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CPHY 122
Class Notes 16
Instructor: H. L. Neal
1 Alternating Current Sources
An alternating current may be generated by a time varying voltage source of the form Vs (t) = Vm ax sin (!t) :
Vs(t)
R
I(t)
The application of Ohmís law to the circuit above implies that
I (t) = Vs (t) R = Vm ax R sin (!t) :
Clearly
Im ax = Vm ax R : Not that Vs (t) and I (t) have minima and maxima at the same values of the time t. They are said to be in phase. The instantaneous power is
P (t) = [I (t)]^2 R:
The average power is deÖned as
Pav (t) =
T
Z T
0
P (t) dt;
where the period T is
T =
Thus
Pav (t) = R
Z 2 !
0
[I (t)]^2 dt
= (Irms)^2 R;
where
(Irms)^2 = (Im ax )^2! 2
Z 2 !
0
sin^2 (!t) dt
=
2 (Im ax^ )
Therefore Irms =
p 2
Im ax :
Similarly
Vrms =
p 2
Vm ax :
2 Inductor in AC circuit
Vs(t)
L
I(t)
Created using UNREGISTERED Top Draw 3.10 Nov 28,'106 10:48:51 AM We must have Vs (t) + EL = 0;
where EL = L dI dt
This means that L dI dt = Vm ax sin (!t) ;
4 The RLC Series AC Circuit
Vs(t)
C
I(t)
R L
Created using UNREGISTERED Top Draw 3.10 Nov 28,'106 12:14:30 PM Applying Kircho§ís rules gives
Vm ax sin (!t) L dI dt = IR + q C or Vm ax sin (!t) = L dI dt
+ IR +
q C
Di§erentiating both sides with respect to t gives
Vm ax cos (!t) = L d^2 I dt^2
+ R
dI dt
C
I:
We may assume that I (t) = Im ax sin (!t ) :
Then dI dt = !Im ax cos (!t ) ; d^2 I dt^2 = !^2 Im ax sin (!t ) :
Inserting this into the di§erential equation gives
Vm ax cos (!t) = L!^2 Im ax sin (!t ) + R!Im ax cos (!t ) +^1 C Im ax sin (!t )
or
Vm ax cos (!t) = Im ax
C
L!^2
sin (!t ) + R!Im ax cos (!t )
Recall that
sin (!t ) = sin (!t) cos ( ) cos (!t) sin ( ) ; cos (!t ) = cos (!t) cos ( ) + sin (!t) sin ( ) :
Inserting this gives
!Vm ax cos (!t) = Im ax
C
L!^2
[sin (!t) cos ( ) cos (!t) sin ( )] +R!Im ax [cos (!t) cos ( ) + sin (!t) sin ( )] = Im ax
C
L!^2
cos ( ) + R! sin ( )
sin (!t)
= Im ax
R! cos ( )
C
L!^2
sin ( )
cos (!t) :
Equating the coe¢ cients of cos (!t) and sin (!t) on both sides of the equation gives 1 C
L!^2
cos ( ) + R! sin ( ) = 0 ;
Im ax
R! cos ( )
C
L!^2
sin ( )
= !Vm ax :
From the Örst equation we get
tan =
L!^2 C^1
R!
!L (!C) ^1
R
Therefore
= tan ^1
!L (!C) ^1
R
Now recall that
cos
tan ^1 x
p^1 x^2 + 1
sin
tan ^1 x
x p x^2 + 1
