Download Chapter 15 Complex Numbers

Chapter 15
Complex Numbers
A few operations with complex numbers
were used earlier in the text and it was
assumed that the reader had some basic
knowledge of the subject. The subject will
now be approached from a broader
perspective and many of the operations will
be developed in detail. To deal with Fourier
analysis in Chapter 16, complex number
operations are required.
1
Complex Numbers and Vectors
In a sense, complex numbers are twodimensional vectors. In fact, some of the
basic arithmetic operations such as
addition and subtraction are the same as
those that could be performed with the
spatial vector forms of Chapter 14 in two
dimensions. However, once the process
of multiplication is reached, the theory of
complex number operations diverges
significantly from that of spatial vectors.
2
Rectangular Coordinate System
The development begins with the twodimensional rectangular coordinate
system shown on the next slide. The two
axes are labeled as x and y. In complex
variable theory, the x-axis is called the
real axis and the y-axis is called the
imaginary axis. An imaginary number is
based on the definition that follows.
i  1
3
Complex Plane
y
Imaginary Axis
Real Axis
x
4
Form of Complex Number
Imaginary Axis
( x, y )
z  x  iy
z
r

Real Axis
5
Conversion Between Forms
Polar to Rectangular:
x  r cos
y  r sin 
Rectangular to Polar:
r x y
2
2
  ang(z )  tan
1
y
x
6
Euler’s Formula
i
e  cos   i sin 
z  r cos   ir sin   r (cos   i sin  )
z  re
i
Common Engineering Notation:
re
i
r 
7
Example 15-1. Convert the following
complex number to polar form:
z  4  i3
r  x  y  (4)  (3)  5
2
2
2
2
3
  tan
 36.87  0.6435 rad
4
1
z  536.87 or
z  5e
i 0.6435
8
Example 15-2. Convert the following
complex number to polar form:
z  4  i3
r  x  y  (4)  (3)  5
2
2
2
2
 3 
1 3
  tan    180  tan
4
 4 
 180  36.87  143.13  2.498 rad
1
z  5e
i 2.498
9
Example 15-3. Convert the following
complex number to rectangular form:
z  4e
i2
x  4cos 2  1.6646
y  4sin 2  3.6372
z  1.6646  i3.6372
10
Example 15-4. Convert the following
complex number to rectangular form:
z  10e
i
x  10 cos(1)  5.4030
y  10sin(1)  8.4147
z  5.4030  i8.4147
11
Addition of Two Complex Numbers
z1  x1  iy1
z 2  x2  iy2
z sum  z1 + z 2
 x1  iy1  x2  iy2
 x1  x2  i ( y1  y2 )
A geometric interpretation of addition is
shown on the next slide.
12
Addition of Two Complex Numbers
Imaginary Axis
z2
z sum
z1
z2
Real Axis
13
Subtraction of Two Complex Numbers
z1  x1  iy1
z 2  x2  iy2
z diff  z1 - z 2
 x1  iy1  ( x2  iy2 )
 x1  x2  i ( y1  y2 )
A geometric interpretation of subtraction
is shown on the next slide.
14
Subtraction of Two Complex Numbers
Imaginary Axis
z1
z 2
z diff
z2
z 2
Real Axis
15
Example 15-5. Determine the sum
of the following complex numbers:
z 1  5  i3
z 2  2  i7
z sum  z1  z 2
 5  i3  2  i 7
 7  i4
16
Example 15-6. For the numbers that
follow, determine zdiff = z1-z2.
z 1  5  i3
z 2  2  i7
z diff  z1 - z 2
 5  i3  (2  i7)
 3  i10
17
Multiplication in Polar Form
z1  r1e
i1
i 2
z 2  r2 e
z prod = z1z 2
  r1e
i1
 r1r2e
 r e 
i 2
2
i (1  2 )
18
Division in Polar Form
z1  r1e
i1
z 2  r2 e
z div
i 2
z1  r1e 
=

i 2
z 2  r2 e 
i1
r1 i (1 2 )
 e
r2
19
Multiplication in Rectangular Form
z prod
z1  x1  iy1
z 2  x2  iy2
 ( x1  iy1 )( x2  iy2 )
 x1 x2  ix1 y2  ix2 y1  i y1 y2
2
z prod  x1 x2  y1 y2  i ( x1 y2  x2 y1 )
20
Complex Conjugate
Start with
z  x  iy  re
i
The complex conjugate is
z  x  iy  re
 i
The product of z and z is
(z)( z )  x  y  r
2
2
2
21
Division in Rectangular Form
z div
z div
z1 x1  iy1


z 2 x2  iy2
( x1  iy1 )( x2  iy2 )

( x2  iy2 )( x2  iy2 )
x1 x2  y1 y2  i ( x2 y1  x1 y2 )

2
2
x2  y2
x1 x2  y1 y2  i ( x2 y1  x1 y2 )

2
r
22
Example 15-7. Determine the product
of the following 2 complex numbers:
 i 0.7
z 2  5e
z1  8e
i2
 i 0.7
i1.3
z 3 = z1z 2   8e  5e
  40e
i2
z 3  40(cos1.3  i sin1.3)
 40(0.2675  i0.9636)
 10.70  i38.54
23
Example 15-8. Repeat previous
example using rectangular forms.
z1  8e  8(cos 2  i sin 2)
i2
 8(0.4162  i0.9093)  3.329  i7.274
z 2  5ei 0.7  5(cos 0.7  i sin 0.7)
 5(0.7648  i0.6442)  3.824  i3.221
z 3 = z1z 2  (3.329  i7.274)(3.824  i3.221)
 12.73  23.43  i (27.82  10.72)
 10.70  i38.54
24
Example 15-9. Determine the quotient
z1/z2 for the following 2 numbers:
z1  8e
z 2  5e
i2
 i 0.7
i2
z1
8e
i 2.7
z4 =
 i 0.7  1.6e
z 2 5e
z 4  1.6(cos 2.7  i sin 2.7)
 1.6(0.9041  i0.4274)
 1.447  i0.6838
25
Example 15-10. Repeat previous
example using rectangular forms.
z1 3.329  i 7.274
z4 =

z2
3.824  i3.221
(3.329  i7.274) (3.824  i3.221)
z4 
(3.824  i3.221) (3.824  i3.221)
12.73  23.43  i(27.82  10.72)

14.62  10.37
36.16  i17.10

 1.446  i0.6840
25.00
26
Exponentiation of Complex
Numbers: Integer Power
zpower = (z)
N
i N
N iN
z power  (re )  r e
 r cos N  ir sin N
N
cos N  Re(e
N
iN
sin N  Im(e
iN
)
)
27
Roots of Complex Numbers
z  re
z roots   re
zroots  r
i (  2 n )

i (  2 n ) 1/ N
  2 n 
i 
1/ N  N N 
e
r
1/ N i ( / N  2 n / N )
e
for n  0,1, 2...., N  1
1/ N i / N
zprincipal  r
e
28
Example 15-11. For the value of z
below, determine z6 = z6.
z  3  i 4  5e
z 6  (z )
i 0.9273
6
 (5e
)  15, 625e
i 0.9273 6
i 5.5638
z 6  15, 625  cos 5.5638  i sin 5.5638 
 11, 753  i10, 296
29
Example 15-12. Determine the 4
roots of s4 + 1 = 0.
s  1  1e
4
se
2 

i  n

4
4


s1  e
i
4
i
3
4
i
5
4
s2  e
s3  e

s4  e
i
for
i (   2 n )
n  0,1, 2,3
 0.7071  i0.7071
7
4
 0.7071  i0.7071
 0.7071  i0.7071
 0.7071  i0.7071
30
MATLAB Complex Number Operations:
Entering in Rectangular Form
>> z = 3 + 4i
z=
3.0000 + 4.0000i
>> z = 3 + 4j
z=
3.0000 + 4.0000i
The i or j may precede the value, but the
multiplication symbol (*) is required.
31
MATLAB Complex Number Operations:
Entering in Polar Form
>> z = 5*exp(0.9273i)
z=
3.0000 + 4.0000i
>> z = 5*exp((pi/180)*53.13i)
z=
3.0000 + 4.0000i
This result indicates that polar to
rectangular conversion occurs automatically
upon entering the number in polar form.
32
MATLAB Rectangular to Polar Conversion
>> z = 3 + 4i
z=
3.0000 + 4.0000i
>> r = abs(z)
r=
5
>> theta = angle(z)
theta =
0.9273
33
Other MATLAB Operations
>> z_conj = conj(z)
z_conj =
3.0000 - 4.0000i
>>real(z)
ans =
3
>> imag(z)
ans =
4
34