Jacobi’s Maximum Power Transfer Theorem
Jacobi’s law (also known as Jacobi’s Maximum Power Transfer Theorem) of nearly 200 years ago stated
Maximum power is transferred when the internal resistance of the source equals the resistance of the load.
Implied is that the internal resistance of the source is held constant, it does not work otherwise. The source must be one that can validly be represented by a Thevenin equivalent circuit. This was in the very early days of harnessing electric current, direct current initially.
Later adaptation dealt with alternating current and it became
Maximum power is transferred when the load impedance is equal to the complex conjugate of the internal impedance of the source.
Again a necessary condition is that the source must be one that can validly be represented by a Thevenin equivalent circuit.
It is one of the principles of basic circuit theory / analysis and the mathematical proof is something a high school student should be able to perform.
High power transmitters
So, if you have a transmitter with a known Thevenin equivalent source impedance, you can seek to provide a load that ensures maximum power transfer.
The practical problem is that it is quite difficult to control the equivalent source impedance of a high power transmitter so that it is a known constant over the entire range of transmitter output power, frequencies etc. So for many purposes, source impedance is not controlled, and the transmitter cannot be validly represented by a Thevenin equivalent circuit, and as a result, the Jacobi Maximum Power Transfer Theorem does not apply.
Do not be confused about transmitter specifications, a requirement that the transmitter load impedance be some value (eg 50Ω) is not specification of a Thevenin equivalent source impedance.
Very few ham transmitters have a controlled equivalent source impedance. Though some claim to have proven by one or few measurements that it is 50Ω, many other valid experiments have shown otherwise and it takes only one valid experiment to disprove the claim.
For the most part, calculations and other claims of mismatch loss in transmitter / antenna systems are misguided, popular but misguided.
The objective with most high powered transmitters is to provide them with a load impedance at their output terminals that complies with the design requirements as embodied in published specifications so as to safely obtain stability, rated power and rated distortion performance.
The obsession with conjugate matching is mostly with the many hams who are devotees of Walt Maxwell’s teachings.
A little knowledge is a dangerous thing; read widely and think.