ET 201 ELECTRICAL CIRCUITS COMPLEX NUMBER SYSTEM q

ET 201 ~ ELECTRICAL CIRCUITS COMPLEX NUMBER SYSTEM q Define and explain complex number q Rectangular form q Polar form q Mathematical operations (CHAPTER 2) 1

COMPLEX NUMBERS 2

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. 3

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. 4

2. 1 Rectangular Form Example 14. 13(a) Sketch the complex number C = 3 + j 4 in the complex plane Solution 5

2. 1 Rectangular Form Example 14. 13(b) Sketch the complex number C = 0 – j 6 in the complex plane Solution 6

2. 1 Rectangular Form Example 14. 13(c) Sketch the complex number C = -10 – j 20 in the complex plane Solution 7

2. 2 Polar Form • The format for the polar form is • Where: Z : magnitude only q : 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. 8

2. 2 Polar Form 9

2. 2 Polar Form Example 14. 14(a) Counterclockwise (CCW) 10

2. 2 Polar Form Example 14. 14(b) Clockwise (CW) 11

2. 2 Polar Form Example 14. 14(c) 12

14. 9 Conversion Between Forms 1. Rectangular to Polar 13

14. 9 Conversion Between Forms 2. Polar to Rectangular 14

2. 3 Conversion Between Forms Example 14. 15 Convert C = 4 + j 4 to polar form Solution 15

2. 3 Conversion Between Forms Example 14. 16 Convert C = 10 45 to rectangular form Solution 16

2. 3 Conversion Between Forms Example 14. 17 Convert C = - 6 + j 3 to polar form Solution 17

2. 3 Conversion Between Forms Example 14. 18 Convert C = 10 230 to rectangular form Solution 18

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: 19

2. 4 Mathematical Operations with Complex Numbers 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 20

2. 4 Mathematical Operations with Complex Numbers Complex Conjugate In rectangular form, the conjugate of: C = 2 + j 3 is 2 – j 3 21

2. 4 Mathematical Operations with Complex Numbers Complex Conjugate In polar form, the conjugate of: C = 2 30 o is 2 30 o 22

2. 4 Mathematical Operations with Complex Numbers Reciprocal • The reciprocal of a complex number is 1 divided by the complex number. • In rectangular form, the reciprocal of: is • In polar form, the reciprocal of: is 23

2. 4 Mathematical Operations with Complex Numbers Addition • To add two or more complex numbers, simply add the real and imaginary parts separately. 24

2. 4 Mathematical Operations with Complex Numbers Example 14. 19(a) Find C 1 + C 2. Solution 25

2. 4 Mathematical Operations with Complex Numbers Example 14. 19(b) Find C 1 + C 2 Solution 26

2. 4 Mathematical Operations with Complex Numbers Subtraction • In subtraction, the real and imaginary parts are again considered separately. NOTE Addition 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° 27

2. 4 Mathematical Operations with Complex Numbers Example 14. 20(a) Find C 1 - C 2 Solution 28

2. 4 Mathematical Operations with Complex Numbers Example 14. 20(b) Find C 1 - C 2 Solution 29

2. 4 Mathematical Operations with Complex Numbers Example 14. 21(a) NOTE Addition 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° 30

2. 4 Mathematical Operations with Complex Numbers Example 14. 21(b) NOTE Addition 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 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: 32

2. 4 Mathematical Operations with Complex Numbers Example 14. 22(a) Find C 1 C 2. Solution 33

2. 4 Mathematical Operations with Complex Numbers Example 14. 22(b) Find C 1 C 2. Solution 34

2. 4 Mathematical Operations with Complex Numbers Example 14. 23(a) Find C 1 C 2. Solution 35

14. 10 Mathematical Operations with Complex Numbers Example 14. 23(b) Find C 1 C 2. Solution 36

14. 10 Mathematical Operations with Complex Numbers 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: 37

2. 4 Mathematical Operations with Complex Numbers Example 14. 24(a) Find Solution 38

2. 4 Mathematical Operations with Complex Numbers Example 14. 24(b) Find Solution 39

2. 4 Mathematical Operations with Complex Numbers Example 14. 25(a) Find Solution 40

2. 4 Mathematical Operations with Complex Numbers Example 14. 25(b) Find Solution 41

2. 4 Mathematical Operations with Complex Numbers • Addition • Subtraction • Multiplication • Division • Reciprocal • Complex conjugate • Euler’s identity 42