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Physics 1C
Winter Quarter
2009
AC circuits
Chapter 31 notes©
W. Gekelman
Consider a capacitor.
This is
an element of an AC circuit.
The charge is related to the
voltage by
Q
=
CV
.
Since
I
=
dQ
dt
then
I
=
C
dV
dt
.
This
is the current through the
capacitor.
We will use complex
notation to express all quantities.
Let
V
=
V
0
e
i
!
t
dV
dt
=
i
V
0
e
i
t
.
All quantities in this
way of looking at things are complex therefore the current is written as
I=I
0
e
i
t
.
Then
I
=
C
dV
dt
becomes
I
0
=
Ci
V
0
.
This is in complex notation and the i really means that
the current and voltage are out of phase.
I
0
=
Ci
V
0
; V
0
=
I
0
i
C
=
iI
0
C
=
I
0
e
i
"
2
C
.
This means that the current and the voltage are out
of phase by 90 degrees in the capacitor.
The voltage LAGs the current by 90 degrees.
For example if the Voltage
is
V
=
Re
V
0
e
i
t
( )
then the current is
I
=
Re
CV
0
e
i
t
e
"
i
#
2
$
%
&
’
(
)
=
CV
0
cos
t
"
2
$
%
&
’
(
)
This is the essence of equation 31.16 in the
text.
This is summarized in a figure 31.9 from the book.
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3
Next let us generalize the concept of Resistance.
For a simple resistor the current and
voltage are in phase V=IR.
For a capacitor (instead of resistance) let us define a quantity
called the capacitive part of the impedance.
Z
c
!
V
0
I
0
=
1
i
"
C
.
Thus instead of R we have Z
C
, but it is complex.
Why are we doing
this?
The goal is that if we have a bunch of elements, R’s, C’s and L’s
we can calculate a
quantity made out of R, Z
C
and an Z
L
and then relate the current to the voltage across the
component.
Now let us consider an inductor only
V
=
L
di
dt
=
L
d
dt
i
0
e
i
!
t
=
i
Li
0
e
i
t
=
V
0
e
i
t
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We are interested in the ratio of voltage to current to find the inductive reactance
V
0
i
0
=
i
!
L
"
Z
L
As shown in the book:
Here the voltage LEADs the current
Let us use this to reexamine the undriven series LRC circuit.
The Kirchoff equation was
LC
d
2
I
dt
2
+
RC
dI
dt
+
I
=
0
.
Using the complex current method
take
I
t
( )
=
I
0
e
i
t
.
When we are finished we will take the real part of the solution.
But
since damping will occur because of the resistor the angular frequency must be complex as
well.
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This note was uploaded on 01/21/2010 for the course PHYS 1C taught by Professor Whitten during the Winter '07 term at UCLA.
 Winter '07
 Whitten
 Charge, Current

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