In electronics, an AC filter is a two-port device that removes a range of frequencies from a signal. Because filters are usually an intermediate components in a larger electric network, it is common to include the load impedance when we analyze the properties of the filter (see Fig. 1). For this reason, the properties of AC filters depend on and are usually specified for a particular load. The input port is usually driven by another component of the network that has a low output resistance (or impendace); for this reason we will often connect either a voltage or a current source when we study the properties of the filter.

AC filters can be either linear or nonlinear however, in this webbook, we are interested only in linear AC filters. Using the same notations for the input and output currents and voltages as for two-port networks a filter can be represented like in Fig. 1.

There are four important transfer functions that used to characterize a filter. These transfer function are summarized in Table 1. Depending on the tranfer function, each transfer function can be calculated by connecting a current or voltage source at the input port and computing the output voltage or current.

Table 1. Transfer functions of filters.

	Transfer function	Definitition	Units
1	Voltage gain	$G_{v}=\frac{V_2}{V_1}$
2	Current gain	$G_{v}=\frac{I_2}{I_1}$
3	Driving point transimpedance	$Z=\frac{V_2}{I_1}$	$\textcolor{gray}{Ω}$
4	Driving point transadmittance	$Y=\frac{I_2}{V_1}$	$\textcolor{gray}{S}$

The transfer functions of AC filters can be represented as a fraction of two polynomials in variable $\textcolor{blue}{s}$ (if you skipped the sections about Laplace transforms, you can think of $\textcolor{blue}{s}$ as a notation for $\textcolor{blue}{s}=\textcolor{blue}{j} w$) $$\begin{equation}H(s)=\frac{P(\textcolor{blue}{s})}{Q(\textcolor{blue}{s}}=\frac{a_m \textcolor{blue}{s}^m + a_{m-1} \textcolor{blue}{s}^{m-1}+...+a_1 \textcolor{blue}{s} +a_0}{b_n \textcolor{blue}{s}^n + b_{n-1} \textcolor{blue}{s}^{n-1}+...+b_1 \textcolor{blue}{s} +b_0}\end{equation}$$ This equation can also be written as $$\begin{equation}H(s)=K_0 \frac{(\textcolor{blue}{s}-z_1)(\textcolor{blue}{s}-z_2)...(\textcolor{blue}{s}-z_m)}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)...(\textcolor{blue}{s}-p_n)}\end{equation}$$ where $K_0$ is a constant, $z_1$,...,$z_m$ are the roots of $P(\textcolor{blue}{s})$ (also called zeros of the transfer function, and $p_1$,...,$p_n$ are the roots of $Q(\textcolor{blue}{s})$ (also called poles of the tranfer function). In addition, it can be shown that the two polynomials must have the following properties:

• The coefficients of $P(\textcolor{blue}{s})$ and $Q(\textcolor{blue}{s})$ must be real and positive.
• If any zero or pole of the transfer is function is complex, then there is also a zero or pole that is the complex conjugate (imaginary poles and zeros must be conjugate).
• The real part of the poles must be negative or zero.
• There should not be any missing term between the highest and lowest degree of $Q(\textcolor{blue}{s})$, unless all the even or odd terms are missing.
• The polynomial $P(\textcolor{blue}{s})$ may have negative terms or even some missing terms between the highest and lowest degree.

Filters can be classified into different types by looking at the frequencies at which the signals can pass or a being significantly atttenuated or rejected:

• Low-pass filter: low frequencies are passed, high frequencies are attenuated.
• High-pass filter: high frequencies are passed, low frequencies are attenuated.
• Band-pass filter: only frequencies in a frequency band are passed.
• Band-stop filter or band-reject filter: only frequencies in a frequency band are attenuated.
• All-pass filter: all frequencies are passed, but the phase of the output is modified.

And, of couse, there are filters that do not fit in any of the above categories. Table 2 gives examples of transfer functions for each of the above types of filters.

Table 2. Filter types with examples of transfer functions.

	Filter type	Examples of fransfer functions ($\textcolor{blue}{s}$-domain)	Examples of transfer functions ($\omega$-domain)
1	Low-pass filter	$$\begin{aligned}H(\textcolor{blue}{s})&=\frac{1}{\textcolor{blue}{s}-p_1} \\ H(\textcolor{blue}{s})&=\frac{1}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)...(\textcolor{blue}{s}-p_n)}\end{aligned}$$	$$H(\textcolor{blue}{j}\omega)=\frac{1}{1 + \textcolor{blue}{j}\omega\tau}$$
2	Heigh-pass filter	$$\begin{aligned}H(\textcolor{blue}{s})&=\frac{\textcolor{blue}{s}}{\textcolor{blue}{s}-p_1} \\ H(\textcolor{blue}{s})&=\frac{\textcolor{blue}{s}^n}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)...(\textcolor{blue}{s}-p_n)}\end{aligned}$$	$$H(\textcolor{blue}{j}\omega)=\frac{\textcolor{blue}{j}\omega\tau}{1 + \textcolor{blue}{j}\omega\tau}$$
3	Band-pass filter	$$\begin{aligned}H(\textcolor{blue}{s})&=\frac{\textcolor{blue}{s}}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)} \\ H(\textcolor{blue}{s})&=\frac{\textcolor{blue}{s}^m}{(\textcolor{blue}{s}-p_1)...(\textcolor{blue}{s}-p_n)}\;\text{such that}\;m \lt n\end{aligned}$$	$$H(\textcolor{blue}{j}\omega)=\frac{\textcolor{blue}{j}\omega\tau}{\textcolor{blue}{j}\left(\omega^2/\omega_0^2-1\right) + \omega\tau}$$
4	Band-stop filter	$$\begin{aligned}H(\textcolor{blue}{s})&=\frac{\textcolor{blue}{s}^2-z_1^2}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)}\;\text{such that}\;|p_1|\ll|z_1|\ll|p_2| \\ H(\textcolor{blue}{s})&=\frac{(\textcolor{blue}{s}-z_1)(\textcolor{blue}{s}-z_2)}{(\textcolor{blue}{s}-p_1)(\textcolor{blue}{s}-p_2)}\;\text{such that}\;|p_1|\ll|z_1|\le|z_2|\ll|p_2|\end{aligned}$$
5	All-pass filter	$$H(\textcolor{blue}{s})=\frac{\textcolor{blue}{s}+s_0}{\textcolor{blue}{s}-s_0}$$

Notice that we can write the following equations for all the examples of transfer functions shown in the Table 2.

For low-pass filters: $$\begin{aligned}\lim_{\textcolor{blue}{s}\to\infty} H(\textcolor{blue}{s})&=0 \\ \lim_{\textcolor{blue}{s}\to 0} H(\textcolor{blue}{s})&=\text{finite}\end{aligned}$$ For high-pass filters: $$\begin{aligned}\lim_{\textcolor{blue}{s}\to\infty} H(\textcolor{blue}{s})&=\text{finite}\\ \lim_{\textcolor{blue}{s}\to 0} H(\textcolor{blue}{s})&=0\end{aligned}$$ For band-pass filters: $$\begin{aligned}\lim_{\textcolor{blue}{s}\to\infty} H(\textcolor{blue}{s})&=0 \\ \lim_{\textcolor{blue}{s}\to 0} H(\textcolor{blue}{s})&=0\end{aligned}$$ For band-stop filters: $$\begin{aligned}\lim_{\textcolor{blue}{s}\to\infty} H(\textcolor{blue}{s})&=\text{finite} \\ \lim_{\textcolor{blue}{s}\to 0} H(\textcolor{blue}{s})&=\text{finite}\end{aligned}$$ $$\begin{aligned}\lim_{\textcolor{blue}{s}\to\infty} H(\textcolor{blue}{s})&\gg H(\textcolor{blue}{s}) \\ \lim_{\textcolor{blue}{s}\to 0} H(\textcolor{blue}{s})&\gg H(\textcolor{blue}{s})\end{aligned}$$ For all-pass filters: $$|H(\textcolor{blue}{s})|\;\text{does not depend on }\: \textcolor{blue}{s}$$