# The transmitter matching problem

In the article The system wide conjugate match stuff crashes out again I worked through an example proffered in an online discussion to show that Walter Maxwell’s teachings on system wide simultaneous conjugate match do not tend to occur in practical systems.

## Why are hams so obsessed with conjugate matching?

The answer is on the face of it quite simple.

Jacobi’s Maximum Power Transfer Theorem extended to alternating current tells us that in a linear circuit, maximum power is transferred from a source to its load when the load impedance is the complex conjugate of the Thevenin equivalent source impedance.

This would seem to have obvious direct application in getting maximum power radiated from a transmitter.

But in most practical cases there are two important points that invalidate this thinking:

1. Ham transmitters are not necessarily well represented by a Thevenin equivalent circuit, ie they do not behave the laws of linear circuits and in those cases Jacobi’s Maximum Power Transfer Theorem is NOT applicable; and
2. Antenna terminals may not be directly connected to the source and even if valid and possible, conjugate matching at source does not necessarily result in maximum power at the antenna.

In the days before common use of VSWR meters, it was common practice to use an antenna current meter or a remote field strength meter to indicate maximum radiation power than transmitter adjustments were made to maximise that (within limits such asdesign or specification limits on PA device current).

Of course maximising power radiated regardless of any and everything else is a fairly inadequate technique, but with its roots in A1 Morse code transmitters and a user base who were not progressive, it survived until the ‘magic’ of VSWR meters penetrated the user base.

## Modern approach to SSB telephony transmitter design and specification

Modern SSB telephony transmitters will usually be solid state and designed to work into some fixed nominal load impedance. Some incorporate a wider range matching network (often known as an Antenna Tuning Unit) for greater flexibility.

For the purpose of this discussion, the term ‘transmitter’ is taken to mean the power amplifier and any necessary filters, but does not include an internal Antenna Tuning Unit.

Transmitter design includes a very wide range of parameters, and whilst it is possible to control source impedance at the output, it has almost no advantage and a lot of disadvantage to other aspects of the implementation… so it is not usually done for this type of transmitter.

Transmitters are usually designed to suit a given range of load impedance, often specified as a nominal value with some notional VSWR tolerance, eg a nominal 50Ω antenna with maximum VSWR of 1.5. It should be safe to operate the transmitter into a load that satisfies that criteria, and the buyer might hope that it should deliver its rated performance including distortion performance. That said, it is not common practice to test at these limits, and in practice specified power, distortion etc might only occur at very close to the nominal load impedance. None of this is to imply that maximum output power occurs in the nominal load (eg 50+j0Ω).

So, to obtain the rated performance, the objective is to deliver a load that is within it rated range, and preferably as close to the nominal value as practicable.

Given the power delivered to that transmitter load, the matter of how much power is radiated falls to the other system elements.

That being the case, the first optimisation objective should be to deliver the transmitter its rated load. Next, address performance of the rest of the system in terms of maximising radiated power.

## Mathematics

Is there room for mathematics and theory here?

The greatest problem in applying conventional linear circuit theory to the problem is that in most cases, ordinary ham SSB telephony transmitters are not well represented by a Thevenin equivalent circuit and analysis based on that is simply invalid.

There is a whole vocabulary that flags woolly thinking, conjugate matching, flywheel effect, mismatch loss, re-reflection, an alternate expression for Γ (the complex reflection coefficient), the new SWR* (conjugate SWR, yes complex conjugate of a scalar) to name a few.

There is discussion in pseudo maths, or junk maths pretending to be science. All grist for the mill for a hobby that is less and less science based with the passage of time.

## Antropomorphism

The proceeding discussion did not need to talk about “happy” transmitters, antennas or other system components. Antropomorphism is a technique often used to ‘communicate’ with hams who do not have a clue… and often by those who also do not understand the problem and often want to preserve their status by having others satisfy their thirst for knowledge with their own inadequate understanding.