Re: basic question about integrators in a loop (circle test)

On Sat, 19 Jul 2025 12:29:38 -0800, Christopher Howard
<christopher@librehacker.com> wrote:

Hi, I'm continuing my exploration of electrical analog computing. I
built one analog computer and have also been studying schematics for
some historical analog computers. An integrator is modeled with an
inverting op amp with a feedback capacitor.
>
In the circle test, a simple loop is constructed modeling the
differential equation y'' = -y. This is implemented with
>
Integrator A -> Integrator B -> inverter -> loop back to input of
Integrator A.
>
In ideal integration, the function y — which is the output of Integrator
B — would be a steady sinusoidal wave that does not increase or decrease
in amplitude. However, in practice the wave will either grow or decay in
amplitude, once the integrators are started. I've found that, with
smaller capacity feedback capacitors, the wave tends to grow, and with
larger ones, it tends to decay. One can see this as a growing or
shrinking circle on an oscilloscope, by feeding the outputs from both
integrators into an XY scope display.
>
Could somebody please explain why this happens? I'm not grasping the
basic cause of this. I've been trying to read up about op amp
(in)stability, like in amplifiers and voltage followers, but I'm not
seeing if/how there is a connection between that and what is going on
here.

I was recently talking to my design center kids about diferential
equations and circuits. All three are CE/EE grads or students. I
mentioned "initial conditions" and one of them recalled hearing the
term.  [1]

Your equation has an infinite number of solutions. One is Y=0, cold
and dead. One is a 1-volt sine wave. Another is a megavolt sine wave.
Which it does depends on the initial condition, what it's already
doing when you walk into the room. It will keep doing that. You can
Spice that.

In real life, there are no ideal integrators, so the loop has
additional phase shifts so there are different solutions to the
equation; specifically decaying or increasing sine waves.

To make a decent oscillator, you need an increasing amplitude circuit
and some sort of active amplitude limiter. Opamp clipping is the
cheapest limiter.


[1] Computer Engineering is a kinda oxymoron.