We define transfer function H(s) as a ratio of the Laplace transform of system output (or response)(s) to the Laplace transform of the input(or forcing function)v(s) when all initial conditions are zero, then

When initial energy is present in a circuit, the Laplace transform method may be used to obtain the complete response. The first is the more fundamental, for it involves writing the differential equations for the network and then taking the Laplace transform of those equations

15-6 The partial-fraction-expansion method

15-5 Several basis theorems for the Laplace transform 1.The linearity theorem{fi(t)+()}=()+(s) Example:

15-4 Laplace transform of some simple time function 1. The unit step function

When x(t), the input to N, is known, and when h(t),the impulse response of N, is known, the y(t), the output or response function, is expressed by

We define the unit impulse as a function of time which is zero when its argument, generally (t-o), is less than zero; which is also zero when argument is greater than zero; which is infinite when its argument is zero; and which has unit area

The two-sided or bilateral Laplace transform The one-sided Laplace transform or direct Laplace transform

14-8 Definition of the Fourier transform The exponential form of the Fourier series

We consider each term of the Fourier aeries representing the voltage as a single source. The equivalent impedance of the network at no is used to compute the current at that harmonic. XL(n) =noL and XC() =-1/noC The sum of these individual responses is the total response i