Transcript Natural Response of an RC Circuit ECE 201 Circuit Theory I 1

Natural Response of an RC Circuit
ECE 201 Circuit Theory I
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Typical RC Circuit
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Circuit Conditions
• Assume that the switch has been in
position a for a very long time.
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The Capacitor is Charged to the Source
Voltage Vg
• The circuit now looks like
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Write the Node Voltage Equation
dv v
C  0
dt R
di
L  Ri  0
dt

t
RC
v(t )  v(0)e , t  0
v(0)  v(0)  Vg  V0
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Substituting back
v(t )  v(0)e

t

RC
,t  0
t
v(t )  V0 e  , t  0
  RC
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Plot of the Natural Response
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Current, Power, and Energy
t
v(t )  V0 e
v(t ) V0 t
i (t ) 
 e
R
R
V02  2t
p  vi 
e
R
t
t
2x
2

V0 
w   pdx   e dx
R
0
0
2t

1
2
w  CV0 (1  e  )
2
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Summary of Procedure
• Determine the initial voltage across the
Capacitor.
• Determine the Time Constant, τ.
• Apply
v(t )  V0e

t

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