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1
ET 201 ~ ELECTRICAL CIRCUITS
COMPLEX NUMBER SYSTEM
Define and explain complex numberRectangular formPolar formMathematical operations
(CHAPTER 2)
3
2. Complex Numbers
• A complex number represents a point in a two-dimensional plane located with reference to two distinct axes.
• This point can also determine a radius vector drawn from the origin to the point.
• The horizontal axis is called the real axis, while the vertical axis is called the imaginary ( j ) axis.
4
2.1 Rectangular Form• The format for the
rectangular form is
• The letter C was chosen from the word complex
• The bold face (C) notation is for any number with magnitude and direction.
• The italic notation is for magnitude only.
5
2.1 Rectangular Form
Example 14.13(a)
Sketch the complex number C = 3 + j4 in the complex plane
Solution
6
2.1 Rectangular Form
Example 14.13(b)
Sketch the complex number C = 0 – j6 in the complex plane
Solution
7
2.1 Rectangular Form
Example 14.13(c)
Sketch the complex number C = -10 – j20 in the complex plane
Solution
8
2.2 Polar Form
• The format for the polar form is
• Where:Z : magnitude only : angle measured
counterclockwise (CCW) from the positive real axis.
• Angles measured in the clockwise direction from the positive real axis must have a negative sign associated with them.
15
Example 14.15
Convert C = 4 + j4 to polar form
543 23 Z
Solution
13.533
4tan 1
13.535C
2.3 Conversion Between Forms
16
Example 14.16
Convert C = 1045 to rectangular form
07.745cos10 X
Solution
07.745sin10 Y
07.707.7 jC
2.3 Conversion Between Forms
17
Example 14.17
Convert C = - 6 + j3 to polar form
71.636 22 Z
Solution
6
3tan180 1
43.15343.15371.6 C
2.3 Conversion Between Forms
18
Example 14.18
Convert C = 10 230 to rectangular form
43.6 230cos10 X
Solution
66.7 230sin10 Y
66.743.6 jC
2.3 Conversion Between Forms
19
2.4 Mathematical Operations with Complex Numbers
• 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:
20
Complex Conjugate
• The conjugate or complex conjugate of a complex number can be found by simply changing the sign of the imaginary part in the rectangular form or by using the negative of the angle of the polar form
2.4 Mathematical Operations with Complex Numbers
21
Complex Conjugate
In rectangular form, the conjugate of:
C = 2 + j3
is 2 – j3
2.4 Mathematical Operations with Complex Numbers
22
Complex Conjugate
In polar form, the conjugate of:
C = 2 30o
is 2 30o
2.4 Mathematical Operations with Complex Numbers
23
Reciprocal
• The reciprocal of a complex number is 1 divided by the complex number.
• In rectangular form, the reciprocal of:
• In polar form, the reciprocal of:
jYX C is jYX 1
ZC is Z1
2.4 Mathematical Operations with Complex Numbers
24
Addition
• To add two or more complex numbers, simply add the real and imaginary parts separately.
2.4 Mathematical Operations with Complex Numbers
25
Example 14.19(a)
13;42 21 jj CC
143221 jCC
55 j
Find C1 + C2.
Solution
2.4 Mathematical Operations with Complex Numbers
26
Example 14.19(b)
36;63 21 jj CC
366321 jCC
93 j
Find C1 + C2
Solution
2.4 Mathematical Operations with Complex Numbers
27
Subtraction• In subtraction, the real and imaginary parts are
again considered separately .
2.4 Mathematical Operations with Complex Numbers
NOTEAddition or subtraction cannot be performed in polar form unless the complex numbers have the same angle ө or unless they differ only by multiples of 180°
28
Example 14.20(a)
41;64 21 jj CC
461421 jCC
23 j
Find C1 - C2
Solution
2.4 Mathematical Operations with Complex Numbers
29
Example 14.20(b)
52;33 21 jj CC
532321 jCC
25 j
Find C1 - C2
Solution
2.4 Mathematical Operations with Complex Numbers
30
Example 14.21(a)
455453452
2.4 Mathematical Operations with Complex Numbers
NOTEAddition or subtraction cannot be performed in polar form unless the complex numbers have the same angle ө or unless they differ only by multiples of 180°
31
2.4 Mathematical Operations with Complex Numbers
06180402
NOTEAddition or subtraction cannot be performed in polar form unless the complex numbers have the same angle ө or unless they differ only by multiples of 180°
Example 14.21(b)
32
Multiplication
• 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.
• In rectangular form:
• In polar form:
2.4 Mathematical Operations with Complex Numbers
33
Example 14.22(a)
105;32 21 jj CC
Find C1C2.
Solution
1053221 jj CC
3520 j
2.4 Mathematical Operations with Complex Numbers
34
Example 14.22(b)
64;32 21 jj CC
Find C1C2.
Solution
643221 jj CC
1802626
2.4 Mathematical Operations with Complex Numbers
35
Example 14.23(a)
3010;205 21 CC
Find C1C2.
Solution
302010521 CC
5050
2.4 Mathematical Operations with Complex Numbers
36
Example 14.23(b) 1207;402 21 CC
Find C1C2.
Solution
120407221 CC
8014
14.10 Mathematical Operations with Complex Numbers
37
Division• 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.
• In rectangular form:
• In polar form:
14.10 Mathematical Operations with Complex Numbers
38
Example 14.24(a)
54;41 21 jj CC Find
Solution
2
1
C
C
5454
5441
54
54
54
41
2
1
jj
jj
j
j
j
j
C
C
27.059.02516
1124j
j
2.4 Mathematical Operations with Complex Numbers
39
Example 14.24(b)
16;84 21 jj CC Find
Solution
2
1
C
C
1616
1684
16
16
16
84
2
1
jj
jj
j
j
j
j
C
C
41.143.0136
5216j
j
2.4 Mathematical Operations with Complex Numbers
40
Example 14.25(a) 72;1015 21 CC Find
Solution
2
1
C
C
7102
15
72
1015
2
1
C
C
33.7
2.4 Mathematical Operations with Complex Numbers
41
Example 14.25(b) 5016;1208 21 CC Find
Solution
2
1
C
C
5012016
8
5016
1208
2
1
C
C
1705.0
2.4 Mathematical Operations with Complex Numbers