Almost no modular synth patch can go without the use of a low-frequency oscillator (LFO). So I decided to build one myself to get my hands dirty on building modules for my Eurorack system. It is based on a fairly simple relaxation oscillator circuit.

The circuit below shows how a simple relaxation oscillator works. We use an operational amplifier as a comparator, feeding it 12V and -12V. Now the maximum range of the output of the op amp is somewhat lower, around -10V to 10V. This is still way too much for our purpose.

A typical LFO in Eurorack format will output -5V to 5V. So we use a voltage divider to bring it in the suitable range with two 4.7k resistors. We call these hysteresis resistors. Combined they set the level on which the op-amp will switch. In our case -5V or 5V. Without them, the op-amp won’t switch reliably.

The output of the voltage divider is fed back into the non-inverting input (+) of the op-amp.

The output of the op-amp is also fed back to the inverting input (-) through a 22k resistor. This resistor, together with the 100nF capacitor, sets the frequency of the oscillation. See below for a calculation example.

If we tap into the circuit on the green node, we get a square wave output. And if we tap into the circuit on the red node we get the desired triangle output.

Well, not an exact triangle wave, more like triangle-ish. But good enough for our purpose.

If you want a straight linear triangle waveform, you can charge the capacitor with a constant current source.

## How does a relaxation oscillator work

An op-amp always tries to keep voltage levels the same at both inputs. It can only do this by changing its output. In this case, the op-amp will switch to the maximum opposite output voltage whenever the voltage at the inverting input reaches the value of the non-inverting input.

So as soon as the voltage at the minus terminal hits 5V, the op-amp will switch to -10V. Therefore, the plus terminal will switch to -5V.

The voltage at the minus terminal will slowly follow, but it takes time to catch up. This is because of the charging and discharging of the capacitor with the value of the resistor combined.

## How to calculate the frequency of the oscillator

If we want to calculate the frequency of the relaxation oscillator, we need to take the values of the voltage divider into account.

This formula can be simplified if we keep the hysteresis resistors equal in value (like in our example circuit).

So the time it takes for the whole cycle to complete is…

A more in-depth explanation on relaxation oscillators can be found in this helpful article on Analog Zoo

So much for the first step in building the LFO module. Next up: putting the LFO on the breadboard to see if the circuit actually works 😬

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