# KL7AJ on the Conjugate Match Theorem – analytical solution – Simsmith

KL7AJ on the Conjugate Match Theorem asked the question Should we have expected this outcome?

Let us solve a very similar problem analytically where measurement errors do not contribute to the outcome.

Taking the load impedance to be the same 10.1+j0.2Ω, and calculating for a T match similar to the MFJ-949E (assuming L=26µH, QL=200, and ideal capacitors) with Simsmith we can find a near perfect match. The capacitors are 177.2 and 92.9pF for the match.

Also calculated is the impedance looking back from the load to the source shown here as L_revZ. The impedance looking back towards the 50Ω load is 17.28-j0.6216Ω, which is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).

Is there some smoke and mirrors in calculation of L_revZ? Lets turn the network around. Now turning the network around by swapping the capacitors and changing the load to 50+j0Ω.

Above, the impedance looking back towards the 50Ω load is 17.28-j0.62Ω, which consistent with the L_revZ calculation and is quite close to the value obtained by measurement, 18.0-j0.8Ω (which is dependent on the actual Q of the ATU elements).

So, in answer to the question Should we have expected this outcome?, the answer is yes, it is not surprising and quite similar to what we might expect from a network of this type.

Walt Maxwell's Conjugate Mirror (Maxwell 2001 24.5) which imbues a magic system wide conjugate match with certain benefits, a utopia, which does not apply to systems that include any loss, it does not apply to real world systems. Maxwell does not state that limitation of his proposition.

Is a ham transmitter conjugate matched to its load? Watch for a follow up post.