The Integrator Circuit - analog computers, Buchla, Serge & eurorack modules

The Op-amp integrator is a very important electronic circuit which I come across again and again.
The West Coast style of synthesis uses it a lot in modules like function generators, and slope generators.
In the synth world, what is possibly the most popular of all modules, the Make Noise Maths I think uses this circuit. (let me know if I'm mistaken).

The Fritz chaos circuits are also made of integrators.

The op-amp integrator is, as the name implies, based on the operational amplifier (op-amp). It performs the mathematical operation of integration (with respect to time). The magnitude of the output voltage is proportional to amplitude as well as the length of time a voltage is present at its input.

This probably sounds like lots of gobbledy gook to most people. Sorry.
To simplify all this nerd stuff.....
Usually operational amplifiers are used as part of a feedback loop circuit involving a resistor.
 Something like this:

 But what if we swap the resistor with a capacitor?
We now have a RC network This new circuit is commonly called a Op-Amp Integrator

Electronic integrator circuits can be found  in old analogue computers:

The COMDYNA GP-6

Integrators, mutipliers, dividers, counters, function generators, operational amplifiers, differential amplifiers, voltmeters... etc etc... These old analog computers had many of the modules you will commonly find in a modular synthesizer. They are arguably the inspiration for many of today's synths.

 Digital systems Lab

Anyway, probably my favorite module, the Buchla 281 QUAD FUNCTION GENERATOR uses it, as does the Buchla 257..... this is a voltage processor. Don experimented with integrators well before the 257. He created the Buchla 155 Dual Integrator module for  the system 100. These are extremely rare and I have never even seen a vintage one.
---------------------------------
"Model 155 Dual Integrator
Produces continuous control voltage functions when used in
conjunction with sources of discrete control voltages (e.g. keyboards, sequencers).
Positive and negative  slopes  may  be  individually  and  continuously varied from
15 volts in .0025 seconds to 15volts in 10 seconds; either or both slopes may be
voltage controlled. Particularly useful for generating complex voltage controlled envelopes,
frequency glides, and repetitive control functions." 

(from a catelog for the 1992 Ars Electronica exhibit Eigenwelt der Apparatewelt.
Pioneers of Electronic Art, edited by David Dunn)

--------------------------------------------------------------------

Buchla Dual voltage processor . Buchla 257.

It's looks a bit confusing. M is a control voltage. "It provides the capability of transferring control from one applied voltage (Vb) to another (Vc). The algebraic manipulations include addition,  subtraction, scaling, inversion, multiplication". You can crossfade & use this module as a VCA.

There is also a intergrator section in the lower right corner of the Buchla 266

 In the Serge world the DUSG (Dual Universal Slope Generator)... often referred to as the "Swiss army knife" is all about the integrator. The Slope generator is an amazing module. 

These Positive & Negative Slews preceded the DUSG

Depending on how one patches it, the DUSG can be a VCLFO,  a envelope generator; a oscillator; a  slew generator; a voltage and audio mixer; an attenuator and inverter; an envelope follower; a comparator; a VC pulse delay, a audio processor,.... whew !!! ........... it's pure genius !!!!

Bananalogue put out the  VCS almost a decade ago I think. This is based on a Serge slope generator

MATHS = DUSG/Buchla 281 and Buchla 257.???

The Befaco Rampage is another variation of the DUSG.

Just released during NAMM 2019 is the Verbos Control Voltage Processor:
It's a bit of DUSG and Buchla 257. Can't wait to get one.

So to sum up....in most op amp circuits, the feedback that is used is usually a resistor. However for the integrator this is not the case - the component providing the feedback between the output and input of the op amp is a capacitor.

It works a bit like this:

When a voltage is initially applied to Vin, the capacitor has almost zero resistance.... it acts like there is a short circuit.... so no current will flow through the op-amp. A virtual earth exists at the op-amp’s inverting input.

As the capacitor starts to charge, negative feedback forces the op-amp to produce an output voltage to maintain that virtual earth at the op-amp’s inverting input. The rate at which the output voltage increases (the rate of change) is determined by the value of the resistor and the capacitor, “RC time constant“.

Once the capacitor fully charges, it acts like a open circuit.

The flow of current then stops.

If we apply a rapidly changing signal to the input then the capacitor will charge and discharge quickly. At higher frequencies, the capacitor has less time to fully charge. This type of circuit is also known as a Ramp Generator. 

A ramp generator is a way of converting a square wave to a sawtooth.... but that's another story.