In this page, we get the transform representation of an inductor and a capacitor. This exercise has been carried out in the previous chapter on network functions. When transform representation of an inductor and a capacitor is obtained in the context of a network function, the initial condition associated with either the inductor or the capacitor is ignored. On the other hand, the initial condition associated with either the inductor or the capacitor has to be taken into account when transform representation of an inductor and a capacitor is obtained in the context of time response of a network.
The voltage across an inductor is expressed by equation (2.1).
In equation (2.1), L is inductance in Henry, and time is expressed in seconds. The Laplace transform for expressions on both sides of equation (2.1) is to be obtained first. The Laplace transform of a derivative of a function is expressed by equation (2.2). Hence the Laplace transform of equation (2.1) is obtained as shown by equation (2.3). The sketch in the middle of Fig. 1 represents equation (2.3). Equation (2.3) can be re-arranged and we get equation (2.4). The last sketch in Fig. 1 represents equation (2.4).
The current through a capacitor is expressed by equation (2.5). The Laplace transform of equation (2.5) is obtained as shown by equation (2.6). Equation (2.6) can be re-arranged and we get equation (2.7). The sketch in the middle of Fig. 2 represents equation (2.7). The last sketch in Fig. 2 represents equation (2.6).
In order to obtain the transform representation of a network, a source has to be replaced by its transform equivalent. The commonly used sources and the corresponding Laplace transform expressions are presented by equations (2.8) to (2.15).
The next page shows how we can obtain the zero-input response of first-order circuits.