Second Order Filters

This page is currently a sandbox for things related to second-order filters.

General Form

For the cases below, we will be looking at specific examples of second-order filters, and in each case we will turn on specific components of a general second-order filter by setting $$K_H$$, $$K_B$$, and $$K_L$$ to a non-zero value in:

$$ \mathbb{H}(s)=\frac{K_H(s^2)+K_B(2\zeta \omega_ns)+K_L(\omega_n^2)}{s^2+2\zeta \omega_ns+\omega_n^2} $$

Band-Pass

For the band-pass filter, with $$K_B$$ set to some non-zero value and $$K_H$$ and $$K_L$$ both set to zero, the transfer function becomes:

$$ \mathbb{H}_B(s)=\frac{K_B(2\zeta \omega_ns)}{s^2+2\zeta \omega_ns+\omega_n^2} $$

or, as a Fourier transform,

$$ \mathbb{H}_B(j\omega)=\frac{K_B(2\zeta \omega_n(j\omega))}{(j\omega)^2+2\zeta \omega_n(j\omega)+\omega_n^2} $$

which is the form we will use here.

Alternate Representation

To analyze this transfer function more easily, we can divide through by the $$2\zeta\omega_n(j \omega)$$ term to get:

$$ \begin{align*} \mathbb{H}_B(j\omega)&=\frac{K_B}{\frac{j\omega}{2\zeta\omega_n}+1+\frac{\omega_n}{2\zeta j\omega}}\\ ~&=\frac{K_B}{1+\frac{j\omega}{2\zeta\omega_n}+\frac{\omega_n}{2\zeta j\omega}}\\ ~&=\frac{K_B}{1+\frac{j}{2\zeta}\left(\frac{\omega}{\omega_n}-\frac{\omega_n}{\omega}\right)}\\ \end{align*} $$

At this point, we introduce a new quantity, the quality factor of the filter $$Q$$, where $$Q=\frac{1}{2\zeta}$$, such that:

$$ \begin{align*} \mathbb{H}_B(j\omega)&=\frac{K_B}{1+jQ\left(\frac{\omega}{\omega_n}-\frac{\omega_n}{\omega}\right)}\\ \end{align*} $$

Magnitude and Phase

This alternate arrangement makes it easier to determine how the magnitude and phase change as the frequency changes:

Magnitude

To find the magnitude of $$\mathbb{H}_B(j\omega)$$, find the magnitude of the numerator and divide it by the magnitude of the denominator:

$$\begin{align*} |\mathbb{H}_B(j\omega)|&=\frac{|K_B|}{\sqrt{1^2+Q^2\left(\frac{\omega}{\omega_n}-\frac{\omega_n}{\omega} \right)^2}} \end{align*}$$

From this, we can see that the numerator has a constant magnitude. Furthermore, the denominator has a constant real part. This means that the magnitude of the denominator is going to be at its smallest when $$\omega=\omega_n$$; any deviation from this will create a non-zero imaginary part and thus increase the size of the denomintor. From this we can assert the following:

• The largest magnitude of this band-pass filter is $$|K_B|$$ and it occurs when $$\omega=\omega_n$$.