Understanding Time-Dependent Behavior in Capacitor–Resistor Circuits

RC circuit

A circuit containing at least one resistor (R) and one capacitor (C), notable because currents and voltages can change with time even with a DC source.

Resistor (R)

A circuit element that relates voltage drop to current; it controls how fast charge can flow in an RC transient.

Capacitor (C)

A device that stores charge on two separated plates; its voltage depends on stored charge and it introduces time dependence in circuits.

Ohm’s law (resistor voltage relation)

The voltage across a resistor is proportional to current: v_R = iR.

Capacitor charge–voltage relation

The charge on a capacitor is proportional to its voltage: $q = C v_C$.

Capacitor current relation

The current through a capacitor is proportional to the rate of change of its voltage: $i = C \frac{dv_C}{dt}$.

Passive sign convention

A sign convention where an element’s voltage is taken positive when current enters the element’s labeled positive terminal, helping keep KVL equations consistent.

Kirchhoff’s loop rule (KVL)

The sum of potential changes around a closed loop is zero; used to set up the differential equations for RC charging/discharging.

RC transient

The time-dependent charging or discharging behavior in an RC circuit, typically described by exponential functions.

Time constant (τ)

The characteristic time scale for an RC circuit: $\tau = RC$ (or $\tau = R_{eq} C$ in more complex circuits).

Meaning of one time constant

After time τ, an exponential quantity has moved about 63% of the way from its initial value toward its final value (or decayed to about 37% of its initial value).

Charging differential equation (series RC with battery)

For capacitor voltage during charging: \frac{dv_C}{dt} = \frac{\text{ℰ} - v_C}{RC}.

Charging capacitor voltage solution

For $v_C(0)=0$ in series with a DC battery: v_C(t) = \text{ℰ}(1 - e^{-t/(RC)}).

Charging capacitor charge solution

During charging from 0 V: q(t) = C\text{ℰ}(1 - e^{-t/(RC)}).

Charging current solution

During charging: i(t) = \frac{\text{ℰ}}{R} e^{-t/(RC)}, which decays exponentially to 0.

Discharging differential equation (RC only)

With no battery: $\frac{dv_C}{dt} = -\frac{v_C}{RC}$.

Discharging capacitor voltage solution

If $v_C(0) = V_0$: $v_C(t) = V_0 e^{-t/(RC)}$.

Discharging current sign interpretation

Using $i = C \frac{dv_C}{dt}$ can yield $i(t) = -\frac{V_0}{R} e^{-t/(RC)}$; a negative sign usually means the true current is opposite the chosen positive direction.

General approach-to-final form (unifying equation)

A first-order RC quantity moves exponentially from initial to final value: $v_C(t)=v_{\infty} + (v_0 - v_{\infty}) e^{-t/\tau}$.

DC steady state (for capacitors)

Long after switching with DC sources, $\frac{dv_C}{dt} = 0$ so $i_C = 0$; an ideal capacitor behaves like an open circuit.

Continuity of capacitor voltage

A capacitor’s voltage cannot change instantaneously, so $v_C(0^+) = v_C(0^-)$ at a switching event.

Equivalent resistance seen by the capacitor (R_eq)

The resistance between the capacitor’s terminals in the post-switch configuration (with sources turned off appropriately), used in $\tau = R_eq C$.

Turning off independent sources (for finding R_eq)

To compute R_eq: replace independent voltage sources with shorts and independent current sources with opens, then find the resistance seen at the capacitor terminals.

Energy stored in a capacitor

Energy in the electric field of a capacitor at voltage $V$: $U_C = \frac{1}{2}CV^2$.

Steady-state capacitor voltage in a divider (capacitor in parallel with R_2)

With $R_1$ and $R_2$ as a series divider across battery \text{ℰ}, and the capacitor across R_2: v_C(∞) = \text{ℰ} \frac{R_2}{R_1 + R_2}.