Multifunction converter using log-antilog

A multifunction converter is a log-antilog circuit whose output is

Vo = Vy · (Vz / Vx)^m

By choosing the inputs and the exponent m, the same circuit performs multiplication, division, squaring, square-root and other powers (e.g. the Burr-Brown 4302 IC).

Circuit realization

Vz -->[LOG AMP]--V1-->(+)
                      [SUBTRACTOR]--V3-->[GAIN m]--V4
Vx -->[LOG AMP]--V2-->(−)                (pot/R ratio)
                                              |
Vy -->[LOG AMP]--V5---------------->[SUMMER]<-+
                                       |
                                       V6
                                       |
                                  [ANTILOG AMP]--> Vo

Working and derivation

Take each log amp as V = V_T ln(Vin/Vref), with Vref = R·Is (signs handled by inverting stages).

1. Log of Vz and Vx:

V1 = V_T ln(Vz/Vref)
V2 = V_T ln(Vx/Vref)

2. Subtractor:

V3 = V1 − V2 = V_T ln(Vz/Vx)

3. Gain stage (voltage divider or amplifier, gain m):

V4 = m·V_T ln(Vz/Vx) = V_T ln[(Vz/Vx)^m]

4. Add log of Vy:

V6 = V4 + V_T ln(Vy/Vref)
   = V_T ln[ (Vy/Vref)·(Vz/Vx)^m ]

5. Antilog amp:

Vo = Vref · e^(V6/V_T)
Vo = Vy · (Vz/Vx)^m

Functions obtained

m < 1 is set with a resistor divider; m > 1 with a non-inverting amplifier. Matched transistors cancel Is, and all inputs must be positive.