The earliest appearance of the "Conjugate Matching Theorem" that I am aware of appeared in the 1937 edition of Everitt's book "Communication Engineering". Everitt says the theorem "...is proposed here."; he may be the originator of the theorem. The relevant page from that book is shown in the image:
http://i42.tinypic.com/dmxs80.jpgThe figure shows a cascade consisting of a generator of Zg ohms impedance, three 4-terminal matching networks (two-ports), and a load impedance Zr. The theorem says that IF the networks are composed of pure reactances (are lossless), and IF there is a conjugate match at any one of the various junctions of the cascade, then there is a conjugate match at all of the junctions.
However, it appears that many unjustified inferences have been drawn from the theorem. For example, at this web page:
http://www.smeter.net/feeding/t-network-antenna-tuner.phpwhich shows how to determine matching networks, under the heading "Network Image Impedances" is the statement "...if there is a conjugate match at the input end of a lossy network there cannot simultaneously be a conjugate match at the other end." This is not true. I must confess that I previously believed that it was true, but I have been researching this problem, and I now know that it isn't true.
Similar assertions can be found on various forums and web pages.
One must carefully consider Everitt's theorem to avoid reading more into it that it says. It does NOT say that a cascade of networks as shown in the figure from his book cannot possess a conjugate match at all the junctions if the networks are lossy. It only says that IF the networks are lossless, a conjugate match is guaranteed.
Strictly speaking, Everitt's conjugate match theorem doesn't apply to lossy networks. It doesn't say anything about lossy networks because one of its assumptions is that the networks are lossless.
But the fact that Everitt's theorem doesn't guarantee a conjugate match at every junction in a lossy cascade (because the networks are lossy so Everitt's theorem doesn't apply), doesn't mean that there CAN'T be a conjugate match at every junction. For a lossy cascade, Everitt's theorem doesn't say there CAN be conjugate matches at every junction, and it doesn't say there CAN'T be conjugate matches; it says nothing about cascades of lossy networks.
Here is an example of a cascade of two lossy matching networks, where there is a conjugate match at all junctions in the cascade. The values of the T networks are some I made up to be lossy.
http://i39.tinypic.com/ay3ec5.pngNext, consider the case of matching a 50 ohm transmitter to the input of a transmission line (with an antenna on the other end) where the impedance is 58.32-j85.85 ohms.
Here is a lossless T network which provides a perfect simultaneous conjugate match to source and load; I say "perfect" to mean perfect to the extent obtainable with 12 digit values. The values of the T network elements are given as impedances rather than inductances and capacitances. Those impedances can be converted to components at a given frequency by the usual method. We would not ordinarily be surprised that such a network can be found.
http://i39.tinypic.com/10mlbwm.pngHere is a quite lossy T network which also provides a perfect (to 12 digit accuracy) simultaneous conjugate match to the same source and load. This is surprising.
http://i41.tinypic.com/rc3h4x.pngMotorola published an application note some years ago:
http://phobos.iet.unipi.it/~barilla/pdf/ARF_AN215A.pdfEquations 10, 11, 12 and 13 in that application note give the source and load impedances which will give a simultaneous conjugate match at the input and output of a given matching network. These equations can fail if the matching network is "singular", such as having only a single series or a single shunt impedance, or is lossless. But, for a typical lossy network, they will work. If the equations work, they will result is a single (unique) solution. This is not the usual computational problem, having a network first and then wanting the source and load impedances providing conjugate matches.
The reverse problem is, given a source and load impedance, find a matching network which will give a simultaneous conjugate match at input and output. In general, there is not a single network solution to this problem. One of the additional parameters associated with this problem which leads to multiple solutions is the attenuation in the network. For different attenuations, different matching networks can be found as demonstrated by the last two examples above.
The examples I have given show that it is not impossible to have a cascade of networks with a simultaneous conjugate match at all junctions even if the networks are lossy. It is also possible to have a single, lossy, network that will provide a simultaneous conjugate match at input and output. The conjugate matching theorem doesn't prohibit this; it doesn't address the issue at all because it only applies to lossless networks. When we have a true theorem, the contrapositive is also true, but the converse and the inverse are not necessarily true, as in this case. See:
http://hotmath.com/hotmath_help/topics/converse-inverse-contrapositive.htmlOne last thing. Even though I can't yet prove it, I think that if we are given a pair of typical impedances we can always find a lossy network that will provide an exact simultaneous conjugate match at input and output. However, if we add constraints to the problem by specifying component Q ahead of time, it may not be possible to get an EXACT simultaneous conjugate match; it may only be possible to get approximate conjugate matches. If the Q of the components is not constrained, it appears that we can get an exact conjugate match.
This is not intended to stir up the conjugate match controversy, and it only applies to linear systems. The output impedance of a modern solid state transmitter is not at issue in this post.
