The term “impedance matching” is used frequently, but the meaning is context sensitive… and sometimes it is misused because the meaning used is not correct for the context.

## Impedance

Understanding impedance matching starts with understanding impedance.

There is no better foundation for understanding the concepts of impedance and applying it to linear circuit analysis that understanding complex numbers. There are plenty of good websites and even Youtube videos on complex numbers, a few of them are good introductions. I would stay away from ham authors, look to introductory maths sites like MathIsFun complex numbers, then learn how to apply those concepts to resistance and reactance in AC circuits. (Note that electrical / electronics applications tend to use operator j as equivalent of operator i to avoid confusion with i for current.)

Put simply, impedance is the ratio of voltage to current, all of which are complex quantities (ie have real and imaginary parts). Being a complex quantity, it can be expressed in rectangular form (eg Z=R+jX) or polar form (eg Z=r∠θ).

## Matching

There are two common meanings used:

- to maximise power transfer from a given source to a given load, or matching source impedance to load impedance; and
- to provide a source with a specified load impedance, or matching load to specification without seeking to maximise power transfer.

## Case 1

Around 1840, Jacobi postulated his maximum power transfer theorem for DC circuits.

The extension of Jacobi’s Maximum Power Transfer Theorem (MPTT) to AC circuits states that in a linear circuit, maximum power is transferred to the load when the load impedance is the complex conjugate of the source equivalent internal impedance.

Some common errors:

- MPTT says nothing of the power dissipated in the source or system efficiency; and
- it is wrong to try and turn the definition around to infer a source impedance that delivers maximum power to the load.

Consider the above circuit of a Thevenin source of 1V with Zsource=1+j1Ω and load Rload+jXload.

Above is a contour plot of the power dissipated in the load as a function of Rload and Xload for a given Thevenin Vsource of 1V and Zsource of 1+j1Ω. Maximum Pload is 0.25W and it can be seen that it occurs where the load impedance is 1-j1Ω, which is the complex conjugate of Zsource.

It is a mistake to apply the Jacobi Maximum Power Transfer Theorem to cases where the source does not have a good Thevenin equivalent circuit, or where the optimisation target is other than maximum power transfer.

This is not applicable to most ham radio transmitters as they are often not well represented as a Thevenin or Norton equivalent source. Nevertheless, it is a widely practiced mistake.

## Case 2

Case 2 covers examples where a source may be specified as compatible with a certain load impedance, or a certain range of load impedances… so the load is being tested for compatibility with the source specifications.

The restrictions on compatible loads for a given source might be for safety (eg to avoid overload, damage, fire etc), it might be for optimal response (eg load on a microphone for optimal or specification amplitude / frequency response), or for any other reason not related to maximum power transfer.

For example, a certain audio power amplifier might be capable of delivering a sine wave output of up to 20Vp and not more than 5Ap, so the maximum instantaneous peak power is 20*5=100W, the average power of a sine wave of that character is 50W. Rather than specifying the amplifier with those values, it is quite likely to be specified as a 50W amplifier to suit speakers of no less that 4Ω (so as to not exceed the maximum safe current capability), though there are other ways that rate such amplifiers with inflated maximum power. Such an amplifier could be used with 8Ω speakers, but with less power available (a result of the voltage limit). The “impedance matching” objective here is to “match” the specification of “no less that 4Ω”, it is not about maximum power transfer. If you were to test such an amplifier with a variable load, you might well obtain more load power at less than 4Ω load, but at risk of damaging the amplifier, or at least degrading its distortion performance.

A common example of impedance matching of this type is that of designing / adjusting a load for a radio transmitter. Transmitter specifications often specify the required load (for example for safe operation and specified distortion) in terms of a VSWR limit, eg load=50Ω and VSWR<2. The 50Ω is actually 50+j0Ω, and the VSWR limit sets a tolerance for the load, for example a load of 65-j33 (VSWR=1.87) would meet the specification. In this case, one is matching the load to the equipment requirement (a stated value with stated tolerance). As mentioned earlier, most ham transmitters are not well represented as a Thevenin source and inferences about maximum power transfer are not sound.

## Conclusions

“Impedance matching” is a term used widely, it does not have a unique meaning, and the actual meaning is important in making sound inferences.

MPTT is often wrongly used to support assertions about optimum source impedance, or system efficiency.