Capacitors

The next component we will look at is capacitors. Fundamentally, a capacitor is two metal plates separated by a dielectric or insulating material. What happens when we apply a voltage source across a capacitor like so?

Before the capacitor is connected, the plates of the capacitor will be neutrally charged. However, once a voltage source is connected, charge will be pushed/pulled as much as possible until one of the plates is positively charged and one of the plates is negatively charged. This happens nearly instantly while the voltage source is being connected. Once this connection is complete, the circuit is in steady state, no charge moves, and no current flows.

The equation for a capacitor in this steady state is shown below.
Q = CV
(Note: Q is technically the absolute value of the charge) As the equation shows, there are two things that allow more charge to accumulate onto a capacitor. Either a higher value of capacitance, that is more room for charge. Or a higher voltage, that is more force pushing the charge onto the plates. As already mentioned, when the voltage across a capacitor is constant, there won't be any current through it, so it won't be doing anything. A capacitor in steady state is basically an open circuit. To find out the transient equation behavior of a capacitor we can differentiate the equation above with respect to time. The derivative of charge with respect to time is current, and the capacitance stays constant. This yields the following equation:
I = C(dV/dt)
As we can see, the current through a capacitor is equal to the derivative of the voltage across it multiplied by its capacitance. The most basic example of this is a ramping voltage across a capacitor. If the ramp is linear, then there will be a constant current through the capacitor. Likewise, a constant current into a capacitor will lead to a linearly ramping voltage across it.

Capacitors in Parallel and Series

Like resistors, you can also reduce series and parallel combinations of capacitors. The series reduction formula is as follows:
Ceq = 1/(1/C1 + 1/C2 + ...)
And the parallel reduction formula:
Ceq = C1 + C2 + ...
As you can see, this is opposite of how resistors are reduced. I won't bother deriving these here, but it is fairly self evident why this is the case for parallel capacitors, and a little less self evident for series. So you can think about it.

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