I read some theory about lowpass prototype filters in the "Handbook of Filter Synthesis" by Zverev and in "Microwave Filters for Communication Systems". I read these two because I want to understand how the coupling matrix of a filter works and how a filter with a given coupling matrix can be realised. However, there is some part which I don't understand. It concerns the lumped element implementation of a filter. First consider this filter.
It is a classical low-pass filter. No special things about it. However, we now insert admittance inverters. This allows the series inductors to be replaced by parallel capacitors. This is nothing special either. Now comes the interesting part. If we do a bandpass transformation, all the parallel capacitors are replaced by LC parallel resonant circuits, as follows:
So far I don't have any trouble to understand the math. Now comes the realisation of the filter. An admittance inverters equivalent circuit consists of two negative parallel capacitors and one positive series capacitor. So, if each of the admittance inverters in the middle of the filter is replaced by its equivalent circuit, the LC resonant circuits are coupled by positive capacitors and the negative capacitors of the admittance inverters are absorbed by the parallel capacitors.
But what happens to the two admittance inverters at the load and the source, J0 and J5? How can these be implemented, especially when lumped elements are considered? There must be a solution, because there is a paper by Seymour B. Cohhn, "Direct coupled resonator filters", where he also has LC parallel resonant circuits coupled by capacitors. Unfortunately, the paper doesn't give a derivation of the circuits.