Difference between revisions of "RC and RL Exponential Responses"

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Revision as of 19:46, 15 June 2006

RC Circuits

Consider the following circuit:

File:RC discharge schematic.jpg

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

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

File:RC charge schematic.jpg

The voltage across the capacitor is given by:

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 circuit 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

In the following circuit, the inductor initally 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:

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

File:RL charge schematic.jpg

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:LC charge voltage.jpg File:LC charge current.jpg
Discharge File:LC discharge voltage.jpg File:LC discharge current.jpg