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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
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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
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Power Systemshase power system
on
se system
rtant Quantities
the distribution of a quantity
[As], [C] (Coulomb)
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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
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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
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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
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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.
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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, ...)
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Calculation of Capacitances
1.)
d
AC r0=
0 - absolutepermittivity = 8.86.10-12As/Vm
r - relativepermittivity
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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
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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
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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
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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
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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]
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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
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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
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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
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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
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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),..
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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
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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
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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
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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
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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
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It is valid:
Using the Ohmic Law follows:
Finally results:
onductance G
ts
tion:
Networksvider
and it follows:
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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
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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
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r
3.2 Kirchhoffs equations
3.2.1 Electric sources
3.2.2 Kirchhoffs node equa
3.2.3 Kirchhoffs loop equat
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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:
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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:
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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
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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:
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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
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3.6 Equivalent circuits of
network
a)
b)
active t
Steps of calculation:
1st step: calculation ofUq ?
2nd step: calculation ofRi ?
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lectric sources
rminal circuit passive termin l circuit
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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
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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.
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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
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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.
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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
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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.
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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
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al function
iftgiven
u = 60(/3); i = -30(-/6);/2)
to the voltage with 90.
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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
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u
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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)
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LL
2
2
3 22
2
2
3 22
2
2
3 22
2
2
3 22
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b) capacity
The voltage over the capacity
The result is
The result is
4.3 Phasor diagrams with p
a)
b)
c)
d)
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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
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to the construction of phasors
a)
b)
c)
d)
e)
f)
g)
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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
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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
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Application: Series connecti
Inserting the given value
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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
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Solution by phasor diagram
Solution by means of a quan
Therefore follows:
We can measure for the phase
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steps of construction
1.)
2.)
3.)
titative phasor diagram
shift:
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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.
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Conclusions for the power components and their calculation withR, L and C
Further on is valid:
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Widerstand
Kapazitt
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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:
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-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:
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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:
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R jLR jL
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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:
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CGY j+=
R
C
1j
j
1
=
C
R
C
1j
j
1
=
C
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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:
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1R
2j
1
C
1R
2j
1
C
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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:
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2R
1j
1
C
2R
1j
1
C
fr
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5.1.5 Phase-turn circuit example: double RC precondition
5.2 Resonant circuits
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CR
1=
R
Cj1
R
Cj1
R
Cj1
R
Cj1
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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 :
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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
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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.
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+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
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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 .
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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
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Alle imReihenschwingkreis erluterten Zusammenhnge und Formeln ergeben
sich in analoger Form fr denParallelschwingkreis, in dem man systematisch
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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
7/30/2019 TB AC DC fundamentals
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
7/30/2019 TB AC DC fundamentals
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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