Can anyone recommend a good resource to understand how a capacitor works as an integrator and an inductor works as differentiator? I'm looking for something to show me how the laplace transform comes into play, resulting in 1/(sC) and (sL) for impedances. Thanks!
Electronic – Laplace Relation to Capacitors and Inductors
capacitorinductorlaplace transform
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Best Answer
Integration is a continuous form of summation. We take infinitesimal pieces of a function and, well, integrate them to make a sum.
A capacitor physically integrates by packing electrons and developing charges across its plates. The current charge on a capacitor is the result of all previous current flow: the integral of the current function from minus infinity to the current time.
Think of the capacitor's charge as a bank account balance. The bank account current balance is the result of the opening balance (initial charge) plus the sum of all of the transactions since opening (all tiny pieces of the current flow this way and that way, added together).
An inductor works differentially because whereas it develops a magnetic field proportional to the current flowing through it, only changes in the magnetic field generate a voltage. An inductor with a steady DC current flowing through it is bathed in a magnetic field, but that field does not change and so the inductor does not develop a voltage. To obtain changes in the magnetic field, we must change the current. The current is a function of time, and a derivative of any such a function gives us another function which informs us of the rate of change in time. For instance, derivative of position is velocity, and the derivative of that is acceleration. If you visualize current flow as a speed, then the inductor voltage indicates its acceleration, so to speak.