2nd Order Resonant Bandpass Filter Design and Simulation

 Background

This active bandpass filter type is of second order due to having two capacitors in the circuit. The filter will have a peak response at its center frequency which is also known as its resonant frequency. It is composed of a high-pass filter stage and a low pass filter stage to achieve bandpass. The difference between this and other filters is that it has an ideal narrow frequency response. Meaning that after the 3dB cutoff from either side of the passband, it will look very narrow. Additionally, it also only uses one op-amp. 

Specifications

In this post I will be designing and simulating a 2nd order resonant narrow bandpass filter with the following specifications:

  • Bandpass filter
  • 2nd order resonant
  • Center frequency at 320Hz
  • Q = 35
  • Peak gain = 25
Filter layout and Transfer Function
The topology of this circuit is shown in Figure 1. From this the transfer function can be extracted. 
We get EQ 1. 
Figure 1: General topology of a 2nd order resonant narrow bandpass filter


EQ. 1

Calculations
To meet our specifications, we must first also extract the transfer function in terms of components. This is given by EQ. 2 below:
EQ. 2

By setting EQ. 1 and EQ. 2 equal to each other, we can match the coefficients of the s terms and notice the following relations:


Capacitor values
To work with accessible capacitor values, I will once again choose all capacitors to be of equal value.

Resistor Values
To calculate the resistor values we compare terms and solve for R1 and R3.
Similarly, we notice that for R2 the following relation applies

By doing some algebraic manipulation to isolate the R2 term:
Simulation
Now that all values were calculated, we can simulate the circuit.
First we simulate the transfer function frequency response in MATLAB using the following code:

Which gives the frequency response plot with a center frequency of 320Hz.

To confirm the result, we simulate it using Tina-TI by setting up the following schematic:
Which outputs the frequency response:
That matches very closely with the theoretical value.

Conclusion
The active bandpass filter with a Q value of 35 was designed and simulated. The transfer function frequency response was simulated with MATLAB and the circuit was behavior was verified with Tina-Ti. Both of the results matched very closely with each other which confirmed that the design met specifications. 





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