RC and RL Exponential Responses

Summary of Equations

Exponential responses of capacitors and inductors

	Discharging	Charging	Time Constant
Capacitor	$v_C(t)=V_0{e}^{-\frac{t}{RC}}$	$v_C(t)=V_0(1-e^{-\frac{t}{RC}})$	$RC$
Inductor	$i_L(t)=\frac{V_0}{R}{e}^{-\frac{R}{L}t}$	$i_L(t)=\frac{V_0}{R}(1-e^{-\frac{R}{L}t})$	$\frac{L}{R}$

RC Circuits

Discharging

Consider the following circuit:

File:RC discharge schematic.jpg

In the circuit, the capacitor is initially charged and has voltage $V_0$ aross it, and the switch is initially open. At time $t=0$, we close the circuit and the capacitor will discharge through the resistor. The voltage across a capacitor discharging through a resistor as a function of time is given as:

$v_C(t)=V_0{e}^{-\frac{t}{RC}}$

Charging

If the capacitor is initially uncharged and we want to charge it by inserting a voltage source $V_s$ in the RC circuit:

File:RC charge schematic.jpg

The voltage across the capacitor is given by:

$v_C(t)=V_0(1-e^{-\frac{t}{RC}})$

where $V_0=V_s$.

The term RC is the resistance of the resistor multiplied by the capacitance of the capacitor, and known as the time constant, which is a unit of time. The value of the function will be 63% of the final value at $t=1RC$, and over 99.99% of the final value at $t=5RC$.

The magnitudes of the voltage and current of the capacitor in the circuits above are shown in the graphs below:

Capacitor

	Voltage (magnitude)	Current (magnitude)
Charge	File:RC charge voltage.jpg	File:RC charge current.jpg
Discharge	File:RC discharge voltage.jpg	File:RC discharge current.jpg

RL Circuits

Discharging

In the following circuit, the inductor initially has current $I_0=V_s/R$ flowing through it; we replace the voltage source with a short circuit at $t=0$.

File:RL discharge schematic.jpg

The current flowing through the inductor at time t is given by:

$i_L(t)=\frac{V_0}{R}{e}^{-\frac{R}{L}t}$

Charging

If the inductor is initially uncharged and we want to charge it by inserting a voltage source $V_s$ in the RL circuit:

File:RL charge schematic.jpg

The inital current will be 0A, but the current at steady state will be equal to $V_0/R$ and in this case, $V_0=V_s$. The current through the inductor is given by:

$i_L(t)=\frac{V_0}{R}(1-e^{-\frac{R}{L}t})$

The time constant for the RL circuit is equal to $L/R$.

The magnitudes of the voltage and current of the inductor for the circuits above are given by the graphs below:

Inductor

	Voltage (magnitude)	Current (magnitude)
Charge	File:RL charge voltage.jpg	File:RL charge current.jpg
Discharge	File:RL discharge voltage.jpg	File:RL discharge current.jpg