Difference between revisions of "RC and RL Exponential Responses"
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!Capacitor !! <math>v_C(t)=V_0e^{-\frac{t}{RC}}</math> || <math>v_C(t)=V_0(1-e^{-\frac{t}{RC}})</math> ||<math>RC\,</math> |
!Capacitor !! <math>v_C(t)=V_0e^{-\frac{t}{RC}}</math> || <math>v_C(t)=V_0(1-e^{-\frac{t}{RC}})</math> ||<math>RC\,</math> |
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!Inductor !! <math>i_L(t)= |
!Inductor !! <math>i_L(t)=\frac{V_0}{R}e^{-\frac{R}{L}t}</math> || <math>i_L(t)=\frac{V_0}{R}(1-e^{-\frac{R}{L}t})</math> || <math>\frac{L}{R}</math> |
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The current flowing through the inductor at time ''t'' is given by: |
The current flowing through the inductor at time ''t'' is given by: |
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<math>i_L(t)= |
<math>i_L(t)=\frac{V_0}{R}e^{-\frac{R}{L}t}</math> |
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===Charging=== |
===Charging=== |
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[[Image:RL_charge_schematic.jpg]] |
[[Image:RL_charge_schematic.jpg]] |
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The current through the inductor is given by: |
The inital current will be 0A, but the current at steady state will be equal to <math>V_0/R</math> and in this case, <math>V_0=V_s</math>. The current through the inductor is given by: |
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<math>i_L(t)= |
<math>i_L(t)=\frac{V_0}{R}(1-e^{-\frac{R}{L}t})</math> |
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The ''time constant'' for the RL circuit is equal to <math>L/R</math>. |
The ''time constant'' for the RL circuit is equal to <math>L/R</math>. |
Revision as of 20:51, 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:
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:
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 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:
Voltage | Current | |
---|---|---|
Charge | File:RL charge voltage.jpg | File:RL charge current.jpg |
Discharge | File:RL discharge voltage.jpg | File:RL discharge current.jpg |