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# The concept of coupling

The concept of coupling is very useful for bandpass filters with narrow passband width. By ‘narrow’ we mean here typically a relative bandwidth of less than 10% of the center frequency. Why is the concept of coupling so useful then ? First let us ask: what actually is coupling ? Coupling is known to exist for example between two inductances (coils) which are not connected to each other but are so closely located to one another that they  share their surrounding magnetic field. The magnetic field caused by the current that flows through one inductance induces a voltage in the secondary inductance via magnetic field coupling. Coupling between distributed element resonators (combline, coaxial, dielectric) exists via their magnetic and/or electric fields. The strength of coupling depends on the distance between the resonators and/or on the size of an aperture in a wall separating the resonators. Coupling can be enhanced by additional means like metallic loops or probes. Such coupling enhancing parts allow for a distance between resonators. They ‘pick up’ on one side and ‘deliver’ to the other.

Coupling is a general concept of energy transfer where a primary circuit passes energy to a secondary circuit. Coupling can also have a capacitive nature. A capacitive coupling in narrow bandpass filters can in fact be treated as a ‘negative inductive coupling’ in terms of the filter port S-parameters.

It is important to understand that even the most complex bandpass filter can be fully described/analysed by very few parameters:

• the resonator frequencies
• the resonator Q’s
• the couplings

So, we really don’t need any particular L’s or C’s or R’s when analysing the circuit of a certain bandpass filter in terms of its transmission and reflection properties (S-parameters). All we need are the resonator resonances, their Q’s and the couplings between the resonators and the input and output coupling. This was first discovered by Milton Dishal in 1949. He analysed fundamental bandpass filter circuits in terms of their forward transconductance ( Y21) and found that only Q’s and the couplings were needed for fully describing the filter transfer function. This discovery led to the so-called ‘k and Q’ approach for the design for lumped element filters and was perhaps re-introduced by Atia & Williams in their classic paper on microwave filter design in the 1970s using the coupling matrix. Chapter-10 (by Fano and Lawson) of the classic MIT Radiation Laboratory series book “Microwave Transmission Circuits” by G.L. Ragan also refers to the coupling concept. The only book – to my knowledge – that correctly refers to Milton Dishal as the ‘father of the coupling concept’ is Anatol Zverev’s classic ‘Handbook of Filter Synthesis’. Filter circuit analysis can be either in terms of a loop impedance matrix or in terms of a nodal admittance matrix following the standard concepts of passive circuit analysis.

It should be noted that the above is strictly limited to analysing the filter in terms of its port parameters. As soon as we want to ‘go inside’ the filter – perhaps we want to analyse resonator voltages and currents – we need to use the actual resonator element values or their equivalents. In the case of an LC filter we would then need the actual L and C values because these determine the actual voltages and currents inside the filter circuit. For other filters, like those using coaxial resonators (sometimes casually referred to as ‘combline’ filters) we would need to use transmission line theory elements in the filter circuits or use true equivalent LC resonator elements.

A direct consequence of Milton Dishal’s 1949 discovery is the possibility of scaling the internal filter impedances to any desired impedance levels. Such scaling does not affect the port S-parameters and is therefore legitimate. In practice this means for example that we can use non-uniform resonator sizes in a filter design as long as the couplings between the resonators are maintained. As far as the resonator Q’s are concerned, it is only the loaded Q’s of the first and the last resonator that needs to be maintained as long as the unloaded resonator Q is appreciably high. The unloaded resonator Q only affects the dissipative insertion loss of the filter and is therefore not of primary importance for the transfer function of the filter. An excellent chapter on coupled resonator filter can be found in the Lancaster-Hong book listed in the text book list.