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- Consider the LC and RC series circuits shown:
- Suppose that at t =0 the capacitor is charged to a value of Q****. Is there is a qualitative difference in the time development of the currents produced in these two cases. Why??
C R C L
++++
- - - - ++++ - - - - L^ C^0
dI Q
V V L
dt C
Kirchoff’s loop rule
C L
I
Q
0
I
(^10 2 4 ) 0 1.01^1
f( x) 0 x 6.
Q
0 x 0 , r1.. n r (^10 2 4 ) 0 1.01^1
f( x) 0 x 6.
t
0
dI
dt
(^10 2 4 ) 0 1.01^1
f( x) 0 x 6. 0
VC
x 0 , r1 n ..r (^10 2 4 ) 0 1.01^1
f( x)
V^0 x^ 6.
t
0 L (^10 2 4 ) 0 1.01^1
f( x) 0 x 6.
Q = Q 0 cos( ω 0 t )
I = − ω 0 (^) Q 0 (^) sin( ω 0 t )
LC
ω 0 =
U E
t
0 x 0 ,r1.. n r 0 2 4 6 0
1 f( x) x
U B
0
t
x 0 ,r1.. n r 0 2 4 6 0
1 f( x) x
Energy in Capacitor
cos( ) 2
( )^1
0 2 2 U (^) E t = (^) CQ 0 ω t + φ
Energy in Inductor
sin( )
U B ( t )= L ω 0 2 Q 02 2 ω 0 t + φ
LC
ω 0 = sin( ) 2
( )= Q 02 2 ω 0 t + φ C U (^) B t
C
Q
U E t UB t
2
+ =^0
Therefore, x 0 ,r1.. n r (^10 5 ) 0 1 f( x) x
R = 0
Q
0
t t
0
Q
R = 0
x 0 , r1..^ r1^10 n^100 n r (^10 5 ) 0 1 f( x) x
- An LC circuit is a natural oscillator.
C L
R
- In a real LC circuit, we must account for the resistance of the inductor. This resistance will damp out the oscillations. x 0 , r1..^ r1^10 n^100 n r (^10 5 ) 0 1 f( x) x inabsenceofresistive loss resonance LC
ω = Q t
- The instantaneous power (for some frequency, ) delivered at time t is given by:
- The most useful quantity to consider here is not the instantaneous power but rather the average power delivered in a cycle. P ( t ) = ε ( t ) I ( t ) = (^) (ε (^) m sin ω t (^) )( Im sin( ω t − φ ) (^) ) = I^2 ( t ) R 〈 P ( t )〉= ε (^) mIm 〈sin ω t sin(ω t − φ) 〉 1 ( ) cos 2 〈 P t 〉 = Vm Im φ ε rms 2 ε m
≡ I^ rms (^) 2 Im ≡^1
〈 P ( t )〉= ε rms Irms cos φ
cos R Z φ = R X (^) L − XC tan φ =
