Need for log-antilog circuits; analog multiplier

Why log–antilog circuits are required

Logarithms turn multiplication into addition and powers into multiplication by a constant:

ln(xy)  = ln x + ln y
ln(x/y) = ln x − ln y
ln(xⁿ)  = n ln x

Adders, subtractors and gain stages are easy to build with op-amps. So log and antilog amplifiers let simple linear circuits perform non-linear operations:

• Analog multiplication and division of signals
• Powers and roots (square, square root, xⁿ), RMS and vector magnitude
• Compression of signals with a very wide dynamic range (5–6 decades) into a small voltage range, for example in sonar, radar, audio and dB meters
• Linearizing sensors with exponential responses, and pH or light measurement

Antilog (analog) multiplier

Building blocks used (all voltages in volts, Vref = 1 V, K = scale factor such as VT or 1 V/decade):

• Log amp (L): output = −K ln(Vin), for Vin > 0
• Antilog amp (E): output = exp(Vin/K)
• Inverting amp of gain −m: output = −m × input

 V1 -> [LOG A1] --Va--[R]--+
                           |       R
 V2 -> [LOG A2] --Vb--[R]--+--+--/\/\/--+
                              |         |
                            (-)\  A3    |
                       GND--(+)  >------+--> Vs
 Vs -> [ANTILOG A4] -> Vo

Working:

1. Log stages: A1 and A2 (each with a transistor in feedback) give:

Va = −K ln V1
Vb = −K ln V2

2. Inverting summer A3 with all resistors R:

Vs = −(Va + Vb) = K (ln V1 + ln V2) = K ln(V1·V2)

3. Antilog stage A4:

Vo = exp(Vs/K) = exp( ln(V1·V2) ) = V1·V2

With a general reference voltage the scaled result is Vo = V1·V2/Vref. For example, V1 = 2 V and V2 = 3 V give Vo = 6 V (Vref = 1 V).

Division: replace the summer with a difference amplifier. Then Vs = K ln(V1/V2) and Vo = V1/V2.

Limitation: the inputs must be positive (one-quadrant operation). Matched transistors and temperature-compensating resistors are needed for accuracy.