B.Sc (Hons, USJ)
(Polymer Science and Technology, Chemistry, Physics)

A band-pass filter is an electronic filter that passes all signals lying within a band between a lower-frequency limit and an upper-frequency limit. So, the band pass filter (BPF) will reject all other frequencies that are outside above specified band. The band-pass filter is a combination of a high pass filter and a low pass filter.

Where, V_{in} is the input voltage and V_{o} is the output voltage, the voltage gain of an active band pass filter can be written as follows,

V_{in} is affected by the impedance given by both the R_{1} resistor and C_{1} capacitor. The total impedance can be written as follows,

According to the Kirchhoff law, V_{in} will be,

Where V+ has been grounded, V- also be zero. Therefore,

V_{out} is affected by the impedance given by both the R_{2} resistor and the C_{2} capacitor. The total impedance can be shown in the following equation,

Output voltage V_{o} is given by,

Where,

- V
_{in}= Input voltage - V
_{o}= Output voltage - R
_{1}= Resistance of the R_{1}resistor - R
_{2}= Resistance of the R_{2}resistor - C
_{1}= Capacitance of the C_{1}capacitor - C
_{2}= Capacitance of the C_{1}capacitor - Z
_{1}= Impedance given by R_{1}resistor and C_{1}capacitor - Z
_{2}= Impedance given by R_{2}resistor and C_{2}capacitor

Therefore, the voltage gain will be,

Let's take,

The above equation is the transfer function of an active band pass filter.

The difference between the upper critical frequency (* f_{c2}*) and the lower critical frequency (

The frequency that the pass band is centered is known as the center frequency, ** f_{o}**. Center Frequency can be defined as the geometric mean of the critical frequencies. The center frequency is calculated as follows.

Passive band pass filters can be either RC circuits or RLC circuits

Where,

- R = resistance of the R resistor
- ฯ = Angular frequency of the signal
- C = capacitance of the C capacitor
- L = inductance of the L inductor

For ease of simplification, letโs take the reciprocal of the voltage gain,

To remove the complex parts of the equation, the 4^{th} power of complex parts is taken separately and the 4^{th} root is taken. So, the transfer function is given by,

When *ฯ = ฯ _{0} *(angular frequency at the center frequency,

To fulfill the above requirement, at the center frequency,

For the 3dB cutoff frequency,

By solving the above equation, upper and lower 3dB frequencies can be obtained.

- Lower critical frequency,

- Upper critical frequency,

- Center frequency,

The quality factor (** Q**) of a band-pass filter can be defined as the ratio of the center frequency to the bandwidth.

If the value of *Q* (quality factor) is higher, the bandwidth will become narrower. So that shows better selectivity for a given value of *f _{o}*. (

The cover image was created using an image by Sergio Stockfleth from Pixabay

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