Is maximum power transfer and conjugate matching simultaneously possible

A reader has asked the question in a transmission line context after reading Walter Maxwell’s teachings on system wide conjugate matching.

In the real world, transmission lines have loss and almost always, the nature of that loss will mean that Zo is not purely real.

The answer to the question depends on whether or not there are standing waves on the transmission line.

Nothing in this article is to imply that a transmitter is well represented by a Thevenin equivalent source.

No standing waves

The analysis is trivially simple.

If a transmission line is terminated in its complex characteristic impedance, all energy in the forward wave is absorbed by the load and there is no reflection.

Since there is no reflection, the impedance (V/I at any point along that line is Zo, ie Zin=Zo=Zl.

A Thevenin equivalent source delivers maximum power into a load which is the complex conjugate of the equivalent internal source impedance, so the source impedance for maximum power transfer must be the conjugate of Zo, Zo*.

With standing waves

If a transmission line is terminated in a load other than its complex characteristic impedance, there will be a reflection and the forward and reflected waves are phasors that add and give rise to standing waves, current and voltage.

In the presence of standing waves, the impedance (V/I) at any point along the line varies in a way that can be predicted by the line parameters, so Zin can be calculated or measured.

The source can be matched for maximum power transfer. The simple solution is that again, the equivalent source impedance for maximum power transfer must be the complex conjugate of the input impedance of the line section.

Obscure and esoteric solutions

There are more esoteric solutions where a grossly mismatched termination on a short line might and source conjugate matched to Zin might maximise power transfer to the load in an overall sense.

Popular misconceptions

It is often stated that the termination of the line must the complex conjugate of Zo.

Let’s look at calculations both ways. We will use a spreadsheet model to see accuracy not available from most line loss calculators which round the displayed results.

Screenshot - 31_12_2015 , 3_43_22 PM

Above is an extract from a spreadsheet of a quite exact calculation of Zo and Zload for a transmission line section at a range of frequencies. The line is 10m in length, Zo nominally 50Ω and has a matched line loss of 1dB/100m at 1MHz. Though the line is a hypothetical one, it is of practical relevance as it has loss characteristics between RG58 and RG8.

In the table above, the line is terminated in the calculated complex Zo, and the input impedance Zin.

Note that Zin=Zo=Zl at all frequencies.

The values are not shown here, but although the source is conjugate matched to Zin, the impedance looking back from the load into the line (47.8618363776457+2.28001675879762i at 1MHz) is not the conjugate of the load impedance. Maxwell’s simultaneous conjugate matching proposition does not worked with lossy lines or networks.

Screenshot - 31_12_2015 , 3_44_03 PMAbove is a repeat but with Zl equal to the conjugate of Zo.

Note that Zin≠Zo.

This latter case is not matched, there are standing waves, albeit quite small in magnitude, but it must be dealt with under the “with standing waves” case above.