2. complex number system (option 2)

1

ET 201 ~ ELECTRICAL CIRCUITS

COMPLEX NUMBER SYSTEM Define and explain complex number

Rectangular formPolar formMathematical operations

(CHAPTER 2)

2

COMPLEXCOMPLEXNUMBERSNUMBERS

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 FormExample 14.13(b)

Sketch the complex number C = 0 – j6 in the complex plane

Solution

7

2.1 Rectangular FormExample 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

180 ZZC

10

2.2 Polar FormExample 14.14(a)

305C

Counterclockwise (CCW)

11

2.2 Polar FormExample 14.14(b)

1207 C

Clockwise (CW)

12

2.2 Polar FormExample 14.14(c)

602.4 C 180602.4

2402.4

180 ZZC

13

14.9 Conversion Between Forms1. Rectangular to Polar

14

14.9 Conversion Between Forms2. Polar to Rectangular

15

Example 14.15Convert C = 4 + j4 to polar form

543 23 Z

Solution

13.5334tan 1

13.535C

2.3 Conversion Between Forms

16

Example 14.16

Convert C = 1045 to rectangular form

07.745cos10 X

Solution

07.745sin10 Y

07.707.7 jC


17

Example 14.17Convert C = - 6 + j3 to polar form

71.636 22 Z

Solution

63tan180 1

43.15343.15371.6 C


18

Example 14.18

Convert C = 10 230 to rectangular form

43.6 230cos10 X

Solution

66.7 230sin10 Y

66.743.6 jC


19

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:

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


21

Complex Conjugate

In rectangular form, the conjugate of:

C = 2 + j3

is 2 – j3


22

Complex ConjugateIn polar form, the conjugate of:

C = 2 30o

is 2 30o


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:

jYX C is jYX 1

ZC is Z1


24

Addition

• To add two or more complex numbers, simply add the real and imaginary parts separately.


25

Example 14.19(a)13;42 21 jj CC

143221 jCC

55 j

Find C1 + C2.

Solution


26

Example 14.19(b)36;63 21 jj CC

366321 jCC

93 j

Find C1 + C2

Solution


27

Subtraction• In subtraction, the real and imaginary parts are

again considered separately .


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°

28

Example 14.20(a)

41;64 21 jj CC

461421 jCC

23 j

Find C1 - C2

Solution


29


532321 jCC

25 j

Find C1 - C2

Solution


30

Example 14.21(a)

455453452



31


06180402


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:


33

Example 14.22(a)105;32 21 jj CC

Find C1C2.Solution

1053221 jj CC

3520 j


34


Find C1C2.

Solution 643221 jj CC

1802626


35

Example 14.23(a) 3010;205 21 CC

Find C1C2.

Solution 302010521 CC

5050


36

Example 14.23(b) 1207;402 21 CC

Find C1C2.

Solution 120407221 CC

8014


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:


38

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


39

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


40

Example 14.25(a) 72;1015 21 CC Find

Solution

2

1

CC

7102

15721015

2

1

CC

33.7


41

Example 14.25(b) 5016;1208 21 CC Find

Solution

2

1

CC

50120168

50161208

2

1

CC

1705.0


42

)()( 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. complex number system (option 2)

Documents

complex number c

polar form c

complex numbersexample

complex numbersreciprocal

complex numberssubtraction

complex numbersmultiplication

complex numbersaddition

convert c