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30huV4 MDDC 770 LADC 389 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 Published for use within the Atomic Energy Commission. Inquir ies for additional copies and any questions regarding reproduction by recipients of this document may be referred to the Documents Distribution Subsection, Publication Section, Technical Information Branch, Atomic Energy Commission, P. O. Box E, Oak Ridge, Tennessee. Inasmuch as a declassified document may differ materially from the original classified document by reason of deletions necessary to accomplish declassification, this copy does not constitute au thority for declassification of classified copies of a similar docu ment which may bear the same title and authors. Document Declassified: 3/26/47 This document consists of 20 pages.  Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/transientrespons00losa MDDC 770 1  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 multistage 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 gainrise time quotient consistent with a monotonic transient response to a step function. MDDC 770 2  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 steadystate 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 socalled indicial admittance) consists of a monotonic rise to a final constant value. For simplicity in presentation only the tran sient response of a lowpass, 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 lowpass 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 unfedback 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 signalcarrying leads. Since we shall be interested in the problem of obtaining the greatest possible gainrise 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 lowfrequency 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. MDDC 770 3  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 splane. 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) = 2j 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 highfre 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 overcompensation 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 cathoderay oscillographs). Video ampli fiers used in television applications evidently are not as critical in this MDDC 770 4 respect since it is customary to overcompensate 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 overcompensated. 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 overcompensated 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 undercompensafed 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 halfvalue 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 MDDC 770 5 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 nstage 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. MDDC 770 6  t (b) (d) (e) FIG. 1. Some typical transient response curves. I_\ MDDC 770  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) ESt dt, (6) where the real part of s is sufficiently positive to make the integral con verge. By expanding est in ascending powers of st, we have that co o o gn(s) 1s 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 MDDC 770 9  Before considering other matters, let us compute values of TD and TR for a singlestage amplifier having a twoterminal plate load impedance of the type shown in Fig. 3a.6 Such an amplifier stage is said to be shuntcompensated. 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 splane. 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 resistancecoupled 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 Rcoupled stage. The quantity S may be termed the risetime figureofmerit. 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 Multistage 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 multistage amplifier depends on the properties of individual stages in it. MDDC 770  10  FIG. 3. Shuntcompensated interstate networks. MDDC 770 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 nstage amplifier should be distributed among the individual stages in order to achieve the shortest possible overall rise time for a given overall gain. Now the rise time of any stage in the am MDDC 770 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 overall 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 nstage 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 V27r RC (18) 1 S where S is the risetime figureofmerit of the stage. By definition S = 1 for a simple Rcoupled stage and we have already shown, for example, that S = V16/7 for a critically shuntcompensated 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 ofmerit 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 nstage amplifier, the according to Eq. (16) the rise time of each stag'emust be T1 = y/n requiring a gain for each stage MDDC 770 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 Rcoupled stage of rise time T whereas G1 is the gain that each stage of the nstage 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. MDDC 770 14  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, twoterminal and fourterminal.11 This distinction is necessary since it is possible to separate the parasitic interstate capacitance into two Strictly speaking, twoterminal and threeterminal networks. portions, the output capacitance of one stage, and the input capacitance of the following stage. If a critically compensated fourterminal 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 twoterminal case. Only twoterminal 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, shuntcompensated 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. 177181 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 sm1 ) 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 n2 n1 l+d s + c2s2 + d3s + +c 2S + d s Z(s) = 1 2 3 n1 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 = n1 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 splane. (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 risetime figureofmerit 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 MDDC 770 16  where p > oo. The following general expressions are found from Eqs. (8) and (5) for the delay time and for the risetime figureofmerit. 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. MDDC 770  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 MDDC 770  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 t5t(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 wellknown, shuntcompensated 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 risetime figureofmerit, 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 figureofmerit possible with a network of the type under consideration. It is conjectured that this network has the greatest figureofmerit possible for a lowpass twoterminal 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 pcoo 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 MDDC 770 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 risetime figureofmerit, 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 7405Eng36 with the Manhattan Project at the Los Alamos Scientific Laboratory of the University of California. UNIVERSITY OF FLORIDA 11IIIHU I 1 111 I111 11111 11111 11 11111111111i I1111 3 1262 08910 5539 
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