Fundamentals of Electrical Engineering 3nts.uni-due.de/downloads/get3-ise/GET3ISE_K2_PO08...

Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 1

FachgebietNachrichtentechnische Systeme

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Fundamentals of ElectricalEngineering 3

Professor Dr.-Ing. Ingolf Willms and

Professor Dr. Dr.-Ing. Adalbert Beyerbased on the script of

Professor Dr. Dr.-Ing. Ingo Wolff



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Fundamentals of Electrical Engineering 3Contents

1 Introduction2 Signal theory of determined signals and applications

2.1 Prefaces2.2 The Fourier series and applications to networks2.3 The Fourier transform and applications to systems

3 Switched circuits and the Laplace transform3.1 The Laplace transform3.2 Properties3.3 Applications



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Literature

• Literature for the lecture:R. Paul Elektrotechnik 2, Grundlagenbuch Netzwerke

Springer-Verlag, Heidelberg 1994

I. Wolff Grundlagen der Elektrotechnik 4Vorlesungs-Script

• Alternative Literature :W. Ameling Grundlagen der Elektrotechnik II

G. Bosse Grundlagen der Elektrotechnik IVB.I. Wissenschaftverlag Mannheim, Wien Zürich 1996

R. UnbehauenGrundlagen der Elektrotechnik ISpringer-Verlag, Heidelberg 1994



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1 Introduction

• GET3 contains predominantly theoretical bases to questinsof information technology

• Information technology: Developed from computer scienceand communications

• IT: Efficient data processing, storage and transport• IT contains 4 groups:

- Bases and technologies (G1)- Structures, procedure, programs (G2)- Devices, mechanisms, plants (G3)

- Applications (G4)



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Chapter overview:• 2.1 Prefaces

- Model for the information transfer- Signal classes- Description and modification of signals

• 2.2 Description of periodic signals (Fourier Series)- Approximation of functions with the Fourier series- Applications to networks

• 2.3 Description of aperiodic signals (Fourier Transform)- The Fourier integral in different forms- Examples- Properties of the Fourier transform- Application to systems (convolution and the Fourier transform)-

2 Signal theory of determined signals and applications



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2.1 Prefaces



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2.1.1 The Exponential Signal

( ) cos sinj ts t e t j t

( )ˆ ˆ ˆ( ) cos( ) Re Re where u uj t jj tuu t u t u e u e u u e

( )j t t j t pte e e e

For voltages it holds:

For increasing/decreasing signals:



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2.1.2 The Dirac Function

0 0( ) ( ) ( )t t t t dt

1( ) ( )at ta

If 1 then: ( ) ( )a t t

( ) ( ) ( )s t t s d

Definition:

Properties:

See formula above.



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2.1.2 The Dirac Function

( ) (0) 1 where ( ) 0 for 0d s

0

1( ) limT

tt rectT T

( )

0



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2.1.3 The Step Function

0 for 0( )

1 for 0t

tt

( ) ( )t

t d

( 1) ( ) ( ) ( )t

t d

1( )t

t0



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2.1.4 Periodic Signals

( ) ( ) where ,..., 1, 1,...,s t s t nT n

2 1 0( ) ( )n

s t s t nT

General formula:

Examples:

2 1 0( ) ( )nn

s t c s t nT



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2.1.4 Periodic Signals

1

02

0

( )

( )

n

n

ts t rectT

t nTs t rectT

Ttrect nT T

t

0n 1n 2n

0T 02T

T2 ( )s t



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2.1.5 Impulse Type Signals

11 for 2( )10 for 2

xrect x

x

1 1

2 0 1

2 1

1

x

( )rect x



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1 for 0( ) ( ) 0 for 0

1 for 0

ts t sign t t

t

t

( )sign t

1

0

1



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1 for ( )

0 otherwise

t t Tts t TT

t

1

T T

( ) ts tT



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2.1.6 Adjustment of Time and FrequencyFunctions

2 1( ) ts t a sb

2 0( )2ts t u rectT

2 1( ) ( )s t s t T

Case1: Compression & expansion

Case2: Shift

Example:



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1

2 0 1

0

( )

( )2

2

ts t rectT

ts t u s

tu rectT

T T t

2 ( )s t0u

Example:

1 12

0 12 1

1

x

( )rect x



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1

2 1

( ) ( )( ) ( )

( )

s t ts t s t

t

t

2 ( )s t

0

1

Case3: Mirroring (b = -1)

Example:

2 1( ) ( )s t s t



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1 2

3 2

( ) ( ) 3

( ) ( )3

t ts t rect s t rectT T

t Ts t s t T rectT

2 1( ) ( )s t s t T

3 2 1

1

( ) Replace by in ( )

t ts t as t s tb btas Tb

Expansion & shift:

Shift & expansion:

3s t

vT

1

t

3T



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1 0( ) ; ; 2ts t rect a u bT

2 ( ) t T Tts t rect rectT T T

3 0( )2

T Tt ts t arect u rectbT T T T

t2 vT

2T0u

3 ( )s t

Example:



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2 1( ) ( )s t s t

3 2 1 1( ) ( ) ( ( )) ( )s t s t T s t T s T t

4 1( ) ( )s t s t T

5 4 1 3( ) ( ) ( ) ( )s t s t s t T s t

Mirroring & shift:

New sequence: Shift & mirroring:



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( )t tr tT T

1( ) ts t rT

2 1( ) ( ) ts t s t rT

3 2( ) ( ) T ts t s t T rT

t

tvT

1( )s t2 ( )s t

3 ( )s t

Example with a ramp function r(t):



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vt T

4 1( ) ( ) t Ts t s t T rT

5 5( ) ( ) t Ts t s t rT

There are 4 cases:

5s t 4s t

vT vT t



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1 2 3

1 2 3

( ) ( ) where ( ) ( ) ( ( ))f x f y y f xf x f f x

01

1

( )f rect

All onsets can be extended to frequency functions:

Example:



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2.2 Description of non-sinusoidal, periodic functions

2.2.1 Approximation of functions– Motivation: Determination of characteristical functions and

parameters, data compression– Starting point: Given is f(t)

Desired is g(t), approximating f(t) in an interval with

t0

g(t)

f(t)

f(t)

1( ) ( ) by given ( )

n

i i ii

g t g t g t



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2.2.1 Approximation of functions

- Requirement: As small an error of the approximation as possible

- Definition of error function: ( ) ( ) ( )t f t g t

- Mean error: 2

1min min

2 1

1 ( ) ( )t

m tf t g t dt

t t

- Mean absolute error: 2

1

min min2 1

1 ( ) ( )t

mat

f t g t dtt t

- Mean square error: 2

1

2

min min2 1

1 ( ) ( )t

mq tf t g t dt

t t

Advantages/disadvantages of the error measures: • Cancelling of errors is possible in the case of mean error• Absolute error results in nonlinearity• Quadratic error measures is most frequent application



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0 1,2,...,

mq

i

i n

2

1

2

12 1

1 ( ) ( ) 0

nt

i itii

f t g t dtt t

2

1 12 1

1 2 ( ) ( ) ( ) 0

nt

i i iti

f t g t g t dtt t

2

2

11

1( ) ( ) ( ) ( )

t nt

i i i itit

f t g t dt g t g t dt

This corresponds to a set of equations, which can be solved forthe coefficients.

- Determination of coefficent:

- Hereby the following steps result:



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2 2 2 2 2

1 1 1 1 1

21 1 1 2 1 2 1 1( ). ( ) ( ) ( ). ( ) ... ( ) ( ) ... ( ) ( )

t t t t t

n nt t t t t

f t g t dt g t dt g t g t dt g t g t dt g t g t dt

2 2 2 2 2

1 1 1 1 1

2 1 1 2 2 2 2 2 2( ). ( ) ( ) ( ) ( ). ( ) ... ( ) ( ) ... ( ) ( ) t t t t t

n nt t t t t

f t g t dt g t g t dt g t g t dt g t g t dt g t g t dt

2 2 2 2 2

1 1 1 1 1

1 1 2 2( ). ( ) ( ). ( ) ( ) ( ) ... ( ) ( ) ... ( ) ( ) t t t t t

n n n n n n nt t t t t

f t g t dt g t g t dt g t g t dt g t g t dt g t g t dt

.

.

.



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2.2.2 Approximation by means of orthogonaler function systems

• Definition of orthogonal functions in interval (t1, t2 ) by means of real functionsgi(t). These functions should be continous in the interval.

• Chronecker‘s delta function is used:

• Onset for the approximation:

• Thus follows for the coefficients:

2

1

( ). ( )

and suitable

t

t

g t g t dt h

h

fürfür

1( ) ( )

n

i ii

g t g t

2

1

2

1

2

( ) ( )

( )

t

it

i t

it

f t g t dt

g t dt



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2.2.2 Approximation by means of orthogonaler function systems

One receives an orthonormal function system by means of the definitions

1 21 2

1 2

( ) ( )( ) ( )( ) , ( ) ,..., ( ) ,..., ( ) nn

n

g t g tg t g tG t G t G t G th h h h

To these applies: 2

1

( ) ( )

t

t

fürG t G t dt

für

Thus a function f(t) in the interval can be developed into a set of orthonormal functions by suitable coefficients. The result of theapproximation is then a function G(t). In summary it applies:

1( ) ( ) ( )

i ii

f t G t AG t

The coefficients Ai are the so-calledgeneralized Fourier coefficients:

2

1

( ) ( )t

i it

A f t G t dt



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2.2.3 Approximation of periodic, non-sinusoidalfunctions

Example of a function with the period T. This function is interpretable as repetition of one period.

f(t)

t0 T 2T

T

For one period it applies: ( ) ( ) 0,1,2,...,f t f t T

After Fourier any function, for which the Dirichlet conditions are fulfilledcan be represented in the following trigonometric form:

0

1( ) cos( ) sin( )

2af t a t b t



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2.2.2 Approximation of periodic, non-sinusoidalfunctions (Fourier series)

Dirichlet conditions (in practice fulfilled)• Function f(t) is either continous in the interval or has finitely many points of

discontinuity• Finite values of f(t) exist in the limit, if t approaches the point of discontinuity

from the right or from the left• The interval can be divided in such a manner that f(t) there is monotonousSentence of Dirichlet• With fulfilment of the Dirichlet conditions the Fourier series converges in the

entire interval• The value of the Fourier series is identical to function f(t) in continous areas• At points of discontinuity the value is alike: • At end points of the interval the value is alike:

0.5 ( 0) ( 0)f t f t

1 20.5 ( 0) ( 0)f t f t



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2.2.3 Fourier seriesAnalogy to the series expansion of orthogonal transforms

• To the series expansion applies usingorthogonal functions

• For the used functions theorthogonality can be shown:

• Otherwise applies as given above:

• Thus it is ensured that Fourier series isthe optimum approximation in thesquare mean sense (also in a terminated series)

2 0 2 0

1 0 1 0

sin( )sin( ) cos( ) cos( )2

t t T t t T

t t t t

Tt t dt t t dt

2

1

( ). ( ) t

t

g t g t dt h

1( ) ( )

n

i ii

g t g t

2

1

2

1

2

( ) ( )

( )

t

it

i t

it

f t g t dt

g t dt

0

1( ) cos( ) sin( )

2af t a t b t

with

2 coefficient sets are necessary, so thateven and odd function parts can berepresented.



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2.2.3 Fourier series

Thus applies to the determination of the Fourier-coeffizients of the trigonometric form:

2 0

2 0

1 0

2 0

1 0

1 0

0

2

( ) 11 ( )

21

t t T

t t Tt t

t t Tt t

t t

f t dta f t dt

Tdt

2 0

2 0

1 0

2 0

1 0

1 0

2

( ) cos( )2 ( )cos( )

cos ( )

t t T

t t Tt tt t T

t t

t t

f t t dta f t t dt

Tt dt

2 0

2 0

1 0

2 0

1 0

1 0

2

( )sin( )2 ( )sin( )

sin ( )

t t T

t t Tt tt t T

t t

t t

f t t dtb f t t dt

Tt dt

This is the DC component (arithm. average value)



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2.2.4 The polar form of the Fourier series(Fourier Cosinus series)

By means of the relationshipthe Fourier series can be rewritten from

2 2cos( ) sin( ) cos( arctan( / ))A x B x A B x B A

01

( ) cos( ) withf t d d t

00 and

2ad

2 2 ; d a b arctan ( / for negative )b a

a

0

1( ) cos( ) sin( ) to

2af t a t b t



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2.2.5 Examples for the determination of theFourier series with symmetrical Functions

• B1: f(t) is an even function with and( ) ( )f t f t

t

TT/2tt-T/2

f(t) f(t)

f(t)

0

2

2

2 ( )sin( ) 0

T

Tt

b f t t dtT

2

0

4 ( )cos( )

T

a f t t dtT

0

1

( ) cos( )2af t a t

The Fourier thus has the form:

Reason:

Representation of even functions is onlypossible by other even functions!



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• B2: f(t) is an odd function with and

f(t)

-T/2T/2

f(t)

f(t)tt

t

( ) ( )f t f t

0 02a a

2

0

4 ( )sin( )

T

b f t t dtT

1( ) sin( )f t b t

Thus it results:



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• B3: f(t) is a completely symmetric function with f(t) = -f(t + T/2)

t

f(t)

f(t+T/2)

t+T/2

tT



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For It applies:

or after split-up of the interval:

for holds:

for holds:

0

2 ( ) cos( )T

a f t t dtT

2

2

2 ( ) cos( ) ( ) cos( )

TT

To

a f t t dt f t t dtT

2k 2

2

2

2 ( )cos(2 ) ( )cos(2 ) 0

TT

kTo

a f t k t dt f t k t dtT

2 1k 2

2 10

4 ( ) cos (2 1)

T

ka f t k t dtT

Reason: Even-numbered k give after T/2 repeating cosine functions. Cancellationof terms result due to parts of f(t) being negative with respect to T/2 and repeatitself.



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- In a similar way the validity of the following statements can be seen(also sin function repeat itself after T/2):

2 0kb 2

2 10

4 ( )sin (2 1)

T

kb f t k t dtT

and

- Therefore only odd-number oscillations occur, for them in this Fourier series2 1k applies:

2 1 2 11

( ) cos (2 1) sin (2 1)k kk

f t a k t b k t



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• B4: The function is completely symmetrically with f(t) = f(t+ T/2) . Fromthis follows:

0

f(t)8t#

TT/2t tt+T/2

2

20

4 ( ) cos(2 )

T

ka f t k t dtT

and 2 1 0ka

2

20

4 ( )sin(2 )

T

kb f t k t dtT

and 2 1 0kb

Reason: After T/2 a repetition of the cos/sin functions with the indices 2k takes place. Cos/sin functions with indices 2k+1 have in each case a different half wave at T/2 distance! The Fourier series of f(t) then has the following form with only even-numbered coefficients:

02 2

1( ) cos (2 ) sin (2 )

2

k kk

af t a k t b k t



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• B5: f(t) is even and completely symmetrical: f(t) = - f( t+ T/2) :

tT

T/2

0

f(t)

02 0

2 ka a and 0b

4

2 10

8 ( ) cos[(2 1) ]

T

ka f t k t dtT

Result: Only odd-number cosine oscillations occur.

Thus the appropriate Fourier series can be written as:

2 10

( ) cos 2 1kk

f t a k t



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• B6: f(t) is odd and completely symmetric with f(t) = -f(t + T/2):

t

T

T/2

0

f(t)

Here only odd-number sine oscillation occur in the Fourier series

0 02a a 2 0kb

4

2 10

8 ( )sin[(2 1) ]

T

kb f t k t dtT

and

The Fourier series can thus be written as:

2 10

( ) sin 2 1kk

f t b k t



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• B7: f(t) is shifted on the time axis:

If the shift amounts to t then it applies with ' :t t t

0

1

( ') ( ) cos[ ( )] sin[ ( )]2

ag t f t t a t t b t t

This expression makes it possible, to determine the Fourier series for arbitrary shifts.

It often is of advantage to shift the origin e.g. if thereby symmetricalcharacteristics of the function result.

( ) results to jv tnf t t c e

A simpler expression results for the complex coefficients:



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2.2.6 Fourier analysis

• It exists the possibility to represent a periodical non-sinusoidal functionregarding its "information content" in two versions:

df(t)

T/40 tA

T/2 T-T/4

3T/4

1) In the time intervall (s. the following picture)

2) In the spectral region (frequency range): Representation of the amplitudes,a b and or the cos amplitude d and the phase

as a function of the frequency.



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2.2.6 Fourier analysisFurther examples

Fourier analysis of the trapezoidal functionThis function is even and completely symmetrically with (s. B5)

02 0, 0

2 k

a a b4

2 10

8 ( ) cos[(2 1) ]

T

ka f t k t dtT

and

To f(t) applies in the first quarter of the period:

04

( )

4 4 4

TA const für t df t

A T T Tt für d t dd

Thus it results: 4 4

2 10

4

8 cos[(2 1) ] ( )cos[(2 1) ]4

T Td

kT d

A Ta A k t dt t k t dtT d



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The final result then is:

2 1 2

4 cos[(2 1) ( )](2 1) 4k

A Ta k dk d

The Fourier series of the trapezoidal function is thereby:

4 1 1( ) [sin( ) cos( ) sin(3 )cos(3 ) sin(5 ) cos(5 )... ...)9 25

Af t d t d t d td

2 1 2

4 sin[(2 1) ] sin[(2 1) ](2 1) 2k

A k da kk d

or



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• Special case 1 of the trapezoidal function : The triangle function (d = T/4) :

t

3T/4

TT/2

T/40-T/4

f(t)

A

d



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• For this function the following amplitudes and phase spectrum results:

2 1 2 1 2 1 2 12 2

2 1:

8 , 0, 0 , / 2(2 1)k k k k k

kEndergebnisse

Aa bk

5 731073210

2

2

v8² ²

vAcv

/ 2



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• Special case 2 of the trapezoidal function : Rectangle function with0d

0-T/4

A

3T/4

tT

T/2

f(t)d

T/4



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• Limiting performed by means of the rule of Bernoulli L'Hospital:

975310

4v

Acv

vc

07

53

1

-

2

2

v

v

v

'

'0 0 0

sin[(2 1) ]sin[(2 1) ]lim lim lim ((2 1) ) (0) 1(2 1) (2 1)

d d d

k dk d si k d sik d k d

Thus it follows: 2 1

2 1 2 1

4 sin[(2 1) ], 0(2 1) 2

4 , sin[(2 1) ](2 1) 2 2

k k

k k

Aa k bkAc k

k



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2.2.7 The complex form of the Fourierseries

• Generally it applies to the Fourierseries:

0

1( ) cos( ) sin( )

2af t a t b t

In addition it applies: cos( )2

j t j te et

sin( )2

j t j te etj

Thus it isobtained

0

1( )

2 2 2

j t j t j t j ta e e e ef t a bj

0

1( )

2 2 2j t j ta a jb a jbf t e e

or:



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• Now also negative values for are included.

With the abbreviations 00 ,

2

for positive 2

for negative 2

ac

a jbc

a jbc

one receives pairs of coefficients.

This can be written in the very compactrepresentation of the Fourier series in complex form:

( ) j tf t c e

*Also: c c



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• For the complex coefficients thereby the conditional equations result:

0

0

0 0

0 0

00

1 ( ) ,2

1 1( )[cos( ) sin( )] ( ) ,2

t T

t

t T t Tj t

t t

ac f t dtT

a jbc f t t j t dt f t e dtT T

0

0

1 ( ) , 0,1,2,...t T

j t

t

c f t eT

In addition it applies: 2 Re( ) 2 Im( ) 0a c b c



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2.2.8 Interpretation of Fourier-coefficients

0

1( ) cos( ) sin( )

2af t a t b t

Usually the following representations of the Fourier series are used:

or

01

( )

or

( ) cos( )

j tf t c e

f t d d t

Appointment off here: c instead of c for better overview of formulas.



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- DC component of the signal: 00 02

a c d

- Peak values or amplitudes of the Fourier components: , and a b d

- Zero-phase (or phase) of the cosine oscillations:

- Basic oscillation at the fundamental frequency: 1 1cos( ) d t

2.2.8 Interpretation of Fourier-coefficients

Summarizing the Fourie series gives the following sets of parameters:



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2.2.9 Application of the Fourier seriesto a network

Given is a cosine-type voltage ˆ( ) cos( ) uvu t u t

A complex pointer is usually assignedlike this: ˆ ˆ uju u e

01

01

ˆ( ) cos( )

ˆ( ) cos( )

v u

v i

u t u u t

i t i i t

Also for electrical networks oneuses the representation (for voltagesand currents) in cosine form:

Now one can set:

*

0

1 ˆ , 120

1 ˆ 12

v

v

v

u für

u u für

u für

( ) withj tvu t u e



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C

Ri(t)

0 ( )u t( )Lu t

tTT/20

0 ( )u t

u

0 21

ˆ ˆ2 4( ) cos(2 )(4 1)k

u uu t k tk

0 ˆ( ) sin( ) mit = 2 /Tu t u t Here is given:

To determine are: ( )Lu t( )i t and

Example of a network: Series oscillator circuit

From this follows:




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• Solution :

- Use of impedance for each frequency k : ˆ

ˆ k

k

k

uZ

i

- Rewriting of the Fourier series of 0 ( )u t in complex form with:

2

ˆ4ˆ(4 1)

k

uuk

20 2

ˆ2( )(4 1)

j k t

k

uu t ek



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• Impedance and/or current of the series oscillator circuit for a certain frequency :k

k kZ R jX with

Then applies to the current:

1 kX k L

k C

22

2

ˆ2 1( ) .(4 1)

j k t

kk k k

ui t i ek R jX

22

ˆ2 1( ) . .[cos(2 ) sin(2 )](4 1)

k k

ui t k t j k tk R jX

Using the Euler' formula it results:

and /k k ki u Z



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2.2.9 Application of the Fourier series to a network

• After rewriting the equation, it results:

2 22 2 2 2 2

2 2

ˆ cos(2 ) sin(2 ) sin(2 ) cos(2 )2( )(4 1)

k k

k k k

R k t X k t R k t X k tui t jk R X R X

If the characteristics of the functions (cos, sin ) for +/- k are used, then it applies

22 2 2

0 2

ˆ cos(2 ) sin(2 )4( )(4 1)

k

k k

R k t X k tui tk R X

with :k kX X



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One receives the voltage at the coil by means of : ( )( )L

di tu t Ldt

22 2 2

0 2

ˆ 2 [ sin(2 ) cos(2 )]4( )(4 1)

kL

k k

k L R k t X k tuu tk R X

2 2 20 22

ˆ4 2( ) . cos[(2 ) arctan( )](2 1)L

k kk

u k L Ru t k tk XR X

The results can be represented also in polar form:

22 2 2

0 2

ˆ4 1( ) . cos[(2 ) arctan( )](2 1)

k

k k

Xui t k tk RR X



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2.2.10 Formulation of Parseval's equation

- Two generally non-sinusoidal periodic functions are regarded- The functions 1( )f t 2 ( )f tand have the same period T: - The 2 appropriate Fourier series formulas then look as follows:

0

0

0

0

1 1

2 2

1( ) with ( )

and

1( ) with ( )

t Tj t j t

t

t Tj t j t

t

f t C e C f t e dtT

f t D e D f t e dtT

- To the product of both functions applies: 1 2( ) ( ) .

j t j tf t f t C e D e

0

0

1 2 1 21( ) ( ) with ( ) ( )

t Tjk t jk t

k kk t

f t f t E e E f t f t e dtT

and at the same time as they are periodical:



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• Further applies: 0

0

1 .t T

j t j t jk tk

t

E C e D e e dtT

0

0

0

0

( )

( )

1

and/or

1

t Tj k t

kt

t Tj k t

kt

E C D e dtT

E C I with I D e dtT

• One can show that I is different from zero only in case of: • (due to orthogonality of cos(nx) and sin(nx))

0k

Thus the integrand becomes identical to 1 and it applies:

1I D T DT



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k kE C D C D

For the fourier-coefficients of the product then results1 2( ) ( )f t f t

Determination of the DC component of the product using k = 0 : 1 2( ) ( )f t f t0E0

0

0 1 21 ( ). ( ) .

t T

t

E f t f t dt C DT

due to 0k or k v

There are various applications of this relationship (determination of the integral in the time or frequency range)!



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All complex Fourier-coefficients possess the characteristic: * *andC C D D

Thus the following formula can be rewritten based on

* *0 Re ReE C D C D

This is Parseval's equation!

0

0

0 1 21 ( ). ( ) . .

t T

vt

E f t f t dt C D C DT

to:



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2.2.11 Power of non-sinusoidalperiodic network functions

The electrical energy E per period (or the power P) concerning an Ohm'sresistance amounts to:

0 0

0 0

2 21 1 1/ ( ) ( )t T t T

t t

P E T u t dt R i t dtT R T

Application of Parseval's equation for the special case

1 2( ) ( ) ( ) ( )f t f t f t i t 21 2( ) ( ) ( ) ( )f t f t f t i t

0

0

2 *0 0 0

1

1 ( ) 2 Re .t T

t

E f t dt C C C CT

220 0

1

( 2 )P E R R C C

results in:

with



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With 00 0 , 1

2 2

a a jbc C c C

it follows:

0

0

2 2 222 2 0

01 1

1 ( ) 22 2

t Tv v

t

a a bf t dt c cT

If f(t) has character of a voltage or a current, the appropriate spectralparameters , anda b c describe the effective power conditions of theappropriate network elements.



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Now a two-pole with non-sinusoidal periodic network functionsu(t) and i(t) is regarded:

U(t) Twopole

i(t)1

1'

1

2

( ) ( )

a n d

( ) ( )

j t

j t

u t f t C e

i t f t D e

0 0

0 0

*1 2 0 0

1

*0 0

1

1 1( ) ( ) ( ) ( ) 2 Re

2 Re

t T t T

t t

P u t i t dt f t f t dt C D C DT T

C D C D

For the power it applies:



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With consideration of the relations

0 0 0 01 1 ˆˆ, , ,2 2

C U D I C u D i

follows:

* *0 0 0 0

1 1

1 1ˆ ˆˆ ˆRe Re2 2

v v v vP U I u i U I u i

Thus the total power is to be determined over the sum of all individual powers for each spectral line!



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2.2.12 Assessing of deviations from thesinusoidal form of periodic functions

Definition of the rms value of a periodic function:

0

0

21( ) ( )

t T

efft

f t f t dtT

2 222 0

1( )

2 2 2

eff

a a bf t c

Parseval's equation permits the determination of the rms value usingFourier coefficients (and/or the associated rms values):



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- The rms value for a periodic voltage u(t) amounts to:

22 20

1 0

ˆ2

eff effuU U U

- The rms value for a periodic current amounts to: 2

2 2 2 20 0

1 1 0

ˆ

2eff eff effiI I I I I I

0U : DC component of u(t)

u : Peak value

ˆ / 2effU u : Rms value of the component (at frequency: )



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Otherwise: f(t) contains both (DC and AC components)

2 2

1 1

2

0

eff eff

effeff

U Us

UU

0 02a

The oscillation content s describes theamount of AC in the total signal.

Thus

For pure alternating currents (AC), without any DC component applies: 0a 0



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The deviation from the sinusoidal form can be described bythe basic oscillation amount g:

1 1

2

1

e f f e f f

e f fe f f

U Ug

UU

2 2

2 2

2

1

eff eff

effeff

U Uk

UU

2 2 1g k

The harmonic content k or distortion factor k amounts to:



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Additionally there are further definitions namedshape factor and amplitude factor :

2

0

0

1 ( )

e f f

f T

Uk

u t d tT

Form factor:

Crest factor for signalswithout DC component:

max

2

1

( )

a

eff

u tkU

For purely sinusoidal form one finds:

1,11 and 2 1,412 2f ak k



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2.2.13 Additional properties of the Fourier series

· Linearity

· Time-shift

· Reflection

( ) gives a series with vk s t k c

1 2 1 2( ) (t) gives a series with v va s t b s a c b c

11( ) gives a series with jv t

vs t t c e

*( ) gives a series with vs t c



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Fundamentals of Electrical Engineering 3

Chapter 2.3Description of aperiodic time operations by means of

the Fourier transform



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2.3.1 Prefaces

Example : A periodic rectangular impulse is regarded

0( ) 2 20

i it tU for tu t

otherwise

A Fourier analysis of this signal is to be accomplished.

u(t) is an even function

Onset: Development of the Fourier transform from the Fourier seriesby transfer of periodic functions to aperiodic impulses

t

f(t)

0

2t

2t

T-2t



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2.3.1 Prefaces

Solution: 2

00 0

2

1

i

i

tt

i

tt

U tc U dtT T

0

2 20

0

22

0 0

1 sin( )cos( )2 2

2

2 2sin sin sin2 2

ii

ii

t tt

ttb t

i i i

i

i

a jb a U tc U t dtT T

withT

t tv v tT TU U t Tc tT T

T



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2.3.1 Prefaces

and:

0 0

sin2 2 2 ( )

i

i i i

i

tU t U t tTa c sitT T T

T

Thus itapplies:

0( ) si

j ti iU t tu t eT T

0,2itTSketch of the spectrum of u(t) for the case

1510

50

1C2C

3C4C

5 0C 10 0C

sin( )xx

15 0C

0C



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2.3.2 The Fourier integral

In the following the Fourier series of a periodic function f(t)is examined. Here the period is:

f(t)

t

T/ 2-To/2 0

The following conditions are assumed: 1) f(t) shall be continous

the function shall2) In each finite period 0 02 2 T Tt

meets the Dirichlet' conditions

3) With infinite period f(t) shall be absolutely integrable

0( )

j tf t c e

00

2T



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The following representation starts from a periodic signal which istransformed into a non-periodical of signal. Note: Enlargement of the period is made by:

0

limT

0

0 2

Tm

Each term in the complex Fourier series corresponds to a line in spectrum. Distances between lines amount to:

In an interval around any point of frequency thereby lie in the following number of lines:

0 02 / T

With sufficient small intervals then only small difference of the mindividual terms of the complex Fourier series result. Consequence: Summarizing of these terms is permitted!



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0 TFor one can select infinite small intervals (if m should remainunchanged)

0 00

2 jv t jv t

v vTm c e c e

Thus arises for the contribution of each interval to the row:

In each interval with m lines applies thereby applies concerning itscontribution to the row:

00 00 with in which it holds:

2 2jv t j t

v vT Td c e v c e d

In addition can be written shortening:

Altogether it results : 1( ) ( )2

j tf t F e d

0 ( )vT c F



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2.3.2 The Fourier integralThus it also applies: *

0 0

1 1( ) ( ) ( )2 2

1( ) ( )2

j t j t

j t

f t F e d F e d

or

f t F e d

0

0

1 ( )

t t T

j t

t t

mit c f t eT

( )f t ( )F

The corresponding symbol:

) ( ) ( )

j tt F f t e dtF

The Fourierspektrum and/or the Fourier transform )tF

represents the function f(t) as follows:

0 ( ) :vT c F Now it consequently follows because of



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2.3.3 The Fourier inverse transformThe function f(t) can thus be represented by means of its spectrum:

1( ) ( )2

j tf t F e d

The inverse transform (from frequncy domain to the time domainis in short written as follows: ( )F ( )f t

x or AmplitudeAmplitude TimeFrequency

( )f t dt S const

( )F has not the property of an amplitude (as in the Fourierseries), but it is an amplitude density function with thedimension:

The existence of the Fourier integral isensured if f(t) is absolutely integrable:



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2.3.4 Interpretation and summary

Now a signal s(t) is considered, from which by ideal BP filtering only portionswithin a certain frequency band are extracted. The filtering takes place so narrow-banded that therein the spectrum (and the exponential function) changes only insignificantly. For this extracted portion g(t) follows:

0 0 0

0 0

0

0

0 0 0

arg( ( )) *0 0 0

0 0

1 1 1( ) ( ) ( ) ( )2 2 2

( ( ) ( ) ) Re{ ( ) }2

Re{ ( ) } wegen ( ) ( )

( ) cos( ar

j t j t j t

j t j t j t

j S j t

g t S e d S e d S e d

S e S e S e

S e e S S

S t

0g( ( ))S



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• The Fourier transform (under given conditions) thus is a measure for theamplitude and the phase of a signal component with reference to the regardedbandwith concerning the regarded frequency.

• The method of the Fourier transform permits: 1) To describe a signal known in the time domain equivalently in the frequency domain2) To determine the function in the time domain from a known Fouriertransform

• The Fourier transform is an important and one of the most powerful tools of electrical engineering, control engineering, physics (optics, mechanics and manymore).

• This method forms at the same time the basis of the Laplace transform, Z-transform and the discrete Fourier transform.

2.3.4 Interpretation and summary

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