Cascaded Bandpass Filter

This page analyzes a cascaded bandpass filter - specifically the one proposed in Alexander & Sadiku Fig. 14.45.

Analysis

The cascaded bandpass filter uses a first-order lowpass filter, followed by a first-order highpass filter, followed by an inverting amplifier to raise or lower to total passband gain to some desired level. The equation given is:

\( \begin{align} H&=\frac{V_o}{V_i}= \left(-\frac{1}{1+j\omega C_1R} \right) \left(-\frac{j\omega C_2R}{1+j\omega C_2R} \right) \left(-\frac{R_f}{R_i}\right) \end{align} \</center>\)

Without inductors, the most likely candidate for such a filter would be the filter on p. 439 of the Rizzoni text which uses a series combination of a resistor and capacitor as \(Z_N\) and a parallel combination as \(Z_F\). This leads to an overall transfer function of:

\(\begin{align} H=-\frac{j\omega C_NR_F}{(1+j\omega C_FR_F)(1+j\omega C_NR_N)} \end{align}\)

or, as re-cast in class,

\(\begin{align} H=\frac{K2\zeta\omega_n j\omega}{(j\omega)^2+2\zeta\omega_n j\omega+\omega_n^2}= -\frac{\left(\frac{C_NR_F}{C_NR_N+C_FR_F}\right)\left(\frac{1}{C_FR_F}+\frac{1}{C_NR_N}\right)}{(j\omega)^2+\left(\frac{1}{C_FR_F}+\frac{1}{C_NR_N}\right)j\omega+\frac{1}{C_FR_FC_NR_N}} \end{align}\)

To make life a little easier, let's call

\(\begin{align} \omega_1&=\frac{1}{C_FR_F} & \omega_2&=\frac{1}{C_NR_N} \end{align}\)

which means

\(\begin{align} H=-\frac{\left(\frac{C_NR_F}{C_NR_N+C_FR_F}\right)\left(\omega_1+\omega_2\right)}{(j\omega)^2+\left(\omega_1+\omega_2\right)j\omega+\omega_1\omega_2} \end{align}\)

This means the bandwidth and natural frequency squared are, respectively,

\(\begin{align} BW&=\omega_1+\omega_2 \\ \omega_n^2&=\omega_1\omega_2 \end{align}\) Good so far...