When I was young the first “computer” I ever owned was an analog computer built from a kit. It had a sloped plastic case which had three knobs with large numerical scales around them and a small center-null meter. To operate it I would dial in two numbers as indicated by the scales and then adjust the “answer” by rotating the third dial until the little meter centered. Underneath there was a small handful of components wired on a terminal strip including two or three transistors.
In thinking back about that relic from the early 1970’s there was a moment when I assumed they may have been using the transistors as logarithmic amplifiers meaning that it was able to multiply electronically. After a few minutes of thought I came to the conclusion that it was probably much simpler and was most likely a Wheatstone Bridge. That doesn’t mean it couldn’t multiply, it was probably the printed scales that were logarithmic, much like a slide rule.
Did someone just ask what a slide rule was? Let me explain further for anyone under 50. If you watch the video footage or movies about the Apollo Space Program you won’t see any anyone carrying a hand calculator, they didn’t exist yet. Yet the navigation guys in the first row of Mission Control known aptly as “the trench”, could quickly calculate a position or vector to within a couple of decimal places, and they did it using sliding piece of bamboo or aluminum with numbers printed on them.
I inherited my first slide rule from my Dad who had used his in college. That slide rule has 32 or 33 scales on it, each a different mathematical computation, and was made of bamboo, a fact my Dad pointed out that during the winter the guys with the aluminum slide rules had a harder time as the cold would cause them to get stuck. I used a slide rule in my 10th grade chemistry class, and by the 12th grade everybody had “four banger” calculators. You couldn’t open a newspaper without seeing advertisement for the calculators as they quickly dropped from $100-$400 down to $29.
Jump forward to today as I sit in front of a breadboard with a couple of analog Op Amp modules and a couple of little voltmeters. There are three basic styles of DC amplification with Op Amps; inverting, non-inverting and differencing depending on which inputs you inject the signal. In the case of differencing, the signal is injected into both inputs.
It’s easy to see then how to add two voltages together, I go so far as to print the formulas on my own PCB’s to help me remember but the determining factor is there will be a voltage divider formed by the resistors that is connected to one or more inputs. Simply put, a circuit that feeds a proportion of the signal back into the inverting input will have a gain set by that proportion.
Quickly becoming bored with addition and subtraction we move up to multiplication. Using a venerable Analog Devices AD633 four quadrant multiplier, it’s easy to show multiplication in action and wiring the same voltage to both inputs results in square/square root. What makes the AD633 so usable is it has been engineered to be somewhat precise about utilizing the internal logarithmic behavior and then includes gain so that the small usable area of the curve becomes a bigger usable area… big enough to see on my little three digit voltage displays.
And finally we take multiplication back to AC where the fun is; injecting a tone in one terminal and a control voltage in the other we recreate a Voltage Controlled Gain block like might be found in an analog synthesizer or as part of an Automatic Gain Control (AGC) system. This is the heart of Amplitude Modulation as used in radios and complex audio synthesis such as ring modulators in 1970’s synthesizers.
The AD633 does a good job of multiplying both DC and AC signals. Basically it uses the inherent logarithmic characteristics of diode/transistors junctions but is also surrounded by carefully trimmed parts (laser trimmed resistors) and built in amplification to make the most use of the desired log responses.
Photo of Science Fair Computer courtesy of [Dan Mathias] of Futurebots