I have written on Walt Maxell’s proposition about simultaneous system wide conjugate matching in antenna systems. I will repeat a little to set the context…

Walt Maxwell (W2DU) made much of conjugate matching in antenna systems, he wrote of his volume in the preface to (Maxwell 2001 24.5):

It explains in great detail how the antenna tuner at the input terminals of the feed line provides a conjugate match at the antenna terminals, and tunes a non-resonant antenna to resonance while also providing an impedance match for the output of the transceiver.

Walt Maxwell made much of conjugate matching, and wrote often of it as though at some optimal adjustment of an ATU there was a system wide state of conjugate match conferred, that at each and every point in an antenna system the impedance looking towards the source was the conjugate of the impedance looking towards the load.

This is popularly held to be some nirvana, a heavenly state where transmitters are “happy” and all is good. Happiness of transmitters is often given in online discussion by hams as the raison d’être for ATUs, anthropomorphism over science.

(Maxwell 2001 24.5) states

To expand on this definition, conjugate match means that if in one direction from a junction the impedance has the dimensions R + jX, then in the opposite direction the impedance will have the dimensions R − jX. Further paraphrasing of the theorem, when a conjugate match is accomplished at any of the junctions in the system, any reactance appearing at any junction is canceled by an equal and opposite reactance, which also includes any reactance appearing in the load, such as a non-resonant antenna. This reactance cancellation results in a net system reactance of zero, establishing resonance in the entire system. In this resonant condition the source delivers its maximum available power to the load. …(1)

Let us look at a very simple example in SimSmith.

The scenario is:

- a Thevenin source at 1MHz with a source impedance of 50+j0Ω;
- a nominal half wave of RG59 transmission line; and
- an adjustable load impedance.

This should not be taken to imply that ham transmitters are commonly well represented as a Thevenin source.

The load impedance has been adjusted for a nearly perfect match at the source.

Above is the SimSmith model. The load R and X were adjusted for extremely low |Γ| at the source. |Γ| at the source is extremely low (0.0000173), Return Loss is 95dB, this is a match better than instruments could ever measure. We have achieved an almost perfect conjugate match at the interface between source and T1.

So let us now examine the impedance looking both ways at the load to T1 interface.

SimSmith has an internal feature to calculate the impedance looking backwards into an element, and it is used to calculate the impedance looking back from the load into element L. It is shown under the generator element as L_revZ.

So at the load to T1 interface:

- Z looking into the load is 38.75+j1.813Ω; and
- Z looking into T1 is 58.43-j1.534Ω.

They are not conjugates of each other, not nearly, in fact the mismatch is characterised by Return Loss (in terms of the load Z) is just 14dB (or VSWR=1.5).

In this very simple configuration, a near perfect match at the source does not result in a similar quality match at the other node in the system.

## Why is Walt wrong?

(Maxwell 2001 24.5) relies on a quotation:

If a group of four-terminal networks containing only pure reactances (or lossless lines) are arranged in tandem to connect a source to its load, then if at any junction there is a conjugate match of impedances, there will be a conjugate match of impedances at every other junction in the system. (Everitt 1937 243) and (Everitt and Anner 1956 407)

The problem is that Maxwell silently dropped from his statement (1) above the requirement that networks and lines must be lossless, and the example calculated here shows that Maxwell’s proposition does not apply to real world networks that have loss.

Recourse to simple linear circuit analysis will reveal that lossy networks do not have the property Everitt ascribed to lossless networks.

Walt Maxwell’s conjugate mirror

does not apply in the real world, and the concept is of limited use in understanding real world antenna systems.

When you see people sprouting the Walt’s conjugate mirror

you can expect that they have not read widely or thought about the subject much.

## References

- Duffy, O. Mar 2013. The failure of lossless line analysis in the real world. https://owenduffy.net/transmissionline/folla/index.htm.
- Everitt, W L. 1937 Communications Engineering, 2nd ed. New York: McGraw-Hill Book Co.
- Everitt, W L, and Anner, G E. 1956 Communications Engineering, 3rd ed. New York: McGraw-Hill Book Co.
- Maxwell, Walter M. 2001. Reflections II. Sacramento: Worldradio books.