# Twin T Bandpass Filter transfer function's derivation

Discussion in 'Homework Help' started by Lieta, Jan 18, 2009.

Jan 18, 2009
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Apr 5, 2008
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3. ### mik3 Senior Member

Feb 4, 2008
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Assume an ideal op amp so you can say that the voltage on the non-inverting terminal (V2) equals the voltage on the inverting terminal (V1). The voltage are measured with respect to ground. First find the transfer function (relationship) between Vin and V2. Then find the transfer function between Vo and V1. Finally, because V1=V2, substitute V1 with the equation relating V2 with Vin to get the transfer function between Vo and Vin.

Jan 28, 2005
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5. ### The Electrician AAC Fanatic!

Oct 9, 2007
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Judging from the subject line, I think he wants you to produce one of those lovely detailed transfer function derivations that you do so well.

But, he hasn't shown any of his own work yet.

6. ### Lieta Thread Starter Member

Jan 18, 2009
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Thanks, bertus, for the document. The twin-T filter without an op-amp is a band-stop filter (see fig. 4).
Denote
Vout - output of whle circuit.
Vin_pas - input of passive twin-T filter
V(-) - input of inverting input of op-amp.
V(+) - input of non-inverting input of op-amp
Vin - input of the whole circuit
Vin_pas = V(-) = V (+) = Vin*k, where k = R5/(R4+R5).
Vout/Vin_pas - band-stop filter transfer funct. and because Vin_pas = Vin*k,
Vout/Vin also band-stop filter transfer funct., but actually the circuit if band-pass filter. How to take an op-amp into account? Where am I wrong?

7. ### The Electrician AAC Fanatic!

Oct 9, 2007
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Are you still interested in deriving the overall transfer function for the opamp plus twin-T feedback network?

Are you required to use any particular method to derive it, or can you use any method you like?

This thread is several months old, so I thought I would ask if your interest level is still high.

8. ### Lieta Thread Starter Member

Jan 18, 2009
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I'm still very interested in deriving the overall transfer function for the opamp plus twin-T feedback network. Any method is fine.

9. ### The Electrician AAC Fanatic!

Oct 9, 2007
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You said "How to take an op-amp into account?". Does this mean that you don't want to make the usual assumption that the opamp is ideal, with infinite gain?

10. ### Lieta Thread Starter Member

Jan 18, 2009
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No, assume op-amp as ideal. I ment how to derive transfer function for whole circuit, including op-amp, because without op-amp it's a notch filter, but with it it's band pass filter.

11. ### The Electrician AAC Fanatic!

Oct 9, 2007
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Let V+ and V- be the voltages at the + and - inputs of the opamp.

Let f(ω) be the transfer function of the twin T network as found in the link Bertus gave you in post #2.

We'll ignore the effect of Cin.

The voltage V+ is given by V+ = Vin * R5/(R4+R5).

The voltage V- is given by V- = Vout * f(ω).

Since the opamp is ideal, V- = V+, so that Vout * f(ω) = Vin * R5/(R4+R5).

From this we have Vout/Vin = (R5/(R4+R5))/f(ω)

12. ### Lieta Thread Starter Member

Jan 18, 2009
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Thanks, The Electritian, now it's clear. The idea is that Vout is actally input of passive twin-T filter and V- is output.

Last edited: May 2, 2009
13. ### Lieta Thread Starter Member

Jan 18, 2009
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Suppose the transfer function of a bandpass filter is
A*(1+Bs)/(1+Cs+Ds*s), where
s=j*ω,
A-D are constants.
How to find
f0 - mid-frequency,
K0 - module of transfer function at mid frequency
Q - filter quality?

14. ### The Electrician AAC Fanatic!

Oct 9, 2007
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The prototypical response of a standard bandpass filter is:

A*(Bs)/(1+Cs+Ds*s)

not:

A*(1+Bs)/(1+Cs+Ds*s)

Then K0 is A*B/C

I think you can find the other two items by searching on the web. For example, on this page: http://en.wikipedia.org/wiki/Q_factor

about halfway down the page, see the expression for H(s).

15. ### Lieta Thread Starter Member

Jan 18, 2009
12
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Unfortunately I have A*(1+Bs)/(1+Cs+Ds*s), and I haven't mistaken, because it matches the one in book. They have also given equations for K0, ω0 and Q, so I'm wondering how they got them. I've printed a graph |K(ω)| and it really is a bandpass filter, though it's not symmetric with respect to ω0, it has bigger values at lower frequencies than at higher frequencies.

16. ### The Electrician AAC Fanatic!

Oct 9, 2007
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I'm not suggesting you're mistaken; I'm just saying that that response is not a standard bandpass response. You can tell that by the unsymmetrical response. You will just have to decide what frequency is the midband frequency by some method that takes into account the asymmetry.

The usual method to determine Q makes use of the 3 dB down frequencies on either side of the midband frequency.

The (1+Bs) numerator is why the twin T feedback network isn't usually used to get a bandpass response.