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RMS detector is kind of a misnomer since we are interested in the power of the waveform and RMS is a voltage. The Mean Square detector outputs a voltage that is proportional the the average power. It is a true power computation that is independent of modulation. The drawback is that it does not have a large dynamic range. A square law diode detector with low-pass filter is a Mean Square detector (as long as the diode is operating in the square law region). If you want an RMS detector then take the square root then you will get a voltage that is proportional to the RMS voltage of the waveform. An example use is an AM demodulator, power meter, etc.. A true MS measurement is really only done with a thermocouple or bolometer; RF energy is converted to heat and measured.
LOG detector has a very high dynamic range but it's output is true for only a CW tone. Measuring Noise or amplitude modulation will yield a false reading. They typically work by measuring the current from several cascaded amplifier cells that are scaled to compress at certain power levels. An example of LOG detector use is an AGC loop.
Spectrum analyzers have both type of detectors for LOG and LINEAR scales, and can apply some corrections to the LOG scale (i.e. noise marker mode). Play around with CW, modulation, and noise and you can see the difference.
Of course there are lots of ICs that give a LOG output of a mean square detector. They have high dynamic range and true power detection. Even some power sensors (some Agilent pulsed sensors) are only true for CW tones. Some powers sensors use thermocouples or bolometers for a true MS power measurement. You really have to be careful with power measurements of modulated signals. If you want a good head scratching exercise measure the power with:
1) A power meter, being carefull of which one you use.
2) A spectrum analyzer in LOG mode.
3) A spectrum analyzer in LINEAR mode.
4) A vector signal analyzer with LOG scale.
5) A vector signal analyzer with LINEAR scale.
Do it first with a CW tone. Use the power meter as the standard and offset the other three measurements to agree.
Now do it with two CW tones combined. Be sure the tone spacing is << than the RBW of your analyzers. Explain the differences. Narrow up the RBW to show two tones. Do they mathematically sum to that of the power meter? Now try it for noise (~0 dBm).