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