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Band-Pass Filters (BPF) - Active Band Pass Filter and Passive Band Pass Filter

Band-Pass Filters (BPF) - Active Band Pass Filter and Passive Band Pass FilterBand-Pass Filters (BPF) - Active Band Pass Filter and Passive Band Pass Filter

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.

Block diagram of a band pass filter
Figure 01: Block diagram of a band- pass filter

Ideal response of a band pass filter
Figure 02: Ideal response of a band-pass filter
The actual response of a band pass filter
Figure 03: Actual response of a band-pass filter

Active band pass filter

Active band pass filter
Figure 04: Active band-pass filter

Calculation of transfer function of an active band-pass filter

Where, Vin is the input voltage and Vo is the output voltage, the voltage gain of an active band pass filter can be written as follows,

voltage gain of an active band pass filter

Vin is affected by the impedance given by both the R1 resistor and C1 capacitor. The total impedance can be written as follows,

Band pass filters eq 2

According to the Kirchhoff law, Vin will be,

Band pass filters eq 3

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

Band pass filters eq 3.1

Vout is affected by the impedance given by both the R2 resistor and the C2 capacitor. The total impedance can be shown in the following equation,

Band pass filters eq 4

Output voltage Vo is given by,

Band pass filters eq 5

Where,

  • Vin = Input voltage
  • Vo = Output voltage
  • R1 = Resistance of the R1 resistor
  • R2 = Resistance of the R2 resistor
  • C1 = Capacitance of the C1 capacitor
  • C2 = Capacitance of the C1 capacitor
  • Z1 = Impedance given by R1 resistor and C1 capacitor
  • Z2 = Impedance given by R2 resistor and C2 capacitor

Therefore, the voltage gain will be,

Band pass filters eq 6

Let's take,

Band pass filters eq 7

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

The difference between the upper critical frequency (fc2) and the lower critical frequency (fc1) is identified as The bandwidth (BW). Upper critical frequency fc2, lower critical frequency fc1, and bandwidth can be calculated as follows.

Band pass filters eq 8

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

Band pass filters eq 9

Passive band pass filter

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

Passive band pass filter - RC band-pass filter
Figure 05: Passive band-pass filter | RC band-pass filter
Passive band pass filter - RLC band-pass filter
Figure 06: Passive band-pass filter | RLC band-pass filter

Calculation of the transfer function of an RLC band pass filter

Band pass filters eq 10

Where,

  • R = resistance of the R resistor
  • ω = Angular frequency of the signal
  • C = capacitance of the C capacitor
  • L = inductance of the L inductor

Band pass filters eq 11

For ease of simplification, let’s take the reciprocal of the voltage gain,

Band pass filters eq 12

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

Band pass filters eq 13

When ω = ω0 (angular frequency at the center frequency, f0), voltage gain shows the highest value which is 1. At the highest gain,

Band pass filters eq 14

To fulfill the above requirement, at the center frequency,

Band pass filters eq 15

For the 3dB cutoff frequency,

Band pass filters eq 16

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

  • Lower critical frequency,
Band pass filters eq 17

  • Upper critical frequency,
Band pass filters eq 18

  • Center frequency,
Band pass filters eq 19

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

Band pass filters eq 20

If the value of Q (quality factor) is higher, the bandwidth will become narrower. So that shows better selectivity for a given value of fo. (Q>10) as a narrow band or (Q<10) as a wide band.


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References and Attributes

Figures:

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


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