1. ECE 3301 General Electrical Engineering
Presentation 32
Complex Numbers

2. Real Numbers
Real numbers consist of:

3. Real Numbers
Real Numbers may be represented on the Real Number Line.

4. Imaginary Numbers
Imaginary Numbers are the square-roots of negative numbers:
The Imaginary Unit is defined as:

5. Imaginary Numbers
The custom for electrical engineers is to use j instead of i (as the mathematicians do) to represent the imaginary unit, because i is used to represent current.

6. Imaginary Numbers
Using the definition for the imaginary unit:

7. Imaginary Numbers
The reciprocal of the imaginary unit number is also of interest.

8. Imaginary Numbers
Any Imaginary number can be expressed using the Imaginary Unit j.

9. Imaginary Numbers
All imaginary numbers may be represented on the Imaginary Number Line.

10. Complex Numbers
Complex Numbers have a Real Part and an Imaginary Part:
z = a + jb
where a is the Real Part and b is the Imaginary Part.

11. Complex Numbers Two complex numbers: z1 = a1 + jb1 and z2 = a2 + jb2
are equal if and only if
a1 = a2 (the Real Parts are equal) and
b1 = b2 (the Imaginary Parts are equal)

12. The Complex Plane

13. The Real Part

14. The Imaginary Part

15. Rectangular Form
Any Complex Number may be expressed in Rectangular Form:

16. Polar Form
The same Complex Number may be expressed in Polar Form:

17. Polar Form
In Polar Form, r is the distance from the origin:

18. Polar Form
In Polar Form,  (in degrees) is the angle formed with respect to the positive Real Axis:

19. Polar Form
The relationship between Rectangular and Polar coordinates can be derived using simple trigonometry :

20. Polar Form
Given a Complex Number in Polar Form, the Rectangular Form is found by:

21. Polar Form
Given a Complex Number in Polar Form, the Rectangular Form is found by:

22. Polar Form
Recall Euler’s Formula:

23. Complex Numbers Any Complex Number is defined in one of two forms:
Real Part and Imaginary Part (Rectangular Form)
Magnitude and Phase Angle (Polar Form)

24. Complex Numbers
Electrical Engineers use a short-hand notation often called the Steinmetz Form:

25. Complex Numbers
For any Complex Number there are four equivalent ways of expressing the number:

26. Complex Numbers Each Complex Number has a Real Part:
Each Complex Number has an Imaginary Part:

27. Complex Numbers Each Complex Number has a Magnitude:
Each Complex Number has a Phase:

28. Conversion From Rectangular to Polar Form
Because the inverse-tangent operation of most calculators returns values in the range
– 90    + 90
care must be taken to place the Complex Number in the appropriate quadrant of the Complex Plane.

29. Conversion From Rectangular to Polar Form

30. Complex Conjugates
For each Complex Number there is a Complex Conjugate that is found by reversing the sign of the Imaginary Part of the Complex number.
z = a + jb
z* = a – jb

31. Complex Conjugates In Trigonometric Form, the conjugate pair is:
z = r cos( ) + j r sin( )
z* = r cos( ) – j r sin( )
z* = r cos( ) + j r sin(– )
recall sin(– ) = – sin( )

32. Complex Conjugates In Exponential Form, the conjugate pair is:
z = r e j
z* = r e – j

33. Complex Conjugates In Polar Form the conjugate pair is: z = r 

34. Complex Conjugates
When a number is multiplied by its complex conjugate:
z z* = (a + jb)(a – jb)
z z* = aa – a jb + jba – jb jb
z z* = a 2 + b 2 = | z | 2

35. Complex Conjugates
When a number is multiplied by its complex conjugate:
z z* = r e j r e – j
z z* = r 2 e j( –  ) = r 2 e 0
z z* = r 2 = | z | 2

36. Addition of Complex Numbers
Given the Complex Numbers
z1 = a1 +jb1 and z2 = a2 +jb2
the sum of the two Complex Numbers is
z1 + z2 = (a1 + a2) + j(b1 + b2).

37. Addition of Complex Numbers

38. Multiplication of Complex Numbers
Given the Complex Numbers
z1 = a1 +jb1 and z2 = a2 +jb2
The product of the two numbers is
z1 z2 = (a1 +jb1) (a2 +jb2)

39. Multiplication of Complex Numbers
z1 z2 = (a1 + jb1) (a2 + jb2)
z1 z2 = a1 a2 + a1 jb2 + jb1a2 +jb1 jb2
z1 z2 = (a1 a2 – b1b2) + j (a1b2 + b1a2)

40. Multiplication of Complex Numbers
This procedure is much easier to carry out in Exponential Form
z1 = r1 e j 1 and z2 = r2 e j 2
z1 z2 =(r1 e j 1)(r2 e j 2)
z1 z2 = r1 r2 e j( 1 +  2)

41. Multiplication of Complex Numbers
Or in Polar Form:
z1 = r1 1 and z2 = r2 2
z1 z2 = (r1 1)(r2 2)
z1 z2 = r1 r2 (1 + 2)

42. Division of Complex Numbers
Given the Complex Numbers
z1 = a1 +jb1 and z2 = a2 +jb2
The quotient of the two numbers is

43. Division of Complex Numbers
Multiplying both top and bottom by the complex Conjugate of the denominator:

44. Division of Complex Numbers
Carrying out the multiplications:

45. Division of Complex Numbers
Using the definition j2:

46. Division of Complex Numbers
Separating into real and imaginary parts:

47. Division of Complex Numbers
This procedure is much easier to carry out in Exponential Form
z1 = r1 e j 1 and z2 = r2 e j 2

48. Division of Complex Numbers
Using the Law of Exponents:

49. Division of Complex Numbers
In Polar Form:

50. Around the Unit Circle Im Re

51. Around the Unit Circle Im Re

52. Around the Unit Circle Im Re

53. Around the Unit Circle Im Re

54. Around the Unit Circle Im Re

55. Around the Unit Circle Im Re

56. Around the Unit Circle Im Re

57. Around the Unit Circle Im Re

58. Around the Unit Circle Im Re

59. Around the Unit Circle Im Re

60. Around the Unit Circle Im Re

61. Around the Unit Circle Im Re

62. Around the Unit Circle Im Re

63. Around the Unit Circle Im Re

64. Around the Unit Circle Im Re

65. Around the Unit Circle Im Re

66. Around the Unit Circle Im Re

67. Around the Unit Circle Im Re

68. Around the Unit Circle Im Re

69. Around the Unit Circle Im Re

70. Around the Unit Circle Im Re

71. Around the Unit Circle Im Re

72. Around the Unit Circle Im Re

73. Around the Unit Circle Im Re

74. Around the Unit Circle Im Re

75. Around the Unit Circle Im Re

76. Around the Unit Circle Im Re

77. Around the Unit Circle Im Re

78. Around the Unit Circle Im Re

79. Around the Unit Circle Im Re

80. Around the Unit Circle Im Re

81. Around the Unit Circle Im Re

82. Around the Unit Circle Im Re