The electrical engineering handbook
Dorf, R.C., Wan, Z., Johnson, D.E. “Laplace Transform” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 6 Laplace Transform 6.1 Definitions and Properties Laplace Transform Integral•Region of Absolute Convergence•Properties of Laplace Transform•Time-Convolution Property•Time-Correlation Property•Inverse Laplace Transform 6.2 Applications Differentiation Theorems•Applications to Integrodifferential Equations•Applications to Electric Circuits•The Transformed Circuit•Thévenin’s and Norton’s Theorems•Network Functions•Step and Impulse Responses•Stability 6.1 Definitions and Properties Richard C. Dorf and Zhen Wan The Laplace transform is a useful analytical tool for converting time-domain signal descriptions into functions of a complex variable. This complex domain description of a signal provides new insight into the analysis of signals and systems. In addition, the Laplace transform method often simplifies the calculations involved in obtaining system response signals. Laplace Transform Integral The Laplace transform completely characterizes the exponential response of a time-invariant linear function. This transformation is formally generated through the process of multiplying the linear characteristic signal x ( t ) by the signal e –st and then integrating that product over the time interval (– ¥ , + ¥ ). This systematic procedure is more generally known as taking the Laplace transform of the signal x ( t ). Definition: The Laplace transform of the continuous-time signal x ( t ) is The variable s that appears in this integrand exponential is generally complex valued and is therefore often expressed in terms of its rectangular coordinates s = s + j w where s = Re( s ) and w = Im( s ) are referred to as the real and imaginary components of s , respectively. The signal x ( t ) and its associated Laplace transform X ( s ) are said to form a Laplace transform pair . This reflects a form of equivalency between the two apparently different entities x ( t ) and X ( s ). We may symbolize this interrelationship in the following suggestive manner: Xs xtedt st () ()= - -¥ +¥ ò Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis David E. Johnson Birmingham-Southern College © 2000 by CRC Press LLC X ( s ) = ᏸ [ x ( t )] where the operator notation ᏸ means to multiply the signal x ( t ) being operated upon by the complex expo- nential e –st and then to integrate that product over the time interval (– ¥ , + ¥ ). Region of Absolute Convergence In evaluating the Laplace transform integral that corresponds to a given signal, it is generally found that this integral will exist (that is, the integral has finite magnitude) for only a restricted set of s values. The definition of region of absolute convergence is as follows. The set of complex numbers s for which the magnitude of the Laplace transform integral is finite is said to constitute the region of absolute convergence for that integral transform. This region of convergence is always expressible as s + < Re( s ) < s – where s + and s – denote real parameters that are related to the causal and anticausal components, respectively, of the signal whose Laplace transform is being sought. Laplace Transform Pair Tables It is convenient to display the Laplace transforms of standard signals in one table. Table 6.1 displays the time signal x ( t ) and its corresponding Laplace transform and region of absolute convergence and is sufficient for our needs. Example. To find the Laplace transform of the first-order causal exponential signal x 1 ( t ) = e –at u ( t ) where the constant a can in general be a complex number. The Laplace transform of this general exponential signal is determined upon evaluating the associated Laplace transform integral (6.1) In order for X 1 ( s ) to exist, it must follow that the real part of the exponential argument be positive, that is, Re( s + a ) = Re( s ) + Re( a ) > 0 If this were not the case, the evaluation of expression (6.1) at the upper limit t = + ¥ would either be unbounded if Re( s ) + Re( a ) < 0 or undefined when Re( s ) + Re( a ) = 0. On the other hand, the upper limit evaluation is zero when Re( s ) + Re( a ) > 0, as is already apparent. The lower limit evaluation at t = 0 is equal to 1/( s + a ) for all choices of the variable s . The Laplace transform of exponential signal e –at u ( t ) has therefore been found and is given by Xs e ute dt e dt e sa at st sat sat 1 0 0 () () () () () == = -+ -- -+ +¥ -¥ +¥ -+ +¥ òò L[ ()] Re() Re()eut sa sa at- = + >- 1 for © 2000 by CRC Press LLC Properties of Laplace Transform Linearity Let us obtain the Laplace transform of a signal, x ( t ), that is composed of a linear combination of two other signals, x ( t ) = a 1 x 1 ( t ) + a 2 x 2 ( t ) where a 1 and a 2 are constants. The linearity property indicates that ᏸ [a 1 x 1 (t) + a 2 x 2 (t)] = a 1 X 1 (s) + a 2 X 2 (s) and the region of absolute convergence is at least as large as that given by the expression TABLE 6.1 Laplace Transform Pairs Time Signal Laplace Transform Region of x(t) X(s) Absolute Convergence 1. e –at u(t)Re(s) > –Re(a) 2. t k e –at u(–t)Re(s) > –Re(a) 3. –e –at u(–t)Re(s) < –Re(a) 4. (–t) k e –at u(–t)Re(s) < –Re(a) 5. u(t)Re(s) > 0 6. d(t) 1 all s 7. s k all s 8. t k u(t) Re(s) > 0 9. Re(s) = 0 10. sin w 0 t u(t)Re(s) > 0 11. cos w 0 t u(t)Re(s) > 0 12. e –at sin w 0 t u(t)Re(s) > –Re(a) 13. e –at cos w 0 t u(t)Re(s) > –Re(a) Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 133. With permission. 1 sa+ k sa k ! ()+ +1 1 ()sa+ k sa k ! ()+ +1 1 s dt dt k k d() k s k ! +1 sgn t t t = ³ < ì í î 10 10 , –, 2 s w w 0 2 0 2 s + s s 2 0 2 +w w w()sa++ 2 0 2 sa sa + ++() 2 0 2 w © 2000 by CRC Press LLC where the pairs (s 1 + ; s + 2 )< Re(s) < min(s – 1 ; s – 2 ) identify the regions of convergence for the Laplace transforms X 1 (s) and X 2 (s), respectively. Time-Domain Differentiation The operation of time-domain differentiation has then been found to correspond to a multiplication by s in the Laplace variable s domain. The Laplace transform of differentiated signal dx(t)/dt is Furthermore, it is clear that the region of absolute convergence of dx(t)/dt is at least as large as that of x(t). This property may be envisioned as shown in Fig. 6.1. Time Shift The signal x(t – t 0 ) is said to be a version of the signal x(t) right shifted (or delayed) by t 0 seconds. Right shifting (delaying) a signal by a t 0 second duration in the time domain is seen to correspond to a multiplication by e –st0 in the Laplace transform domain. The desired Laplace transform relationship is where X(s) denotes the Laplace transform of the unshifted signal x(t). As a general rule, any time a term of the form e –st0 appears in X(s), this implies some form of time shift in the time domain. This most important property is depicted in Fig. 6.2. It should be further noted that the regions of absolute convergence for the signals x(t) and x(t – t 0 ) are identical. FIGURE 6.1Equivalent operations in the (a) time-domain operation and (b) Laplace transform-domain operation. (Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 138. With permission.) FIGURE 6.2Equivalent operations in (a) the time domain and (b) the Laplace transform domain. (Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 140. With permission.) max( ; ) Re() min( ; )ss ss ++ -- << 12 12 s ᏸ dxt dt sXs () () é ë ê ù û ú = ᏸ[( )] ()xt t eXs st -= - 0 0 © 2000 by CRC Press LLC Time-Convolution Property The convolution integral signal y(t) can be expressed as where x(t) denotes the input signal, the h(t) characteristic signal identifying the operation process. The Laplace transform of the response signal is simply given by where H(s) = ᏸ [h(t)] and X(s) = ᏸ [x(t)]. Thus, the convolution of two time-domain signals is seen to correspond to the multiplication of their respective Laplace transforms in the s-domain. This property may be envisioned as shown in Fig. 6.3. Time-Correlation Property The operation of correlating two signals x(t) and y(t) is formally defined by the integral relationship The Laplace transform property of the correlation function f xy (t) is in which the region of absolute convergence is given by FIGURE 6.3Representation of a time-invariant linear operator in (a) the time domain and (b) the s-domain. (Source: J. A. Cadzow and H. F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 144. With permission.) yt hxt d() ()( )=- -¥ ¥ ò ttt Ys HsXs() ()()= ft t xy xtyt dt() ()( )=+ -¥ ¥ ò F xy sXsYs() ()()=- max( , ) Re() min( , )-<<- -+ +- ss ss xx yy s © 2000 by CRC Press LLC Autocorrelation Function The autocorrelation function of the signal x(t) is formally defined by The Laplace transform of the autocorrelation function is and the corresponding region of absolute convergence is Other Properties A number of properties that characterize the Laplace transform are listed in Table 6.2. Application of these properties often enables one to efficiently determine the Laplace transform of seemingly complex time functions. TABLE 6.2Laplace Transform Properties Signal x(t) Laplace Transform Region of Convergence of X(s) Property Time Domain X(s) s Domain s + < Re(s) < s – Linearity a 1 x 1 (t) + a 2 x 2 (t) a 1 X 1 (s) + a 2 X 2 (s) At least the intersection of the region of convergence of X 1 (s) and X 2 (s) Time differentiation sX(s) At least s + < Re(s) and X 2 (s) Time shift x(t – t 0 )e –st0 X(s) s + < Re(s) < s – Time convolution H(s)X(s) At least the intersection of the region of convergence of H(s) and X(s) Time scaling x(at) Frequency shift e –at x(t) X(s + a) s + – Re(a) < Re(s) < s – – Re(a) Multiplication (frequency convolution) x 1 (t)x 2 (t) Time integration At least s + < Re(s) < s – Frequency differentiation (–t) k x(t) At least s + < Re(s) < s – Time correlation X(–s)Y(s)max(–s x– , s y+ ) < Re(s) < min(–s x+ , s y– ) Autocorrelation function X(–s)X(s)max(–s x– , s x+ ) < Re(s) < min(–s x+ , s x– ) Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985. With permission. ft t xx xtxt dt() ()( )=+ -¥ ¥ ò F xx sXsXs() ()()=- max( , ) Re() min( , )-<<- - ++- ss ss xy x s y dxt dt () hxt dt()( )tt- -¥ ¥ ò 1 ΈΈa X s a æ è ç ö ø ÷ ss +- < æ è ç ö ø ÷ <Re s a 1 2 12 pj XuXsud cj cj ()( ) -¥ +¥ ò - ss ss ss ss ++ -- ++ -- +<<+ +<<+ () () () () () () () () Re() 12 12 12 12 s c xd t ()tt -¥ ò 1 0 s Xs X() ()for = dXs ds k k () xtytzdt()( )+ -¥ +¥ ò xtxtzdt()( )+ -¥ +¥ ò © 2000 by CRC Press LLC Inverse Laplace Transform Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(t) that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as The signal x(t) can be recovered by means of the relationship In this integral, the real number c is to be selected so that the complex number c + jw lies entirely within the region of convergence of X(s) for all values of the imaginary component w. For the important class of rational Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly evaluating this integral. This procedure is generally known as the partial-fraction expansion method. Partial Fraction Expansion Method As just indicated, the partial fraction expansion method provides a convenient technique for reacquiring the signal that generates a given rational Laplace transform. Recall that a transform function is said to be rational if it is expressible as a ratio of polynomial in s, that is, The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e., generating signals) are readily found in standard Laplace transform pair tables. This method entails the simple five-step process as outlined in Table 6.3. A description of each of these steps and their implementation is now given. I. Proper Form for Rational Transform.This division process yields an expression in the proper form as given by TABLE 6.3Partial Fraction Expansion Method for Determining the Inverse Laplace Transform I. Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of the denominator polynomial. II. Factor the denominator polynomial. III. Perform a partial fraction expansion. IV. Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose. V. Using a Laplace transform pair table, obtain the inverse Laplace transform. Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 153. With permission. xt Xs() [()]=ᏸ ±1 xt j Xseds st cj cj () ()= -¥ +¥ ò 1 2p Xs Bs As bs bs bsb sas asa m m m m n n n () () () == + +×××+ + + +×××+ + - - - - 1 1 10 1 1 10 Xs Bs As Qs Rs As () () () () () () = =+ © 2000 by CRC Press LLC in which Q(s) and R(s) are the quotient and remainder polynomials, respectively, with the division made so that the degree of R(s) is less than or equal to that of A(s). II. Factorization of Denominator Polynomial. The next step of the partial fraction expansion method entails the factorizing of the nth-order denominator polynomial A(s) into a product of n first-order factors. This factorization is always possible and results in the equivalent representation of A(s) as given by The terms p 1 , p 2 , . . ., p n constituting this factorization are called the roots of polynomial A(s), or the poles of X(s). III. Partial Fraction Expansion. With this factorization of the denominator polynomial accomplished, the rational Laplace transform X(s) can be expressed as (6.2) We shall now equivalently represent this transform function as a linear combination of elementary transform functions. Case 1: A(s) Has Distinct Roots. where the a k are constants that identify the expansion and must be properly chosen for a valid representation. and a 0 = b n The expression for parameter a 0 is obtained by letting s become unbounded (i.e., s = +¥) in expansion (6.2). Case 2: A(s) Has Multiple Roots. The appropriate partial fraction expansion of this rational function is then given by As spsp sp n ( ) ( )( ) .( )=- - - 12 Xs Bs As bs b s b spsp sp n n n n n () () () ( )( ) ( ) == + +×××+ - - ××× - - - 1 1 0 12 Xs sp sp sp n n ()=+ - + - +×××+ - a aa a 0 1 1 2 2 a kk spXs k n sp k =- = = ( ) ( ) , , .,for 1 2 Xs Bs As Bs spAs q () () () () ()() == - 11 Xs sp sp nq As q q () () () ()=+ - +×××+ - +- ( ) a a a 0 1 1 1 1 other elementary terms due to the roots of 1 © 2000 by CRC Press LLC The coefficient a 0 may be expediently evaluated by letting s approach infinity, whereby each term on the right side goes to zero except a 0 . Thus, The a q coefficient is given by the convenient expression (6.3) The remaining coefficients a 1 , a 2 , … , a q–1 associated with the multiple root p 1 may be evaluated by solving Eq. (6.3) by setting s to a specific value. IV. Causal and Anticausal Components.In a partial fraction expansion of a rational Laplace transform X(s) whose region of absolute convergence is given by it is possible to decompose the expansion’s elementary transform functions into causal and anticausal functions (and possibly impulse-generated terms). Any elementary function is interpreted as being (1) causal if the real component of its pole is less than or equal to s + and (2) anticausal if the real component of its pole is greater than or equal to s – . The poles of the rational transform that lie to the left (right) of the associated region of absolute convergence correspond to the causal (anticausal) component of that transform. Figure 6.4 shows the location of causal and anticausal poles of rational transform. V. Table Look-Up of Inverse Laplace Transform.To complete the inverse Laplace transform procedure, one need simply refer to a standard Laplace transform function table to determine the time signals that generate each of the elementary transform functions. The required time signal is then equal to the same linear combi- nation of the inverse Laplace transforms of these elementary transform functions. FIGURE 6.4Location of causal and anticausal poles of a rational transform. (Source: J.A. Cadzow and H.F. Van Landing- ham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 161. With permission.) a 0 0== ®+¥ lim () s Xs a q q sp spXs Bp Ap =- = = ()() () () 1 1 11 1 ss +- <<Re()s