. I .... ......... I
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
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.
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.
Most investigators have used autocorrelation and crosscorrelation measurements to obtain the transfer functions of single region reactor systems. This method
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)
The definition of the crosacorrelation function,
(T lm) f fl (t)f2 (t+T)dt, (1)
T-12 T-oo 2 1 2
may be substituted into (17) to give,
lim 1 f(t) f(t+T)dt 12 (j) eJ dlJ T-* oo2T 2
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 =
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,
o fl (t) f2 (t+ 7) 1
dt = 12(u1)e 1 A1L
The power spectrum measurement, when fl (t)=f2(t), is considered first; then,
TIJ fl(t) f f (t+T) I1dt = 1lwl)eil Awl
If 7' is set equal to zero, equation (76) is then written,
dt = t11(j)Awi1 I
where the left-hand side is the mean square value of
* This expression is solved for 11I(1)
11 (L/1) =
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
to reduce equation (75) to
Re 2cos WlT-Iml2sinLJlT+j(ImcosLWT+Re sinLJ ) Aw1.
1 (t) I i 2 (t*F-) I
(11 (() TA(1J
Sequence of Operations Necessary for Experimental Determination of Power Spectrum
Re z2(LJ1) TALJ1
Iml2 (l) T AL
f(t) Band pass 90 Multiplier Integratorf2 filter Phase Shifter
Figure 5. Sequence of Operations Necessary for Experimental
Determination of Cross-Power Spectrum
When T is set equal to zero, equation (80) reduces to,
f1 fl(t) f2(t) dt = Rel2 + jIml2 A l.
The integral on the left side of this equation is a real quantity so that when the real parts are equated,
Sfl(t) U1f2(t) 1 dt
Rel2 (J1) = (82) TA /I
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)
Thus equation (80) becomes,
1 fl(t) f2(t + 900) dt = -Iml2+jRel2 AJl"
0 1 i1
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,
-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
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
0.1 to 1000 cps
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
Coil is #28 Wire Wound in a Double Layer on 1" Dia. 3" Long Form Tuned to Approx.
Figure 8. Modulator Schematic
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
set at fl
Cross-Power Sectrum Analyzer
From Tape Recorder 0.l f
Demod #1 Demod #2
R1 6.8KQ R1 5.6K
R2 8.2K R2 10K
Figure 10. Demodulator Schematic
insertion losses in the filters become; however, the resolution also becomes poorer.
The two signals, fl(t)] and f2(t) were
then given a phase lag in a low pass filter and f2(t) I was also given a phase lead in a high pass
filter such that there was a +90 degree shift between it and the lagging fl(t)] and f2(t)J The phase
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,
fAIL + (86) (R + K)2 + 2y2(j2C2
PL = tan-1 [ RKUC (87) [-R + K
and for the high pass filter,
A/ RKL)C (88) (R + K)2 + R2K2W2C2
= 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
R = 2 c (90) WC 27Tfc
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
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,
Rel2 (f1) = T A (91)
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
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,
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)
Im R12 (f1) (95)
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
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
Power Amplifier Schematic
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.
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.
Performance of the cross-power spectrum analyzer and the data transcribing system were checked with the
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
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)
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,
/A/G (86) )(R + K)2 + R2K2LJ2C2
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).
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
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
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
44 (D Run L G~
0 Run L
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
-20 Theoretical Curve
0 Run L o Run M S-50
44 % o0
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
Normalized Power Spectrum of Random Step from Run L
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
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,
(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
=. (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.
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
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.
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
O Theoretical o o
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)
from Run P
-50 Theoretical Curve o -60
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
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
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
2.0 2.5 3.0
6.0 7.0 8.0
10.0 12.5 15.0 20.0 25.0 30.0
70.0 80.0 100.0
102.7 99.3 98.3 100.1
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
78.9 88.3 89.2 99.6
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
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
90 80 70
a 0 e
-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
w I1 2 3 4 5 6
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
90 80 70 60
50 40 30
00 0 0o
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
0 0 ee 0
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.
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
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
2.0 2.5 3.0
4.0 5.0 6.0 7.0
10.0 12.5 15.0
20.0 25.0 30.0
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
101.4 102.1 100.7
97.0 95.8 9'4.7
90.9 89.7 88.3 85.7
78.0 71.5 66.3 59.3 58.0 52.8 52.3 53.0
48.7 48.9 48.6
2.0 5.0 10 20 50 Frequency, cycles per second
Figure 21. Normalized Amplitude Component of R12 (J) from Run CC
110 100 100
90 80 70
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
0 \ Od 0 S-70 0
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
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
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.