# RC Circuit

“RC” refers to the resistor (R) and capacitor (C) making up the circuit. The circuit diagram shows how the resistor and capacitor are connected – the resistor between $V_{in}$ and $V_{out}$ and the capacitor from $V_{out}$ to $Gnd$. Different assignments of $V_{in}$ and $V_{out}$ would lead to different (but related) input/output relationships.

## Qualitative Behavior

The time series plot shows how the RC circuit behaves when $V_{in}$ steps from 0V to 1V. In general:

The output voltage of the RC circuit will decay exponentially towards the input voltage.

Current flows through the resistor and into (or out of) the capacitor as long as the capacitor voltage ($V_{out}$) differs from $V_{in}$. The greater the difference between $V_{in}$ and $V_{out}$, the more current flows, and the more quickly $V_{out}$ approaches $V_{in}$.

## Quantitative Behavior

The capacitor law $\left(i = C\frac{dV}{dt}\right)$ tells us that the current through the capacitor is proportional to the change in voltage across the capacitor. Integrating the capacitor law gives us

which shows that the voltage across the capacitor is proportional to the total current that has flowed into the capacitor over time. Current will flow through the capacitor until it has charged to the input voltage.

The current through the resistor is the same as the current through the capacitor, so we can write

This yields the differential equation

If we take $V_{in}$ to be constant with the initial condition $V_{out}(t=0) = 0$, we can solve this equation for $V_{out}(t)$:

Solving this differential equation in the time domain is outside the scope of the class (since we’re going to adopt a simpler and more powerful method of analyzing circuits soon), but one method of solving this equation is at the end of this recap.