basic question about integrators in a loop (circle test)

Christopher Howard · 2025-07-19T20:29:38+00:00

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