# Ring Rant

I’ve been reading about ring modulators, and now I want to put on my Maximum Insufferable Pedantry hat and write about them.

That? That’s not a ring modulator.

I love the ARP 2600. But its “ring modulator” is no such thing. I’m not sure what it is, the schematics don’t seem to be available, but it’s not a ring modulator.

I’ve read the ARP Odyssey “ring modulator” is based on NAND chips, so it isn’t one either.

All right, all right, the descriptivist viewpoint is, sure, these things are called ring modulators, so they’re ring modulators. But the pedant prescriptivist says no, a ring modulator is a specific thing, and these aren’t them.

Now, some people say ring modulators are four quadrant amplifiers, but not all four quadrant amplifiers are ring modulators. But that’s not strictly true, either. So what is true?

Consider a VCA. You put a signal, maybe in the range -5 V to 5 V, into one input, and a control voltage, 0 to 5 V, into the other. What you get out is the signal, but it’s reduced by a factor of CV/5: If the CV is 0 you get nothing, if it’s 5 you get the full amplitude signal, if it’s in between you get an attenuated signal.

If you think about it, what it’s doing is multiplying the signal times the CV — or rather, times 1/5 of the CV. So if the signal is 2 V and the CV is 3 V the output is $2\times 3/5 = 1.2$ V, and if the signal is -2 V the output is $-2\times 3/5 = -1.2$ V.

But this description breaks down if you try to feed it a negative CV. Then you still get nothing on the output. So this is what’s known as a two quadrant amplifier: If you plot the valid range of CV versus the valid range of signal, it’s in the first two quadrants of Cartesian coordinates.

A four quadrant amplifier, on the other hand, is something whose output is (proportional to) signal times CV in all four quadrants. You could put in -2 V signal and -3 V CV and get $-2\times -3/5 = 1.2$ V out, for instance. And since both inputs can be positive or negative, and since the output is just their product, you could just as well put CV into the first input and signal into the second, or audio signals into both, or whatever.

Suppose you put sine waves into both: $A_1\sin(2\pi f_1t)$ into the first and $A_2\sin(2\pi f_2t)$ into the second; then the output is proportional to $A_1A_2\sin(2\pi f_1t)\sin(2\pi f_2t)$. But there’s a trig identity, $\sin(a)\sin(b) = (\cos(a-b)-\cos(a+b))/2$, so that output really is a combination of two sine (with a phase) waves with frequencies $f_1-f_2$ and $f_1+f_2$. In other words, the output has no components with the same frequencies as the inputs, but it does have the sum and difference of the input frequencies. And if the two inputs are square or ramp or triangle waves, or any periodic function other than a sine, then the output contains none of the frequencies in the harmonic spectrum of either (unless $f_1/f_2$ is a simple ratio like $1/2$) but all of the differences and sums of the harmonic frequencies of the two inputs. Which is a complicated mess of frequencies, not in general all whole number multiples of a fundamental frequency, so it’s not a simple tone — it’s a complicated sound, often bell-like.

People sometimes use “four quadrant amplifier” and “ring modulator” as synonyms. But, strictly, they’re not. In fact, strictly, they’re distinct: a 4QA is not a RM, and a RM is not a 4QA.

## Ringing

Here’s what a ring modulator is: A circuit like this:

Two transformers, four diodes. Pretty simple. Not even a power supply! You put two signals in — they’re usually called signal and carrier — and output comes out.

How does it work? The carrier is supposed to be significantly larger in amplitude than the signal. So, if we neglect the signal, then when the carrier is positive, diodes D1 and D3 are forward biased — so they’re conducting — and D2 and D4 are reverse biased — they’re not conducting. In that case an oscillating signal input produces a similar signal in the XFMR1 secondary coil, which is connected parallel to the XFMR2 primary so current flows the opposite direction in the latter, and so the output is the input, inverted. On the other hand, if the carrier is negative, then D2 and D4 are conducting and D1 and D3 are not. Then if you trace the conducting paths you find the currents in the XFMR1 secondary and XFMR2 primary are in the same direction, so the output is the input, non inverted.

Then if the carrier frequency is higher than the signal frequency, a sine wave signal like this:

with a carrier like this:

becomes an output signal like this:

Which if you think about it is always either 1 or -1 times the input signal, so it’s the input signal times a square wave at the carrier frequency. Which is not the input times the carrier, unless the carrier actually is a square wave, so it’s not really a four quadrant multiplier. The output always contains whatever harmonics are present in the signal plus or minus all odd harmonics of the carrier frequency — and no even harmonics of the carrier — regardless of whether they’re present in the carrier signal.

Well, that’s the simple version. One complication is that we’ve assumed ideal rectifiers, and real diodes have a forward voltage; another is that if the carrier isn’t a square wave it’s sometimes small and the diodes may be more influenced by the signal than by the carrier. When both carrier and signal are small you might have all four diodes non conducting. And if the diodes aren’t well matched you can get some asymmetries in the response. With real world components you can get the signal and/or carrier frequencies bleeding through into the output, and you can get some amplitude loss.

Which is why other 4QA or 4QA-like circuits have largely displaced ring modulators, both in synthesizers and in other modulation applications. But true ring modulation, imperfections and all, is a classic piece of the analog synthesis picture, and RMs are ridiculously simple circuits (at least on paper!), so there’s still a place for them.