Complex NumbersComplex Numbers

1 August 2006 Slide 2

DefinitionDefinition• A complex number z is a number of the form

where

• x is the real part and y the imaginary part, written as x = Re z, y = Im z.

• j is called the imaginary unit

• If x = 0, then z = jy is a pure imaginary number.

• The complex conjugate of a complex number, z = x + jy, denoted by z* , is given by

z* = x – jy.

• Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

jyx 1j

1 August 2006 Slide 3

Complex PlaneComplex PlaneA complex number can be plotted on a plane with two

perpendicular coordinate axes The horizontal x-axis, called the real axis

The vertical y-axis, called the imaginary axis

P

z = x + iy

x

y

O

Represent z = x + jy geometrically as the point P(x,y) in the x-y plane, or as the vector from the origin to P(x,y). OP

��������������

The complex planex-y plane is also known as the complex plane.

1 August 2006 Slide 4

Polar CoordinatesPolar Coordinates

With siny r cos ,x r

z takes the polar form:

r is called the absolute value or modulus or magnitude of z and is denoted by |z|.

*22 zzyxrz

22

* ))((

yx

jyxjyxzz

Note that :

)sin(cos jrz

1 August 2006 Slide 5

Complex plane, polar form of a complex number

Pz = x + iy

x

y

O

Im

Re

θ

Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x-axis to OP in the above figure.

x

y1tanFrom the figure,

1 August 2006 Slide 6

θ is called the argument of z and is denoted by arg z. Thus,

For z = 0, θ is undefined.

A complex number z ≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually

The value of θ that lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.

0tanarg 1

z

x

yz

,...2,1,0,2tan 1

nn

x

y

1 August 2006 Slide 7

Euler Formula – Euler Formula – an alternate polar forman alternate polar form

The polar form of a complex number can be rewritten as :

This leads to the complex exponential function :

jre

jyxjrz

)sin(cos

jj

jj

eej

ee

2

1sin

2

1cos

Further leads to :

yjye

eeeex

jyxjyxz

sincos

1 August 2006 Slide 8

In mathematics terms, is referred to as the argument of z and it can be positive or negative.

In engineering terms, is generally referred to as phase of z and it can be positive or negative. It is denoted as

The magnitude of z is the same both in Mathematics and engineering, although in engineering, there are also different interpretations depending on what physical system one is referring to. Magnitudes are always > 0.

The application of complex numbers in engineering will be dealt with later.

z

1 August 2006 Slide 9

x

+1

x

z1

z2

Im

Re-2

r1

r2

111

jerz

222

jerz 0,,, 2121 rr

1 August 2006 Slide 10

A complex number, z = 1 + j , has a magnitude

2)11(|| 22 z

Example 1Example 1

rad24

21

1tan 1

nnzand argument :

Hence its principal argument is : Arg / 4z rad

Hence in polar form :

424

sin4

cos2 j

ejz

1 August 2006 Slide 11

A complex number, z = 1 - j , has a magnitude

2)11(|| 22 z

Example 2Example 2

rad24

21

1tan 1

nnzand argument :

Hence its principal argument is : rad

Hence in polar form :

In what way does the polar form help in manipulating complex numbers?

4

zArg

4sin

4cos22 4

jezj

1 August 2006 Slide 12

What about z1=0+j, z2=0-j, z3=2+j0, z4=-2?

Other ExamplesOther Examples

5.01

1

105.0

1

je

jz

5.01

1

105.0

2

je

jz

02

2

020

3

je

jz

2

2

024

je

jz

1 August 2006 Slide 13

●

●

●

Im

Re

z1 = + j

z2 = - j

z3 = 2z4 = -2●

5.0

1 August 2006 Slide 14

Arithmetic Operations in Polar FormArithmetic Operations in Polar Form

• The representation of z by its real and imaginary parts is useful for addition and subtraction.

• For multiplication and division, representation by the polar form has apparent geometric meaning.

1 August 2006 Slide 15

Suppose we have 2 complex numbers, z1 and z2 given by :

2

1

2222

1111

j

j

erjyxz

erjyxz

2121

221121

yyjxx

jyxjyxzz

))((

21

2121

21

21

j

jj

err

ererzz

Easier with normal form than polar form

Easier with polar form than normal form

magnitudes multiply! phases add!

1 August 2006 Slide 16

For a complex number z2 ≠ 0,

)(

2

1))((

2

1

2

1

2

1 2121

2

1

jj

j

j

er

re

r

r

er

er

z

z

magnitudes divide!phases subtract!

2

1

2

1

r

r

z

z

2121 )( z

1 August 2006 Slide 17

Given the transfer function model :

Generally, this is a frequency response model if s is taken to be .

20( )

1H s

s

A common engineering problem involving A common engineering problem involving complex numberscomplex numbers

js

In Engineering, you are often required to plot the frequency response with respect to the frequency, .

1 August 2006 Slide 18

1

10tan

0

1

101

20

110

20)10(

j

j

e

e

jjH

Let’s calculate H(s) at s=j10.

01

10

3.84rad47.1)10(tan0)10(

dB98.5dB2log202101

20)10(

jH

jH

x

03.84 Re

Im

2

47.12 ie

0201

20)0(

0

0

j

j

e

esHFor a start :

1 August 2006 Slide 19

1

1tan

0

1

2

20

11

20)1(

j

j

e

e

jjH

Let’s calculate H(s) at s=j1.

01

10

45rad7854.0)1(tan0)1(

dB23dB142.14log20142.142

20)1(

jH

jH

x

03.84 Re

Im

2

)10(iH

x

045

)1(iH

1 August 2006 Slide 20

What happens when the frequency tends to infinity?

?01

20)(

jsH

js

0

tan

900

20

1

20)( 1

jjs ej

sH

When the frequency tends to infinity, H(s) tends to zero in magnitude and the phase tends to -900!

1 August 2006 Slide 21

Polar Plot of H(s) showing the magnitude and phase of H(s)

1 August 2006 Slide 22

Frequency response of the systemFrequency response of the system

Alternate view of the magnitude and phase of H(s)