ET 242

I want to express my gratitude to Prentice Hall giving me the permission to use instructor’s material for developing this module. I would like to thank the Department of Electrical and Telecommunications Engineering
Technology of NYCCT for giving me support to commence and complete this module. I hope this module is helpful to enhance our students’ academic performance.

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Key Words : Complex Number, Phasor, Time Domain, Phase Domain
ET 242 Circuit Analysis II – Phasors
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Complex numbers lend themselves readily to the basic mathematical operations of addition , subtraction , multiplication , and division . A few basic rules and definitions must be understood before considering these operations.
Let us first examine the symbol j associated with imaginary
By definition , and with j
 j
3 
1 Thus , j
2  j
2 j
 
1
 j
  j

1 j
4  j
5  j
2  j
2 
(

1 )(

1 )
 
1 j and so on .
Further ,
1 j

( 1 )


1 j




 j j




1 j




 j j
2


 j

1
  j numbers,
ET 242 Circuit Analysis II – Phasors
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The conjugate or complex conjugate of a complex number can be found by simply changing the sign of imaginary part in rectangular form or by using the negative of the angle of the polar form. For example, the conjugate of
C = 2 + j3 is 2 – j3



o is
 
o

Figure 14.53
Defining the complex conjugate of a complex number in rectangular form.
Figure 14.54
Defining the complex conjugate of a complex number in polar form.
ET 242 Circuit Analysis II – Average power & Power Factor
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The reciprocal of a complex number is 1 devided by the complex number. For example, the reciprocal of
C

X
 jY is
X
1
 jY and of Z
 
,
1
Z
 
We are now prepared to consider the four basic operations of addition , subtraction , multiplication , and division with complex numbers.
To add two or more complex numbers, add the real and imaginary parts separately. For example, if then
C
1
= ± X
1
± jY
1 and C
2
= ± X
2
± jY
2
C
1
+ C
2
= (
±
X
1
±
X
2
) + j(
±
Y
1
±
Y
2
)
There is really no need to memorize the equation. Simply set one above the other and consider the real and imaginary parts separately, as shown in Example 14-19.
ET 242 Circuit Analysis II – Phasors
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Ex. 14-19 a.
Add b.
Add
C
1
C
1
= 2 + j4
= 3 + j6 and and
C
2
C
2
= 3 + j1
= –6 – j3 a. C
1
+ C
2
= (2 + 3) + j(4 + 1) = 5 + j5 b. C
1
+ C
2
= (3 – 6) + j(6 + 3) = –3 + j9
Figure 14.55
Example 14-19 (a)
ET 242 Circuit Analysis II – Phasors
Figure 14.56
Example 14-19 (b)
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In subtraction, the real and imaginary parts are again considered separately. For example, if then
C
1
=
±
X
1
± jY
1 and C
2
=
±
X
2
± jY
2
C
1
– C
2
= [(
±
X
1
– ( ±
X
2
)] + j[(
±
Y
1
– ±
Y
2
)]
Again, there is no need to memorize the equation if the alternative method of
Example 14-20 is used.
Ex. 14-20 a.
Subtract C
2 b.
Subtract C
2
= 1 + j4 from C
1
= –2 + j5 from C
1
= 4 + j6
= +3 + j3 a.
C
1
–
C
2
= (4 – 1) + j(6 – 4)
= 3 + j2 b.
C
1
–
C
2
= (3 – (–2)) + j(3 – 5)
= 5 – j2
Figure 14.58
Example 14-20 (b)
Figure 14.57
Example 14-20 (a)
ET 242 Circuit Analysis II – Phasors
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Ex.
14

21 a.
b.
2

45 o 
3

45 o
2

0 o 
4

180 o

5

45 o
.

6

0 o .
Note
Note
Fig.
14

59.
Fig.
14

60.
Figure 14.59
Example 14-21 (a)
ET 242 Circuit Analysis II – Phasors
Figure 14.60
Example 14-21 (b)
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To multiply two complex numbers in rectangular form, multiply the real and imaginary parts of one in turn by the real and imaginary parts of the other. For example, if then and
C
1
= X
1
+ jY
1
C
1
· C
2
: and C
2
= X
2
+ jY
2
X
1
+ jY
1
X
2
+ jY
2
X
1
X
2
+ jY
1
X
2
+ jX
1
Y
2
+ j 2 Y
1
Y
2
X
1
X
2
+ j(Y
1
X
2
+ X
1
Y
2
) + Y
1
Y
2
(–1)
C
1
· C
2
= (X
1
X
2
– Y
1
Y
2
) + j(Y
1
X
2
+ X
1
Y
2
)
In Example 14-22(b), we obtain a solution without resorting to memorizing equation above. Simply carry along the j factor when multiplying each part of one vector with the real and imaginary parts of the other.
ET 242 Circuit Analysis II – Phasors
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Ex. 14-22 a.
Find C
1 b.
Find C
1
· C
2
· C
2 if if
C
1
C
1
= 2 + j3
= –2 – j3 and and
C
2
C
2
= 5 + j10
= +4 – j6 a.
Using the format above, we have
C
1
· C
2
= [(2) (3) – (3) (10)] + j[(3) (5) + (2) (10)]
= – 20 + j35 b.
Without using the format, we obtain
– 2 – j3
+4 – j6
–8 – j12 and C
1
+ j12 + j 2 18
–8 + j(–12 + 12) – 18
· C
2
= –26 = 26ے180
In polar form, the magnitudes are multiplied and the angles added algebraically. For example, for C
1

Z
1
 θ
1 and C
2

Z
2
 θ
2 we write
C
ET 242 Circuit Analysis II – Phasors
1

C
2

Z
1
Z
2
ET 242 Circuit Analysis II – Average power & Power Factor

(θ
1
 θ
2
)
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Ex.
14

23 a.
b.
Find
Find a.
b.
C
1
C
1


C
C
2
2 if if
C
1

5

20 o
C
1

2
 
40 o and and
C
2

C
2
10

30 o

7
 
120 o
C
1

C
2

(5

20 o
)(10

30 o
)

(5)(10)

(20 o 
30 o
)

50

50 o
C
1

C
2


(2
 
40 o
)(7
 
120 o
)
(2)(7)

(

40 o 
120 o
)

14
 
80 o
To multiply a complex number in rectangular form by a real number requires that both the real part and the imaginary part be multiplied by the real number. For example,
( 10 )( 2
 j 3 )

20
 j 30 and 50

0 o
( 0
 j 6 )
 j 300

300

90 o
ET 242 Circuit Analysis II – Phasors
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To divide two complex numbers in rectangular form, multiply the numerator and denominator by the conjugate of the denominator and the resulting real and imaginary parts collected. That is, if then and
C
1
C
1
C
2
C
1
C
2




X
1
 jY
1 and C
2

X
2
(
(
(
X
X
X
1
1
2


X jY
1
)( X
2

2 jY
2

)( X
Y
1
Y
2
)
2


X
2
2  jY jY j (
Y
2
2
2
X
2
)
)
2
Y
1
X
1
X
X
2
2
2


Y
Y
2
2
1
Y
2
 j
X
2
X
Y
1
2
2




X
1
Y
2
Y
2
2 jY
2
X
1
Y
2
)
The equation does not have to be memorized if the steps above used to
obtain it are employed.
That is, first multiply the numerator by the complex conjugate of the denominator and separate the real and imaginary terms. Then divide each term by the sum of each term of the denominator square.
ET 242 Circuit Analysis II – Phasors
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Ex. 14-24 a.
Find C
1 b.
Find C
1
/ C
2
/ C
2 if if
C
1
C
1
= 1 + j4
= –4 – j8 and and
C
2
C
2
= 4 + j5
= +6 – j1 a .
C
C
By
1
2
 preceding equation,
(1)(4)
4 2

(4)(5)

5 2

(4)(4) j
4 2 
5 2

24
41
 j
11
41

0.59
 j0.27
ET 242 Circuit Analysis II – Phasors
.
and
Using an alternativ

4
 j8 e method,

6


24
 j1 j48 we obtain


24
 j4
 j
2
8 j52

8
 
16
 j52

6
 j1

6
 j1
36
 j6
 j6
 j
2
1
36

0

1

37
1


16
37
2
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 j
52
37
 
0.43
 j1.41
13

To divide a complex number in rectangular form by a real number, both the real part and the imaginary part must be divided by the real number. For example,
8

2 j 10

4
 j 5 and 6 .
8

2 j 0

3 .
4
 j 0

3 .
4

0 o
In polar form, division is accomplished by dividing the magnitude of the numerator by the magnitude of the denominator and subtracting the angle of the denominator from that of the numerator. That is, for
C
1

Z
1
 
1 and C
2

Z
2
 
2 we write C
1
C
2

Z
1
Z
2

(

1
 
2
)
Ex.
14

25 a.
b.
Find
Find a .
b .
C
1
C
1
/
/ C
2
C
2 if if
C
1

1 5

10 o
C
1

8

120 o and and
C
2

2

7 o
C
2

1 6
 
50 o
C
C
2
1

15

10 o
2

7 o

15

( 10 o
2
C
1

8


120
 o

8
C
2
16 50 o
16

(

120
7 o
)

7 .
o 
(

50
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5 o

))
3 o

0 .
5

170 o
14

The addition of sinusoidal voltages and current is frequently required in the analysis of ac circuits . One lengthy but valid method of performing this operation is to place both sinusoidal waveforms on the same set of axis and add a algebraically the magnitudes of each at every point along the abscissa, as shown for c = a + b in Fig. 14-71. This, however, can be a long and tedious process with limited accuracy.
Figure 14.71
Adding two sinusoidal waveforms on a point-by-point basis.
ET 242 Circuit Analysis II – Phasors
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A shorter method uses the rotating radius vector . This radius vector , having a constant magnitude (length) with one end fixed at the origin , is called a phasor when applied to electric circuits. During its rotational development of the sine wave, the phasor will, at the instant = 0, have the positions shown in Fig. 14-72(a) for each waveform in Fig. 14-72(b).
Figure 14.72
Demonstrating the effect of a negative sign on the polar form.
ET 242 Circuit Analysis II – Phasors
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Note in Fig. 14-72(b) that v
2 passes through the horizontal axis at t = 0 s, requiring that the radius vector in Fig. 14-
72(a) is equal to the peak value of the sinusoid as required by the radius vector. The other sinusoid has passed through
90 ° of its rotation by the time t
= 0 s is reached and therefore has its maximum vertical projection as shown in Fig. 14-
72(a). Since the vertical projection is a maximum, the peak value of the sinusoid that it generates is also attained at t
= 0 s as shown in Fig. 14-
72(b).
16

It can be shown [see Fig. 14-72(a)] using the vector algebra described that
1 V

0 o 
2 V

90 o 
2 .
236 V

63 .
43 o
In other words, if we convert v
1 v

V m sin(
 t
 and v
2

) to the phasor form using

V m
  
And add then using complex number algebra, we can find the phasor form for v
T with very little difficulty. It can then be converted to the time-domain and plotted on the same set of axes, as shown in Fig. 14-72(b). Fig. 14-72(a), showing the magnitudes and relative positions of the various phasors, is called a phasor diagram .
In the future, therefore, if the addition of two sinusoids is required, you should first convert them to phasor domain and find the sum using complex algebra. You can then convert the result to the time domain.
The case of two sinusoidal functions having phase angles different from 0 ° and 90 ° appears in Fig. 14-73. Note again that the vertical height of the functions in Fig. 14-
73(b) at t = 0 s is determined by the rotational positions of the radius vectors in Fig.
14-73(a).
ET 242 Circuit Analysis II – Phasors
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In general, for all of the analysis to follow, the phasor form of a sinusoidal voltage or current will be
V

V
  and I

I
  where V and I are rms value and
θ is the phase angle. It should be pointed out that in phasor notation, the sine wave is always the reference, and the frequency is not represented.
Phasor algebra for sinusoidal quantities is applicable only for waveforms having the same frequency.
Figure 14.73
Adding two sinusoidal currents with phase angles other than 90 ° .
ET 242 Circuit Analysis II – Phasors
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Ex. 14-27 Convert the following from the time to the phasor domain: b. c.
Time Domain a. √2(50)sinωt
69.9sin(ωt + 72°)
45sinωt
Phasor Domain
50 ∟ 0°
(0.707)(69.6) ∟ 72° = 49.21
∟ 90°
(0.707)(45)
∟
90° = 31.82
∟
90°
Ex. 14-28 Write the sinusoidal expression for the following phasors if the frequency is 60 Hz:
Time Domain a. I = 10∟30° b. V = 115∟–70°
Phasor Domain i = √2 (10)sin(2π60t + 30°) and i = 14.14 sin(377t + 30°) v = √2 (115)sin(377t – 30°) and v = 162.6 sin(377t – 30°)
ET 242 Circuit Analysis II – Phasors
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Ex. 14-29 Find the input voltage of the circuit in Fig. 14-75 if v a
= 50 sin(377t + 30
°
) f = 60 Hz v b
= 30 sin(377t + 60
°
)
Applying e m
Kirchhoff
 v a
 v b
' s voltage law , we have
Converting from the time to the phasor domain v a v b

50 sin( 377 t

30 o
)

V a

50 sin( 377 t

60 o
)

V b

35 .
35 V

30


21 .
21 V

60
 yields
Converting
V a from polar

35 .
35 V

30
  to rectangula r
30 .
61 V
 j 17 form
.
68 V
V b

21 .
21 V

30
 
10 .
61 V
 j 18 .
37 V for addition yields
Figure 14.75
ET 242 Circuit Analysis II – Phasors
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Figure 14.76
ET 242 Circuit Analysis II – Phasors
Then
E m


V a

V b

41.22
V

(30.61
V
 j36.05
V j17.68)

(10.61
V
 j18.37)
Converting
E m from

41.22
V
 rectangula j36.05
V r to polar form, we

54.76
V

41.17
 have
Converting from the phasor to the time domain, we obtain and
E m

54.76
V

41.17
  e m

77.43sin(3 77t e m


2 (54.76)sin
41.17

)
(377t

41.17

)
A plot of the three waveforms is shown in Fig. 14-76. Note that at each instant of time, the sum of the two waveform does in fact add up to e m
. At t = 0 (ωt = 0), e m is the sum of the two positive values, while at a value of ωt, almost midway between
π/2 and π, the sum of the positive value of v a v b and the negative value of results in e m
= 0 .
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Ex. 14-30 Determine the current i
2 for the network in Fig. 14-77.
Figure 14.77
T
'

1

2
2

T
,

1
T

120

10

3 sin(


60 o
)

84 .
84

60

1

80

10

3 sin


56 .
56

0

T

84 .
84

60
1

56 .
56
ET 242 Circuit Analysis II – Phasors

0




56
42 .
.
56
42


73 .
47
0
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22

Then I
2

I
T

I
1

( 42 .
42 mA

 
14 .
14 mA
 j 73 .
47 mA )

( 56 .
56 mA
 j 73 .
47 mA j 0 )
Converting
I
2 from rectangula r to

74 .
82 mA

100 .
89
 polar form , we have
Converting from the phasor to the time domain , we have
I
2 and

74 .
82 mA

100 .
89
  i
2 i
2

105 .
8

10

3 sin(
 t

2 ( 74 .
82

10

3
) sin(

100 .
89

)
 t

100 .
89

)
A plot of the three waveforms appears in Fig. 14-78. The waveforms clearly indicate that i
T
= i
1
+ i
2 .
ET 242 Circuit Analysis II – Phasors
Figure 14.78
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