Transcript Diode with an LC Load No initial conditions.

Diode with an LC Load
No initial conditions.
Close the switch @ t = 0.
Solve for the current i.
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Solution
t
di 1
Vs  L   idt  vC (t  0)
dt C t0
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t
di 1
Vs  L   idt  vC (t  0)
dt C t0
Vs
1
 LsI ( s ) 
I (s)
s
Cs
1
2
Vs  Ls I ( s )  I ( s )
C
I ( s) 
Vs
1
Ls 
C
2
i (t )  Vs

Vs
 2 1 
L s 

LC



1
Vs
LC
1
 2 1 
L s 

LC 
LC 
C
sin 0t
L
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i (t )  Vs
C
sin 0t
L
di
C
 0Vs
cos 0t 
dt
L
di Vs
 cos 0t
dt L
Vs
di
t  0 
dt
L
1
C
Vs
cos 0t
L
LC
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Capacitor Voltage
t
t
1
1
C
vC (t )   idt   Vs
sin 0t
C0
C0
L

t
Vs 
1
C
vC (t )  Vs
sin 0t 


C
L0
LC 


vC (t )  Vs   cos 0t   cos(0) 
t


1
cos 0t 
1


LC
0
vC (t )  Vs 1  cos 0t 
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What happens at time t1=π√LC?
t  t1   LC
C
C
 1

i (t1 )  Vs
sin 0t1  Vs
sin 
 LC   0
L
L
 LC

1


vC (t1 )  Vs 1  cos 0t1   Vs 1  cos(
 LC   2Vs
LC


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Waveforms of Current and Voltage
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