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 in Alexander & Sadiku is:

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

and as noted in Alexander & Sadiku, defining constants:

\( \begin{align} \omega_1&=\frac{1}{RC_2} & \omega_2&=\frac{1}{RC_1} \end{align} \)

the transfer function can be written as:

\( \begin{align} H&=-\frac{R_f}{R_i}\frac{j\omega \omega_2}{(j\omega+\omega_1)(j\omega+\omega_2)}\\ &=-\frac{\frac{R_f}{R_i}j\omega \omega_2}{(j\omega)^2+(\omega_1+\omega_2)(j\omega)+\omega_1\omega_2} \end{align} \)

or, as re-cast in class,

\(\begin{align} H=(-1)\frac{K2\zeta\omega_n j\omega}{(j\omega)^2+2\zeta\omega_n j\omega+\omega_n^2}=(-1) \frac{\left(\frac{R_f}{R_i}\frac{\omega_2}{\omega_1+\omega_2}\right)(\omega_1+\omega_2)j\omega}{(j\omega)^2+\left(\omega_1+\omega_2\right)j\omega+\omega_1\omega_2} \end{align}\)

This means the passband gain (which is an absolute value), natural frequency, and damping ratio are, respectively,

\(\begin{align} K&=\frac{R_f}{R_i}\frac{\omega_2}{\omega_1+\omega_2}\\ \omega_n&=\sqrt{\omega_1\omega_2}\\ \zeta&=\frac{\omega_1+\omega_2}{2\sqrt{\omega_1\omega_2}} \end{align}\)

From this, you can determine the quality and the bandwidth:

\( \begin{align} Q&=\frac{1}{2\zeta}=\frac{\sqrt{\omega_1\omega_2}}{\omega_1+\omega_2}\\ BW&=\frac{\omega_n}{Q}=2\zeta\omega_n=\omega_1+\omega_2 \end{align} \)

This is where disagreement with the book comes in; the book assumes that the bandwidth is the difference between the highpass stage's cutoff and the lowpass stage's cutoff. This fails to take into account, however, the fact that these are not ideal filters. The highpass filter's gain is not just 1 for frequencies higher than \(\omega_1\) nor is the lowpass filter's gain just 1 for frequencies lower than \(\omega_2\). Because of that, the actual half-power frequencies are different from the corners of the first-order filters. Specifically,

\( \begin{align} \omega_{cen,lin}&=\omega_n\frac{\sqrt{1+4Q^2}}{2Q}=\frac{1}{2}\sqrt{\omega^2_1+6\omega_1\omega_2+\omega^2_2}\\ \omega_{hp}&=\omega_{cen,lin}\pm\frac{BW}{2}\\ &=\frac{1}{2}\left(\sqrt{\omega^2_1+6\omega_1\omega_2+\omega^2_2} \pm (\omega_1+\omega_2)\right) \end{align} \)

Design Limitations

This particular circuit has a design limitation: the damping cannot be less than 1. As proof, given the above, we can substitute the bandwidth equation into the natural frequency equation:

\(\begin{align} \omega_2&=BW-\omega_1\mbox{, so}\\ \omega_n^2&=\omega_1\omega_2=\omega_1(BW-\omega_1)=BW\cdot\omega_1-\omega_1^2 \end{align}\)

We can then re-write this to find \(\omega_1\):

\(\begin{align} \omega_1^2-BW\cdot\omega_1+\omega_n^2 &= 0\\ \omega_1&=\frac{BW+\sqrt{BW^2-4\omega_n^2}}{2}\\ \omega_1&=\frac{BW+\sqrt{(BW)^2-(2\omega_n)^2}}{2} \end{align}\)

In order for this to be a real number - and it must be for this circuit, because we want to use real capacitors and resistors - the bandwidth must be at least twice the natural frequency or else the discriminant is negative. And since the damping ratio is

\( \begin{align} \zeta&=\frac{1}{2Q}=\frac{BW}{2\omega_n} \end{align} \)

the damping ratio must be greater than 1 and the quality factor must be less than 0.5.