Matlab complex numbers

11
Complex Numbers in MATLAB
Lecture Series - 2
by
Shameer Koya

2
Complex Plane
Real Axis x
y
Imaginary Axis

3
Form of Complex Number
Real Axis
Imaginary Axis
( , )x y

z
r
x iy z

4
Conversion Between Forms
cosx r 
siny r 
Polar to Rectangular:
Rectangular to Polar:
2 2
r x y 
1
ang( ) tan
y
x
 
 z

5
Euler’s Formula
cos sin (cos sin )r ir r i      z
i
re
z
i
re r

Common Engineering Notation:
cos sini
e i
  

6Convert the complex number to polar form:
4 3i z
2 2 2 2
(4) (3) 5r x y    
1 3
tan 36.87 0.6435 rad
4
 
  
5 36.87 or z
0.6435
5 i
ez

7Convert the complex number to polar form:
4 3i  z
2 2 2 2
( 4) (3) 5r x y     
1 13 3
tan 180 tan
4 4
180 36.87 143.13 2.498 rad
   
   
 
   
2.498
5 i
ez

8
Convert the complex number to
rectangular form:
2
4 i
ez
4cos2 1.6646x   
4sin2 3.6372y  
1.6646 3.6372i  z

9Addition of Two Complex Numbers
1 1x iy 1z
2 2x iy 2z
1 1 2 2
1 2 1 2( )
x iy x iy
x x i y y

   
   
sum 1 2z z + z
A geometric interpretation of addition is
shown on the next slide.

10
Addition of Two Complex Numbers
Real Axis
Imaginary Axis
1z
2z
2z
sumz

11
Subtraction of Two Complex Numbers
1 1x iy 1z
2 2x iy 2z
A geometric interpretation of subtraction
is shown on the next slide.
1 1 2 2
1 2 1 2
( )
( )
x iy x iy
x x i y y

   
   
diff 1 2z z - z

12
Subtraction of Two Complex Numbers
Real Axis
Imaginary Axis
1z
2z
 2z
diffz
 2z

13Multiplication in Polar Form
1
1
i
re
1z
2
2
i
r e
2z
  1 2
1 2
1 2
( )
1 2
i i
i
re r e
rr e
 
 


prod 1 2z = z z

14Division in Polar Form
1
1
i
re
1z
2
2
i
r e
2z
 
 
1
2
1 2
1
2
( )1
2
i
i
i
re
r e
r
e
r


 


1
div
2
z
z =
z

15
Multiplication in Rectangular Form
1 1x iy 1z
2 2x iy 2z
1 1 2 2
2
1 2 1 2 2 1 1 2
( )( )x iy x iy
x x ix y ix y i y y
  
   
prodz
1 2 1 2 1 2 2 1( )x x y y i x y x y   prodz

16
Complex Conjugate
Start with
i
x iy re
  z
The complex conjugate is
i
x iy re 
  z
2 2 2
The product of and is
( )( )
z z
x y r  z z

17
Division in Rectangular Form
1 1
2 2
x iy
x iy

 

1
div
2
z
z
z
1 1 2 2
2 2 2 2
1 2 1 2 2 1 1 2
2 2
2 2
1 2 1 2 2 1 1 2
2
( )( )
( )( )
( )
( )
x iy x iy
x iy x iy
x x y y i x y x y
x y
x x y y i x y x y
r
 

 
  


  

divz

18
Exponentiation of Complex Numbers:
Integer Power
N
powerz = (z)
( )
cos sin
i N N iN
N N
re r e
r N ir N
 
 
 
 
powerz
cos Re( )iN
N e 
 
sin Im( )iN
N e 
 

MATLAB Complex Operations
 Complex number
 Construct complex data from real and imaginary
components
>> c = complex(a,b)
>> z = 3 + 4i
z =
3.0000 + 4.0000i
>> z = 3 + 4j
z =
3.0000 + 4.0000i
19

20
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.

21
Rectangular to Polar Conversion
>> z = 3 + 4i
z =
3.0000 + 4.0000i
>> r = abs(z)
r =
5
>> theta = angle(z)
theta =
0.9273

22
Real and Imaginary and Conjugate
>>real(z)
ans =
3
>> imag(z)
ans =
4
>> z1 = conj(z)
z1 =
3.0000 - 4.0000i

Complex Algebra
 Z1 = 3+4i
 Z2 = 2-5i
 Z3 = Z1+Z2
 Z4 = Z1-Z2
 Z5 = Z1*Z2
 Z6 = Z1/Z2
23

Plotting complex number
 Use simple ‘plot’ function
 Plot (real, imaginary)
 Use ‘compass’ function
 Compass (z)
24

Exercise
 a= 3+2i b= 4+5i
 Find
 Magnitude of a
 Angle of a
 Real part of a
 Imaginary part of a
 Conjugate of a
 Plot a and b using ‘plot’ and ‘compass’
 a+b, a-b, a/b, a*b, a2
25

Matlab complex numbers

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Matlab complex numbers