2. complex number system (option 2)

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ET 201 ~ ELECTRICAL CIRCUITS

COMPLEX NUMBER SYSTEM Define and explain complex number

Rectangular formPolar formMathematical operations

(CHAPTER 2)

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COMPLEXCOMPLEXNUMBERSNUMBERS

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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.

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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.

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2.1 Rectangular Form

Example 14.13(a)

Sketch the complex number C = 3 + j4 in the complex plane

Solution

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2.1 Rectangular FormExample 14.13(b)

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

Solution

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2.1 Rectangular FormExample 14.13(c)

Sketch the complex number C = -10 – j20 in the complex plane

Solution

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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.

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2.2 Polar Form

180 ZZC

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2.2 Polar FormExample 14.14(a)

305C

Counterclockwise (CCW)

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2.2 Polar FormExample 14.14(b)

1207 C

Clockwise (CW)

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2.2 Polar FormExample 14.14(c)

602.4 C 180602.4

2402.4

180 ZZC

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14.9 Conversion Between Forms1. Rectangular to Polar

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14.9 Conversion Between Forms2. Polar to Rectangular

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Example 14.15Convert C = 4 + j4 to polar form

543 23 Z

Solution

13.5334tan 1

13.535C

2.3 Conversion Between Forms

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Example 14.16

Convert C = 1045 to rectangular form

07.745cos10 X

Solution

07.745sin10 Y

07.707.7 jC

2.3 Conversion Between Forms

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

71.636 22 Z

Solution

63tan180 1

43.15343.15371.6 C

2.3 Conversion Between Forms

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Example 14.18

Convert C = 10 230 to rectangular form

43.6 230cos10 X

Solution

66.7 230sin10 Y

66.743.6 jC

2.3 Conversion Between Forms

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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:

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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

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Complex Conjugate

In rectangular form, the conjugate of:

C = 2 + j3

is 2 – j3

2.4 Mathematical Operations with Complex Numbers

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Complex ConjugateIn polar form, the conjugate of:

C = 2 30o

is 2 30o

2.4 Mathematical Operations with Complex Numbers

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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

2.4 Mathematical Operations with Complex Numbers

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Addition

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

2.4 Mathematical Operations with Complex Numbers

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Example 14.19(a)13;42 21 jj CC

143221 jCC

55 j

Find C1 + C2.

Solution

2.4 Mathematical Operations with Complex Numbers

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Example 14.19(b)36;63 21 jj CC

366321 jCC

93 j

Find C1 + C2

Solution

2.4 Mathematical Operations with Complex Numbers

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Subtraction• In subtraction, the real and imaginary parts are

again considered separately .

2.4 Mathematical Operations with Complex Numbers

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°

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Example 14.20(a)

41;64 21 jj CC

461421 jCC

23 j

Find C1 - C2

Solution

2.4 Mathematical Operations with Complex Numbers

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Example 14.20(b)52;33 21 jj CC

532321 jCC

25 j

Find C1 - C2

Solution

2.4 Mathematical Operations with Complex Numbers

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Example 14.21(a)

455453452

2.4 Mathematical Operations with Complex Numbers

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°

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2.4 Mathematical Operations with Complex Numbers

06180402

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°

Example 14.21(b)

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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

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Example 14.22(a)105;32 21 jj CC

Find C1C2.Solution

1053221 jj CC

3520 j

2.4 Mathematical Operations with Complex Numbers

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Example 14.22(b)64;32 21 jj CC

Find C1C2.

Solution 643221 jj CC

1802626

2.4 Mathematical Operations with Complex Numbers

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Example 14.23(a) 3010;205 21 CC

Find C1C2.

Solution 302010521 CC

5050

2.4 Mathematical Operations with Complex Numbers

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Example 14.23(b) 1207;402 21 CC

Find C1C2.

Solution 120407221 CC

8014

14.10 Mathematical Operations with Complex Numbers

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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:

14.10 Mathematical Operations with Complex Numbers

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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

2.4 Mathematical Operations with Complex Numbers

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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

2.4 Mathematical Operations with Complex Numbers

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Example 14.25(a) 72;1015 21 CC Find

Solution

2

1

CC

7102

15721015

2

1

CC

33.7

2.4 Mathematical Operations with Complex Numbers

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Example 14.25(b) 5016;1208 21 CC Find

Solution

2

1

CC

50120168

50161208

2

1

CC

1705.0

2.4 Mathematical Operations with Complex Numbers

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)()( 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

2.4 Mathematical Operations with Complex Numbers