ECE 2100
Circuit Analysis
Lesson 24
Chapter 9 & App B:
Passive circuit elements
in the phasor representation
Daniel M. Litynski, Ph.D.
http://homepages.wmich.edu/~dlitynsk/
ECE 2100
Circuit Analysis
Lesson 23
Chapter 9 & App B:
Introduction to Sinusoids & Phasors
Sinusoids and Phasor
Chapter 9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Motivation
Sinusoids’ features
Phasors
Phasor relationships for circuit elements
Impedance and admittance
Kirchhoff’s laws in the frequency domain
Impedance combinations
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9.1 Motivation (1)
How to determine v(t) and i(t)?
vs(t) = 10V
How can we apply what we have learned before to
determine i(t) and v(t)?
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9.2 Sinusoids (1)
• A sinusoid is a signal that has the form of the
sine or cosine function.
• A general expression for the sinusoid,
v(t )  Vm sin(t   )
where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
Ф = the phase
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9.2 Sinusoids (2)
A periodic function is one that satisfies v(t)
= v(t + nT), for all t and for all integers n.
T
2

f 
1
Hz
T
  2f
• Only two sinusoidal values with the same frequency can be compared
by their amplitude and phase difference.
• If phase difference is zero, they are in phase; if phase difference is not
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zero, they are out of phase.
9.2 Sinusoids (3)
Example 1
Given a sinusoid, 5 sin(4t  60 o ), calculate its
amplitude, phase, angular frequency, period, and
frequency.
Solution:
Amplitude = 5, phase = –60o, angular frequency
= 4 rad/s, Period = 0.5 s, frequency = 2 Hz.
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9.2 Sinusoids (4)
Example 2
Find the phase angle between i1  4 sin(377t  25 o )
and i2  5 cos(377t  40 o ), does i1 lead or lag i2?
Solution:
Since sin(ωt+90o) = cos ωt
i2  5 sin(377t  40o  90o )  5 sin(377t  50o )
i1  4 sin(377t  25o )  4 sin(377t  180 o  25o )  4 sin(377t  205o )
therefore, i1 leads i2 155o.
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9.3 Phasor (1)
• A phasor is a complex
number that represents the
amplitude and phase of a
sinusoid.
• It can be represented in one
of the following three forms:
Rectangular z  x  jy  r (cos  j sin  )
z  r 
j
z

re
Exponential
Polar
where
r  x2  y2
  tan 1
y
x
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9.3 Phasor (2)
Example 3
• Evaluate the following complex numbers:
a.
[(5  j2)( 1  j4)  5 60o ]
b.
10  j5  340 o
 10 30 o
 3  j4
Solution:
a. –15.5 + j13.67
b. 8.293 + j2.2
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9.3 Phasor (3)
Mathematic operation of complex number:
1. Addition
z1  z 2  ( x1  x2 )  j ( y1  y 2 )
2. Subtraction
z1  z2  ( x1  x2 )  j ( y1  y2 )
3. Multiplication
z1 z2  r1r2  1  2
4. Division
z1 r1
 1  2
z 2 r2
5. Reciprocal
1 1
  
z
r
6. Square root
z  r  2
7. Complex conjugate
z   x  jy  r     re  j
8. Euler’s identity
e  j  cos  j sin 
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9.3 Phasor (4)
• Transform a sinusoid to and from the time
domain to the phasor domain:
v(t )  Vm cos(t   )
(time domain)
V  Vm 
(phasor domain)
• Amplitude and phase difference are two principal
concerns in the study of voltage and current sinusoids.
• Phasor will be defined from the cosine function in all our
proceeding study. If a voltage or current expression is in
the form of a sine, it will be changed to a cosine by
subtracting from the phase.
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9.3 Phasor (5)
Example 4
Transform the following sinusoids to phasors:
i = 6cos(50t – 40o) A
v = –4sin(30t + 50o) V
Solution:
a. I  6  40 A
b. Since –sin(A) = cos(A+90o);
v(t) = 4cos (30t+50o+90o) = 4cos(30t+140o) V
Transform to phasor => V  4140
V
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9.3 Phasor (6)
Example 5:
Transform to the sinusoids corresponding to
phasors:
a. V   1030 V
b. I  j(5  j12) A
Solution:
a) v(t) = 10cos(t + 210o) V
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)  13 22.62 
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b) Since I  12  j5  12 2  52  tan 1 (
i(t) = 13cos(t + 22.62o) A
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9.3 Phasor (7)
The differences between v(t) and V:
•
•
•
v(t) is instantaneous or time-domain
representation
V is the frequency or phasor-domain
representation.
v(t) is time dependent, V is not.
v(t) is always real with no complex term, V is
generally complex.
Note: Phasor analysis applies only when frequency is
constant; when it is applied to two or more sinusoid
signals only if they have the same frequency.
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9.3 Phasor (8)
Relationship between differential, integral operation
in phasor listed as follow:
v(t )
V  V
dv
dt
jV
 vdt
V
j
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9.3 Phasor (9)
Example 6
Use phasor approach, determine the current i(t)
in a circuit described by the integro-differential
equation.
di
4i  8 idt  3  50 cos(2t  75)
dt
Answer: i(t) = 4.642cos(2t + 143.2o) A
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