Difference between revisions of "RC and RL Exponential Responses"

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[[Image:RC_discharge_schematic.jpg]]
[[Image:RC_discharge_schematic.jpg]]


In the circuit, the capacitor is initally charged and has voltage <math>V_0</math> aross it, and the switch is initially open. At time <math>t=0</math>, 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:
In the circuit, the capacitor is initially charged and has voltage <math>V_0</math> aross it, and the switch is initially open. At time <math>t=0</math>, 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:


<math>v_C(t)=V_0e^{-\frac{t}{RC}}</math>
<math>v_C(t)=V_0e^{-\frac{t}{RC}}</math>


===Charging===
===Charging===
If the capacitor is initially uncharged and we want to charge it by inserting a voltage source <math>V_s</math> in the RC cicuit:
If the capacitor is initially uncharged and we want to charge it by inserting a voltage source <math>V_s</math> in the RC circuit:


[[Image:RC_charge_schematic.jpg]]
[[Image:RC_charge_schematic.jpg]]
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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 <math>t=1RC</math>, and over 99.99% of the final value at <math>t=5RC</math>.
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 <math>t=1RC</math>, and over 99.99% of the final value at <math>t=5RC</math>.


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


{| border="1" cellspacing="0" cellpadding="5" align="center"
{| border="1" cellspacing="0" cellpadding="5" align="center"
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==RL Circuits==
==RL Circuits==
===Discharging===
===Discharging===
In the following circuit, the inductor initally has current <math>I_0=V_0/R</math> flowing through it; we replace the voltage source with a short circuit at <math>t=0</math>.
In the following circuit, the inductor initially has current <math>I_0=V_0/R</math> flowing through it; we replace the voltage source with a short circuit at <math>t=0</math>.


[[Image:RL_discharge_schematic.jpg]]
[[Image:RL_discharge_schematic.jpg]]
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===Charging===
===Charging===
If the inductor is initially uncharged and we want to charge it by inserting a voltage source <math>V_s</math> in the RL cicuit:
If the inductor is initially uncharged and we want to charge it by inserting a voltage source <math>V_s</math> in the RL circuit:


[[Image:RL_charge_schematic.jpg]]
[[Image:RL_charge_schematic.jpg]]

Revision as of 19:03, 15 June 2006

Summary of Equations

Exponential responses of C and L
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_s(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 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}} 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})}

RC Circuits

Discharging

Consider the following circuit:

File:RC discharge schematic.jpg

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 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} , 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:

File:RC charge schematic.jpg

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_s(1-e^{-\frac{t}{RC}})}

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 .

File:RL discharge schematic.jpg

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 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 RL circuit:

File:RL charge schematic.jpg

The current through the inductor 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_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
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