The AM Forum
July 24, 2024, 05:37:03 PM *
Welcome, Guest. Please login or register.

Login with username, password and session length
 
   Home   Help Calendar Links Staff List Gallery Login Register  
Pages: [1]   Go Down
  Print  
Author Topic: What the Conjugate Matching Theorem doesn't say.  (Read 7998 times)
0 Members and 1 Guest are viewing this topic.
N7TCY
Guest
« on: February 13, 2012, 12:24:17 AM »

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.jpg

The 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.php

which 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.png

Next, 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.png

Here 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.png

Motorola published an application note some years ago:

http://phobos.iet.unipi.it/~barilla/pdf/ARF_AN215A.pdf

Equations 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.html

One 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.
Logged
IN3IEX
Member

Offline Offline

Posts: 128


« Reply #1 on: February 13, 2012, 08:11:00 AM »

Thank you.
I just want to add a couple of examples.
Here you will find a method for the design of Pi networks for matching a tube to a range of antenna impedances.
http://www.ing.unitn.it/~fontana/SimpleP.pdf

Looking at Smith chart simulations we see that it is not possible to match (complex conjugate match) the tube to any antenna impedance. A range of resistive impedances must be choosen if using this method.
Anyway the range of complex admissible antenna (plus cable) impedances is even more difficult to simply describe in words, look at the pictures: it is not the full Smith chart.

Smith chart simulations show that the T network is much, much more versatile than the pi network, anyway unmatchable patches still do exist on the Smith chart.   see "Rete a T(CLC). L variabile, C variabili. ZSTART 50ohm" second plot on http://www.ing.unitn.it/~fontana/antmatch.pdf .
Quite interesting is that adding some cable lengh to the antenna cable you can exit the unmatchable patches (the point on the Smith chart that represents the complex conjugate impedance of your antenna plus cable "rotates" around the center of the chart).

None of these studies consider the losses in the matching networks. With similar components costs and quality, it usually happens that the pi network exhibits less losses than the T network.


 
Logged
Steve - K4HX
Guest
« Reply #2 on: February 13, 2012, 10:15:56 AM »

Interesting posts. Thanks for the knowledge!

It seems to boil down to,  When is "close enough" "good enough"?
Logged
W4AMV
Member

Offline Offline

Posts: 671


« Reply #3 on: February 13, 2012, 12:34:11 PM »

I looked into this problem a number of years ago. The following paper attached should be of interest.

* Gilbert_Lossy Matching_01084016.pdf (622.44 KB - downloaded 338 times.)
Logged
The Slab Bacon
Member

Offline Offline

Posts: 3929



« Reply #4 on: February 13, 2012, 02:02:46 PM »

It seems to boil down to,  When is "close enough" "good enough"?

At the point of diminishing return.
Logged

"No is not an answer and failure is not an option!"
N7TCY
Guest
« Reply #5 on: February 13, 2012, 02:19:10 PM »

I looked into this problem a number of years ago. The following paper attached should be of interest.

Thanks for that reference.  I'll digest it over the next few days.

A more recent, quite relevant, paper is this:

http://faraday.ee.emu.edu.tr/eeng224/Papers/Maximum%20Power%20Transfer.pdf
Logged
KM1H
Contributing
Member
*
Offline Offline

Posts: 3514



« Reply #6 on: February 13, 2012, 02:36:35 PM »

Quote
It seems to boil down to,  When is "close enough" "good enough"?

It really boils down to "Why bother"? Time spent to achieve perfection should be left to PhD's and academics who rarely accomplish much at the equipment level. The rest of us can live with what small error or loss exists since it is undetectable at the other end.

Rarely does an engineer reach perfection since his job is to help the company turn a profit not spend years on something that may not be reproducable in manufacturing..... if it meets spec then ship it!

A good part of my R&D career was as a direct report to a PhD. It was always fun balancing the anal demands of the PhD with the needs of the department manager who was getting his butt reamed by the VP of Engineering who was getting his butt reamed by...... and so on. It all flows downhill Grin

Carl
Logged
W4AMV
Member

Offline Offline

Posts: 671


« Reply #7 on: February 13, 2012, 03:36:49 PM »

Hi Carl. For the most part I agree. However, it is alway useful to understand where the limits are and their bounds. What can and cannot be accomplished and unfortunate it is soemtimes counter intuitive and not obvious. So the academic items and approaches can be quite useful. However, I agree they need to be tempered and in the end... get the job done.
Logged
IN3IEX
Member

Offline Offline

Posts: 128


« Reply #8 on: February 15, 2012, 04:25:59 AM »

I just add my FINAL conclusion.
Pure T networks (C series-L to ground-C series) are much much better than ANYTHING else. In theory and in practice.
This points to the well known firm P...

I also suggest EXPERIMENTATION of pure low pass T networks for tube transmitters:
(L series - C to ground - L series) instead of the usual pi networks.
Two roller inductors and a variable capacitor - no band switching. This thing may match anything... and it is big enough for a big transmitter. With three knobs you can also optimize the Q and minimize losses. If it is possible to do it... just do it.

Giorgio
Logged
KM1H
Contributing
Member
*
Offline Offline

Posts: 3514



« Reply #9 on: February 15, 2012, 08:46:30 PM »

Quote
Hi Carl. For the most part I agree. However, it is alway useful to understand where the limits are and their bounds.

When the silliness goes on for decades its time to call a halt, this subject is as bad as the parasitic debate that has been going on about 30 years now.

I thought Steve called a halt to it on here last year and stuffed it off on its own forum which most nobdy cares about. It wore out its welcome on QEX ages ago.

Carl
Logged
W4AMV
Member

Offline Offline

Posts: 671


« Reply #10 on: February 16, 2012, 10:47:57 AM »

Carl, thanks for the comments. However, the general question(s) covering any facet of impedance matching will continue. As a sanity check on just this point, I looked at the number of publications spanning 2010-2012 in the IEEE xplore web site. The total current to date is 1,870 results and the year is not over! Sorry. 
Logged
KM1H
Contributing
Member
*
Offline Offline

Posts: 3514



« Reply #11 on: February 16, 2012, 12:06:00 PM »

And thats likely 1870 individuals who dont produce much in the real world. I dropped IEEE years ago but still have access for what I need thru a friend.
Logged
Pages: [1]   Go Up
  Print  
 
Jump to:  

AMfone - Dedicated to Amplitude Modulation on the Amateur Radio Bands
 AMfone 2001-2015
Powered by SMF 1.1.21 | SMF © 2015, Simple Machines
Page created in 0.076 seconds with 18 queries.