Analog Multipliers

[John Calhoun]

If you convert a pair of voltages to their logarithm, you can use a simple adder (summer) op-amp circuit to "multiply" the logs of the voltages—required you "anti-log" the result.

(Related: slide-rules multiplied exactly this way: by taking advantage of adding the logs of two numbers, then getting the anti-log of this sum when reading from the rule.)

Gemini tells me that using a log/anti-log circuit is problematic for a number of reasons. You cannot get the logarithm of a negative number for one. (I would have to run the numbers to see if this mattered if you assume a positive "virtual ground" that represents machine-zero for your analog circuit.)

Gemini also suggests that because the transistors/diodes in the circuit are affected by temperature you'll discard precision. For a "hobbyist" analog computer I wonder how significant temperature is. For example, if I want to demonstrate a Lorenz Attractor (a chaos function that results in a rat's nest of traces racing around on the oscilloscope), would it matter the multiplier were a little off? I think you would still get "chaos".

I've recovered a slide preso (lecture notes) on using the Log/Anti-log circuit to multiply two voltages. I have attached it here. (Original url was: http://www.electronics.dit.ie/staff/ypanarin/Lecture%20Notes/DT021-4/6LogAntiLogAmplifiers.pdf)

It goes into the log/anti-log circuit and even addresses temperature compensation in a proposed circuit. So perhaps positive/positive quadrant multiplication is perfectly fine to perform with an log/anti-log circuit.

[John Calhoun]

Forrest M. Mims III had a regular column in the electronic hobbyist magazine: Popular Electronics.

He discussed but did not build a log/anti-log multiplier in his Experimenter's Corner in the February 1979 issue. (It begins on page 80.)

It does no temperature compensation.

Part one of his discussion of analog computing was in the previous Experimenter's Corner in the January 1979 issue. (It begins on page 81.)

[John Calhoun]

A so-called low-cost analog multiplier on a chip is the Analog Devices AD633.

It is not cheap though—about $15 a chip is sourced from LCSC for PCB manufacture at the time I am posting this.

(A hobbyist computer that I am building though is powered from 0V—5VDC USB power and as such the AD633 is not suitable.)

[John Calhoun]

Gemini also suggests a Gilbert Cell circuit to multiply voltages. I didn't dig into the Gilbert Cell as Gemini claims it "requires precise transistor matching".

In any event here is a YouTube video on the principle.

[John Calhoun]

Gemini seemed to push the Quarter-Square Multiplier topology for performing multiplication. It's interesting, I did not know that:

X*Y = ​[(X+Y)^2−(X−Y)^2] / 4

There is curiously little online about the Quarter-Square Multiplier, but this YouTube video was recommended. I may try to breadboard the circuit.

[John Calhoun]

Finally Gemini mentioned the PWM Multiplier. Again I did little research into this. There is some brief discussion on the EEVBlog.

[John Calhoun]

Dr. Bernd Ulmann refers to the Quarter-Square Multiplier as a Parabola Multiplier on page 42 of his Analog and Hybrid Computing slide deck.

[John Calhoun]

A PDF from Analog Devices that talks generally about Analog Multiplier Basics including the Gilbert Cell, log/anti-log, etc.

[John Calhoun]

For a hobbyist analog computer that operates 0-5VDC (USB), it looks as though the Gilbert Cell is a non-starter (as an LLM explained to me: the transistors and their activation voltages will eat the 5V quickly leaving no operating range).

Nonetheless, I created a KiCad schematic (PDF attached) of the Gilbert cell demonstrated in the previous YouTube video.

[John Calhoun]

As chance would have it, an article about the Gilbert cell multiplier just appeared.