Complex Numbers S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of...

ComplexNumbers

S. Awad, Ph.D.

M. Corless, M.S.E.E.

E.C.E. Department

University of Michigan-Dearborn

Math Review with Matlab:

Complex Number Theory

U of M-Dearborn ECE DepartmentMath Review with Matlab

2

Complex Numbers: Complex Number Theory

Complex Number Theory

General Complex Numbers General Complex Numbers in Matlab Argand Diagrams Exponential Form Polar Form Polar Form in Matlab


3


Representing Complex Numbers The conventional representation of a complex number z

is the sum of a real part x and an imaginary part y

iyxz

i is the imaginary unit where: 12 i

Complex Real Imaginary

Conventional representation is also referred to as the rectangular form of a complex number


4


Functional Notation In different applications, the imaginary unit may be

represented as i or j and may be placed before or after the imaginary part

yjxzyixz

jyxziyxz

In functional notation, it is sometimes convenient to write:

zx Re zy Im


5


Matlab Complex Numbers Matlab complex numbers consist of a real portion plus an

imaginary portion

For example, use Matlab to find the square root of -1

» sqrt(-1)

ans = 0 + 1.0000i

1i

Real Portion = 0 Imaginary Portion = -1


6


Entering Complex Numbers Both the variables i and j in the Matlab workspace are

reserved for representing imaginary numbers

Placing i or j after a number defines it as imaginary

Example: A complex number with real portion = 1 and imaginary portion = 2 can be entered in two ways:

» z = 1+2i

z = 1.0000 + 2.0000i

» z = 1+2j

z = 1.0000 + 2.0000i

OR


7


Real and Imag Commands The real and imag functions extract the real and

imaginary portion of a complex number respectively

Example: » z1= 2+4iz1 = 2.0000 + 4.0000i

» z1_re = real(z1)z1_re = 2

» z1_im = imag(z1)z1_imag = 4


8


Argand Diagram A complex number can be

represented as a Point P(x,y) in the xy-plane (Cartesian plane)

Ordered Pair Notation

yxz ,

This representation is called an Argand Diagram

The xy-plane is often referred to as the Complex Plane or the z-plane

zIm

zRex

y yxz ,


9


Feather Command The feather command draws complex numbers as

arrows in the xy plane

» z1=1+2j;» z2=3+3j;» z3=3+j;» feather(z1,'r');» hold on» feather(z2,'b');» feather(z3,'g');» feather(z3,'k');» ylabel('Imaginary');» xlabel('Real');


10


Exponential Representation Sometimes an exponential representation of a complex

number is easier to manipulate than the rectangular sum of the real and imaginary part

The Complex Exponential Function ez will be used to represent a complex number whose Taylor Series Expansion is:

!3!2

1)exp(32 zz

zze z

We can assume that ez will be valid for all z


11


Exponential Series If z = i, purely imaginary, where is real, the complex

exponential function can be written using a Taylor Series Expansion:

!3

)(

!2

)(1

32 iiiei

Which can be separated into real and imaginary components since i raised to even powers of n will be -1

!5!3!4!21

5342 iei


12


Derivation of Euler’s Equation

Knowing that the Taylor Series Expansions for sine and cosine functions are:

Through substitution, Euler’s Equation is derived:

!5!3!4!21

5342 iei

sincos iei

!4!2

1cos42

!5!3sin

53


13


Exponential Polar Form

Through Euler’s Equation we see that:

iyxre

irrei

i

sincos

This representation of a complex number is referred to as the Exponential Polar Form:

irez Sometimes Polar Form is written

using the shortened notation: rz


14


Plotting Polar Form Graphically depicting Euler’s

Equation in the First Quadrant of an Argand Diagram we see:

zIm

zRex

y

r yxz ,

sincos rirrez i

r refers to the radius from z to the origin, commonly called the Magnitude of z

is the Angle of z with respect to the Real Axis measured in degrees or radians

sin)Im(

cos)Re(

rzy

rzx


15


Magnitude Through use of the Pythagorean theorem, the Magnitude

(radius) of a complex number always has the relationship:

22 yxr This relationship always holds true regardless of which

quadrant of the Argand diagram that the number lies in

The Magnitude of a complex number z=x+iy is denoted using Absolute Value notation

22 yxrz


16


Angle

Determining the angle of a complex number depends on the quadrant of the Argand diagram

The angle, , of a complex number z denoted by: z

Measuring the angle counter-clockwise from the x-axis gives a positive

Measuring the angle clockwise from the x-axis gives a negative

2702

390

2

z

Example: The angle of z=0+j can be represented in multiple ways


17


Angle of Quadrants I and IVQuadrant I

zIm

zRex

y

r yxz ,

zIm

zRe

x

y

r

yxz ,

x

y1tan

x

y

x

y 11 tantan

Quadrant IV


18


Angle of Quadrants II and IIIQuadrant II

zIm

zRex

y

r

yxz , zIm zRex

y

r

yxz ,

x

y1tan

x

y1tan

Quadrant III


19


Purely Real or Imaginary Numbers

Rectangular Rectangular Polar Polar

Short Long Degrees Radians

iiji-ji


20


Abs and Angle Commands The abs command in Matlab returns the magnitude of a

complex number The angle command returns the angle of a complex

number in radians

Remember that the conversion between radians and degrees is:

rad

180rad


21


» z=1+i;» r=abs(z)r = 1.4142

» theta_rads=angle(z)theta_rads = 0.7854

» theta_degs=theta_rads*180/pitheta_degs = 45

Polar Example Use Matlab to find the magnitude and angle of z=1+i

4142.1

211 22

r

r

7854.04

1

1tan 1

RADS

45DEGREES


22


Entering Polar Form Complex numbers can be directly entered into Matlab

using the exponential polar form

» z=2*exp(i*pi/2)

z = 0.0000 + 2.0000i

Example: Create a complex number z with a magnitude of 2 and an angle of /2 radians

iz

z

z

ezi

2

902

2

2

2

2


23


Polar Plotting Matlab’s compass(z) or compass(x,y) command can

be used to draw complex numbers on a polar plot Note that z is a complex number in rectangular form

» z1=3+3i;» compass(z1)» hold on» compass(4,-3)

Angles are displayed in degrees


24


Plotting Example Example: Plot the following complex numbers by hand.

Use Matlab’s compass function to verify your results.

21505.2

488

302

65

243

6021

izz

ezjz

ezzi

i


25


Matlab Verification

z(1) = 2;z(2) = 3*exp(i*60*(pi/180));z(3) = -(8^0.5) + j*(8^0.5);z(4) = 4*exp(-i*pi/2);z(5) = 2.5*exp(-i*150*(pi/180));z(6) = 2i;compass(z)

21505.24

88302

652

4

360

21

izzez

jzezzi

i


26


Polar Plot

2

1505.2

4

88

3

02

6

5

24

3

602

1

iz

z

ez

jz

ez

z

i

i

1z

2z3z

4z

6z

5z


27


Summary Representing complex numbers in rectangular,

exponential, and polar forms

Using Euler’s Equation to represent real and imaginary parts of complex numbers

Determining magnitude and angles of complex numbers

Graphing complex numbers using Argand Diagrams

Complex Numbers S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of...

Documents

Transcript of Complex Numbers S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of...