RC and RL Exponential Responses

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Summary of Equations

Exponential responses of capacitors and inductors
Discharging Charging Time Constant
Capacitor
Inductor

RC Circuits

Discharging

Consider the following circuit:

File:RC discharge schematic.jpg

In the circuit, the capacitor is initially charged and has voltage aross it, and the switch is initially open. At time , 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:

Charging

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

File:RC charge schematic.jpg

The voltage across the capacitor is given by:

where .

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 , and over 99.99% of the final value at .

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 flowing through it; we replace the voltage source with a short circuit at .

File:RL discharge schematic.jpg

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

Charging

If the inductor is initially uncharged and we want to charge it by inserting a voltage source 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 and in this case, . The current through the inductor is given by:

The time constant for the RL circuit is equal to .

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