Filters designed using the image impedance methodology suffer from a peculiar flaw in the theory. The predicted characteristics of the filter are calculated assuming that the filter is terminated with its own image impedances at each end. This will not usually be the case; the filter will be terminated with fixed resistances. This causes the filter response to deviate from the theoretical. This article explains how the effects of image filter end terminations can be taken into account. Generally, the effect of the terminations is to cause a rounding of the frequency response at cut-off. The image method predicts a sharp discontinuity in the slope of the response at cut-off which is not realised in practice, although a well designed image filter may get close to this. Another prediction of the image method is zero loss in the passband (assuming ideal lossless components). Again, this cannot be achieved in practice because reflections from the end terminations always cause some loss.
Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.
Symbols used in this article
Impedances
Z I 1 {\displaystyle Z_{I1}\,\!} the image impedance at end 1
Z I 2 {\displaystyle Z_{I2}\,\!} the image impedance at end 2
Z I {\displaystyle Z_{I}\,\!} the image impedance when both ends are identical
R 1 {\displaystyle R_{1}\,\!} the terminating resistance at end 1
R 2 {\displaystyle R_{2}\,\!} the terminating resistance at end 2
R {\displaystyle R\,\!} the terminating resistance when both ends are identical
Coefficients
r I 1 {\displaystyle r_{I1}\,\!} the reflection coefficient at end 1
r I 2 {\displaystyle r_{I2}\,\!} the reflection coefficient at end 2
r I {\displaystyle r_{I}\,\!} the reflection coefficient when both ends are identical
τ I 1 {\displaystyle \tau _{I1}\,\!} the transmission coefficient at end 1
τ I 2 {\displaystyle \tau _{I2}\,\!} the transmission coefficient at end 2
γ {\displaystyle \gamma \,\!} the complex propagation coefficient of the filter
α {\displaystyle \alpha \,\!} the attenuation coefficient of the filter
β {\displaystyle \beta \,\!} the phase coefficient of the filter Note that all of these coefficients are defined relative to the image impedance and not the actual input impedance of the filter.
General case
The transfer function of any filter connected as shown in the diagram above is given by the expression
A ( i ω ) = V o V i = Z I 2 Z I 1 e − γ [ τ I 1 τ I 2 1 − e − 2 γ r I 1 r I 2 ] {\displaystyle A(i\omega )={\frac {V_{o}}{V_{i}}}={\sqrt {\frac {Z_{I2}}{Z_{I1}}}}e^{-\gamma }\left[{\frac {\tau _{I1}\tau _{I2}}{1-e^{-2\gamma }r_{I1}r_{I2}}}\right]}
where
r I 1 = R 1 − Z I 1 R 1 + Z I 1 {\displaystyle r_{I1}={\frac {R_{1}-Z_{I1}}{R_{1}+Z_{I1}}}}
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