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# How do Filters Work ?

How does a passive filter work ? How does it reject signals and pass others ? In order to answer this question, we first need to look at the so-called concept of mismatch. Mismatch is usually understood as imperfect match between the internal impedances of a signal source and a load in the sense that most energy is transferred to the load, but a small part of it is not – due to a mismatch. A measure for this ‘reflective transmission loss’ is the reflection coefficient and the related return loss. Both quantities derive from the source and load impedance. However, extreme mismatch can result in so-called ‘reflective attenuation’ amounting to 10’s of deciBels. In such a case, most energy is reflected back to the source and only a very small amount is actual being passed through to the load.

In order for energy to exit from an rf generator (eg. transmitter) having a certain internal impedance and reach a load (eg. antenna) having its own impedance there, certain conditions must exist. Let’s assume here that both the generator’s impedance and the load are 50 Ohms real. For the case of a direct connection of the load to the transmitter  maximum energy transfer occurs as both impedances are the same. If however, a frequency dependant mismatch exists due to another 2-port device inserted between source and load, then signals at those frequencies where the mismatch exists will experience reflection caused by the mismatch. That means that effectively these signals will be attenuated due to reflection caused by the mismatch. If the mismatch is substantial, then the reflective attenuation will be large. Extreme mismatches are caused by open and short circuits. Filters approach open or short circuit impedances in their stopbands – implying near total reflection. It is for these reasons, that the attenuation of a lossless filter can be calculated from the reflection coefficient. Lossless filters play and important role in the filter synthesis process.

In the passband of a bandpass filter, the resonators and the couplings are arranged in such a way, that the filter is transparent for passband signals. It can here almost be replaced by a length of coaxial line in terms of its attenuation and its delay properties. If the filter is designed for 50 Ohms port impedances and if it is terminated in 50 Ohms then the passband impedance seen from the input of the filter is indeed very close to 50 Ohms – a near perfect match. In the stopbands of the filter the mismatch will cause reflection and thereby attenuation/rejection. The further we move away from the passband the larger the mismatch becomes and so does the attenuation. Where do the reflected signals go ? They are reflected back to the signal source and may experience dissipative loss on the way back to the source if a transmission line is used between source and filter.

Take series resonant circuit between transmitter and load. At resonance its impedance reduces to a small resistive part given by resistive losses in the circuit – its finite Q. Signals at the resonant frequency can travel through the series resonant circuit without being significantly attenuated and go straight into the load. Away from the resonant frequency the impedance of the series tuned circuit changes rapidly approaching infinity at DC (capacitor does not pass DC) and infinity at infinitely high frequencies (inductance does not pass infinitely high frequency signals).

Passive non-resistive filters work by reflection caused by a mismatch condition introduced by the frequency dependent nature of the input impedance.