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Richards' Theorem

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

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  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
    microstrip). To find out more about this methodology, look up Kuroda
    identities in any reasonable RF design book.

    Alex
     
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