juliusz b. gajewskifluid.itcmp.pwr.wroc.pl/elektra/l 05.pdfconsidered as a d i s t r i b u t e d –...

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Juliusz B. GajewskiProfessor of Electrical Engineering

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2222

FACULTY OF MECHANICAL AND ELECTRIC POWER ENGINEERING

Process Engineering and Equipment, Electrostaticsand Tribology Research Group

Wybrzeże S. Wyspiańskiego 2750-370 Wrocław, POLAND

Building A4 „Stara kotłownia”, Room 359Tel.: +48 71 320 3201; Fax: +48 71 328 3218

E-mail: [email protected]: www.itcmp.pwr.wroc.pl/elektra

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Contents

1. Terms. Fundamental Definitions and Units.2. Electrostatics. Electrostatic and Electric Fields.3. Electrodynamics. DC Current.4. Electromagnetism. Magnetic Field of DC Current.5. Electric Circuit Elements.6. Sinusoidal AC Voltage.7. Complex Frequency Concept.8. Electric Filters.9. Electrical Measurements.

10. Three-Phase Circuits.11. Electrical Signals.12. Electric Switches.

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Electric Circuit Elements

Vacuum0 8.854×1012 F/m0 4×107 H/m

0 S/mc2 (0×0)1 c (0×0)0.5 2.997956×108 m/s

relative permittivity relative permeability conductivity

Physical Constants

5


If the physical (electrical) constants: the relative permittivity ,the relative permeability , and the conductivity do not dependon the electric field strength and the magnetic field strength, thena given medium is l i n e a r.

Physical Constants

6


Circuit and its ElementsE l e c t r i c c i r c u i t is a closed path in space along whichelectrons can flow or an electromagnetic wave can travel. A fragmentof space, where one can isolate phenomena related to conversion ofthe electric energy, is called an e l e m e n t of an electric circuit. Theelements can be active — the sources of electric energy for the circuit;they may be batteries, direct-current generators, or alternating-currentgenerators, and passive — resistors, inductors, and capacitors.

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P a r a m e t e r s of elements of an electric circuit are thecoefficients, that combine a current flowing through a given elementwith a voltage at its terminals, or their derivatives. The essentialparameters are: resistance R, capacitance C and inductance L.The values of the parameters of electric circuit’s elements depend ontheir geometrical size (length, cross-sectional area) and the physicalproperties of a medium (relative permittivity, relative permeability andconductivity) in which an electric current flows or an electromagneticwave travels.The values can be determined experimentally by studying the voltage-current or current-voltage characteristics of an element.

Parameters of Circuit

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Parameters of CircuitAn element has l u m p e d p a r a m e t e r s if its geometricallinear dimensions are m u c h smaller, e.g. ten times, than the lengthof an electromagnetic wave travelling over the element. Otherwise,the larger physical dimensions cause the model of an element to beconsidered as a d i s t r i b u t e d–p a r a m e t e r m o d e l (ordistributed-element model).

f

l

22 c speed of wave in any medium

wavelength in any medium

element’s length

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S t a t i o n a r y e l e m e n t is such an element whose parametersdo not vary with time, otherwise the element is n o n-s t a t i o n a r y.

Stationary Elements

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Multiport ElementsEach element of an electric circuit has terminals or ports that permitan electric current to flow into or leave them (energy can be suppliedor withdrawn). If the same current enters and leaves two terminals,then the terminals constitute a port. An element that has two terminals(one port) is called a t w o–t e r m i n a l element (resistors, capaci-tors, inductors, diodes, batteries, direct-current generators). Elementsthat have more terminals are called t h r e e–t e r m i n a l elements(transistors) or f o u r–t e r m i n a l (two-port) elements (transfor-mers, filters).

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Multiport Elements

Two-

term

inal

ele

men

t

Four

-ter

min

al e

lem

ent

12


Multiport Elements

resistortransistor

transformerwith iron core

amplifier

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Symmetrical ElementsS y m m e t r i c a l (r e v e r s i b l e) e l e m e n t has the sameproperties regardless of the way an element is connected in a cir-cuit and of the polarity of the voltage applied — the direction of acurrent flow.

I

U

diode

resistor

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AUI

~0)( uUtu

~0~0~0)( iIAuAUuUAti

current-voltage characteristic

linear element response

Linear Elements

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Linear Elements

I

U

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I

U

i(t)

t

u(t)

t

t1 t2 t3 t4

t8 t7

t5 t6

t1 t2

t3 t4

t8 t7 t5

t6

Linear Elements

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Non-Linear Elements

2AUI

~0)( uUtu

~0

2~

20

2~~0

20

2~0

22)(

uAUiIAuuAUAUuUAti

current-voltage characteristic

non-linear element response

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Non-Linear Elements

I

U

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Non-Linear Elements

I

U

i(t)

t

u(t)

t

t1 t2 t3 t4

t8 t7

t5 t6

t1 t2

t3 t4

t8 t7 t5

t6

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Active Elements

ttte

d)(d)(

SB dd

cosBS SB

Generator. Faraday’s Law

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Active ElementsGenerator. Faraday’s Law

B

l

d

cos2cos rBlBS SB tStS cos)(

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Active ElementsGenerator. Faraday’s Law

tBSt cos)(

tBStt sin

d)(d

tEtBSte sinsin)( m

BSUE mm

tUtu sin)( m

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Active ElementsVoltage sources

E E

E

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Ri

E

U EIRU i

EUUIREIIU

0

is

00

real voltage source

short circuit state

no load state

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U

E

I

ideal

real

0 Is E/Ri

Voltage-current characteristics

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Active ElementsCurrent sources

I I Is Ri

si IUGI ii 1 RG

s

si0

00

IIUIRUUI

no-load state

short-circuit state

real current source

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Active ElementsCurrent sources

I

Is

U

ideal

real

0 U0 I/Gi

Current-voltage characteristics

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Passive ElementsrErF ddd qW

tptuiquW dddd

t

tpW 0d

)()()( titutp

t t

ttituttituWtW0 0

0d)()(d)()(0)(

condition for element’s passiveness

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photoresistor otorezystor

varistor

resistor resistor

resistor

variable resistor nastawny

variable resistor—potentiometer potencjometryczny

Passive ElementsResistor

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)()(or)(1)( tRitutuR

ti

I

U

I

U

0d)(d)()()( 2R

ttRittitutWtt


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S

l

lS

lSG

Sl

SlR

1or1

[] = ·m or ·m2/m (·mm2/m)[] = S/m or S/(m2/m) S·m/m2


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S

l

TRTTRTRR TT 11)( 00 0


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Types of windings in wire resistors:1. common,2. bifilar,3. common on a thin former,4. Ayrton-Perry.

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Resistivity of some materialsMaterial [10-6·W·m]

SilverCuprumGoldAluminumTungstenBronze MS 63IronPlatinumTinManganin WM 43Constantan WM 50MercuryCoal (graphite)Cable paper — impregnatedRubberTransformer oilPolyethylene

0.0156 (162)0.017250.0220.0283 (290)0.05480.0740.09780.1110.1140.430.490.95820–10010131013–10141011–10121020–1022

Passive Elements

Resistor

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capacitor capacitorpolarized

capacitor

trimmer capacitor

variable capacitor

electrolytic capacitor

Passive ElementsCapacitor

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uqC

Q

U

Q

U

tqti

dd)(

tuCtid(t)d)(

ttiC

utut

d)(1)0()(0


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0)(21

d)(ddd)(d)()()(

2

C

tCu

utuCttuCtuttitutW

ttt

)(21d)(d)(d)()( 2

00

0

C tCuutuCutuutuCtWtt


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S

d

dSC 0

[] = – (dimensionless number)0 = 8.854×1012 F/m


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Pemittivity of some materialsMaterial [–]

VacuumAirKerosene, teflonIcePolyethyleneTransformer oilRubberWoodPaper – impregnatedBakeliteGlassPetrolPVCSiliconeMethanolWater – distilledCeramics X7RCeramics Z5U

1.00001.00062.02.0–3.02.22.2–2.52.5–2.83.3–3.53.5–84.4–5.45.0–165.76.112338015005000

Passive Elements

Capacitor

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air core inductor iron-core inductor (ferromagnetic core)

nonlinear inductor

variable inductor

slug-tuned inductorwith ferrite core

Passive ElementsInductor

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I I

iL

tz

ttetu

dd

dd)()(

tiLtud(t)d)(

ttuL

itit

d)(1)0()(0


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0)(21

d)(d)(ddd)()()(

2

L

tLi

itiLttitiLttitutW

ttt

)(21d)(d)(d)()( 2

00

0

L tLiitiLitiitiLtWtt


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S

l

z

lSzL 20

[] = – (dimensionless number)0 = 4×107 H/m


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Permeability of some materialsMaterial [–]

BismuthCuprumWaterVacuumAirAluminumNickelMild steelSpecialty (special-purpose) steel

0.999840.999990.9999911.000001.0000041.000004< 300< 5000< 10 000


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BlF dd I

lrBrBlrF

dddd=ddd

IIW

SB dd IW

d 2r

F

F I I d

B

N

S

work equivalent to torque

Electric Motor. Ampere’s ForcePassive Elements

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