Bilinear Transfer function (one pole, one zero)

Transfer function that has its numerator and denominator. It has a linear function of ‘s’. The transfer function is given by

\[
T(s) = \frac{b_0 s^n + b_1 s^{n-1} + \cdots + b_n}{a_0 s^m + a_1 s^{m-1} + \cdots + a_m}
\]

For m = n = 1, then the T(s) of equation is said to bilinear, where a1, a2, b1, b2 must be real and positive

\[
T(s) = \frac{N(s)}{D(s)} = \frac{a_1 s + a_0}{b_1 s + b_0}
= \frac{a_1 (s + \frac{a_0}{a_1})}{b_1 (s + \frac{b_0}{b_1})}
= G \cdot \frac{(s + Z_1)}{(s + P_1)}
\]

\[
\text{Where:} \quad G = \frac{a_1}{b_1}, \quad Z_1 = -\frac{a_0}{a_1}, \quad P_1 = -\frac{b_0}{b_1}
\]

Properties

• Zeros and poles are real and complex.

• Location of zeros and poles give characteristics and responses

• Gain is independent of poles and zeros

• Poles is in negative Real axis and zeros can be anywhere