The following circuit is an example of a band pass filter:

First we will consider a qualitative analysis of the circuit. Recall that the impedance of the inductor and capacitor are:

and .

Hence if the frequency is zero (i.e. D.C.) the impedance of the inductor is zero (i.e. a short circuit) and the impedance of the capacitor is infinite (i.e. an open circuit), this is shown in the circuit below:

Now if the frequency is infinite, the impedance of the inductor is infinite (i.e. an open circuit) and the impedance of the capacitor is zero (i.e. a short circuit), this is shown in the circuit below:

Now we will consider the quantitative analysis.

Using Kirchoffs' voltage law gives:

and ohm's law:

we can calculate the gain of the circuit by:

The following graph is of the gain of the band pass filter circuit shown above:

The gain of the circuit is:

and the following graph shows the phase as a function of frequency:

A bandpass filter has five characteristic parameters. These are listed in the following table:

Name of Variable	Description	Symbol
Center Frequency	This is the frequency at which the transfer function is at a maximum
Cut off frequency 1	This is the lower frequency at which the transfer function equals of the maximum value
Cut off frequency 2	This is the higher frequency at which the transfer function equals of the maximum value
Bandwidth	This variable is the width of the pass band. (see graph below)
Quality factor	This parameter is the ratio of the center frequency to the bandwidth. This gives a measure of the pass band and can be used to describe the shape of the transfer function graph	Q

 

Calculation of the center frequency

The center frequency is when the impedance of the whole circuit is real, hence:

Calculation of the cut off frequencies

By definition, the cut off frequency is when the transfer function is of the maximum value. Hence to find the cut off frequencies we set the gain to equal to and solve for:

which then gives:

The solution of this yields four values for the cutoff frequencies. Only two are positive and have physical significance; they identify the pass pand of the filter:

Calculation of the bandwidth

The bandwidth is simply the difference in the two cutoff frequencies:

Calculation of the quality factor

The quality factor is defined as the ratio of center frequency to bandwidith, hence: