It is necessary to know how to represent an inductor and a capacitor in terms of Laplace transforms. Given that our interest centres around network functions, the we follow the principle applied in evaluating a network function. A network function is the ratio of an output variable to an input variable, subject to the conditions that both the output and input variables are expressed in terms of Laplace transforms and that the initial conditions are assumed to be of zero value.
Next we find out how the Laplace-transformed reactance of an inductor can be obtained.
The voltage across an inductor is expressed as follows:
where L is the inductance in Henry, i is the current through the inductor in amps, t is time in seconds and vL is the inductor voltage in Volts.
We can obtain Laplace transform expression for equation (1.1). Given function f(t) and its Laplace transform F(s), the Laplace transform of derivative of f(t) is expressed as follows.
When our interest is focused on network functions, the initial condition is ignored. Then we get the following equations.
Equation (1.3) presents equation (1.1) in terms of Laplace transform. We get equation (1.4) from equation (1.3) by ignoring the initial condition. The transform reactance of an inductor is expressed by equation (1.5). By definition, reactance has the unit of Ohms, and it is the ratio of voltage to current. The transform reactance of an inductor is the Laplace transform of its voltage to that of its current.
The sketches in Fig. 1 show how an inductor is represented in a circuit. Here sL is the transform reactance of an inductor. In the case of an circuit excited by a sinusoidal signal, the reactance of an inductor is expressed as jwL, where s is replaced by jw.
In the next sub-section, we find out how the transform reactance of a capacitor can be obtained.
The current through a capacitor is expressed as follows:
In equation (1.6), C is the capacitance in Farad, iC is the current through the inductor in amps, t is time in seconds and the capacitor voltage in Volts. Using Laplace transforms, equation (1.6) can be represented as shown below.
Equation (1.7) presents equation (1.6) in terms of Laplace transform. We get equation (1.8) from equation (1.7) by ignoring the initial condition. The transform reactance of a capacitor is expressed by equation (1.9). By definition, reactance has the unit of Ohms, and it is the ratio of voltage to current. The transform reactance of a capacitor is the Apace transform of its voltage to that of its current.
The sketches in Fig. 2 show how a capacitor is represented in a circuit. Here (1/sC) is the transform reactance of a capacitor. In the case of a capacitor, the label above or below a capacitor may express its transform susceptance instead of its transform reactance. In Fig. 1.2, (sC) is the transform susceptance of a capacitor. In the case of an circuit excited by a sinusoidal signal, the reactance of a capacitor is expressed as (1/jwC) or (- j/wC). where s is replaced by jw.
This page has explained how an inductor and a capacitor can be represented using Laplace transforms. The next page explains how driving point impedance of single port network can be obtained.