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If an RF transmitter is designed to transmit into a 50 Ohm load, is the impedance of the transmitter, when looking from the load towards the transmitter, always 50 Ohms?

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No.

Generally though, it's not far off.

I well remember our QA guy using a 50W transmitter called a 'Sierra', to test the input protection on the test set we'd just designed (Hi Roy!). The output impedance was only 10 ohms or so, so when our series protection switch opened, if there was a tuned length of transmission line between the source and our switch (there always was, Roy made sure of that), then the opening switch was subjected to 5x the voltage that you would expect for a matched source. But as a result, we tended to produce the most rugged RF test sets on the market!

There's another distinction, instantaneous or wideband impedance, and long term tuned frequency impedance. A common topology for a precision power source is a fed-back levelled voltage source, followed by a 50 ohm resistor. Instanteously, if the amplifier feeding the voltage source is 50ohms, then the whole thing looks like 100 ohms output impedance. However, after a few cycles of levelling, the voltage at the output is consistent with being fed from a 50 ohm source, at the generated frequency.

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