Fundamentals of Electrical Engineering - Kierunki...

1

Fundamentals of Electrical Engineering

Juliusz B. GajewskiProfessor of Electrical Engineering

1 2

INSTITUTE OF HEAT ENGINEERINGAND FLUID MECHANICS

Electrostatics and Tribology Research GroupWybrze e S. Wyspia skiego 27

50-370 Wroc aw, POLAND

2

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

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

33

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. Sinusoidal AC Voltage.6. Electrical Measurements.7. Three-Phase Circuits.

44

For electrical engineering the science of electricity is fundamental andis the branch of physics. Physics studies, finds, and explains the prin-ciples of electrical phenomena, while electrical engineering explains the applications of those phenomena to engineering and technology.

Terms.Fundamental Definitions and Units

E l e c t r i c a l e n g i n e e r i n g is engineering that deals with practical applications of e l e c t r i c i t y; generally restricted to applications involving current flow through conductors, as in motors and generators.E l e c t r i c a l e n g i n e e r i n g is an engineering discipline that deals with the study and practical application of e l e c t r i c i t y ande l e c t r o m a g n e t i s m.

Electrical Engineering

55

E l e c t r i c c h a r g e or c h a r g e is a basic property of elementary particles of matter. One does not define charge but takes it as a basic experimental quantity and defines other quantities interms of it.

The early Greek philosophers were aware that rubbing amber withfur produced properties in each that were not possessed before the rubbing. For example, the amber attracted the fur after rubbing, butnot before. These new properties were later said to be due to “charge.” The amber was assigned a negative charge and the fur wasassigned a positive charge.


Electric Charge

66


Thales of Miletus (ca. 624–ca. 546 BC), a Greek, found that amber attracted different lightobjects when rubbed with silk (fur). He is believed to be a discoverer of static electricity and could be generally named a father of electricity. The Greek word for amber is (élektron) ëelectron (English electron) from which one can get ëelectricity and ëelectronics.The English word electric is based on the Greek amber. Both words derive from the electro-static properties of amber. It is also said that “a first usage of the word e l e c t r i c i t y isascribed to Sir Thomas Browne in his 1646 work Pseudodoxia Epidemica”.Ancient and medieval awareness of electrical effects includes lightning, electric fish, St.Elmo’s fire, the amber effect, and, especially in early China, the lodestone (magnet). Little (or even nothing) is known about the discoveries or inventions in the field of electricity between ancient Greece and the Early Modern Times that is times after the development of printing —Gutenberg’s moveable type printing machine — in 1452 and the increasing dispersion ofknowledge in the Renaissance and especially later in the Enlightenment. Those were the Dark (Early) (AD 476–1000) and Middle Ages (AD 1000–1300).

Electric Charge

77

A charge can be p o s i t v e or n e g a t i v e, or z e r o. In nature there occurs only an integral multiple of a universal basic charge of proton —a positively charged particle that is the nucleus of the lightest chemical element, hydrogen.The term „charge” is a primitive notion and an independent quantity (variable) in physics. Its unit is coulomb [C].The charge of e l e c t r o n is conventionally n e g a t i v e, while that of proton is p o s i t i v e. Both charges are the charged constituents ofordinary matter and the smallest known particles (portions) of charge innature. They are referred to as e l e m e n t a r y and are marked as e i

e, where e 1.6021892 0.0000046×10 19 C. They are exactly equal to each other as to their absolute value and are the smallest undivided „amount” of electricity. Each atom has an equal number of electrons and protons, and therefore is electrically neutral as a whole.

Electric Charge


88


Balance of electric charges is one of the most fundamental laws of nature.The electric charge can be neither c r e a t e d nor d e s t r o y e d. One can only transfer some number of elementary charges, for ex-ample, electrons, from one body to another body which causes the first body to be positively charged while the second body has a negative charge of the same absolute value. This process is strictly related to:

Charge quantization — the principle that the electric charge of an object must equal an integral multiple of a universal basiccharge.

Conservation of charge — a law which states that the total charge or the total algebraic sum of charges of an isolated system is constant; no violation of this law has been discovered.

Electric Charge — Laws and Principles

– 2 –

99


Isolated system is s u c h a system through which boundaries n ocharges can pass

or

is a system which is s o i s o l a t e d that it c a n n o t exchangecharges with its surroundings and therefore the total charge inside the system is p r e s e r v e d.

Therefore the charge is indestructible: never can be c r e a t e d or d e s t r o y e d. The charges then can transfer from one place to another one, but never come from nowhere. We therefore say that the charge is p r e s e r v e d.


1010


Transfer of electrons from one body to the other causes the bodies to be charged as a result of an e x c e s s or a d e f i c i e n c y ofcharges.

Such a process is called e l e c t r i f i c a t i o n or c h a r g i n g and is a physical proof of the law of charge conservation.


11

Charge PropertiesThere are negative charges as e l e c t r o n s or n e g a t i v ei o n s and positive charges as i o n s which always are the integral multiples of the smallest charge, that is an electron or a proton.

Opposite charges a t t r a c t and like charges r e p e l.

Charges can be s t a t i c, i m m o b i l e and i n v a r i a b l eor they can be in m o t i o n, or can v a r y with time.


12

Current


c o n d u c t o r s: class I — metals and coal; class II —electrolytes (water solutions of acids, salts and bases);

i n s u l a t o r s (dielectrics, or non-conductors) — gases,insulating liquids (water without additives, distilled water),insulating oil, glass, porcelain, paper, cotton, silk, isinglass, plastics, etc.;

s e m i c o n d u c t o r s — germanium, silicon, oxides of different metals and other bodies of complex structure.

E l e c t r i c c u r r e n t is connected with the motion or time-variations of electric charges; it is strictly related to the classifica-tion (division) of bodies which is as follows:

– 3 –

13

Current


Conduction current in conductors — in a crystal lattice free electrons are loosely bound with atomic nuclei (positive ions) located in the lattice points and can move about in the space of a lattice between at very high velocities of about 105 m/s at room temperature and at almost as twice as great velocity at atemperature of 1000 K.

Displacement current in insulators — there are few or no free electrons at all and hence the insulator (dielectric) ability to carry electric current is minimal or it does not conduct the current; electrons are strongly bound with the atomic nuclei andcan move only within a given atom. In an ideal (perfect) dielectric charges can move in its interior without disturbing itsstructure, and the so-called dielectric polarization occurs.

14

International System of Units SI


SI base unitslength l, s metre mmass m kilogram kgtime t, second scurrent I, i amper Athermodynamic temperature T kelvin K, degluminous intensity j candela cd

Derived unitsangle , , radian rad

solid angle , steradian sr

15

Standard Prefixes for the SI Units of Measure


Multiples SubdivisionsName Symbol Factor Name Symbol Factoryottazettaexapetateragigamegakilohectodeca

YZEPTGMkhda

1024

1021

1018

1015

1012

109

106

103

102

10

decicentimillimicronanopicofemtoattozeptoyocto

dcn

npfazy

10 1

10 2

10 3

10 6

10 9

10 12

10 15

10 18

10 21

10 2416

Selected Quantities in Electrical Engineering


electric charge Q coulomb Cpotential V, , volt Vvoltage, SEM U, E volt Velectric field strength E volt per metre V/melectric displacement D coulomb per square metre C/m2

permittivity farad per metre F/mcapacitance C farad Fresistance R ohmresistivity ohm metre ·mconductance G siemens Sconductivity siemens per metre S/mmagnetic flux density B tesla Tmagnetic flux weber Wbmagnetic field strength H ampere per metre A/m

– 4 –

17

Selected Quantities in Electrical Engineering


magnetic permeability henry per metre H/minductance L henry Hmagnetic resistance R turns per henry 1/Hfrequency f hertz Hzangular velocity radians per second rad/swork, energy A, W joule Jpower P watt Wreactive power Q war varapparent power S volt-ampere VAvelocity metre per second m/sacceleration a metre per second squared m/s2

force F newton Ntorque, moment of force M newton metre N·mother … … … 18

Electrostatics.Electrostatic and Electric Fields

1919


E l e c t r o s t a t i c s — The class of phenomena recognized by the presence of electrical charges, either stationary or moving,and the interaction of these charges, this interaction being solely by reason of the charges and their positions and not by reason of their motion.

Electrostatics

2020


Electrostatics

VQ

VQq

V ddlim

0v

ShQq

Sh

00v lim

q hq QS

QSh S

s vddlim lim

0 0

– 5 –

2121


Electrostatics

q Ql

Qll

lddlim

0

VqQV

dv

SqQS

ds

lqQl

dl2222


Electrostatic and Electric FieldsE l e c t r i c f i e l d*) is space where positive and negative electric charges are and interact with each other.

E l e c t r o s t a t i c f i e l d is such an electric field which is time-independent and in which stationary, not time-varying and immobile with respect to the earth positive and negative electric charges are and interact with each other.

Both fields belong to vector fields.

*) One of the fundamental fields in nature, causing a charged body to be a t t r a c t e dto or r e p e l l e d by other charged bodies.

2323


Coulomb’s Law (Coulomb’s Force)

F q qr

1 2

024

0 — permittivity of empty (free) space ( 8.854×10 12 F/m)

— relative permittivity, dielectric constant [–]

0 — absolute permittivity, permittivity [Fm 1]

rrqq rF

20

214

2424


Electric Field Strength*)

E F rq

Qr4 0 3

F rQqr4 0 3

*) Also known as electric field intensity; electric field vector; electric vector.

– 6 –

2525


Work in Electric Field

Q

r

E

A

O( )

B

rA

rB

drd l

2626


Work in Electric Field — Voltage

d d d dW q qE lF l E l cos

B

A

B

AAB cosdd lEqqW lE

AB

B

A

B

A

AB cosdd UlEq

W lE

scalar product

electric voltage

2727


Work in Electric Field — Potential

B0A02

0AB 44

d4

B

Ar

Qr

QrrQU

r

r

)A(ddA

O

O

AAO lElEU

204 rQE

electric potential

For and dlcos dr

2828


Electric Potential — Potential DifferenceO

B

B

A

O

A

ddd lElElE

BABA U

BAABU

– 7 –

2929


Conclusions:In an irrotational electric field

the voltage between two points isequal to a difference of theirpotentials:

UAB A – B.Surfaces in space with the same

electric potential [ = (x, y, z) = const] at every point are callede q u i p o t e n t i a l surfaces.

Electric Potential — Potential Difference

Q E

const

A

B

Q

3030


0ddA

AAA

L

qqW lElE

0E


3131



SL

SElE drotd

rot E 0irrotational electric field

From Stokes’ Theorem

3232


Potential Gradient

An irrotational vector electric field E whose curl is identically zero: rotE = 0 is always the gradient*) of a scalar function, here the electric potential or simply potential . It is called a potential gradient.

rrE grad

*) Potential gradient is the potential difference per unit length, as measured in thedirection in which it is a maximum at a point.

– 8 –

3333


Gauss’ Law — Gauss’ Flux Theorem*)

E

dS

rB

S

dS

E

d dE S

*) Also known as the integral form of Gauss’ Law.3434


Gauss’ Law — Gauss’ Flux Theorem

SS

SE cosddSE

For and dScos d r2

4d

4d QQ

S

SE

204 rQE

3535



Q

S

SE d

For 4

The electric flux through any closed surface S is proportional to the total electric charge Qenclosed by S and divided by the absolute permittivity 0.

3636



QSS

SDSE dd

D E0electric induction*)

*) Also known as dielectric displacement; dielectric flux density; displacement;electric displacement density; electric flux density.

– 9 –

3737


Divergence Theorem

VS

Vddivd DSD

If exists, then

VV

VqV dddiv vD div vD

From the Gauss-Ostrogradsky Theorem

q

source electric field

sourceof electric fieldV

VqQ dv

3838


Conductor in External Electric Field

0ie EEE

E 0Ee Ei

E

E

1

2 1

3939


Conductor in External Electric Field0E

0div.1 0v Eq

const0grad.2 E

0tn EEE i

0

sqE4040


Capacitance

kQQC

RQ

04potential of sphere

capacitance of sphere

Capacitance of Isolated Conductor

RC 04

– 10 –

4141


CapacitanceMutual Capacitance of Two Isolated Conductors

E

1

2Q

4242


CapacitanceMutual Capacitance — Capacitor

QC1

21

21

QC

21

QC

4343


CapacitanceCapacitors in Series

Q1 Q2

U1 U2

U

C1 C2

4444


CapacitanceCapacitors in Series

2

2

1

121

2

22

1

11 and;

CQ

CQUUU

CQU

CQU

QQQ 21

CQ

CCQ

CQ

CQU

212

2

1

1 11

21

21

21or111

CCCCC

CCC

n

i iCC 1

11

– 11 –

4545


CapacitanceCapacitors in Parallel

C2U U1 U2C1

Q2Q1

4646


CapacitanceCapacitors in Parallel

22112121 and UCUCQQQUUU

212211 CC

UUCUC

UQC

n

iiCC

1

4747


CapacitanceEnergy of Isolated Conductor; Energy of Electric Field

CQQ

CQWQ

CQQW

Q

2dddd

2

0

222

22 QCC

QW

Energy of Capacitor

CQQUCUUCUW

U

222d

22

04848

Elektrodynamika. Pr d sta y

– 12 –

4949

Elektrodynamika. Pr d sta y

Thomas Alva Edison (1847–1931)

…promoted direct current

Electrodynamics. DC Current5050

Electrodynamics.DC Current

E l e c t r i c c u r r e n t or c u r r e n t — A net ordered (directed) motion of electrically charged particles or charged macroscopicbodies in space through the cross-section of a medium (solids, liquids, gases, or free space) under an electric field — it is the phenomenon arising from the presence of this field.The electric field is given vectorially by the electric field strength E orscalarly by the voltage U.

Electric Current

DC or d i r e c t c u r r e n t — Such an electric current which flows in one direction only (the unidirectional flow of electric charge), as opposed to alternating current.

5151


Schematic Circuit Diagram

C2U1 U4

R1

R4

C3I1

I2

I3

I4

I II

branches

loop

node

elementsE1

U2port III

Loops I and II are c l o s e d ones, while Loop III is o p e n because I3 I4.Element E1 is an a c t i v e one, while other elements are p a s s i v e. 5252


Ohm’s LawGeneral Ohm’s Law

eC EEEB

Ae

B

AC

B

A

ddd lElESlI

The conductor of a lengthfrom A to B with the same current intensity in all itscross sections.

A resultant electric field in a conductor is a vector sum of Coulomb’s field EC and external forces Ee

– 13 –

5353



For E·dl d

ABBA

B

AC d UlE

BAAB

B

Ae d EElE

potential differencebetween A and B

electromotive force EMFbetween A and B

5454



ABAB

B

A

B

AeC

*AB dd EUU lElEE

For J const, const and S const

IRIS

llSI

ABAB

B

A

B

A

ddlJ

5555



SlR AB

AB resistance of conductor

ABABAB

AB*AB

or

EUIR

IRU

5656



IA

B

A

UAB

EAB

*ABU

B

RAB

– 14 –

5757



For A B, RAB R, EAB E

EIR

EI R

5858


Energy, Power, Heat — Joule’s Lawd dW UI t W UIt

]W[d

d= UIt

WP

Since1 J 0.24 cal

electric power ofDC current

electric energy

work of DC current

heat

]J[2tRIW

]cal[24.0 2tRIQ

5959

01

n

kkI


Kirchhoff’s Laws of Electric Circuits

1I w

2I3I

4I

5I

53241

54321

or

0

IIIII

IIIII

Kirchhoff’s Current or First Law

6060


Kirchhoff’s Laws of Electric CircuitsKirchhoff’s Voltage or Second Law

U

w1

w4

w2

w3R3

R3I3

E1 R1

R2

R1I1

R2I2

E2

I1

I3

I2

– 15 –

6161


Kirchhoff’s Laws of Electric CircuitsKirchhoff’s Voltage or Second Law

33221121

33222111 0

IRIRIREEU

IRIREIREU

n

kk

n

kkk EIR

11

6262


ResistorsSeries Circuit

U

I R1

R2

R3

U1

U2

U3

6363


ResistorsSeries Circuit

IRUIRUIRU 332211 ;;

IRIRRRUUUU e321321

321e RRRR

n

kkRR

1e equivalent resistance

6464


ResistorsParallel Circuit

U

I

R1 R2 R3U1 U2 U3

I1 I2 I3

– 16 –

6565



33

22

11 ;;

RUI

RUI

RUI

e321321 R

URU

RU

RUIIII

321e

1111RRRR

6666



n

k kRR 1e

11

n

kkGG

GGGG

1e

321e

equivalent resistance

Since G 1 R — conductance

equivalent conductance

6767

Electromagnetism.Magnetic Field of DC Current 6868

Electromagnetism.Magnetic Field of DC Current

Magnetic FieldM a g n e t i c f i e l d is one of the elementary fields in nature; it is found in the vicinity of a magnetic body or current-carrying medium and, along with an electric field, in a light wave. It one ofthe many field existing in nature in which electric charges are affected by forces by magnets or by currents in conductors. Thisfield in turn acts on other magnets or conductors with currents being in it.

A magnetic field is characterized by energy and inertia, and to some extent is material and similar to an electric field. It possesses two poles: positive (North — N) and negative (South — S) and the opposite poles attract and the like poles repel.

– 17 –

6969


Magnetic FieldEach point in space around a current-carrying wire is described by such a vector of magnetic induction and therefore a wire or currentcircuit generates a magnetic field. The sources of such a field arenot only wires or circuits but also magnetic materials, the so-called ferromagnetic materials or ferromagnetics, and strictly currentmicrocircuits in their atoms.

Between the electric and magnetic fields there is difference in the interactions of both fields. It is a result of their character: an electric field has a central, radial character and its lines of force are open, while a magnetic field has a crosswise character — theforce acts on a charge in motion perpendicularly to its direction and the lines of force are closed. Both fields are complementary in the description of a general e l e c t r o m a g n e t i c f i e l d. 7070


Magnet(ostat)ic Field. Ampere’s Force

S

N N

S

F F II

B B

7171


F I l B I l Bsin sin( , )l B

F l BI

F

B

l

Magnet(ostat)ic Field. Ampere’s Force

”left hand rule”

7272


Magnet(ostat)ic Field. Oersted

B

I

”right hand grip rule”

– 18 –

7373


Biot–Savart–Laplace’s Law

I

dl

rP90°

dB7474


Biot–Savart–Laplace’s Law

30 d

4d

rI rlB

20

20 sind

4,dsind

4d

rlI

rlIB rl

0 — permeability of free space (vacuum), magnetic constant( 4 ×10 7 H/m)

— relative permeability [–]

0 — absolute permeability, permeability of a specific medium,permeability [Hm 1]

Magnetic Induction, Magnetic Flux Density

7575


Biot–Savart–Laplace’s LawMagnetic Field Strength

3

d4

dr

I rlH

2

,dsind4

dr

lIH rl

HB 0

7676


Ampere’s Law

H

rI

L

dl

– 19 –

7777


Ampere’s Law

IrrI

lrIl

rI rr

L

2241

d241=dd,cos2

41d

2

0

2

0

lHlH

n

kk

r

L

IlH1

2

0

=d,cosdd lHlH

rotational field(nonpotential)

7878


Lorentz’s Force

FB

+q 0

7979


Lorentz’s Force and Electromagnetic Forced dF l BI

I q nd dl

d dF Bq n

F F BLddn q

F F F E Be L q q electromagnetic forceor Lorentz’s equation

Lorentz’s force or magnetic force

8080


Faraday’s Law of Induction

tkE

dd m

i

dS

EiB

I

R

S

SB dd

– 20 –

8181


Faraday’s Law of Induction

tkE

dd m

i

Lenz’s Law

S

B

I

BiIiEi

tE

dd m

i1k

8282


Self-Induction*)

S

Es

B

Ii

E

Bi

I

IiBn

dS

90°

SB ddd nm SB*) The production of a voltage in a circuit by a varying current in that same circuit

8383


Self-Inductance or Inductance

S l

ILr

SI n30

m dd4

rl

S l rSL n3

0 dd4

rl

8484


Electromotive Force*) of Self-Induction

LIt

Edd

s

E L Its

dd

L f(t)

*) Also known as induced voltage; induced electromotive force

– 21 –

8585

Nikola Tesla (1856–1943) with one of his early electrical generators…

…advocated alternating current while Thomas A. Edison (1847–1931) promoted direct current

Sinusoidal AC Voltage

8686


A l t e r n a t i n g v o l t a g e — Periodic voltage, the average value of which over a period is zero.

The time variations of periodic voltages can be waves of different shapes: square, rectangular, triangular, sine, and so forth. Their distinctive feature is a cycle of changes repeated within the time Tcalled a period. Its reciprocal is the frequency of voltage f.

21T

f

f — voltage frequency [Hz]T — period [s]

— angular velocity of rotation of an electromotive force(emf) vector Em [rad·s 1] or else angular frequency [1/s]

8787


B

l

d

AC Voltage Generation

8888


AC Voltage Generation

e Bl m Bld

B Bm sin Bld tcos cosm

e B l E

E t

m m

m

sin sin

sin tEtzt

tzt

ze

sinsindcosd

dd

mm

m

i eR

ER t I tm

msin sin

B = var B = const

– 22 –

8989


AC Voltage

u u t U t( ) sinm

tUtuu sin)( m

— voltage angular frequency [1/s]— phase angle [rad]

In general

u(t) e(t) and hence Um Em

9090


AC Voltage

t, t

u(t)

T

Um

t 0

Conclusion: Any variable sinusoidal physical quantities can be presented e x p l i c i t l y by means of threequantities: amplitude, frequency and phase angle.

amplitude

angularfrequency

phase angle

period

9191

AC Voltage

iu tItitUtu sin)(sin)( mm

iuiu tt

Phase Shift

u

t

u, i

0

u

i

i


9292

AC VoltageRotating Vector*) — Phasor Diagram

t

u, i

0

u

i

i

u

i

u

*) Also known as phasor.


– 23 –

9393


AC VoltageRMS Value*)

*) Also known as root-mean-square value, effective value.

tRiA dd 2

TT

T tiRtRiA0

2

0

2 dd

RTIAT2

mm

0

22m

0

2 707.02

dsin1d1 IIttIT

tiT

ITT

9494


AC VoltageRMS Value

U U E Em mi2 2

i(t)

i(t)

t0 T/2

I

T

Im

i2(t)

2mI

9595


AC Power CircuitResistance R

uip

mmmm sinsin RIURiutIitUu

tPtIUp 2m

2mm sinsin

mmm IUPT

T tpA0

d

PTAT

Ideal resistor R const, L C 0instantaneous power

T

ttPT

P0

2m dsin1

9696

Resistance R


AC Power Circuit

IUIUIUPPP2222mmmmm

i

u P R

[P] W

active (real) power

u(t)

i(t)t0 T/2

P UI

T

p(t)

– 24 –

9797


AC Power CircuitInductance L

i Imsin tIdeal inductor L const, R C 0

i

u LeL

LL eueu 0

2sincos

dd

mm tLItLI

tiLeL

9898



2sin

mtEe LL

u e u eL L0

2sin

2sincos mmm tUtLItLIu

Conclusion: the phase of the current l a g s that of the voltage by /2.

9999



I

EL

U LIu, i

2

i

eLuL

0 /2 t

100100



LIULIU22mm

LX L

LL X

UIIXU

fLLX L 2

[XL] [ ] [L] (1 s) 1·1 H (1 s) 1·( ·s)

inductive reactance

– 25 –

101101



tUIttIUuip 2sin2

sinsinmm

t

u, i, p

i p

2

uL

0 /2

102102



T

T tpA0

d

00TAPA T

T

4/

0 0m

2m

4/

04/

m

2dd

ddd

T IT

T WLIiLititiLtuiA

103103



2,IUUIQ

[Q] var

QtAb

[Ab] var·s

reactive power

reactive energy

104104


AC Power CircuitCapacitance C

tuCi

uCqtiq

dd

dddd

Ideal capacitor C const, R L 0u Umsin t

i

u CuC

tCUt

tCUtuCi cos

d)d(sin

dd

mm

– 26 –

105105



2sin

2sincos ttt

2sin

2sin mm tItCUi

Conclusion: the phase of the voltage l a g s that of the current by /2.106106

Capacitance C


AC Power Circuit

I U C I CUm m2 2

t

u, i

2

i

uC

0 /2

I/2

U I/ C

107107



CX C

1

CC

IXUXUI

fCCX C 2

11

[XC] [ ] 1 [C] 1 1 s:1 F 1s 1F 1s:(1C:1V 1V:1A

capacitive reactance

108108



tUIttIUuip 2sin2

sinsinmm

t

u, i, p

ip

2

uC

0 /2

– 27 –

109109



T

T tpA0

d

00TAPA T

T

4/

0 0e

2m

4/

04/

m

2dd

ddd

T UT

T WCUuCuttuuCtuiA

110110



tQA cb

2CUQc2

,IUUIQ

[Q] var

[Ab] var·s

reactive power

reactive energy

111111


Resonance

i

u L

C

RuR

uL

uC

Series RLC Circuit

112112


ResonanceSeries RLC Circuit

t

CLR tiCt

iLRiuuuu0

d1dd

IC

IXULIIXURIU CCLLR1;;

tIi sinm

?sinm tUu 2;

2mm IIUU

– 28 –

113113



IU

UL

UC

UL

UC

UR

LCR UUUU114114



IZXXRIC

LRI

IC

LIRIUUUU

CL

CLR

222

2

2222

1

1

115115



22222

2 1 XRXXRC

LRZ CL

Z — impedance [ ]XL — inductive reactance [ ]XC — capacitive reactance [ ]X — reactance [ ]

116116



R

ZX

RC

L

RXX

RX CL

1

tg

impedance triangle

– 29 –

117117



RC

L

RXX

RX CL

1

tg

X 0 XL XC u i 0 — inductive character;

X 0 XL XC u i 0 — capacitive character;

X 0 XL XC u i 0 UL UC — resistivecharakter series (voltage) r e s o n a n c e.

118118



LCf

21

20

0

CL 1

I0

UL UL

UC

UC

U UR

119119


ResonanceParallel RLC Circuit

i

u L CR

iR iCiL

120120



tuCtu

LRuiiii

t

CLR ddd1

0

CUXUI

LU

XUI

RUI

CC

LLR ;;

tUu sinm

?sinm tIi 2;

2mm UUII

– 30 –

121121



U

I

IC

IL

IR

IC

IL

LCR IIII122122



UYBBGUL

CR

U

LUCU

RUIIII

LC

LCR

2222

2222

11

123123



Y — admittance [S]BC — capacitive susceptance [S]BL — inductive susceptance [S]B — susceptance [S]

222222 11 BGBBG

LC

RY LC

124124



G

BY

admittance triangle

G

CL

GL

C

GBB

GB LC

1

tg

1

tg

– 31 –

125125



G

CL

GL

C

GBB

GB LC

1

tg

1

tg

B 0 BC BL u i 0 — capacitive character;

B 0 BC BL u i 0 — inductive character;

B 0 BC BL u i 0 UL UC — resistivecharakter parallel (current) r e s o n a n c e.

Comparison of series and parallel circuitsX 0 B 0 X 0 B 0 126126



LCf

21

20

0

L C1

U0

IC

IL

IR

IC

IL

127127


AC Network Analysis — Complex NumbersVoltage and Current Relationships

in The Time and Frequency Domains

ttiC

tu

ttiLtu

tRitu

d)(1)(

d)(d)(

)()(

ttuCti

ttuL

ti

tGutuR

ti

d)(d)(

d)(1)(

)()(1)(

tjtjtjtj IItiUUtu e2e)(e2e)( mm128128


AC Network Analysis — Complex Numbers

U RI

U j LI

U j C I1

I GU

I j LU

I j CU

1

UYIIZU

Voltage and Current Relationshipsin The Time and Frequency Domains

– 32 –

129129



jZjXRC

LjRZ e1

— modulus of the complex impedance22 XRZ

RXarctg

cosRe ZRZ

sinIm ZXZ

— argument of the impedance (phaseshift)

— resistance of a circuit

— reactance of a circuit


130130



— modulus of the complex admittance22 BGY

GBarctg

cosRe YGY

sinIm YBY

— argument of the admittance (phaseshift)

— conductance of a circuit

— susceptance of a circuit

jYjBGL

CjGY e1


131131



1Y

Z

2222

22

1

BGBX

BGGR

BGjBG

jBGjXR

2222

22

1

XRXB

XRRG

XRjXR

jXRjBG


132132



iu

i

uj

j

jj

IU

IU

IUZZ e

eee

iuIU

IUZ

m

m


– 33 –

133133



iu jj IIUU ee mmmm

iu jj IIUU ee

)(mmm

)(mmm

eeee)(

eeee)(ii

uu

tjjtjtj

tjjtjtj

IIIti

UUUtu


134134



IZU

jXRC

LjRZ 1

Cj

C

jCj

jCjC

j 221

1since,1

Ohm’s Law

135135



21 ZZI

UZn

i i

n

ii YY

ZZ11

11

I

UU1 U2

Z1 Z2

Ohm’s Law — Series Circuit

136136


AC Network Analysis — Complex NumbersOhm’s Law — Series Circuit

n

ii

n

ii XXjRRXjRZ

12121

1

21

21

221

221

arctgRRXX

XXRRZZ

– 34 –

137137


AC Network Analysis — Complex NumbersOhm’s Law — Parallel Circuit

I

U Z2Z1

I1 I2

21 YYUIY

n

i i

n

ii ZZ

YY11

11

138138


AC Network Analysis — Complex NumbersOhm’s Law — Parallel Circuit

n

ii

n

ii BBjGGBjGY

12121

1

21

21

221

221

arctgGGBB

BBGGYY

139139


AC Network Analysis — Complex NumbersOhm’s Law

0jIeI

jXR UejUUjXIRIIjXRIZU

— modulus of voltage22XR UUU

RX UUarctg — argument of voltage (phaseshift)

140140


AC Network Analysis — Complex NumbersOhm’s Law

L (1/ C) X 0, UX 0 u i 0 — inductivecharacter;

L < (1/ C) X < 0, UX < 0 u i < 0 — capacitivecharacter;

L = (1/ C) X = 0, UX = 0 u i = 0 — resistivecharacter v o l t a g e r e s o n a n c e.

– 35 –

141141


AC Network Analysis — Complex NumbersAC Power

ii

u

jj

j

IeIIeIUeU

*

jQPjUIUIeUIeIeUeIUS jjjj iuiu

)sin(cos

ijIeI *Remark: is the conjugate of the complex current

sincos

UIQUIP

142142


AC Network Analysis — Complex NumbersAC Power

UIQPSS 22

S — apparent power [VA]S — complex power (absolute value of complex power) [VA]P — active (real, true) power [W]Q — reactive power [var]

143143


AC Network Analysis — Complex NumbersAC Power — Power Triangle

Re S

jS

P

Im SjQ

Remark: cos is called power (phase) factor144144

Electrical Measurements

– 36 –

145

Pomiar oporu czynnego (rezystancji)

U

A

V

I

RxRv

Iv

I Iv vv R

UI

v

vx

RUI

UII

UR

IUR*

x

v

*x

x

x*x

RR

RRR

vx RR Rx 1.0

vR


Technical Method of Resistance MeasurementAccurate Measurement of Voltage

146


IUR*

x

a*x

a

x

x*x

RRR

RRR

ax RR Rx 1.0

U

A

V

I

Rx

Raax R

IUR

0aR

Technical Method of Resistance MeasurementAccurate Measurement of Current


147


0.1 (1.0) Rx 106

D

V

A B

C

E

Rx Rn

R1 R2

W2

W1

DBCBADAC i UUUU

12x1 RIRI

22n1 RIRI

2

1x R

RRR n

Rx 0.1 (1.0) — Thomson (Kelvin) bridge


Resistance MeasurementWheatstone Bridge

148

RU

IW

UIP


Measurement of Active and Apparent Powersand Power Factor

Active Power and Resistance

– 37 –

149



Active Power and Impedance

ZU

IW

PcUIcUIP ww cossince,cos150

ZU

IA

V

W

SPUISUIP cos,cos



Apparent Power, Power Factor and Impedance

151

Three-Phase Circuits152

Three-Phase Circuits

Three-Phase Voltage and Current

2sin4sin2sin

sin

mm3

m2

m1

tUtUutUu

tUu

L

L

L

3/4m3

3/2m2

m1

sinsinsin

tIitIitIi

L

L

L

ZL1 ZL2 ZL3UL1 UL2 UL3 IL1 IL2 IL3

– 38 –

153


Three-Phase Star (Y) Configuration

U

XYZ

W V

L1

L2L3N

ppp 323230cos2 UUUU

UUUU 312312

60°

L1

L2L3

30°U12U31

U23

UL1

UL3

UL2

U12

UL1

UL2

154


Three-Phase Delta ( ) Configuration

U

X

Y

Z

W

V

L1

L2

L3

UL1

UL2

UL3

331

223

112

L

L

L

UUUUUU

pp 3and IIUU

155


Power in Star Configurationcos33cos pppppp IUPPIUP

L1

L2

L3

UpZp

I Ip

U 3 p

star

156


Power in Delta Configurationcos33cos pppppp IUPPIUP

L1

L2

L3

I 3 If

U Uf

If

Zf

delta

– 39 –

157


Power of Symmetric Three-Phase System

IIUU pp ,3 3

, ppIIUU

]VA[3

]var[sin3

]W[cos3

UIS

UIQ

UIP

star delta

cos33cos pppppp IUPPIUP

158


Measurement of Power and Energyin Three-Phase System

L1

L2

L3

N

W

W

W

P1

P3

ZL1

P2 ZL2

ZL3

321 PPPP pp321 3PPPPPPAsymmetric load Symmetric load

159



L1

L2

L3

W

W

Load

P

P

IL1

IL3

U12

U32

Aaron’s System

160



Aaron’s System

30cos30cos

UIPUIP

cos330cos30cos UIUIPPP

coscos 112332 LL IUIUPPPAsymmetric load

Symmetric load

– 40 –

161

Pomiary mocy i energii pr du trójfazowego

[°]

-50

50

150

-100

0

100

200

P [%

]

0

P

P

P

90 60 30 30 60 90

capacitivecharacter

inductivecharacter



Aaron’s System

162


UIUIPP2330cos

When 0 cos 1

When 60° cos 0,5

UIPPP 3

UIPPP23;0



Aaron’s System

163


L1

L2

L3

WIL1

U23

ZL1

ZL2

ZL3

Wattmeter or watt-hourmeter



Power and Reactive Energy in Symmetric System

164

Pomiary mocy i energii pr du trójfazowegoThree-Phase Circuits



90°0

IL1

U23

UL1

UL3 UL2

sin111 LLL IUQ

sin90cos 123123 LL IUIUP

reactive power by definition

active power measured

– 41 –

165

Pomiary mocy i energii pr du trójfazowegoThree-Phase Circuits



123 3 LUU

PQQPQ LL 333 11

QtAbreactiveenergy

reactivepower

166


Thank you for your attention!

©© 20102010 Juliusz B. GajewskiJuliusz B. Gajewski

– 42 –

Date post:	08-Jul-2018
Category:	Documents
Upload:	voliem
View:	249 times
Download:	1 times

Fundamentals of Electrical Engineering - Kierunki...

Documents