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

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

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

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

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

14-5 Waveform synthesis Synthesis is a combination of so as form a whole

1.A functionf(t) is said to be even iff(t)=(-t). They are symmetrical with respect to the vertical axis. (bn=0)

Any periodic waveform f(t)f(t) can be expressed by a Fourier series provided that (1) If it is discontinuous, there are only a finite number of discontinuous in the period T (2)It has a finite average value over the period T (3)It has a finite number of positive and negative maximums in the period T

current of one port into the voltage of the otherport and vice verse. U1 =-ais \2 =ai..gyration constant

the open-circuit reverse voltage gainthe open-circuit transfer admittancethe short-circuit transfer impedancethe short-circuit reverse current gain