Passive Integrator and Differentiator Circuits

One of the fundamental principles of calculus is a process called integration. This principle is important to understand because it is manifested in the behavior of inductance. Thankfully, there are more familiar physical systems which also manifest the process of integration, making it easier to comprehend.

If we introduce a constant flow of water into a cylindrical tank with water, the water level inside that tank will rise at a constant rate over time:

In calculus terms, we would say that the tank integrates water flow into water height. That is, one quantity (flow) dictates the rate-of-change over time of another quantity (height).

Like the water tank, electrical inductance also exhibits the phenomenon of integration with respect to time. Which electrical quantity (voltage or current) dictates the rate-of-change over time of which other quantity (voltage or current) in an inductance? Or, to re-phrase the question, which quantity (voltage or current), when maintained at a constant value, results in which other quantity (current or voltage) steadily ramping either up or down over time?

Describe what happens to the capacitor voltage in this circuit over time as it is charged by the constant-current source:

Now, determine the ideal values for V and R that will result in similar behavior in a capacitor circuit powered by a voltage source rather than a current source:

Your answers will, of course, be qualitative rather than quantitative. Explain whether or not the time constant for the voltage-source-powered circuit ought to be large or small, and why.

It is relatively easy to design and build an electronic circuit to make square-wave voltage signals. More difficult to engineer is a circuit that directly generates triangle-wave signals. A common approach in electronic design when triangle waves are needed for an application is to connect a passive integrator circuit to the output of a square-wave oscillator, like this:

Anyone familiar with RC circuits will realize, however, that a passive integrator will not output a true triangle wave, but rather it will output a waveshape with “rounded” leading and trailing edges:

What can be done with the values of R and C to best approximate a true triangle wave? What variable must be compromised to achieve the most linear edges on the integrator output waveform? Explain why this is so.