Realization of Biquad Transfer Function II

Let us now see how the transfer function and its various special cases can be realized with passive elements.

fig: Biquad Circuit

From Fig:

\[
Z_1 = R_1 + \frac{1}{C_1 s}
\]

The Transfer function can be represented as

\[
T(s) = \frac{Z_1}{R_1 + Z_1}
\]

\[
Z_1 = \frac{R_2 C_1 s + 1}{C_1 s}
\Rightarrow T(s) = \frac{\frac{R_2 C_1 s + 1}{C_1 s}}{R_1 + \frac{R_2 C_1 s + 1}{C_1 s}}
\]

\[
T(s) = \frac{\frac{R_2 C_1 s + 1}{C_1 s}}{\frac{R_2 C_1 s + 1 + R_1 C_1 s}{C_1 s}}
= \frac{R_2 C_1 s + 1}{R_2 C_1 s + 1 + R_1 C_1 s}
\]

\[
= \frac{R_2 C_1 \left(s + \frac{1}{R_2 C_1 s} \right)}{R_2 C_1 \left(s + \frac{R_1}{R_2} s + \frac{1}{R_2 C_1} \right)}
\]

\[
T(s) = \frac{s + \frac{1}{R_2 C_1 s}}{s \left(1 + \frac{R_1}{R_2} \right) + \frac{1}{R_2 C_1}}
= \frac{s + \frac{1}{R_2 C_1 s}}{s \cdot \frac{R_1 + R_2}{R_2} + \frac{1}{R_2 C_1}}
\]

\[
= \frac{R_2}{R_1 + R_2} \cdot \frac{s + \frac{1}{R_2 C_1 s}}{s + \frac{1}{C_1(R_1 + R_2)}}
\]

Magnitude Plot

\[
|T(s)| = \left| \frac{R_2}{R_1 + R_2} \cdot \frac{s + \frac{1}{R_2 C_1 s}}{s + \frac{1}{C_1(R_1 + R_2)}} \right|
\]

\text{At } \omega = 0:
\[
|T(j\omega)| = \left| \frac{R_2}{R_1 + R_2} \cdot \frac{\frac{1}{R_2 C_1 j\omega}}{\frac{1}{C_1 (R_1 + R_2)}} \right|
= \frac{R_2}{R_1 + R_2} \cdot \frac{1/(R_2 C_1 \omega)}{1/(C_1 (R_1 + R_2))}
= 1
\]

\text{At } \omega = \infty \Rightarrow s = j\omega, \text{ and } \frac{1}{s} \to 0

\[
\Rightarrow |T(j\omega)| = \left| \frac{R_2}{R_1 + R_2} \cdot \frac{j\omega + 0}{j\omega + 0} \right|
= \left| \frac{R_2}{R_1 + R_2} \right|
\]

\[
\therefore |T(j\omega)| = \frac{R_2}{R_1 + R_2}
\]

fig: Magnitude Plot

Phase Response

\[
\theta(j\omega) = 180^\circ + \tan^{-1}\left( \frac{\omega}{\omega_0} \right) - \tan^{-1}\left( \frac{\omega}{\omega_0} \right)
\]

\[
\text{At } \omega = 0: \quad \theta(j\omega) = 0^\circ
\]

\[
\text{At } \omega = \infty: \quad \theta(j\omega) = 0^\circ
\]

\[
\text{At } \omega = \frac{1}{R_2 C_1}: \quad 
\theta(j\omega) = 45^\circ - \tan^{-1} \left( \frac{R_1 + R_2}{R_2} \right)
\]

\[
\text{At } \omega = \frac{1}{C_1(R_1 + R_2)}: \quad 
\theta(j\omega) = \tan^{-1} \left( \frac{R_1 + R_2}{R_2} \right) - 45^\circ
\]

fig: Phase Plot