Transient response of damped linear network with particular regard to wideband amplifiers

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Transient response of damped linear network with particular regard to wideband amplifiers
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United States. Atomic Energy Commission. MDDC ;
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Los Alamos Scientific Laboratory
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MDDC 770
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UNITED STATES
ATOMIC ENERGY COMMISSION
OAK RIDGE
TENNESSEE



"TRANSIENT RESPONSE OF DAMPED LINEAR NETWORK WITH
PARTICULAR REGARD TO WIDEBAND AMPLIFIERS"

by

W. C. Elmore


Los Alamos Scientific Laboratory



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ABSTRACT


When the transient response of a linear network to an applied
unit step function consists of a monotonic rise to a final constant value,\
it is found possible to define delay time and rise time in such a way
that these quantities can be computed very simply from the Laplace sys-
tem function of the network. The usefulness of the new definitions is
illustrated by applications to multi-stage wideband amplifiers for which
a number of general theorems are proved. In addition an investigation
of a certain class of two -terminal interstate networks is made in an
endeavor to find the network giving the highest possible gain-rise time
quotient consistent with a monotonic transient response to a step function.









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I. Introduction.

The transient behavior of any linear system (or network) is con-
tained implicitly in the system function F(s) which expresses directly
the steady-state sinusoidall) response of the system. The variable in
the system function, S = a + j ( is the complex angular frequency:
w is the ordinary (real) angularfrequency andais a real variable in-
troduced for the purpose of facilitating the transient analysis of the sys-
tem.1 In the present paper we shall be concerned primarily with the
class of linear systems in which the transient response to a unit step
function(the so-called indicial admittance) consists of a monotonic rise
to a final constant value. For simplicity in presentation only the tran-
sient response of a low-pass, wideband amplifier will be considered.
Many of the results obtained, however, apply equally well to other elec-
trical systems, as well as to mechanical, acoustical, thermal, and to
mixed systems, provided only that they are linear and have a monotonic
transient response to a unit step function.

The most important system function of an amplifier is the com-
plex gain, G(s), connecting input and output voltages of the form Eest.
In the case of a low-pass amplifier, G(s) can always be separated into
two factors, G9(s), which governs the response at low frequencies, and
G2(s), which governs the response at high frequencies In an unfed-back
amplifier, G1(s) owes its origin to various RC networks which couple the
plate of one tube to the grid of the next, and which furnish bias voltages
to various points in the amplifier. The system function G2(s) owes its
origin primarily to parasitic interstate capacitances which shunt the
signal-carrying leads. Since we shall be interested in the problem of
obtaining the greatest possible gain--rise time quotient for an amplifier,
G2(s) may reflect the presence of compensating inductances, of feedback.
or of any other circuit arrangements used to shorten the rise or to im-

2The definition of rise time is considered in Sec. 2.
prove the transient properties of the amplifier. The portion G2(s) of the
system function may be considered as that of an equivalent amplifier
idealized to have perfect low-frequency response.

For convenience in analysis, we shall use the normalized system
function_g2(s) = G2(s)/G2(O), where G2(O) is the gain of the idealized
amplifier at zero frequency. Normalization evidently makes the final
value of the response to a unit step function also unity.

*I
The notation and terminology adopted here is that found in Gardner
and Barnes, Transients in Linear Systems, Vol. 1.








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It is not difficult to show that the normalized system function
g2(s) of a stable amplifier containing a finite number of lumped circuit
elements takes the form

2(s) = 1 als +a2s2 +...+ans (1)
2 m
1 + bls + b2s +...bm s
+ 2 m

where the coefficientsai and bi are real, m > n and the poles of -2(s)
all lie in the left half of the complex s-plane. The transient response
of the amplifier to the unit step function u(t) can be computed by means
of the inverse Laplace transformation
c + j a
1 C 1 st
e(t) = 2-j s g2(s) eds. c>0. (2)

c j OD

Transient response curves computed from Eq. (2) for various
amplifiers have a variety of shapes, some common forms of which are
illustrated in Fig. 1. The input signal, u(t), is shown in (a). The tran-
sient response shown in (b) consists of a delayed rise, followed by a
train of damped oscillations. The response shown in (c) is similar to
that in (b) except only a finite number of oscillations occur, preceding
a gradual approach of the curve to the final value unity. In (d) and (e)
are illustrated monotonic transient response curves having different
amounts of damping. The response in (e) is supposed to be that of an
amplifier having certain adjustable circuit parameters which have been
chosen to achieve the shortest possible monotonic rise for a given am-
plifier gain.

Any circuit elements introduced in an amplifier for the purpose
of controlling the shape of the transient response curve may be termed
compensating elements. In the present instance they afford high-fre-
quency compensation to the response of the amplifier. When the fastest
possible monotonic rise has been obtained with the particular type of
compensation used, the amplifier is said to be critically compensated.
Any other degree of compensation may be referred to as under- or over
compensation.3

For many applications it is important to avoid over-compensation in an
amplifier. This is particularly true for pulse amplifiers used in nuclear
physics (to amplify pulses obtained from an electrical detector of radia-
tion) an 1 for wideband amplifiers used in studying fast electrical tran-
sients (such as amplifiers for cathode-ray oscillographs). Video ampli-
fiers used in television applications evidently are not as critical in this








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-4-

respect since it is customary to over-compensate them.

It is evident that the various types of transient curves illustrated
in Fig. 1 possess certain common features, in particular, a finite time
of rise which occurs delayed with respect to the input step signal. For
many purposes each curve can be sufficiently well characterized by its
delay time and rise time, which can be defined in several different, but
approximately equivalent ways. One of the purposes of the present paper
is to propose useful definitions for these quantities, with a view to facil-
itating their computation from the system functiong2(s). The new defini-
tions, unfortunately, are of such a nature that they apply only to systems
which are not over-compensated. Their utility for all systems having a
monotonic transient response, however, appears to be great enough to
outweigh this defect. It is possible that an equally useful method for
treating the over-compensated case can be discovered.

2. The Definition of Delay Time and of Rise Time

A number of definitions of delay time and of rise time appear to
be in practical use. Two of these will be illustrated by reference to Fig.
2, which shows the transient response e(t) to the unit step function, and
its derivative e'(t), of an under-compensafed amplifier.4

4The curve e'(t) may be considered to be the response of the amplifier
to a unit impulse applied at time t = 0.

The delay time,_TD, is usually defined as the time required for
the response to reach half its final value, as illustrated in Fig. 2a. The
rise time, TR, is sometimes defined as the reciprocal of the slope of the
tangent drawn to the response curve at its half-value point, again as illus-
trated in Fig. 2a. A somewhat more practical definition results if TR
is taken to be the time required for the response to increase from 10 to
90 percent of its final value. Although these definitions are useful in the
laboratory, they are extremely awkward for making computations, or for
entering upon a theoretical investigation of the relative merits of various
methods of compensating an amplifier to improve its rise time. The
difficulty, of course, lies in the necessity for computing the transient re-
sponse curve for each case under consideration, a formidable undertak-
ing. It is practically impossible to obtain values of TD and TR, as de-
fined, by a simple method of analysis.

Let us now consider alternative definitions for delay time and rise
time. Evidently the delay time should be measured from t = 0 to some








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time at which the transient rise is about one half over. It is reasonable,
therefore, to measure TDto the centroid of area of the curve e' (t), that
is, 0

TD = f t e' (t)dt (3)


This definition of delay time is illustrated in Fig. 2b, and it is seen to
give a result which differs but little from that obtained from the custom-
ary definition. The two values of delay time depart most markedly in
the case of a very asymmetrical response curve. It is easy to convince
oneself that the new definition becomes meaningless if the curve e' (t)
possesses a negative portion i.e., e(t) is not monotonic. It will be shown
presently that it is a simple matter to obtain a value of the integral in
Eq. (3) directly from the system function g2 (s).

The rise time TR should express in a prescribed manner the time
required for the transient rise to occur. Now the shorter the rise time,
the narrower (and higher) the curve of e '(t). It is reasonable, therefore
to define TR as proportional to the radius of gyration of the area under
the curve, that is,
o0
TR2 = Const. (t TD)2 e '(t) dt. (4)

o

The constant of proportionality is chosen to be 2 7r for the following
reason: it is possible to show that the curve e '(t) for any n-stage ampli-
fier5 approaches more and more closely the form of a Gauss error curve
with increasing n. To make the new definition of rise time agree with the
definition based on the slope of the transient response curve (Fig. 2a),
the value of TR should therefore be

TR = 1 = /2- [radius of gyration of
e' (t)I e' (t)]
max


5. The individual stages in the amplifier must each have a monotonic
transient response to the unit step function.








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6 -


t (b)


(d)


(e)


FIG. 1. Some typical transient response curves.


I_\







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- 7 -


- TD)2 e'(t)dt


FIG. 2. Curves illustrating the definitions of delay time and of rise
time.


e(t)
1

0.5


e' (t)


.. '. -"r


I _








MDDC 770
-8 -

which expresses the relation between the height and the radius of gyra-
tion of a Gauss error curve of unit area, here denoted temporarily by
e (t). Equation (4) can now be written


TR = 27r [ t2 e '(t) dt TD2 1/ (5)



where the integral has been expressed in terms of moments about the
time origin. It is found in most instances that rise times computed from
Eq. (5) differ by less than ten percent from the rise times defined earlier,
which can continue to be used for most laboratory work.

The great usefulness of the new definitions of delay time and of
rise time will now be demonstrated. The system function g2(s) aud the
transient response e '(t) are related by the direct Laplace traniorma-
tion '

g2(s) = e '(t) E-St dt, (6)


where the real part of s is sufficiently positive to make the integral con-
verge. By expanding e-st in ascending powers of st, we have that
co


o o
gn(s) 1-s t e '(t)dt + s 2 t2 e'(t)dt-... (7)
0 0
Hence, if a given system function is expanded in ascending powers of s,
it is a simple matter to obtain by inspection the first and second moments
of e (t) about the time origin, and therefore to obtain values of TD and
TR defined by Eqs. (3) and (5), respectively. Part of the virtue of the
proposed definitions lies in the ready way in which delay times and rise
times can be computed. Other advantages of the definitions will be made
use of in Sec. 3.

It is useful to obtain expressions for TD and TR for a system
function of the form given in Eq.(1). By expanding Eq. (1) in ascending
powers ofs, it is found that
TD = bl a. (8)

and that

22 2
R = al +2(a2- b2) (9)
2f








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9 -

Before considering other matters, let us compute values of TD
and TR for a single-stage amplifier having a two-terminal plate load
impedance of the type shown in Fig. 3a.6 Such an amplifier stage is said
to be shunt-compensated. The system function of the single stage is
identical to the driving point impedance of the plate load. Hence, we

6 2
By setting C = 1, R = 1, and expressing L in units of R 2~ values of
TD and TR are obtained in units of RC. This device enables the system
function to be written immediately in a simple, normalized form.

have that
1 + Ls
2(s) = s + Ls2 (10)

In order that no transient oscillation of the type shown in Fig. lb shall
exist, the poles of_g2(s) must lie on the negative real axis of the s-plane.
This requires that in Eq. (10) L 4 1/4. The values of TD and TR (com-
puted using Eqs. (8) and (9) are

TD = 1 L,
T = 2 (1 2L L2). (11)

When L = 0, corresponding to a simple resistance-coupled amplifier
stage, TD = 1, and TR = 2f= 2.51. When L = 1/4, corresponding to
critical shunt compensation, TD = 0.75 and TR = V --t 16 = 1.66.
The critically compensated stage is S = R = V 16/7 (=1.51)
times faster in its transient rise thail the simple R-coupled stage. The
quantity S may be termed the rise-time figure-of-merit. In Sec. 4 an
attempt will be made to discover an interstate network which gives the
smallest value of TR(or largest value of S) with a given interstate par-
asitic capacitance and load resistance. The problem is somewhat analo-
goustothat of discovering the network which leads to the maximum band
width (without regard.to good transient response.)7.

7See, for instance Bode, Network Analysis and Feedback Amplifier
Design, pp. 408 et seq.

3. Some Theorems Regarding Multi-stage Amplifiers.

We have just seen how the delay time and rise time of a single
amplifier stage can be computed. Let us now consider how the delay
time and rise time of an unfedback multi-stage amplifier depends on the
properties of individual stages in it.









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- 10 -


FIG. 3. Shunt-compensated interstate networks.








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11 -

If the amplifier contains n stages in tandem, the system function
of the entire an lifier is the product of the system functions of the in-
dividual stages. Let the system function of the ith stage be g2i(s), and

8This statement is true provided that no coupling between stages exists
except through the electron stream in the constituent amplifying tubes.
This situation can be realized in practice if the tubes in the amplifier
are pentodes.

let the corresponding values of delay time and rise time be TDi and TRi
respectively. The functiong2i (s) can be expanded in the series
s2 T2R1i
g2i(s) = 1 sTDi + + T2Di) -", (12)

which is obtained directly from Eqs. (3),(5) and (7). The system function
of the entire amplifier therefore becomes
n
g2(s) = r g2is)
1 2
s2 T2Ri 2
= sZTDi + 2 2 T2 +ZT Di + 2 TDi

TDi]-... i>j

By again using Eqs. (3), (5), and (7), the delay time and the rise time ol
the entire amplifier are found to be

TD TDi (13)

1
and
n
2 T2 (14)
R Ri

The result expressed by Eq. (13) is intuitively obvious, since it
is to be expected that the total delay is the sum of the delays of the in-
dividual stages. The manner of combining rise times indicated by Eq.
(14) is not as evident, although the fact that this simple mode of combine
tion is the correct one has been proposed by several of the author's col-
leagues. prior to the present proof of the theorem.

Another theorem of practical importance concerns the manner in
which the gain of an n-stage amplifier should be distributed among the
individual stages in order to achieve the shortest possible over-all rise
time for a given over-all gain. Now the rise time of any stage in the am-








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12 -

plifier varies directly with the gain of the stage, since both quantities
are proportional to the value of the plate load resistor. It is desired,
therefore, to minimize the expression (14) subject to the condition that
n
7 TR = Constant. (15)
1 Ri
It is easy to prove from Eqs. (14) and (15) that the over-all rise time
is a minimum when the rise times of all stages are made the same. If
TR1 is the rise time of each stage, the rise time of an n-stage amplifier
becomes
TR = T n1 / (16)

Let us now consider certain matters regarding the design of ax
amplifier consisting of n identical stages. We shall treat the simple
case where the interstate couplings are of the general type illustrated
in Fig. 3, i.e., a parasitic capacitance C, and a resistance R in series
with some sort of compensating reactance whose impedance becomes
zero at zero frequency. At frequencies where 1/w C> R, the gain of
each stage is
G =gmR, (17)

where._m is the transconductance of the amplifying tube.9 The rise time
of each stage can be written in the form

It is assumed that R << rp, the plate resistance of the tube.


1 V-27r RC (18)
1 S
where S is the rise-time figure-of-merit of the stage. By definition S
= 1 for a simple R-coupled stage and we have already shown, for example,
that S = V16/7 for a critically shunt-compensated stage. Eliminating
the resistance R between Eqs. (17) and (18), we have that

G = ( gm ) T
G1 (m ) T1 (19)

The quantitye/ m f C expresses the figure -of-merit of the amplifier
tubes, and may be conveniently stated as so much gain per microsecond
rise time.

If T is the rise time of the n-stage amplifier, the according to
Eq. (16) the rise time of each stag'emust be T1 = y/n requiring a
gain for each stage








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13 -


G S /gE__ T (20)
1 \ / I n

Equation (20) can be written as a pair of equations,
S
G1 = G (20a)

where J
-C
o -2TT C T

The quantity Go is the gain of a single R-coupled stage of rise time T
whereas G1 is the gain that each stage of the n-stage amplifier must
have In order that the entire amplifier shall have the rise time T.10

1The pair of equations (20a) can be made the basis of a convenient
nomograph to aid in the design of an amplifier of assigned rise time and
total gain.

The total gain of the amplifier is
n
Gt = G1 (21)

Let us now investigate what gain per stage will result in the
shortest rise time for a given total amplifier gain. From Eqs. (20) and
(21), we have that

T= ( 2, C Gt (22)
S g n

On minimizing T with respect to n while keeping Gt constant, it
is found that n In Gt or that
1 `2 /2
G1 = E = (2.72 ..)1/2 = 1.65 *". (23)

This result is independent of the degree of compensation used, provided,
of course, that critical compensation is not exceeded. The minimum rise
time which can be obtained for a total gain Gt is found from Eqs. (20),
(21) and (23) to be

min S C( m ) 2ElnGt, (24)
v m


requiring a total of n = 2 In Gt stages.








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4. Some Critically Compensated Interstage Networks

There are two matters of considerable interest concerning inter-
stage networks of a critically compensated wideband amplifier. The
first is primarily of theoretical importance and concerns the maximum
value that can be obtained for the quantity S (the ratio of the rise time
obtained with a simple RC network to that obtained with a compensated
network). The second matter is of practical importance and concerns
the design of networks whose performance approaches as nearly as pos-
sible the theoretical limit.

Interstage coupling networks of two types must be distinguished,
two-terminal and four-terminal.11 This distinction is necessary since
it is possible to separate the parasitic interstate capacitance into two

Strictly speaking, two-terminal and three-terminal networks.

portions, the output capacitance of one stage, and the input capacitance
of the following stage. If a critically compensated four-terminal network
is based on the two capacitances, as separate entities, it would seem
likely that a shorter rise time can be achieved than for the two-terminal
case. Only two-terminal networks of a simple type will be discussed in
the present paper, mainly because a treatment of other cases is beset
with algebraic difficulties.

Let us then consider the generalized, shunt-compensated inter-
stage network illustrated in Fig. 3b, where the pure reactance X has a
value zero at zero frequency, but is otherwise unrestricted in form. Ac-
cording to Foster's reactance theorem12, a possible formula for any
reactance of this type can be written

12
1See Refr. 7, pp. 177-181 for a discussion of Foster's reactance
theorem and of the various networks which can be used to realize any
reactance.


2 2 2... 2 2
X(s) = k s(s s2 ) (s2 s4 (s m) (25)
(2 2 2 2) 2 2*
(s s1 ) (s s3 ) (s sm-1 )
2
where the si2 (i = 1 ""m) are negative real numbers, k is a positive con-
stant, and m is an even integer. The general reactance can be realized
physically by a variety of equivalent networks made up of inductances
and capacitances.12 It is convenient here to adopt the form of network
shown in Fig. 4 to represent the general reactance X(s). If the inductance









- 15 -


MDDC 770


1o vanishes, i.e., the general reactance becomes zero at infinite fre-
quency, it is necessary to omit the factor (s2s2m) from the right-
hand member of Eq. (25).

By writing Eq. (25) in the form

X(s) = ds + d3s3 +.. d + m sm + (25a)
1 + c2s2 + c4s4 + + cms

where the new, real constants, ci and di, are uniquely related to the con-
stants appearing in E 1. (25), and to the circuit constants defined in Fig.
4, we find that the driving point impedance becomes
2 3 n-2 n-1
l+d s + c2s2 + d3s + +c 2S + d -s
Z(s) =
1 2 3 n-1 n
1+s+(dI + c2)s +c2s + ** +c n2s +dnl s
(26)

where n = m + 2. It should be noticed that when o = 0, the coefficient
4m+1 = n--1 vanishes. Equation (26), of course, has the form of Eq.(l).

To realize a monotonic transient response to the unit step func-
tion, it is necessary, but not always sufficient, to require that the poles
of Z(s) all lie on the negative real axis of the s-plane. (Otherwise the
transient response will contain oscillatory terms.) We shall assume
that the most desirable arrangement of poles is to have but one multiple
pole, and then show that this assumption leads to useful results.13

13It is not difficult to prove for the network under consideration that
this arrangement of poles on the negative real axis gives a shorter rise
time than any other arrangement of poles there.

The denominator of Eq. (26) can be an even or an odd polynomial
of order n, or n 1, respectively, depending on whether or not o1 occurs
in the network of Fig. 4. The treatment for both cases follows similar
lines, and will be illustrated for the case where o4, 0. In this case we
require that 7(s) have one multiple pole of order n (n is always even),
and the denominator of Eq. (26), accordingly, must be the binomial ex-
pansion of [ + (s/o) ] n, giving a set of n equations from which the n
quantities so, ,, 3, 34,, 4", dn- can be determined. The values of
the components in the network of Fig. 4 can then be computed, as well as
a value for the rise-time figure-of-merit and an expression for the
transient response to a unit impulse applied at t = 0.

The computation suggested has been carried out for cases where
Z(s) has poles of order p = 1,2,3,4, and 5, as well as for the limiting case








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16 -

where p -> oo. The following general expressions are found from Eqs.
(8) and (5) for the delay time and for the rise-time figure-of-merit.

S 2 1
TD 3 + 3p2
and (j 27)

S =
2 + 8/p 1/p


where p is the order of the multiple pole of Z(s). A summary of the re-
sults derived from the computations is presented in Table I. The analy-
sis employed for the limiting case is given in Appendix I.









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- 17 -


FIG. 4. A general driving point reactance having zero reactance at zero
frequency. The inductance / will vanish if the reactance must be zero
at infinite frequency.
0








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- 18 -


TABLE I. Some Critically Compensated Networks

P Circuit Constants S Response e' (t) to Unit
Impulse



-2t
2 A, = 1/4 1.512 (1 +2t)

3 L1=8/27; C1 = 1/8 1.769 -3t(1+2t + 6t2)

4 o = 1/4;
1 = 1/16; c1 = 1 1.899 -4t(1 + 4t + (64/3)t3

5 1 = (4/125) (5 + V5) 1.970 t-5t(1+4t 20t2
(100/3)t +(250/3
cl = (1/16)(3 +f5) t4)
2 = (4/125) (5- 45)
2 = (1/16) (3 ))


(k =2.121 1 -t/2, Ot 2

(k = 1, 2,3,









MDDC 770
19 -

The RC network (p = 1) has been included in Table I to serve as
a basis for comparing the other critically compensated networks. The
network for p = 2 is the well-known, shunt-compensated network, used
as an illustration in Sec. 2. By increasing the value of the inductance
from 0.25 to 0 290, and shunting it 'ith a capacitance of 0.125 (p = 3),
a decrease in rise time of about 17 percent is obtained. Adding a second
inductance (p 4) results in a further decrease in rise time of about 7
percent. By adding more and more components, the limiting value for
the rise-time figure-of-merit, Smax = 3 1 2 = 2.12 is approached
The remaining improvement possible in the transient behavior after a
few inductances and capacitances are incorporated m the network is not
very marked. These cases, therefore, are not of great practical impor-
.ance. The limiting case (p -*j> ) is of interest primarily because it pos-
sesses the greatest figure-of-merit possible with a network of the type
under consideration. It is conjectured that this network has the greatest
figure-of-merit possible for a low-pass two-terminal interstate network.
No completely adequate proof, however, has been found for this theorem.

The transient response to a unit step function for all the cases
listed in Table I has a monotonic form, which, of course, is necessary
in order that the method used for computing delay time and rise tune be
applicable. The general proof that the transient response is monotonic
for arbitrary values of p appears to present considerable algebraic dif-
ficulties.

APPENDIX I. Case Where p-coo

The analysis for the case where the reactance X(s) in Fig. 3b has
an infinite number of poles can be made by setting the denominator in
Eq. (26) equal to [ 1 + (s,'p) ] P and then writing the resulting expression
in the algebraically equivalent form

1-
7(s) = s 2sI ]

1+ s
P

In the limit where poo this expression becomes
I 1 -2s
& (s) s (0 ) (28)

14
It is of interest to note that if a switch is inserted,in series with the
capacitor C in Fig. 3b, and the capacitor is initially given a unit charge,
then Eq. (28) is the Laplace transform of the voltage developed across
the network when the switch is closed at t = 0. Since the voltage across
the capacitor decreases linearly while it is being discharged into the








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20 -

remaining branch of the network, the current flowing through the re-
sistor must have the form of a rectangular pulse (of amplitude 1/2). The
network can evidently be used (ideally, at any rate) to convert either a
current impulse, or the sudden discharge of a capacitor, into a rectan-
gular voltage pulse across a resistive load.

From Eq.(28) and the network of Fig. 3b, the reactance X(s) is
found to have the form
D(s) = coth s 1/s (29)
The zeros and poles of X(s) are located, of course, on the real frequency
axis, and are given by the roots of tanw =w, and sin w /w = 0, respec-
tively.

To determine values of k andc the expression for the re-
actance, Eq. (29), can be expanded in the infinite series15
2s
X(s) = coth s /s = I 2 2 2 (30)
1 s +k I

1See, for instance, Whittaker and Watson, Modern Analysis, Fourth
Edition, p. 136, Example 7.

Each term in the infinite series can be interpreted as the reactance of
a parallel combination of inductance and capacitance, where


Ik = 1/,2k2
k = 1/2. (k = 1,2,3, '") (31)

It is evident from the ature of the terms occurring in the infinite series
that the inductance o must vanish.

The formulas for delay time and rise-time figure-of-merit, Eqs.
(27), hold in the limit when p-+oo so no separate computation need be
made for these quantities.

This paper is based on work performed under Contract No. W-
7405-Eng-36 with the Manhattan Project at the Los Alamos Scientific
Laboratory of the University of California.







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