CHAPTER III LINEAR SYSTEM RESPONSE 3.1 OBJECTIVES The output produced by an operational amplifier (or any other dynamic system) in response to a particular type or class of inputs normally pro­ vides the most important characterization of the system. The purpose of this chapter is to develop the analytic tools necessary to determine the re­ sponse of a system to a specified input. While it is always possible to determine the response of a linear system to a given input exactly, we shall frequently find that greater insight into the design process results when a system response is approximated by the known response of a simpler configuration. For example, when designing a low-level preamplifier intended for audio signals, we might be interested in keeping the frequency response of the amplifier within i 5% of its mid- band value over a particular bandwidth. If it is possible to approximate the amplifier as a two- or three-pole system, the necessary constraints on pole location are relatively straightforward. Similarly, if an oscilloscope vertical amplifier is to be designed, a required specification might be that the over­ shoot of the amplifier output in response to a step input be less than 3 % of its final value. Again, simple constraints result if the system transfer function can be approximated by three or fewer poles. The advantages of approximating the transfer functions of linear systems can only be appreciated with the aid of examples. The LM301A integrated- circuit operational amplifier' has 13 transistors included-in its signal-trans­ mission path. Since each transistor can be modeled as having two capaci­ tors, the transfer function of the amplifier must include 26 poles. Even this estimate is optimistic, since there is distributed capacitance, comparable to transistor capacitances, associated with all of the other components in the signal path. Fortunately, experimental measurements of performance can save us from the conclusion that this amplifier is analytically intractable. Figure 3.la shows the LM301A connected as a unity-gain inverter. Figures 3.1b and 3.lc show the output of this amplifier with the input a -50-mV step 1This amplifier is described in Section 10.4.1. 63 64 Linear System Response 4.7 kn 4.7 k V F -+ V LM301 A 0 - Compensating capacitor (a) 20 mV (b) 2 AsK Figure 3.1 Step responses of inverting amplifier. (a) Connection. (b) Step response with 220-pF compensating capacitor. (c) Step response with 12-pF compensating capacitor. 2 for two different values of compensating capacitor. The responses of an R-C network and an R-L-C network when excited with +50-mV steps supplied from the same generator used to obtain the previous transients are shown in Figs. 3.2a and 3.2b, respectively. The network transfer func­ tions are V0(s) 1 Vi(s) 2.5 X 10- 6 s + I Objectives 65 20 mV (c) 0.5 yAs Figure 3.1-Continued for the response shown in Fig. 3.2a and V0(s) 1 1 Vi(s) 2.5 X 10 4 s2 + 7 X 10- 8s + I for that shown in Fig. 3.2b. We conclude that there are many applications where the first- and second-order transfer functions of Eqns. 3.1 and 3.2 adequately model the closed-loop transfer function of the LM301A when connected and compensated as shown in Fig. 3.1. This same type of modeling process can also be used to approximate the open-loop transfer function of the operational amplifier itself. Assume that the input impedance of the LM301A is large compared to 4.7 ki and that its output impedance is small compared to this value at frequencies of interest. The closed-loop transfer function for the connection shown in Fig. 3.1 is then V0(s) -a(s) Vi(s) 2 + a(s) 2 Compensation is a process by which the response of a system can be modified advan­ tageously, and is described in detail in subsequent sections. 20 mV T (a) 2 S 20 mV (b) 0.5 /s Figure 3.2 Step responses for first- and second-order networks. (a) Step response 6 for V.(s)/Vi(s) = ]/(2.5 X 10- s + 1). (b) Step response for V(s)/Vi(s) = 1/(2.5 X 10 14 S2 + 7 X 10 8s + 1). 66 Laplace Transforms 67 where a(s) is the unloaded open-loop transfer function of the amplifier. Substituting approximate values for closed-loop gain (the negatives of Eqns. 3.1 and 3.2) into Eqn. 3.3 and solving for a(s) yields a~) 8 X 105 a(s) 8X(3.4) and 2.8 X 107 a(s) 2. 0 (3.5) s(3.5 X 10- 7 s + 1) as approximate open-loop gains for the amplifier when compensated with 220-pF and 12-pF capacitors, respectively. We shall see that these approxi­ mate values are quite accurate at frequencies where the magnitude of the loop transmission is near unity. 3.2 LAPLACE TRANSFORMS3 Laplace Transforms offer a method for solving any linear, time-invariant differential equation, and thus can be used to evaluate the response of a linear system to an arbitrary input. Since it is assumed that most readers have had some contact with this subject, and since we do not intend to use this method as our primary analytic tool, the exposure presented here is brief and directed mainly toward introducing notation and definitions that will be used later. 3.2.1 Definitions and Properties The Laplace transform of a time functionf(t) is defined as 2[f(t)] A F(s) A f f(t)e-" dt (3.6) where s is a complex variable o + jw. The inverse Laplace transform of the complex function F(s) is -1[F(s)]A f(t) A F(s)e" ds (3.7) 2.rj ,1-J Fs d A complete discussion is presented in M. F. Gardner and J. L. Barnes, Transientsin LinearSystems, Wiley, New York, 1942. In this section we temporarily suspend the variable and subscript notation used else­ where and conform to tradition by using a lower-case variable to signify a time function and the corresponding capital for its transform. 68 Linear System Response The direct-inverse transform pair is unique4 so that 2-12[ftt)] =f (t (3.8) iff(t) = 0, t < 0, and if f f(t) | e- dt is finite for some real value of a1. A number of theorems useful for the analysis of dynamic systems can be developed from the definitions of the direct and inverse transforms for functions that satisfy the conditions of Eqn. 3.8. The more important of these theorems include the following. 1. Linearity 2[af(t) + bg(t)] = [aF(s) + bG(s)] where a and b are constants. 2. Differentiation Ldf(t) sF(s) - lim f(t) dt =-.0+ (The limit is taken by approaching t = 0 from positive t.) 3. Integration 2 [ f(T) d-] F(s) 4. Convolution 2 [f tf(r)g(t - r) d] = [ ff(t - r)g(r) dr] = F(s)G(s) 5. Time shift 2[f(t - r)] = F(s)e-" if f(t - r) = 0 for (t - r) < 0, where r is a positive constant. 6. Time scale 1 Fs] 2[f(at)] = F -­ a a where a is a positive constant. 7. Initialvalue lim f(t) lim sF(s) t-0+ 8-.co 4 There are three additional constraints called the Direchlet conditions that are satisfied for all signals of physical origin. The interested reader is referred to Gardner and Barnes. Laplace Transforms 69 8. Final value limf (t) = lim sF(s) J-oo 8-0 Theorem 4 is particularly valuable for the analysis of linear systems, since it shows that the Laplace transform of a system output is the prod­ uct of the transform of the input signal and the transform of the impulse response of the system. 3.2.2 Transforms of Common Functions The defining integrals can always be used to convert from a time func­ tion to its transform or vice versa. In practice, tabulated values are fre­ quently used for convenience, and many mathematical or engineering ref­ erencesI contain extensive lists of time functions and corresponding Laplace transforms. A short list of Laplace transforms is presented in Table 3.1. The time functions corresponding to ratios of polynomials in s that are not listed in the table can be evaluated by means of a partialfraction ex­ pansion. The function of interest is written in the form F(s) = p(s) _ p(s) q(s) (s + s1 )(s + s2) . (s + s.) It is assumed that the order of the numerator polynomial is less than that of the denominator. If all of the roots of the denominator polynomial are first order (i.e., s, / si, i # j), F(s) = (3.10) k=1 S + sk where Ak = lim [(s + sk)F(s)] (3.11) k If one or more roots of the denominator polynomial are multiple roots, they contribute terms of the form "ZBk k- BsI~k (3.12) k1(S + Si) k See, for example, A. Erdeyli (Editor) Tables of Integral Transforms, Vol. 1, Bateman Manuscript Project, McGraw-Hill, New York, 1954 and R. E. Boly and G. L. Tuve, (Editors), Handbook of Tables for Applied Engineering Science, The Chemical Rubber Company, Cleveland, 1970. 70 Linear System Response Table 3.1 Laplace Transform Pairs f(t), t > 0 F(s) [f(t) = 0, t < 0] 1 Unit impulse uo(t) 1 Unit step u-1 (t) S2 [f(t) = 1, t > 0] 1 Unit ramp u- 2(t) [f(t) = t, t > 0] 1 tn s+1a n! e-at s + 1 a 1 tn Se-at (s + a)"+1 (n)! 1 1 - e-t/ s(rs + 1 e-a sin wt (s + a)2 + W2 s + a e-a cos Wt (s + a)2 + W22 1 on 2 2 e-t-wn(sin 1 - -2 t), < 1 s /, + 2 s/n + 1 1 - 2 1 2 1 - sin coV/1 - 2 t + tan-' 2, s(s /O,,2 + 2 s/Wn + 1) where m is the order of the multiple root located at s = -si.