A thinking exercise on Jacobi Maximum Power Transfer

At The system wide conjugate match stuff crashes out again I discussed the failure of Walt Maxwell’s teachings on system wide simultaneous conjugate match using an example drawn from an online expert’s posting.

The replicated scenario with matching with an L network where the inductor has a Q of 100, no other loss elements is shown below. (Quality real capacitor losses are very small, and the behavior will not change much, the inductor loss dominates.)

Above is a model in Simsmith where I have adjusted the lossy L network for a near perfect match. I have used a facility in Simsmith to calculate the impedance looking back from L1, often known as the source impedance at a node but in Simsmith speak the calculated L1_revZ on the form, (ie back into the L network)  from the equivalent load.

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).

Complex conjugate, shortened to “conjugate”, has a well known meaning in mathematics, if x=a+jb then x*=a-jb. (Notwithstanding that fact, I see ham discussion redefining the meaning by talking about SWR* which in mathematical terms, because SWR is a scalar quantity (ie its imaginary part is zero), means SWR*=SWR.)

The previous discussion showed that even though the source G was conjugate matched to its load, along the network at the L2L1 interface L2L1_lZ is significantly different to L2L1_sZ* and so there was not a system wide conjugate match.

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?

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.

What about Q2, is it a no-brainer?