RC and RL Exponential Responses

Summary of Equations

Exponential responses of capacitors and inductors

	Discharging	Charging	Time Constant
Capacitor	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_C(t)=V_0e^{-\frac{t}{RC}}}	$v_{C}(t)=V_{0}(1-e^{-{\frac {t}{RC}}})$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RC\,}
Inductor	$i_{L}(t)=I_{0}e^{-{\frac {R}{L}}t}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_L(t)=I_0(1-e^{-\frac{R}{L}t})}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{L}{R}}

RC Circuits

Discharging

Consider the following circuit:

In the circuit, the capacitor is initially charged and has voltage Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_C(t)=V_0e^{-\frac{t}{RC}}}

Charging

If the capacitor is initially uncharged and we want to charge it by inserting a voltage source Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_s} in the RC circuit:

The voltage across the capacitor is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=1RC} , and over 99.99% of the final value at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=5RC} .

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

Capacitor

	Voltage	Current
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0=V_0/R} flowing through it; we replace the voltage source with a short circuit at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0} .

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_L(t)=I_0e^{-\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:

The current through the inductor is given by:

$i_{L}(t)=I_{0}(1-e^{-{\frac {R}{L}}t})$

The time constant for the RL circuit is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L/R} .

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

Inductor

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