TB AC DC fundamentals

7/30/2019 TB AC DC fundamentals

1/47

Fachhochschule Frankfurt am Main

University of Applied Sciences

Faculty of Computer Science and Engineering

Vietnamese German University

Ho Chi Minh City, Vietnam

Fundamentals of Electrical Engineering

DC- and AC-current networks

Prof. Dr.-Ing. habil. Joachim Lmmel

Academic Year 2012/2013


2/47


3/47

5.2 Resonant circuits

6 Three-Phase Electric6.1 Generation of a three-

6.2 Star and delta connecti

6.3 Power of the three-pha

1 Electric Field and Imp1.1 Fundamentals of fields

In common the field describes

in a space. There are:

1.2 Electric charge Q

VGU 2012/3/Lae

Power Systemshase power system

on

se system

rtant Quantities

the distribution of a quantity

[As], [C] (Coulomb)


4/47


5/47

1.4 Electric potential an

dependence between voltage

common is valid:

1.5 Electric flux and ele

The relation between electric

material equation of the elec

VGU 2012/5/Lae

electric voltage U scalars; [

nd field strength:

in case of homogeneous

field is valid:

tric flux densityD : scalar;[As];

Demonstrative solution:

lux densityD and electric field strengthE is

trostatic field.

V]

:vector; [As/m2

]

realized by the


6/47

1.6 Capacitor and capacit

For the dimensioning of a cap

we can use the size and the m

It is valid:

Features of capacitors: See r

Calculation of capacitances:

The capacitor storages electric

means of varying the voltageThe accumulated electric ener

VGU 2012/6/Lae

C [As/V] = [F] (Farad)

Consequential the equation fot the capacity Cof a capacitor f

acitor Therefore the design equationterial. a homogeneous field is the fol

marks page 7

ee remarks page 8

energy by charging. We can use the accumu

between the electrodes. By discharging we cgy can be calculated according to:

r the definition ofllows:

of a capacitor with

lowing.

lated energy by

an use the energy.


7/47

Capacitors

1. Properties of Differentiation of Capacitors:

Capacitors differ according to the following properties,

- if there is afixed or variable capacitance

(e. g. capacitors with variable capacitance are: rotary capacitors, flat trimming capacitors)

- which kind of dielectric is used between the electrodes

(e. g. paper and glimmer capacitors, foil capacitors, porcelain and electrolytic capacitors)

or

- which kind of construction is used

(e. g. capacitors with plane electrodes, tube capacitors, cup capacitors, )

2. Relative Permittivityr of Selected Materials :

(absolute permittivity 0 = 8.85410-12 As/Vm)

acetone 21.5 polyethylene 2.3

barium titanate 1000...2000 polypropylene 2.3

glass 5...12 polystyrene 2.5

glimmer 5...8 polyvinyl chloride 3...4

hard paper 5 porcelain 6cable paper in oil 4.3 sulphur hexafluoride 2.4

air 1.0006 transformer oil 2.3

paper, dry 2 water 80.8

acrylic glass 3 special ceramic for

hf-technology 10000

special materials for electrolytic capacitors:

Al2O3 8

Ta2O5 26

3. Standardization of Capacitors:

Capacitors are produced in classes of capacitance values between 0.6 pF and 10000 F

resp. for operation voltages from 3V until more than 100 kV.

There are German DIN-standards: DIN 42007 (variable capacitors), DIN 45910

(porcelain-, synthetic- and electrolytic capacitors, ...)

VGU 2012/7/Lae


8/47

Calculation of Capacitances

1.)

d

AC r0=

0 - absolutepermittivity = 8.86.10-12As/Vm

r - relativepermittivity

VGU 2012/8/Lae

2.)

20 4

111

r

dr

A

dr

Cd

r ==

==

=a

i

a

i

r

r ai

r

rrrr

dr

Cd

C

11

4

1

4

1112

C

rr

r rri a

a i=

0

4

(Vergleich mit 1.)_____________________________________________________________________

3.) Q C U= , =a

i

r

r

drrEU )(

E rD r

( )( )

,=

D rQ

A r

Q

rl

( )

( )

= =

2

==a

i

r

r i

a

r

r

l

Q

r

dr

l

QU ln

22

CQ

U

l

r

r

l

r

r

a

i

ra

i

= = =2 2

0

ln ln

(Vergleiche mit 1.)

r

ra

i

dr

Kugelkondensator

rr

ia

l

Zylinderkondensator

spheric capacitance

cylindric capacitance

compare with 1.

compare with 1.

distance darea A

plate capacitance


9/47

1.7 Parallel and series con

Using the equation of definiti

e. g.: C1=100nF; C2=0,1F

follows

or

e. g.: C1=100nF; C2=0,1F

VGU 2012/9/Lae

ection of capacitors

It is valid:

and

n of the capacity C=Q/Uresults:

or common for an arbitr

number n of capacitors

It is valid:

and with

or common for an arbit

number m of

capacitors

r

ry

rary


10/47

1.8 Capacitor with layered

For the electric flux intensity i

Therefore the following depen

The ratios of electric field stre

2 Electric Flow Field an2.1 Electric currentIand

The definition of the current I

To the definition of the Tec

The current is positive in the

of moving of negative charges

VGU 2012/10/Lae

dielectric

- The arrangement is identical

series connection of two capa

- The electric field strengthEa

voltage drop Uover the two l

technical meaning especially i

arrangements e. g. as cables.

n both layers is valid:

dence results:

ngth and voltage drops over the layers can b

Resistanceurrent densityJ I scalar;[A];J vector

is described by:

generally written:

resp. in homogeneous fields

nical current direction

pposite direction

.

ith the

citors.

d the

yers are from

n case of coaxial

calculated.

[A/m]


11/47


12/47

In the electric flow field a de

field strengthEby the materia

From this equation the dimens

2.2 Electric resistanceR a

This results in the definition

VGU 2012/12/Lae

endency exist between the current densityJ

l equation of the electric flow field (compare

(Kappa) electric co

ion of the electric conductivity follows:

d Ohms Law

quation for resistanceR.

and the electric

D = E).

nductivity


13/47

OHMIC LAW:

The material and size (length

resistor. The following relatio

Features of resistors: See re

Calculation of resistances: S

Sometimes it could be easier t

it the electric conductance G

2.3 Electrical energy Wan

We use

VGU 2012/13/Lae

nd cross section) can be changed to dimensi

results:

arks page 14

e remarks page 15

Remark to the notation of the dimen

o calculate with the reciprocal value of the re

d powerP W [VAs]; P [VA]; sc

on the value of a

sion in the tables:

sistanceR. We call

lars


14/47

Resistors

1. Properties of Differentiation of Resistors:

Resistors differ according to the following properties and their design. There are

film resistors, which has a thin conductive layer at a porcelain body,

fixed wire- or band resistors, where the wire or the band is wound around a porcelain body,

temperature-dependent resistors(thermistors), changing the resistance with the temperature,

consisting of sintered ceramic on the basis of semiconductors,

voltage-dependent resistors (varistors), changing the resistance with the voltage level,

consisting of sintered ceramic on the basis of SiC (e. g.) and

adjustable resistors, where the resistance can be changed by a sliding contact.

2. Resistivity, Conductivity and Cofficient of Temperature of Selected Materials

in in 20 inmm m2/ m mm/ 2 1/C

aluminium Al 0,028 36 0,00377

argent Ag 0,016 63 0,0038

copper Cu 0,018 56 0,00393

gold Au 0,023 44 0,004

platinum Pt 0,11 9 0,002

iron Fe 0,125 8 0,0046

manganinNiMn,

Fe,Cu,0,4 2,5 0,00001

chromium- FeNi,Cr, 1 1 0,00005

nickel

3. Standardization of Resistors:

Resistors are produced in classes of resistance values between 10-3 and 1014 for different

voltage and power values

There are German DIN-standards e.g. DIN 40712 (symbols), DIN 41450 (adjustable

resistances), DIN 44080 (temperature dependent resistors), DIN 44050...55 ( fixed film

resistors), DIN 45921 (metallic and nonmetallic fixed resistors),..

VGU 2012/14/Lae


15/47

Calculation of Resistances

1.) Rl

A=

- conductivity

--------------------------------------------------------------------------------------------------------------

2.) ES

=

, SI

A

I

r= =

4 2

EI

r=

4 2

==

ai

r

rrr

I

r

drIU

a

i

11

44

==

airrI

UR

11

4

1

--------------------------------------------------------------------------------------------------------------

3.) dR drA

drlr

=

= 2

==a

i

r

r i

a

r

r

lr

dr

lR ln

2

1

2

1

VGU 2012/15/Lae

r

r

i

Widerstand (Kugelfeld)

a

rr

ia

l

Widerstand (Zylinderfeld)

dr

spheric resistance

cylindric resistance

compare with 1.

compare with 1.

length l areaA


16/47

respectively considering the r

The power can be calculated i

2.4 Temperature depende

It is valid:

For the calculation of the tem

measured at the terminals can

2.5 Series- and parallel- c

We know that the voltage dro

total voltage drop. Using the

currentI

VGU 2012/16/Lae

sistanceR

fwe divide the energy Wby the time t:

ncy of the resistance

(related to 20C

erature inside of windings out of the changi

be used the equation:

nnected resistors

s in a series connection of components hav

hmic Law the following derivation results

Thereby for the ratio o

results:

)

g of the resistance

to be added to the

y dividing by the

the voltage drops


17/47

and expressed by the electric

The ratio of the parallel curre

can be formulated by the equa

3 Analysis of Linear DC3.1 Voltage and current di

resistive voltage divider

VGU 2012/17/Lae

It is valid:

Using the Ohmic Law follows:

Finally results:

onductance G

ts

tion:

Networksvider

and it follows:


18/47

Loaded Voltage Divider

For the voltage-divider ratio results from the

circuit

)()1( 22

xxRR

Rx

RRxRx

RRx

U

U

sv

v

vss

vs

+

=

+

=

=+

x

R

Rx xs

v

1 1( )

and for the load current follows

)1(2

xxRR

xU

R

UI

svv

v +==

With x = 1 the maximum load current results according to the equation V

VmaxR

UI =

and

damit

)1(1

2

max xxR

R

x

U

U

I

I

v

sv

v

+==

Border cases are: Rv (unloaded voltage divider, linear dependency) xU

U=2

and Rv = 0 (short circuit at the output, Ux = 0) IU

R xv

s

=( )1

Beetween the border cases the nonlinarity increases with increasing load or load current.

Diagram: Related values of

load voltage U2 (UX) and

load current IVas function of

the slider position x

of the voltage divider

VGU 2012/18/Lae

I

U

U2

R1= (1-x)RS

R2= xRS RV

I

U

U2

R1= (1-x)RS

R2= xRS RV

Circuit of the voltage divider loaded with

the resistor RV


19/47

r

3.2 Kirchhoffs equations

3.2.1 Electric sources

3.2.2 Kirchhoffs node equa

3.2.3 Kirchhoffs loop equat

VGU 2012/19/Lae

and it follows:

esistive current divider

ion

e can simply conclude:

nd for the example beside:

ion

e can conclude again:

nd for the circuit on the left:


20/47

3.2.4 Analysis of networks

Example for solution by Kir

Result forI3:

In the following chapters we

have their distinct advantages

3.3 Superposition method

The final result is

the addition of both currents:

VGU 2012/20/Lae

y Kirchhoffs equations

hoffs equation

ill introduce more techniques for the analys

and handicaps.

I3 : current by means of only Uq1

I3 : current by means of only Uq2

s of networks. All


21/47

3.4 Mesh current method

If you solve the equation syste

for the unknown mesh current

e. g. for the currentI3:

3.5 Real voltage and currereal voltage sour

precondition for equivalence

of both sources:

VGU 2012/21/Lae

The first step is to find the number of

currents in. In our case there are two o

m

s, we find with:

nt sourcese real current so

independent mesh

f it.

rce


22/47

3.6 Equivalent circuits of

network

a)

b)

active t

Steps of calculation:

1st step: calculation ofUq ?

2nd step: calculation ofRi ?

VGU 2012/22/Lae

lectric sources

rminal circuit passive termin l circuit


23/47

3rd step: calculation ofRa ?

4th step: calculation ofIRa an

BASIC CURRENT CIRCU

Example:

Example:maximum availabl

mesh equation:

for the power delivered to the

Imagine the following: Ra =

That means, between both the

By differentiating the power e

VGU2012/23/Lae

URa

T

power at load resistor Ra

load Ra is valid:

or Ra = . In both cases the power at the lo

power should have a maximum.

quation we get:

d resistor is zero.


24/47

The last equation shows that t

has to have the same value as

That means, the maximum av

Example: maximum efficien

Now we will see the circuit fr

Due to the internal resistorRi

load resistorRa. Because of th

The efficiency of each transfe

VGU 2012/24/Lae

Technical meaning:

Technical meaning:

e load resistance

the internal resistance:

ilable power is:

y of transmission

m another point of view.

ot the total power Pqof the source will find

currentIRa there will be a dissipation of po

of energy is defined as

the way to the

er P overRi.


25/47

4 Representation of Periodical Quantities4.1 Parameter of periodical quantities

e. g.: power gridf= 50Hz (USAf= 60Hz; train grid in Germanyf= 16 2/3Hz; example for

radio frequency f= 107,2MHz

Therefore we find:

aufgetragen.

and

with the angular frequency

VGU 2012/25/Lae

Eine zeitabhngige Gre wird als periodisch bezeichnet, wenn sich ihr zeitlicher Verlauf

nach einer bestimmten Zeitspanne Periodendauer T wiederholt.

Nach FOURIER lsst sich eine periodische Funktion durch die Summe bestehend aus einemGleichanteil und Sinusfunktionen unterschiedlicher Frequenz und Amplitude ersetzen.

Die Grundfunktion aller periodischen Funktionen ist deshalb die Sinusfunktion.

Die Anzahl der Schwingungen, die eine periodische Funktion in einer Sekunde durchfhrt, ist

dieFrequenz f.

[s- ] = [Hz] = [Hertz]

DiePeriodendauer Tist der Kehrwert der Frequenz f.


26/47

Representation of the sinusoi

The definition of the phase sh

between voltage and current i

by the equation:

e. g. the characteristic above:

= u-i = 60-(-30) = 90(We say: The current is laggin

VGU 2012/26/Lae

al function

iftgiven

u = 60(/3); i = -30(-/6);/2)

to the voltage with 90.


27/47

More parameter can be calculeted of importance for different technical processes.

For the description of voltage shapes two further parameter are useful.

for sinusiodal shape

for sinusiodal shape

VGU 2012/27/Lae

u


28/47

4.2 Current and voltage at the elements R, L and C

a) resistor preconditionThe voltage over the resistor is calculated:

The result is a1) current-voltage relation

a2) phase and phase shift

a3) impedance (AC resistance)

(resistance)

b) Inductivity precondition

The voltage over the inductivity is calculated:

The result is a1) current-voltage relation

a2) phase and phase shift

a3) impedance (AC resistance)

VGU 2012/28/Lae (inductive reactance)

LL

2

2

3 22

2

2

3 22

2

2

3 22

2

2

3 22


29/47

b) capacity

The voltage over the capacity

The result is

The result is

4.3 Phasor diagrams with p

a)

b)

c)

d)

VGU 2012/29/Lae

ii

2

2

2

2

precondition

is calculated:

a1) current-voltag

a2) phase and pha

a3) impedance (A

(capacitive reactriodical quantities

C

u

C

u

=i

2

3 22

3 2

relation

e shift a2)

resistance)

nce)

0,sin i =t


30/47

to the construction of phasors

a)

b)

c)

d)

e)

f)

g)

VGU/2012/30/Lae


31/47

4.4 Complex operatorZ

The definition of the complex

The exponential form is:

Written in the Cartesian form:

The equations for the calculati

phase shift are:

Survey over the known depe

Preconditions:

Resistor

VGU/2012/31/Lae

operator is given by the equation:

on of the absolute value (length of the phaso

ndencies for the components R, L and C

Inductivity Cap

r) and angle of the

citor


32/47

Application: Series connecti

Inserting the given value

VGU2012/32/Lae

n ofR,L and C

uq = 10Vsint, f= 50Hz,R = 1k

esh equation

solution of the differential equation

or direct approach by complex ope

solution of the algebra

the current i

s:

,L = 10H, C= 6F

of the current i

rators:

ic equation for


33/47

Solution by phasor diagram

Solution by means of a quan

Therefore follows:

We can measure for the phase

VGU2012/33/Lae

steps of construction

1.)

2.)

3.)

titative phasor diagram

shift:


34/47

4.4 Power in AC networks

In DC networks the power is calculated by the formula

The product of the time-depending voltage and current

has no practical meaning.

VGU2012/34/Lae


35/47

Conclusions for the power components and their calculation withR, L and C

Further on is valid:

VGU/2012/35/Lae

Widerstand

Kapazitt


36/47

Fr die Auswertung von Leistungsmessungen ist die Tatsache wichtig, dasssich in einem beliebigen Netzwerk die Gesamtleistung aus der Summe derTeilleistungen an den einzelnen Bauelementen ergibt. Das gilt fr Wirk-und fr Blindleistungen.

or

From the calculation follows: We have always to calculate the total apparent power of anarbitrary network with the following equation.

Never

e.g.: Three units with different components of active and reactive power:

VGU2012/36/Lae

-j1/Cz.B.

R1

R2

jL

I1

I2

I3

geg.:R1=R2=1; jL=1; -j1/C=-j1

I =I =1A I =1,41A

-j1/Cz.B.

R1

R2

jL

I1

I2

I3

-j1/Cz.B.

R1

R2

jL

I1

I2

I3

z.B.

R1

R2

jL

I1

I2

I3

geg.:R1=R2=1; jL=1; -j1/C=-j1

I =I =1A I =1,41A

W2

1A)1(1A)1(

Gesamt

22

2221

21Gesamt

=

+=

+=

P

RIRIP

var1

1)A1(1A)2(

1

Gesamt

22

22

23Gesamt

=

=

=

Q

CILIQ

2,24VA

VA522

=

=+=

S

QPS

e.g.

given:


37/47

5 Analysis of AC Networks5.1 Basic circuits5.1.1 Series connection ofR andL (real coil)Circuit Phasor

diagram

There are

the wellknown equations:

Further on is valid: complex impedance = resistance + j reactance

and the impedance (absolute value of the complex impedance)

In consequence of the both triangles follows for the phase

After the multiplication of the edges of the impedance triangle with I2, we get the triangle of

the power components.

The relations between the power components is described by:

For the phase follows:

VGU2012/37/Lae

R jLR jL


38/47

5.1.2 Parallel connection ofR and C(real capacitor)

Circuit Phasor diagram

Again is wellknown:

Further on is valid: admittance = conductance + j susceptance

and the conductance (absolute value of complex admittance)

In consequence of the both triangles follows for the phase

After the multiplication of the edges of the admittance triangle with U2, we get the triangle of

the power components.

The relations between the power components is described by:

For the phase follows:

VGU2012/38/Lae

CGY j+=

R

C

1j

j

1

=

C

R

C

1j

j

1

=

C


39/47

5.1.3 Low pass

circuit phasor diagram

calculation of the ratio U2/U1

and the phase 21

absolute value

phase shift

step response u(t)

amplitude characteristic U2/U1= f() phase characteristic 21= f()

cutoff frequency:

VGU2012/39/Lae

1R

2j

1

C

1R

2j

1

C


40/47

5.1.4 High pass

Circuit phasor diagram

calculation of the ratio U2/U1 and

the phase 21

absolute value

phase shift

step response u(t)

amplitude characteristic U2/U1 = f() phase characteristic 21= f()

cutoff frequency:

VGU2012/40/Lae

2R

1j

1

C

2R

1j

1

C

fr


41/47

5.1.5 Phase-turn circuit example: double RC precondition

5.2 Resonant circuits

VGU2012/41/Lae

CR

1=

R

Cj1

R

Cj1

R

Cj1

R

Cj1


42/47

circuit phasor diagram impedance

At the resonance point are UandI in phase. That

means, only the resistance come into effect for

the impedance.

That means

so the angular resonance frequencyis calculated by

resp. the resonance frequency is

In the case of resonance the impedance, its absolute value and the phase angle between

voltage and current phasor result to

Fr die Impedanz, deren Betrag und den Phasenwinkel zwischen Spannung und Strom ergibtsich im Zustand der Resonanz

Phasor diagram ofZ :

VGU2012/42/Lae

Cj1

jLR Cj1

jLR

)1

(jC

LRZ

+=

22 )1(C

LRZ

+=

)

1

arctanR

CL

=

Impedanz

Betrag

Phasenwinkel

)1

(jC

LRZ

+=

22 )1(C

LRZ

+=

)

1

arctanR

CL

=

Impedanz

Betrag

Phasenwinkel

Impedance

absolute value

phase shift


43/47

Characteristic frequencies for the evaluation of diagrams of resonance networks are the so-

called 45-frequencies. The name results of the shape of the phasor diagram.

The following diagrams of the related parameterZ/R =f() and the phase angle=f() showthe dependence from the angular frequency.

VGU2012/43/Lae

+45 -45

1RZ

0

L

1/C

+45-45

1RZ

0

L

1/C

+45-45

0

+45

-45

90

-90

-45

45

0

0

+45

-45

90

-90

-45

45

0


44/47

From the behaviour of the impedanceZ we are able to conclude the diagram ofI=f().

Die Teilspannungen eines Reihenschwingkreises knnen die Gesamtspannung Uwesentlich

bersteigen. Die berhhung im Resonanzpunkt ergibt sich aus derKreisgte Q .

VGU2012/44/Lae

1RI

I

0+45-45

0,5

1

IR1 = I beiR1 und0

Z

UI= bei U= konstant1RI

I

0+45-45

0,5

1

IR1 = I beiR1 und0

Z

UI= bei U= konstant

0

U= konstant

U

0

U= konstant

U


45/47

Alle imReihenschwingkreis erluterten Zusammenhnge und Formeln ergeben

sich in analoger Form fr denParallelschwingkreis, in dem man systematisch

VGU2012/45/Lae

0

kapazitivinduktiv

I= konstant

I

ICIL

IG

0

kapazitivinduktiv

I= konstant

I

ICIL

IG

Cj

1jLR

UR UC

U

I

UL

Cj

1jLR

UR UC

U

I

UL

Cj1

jL

R , GIR

U

I

IL

IC Cj1

jL

R , GIR

U

I

IL

IC

0

kapazitiv induktiv

U= konstant

U

ULUC

UR

0

kapazitiv induktiv

U= konstant

U

ULUC

UR


46/47

6 Three-Phase Electric Power Systems6.1 Generation of a three-phase power systemIn 1891 the first three-phase power transmission was realized between Lauffen am Neckar

nach Frankfurt am Main . Three-phase current is the most effective kind to transmit electric

power over long distances.

Grundlage ist Bewegungsinduktion und insbesondere die Bewegung einer Leiterschleife imMagnetfeld (siehe Kapitel 5.4.2)

For the three characteristics we find the equations:

Aus Leistungsgrnden wird die praktischeAnordnung umgekehrt realisiert.

Dabei wird das rotierenden Gleichfelddurch den Gleichstrom in einer Wicklungdes sich drehenden Lufers (Rotor)

hervorgerufen.

Die drei Wicklungen U, V und W sindfest im Stnder der Maschine angeordnet.In ihnen werden diedrei um 120phasenverschobenen Wechselspannungeninduziert.

Fr die Energiebertragung vom Kraftwerk (Generator) zum Verbraucher (z.B. Motor)

FHF 2007/GET/L68_Drehstrom1

VGU2012/46/Lae

tUu sin= )120sin( = tUu )240sin( = tUu

ROTOR

STNDER

GENERATOR

ROTOR

STNDER

ROTOR

STNDER

GENERATOR


47/47

9.2 Stern- und Dreieckschaltung

Star connection (Y, y, Y) Delta connection(D, d, )

9.3 Power of the three-phase system

The power of the three-phase system must be consider the kind of the internal connections of

the windings:

StrNetz

StrNetz

3IIUU

==

StrNetz

StrNetz

3 IIUU

==

S=3UStrIStr S=3UStrIStr

Star connection Delta connection

(3 Strnge) (3 Strnge)

INetz= IStr UNetz= 3 UStr INetz= 3 IStr UNetz=UStr

Y

Stran ren

Netzgren

L1 L2 L3L1 L2 L3 L1 L2 L3L1 L2 L3

TB AC DC fundamentals

Documents

Transcript of TB AC DC fundamentals