How to design Multiple-feedback band-pass filter?

 There are couples of methods to create a band-pass filter such as cascaded band-pass filter, multiple-feedback band-pass filter, state variable band-pass filter, biquad band-pass filter etc. The process of designing a cascaded band-pass filter was explained in the earlier tutorial How to Design Active Band Pass Filter - Cascaded HPF and LPF Band Pass Filter. Here the process of designing a multiple-feedback band-pass filter is explained with online calculator to calculate the component values.

The multiple-feedback band-pass filter circuit diagram is shown below.

The above multiple feedback band-pass filter circuit has two feedback, the path from output via R2 back to input and the path from output via C1 back to the input. That is why it is referred as multiple-feedback. The circuit has both low pass filter and high pass filter. The capacitor C1 and resistor R1 forms the LPF while the capacitor C2 and R2 forms the HPF.

The frequency response graph of a band-pass filter is shown below.

The center frequency $f_{0}$ of the band-pass filter can be expressed as,

$$f_{c}= \frac{1}{2 \pi \sqrt{(R_{1} ||  R_{3}) R_{2}C_{1} C_{2}} }$$

because as viewed from capacitor C1 feedback R1 is parallel with R3.

Lets take C=C1=C2 then,

$$f_{c}= \frac{1}{2 \pi\sqrt{(R_{1} R_{3}) R_{1} C^2} }$$

Resolving we can write expression for center frequency as,

$$f_{c}= \frac{1}{2 \pi C }  \sqrt{\frac{R_{1} + R_{3}}{R_{1} R_{2} R_{3}}}$$  

The Quality factor or Q-factor is,

  $$Q= \frac{f_{0}}{BW}$$

or,

  $$Q= \frac{f_{0}}{f_{c2}-f_{c1}}$$

 Without derivation, the value of the three resistors R1,R2 and R3 are as follows,

$$R_{1}= \frac{Q}{2 \pi f_{0} C A_{0}}$$

$$R_{2}= \frac{Q}{\pi f_{0} C}$$

$$R_{3}= \frac{Q}{2 \pi f_{0} C (2Q^2 - A_{0})}$$

Solving for gain $A_{0}$ at center frequency using relation for R1 and R2 we get,

$$A_{0}= \frac{R_{2}}{2 R_{1}}$$

But from the expression for R3 above, the denominator has to be positive. For denominator to be positive we have the condition,

  $$A_{0} \lt 2 Q^2$$

It should be noted that the maximum gain $A_{0}$ occurs at the center frequency $f_{0}$.

 Online Calculator for Multiple-Feedback Band-Pass Filter

Inputs:
f_c: Hz KHz MHz
Q:
A_o:($A_{0} \lt 2 Q^2$)
C(=C₁=C₂): µF nF pF

 
Theoretical Results:
BW: R₁: R₂: R₃:

Multi-Feedback Band-Pass Filter Design Example

Consider the design of a multi-feedback BPF with center frequency at 1KHz and bandwidth of 1KHz. Let the maximum gain of the filter which occurs at the center frequency be 1.5. Let us select value of the capacitor C1 and C2 of 0.01uF. Then using the above online multi-feedback band pass filter calculator we get,

Q = 1, R1 = 10.62KOhm, R2 =  31.85KOhm, R3 = 31.85KOhm

Let us use LM324 operational amplifier. The circuit diagram of multiple feedback BPF designed with calculated values is shown below,

The frequency response of the above multiple feedback band pass filter is shown below.

Following are related tutorials and online filter calculators.

- How to design Active Filters- Low Pass Filter & High Pass Filter

- How to design Twin T Active Notch Filter?