[SI-LIST] Limits of Wide Band Match to Reactive Impedances

  • From: "Christian Schuster" <cschuste@xxxxxxxxxx>
  • To: si-list@xxxxxxxxxxxxx
  • Date: Fri, 3 May 2002 13:40:07 -0400


Dear SI-Listers,

imagine the following situation:

            +-----------------------------+
        o---|                             |--- C ---+
 50 Ohm     | "Optimum" Matching Network  |         |
        o---|                             |--- R ---+
            +-----------------------------+

An RC series element should be matched to 50 Ohm over all
frequencies from 0 to infinity (or whatever is possible).

Question:
---------
What is the upper limit of (passive) matching?
In other words, how good can the "optimum" matching network be
if we are clever enough to find it?

Answer:
-------
I have come the following formula in a text book (supposed to be from
Bode):

Integral 0-infinity of (2*pi*ln (1/r) df) <= pi /RC

The formula gives an upper limit for the integral of the reflection
coefficient r
over all frequencies f from 0 to infinity.

Now my questions to you:
------------------------
1. My text book does not explain or derive the formula.
   Do you have more detailed references?
2. What is the corresponding formula for an LR network,
   for an LRC network, for an arbitrary network?
3. Have you ever made use of that formula?

Any input is very much appreciated!

Regards,


Christian Schuster
IBM


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