Appendix AComplex Numbers

Basic Complex-Number Concepts

• Complex numbers involve the imaginary number( 虛數 ) j =

• Z=x+jy has a real part (實部 ) x and an imaginary part (虛部 ) y , we can represent complex numbers by points in the complex plane (複數平面 )

• The complex numbers of the form x+jy are in rectangular form (直角座標 )

1

y

x

Imaginary

Real

Z

• The complex conjugate (共軛複數 ) of a number in rectangular form is obtained by changing the sign of the imaginary part.

• For example if then the complex conjugate of is

•

432 jZ

43*2 jZ

12 j

2Z

Example A.1 Complex Arithmetic in Rectangular Form

• Solution:

/Z, ,Z

reduce ,43 and 5 5 Given that

2121 21

21

ZZZZ

jZjZ

924355

is difference The

184355

have wesum, For the

21

21

j)-j)-(j(ZZ

j)-j()j(ZZ

to rectangular form

Example A.1 Complex Arithmetic in Rectangular Form

535

20152015

20152015

4355

get we,product For the

2

21

j

jj

-jj-j

)-j)(j(ZZ

4.12.0 25

355

1612129

20152015

1612129

20152015

43

43

43

55

43

55

r denominato theof conjugatecomplex by ther denominato and

numberator thegmultiplyinby formr rectangula toexpression thisreduce can We

43

55

obtain we, numbers thedivide To

2

2

*2

*2

2

1

2

1

j

j

jj

jj

jjj

jjj

j

j

j

j

Z

Z

j

j

Z

Z

j

j

Z

Z

Complex Numbers in Polar Form( 極座標 )

• Complex numbers can be expressed in polar form (極座標 ). Examples of complex numbers in polar form are :

The length of the

arrow that represents a complex

number Z is denoted as |Z|

and is called the magnitude

( 幅值 or 大小 ) of the complex number.

• Using the magnitude |Z| , the real part x, and the imaginary part y form a right triangle ( 直角三角形 ).

• Using trigonometry, we can write the following relationships:

These equations can be used to convert numbers from polar to rectangular form.

(A.3)

(A.4)

(A.1)

(A.2)

Example A.2 Polar-to Rectangular Conversion

formr rectangula to305 Convert Z3

j2.5 4.33 jy x 305 Z

2.5 )sin(30 5 )sin( Z y

4.33 )cos(30 5 )cos( Z x

page) (pre. A.4 and A.3Equation Using

:Solution

3

Example A.3 Rectangular-to-Polar Conversion

form.polar toj510- Zand j5 10 Convert Z 65

11.185(-10)yx Z

11.18510yx Z

numbers theof

each of magnitudes thefind

toA.1Equation using First,

:Solution

2226

266

2225

255

10-10

55

Z6 Z5

11.18153.43o

26.57o

Figure A.4

153.4311.18

j510 Z

153.43 57.26180

)x

yarctan(180

0.5- 10-

5

x

y)tan(

For Z

6

6

66

6

66

6

A.2.Equation use weangles, thefind To

26.5711.18

j510Z

26.57)arctan(0.5

0.5 10

5

x

y)tan(

For Z

5

5

5

55

5

• The procedures that we have illustrated in Examples A.2 and A.3 can be carried out with a relatively simple calculator. However, if we find the angle by taking the arctangent of y/x, we must consider the fact that the principal value of the arctangent is the true angle only if the real part x is positive. If x is negative,

we have:

Euler’s Identities

• The connection between sinusoidal signals and complex number is through Euler’s identities, which state that

and

• Another form of these identities is and)sin()cos( je j

2)cos(

jj ee

j

ee jj

2)sin(

)sin()cos( je j

• is a complex number having a real part of and an imaginary part of

• The magnitude is

je)cos( )sin(

1)(sin)(cos 22 je

• The angle of

• A complex number such as can be written as

• We call the exponential form ( 指數形式 )of a complex number.

)sin()cos(1

)sin()cos(1

je

jej

j

ise j

A)sin()cos()1( jAAAeAA j

jAe

• Given complex number can be written in three forms:– The rectangular form– The polar form– Exponential form

Example A.4 Exponential Form of a Complex Number

• Solution:

66.85)]60sin()60[cos(10)(10

10601060

60

jjeZ

eZj

j

planecomplex in thenumber Sketch the

forms.r rectangula and lexponentiain 6010number Zcomplex theExpress o

Arithmetic Operations in Polar and Exponential Form

• To add (or subtract) complex numbers, we must first convert them to rectangular form. Then, we add (or subtract) real part to real part and imaginary to imaginary.

• Two complex numbers in exponential form:

• The polar forms of these numbers are

22211 and 1 jj eZZeZZ

222111 and ZZZZ

• For multiplication of numbers in exponential form, we have

• In polar form, this is

2121212121

jjj eZZeZeZZZ

2121221121 ZZZZZZ

))sin()(cos(

))cossincos(sinsinsincos(cos

)sin(cos)sin(cos

212121

1221212121

22211121

jZZ

jZZ

jZjZZZProof:

( )

• Now consider division:

• In polar form, this is

21

2

1

2

1

2

1

2

1

j

j

j

eZ

Z

eZ

eZ

Z

Z

212

1

22

11

2

1

Z

Z

Z

Z

Z

Z

Example A.5Complex Arithmetic in Polar Form

• Solution:

Before we can add (or subtract) the numbers, we must convert them to rectangular form.

form.polar in and ,ZZ , find ,455 and 6010Given 21

2

12121 ZZZZZZ

152455

6010

105504556010

2

1

21

Z

Z

ZZ

54.354.3)45sin(5)45cos(5455

66.85)60sin(10)60cos(106010

2

1

jjZ

jjZ

The sum as Zs:

Convert the sum to polar form:

Because the real part of Zs is positive, the correct angle is the principal value of the arctangent.

2.1254.854.354.366.8521 jjjZZZ s

55)43.1arctan( 43.154.8

2.12tan

9.14)2.12()54.8( 22

ss

sZ

559.1421 ZZZ s