# Richards' Theorem

Discussion in 'Electronic Design' started by [email protected], Dec 1, 2006.

1. ### Guest

Can anyone here explain what is meant by Richards' Theorem in the field
of electronics and networks?

hugo

2. ### Guest

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