The transfer function \(\mathbf{H}(\omega)\) (also called the network function) is a useful analytical tool for finding the frequency response of a circuit. The frequency response of a circuit is the plot of the circuit’s transfer function \(\mathbf{H}(\omega)\) versus \(\omega\), with \(\omega\) varying from \(\omega = 0\) to \(\omega = \infty\).
A transfer function is the frequency-dependent ratio of a forced function to a forcing function (or of an output to an input). The transfer function of a network describes how the output behaves with respect to the input.
The transfer function \(\mathbf{H}(\omega)\) of a circuit is the frequency-dependent ratio of a phasor output \(\mathbf{Y}(\omega)\) (an element voltage or current) to a phasor input \(\mathbf{X}(\omega)\) (source voltage or current), assuming all initial conditions are zero.
\(\boxed{\mathbf{H}(\omega) = H(\omega)\angle\phi = \frac{\mathbf{Y}(\omega)}{\mathbf{X}(\omega)}}\)
where \(H(\omega)\) is the magnitude, \(\phi\) is the phase, and \(\omega\) is the angular frequency.
Since the input and output can be either voltage \(\mathbf{V}\) or current \(\mathbf{I}\) at any place in the circuit, there are four possible transfer functions.
\(\boxed{\mathbf{H}(\omega) = \textrm{Voltage Gain} = \frac{\mathbf{V}_{o}(\omega)}{\mathbf{V}_{i}(\omega)}}\)
\(\boxed{\mathbf{H}(\omega) = \textrm{Current Gain} = \frac{\mathbf{I}_{o}(\omega)}{\mathbf{I}_{i}(\omega)}}\)
\(\boxed{\mathbf{H}(\omega) = \textrm{Transfer Impedance} = \frac{\mathbf{V}_{o}(\omega)}{\mathbf{I}_{i}(\omega)}}\)
\(\boxed{\mathbf{H}(\omega) = \textrm{Transfer Admittance} = \frac{\mathbf{I}_{o}(\omega)}{\mathbf{V}_{i}(\omega)}}\)
where subscripts \(i\) and \(o\) denote input and output values, respectively.
To obtain the transfer function \(\mathbf{H}(\omega)\), first obtain the frequency-domain equivalent of the circuit by replacing resistors, inductors, and capacitors with their impedances. Then use any circuit techniques to obtain the transfer function. The frequency response of the circuit can be obtained by plotting the magnitude and phase of the transfer function as the frequency varies.
A transfer function \(\mathbf{H}(\omega)\) can be expressed in terms of its numerator polynomial \(\mathbf{N}(\omega)\) and denominator polynomial \(\mathbf{D}(\omega)\).
\(\boxed{\mathbf{H}(\omega) = \frac{\mathbf{N}(\omega)}{\mathbf{D}(\omega)}}\)
where \(\mathbf{N}(\omega)\) and \(\mathbf{D}(\omega)\) are not necessarily the same expressions for the input and output functions, respectively.
The roots of \(\mathbf{N}(\omega) = 0\) are called the zeros of \(\mathbf{H}(\omega)\), and the roots of \(\mathbf{D}(\omega) = 0\) are the poles of \(\mathbf{H}(\omega)\).
A zero, as a root of numerator polynomial \(\mathbf{N}(\omega)\), is a value for which the transfer function \(\mathbf{H}(\omega)\) is zero. A pole, as a root of denominator polynomial \(\mathbf{D}(\omega)\), is a value for which the transfer function \(\mathbf{H}(\omega)\) is infinite.
To avoid complex algebra, it is expedient to replace \(j\omega\) temporarily with \(s\) when working with \(\mathbf{H}(\omega)\) and replace \(s\) with \(j\omega\) at the end.
In general, a transfer function \(\mathbf{H}(\omega)\) in the s-domain can be expressed in the form
\(\boxed{\mathbf{H}(s) = \frac{\mathbf{N}(s)}{\mathbf{D}(s)} = K\frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}}\)
where \((s = \sigma + j\omega)\) is the complex frequency, K is a constant, \(z_1,\,z_2,\,\dots,\,z_m\) are the zeros of the transfer function, and \(p_1,\,p_2,\,\dots,\,p_n\) are the poles of the transfer function.
The complex frequency \(s\)-plane plot of the poles and zeros graphically portrays the character of the natural transient response of the system.
