Dorf, R C, Wan, Z, Johnson, D.E. "Laplace Transform The electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Dorf, R.C., Wan, Z., Johnson, D.E. “Laplace Transform” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
6 Laplace Transfor 6.1 Definitions and Properties Richard C. dorf aplace Transform Integra University of California, davis Convergence. Properties of Laplace Transform. Time-Co Property. Time-Correlation Property. Inverse Lapla Transfo en wa 6.2 Applications Differentiation Theorems. Applications to Integrodifferential David E. Johnson Equations. Applications to Electric Circuits.The Transformed Circuit Thevenin's and Nortons Theorems . Network Birmingham-Southern College 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 procedure is more generally known as taking the Laplace transform of the sigma dd+oo). This systematic x n by the signal e-st and then integrating that product over the time interval (-oo, too). This systematic 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=0+10 where o= Re(s) and (=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. Th reflects a form of equivalency between the two apparently different entities x r and X(s). We may symbolize this interrelationship in the following suggestive manner c 2000 by CRC Press LLC
© 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 + jw 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: X s x t e dt st ( ) = ( ) - -• +• Ú Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis David E. Johnson Birmingham-Southern College
X(s)=9[x(t) where the operator notation s means to multiply the signal x) being operated upon by the complex expo- nential e-st and then to integrate that product over the time interval (oo, too) 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 σ+<Re(s)<o where o, and o denote real parameters that are related to the causal and anticausal components, respectively, of the sigr 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 ignal 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,(t)=e-atu(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 x1(s) eu(t)e dt In order for X,(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= too 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/(5+ 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 ∠[e-"u(t) Rels)
© 2000 by CRC Press LLC X(s) = +[x(t)] where the operator notation + means to multiply the signal x(t) being operated upon by the complex exponential 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 x1(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 X1(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 X s e u t e dt e dt e s a at st s a t s a t 1 0 0 ( ) ( ) ( ) ( ) ( ) = = = - + - - - + +• -• +• - + +• Ú Ú L [e u(t)] Re( ) Re( ) s a s a -at = + > - 1 for
TABLE 6.1 Laplace Transform Pairs Time Signal Laplace Transform Region of 2.re-"u(-n) Re(s)>-Refa) 3. Re(s<-Re(a 4.(-n)e-"a(- Re(s)<-Re(a) 5. un Re(s) 6.() 8.r(t) 2 9. 11. cos o,t u(r Re(s)>0 12. ea sin o,t u(r) Re(s)>-Re(a (s+a)2+ 13. e-a cos t u(n) s+a Re(s)>-Re(a) (s+a)2+o Source: JA Cadzow and H.E. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J Prentice-Hall, 1985, P. 133. with permission. Properties of Laplace Transform Li Let us obtain the Laplace transform of a signal, xd n), that is composed of a linear combination of two other x(1)=1x(t)+2x2(t) The linearity property indicates that [α1x1(1)+a2x2(]=a1X1(s)+2X2(s)
© 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) = a1x1(t) + a2x2(t) where a1 and a2 are constants. The linearity property indicates that + [a1x1(t) + a 2x2(t)] = a 1X1(s) + a2X2(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 –atu(t) Re(s) > –Re(a) 2. tke –atu(–t) Re(s) > –Re(a) 3. –e –atu(–t) Re(s) < –Re(a) 4. (–t)ke –atu(–t) Re(s) < –Re(a) 5. u(t) Re(s) > 0 6. d(t) 1 all s 7. sk all s 8. tk u(t) Re(s) > 0 9. Re(s) = 0 10. sin w0t u(t) Re(s) > 0 11. cos w0t u(t) Re(s) > 0 12. e –at sin w0t u(t) Re(s) > –Re(a) 13. e –at cos w0t 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 s a + k s a k ! ( ) + +1 1 ( ) s a + k s a k ! ( ) + +1 1 s d t dt k k d( ) k s k ! +1 sgnt t t = ³ < Ï Ì Ó 1 0 1 0 , – , 2 s w w 0 2 0 2 s + s s 2 0 2 + w w ( ) s a + +w 2 0 2 s a s a + ( ) + +2 0 2 w
x(t) Differentiation x(t) X(s) Multiplication FIGURE 6.1 Equivalent operations in the (a)time-domain operation and(b) Laplace transform-domain operation. ( Source: J.A. Cadzow and H.E. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J Prentice-Hall 985,P. 138. With permission. max (o+;o2)<Re)<min(2;σ2) where the pairs(ol; 02)< Re(s)< min(o o2)identify the regions of convergence for the Laplace transforms X(s) and X,(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 n)/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(o) This property may be envisioned as shown in Fig. 6.1 Time Shift The signal x(t-to) is said to be a version of the signal x n right shifted (or delayed) by to seconds. Right shifting (delaying)a signal by a to second duration in the time domain is seen to correspond to a multiplication by e-sro in the Laplace transform domain. The desired Laplace transform relationship [x(t-t0)=c-X(s) 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-sto 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 n)and xt- to) are ide Delay by o FIGURE 6.2 Equivalent 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
© 2000 by CRC Press LLC where the pairs (s1 +; s + 2 ) < Re(s) < min(s– 1 ; s– 2 ) identify the regions of convergence for the Laplace transforms X1(s) and X2(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 – t0) is said to be a version of the signal x(t) right shifted (or delayed) by t0 seconds. Right shifting (delaying) a signal by a t0 second duration in the time domain is seen to correspond to a multiplication by e–st 0 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–st 0 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 – t0) are identical. FIGURE 6.1 Equivalent 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 6.2 Equivalent 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(s s; ) Re( ) min(s s; ) + + < < - - 1 2 1 2 s + dx t dt sX s ( ) ( ) È Î Í ˘ ˚ ˙ = + [ (x t t )] e X(s) st - = - 0 0