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EGR 272 – Circuit Theory II
Lecture #6
Read: Chapter 9 and Appendix B in Electric Circuits, 6th Edition by Nilsson
Using Phasors to Add Sinusoids
Sinusoidal voltages or currents could be added using various trigonometric identities;
however, they are more easily combined using phasors.
Example: If v1 = 10cos(200t + 15), v2 = 15cos(200t + -30), and v3 = 8sin(200t),
find v4.
+
V2
_
+
+
V1
V3
_
_
+
V4
_
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Lecture #6
EGR 272 – Circuit Theory II
sin(wt) or cos(wt)?
Since phasor analysis is used to calculate results that are relative to the sources, it
generally doesn’t matter whether the sinusoidal source is expressed using sin(wt) or
cos(wt). If the sources are expressed using sin(wt), then the results will also be in
terms of sin(wt) and if the sources are expressed using cos(wt), then the results will
also be in terms of cos(wt). However, the approach for a given circuit containing
multiple sources must be consistent – either using cos(wt) or sin(wt).
Review of DC Circuit Analysis Techniques
Analyzing AC circuits is very similar to analyzing DC resistive circuits. Several
examples are presented below which will also serve to review many DC analysis
techniques, including:
• Source transformations
• Mesh equations
• Node equations
• Superposition
• Thevenin’s and Norton’s theorems
• Maximum Power Transfer theorem
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Lecture #6
EGR 272 – Circuit Theory II
Source transformations
A phasor voltage source with a series impedance may be transformed into a phasor
current source with a parallel impedance as illustrated below. The two sources are
identical as far as the load is concerned.
Notes:
• Not all sources can be transformed. Discuss.
• The two sources are not equivalent internally. For example, the voltage across
Zs is not equivalent to the voltage across Zp.
• Dependent sources can be transformed.
I
Zs
Vs
+
_
I
+
V
+
Load
Ip
Zp
_
Converting a real current source
to a real voltage source:
V s  I p  Z p and
Zs  Z p
V
Load
_
Converting a real voltage source
to a real current source:
Ip 
Vs
and
Zs
Z p  Zs
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EGR 272 – Circuit Theory II
Lecture #6
Example: Solve for the voltage V using source transformations.
+
+
100cos(20t) V
8H
V _
100
240
500 uF
3sin(20t)
_
4
EGR 272 – Circuit Theory II
Lecture #6
Mesh equations:
Example: Solve for the voltage V using mesh equations.
+
50cos(400t) V
_
10
50 mH
100 uF
30
+
50 uF
V
_
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Lecture #6
EGR 272 – Circuit Theory II
Node equations:
Example: Solve for the current i(t) using node equations.
+
3sin(4t)
V1
_
6
1
F
8
4
i(t)
2H
+
_
V1
2
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Lecture #6
EGR 272 – Circuit Theory II
Superposition:
Superposition can be used to analyze multiple-source AC circuits in a manner very
similar to analyzing DC circuits. However, there are two special cases where it is
highly recommended that superposition be used:
• Circuits that include sources at two or more different frequencies
• Circuits that include both DC and AC sources (Note: you could think of DC
sources as acting like AC sources with w = 0.)
Example 1 (sources with different frequencies): Solve for the voltage V using superposition.
+
+
2cos(4t) V
V
_
4
4
_
1 F
8
+
3cos(2t) V
_
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Lecture #6
EGR 272 – Circuit Theory II
Example 2 (AC and DC sources): Solve for the voltage V using superposition.
+
+
6cos(5t) V
V
_
2
3
0.2 F
0.8 H
+
_
10 V
_
8