The loaded resonator quality factor

The first and the last resonators of a bandpass filter may appear physically as being not very different from the other resonators. However, these outermost resonators form the transition from the inner filter resonators to the connected signal source and load impedance. This function makes them very different from inner resonators in terms of their operational quality factor.

The outermost resonators are loaded by the source- and load impedance via the input- and output coupling. These couplings are of such strength, that the required loaded Q for the given bandpass filter response exists. Therefore, the loaded Q is a calculated design parameter. In effect, the first resonator contains the transformed signal source with its internal impedance and the last resonator contains a transformed load impedance. Naturally, such loading results in a relatively low resonator Q – the loaded Q (also called Q_end). Typical loaded Q values are in the order 10 … 100 depending mainly on the relative filter bandwidth (BW/fc). In contrast to that, unloaded Q factors of metal and ceramic filter resonators are in 1000 … 10,000 range. 

While input- and output coupling arrangements can have many different forms, their purpose is always to produce the correct transformation of the signal source impedance and load impedance so as to create the prescribed loaded Q factors of the end resonators.

As a consequence, the voltages and power dissipation in the outermost resonators are significantly lower than in the other filter resonators. That is why the outermost resonators have a lesser effect on the filter insertion loss that the inner resonators. The contribution of the “end-loaded” resonators to the filter selectivity is also significantly lower than that of the inner filter resonators. 

Of course, the outermost resonators of a bandpass filter also possess an unloaded Q as a result of the losses in the resonator. That Qu is however not their operational Q. The operational Q is dominated by the loading described above.