Your design task, should you decide to accept it: given an input voltage, square it. Ok, that’s too hard since squaring 8 volts would give you 64 volts, so let’s say the output should be 10% of the square, so 8 volts in would result in 6.4V. How do you do it? [Engineering Prof.] knows how and will show you what you can do in the video below.
The circuit uses two op amps and some transistors. However, the transistors are used in a way that depends on the temperature, so it is important to use a transistor array so they are matched and will all be at the same temperature.
The math depends on the fact that the transistor response has a natural log term in it, and the property that the sum of two logs is the same as the log of the product of the numbers.
Because of the matching transistors, many of the terms in the equation cancel out. Because the transistors are current devices, the transistor circuit’s output current is the input current squared divided by the output transistor’s collector current. Then it is just a matter of converting the voltage to a current and back again using the right scaling.
There’s more to it, of course, but that’s the gist of it. You can dig into the math by watching the video. If the KCL references are fuzzy for you, here’s a refresher. Squaring a voltage would be pretty important for an analog computer.
If you want to use transistors, an LM394 is the way to go. Motorola used to make an analog multiplier IC; not sure if they still do.
I used that exact squaring core (but with a current out, not voltage) in a Ku band PiN diode attenuator to control satellite uplink power (~1989 perhaps?). The shunt PiN diode attenuation varied non-linearly with the current, become less sensitive (in terms of dB/unit current) at higher currents; this circuit compensated for that and allowed me to get a roughly constant dB per DAC step from my 12 bit DAC.
BTW, higher resolution DACs were prohibitively expensive back then.
I think I used an LM3046 for the matched transistors.
I didn’t spot it in my quick run through the video, but that circuit really needs protection diodes for the BJT BE junctions, to prevent damage if the opamp outputs (for whatever reason) go to the wrong rail.
Looking at THAT 300P14-U, ADI MAT14ARZ-R7 and friends at 10+ bucks each, this starts smelling a lot like strapping needle probes to your 3D printer to turn it into a transistor binning device, heated bed and all.
Well, that would be a good hack.