On Dec 24, 2:38*pm, Cecil Moore wrote:
Keith Dysart wrote:
And for a challenging use case, please consider two
circuits connected together. The circuits are in black
boxes so you do not know their details, but the voltage
on the connection between the circuits is measured as
10 V RMS at 4 MHz. The current is measured as 0.

How much energy is being transferred between the
circuits?

Inside each black box is a 50 ohm signal generator equipped
with a circulator and a 50 ohm load resistor. The signal
generators are outputting identical 10 V RMS phase-locked
signals.

Signal generator #1 "sees" the 50 ohm resistor in
signal generator #2 as it's load and supplies 200
milliamp, thus heating up the load resistor.

Signal generator #2 "sees" the 50 ohm resistor in
signal generator #1 as it's load and supplies 200
milliamp, thus heating up the load resistor.

The net voltage is measured as 10 V RMS at 4 MHz.
The net current is measured as 0.

How much energy is being transferred between the
circuits?

Install a one wavelength 50 ohm lossless transmission
line between the two signal generators. Nothing changes
but the standing waves become very visible and measurable.

An excellent example.

Allow me to provide an alternate analysis.

The current between the black boxes is 0.

From circuit analysis theory, any branch with a current
that is always zero can be cut without altering the
conditions anywhere in the rest of the circuit.

Cut the branches connecting the two black boxes.

No conditions change in either of the now two cicuits.

It is obvious that the power dissipated in circuilator
1 must be provided by signal generator 1 and the power
dissipated in circulator 2 must be provided by signal
generator 2.

...Keith