below is the schematic of twin-t filter, and the transfer function of it. in actual design, it can also be used for bandpass filter if connected between negative input and output of opamp. now, the twin-t network is a three order system, so, how to get the expression of center frequency f0 and quality factor Q? for two order system, we use 's^2+(2*pi*f0/Q)*s+(2*pi*f0)^2' to calculate f0 and Q, but for the three order system, how to calculate the parameter f0 and Q?
thanks all.
My gut feel is that Q can't be easily be calculated (for a twin-t + opamp bandpass filter) as it depends entirely on the accuracy of the component values.
You could guarantee a certain minimum Q by using sufficiently close-tolerance components, but without measurement you wouldn't know how much better than the minimum it actually is.
Similarly, you could deliberately degrade the Q by mismatching components.
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On the other hand, it's not totally unpredictable. According to Horowitz&Hill:
...a twin-T driven by a perfect voltage source is down 10dB at twice (or half) the notch frequency and 3dB at four times (or one-fourth) the notch frequency.
The twin-T network is used as band stop (notch) filter for a particular selection of component values.
You get an ideal notch (neglecting tolerances) for C1=C2+C3 and R3=R1||R2.
In this case you arrive at a second-order transfer function due to pole-zero cancellation.
Thus, you can apply the classical procedure to find the pole frequency and the pole Q.
ok, i know. but if the relationship C1=C2+C3 and R3=R1||R2 is not satisfied (there is mismatch), such as R1=R2=2*R3 and C2=C3=a*C1(a is not 1/2), so, how to get the expression of f0 and Q? or is there some simple expression here?
besides,
Similarly, you could deliberately degrade the Q by mismatching components.
Unless someone has already solved the problem, you can't seriously expect others do the calculation for you, I think.
Strictly speaking, Q is a parameter of the second order system and undefined for third order. Calculating parameter sensitivity of the ideal filter makes sense however. I'm sure, it can be derived analytically, but I also guess you won't want to do the calculation. Everybody is calculating sensitivities by numerical methods these days. You can do with a spread sheet calculator, by stepping parameters in a basic SPICE simulation, or using a tool like PSpice advanced analysis that does the calculation for you automatically.
with regard to FvM`s contribution (Q value) perhaps it is good to give the following explanation:
The Q expression in my last post is defined as the ratio center frequency-to-3dB bandwidth B.
B is the bandwidth measured for 3 dB attenuation.
if C1=C2=2*C3(C1 and C3 inter-changed in my posted schematic), R1=R2=0.5*R3, then, a=1, b=c=0.5, so, Q=0.25, not 0.5.
besides, the Q expression does not include C3, but in my simulation, Q can be influenced by C3. am i wrong?
thanks.
if C1=C2=2*C3(C1 and C3 inter-changed in my posted schematic), R1=R2=0.5*R3, then, a=1, b=c=0.5, so, Q=0.25, not 0.5.
besides, the Q expression does not include C3, but in my simulation, Q can be influenced by C3. am i wrong?
thanks.