Richards' Theorem

Richards' theorem states that one can implement any reactance (* see more on reactance below) by cascading sections of ideal transmission lines with the last section either open ended or short circuited.

The mathematical content of Richards' result is pretty trivial (it is a simple fact on complex rational functions, even simpler than Nyiquist stability criterion), however, the engineering implications are quite significant.

Now the meaning of reactance here is a bit subtle and involves the so-called Richards (nonlinear) frequency transform which is defined as \\lambda=\\tanh(sl/v) where s is the complex frequency considered in limped filter design (where everything is essentially based on Laplace/Fourier transforms of time-domain equations). Now the terms `reactance' `inductance' and such are meant with respect to this new `frequency' which is usually not a problem since a filter is supposed to work in a *band* of frequencies and Richards transform wil simply shift and stretch such a band ... almost. In reality, since all the functions considered are complex, the transform is periodic and the characteristic of such filters will repeat as the frequency goes higher. This generally considered not a bad thing since the same machinery that is used to develop low-pass filters applies to band pass, etc. SInce most microwave filters are relatively narrow band, Richards' approach works fine.

Finally, the reason to do this at all is also simple. Richards' methodology allows one to design a lumped filter (using the usual tricks including all that Butterworth, Chebyshev, Ellipltic, and similar stuff engineers are so fond of) but implement it as a distributed device built out of sections of transmission lines (say, a microstrip). To find out more about this methodology, look up Kuroda identities in any reasonable RF design book.

Alex

snipped-for-privacy@gmail.com wrote: