Understanding a Bainter Circuit (chebychev filter)

Question

As part of a class in Signal Processing, I am building a 3rd order Chebychev band reject filter. We implement this by using three cascading Bainter circuits. Although not part of the class, I have a question about the gain of the Bainter circuit.

I am trying to write a script which will automate the component selection using the corner frequencies and maximum overall gain as design rules, but have some issue with calculating the overall gain.

To calculate the overall gain of a Bainter stage, would I simply work out the individual gains of the three op-amp sections? The overall gain would then be the product of the three individual gains?

Answer 1

To calculating the overall gain of a Bainter stage, would I simply work out the individual gains of the three op-amp sections. The overall gain would then be the product of the three individual gains?

The short answer is: Yes, you can (probably) analyze them individually.

When asking what happens when you cascade multiple analog filter stages, the questions to ask are: what is the source impedance of the first stage, and what is the load impedance of the second stage? If a circuit stage has a large and complicated output impedance, then loading it with another stage can modify its behavior. When working with passive filters, this is a big problem: unless the load impedance of each stage is significantly greater than the source impedance of the prior stage, cascading passive filter sections will result in complicated changes to the behavior of each stage.

One of the attractions of op-amp based circuits is that op-amps generally have a very low output impedance; for the ideal op-amp, there is zero output impedance. Moreover, the op-amp inputs themselves generally have very high input impedance, ideally infinite. This means that circuit sections whose outputs are op-amp driven can generally be cascaded without having one stage change the behavior of another.

Consider this schematic of a Bainter notch (taken from an Analog Devices publication):

The "notch out" is driven by the output of an op-amp. Thus this circuit will have very small output impedance. In other words, the voltage at "notch out" will be relatively insensitive to the load that is connected. This output impedance will almost certainly be much lower than the input impedance.

Thus, in the design phase, you can analyze several cascaded notch circuits separately, and simply multiple their transfer functions together. After producing a design in such a way, you might want to simulate the entire circuit in SPICE to check for behaviors due to op-amp nonidealities, etc.

References

• "Active filter has stable notch, and response can be regulated", in Electronics, October 2, 1975. Original article describing the circuit.
• "How to Write Node Equations for Active Circuits", by Mr. Bainter himself.

Answer 2

Here's what I did in the end.

When building one stage of the Bainter, I knew the first opamp was a unity inverting buffer. So I could easily check its performance. I knew the next two stages were a high pass and low pass respectively. I did not know exactly what frequency they would break at, but I could roughly check their performance.

Once the Bainter was put together, I was able to calculate the DC gain and step response using Matlab. I measured these two characteristics on the actual Bainter and compared. If they were reasonably close, I moved on to the next Bainter stage and repeat.

Once all three Bainter stages were built (for a 3rd order filter) I wired them in order of lowest to highest DC gain.

In the end, I had a fairly accurate Chebyshev filter.

Thanks for the input.