Transfer Function

The transfer function H(ω) (also called the network function) is a useful analytical tool for finding the frequency response of a circuit. In fact, the frequency response of a circuit is the plot of the circuit’s transfer function H(ω) versus ω, with ω varying from ω = 0 to ω = ∞. A transfer function is the frequency-dependent ratio of a forced function to a forcing function (or of an output to an input). The idea of a transfer function was implicit when we used the concepts of impedance and admittance to relate voltage and current. In general, a linear network Input Output can be represented by the block diagram shown in Fig. 1. Thus, assuming zero initial conditions. Since the input and output can be either voltage or current at any place in the circuit, there are four possible transfer functions: where subscripts i and o denote input and output values. Being a complex quantity, H(ω) has a magnitude H (ω) and a phase φ; that is, $H(ω) = H (ω) \angle φ$. To obtain the transfer function using Eq. (1), we first obtain the frequency-domain equivalent of the circuit by replacing resistors, inductors, and capacitors with their impedances $R$, $jωL$, and $1/jωC$. We then use any circuit technique(s) to obtain the appropriate quantity in Eq. (1).
We can obtain the frequency response of the circuit by plotting the magnitude and phase of the transfer function as the frequency varies. Some authors use H( jω) for transfer instead of H(ω), since ω and j are an inseparable pair. The transfer function H(ω) can be expressed in terms of its numerator polynomial N(ω) and denominator polynomial D(ω) as where N(ω) and D(ω) are not necessarily the same expressions for the input and output functions, respectively. The representation of H(ω) in Eq. (2) assumes that common numerator and denominator factors in H(ω) have canceled, reducing the ratio to lowest terms. The roots of N(ω) = 0 are called the zeros of H(ω) and are usually represented as jω = z1, z2, . . . . Similarly, the roots of D(ω) = 0 are the poles of H(ω) and are represented as jω = p1, p2, . . . . To avoid complex algebra, it is expedient to replace jω temporarily with s when working with $H(ω)$ and replace s with $jω$ at the end.