complex numbers and complex-valued functions

7/31/2019 Complex Numbers and Complex-Valued Functions

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EECE 301

Signals & Systems

Prof. Mark Fowler

Discussion #1

Complex Numbers and Complex-Valued Functions Reading Assignment: Appendix A of Kamen and Heck


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Complex Numbers

Complex numbers arise as roots of polynomials.

1))((

1))((

11 2

=

=

==

jj

jj

jj

Rectangular form of a complex number:

}Im{

}Re{

zb

zajbaz

=

=+=

real numbers

Recall that the solution of

differential equationsinvolves finding roots of the

characteristic polynomial

So

Definition of

imaginary #j

and some

resulting

properties:

)()())((:

)()()()(:

bcadjbdacjdcjbaMultiply

dbjcajdcjbaAdd

++=++

+++=+++

The rules of addition and multiplication are straight-forward:


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Polar Form

jrez =

Polar form an alternate way to express a

complex number

Polar Form

good for multiplication and division

r> 0

Note: you may have learned polar form as r we will NOT use that here!!The advantage of therej is that when it is manipulated using rules of

exponentials and it behaves properly according to the rules of complex #s:

yxyxyxyx

aaaaaa+

== /))((

Dividing Using Polar Form

( )( ))(

2

1

2

1 21

2

1

=j

j

j

err

erer

2

2

222

111 jj ererz==

Multiplying Using Polar Form

( )( ))(

2121 2121 +

=jjj

errerer

( )njnn

jnnnjn

erz

errez

//1/1

=

==


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b

a

z = a + jb

Im

Re

r

Geometry of Complex Numbers

We need to be able convert between Rectangular and Polar Forms this is made

easy and obvious by looking at the geometry (and trigonometry) of complex #s:

r

a

b

cos

sin

ra

rb

=

=

=

+=

a

b

bar

1

22

tan

Conversion Formulas


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Since cos +jsin has the same expansion as e j we can say that:

jej =+ sincos

From Calc II: ...!6!4!2

1cos642

++=

...!7!5!3

sin753

++=

jjjjj

...!4!3!2

1sincos432

+++=+

j

jj

...

!4!3!2

1432

+++++=xxx

xex

Also From Calc II:

...

!4!3!2

1432

+++=

j

jej

Q: Why the formrej for polar form?? Start with trigonometry:

[ ] sincos)sin(cos jrrjrjbaz +=+=+=


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Complex Exponentials vs. Sines and Cosines

Eulers Equations:

(A)

(B)

(C)

(D)


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Summary of Rectangular & Polar Forms

Warning: Your calculatorwill give you the wrong answerwhenever you have a < 0. In other words, forz values that lie in

the II and III quadrants.

You can always fix this by either adding or subtracting .

Use common sense looking at the signs ofa and b will tell

you what quadrantz is in make sure your angle agrees with

that!!!

Summary of Rectangular & Polar Forms

==

+==

=

abz

barz

rrez j

1

22

tan

],(0

Polar Form:

sin}Im{

cos}Re{

rbz

raz

jbaz

==

==+=

Rect Form:


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Conjugate ofZ

jjrezrez

jbazjbaz

==

=+=

*

*

Denoted as

zz or*

Properties ofz*

222*

*

))((.2

}Re{2.1

zbajbajbazz

zzz

=+=+=

=+

U it Ci l


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Unit Circle

Im

Re

1

Unit Circle: A set of complex numbers with

magnitude of 1 (|z| =1)

Allz on the unit circle look like: je1

Re

Im

Four special points on the unitcircle:

2j

ej =

01 je=

2j

ej

=

je=1

=

=

=

=

=

=

=

,...10,6,2,1

,...8,4,0,1

...,11,7,3,

...,9,5,1,

integersallfor1

integereven1,

integerodd,1

2

2

n

n

nj

nj

e

e

n

ne

jn

jn

jn

Know these!!!


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A sinusoid is completely defined by its three parameters:

-AmplitudeA (for EEs typically in volts or amps or other physical unit)

-Frequency in radians per second

-Phase shift in radians

Tis the period of the sinusoid and is related to the frequency


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Phase shift (often just shortened to phase) shows up explicitly in the equation but

shows up in the plot as a time shift (because the plot is a function of time).

Q: What is the relationship between the plot-observed time shift and the equation-

specified phase shift?

A: We can write the time shift of a function by replacing tby t + to (more on this

later, but you should be able to verify that this is true!) Then we get:

)sin())(sin()( ooo ttAttAttf +=+=+

=

So we get that: = to (unit-wise this makes sense!!!)

Frequency can be expressed in two common units:

-Cyclic frequency: f= 1/T in Hz (1 Hz = 1 cycle/second)

-Radian Frequency: = 2/T(in radians/second)

From this we can see that these two frequency units have a simple conversion

factor relationship (like all other unit conversions e.g. feet and meters):

f 2=


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In circuits you used phasors (well call them static phasors here) the point of

using them is to make it EASY to analyze circuits that are driven by a single sinusoid.

Here is an example to refresh your memory!!

Find output voltage of the following circuit:

R = 1

L = 2mH4

1000cos5)( += ttx ?)( =ty

Use phasor and impedance ideas:

45:InputofPhasor

2:InductorofImpedance

j

L

ex

jLjZ

=

==

Use voltage divider to find output:

[ ]46.04 89.0521

2 j

jee

j

jxy

=

+=

Output phasor:

25.145.4 jey =

Output signal: )25.11000cos(45.4)( += tty


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Note that in using static phasors there was no need to carry around the

frequency it gets suppressed in the static phasor

BUT if you have multiple driving sinusoids (each at its own unique frequency)then youll need to keep that frequency in the phasor representation that leads to:

{ } )cos(Re)sin()cos(

oootjj

o

oo

tj

tAeeA

tjte

o

o

+=

+=

system)cos( ooo tA +

input

rotating

phasorModified

system

tjj

ooo eeA

Rotating phasor

output

rotating

phasor

Rotating Phasors

Keeping the frequency part

static phasor part

tjj

o

tj

oooo

oo

oo

eeA

eAtA

=

+ + )()cos(

[ ]


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[ ]

( )

[ ]

{ }

)cos(2

)(~Re22

)(*~)(~

)(*~:isWhat)cos()(

)(~If

oo

tjj

tj

oo

tj

tA

txtxtx

eAe

Ae

txtAtx

Aetx

oo

oo

oo

+=

=+

=

=

+=

=

+

++

Because rotating phasors take the value of a complex number at each

Instant of time they must follow all the rules of complex numbersEspecially: EULERS EQUATIONS!!

Rotating Phasors


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Rotating Phasors

Eulers Equations


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Viewing rotating phasor on the complex plane

complex numbers and complex-valued functions

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