Short note: RMS of sine wave using log-antilog

The RMS value of a signal is the square root of the mean of its square:

Vrms = √( (1/T) ∫₀^T v² dt )

For a sine wave v = Vm sin ωt, Vrms = Vm/√2 = 0.707 Vm. Log and antilog amplifiers can compute this directly. Squaring and square-rooting become multiplication of a log by 2 and by ½.

Block diagram (explicit method):

 v(t) -> [Precision rectifier |v|]
          -> [LOG] -> [x(-2)] -> [ANTILOG] = v^2
          -> [LPF / averager]              = mean(v^2)
          -> [LOG] -> [x(-1/2)] -> [ANTILOG]
          -> Vrms

Steps:

1. A precision full-wave rectifier gives |v|, because a log amp accepts only one polarity.
2. The log amp gives −K ln|v|. An inverting amplifier of gain −2 makes it 2K ln|v|. The antilog amp then gives v².
3. A low-pass filter or averager, with a time constant much longer than the signal period, gives the mean of v². For a sine wave this is Vm²/2.
4. A second log amp, a gain of −½ and an antilog amp take the square root:

Vo = √( mean(v²) ) = √(Vm²/2) = Vm/√2 = 0.707 Vm

Implicit method (fewer stages): the output is fed back into the log–antilog block to form Vo = mean(v²/Vo). At steady state Vo² = mean(v²), so Vo = Vrms directly. This gives a wider dynamic range.

The result is a true-RMS reading, correct for any waveform and not only sine waves. It is used in true-RMS voltmeters and power measurement. Example IC: AD536.