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

2 COMPLEX NUMBERS

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.

9 2.2 Polar Form

Polar Form Example 14.14(a) Counterclockwise (CCW)

Polar Form Example 14.14(b) Clockwise (CW)

Polar Form Example 14.14(c)

Conversion Between Forms 1. Rectangular to Polar

Conversion Between Forms 2. Polar to Rectangular

15 Example Convert C = 4 + j4 to polar form Solution 2.3 Conversion Between Forms

16 Example Convert C = 10  45  to rectangular form Solution 2.3 Conversion Between Forms

17 Example Convert C = j3 to polar form Solution 2.3 Conversion Between Forms

18 Example Convert C = 10  230  to rectangular form Solution 2.3 Conversion Between Forms

19 2.4Mathematical 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  30 o is 2  30 o 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: is 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) Find C 1 + C 2. Solution 2.4 Mathematical Operations with Complex Numbers

26 Example 14.19(b) Find C 1 + C 2 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 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°

28 Example 14.20(a) Find C 1 - C 2 Solution 2.4 Mathematical Operations with Complex Numbers

29 Example 14.20(b) Find C 1 - C 2 Solution 2.4 Mathematical Operations with Complex Numbers

30 Example 14.21(a) 2.4 Mathematical Operations with Complex Numbers 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°

Mathematical Operations with Complex Numbers 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° 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) Find C 1  C 2. Solution 2.4 Mathematical Operations with Complex Numbers

34 Example 14.22(b) Find C 1  C 2. Solution 2.4 Mathematical Operations with Complex Numbers

35 Example 14.23(a) Find C 1  C 2. Solution 2.4 Mathematical Operations with Complex Numbers

36 Example 14.23(b) Find C 1  C 2. Solution 14.10Mathematical 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: 14.10Mathematical Operations with Complex Numbers

38 Example 14.24(a) Find Solution 2.4 Mathematical Operations with Complex Numbers

39 Example 14.24(b) Find Solution 2.4 Mathematical Operations with Complex Numbers

40 Example 14.25(a) Find Solution 2.4 Mathematical Operations with Complex Numbers

41 Example 14.25(b) Find Solution 2.4 Mathematical Operations with Complex Numbers

42 Addition Subtraction Multiplication Division Reciprocal Complex conjugate Euler’s identity 2.4 Mathematical Operations with Complex Numbers