appendix a complex numbers. basic complex-number concepts complex numbers involve the imaginary...
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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
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