Lesson 34 – AC Circuit Analysis II (Sections 82 and 83) (CLO 83)
This lesson begins to apply all of the theorems learned back in Chapters 2 and 3 to ac circuits.
But, before we
start we bring in one very important concept involving impedance.
It is very important that students understand the relationship between frequency and impedance.
Once they do the
whole idea of how
RLC
elements behave and why such things as filters work becomes clear.
They may not really
fully appreciate this concept now but it will be a useful landmark to return to often in subsequent lectures.
The figure on the left shows how the magnitude of the impedance of each
element varies with frequency.
A resistor is not affected (in the ideal) by
frequency, hence a 1k
resistor at dc is a 1k
Ω
Ω
resistor at 10 krad/s and at
1 Grad/s or whatever.
This is not so for the other two elements.
At dc, an
inductor has 0
– a short circuit.
We used this concept in Chapter 7 when
Ω
we found the initial and final conditions of a circuit excited by a step.
An
inductor, however, changes its character as frequency is increased from a
short at dc to an open as
.
A capacitor has the opposite reaction.
At
ω→ ∞
dc it behaves like an open circuit or
. As the frequency increases its
∞Ω
impedance decreases so that as
, 
ω→ ∞
Z
C

0
, a short circuit.
→
Ω
In a circuit with inductors and capacitors there could exist a frequency where the impedance of the inductor
equals that of the capacitor, at that frequency, the reactance of the circuit is zero.
When that occurs we say that
the circuit is in
resonance
and the frequency at which that occurs is called the
resonant frequency
ω
0
.
We are now ready to start ac circuit analysis. Start with an
RC
voltage divider circuit.
Do this with two purposes
in mind.
First is to simply show that voltage division works quite well in the Phasor domain and second to
demonstrate how the circuit’s behavior changes as we change the frequency of the
source.
Do this latter analysis qualitatively explaining how the impedance of the
capacitor changes from an open to a short as we increase the frequency.
The focus
here is not filters but rather that the capacitor and inductor have an ac “resistance” or
impedance that depends on the frequency of the source.
True it will serve us well
when we discuss filters in later discussions but for now the frequencydependence nature of these new devices is
what is important.
Now apply some values to the various parameters – say
R
=100 k
Ω
,
C
=10 μF,
°
∠
=
0
10
i
V
~
V and
ϖ
=
2
rad/s, and
solve for
O
V
~
and
i
I
~
.
Repeat for
ω
=
0 (dc) and for
ω
=
20 rad/s.
ϖ
O
V
~
0(dc)
S
V
~
1/
RC
2
S
V
~
∞
0
AC
v
O
(t)
C
R
v
i
(t)
+
_
RC
j
V
RC
CR
j
R
C
j
C
j
V
1
1
1
1
1
1
i
O
+
ϖ
=
+
ϖ
=
+
ϖ
ϖ
=
~
)
(
~
(
29
(
29
A
CR
j
C
R
C
j
V
I
μ
°
∠
=
°
∠
°
∠
×
=
∠
°
∠
×
=
∠
+
ϖ
°
∠
×
°
∠
×
=
+
ϖ
°
∠
×
°
∠
ϖ
=
+
ϖ
=



ϖ


6
26
4
89
4
63
236
2
90
10
2
5
90
10
2
1
0
10
90
10
2
1
0
10
90
1
4
1
2
1
4
1
1
2
2
5
i
i
.
.
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 Spring '08
 MAURICIODEOLIVERIA
 Frequency, Voltage divider, RC circuit, Electrical impedance, Thévenin's theorem

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