1 ET 201 ~ ELECTRICAL CIRCUITS COMPLEX NUMBER SYSTEM Define and explain complex number Rectangular...

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ET 201 ~ ELECTRICAL CIRCUITS

COMPLEX NUMBER SYSTEM

Define and explain complex numberRectangular 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

6


Example 14.13(b)

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

Solution

7


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.

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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 Form

Example 14.14(b)

1207 C

Clockwise (CW)

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2.2 Polar Form

Example 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 14.15

Convert C = 4 + j4 to polar form

543 23 Z

Solution

13.533

4tan 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


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Example 14.17

Convert C = - 6 + j3 to polar form

71.636 22 Z

Solution

6

3tan180 1

43.15343.15371.6 C


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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


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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


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Complex Conjugate

In rectangular form, the conjugate of:

C = 2 + j3

is 2 – j3


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Complex Conjugate

In polar form, the conjugate of:

C = 2 30o

is 2 30o


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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


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Addition

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


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Example 14.19(a)

13;42 21 jj CC

143221 jCC

55 j

Find C1 + C2.

Solution


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Example 14.19(b)

36;63 21 jj CC

366321 jCC

93 j

Find C1 + C2

Solution


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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°

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Example 14.20(a)

41;64 21 jj CC

461421 jCC

23 j

Find C1 - C2

Solution


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Example 14.20(b)

52;33 21 jj CC

532321 jCC

25 j

Find C1 - C2

Solution


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Example 14.21(a)

455453452



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06180402


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:


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Example 14.22(a)

105;32 21 jj CC

Find C1C2.

Solution

1053221 jj CC

3520 j


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Example 14.22(b)

64;32 21 jj CC

Find C1C2.

Solution

643221 jj CC

1802626


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Example 14.23(a)

3010;205 21 CC

Find C1C2.

Solution

302010521 CC

5050


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Example 14.23(b) 1207;402 21 CC

Find C1C2.

Solution

120407221 CC

8014


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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:


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Example 14.24(a)

54;41 21 jj CC Find

Solution

2

1

C

C

5454

5441

54

54

54

41

2

1

jj

jj

j

j

j

j

C

C

27.059.02516

1124j

j


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Example 14.24(b)

16;84 21 jj CC Find

Solution

2

1

C

C

1616

1684

16

16

16

84

2

1

jj

jj

j

j

j

j

C

C

41.143.0136

5216j

j


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Example 14.25(a) 72;1015 21 CC Find

Solution

2

1

C

C

7102

15

72

1015

2

1

C

C

33.7


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Example 14.25(b) 5016;1208 21 CC Find

Solution

2

1

C

C

5012016

8

5016

1208

2

1

C

C

1705.0


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)()( 212121 yyjxxzz

)()( 212121 yyjxxzz

212121 rrzz

212

1

2

1 r

r

z

z

rz

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jrerjyxz

sincos je j

• Addition

• Subtraction

• Multiplication

• Division

• Reciprocal

• Complex conjugate

• Euler’s identity


1 ET 201 ~ ELECTRICAL CIRCUITS COMPLEX NUMBER SYSTEM Define and explain complex number Rectangular...

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