A thinking exercise on Jacobi Maximum Power Transfer #2

At A thinking exercise on Jacobi Maximum Power Transfer I posed an unanswered Q2:

Keeping in mind that C2 and L2 are an adjustable matching network, usually adjusted for minimum VSWR as seen at the source G. So, the questions are:

  1. Does the system take maximum available power from the source G when the load impedance seen by source G is equal to the conjugate of its Thevenin equivalent source impedance (ie C2.Z=G.Zo in Simsmith speak)?

  2. Does that ‘matched’ condition result in maximum power in the load L?

Above for reader’s convenience is the model conjugate matched at the GC2 interface. The calculated Po figure (lower right) is the power in the load L to high resolution.

I have already stated:

The answer to Q1 is easy, Jacobi’s Maximum Power Transfer Theorem tells us that for a valid Thevenin source (as in this simulation), then the maximum available power is obtained from the source when the Zl=Zs*. The mathematical proof of this is simple and in good text books.

Now to Q2.

Tweaking the matching components C2 and L2 by hand for best Po we get:

Po is very slightly greater and yet the input VSWR is higher.

This should be no surprise in this case as the matching inductor L2 is lossy, not grossly so but typical of an ATU, and adjusting it also changes the loss it brings to the system and the better Po solution above finds that although the entire network accepts less than available power from the source G, the reduction in L2 loss more than offsets it.

The difference in maximum power is very small in this scenario, but the maths of it proves the case that optimising VSWR seen by the source does not necessarily maximise power to the load.

Again the calculated values L2L1_lZ are the load impedance at the L2L1 junction (looking left as Simsmith is unconventional), and L2L1_sZ is the source impedance at the L2L1 junction (looking right).

The next installment will discuss a metric for the mismatch at this point… think about it.