THE LAPLACE TRANSFORM METHOD
THE NETWORK FUNCTION AND LAPLACE TRANSFORMS
In Chapter 8 we obtained responses of circuit elements to exponentials est , based on which we introduced the concept of complex frequency and generalized impedance. We then developed the network function HðsÞ as the ratio of input-output amplitudes, or equivalently, the input-output diﬀerential equation, natural and forced responses, and the frequency response. In the present chapter we used the Laplace transform as an alternative method for solving diﬀerential equations. More importantly, we introduce Laplace transform models of R, L, and C elements which, contrary to generalized impedances, incorporate initial conditions. The input-output relationship is therefore derived directly in the transform domain. What is the relationship between the complex frequency and the Laplace transform models? A short answer is that the generalized impedance is the special case of the Laplace transform model (i.e., restricted to zero state), and the network function is the Laplace transform of the unit-impulse response. EXAMPLE 16.17 Find the current developed in a series RLC circuit in response to the following two voltage sources applied to it at t ¼ 0: (a) a unit-step, (b) a unit-impulse. The inductor and capacitor contain zero energy at t ¼ 0 . Therefore, the Laplace transform of the current is IðsÞ ¼ VðsÞYðsÞ. (a) VðsÞ ¼ 1=s and the unit-step response is 1 Cs 1 1 ¼ s LCs2 þ RCs þ 1 L ðs þ Þ2 þ !2d 1 t iðtÞ ¼ e sin ð!d tÞuðtÞ L!d
where R ; ¼ 2L
s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ R 2 1 !d ¼ 2L LC
(b) VðsÞ ¼ 1 and the unit-impulse response is 1 s L ðs þ Þ2 þ !2d 1 t iðtÞ ¼ e ½!d cos ð!d tÞ sin ð!d tÞuðtÞ L!d
The unit-impulse response may also be found by taking the time-derivative of the unit-step response. EXAMPLE 16.18 Find the voltage across terminals of a parallel RLC circuit in response to the following two current sources applied at t ¼ 0: (a) a unit-step, (b) a unit-impulse. Again, the inductor and capacitor contain zero energy at t ¼ 0 . Therefore, the Laplace transform of the current is VðsÞ ¼ IðsÞZðsÞ.