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I try to design a Chebyshev bandpass filter for 6 GHz as a coaxial cavity filter. I know that the filter can be designed with the aid of the coupling coefficients between the resonators and the loaded Q of the first and last resonator, and all resonators need to be tuned to the centre frequency of the filter.

First, I simulated the eigenfrequency of a single cavity, seen below:

single cavity

I adjusted the length of the centre rod such that the cavity has its first eigenfrequency exactly at the centre frequency of the filter, 6 GHz.

Afterwards I tried to derive the coupling coefficients. For this, I placed two idntical cavities next to each other, as seen below:

coupled cavities

I then do again an eigenfrequency simulation to find the even and odd mode eigenfrequency. Then I calculate the coupling coefficient as follows: $$ k = \frac{ f_2^2 - f_1^2 }{f_2^2 + f_1^2} $$ where I use the higher of the two eigenfrequencies for f2 and the lower one for f1, of course. To vary the coupling coefficient, I change the width of the iris between the two cavities.

I always thought this is a possible approach for finding the coupling coefficients from simulations, but it's not accurate because the resonators get heavily detuned due to the iris. So it would be necessary to re-tune them as the iris width is changed, but I don't know the relation between the iris width and the resonator detuning.

I am also unsure about how I can actually determine the loaded Q of the resonators. If I tap the centre rod of a resonator with the inner conductor of a coaxial cable, the resonator gets detuned again, of course - then I should adjust the centre rod length, but this will affect loaded Q again.

At the end I try to design a filter as shown below:

complete filter

So my questions are: a) is my method of calculating the coupling coefficients accurate enough, or are there better methods? b) how could I determine the loaded Q? c) how I deal in the simulation with the fact that the resonators are detuned due to coupling or tapping?

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When designing coupled resonator filters I have always used tables of k and q from "Handbook of Filter Synthesis" by Zverev and scaled them for centre frequency, bandwidth and impedance. There's probably an online resource these days. To determine the loaded Q of a resonator in isolation, you can look at the bandwidth at a given return loss figure. There is a relationship between Q and, say, 10 dB return loss bandwidth but I can't remember what the numbers are. This only works if matching is close at the centre frequency. For tuning a physical filter, Dishal's method works well. I'm not sure how well that can be applied to a simulation.

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