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RMS of sinusoid using log-antilog amplifiers

Short answer

The circuit squares the input by taking 2·ln|Vi|, subtracts ln(Vo), antilogs to get Vi²/Vo, which is averaged by a low-pass filter. Feedback forces Vo = √(avg(Vi²)), giving true RMS. For a sinusoid Vi = Vm sin ωt, Vo = Vm/√2 = 0.707·Vm, e.g. 7.07 V for Vm = 10 V.

How can you find the RMS value of a sinusoidal signal using log and antilog amplifiers? Explain with necessary derivations and circuit diagrams.

The RMS value is Vrms = √( average of Vi² ). A log-antilog circuit computes this implicitly: it forms Vi²/Vo, averages it and feeds the result back as Vo. This avoids a separate squarer whose output range would be very large.

Principle

Vo = avg( Vi² / Vo )
Vo is DC, so Vo = avg(Vi²)/Vo
Vo² = avg(Vi²)
Vo  = √avg(Vi²) = Vrms

Circuit (block form)

Vi -->[ABSOLUTE ]--|Vi|-->[LOG]--V1--x2--+
      [VALUE CKT]                         |
                                     [SUBTRACT]--V3
         +-->[LOG]--V2------------->(−)   |
         |                           [ANTILOG]--V4
         |                                |
         |                         [LOW-PASS RC]
         |                         [ (averager) ]
         |                                |
         +--------------------------------+--> Vo

Derivation

  1. Precision full-wave rectifier gives |Vi| (log amp needs a positive input; |Vi|² = Vi²).
  2. Log amp and gain of 2:
V1 = 2V_T ln(|Vi|/Vref) = V_T ln(Vi²/Vref²)
  1. Log of the output (feedback):
V2 = V_T ln(Vo/Vref)
  1. Subtractor:
V3 = V1 − V2 = V_T ln( Vi² / (Vref·Vo) )
  1. Antilog:
V4 = Vref·e^(V3/V_T) = Vi² / Vo
  1. Low-pass filter (time constant much larger than signal period) takes the average:
Vo = avg(Vi²/Vo) = avg(Vi²)/Vo
Vo = √avg(Vi²)

For a sinusoid Vi = Vm sin ωt

avg(Vi²) = (1/T)∫₀ᵀ Vm² sin²ωt dt
         = (Vm²/T)∫₀ᵀ (1 − cos2ωt)/2 dt
         = Vm²/2
Vo = √(Vm²/2) = Vm/√2 = 0.707·Vm

Example: for Vi = 10 sin ωt V, the circuit gives Vo = 7.07 V DC.

Advantages: it is a true RMS converter (correct for any waveform, not only sine), and its dynamic range is wide because the internal signal Vi²/Vo stays close to Vi in size.

✓ Answer checked by Claude Opus 5.5 (high effort)

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