Difference between revisions of "RC and RL Exponential Responses"
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==Summary of Equations== |
==Summary of Equations== |
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|+Exponential responses of |
|+'''Exponential responses of capacitors and inductors''' |
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! !!Discharging !! Charging !! Time Constant |
! !!Discharging !! Charging !! Time Constant |
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!Capacitor !! <math>v_C(t)=V_0e^{-\frac{t}{RC}}</math> || <math>v_C(t)=V_s(1-e^{-\frac{t}{RC}})</math> ||<math>RC\,</math> |
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!Inductor !! <math>i_L(t)=I_0e^{-\frac{R}{L}t}</math> || <math>i_L(t)=I_0(1-e^{-\frac{R}{L}t})</math> || <math>\frac{L}{R}</math> |
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Revision as of 20:08, 15 June 2006
Summary of Equations
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:
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 circuits above are shown in the graphs below:
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:
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:
Voltage | Current | |
---|---|---|
Charge | File:RL charge voltage.jpg | File:RL charge current.jpg |
Discharge | File:RL discharge voltage.jpg | File:RL discharge current.jpg |