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ET 201 ~ ELECTRICAL CIRCUITS
COMPLEX NUMBER SYSTEM Define and explain complex number
Rectangular formPolar formMathematical operations
(CHAPTER 2)
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COMPLEXCOMPLEXNUMBERSNUMBERS
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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.
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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.
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2.1 Rectangular Form
Example 14.13(a)
Sketch the complex number C = 3 + j4 in the complex plane
Solution
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2.1 Rectangular FormExample 14.13(b)
Sketch the complex number C = 0 – j6 in the complex plane
Solution
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2.1 Rectangular FormExample 14.13(c)
Sketch the complex number C = -10 – j20 in the complex plane
Solution
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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.
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2.2 Polar Form
180 ZZC
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2.2 Polar FormExample 14.14(a)
305C
Counterclockwise (CCW)
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2.2 Polar FormExample 14.14(b)
1207 C
Clockwise (CW)
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2.2 Polar FormExample 14.14(c)
602.4 C 180602.4
2402.4
180 ZZC
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14.9 Conversion Between Forms1. Rectangular to Polar
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14.9 Conversion Between Forms2. Polar to Rectangular
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Example 14.15Convert C = 4 + j4 to polar form
543 23 Z
Solution
13.5334tan 1
13.535C
2.3 Conversion Between Forms
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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
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Example 14.17Convert C = - 6 + j3 to polar form
71.636 22 Z
Solution
63tan180 1
43.15343.15371.6 C
2.3 Conversion Between Forms
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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
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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:
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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
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Complex Conjugate
In rectangular form, the conjugate of:
C = 2 + j3
is 2 – j3
2.4 Mathematical Operations with Complex Numbers
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Complex ConjugateIn polar form, the conjugate of:
C = 2 30o
is 2 30o
2.4 Mathematical Operations with Complex Numbers
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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
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Addition
• To add two or more complex numbers, simply add the real and imaginary parts separately.
2.4 Mathematical Operations with Complex Numbers
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Example 14.19(a)13;42 21 jj CC
143221 jCC
55 j
Find C1 + C2.
Solution
2.4 Mathematical Operations with Complex Numbers
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Example 14.19(b)36;63 21 jj CC
366321 jCC
93 j
Find C1 + C2
Solution
2.4 Mathematical Operations with Complex Numbers
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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°
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Example 14.20(a)
41;64 21 jj CC
461421 jCC
23 j
Find C1 - C2
Solution
2.4 Mathematical Operations with Complex Numbers
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Example 14.20(b)52;33 21 jj CC
532321 jCC
25 j
Find C1 - C2
Solution
2.4 Mathematical Operations with Complex Numbers
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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°
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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)
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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
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Example 14.22(a)105;32 21 jj CC
Find C1C2.Solution
1053221 jj CC
3520 j
2.4 Mathematical Operations with Complex Numbers
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Example 14.22(b)64;32 21 jj CC
Find C1C2.
Solution 643221 jj CC
1802626
2.4 Mathematical Operations with Complex Numbers
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Example 14.23(a) 3010;205 21 CC
Find C1C2.
Solution 302010521 CC
5050
2.4 Mathematical Operations with Complex Numbers
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Example 14.23(b) 1207;402 21 CC
Find C1C2.
Solution 120407221 CC
8014
14.10 Mathematical Operations with Complex Numbers
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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
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Example 14.24(a)54;41 21 jj CC Find
Solution
2
1
CC
5454
54415454
5441
2
1
jjjj
jj
jj
CC
27.059.025161124 jj
2.4 Mathematical Operations with Complex Numbers
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Example 14.24(b)16;84 21 jj CC Find
Solution
2
1
CC
1616
16841616
1684
2
1
jjjj
jj
jj
CC
41.143.01365216 jj
2.4 Mathematical Operations with Complex Numbers
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Example 14.25(a) 72;1015 21 CC Find
Solution
2
1
CC
7102
15721015
2
1
CC
33.7
2.4 Mathematical Operations with Complex Numbers
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Example 14.25(b) 5016;1208 21 CC Find
Solution
2
1
CC
50120168
50161208
2
1
CC
1705.0
2.4 Mathematical Operations with Complex Numbers
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)()( 212121 yyjxxzz
)()( 212121 yyjxxzz
212121 rrzz
212
1
2
1 rr
zz
rz11
jrerjyxz
sincos je j
• Addition• Subtraction• Multiplication
• Division• Reciprocal
• Complex conjugate• Euler’s identity
2.4 Mathematical Operations with Complex Numbers