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Imaginary axis jb (a + jb) = r
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a Real axis
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Figure A1 Polar form representation of complex numbers
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It is important to note that a and b are both real numbers The complex number a + j b can be represented on a rectangular coordinate plane, called the complex plane, by interpreting it as a point (a, b) That is, the horizontal coordinate is a in real axis and the vertical coordinate is b in imaginary axis, as shown in Figure A1 The complex number A = a + j b can also be uniquely located in the complex plane by specifying the distance r along a straight line from the origin and the angle , which this line makes with the real axis, as shown in Figure A1 From the right triangle of Figure A1, we can see that: r = a 2 + b2 b = tan 1 (A12) a a = r cos b = r sin Then, we can represent a complex number by the expression: A = rej = r (A13)
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which is called the polar form of the complex number The number r is called the magnitude (or amplitude) and the number is called the angle (or argument) The two numbers are usually denoted by: r = |A| and = arg A = A Given a complex number A = a + j b, the complex conjugate of A, denoted by the symbol A , is de ned by the following equalities: Re A = Re A Im A = Im A (A14)
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That is, the sign of the imaginary part is reversed in the complex conjugate Finally, we should remark that two complex numbers are equal if and only if the real parts are equal and the imaginary parts are equal This is equivalent to stating that two complex numbers are equal only if their magnitudes are equal and their arguments are equal The following examples and exercises should help clarify these explanations
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EXAMPLE A1
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Convert the complex number A = 3 + j 4 to its polar form
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Solution
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32 + 4 2 = 5
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= tan 1
= 5313
A = 5 5313
Appendix A
Linear Algebra and Complex Numbers
EXAMPLE A2
Convert the number A = 4 60 to its complex form
Solution
b = 4 sin( 60 ) = 4 sin(60 ) = 2 3 Thus, A = 2 j 2 3
a = 4 cos( 60 ) = 4 cos(60 ) = 2
Addition and subtraction of complex numbers take place according to the following rules: (a1 + j b1 ) + (a2 + j b2 ) = (a1 + a2 ) + j (b1 + b2 ) (a1 + j b1 ) (a2 + j b2 ) = (a1 a2 ) + j (b1 b2 ) (A15)
Multiplication of complex numbers in polar form follows the law of exponents That is, the magnitude of the product is the product of the individual magnitudes, and the angle of the product is the sum of the individual angles, as shown below (A)(B) = (Aej )(Bej ) = ABej ( + ) = AB ( + ) (A16)
If the numbers are given in rectangular form and the product is desired in rectangular form, it may be more convenient to perform the multiplication directly, using the rule that j 2 = 1, as illustrated in equation A17 (a1 + j b1 )(a2 + j b2 ) = a1 a2 + j a1 b2 + j a2 b1 + j 2 b1 b2 = (a1 a2 + j 2 b1 b2 ) + j (a1 b2 + a2 b1 ) = (a1 a2 b1 b2 ) + j (a1 b2 + a2 b1 ) Division of complex numbers in polar form follows the law of exponents That is, the magnitude of the quotient is the quotient of the magnitudes, and the angle of the quotient is the difference of the angles, as shown in equation A18 A A Aej A = = = ( ) j B Be B B (A18) (A17)
Division in the rectangular form can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator Multiplying the denominator by its complex conjugate converts the denominator to a real number and simpli es division This is shown in Example A4 Powers and roots of a complex number in polar form follow the laws of exponents, as shown in equations A19 and A20 An = (Aej )n = An ej n = An n (A19)
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