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20 December 2013
■ WITH RUSSELL KINCAID Q & A
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• Bandpass Filter Calculations
• 12 Volt Control
• NiCad Battery Chargers
Bandpass Filter Calculations
QHere is a simple bandpass filter (see Figure 1). I would like to know how to calculate the values of R and C for a specific center frequency. Any help is greatly appreciated.
— Bob Wojcik
AI don’t think I have the mathematical ability to do the calculation, but I can tell you how I would approach the problem. Using Kirchoff’s law that the sum of currents at a point is zero,
and defining the currents as in Figure 1, then:
I1 = (Vin-Va)/R1, I2 = (Vout-Va)jwC2, I3 = (Vb-Va) jwC1
Since the op-amp input is virtual ground, Vb = 0 and,
I3 = I4 therefore Va = Vout/R2 jwC1
At hmc.edu, I find that Z = es = e(σ+jω)
Taking the antilog of both sides: s= σ +jω. If the analysis is
limited to sine waves, σ = 0.
Adding I1 + I2 + I3 and I3 = I4 and solving for Va:
Va = (sC2Vout + Vin/R1)*(R1/(1 + sR1(C2+C1))) =
Eliminating Va and solving for Vout/Vin:
Vout/Vin = -(s/R1C2)/(s2 + s(C2 + C1)/(R2C1C2) +
The equation for a bandpass filter is:
Vout/Vin = H(s) = s/(s2 + sω0/Q + ω02) ( 2)
When s2 = ω02 those terms cancel and the gain is: Q/ω0
ω0 = 2πF0
Q = F0/(F2 – F1)
F2 = upper - 3 dB frequency
F1 = lower - 3 dB frequency
F0 = center frequency = (F1F2)1/2
Equating like terms of (1) and ( 2) and making C1 = C2 = C,
I get the values of R1 and R2:
R1 = Q/ω0/C/Gain
R2 = 2*Q/C/ω0
To test the results, I designed a filter with a gain of
100 ( 40 dB), F1 = 7.0 kHz, F2 = 8.0 kHz, and C = 1.0 nF.
The peak frequency should be ( 56)1/2 = 7. 48 kHz. The
response is shown in Figure 2.
I spent an inordinate amount of time on this because
it is very easy to make an error that carries through and
then I end up with gibberish. However, what I lack in
accuracy I make up for in persistence.
The response in Figure 2 is not exact; the gain, peak
Can’t figure out that pesky
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■ FIGURE 1.