Juliusz B. Gajewskifluid.itcmp.pwr.wroc.pl/elektra/L 06.pdfR e s o n a n c e is generated o n l y...

Juliusz B. GajewskiProfessor of Electrical Engineering

1

222

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

3

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.

4

Sinusoidal AC Voltage

Nikola Tesla (1856–1943) George Westinghouse (1846–1914)

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

5


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

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

21 T

f

f — voltage frequency [Hz]T — period [s] — angular velocity of rotation of an electromotive force

(emf) vector Em [rad·s1] or else angular frequency [1/s]

6


B

l

d

AC Voltage Generation

7



Synchronous AC generator

8



e Bl m Bld

B B m sin Bld tcos cos m

e B l E

E t

m m

m

sin sin

sin

tEtzt

tzt

ze

sinsindcosd

dd

mm

m

i eRER t I t m

msin sin

B = var B = const

9


AC Voltage

u u t U t ( ) sinm

tUtuu sin)( mUm — amplitude (peak-to-peak, maximum absolute value) [V] — angular frequency [1/s] — phase angle [rad]

In general

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

10


AC Voltage

t, t

u(t)

T

Umt 0

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

amplitude

angular frequency

phase angle

period

11

AC Voltage

iu tItitUtu sin)(sin)( mm

iuiu tt

Phase Shift

u

t

u, i

0

u

i

i


12

AC VoltageRotating Vector*) — Phasor Diagram

t

u, i

0

u

i

i

u

i

u

*) Also known as phasor.


13

AC VoltageRotating Vector

1m11 sin tUu

2m22 sin tUu tUuuu sinm21


14


t

u

0

u1 u2

u

1

2


15



U2

U11

2

180° (2 1)

U

O A B

C

D

16



12m2m12m22m1

12m2m12m2

2m1m

cos2

180cos2

UUUU

UUUUU

2211

2211

coscossinsinarctg

ABOACDBCarctg

UUUU

tUuuu sinm21

law of cosines

17


AC VoltageMean (Average) Value

2/

0/2 d

T

T tiQ

2a/2TIQT

m

2/

0mm

2/

0a 637.0

2dsin2d2 IIttIT

tiT

ITT

18


AC VoltageMean (Average) Value

i(t) |i(t)|

t0 T/2

Ia

T

Imi(t)

19


AC VoltageRMS Value*)

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

tRiA dd 2

TT

T tiRtRiA0

2

0

2 dd

RTIAT 2

mm

0

22m

0

2 707.02

dsin1d1 IIttIT

tiT

ITT

20


AC VoltageRMS Value

U U E E m mi2 2

i(t)

i(t)

t 0 T/2

I

T

Im

i2(t)

2mI

21


AC Power CircuitResistance R

uip

mmmm sinsin RIURiutIitUu

tPtIUp 2m2mm sinsin

mmm IUP

T

T tpA0

d

PTAT

Ideal resistor R const, L C 0instantaneous power

T

ttPT

P0

2m dsin

1

22


AC Power CircuitResistance R

IUIUIUPPP 2222mmmmm

i

u P R

[P] W

active (real) power

u(t)

i(t)t 0 T/2

P UI

T

p(t)

23


AC Power CircuitInductance L

i ImsintIdeal inductor L const, R C 0

i

u L eL

LL eueu 0

2sincos

dd

mm tLItLI

tiLeL

24



2sin

mtEe LL

u e u eL L 0

2sin

2sincos mmm tUtLItLIu

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

25



I

EL

U LI

u, i

2

i

eLuL

0 /2 t

26



LIULIU 22mm

LX L

LL X

UIIXU

fLLX L 2[XL] [] [L] (1 s)1·1 H (1 s)1·(·s)

inductive reactance

27



tUIttIUuip 2sin2

sinsinmm

t

u, i, p

i p

2

uL

0 /2

28



In an ideal inductive element — an inductor (L = const, R C = 0)— there is periodical exchange of energy between a receiver and asource without a one-directional energy flow strictly related toirreversible conversion of the electrical energy into another form ofenergy, for instance the thermal energy, as in a resistor.As a result, electrical energy used by an inductor within a given timeconsisting of some periods equals z e r o (in joules), and electricpower, equal to the energy taken within a unit time (in watts), is alsoz e r o.In consequence of periodical exchange of electrical energy throughan inductor reactive current flows whose effective value (RMS) is Iand at its terminals is effective (RMS) voltage U.

29



T

T tpA0

d

00 TAPA TT

4/

0 0m

2m

4/

04/

m

2dd

ddd

T IT

T WLIiLiti

tiLtuiA

30



2,IUUIQ

[Q] var

QtAb

[Ab] var·s

reactive power

reactive energy

31


AC Power CircuitCapacitance C

tuCi

uCqtiq

dd

dddd

Ideal capacitor C const, R L 0u Umsint

i

u C uC

tCUt

tCUtuCi cos

d)d(sin

dd

mm

32



2sin

2sincos ttt

2sin

2sin mm tItCUi

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

33



I U C I CUm m2 2

t

u, i

2

i

uC

0 /2

I/2

U I/C

34



CX C

1

CC

IXUXUI

fCCX C

211

[XC] [] 1 [C] 1 1 s:1 F 1s1F 1s:(1C:1V 1V:1A

capacitive reactance

35



tUIttIUuip 2sin2

sinsinmm

t

u, i, p

ip

2

uC

0 /2

36



In an ideal capacitive element — a capacitor (C = const, R L = 0)— there is periodical exchange of energy between a receiver and asource without a one-directional energy flow strictly related toirreversible conversion of the electrical energy into another form ofenergy, for instance the thermal energy, as in a resistor.As a result, electrical energy used by a capacitor within a given timeconsisting of some periods equals z e r o (in joules), and electricpower, equal to the energy taken within a unit time (in watts), is alsoz e r o.In consequence of periodical exchange of electrical energy througha capacitor reactive current flows whose effective value (RMS) is Iand at its terminals is effective (RMS) voltage U.

37



T

T tpA0

d

00 TAPA TT

4/

0 0e

2m

4/

04/

m

2dd

ddd

T UT

T WCUuCut

tuuCtuiA

38



tQA cb

2CUQc

2,IUUIQ

[Q] var

[Ab] var·s

reactive power

reactive energy

39


ResonanceThe phenomenon of r e s o n a n c e occurs in various physicalsystems and comes out when a system is underwent the periodicalexcitation with frequency equal to the natural frequency of freevibration of a system — the r e s o n a n c e frequency.R e s o n a n c e is generated o n l y when the transient responseof a circuit (system) is of an o s c i l l a t o r y character whichrequires energy to be stored in two different ways: in an electric field as the capacitor is charged, and in a magnetic field as current flows through the inductor.Energy can be transferred from one to the other within the circuitand this can be oscillatory. Therefore the resonant circuit musthave the p a s s i v e elements that store energy of an electric field— capacitors, and of a magnetic field — inductors.

40


ResonanceResonance of a circuit involving capacitors and inductors occursbecause the collapsing magnetic field of the inductor generates anelectric current in its windings that charges the capacitor, and thenthe discharging capacitor provides an electric current that buildsthe magnetic field in the inductor, and the process is repeatedcontinually. An analogy is a mechanical pendulum. In some cases,resonance occurs when the inductive reactance and the capacitivereactance of the circuit are of equal magnitude, causing electricalenergy to oscillate between the magnetic field of the inductor andthe electric field of the capacitor.

41


ResonanceAccording to the way the elements L and C are connected with asource of energy there are two possible cases for resonance tooccur when electric circuits are as follows: s e r i e s ones — the v o l t a g e resonance; p a r a l l e l ones — the c u r r e n t resonance.

42


Resonance

i

u L

C

RuR

uL

uC

Series RLC Circuit

43


ResonanceSeries RLC Circuit

t

CLR tiCtiLRiuuuu

0

d1dd

IC

IXULIIXURIU CCLLR 1;;

tIi sinm

?sinm tUu 2;

2mm IIUU

44



I U

UL

UC

UL

UC

UR

LCR UUUU

45



IZXXRIC

LRI

IC

LIRIUUUU

CL

CLR

222

2

2222

1

1

46



22222

2 1 XRXXRC

LRZ CL

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

47



R

Z X

RC

L

RXX

RX CL

1

tg

impedance triangle

48



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 — resistivecharacter series (voltage) r e s o n a n c e.

49



LCf

21

20

0

CL

1

I

0

UL UL

UC

UC

U UR

50



f

XXC

0

XL

f0

51



f f0 XL XC — inductive;

f f0 XL XC — capacitive;

f = f0 XL = XC — resistive s e r i e s r e s o n a n s ef

I

R2

0

R1

f0

U/R1

R2 R1

U/R2

52


ResonanceParallel RLC Circuit

i

u L C R

iR iC iL

53



tuCtu

LRuiiii

t

CLR ddd1

0

CUXUI

LU

XUI

RUI

CC

LLR

;;

tUu sinm

?sinm tIi 2;

2mm UUII

54



U

I

IC

IL

IR

IC

IL

LCR IIII

55



UYBBGUL

CR

U

LUCU

RUIIII

LC

LCR

2222

2222

11

56



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

222222 11 BGBBG

LC

RY LC

57



G

B Y

admittance triangle

G

CL

GL

C

GBB

GB LC

1

tg

1

tg

58



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 — resistivecharacter parallel (current) r e s o n a n c e.

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

59



LCf

21

20

0

L C1

U 0

IC

IL

IR

IC

IL

60



f

BBL

0

BC

f0

61



f f0 BC BL — capacitive;

f f0 BC BL — inductive;

f = f0 BC = BL — resistive p a r a l l e l r e s o n a n c e

f

U

G2

0

G1

f0

I/G1

G2 G1

I/G2

62


ResonanceEnergy in Series RLC Circuit

t

wLit

itiLuip m

dd

dd

dd 2

21

Inductor: i Imsint

tLItLILiw mm 2222

212

21 sinsin

Capacitor: uC UCmsin(t /2 UCmcost

tCUtCUCuw CCmCe 2222

212

21 coscos

63


Resonance

Sum of energies in resonanse

constcossin 2022

022 LItCUtLIww Cem

CLI

CIU

CL

CL C

000 and

1

— characteristic impedance []

Energy in Series RLC Circuit

64



Conclusion: At resonance an exchange of energy between themagnetic field of an inductor and the electric field of a capaci-tor occurs.

65


Practical Applications of ResonanceAntenna

66


Practical Applications of ResonanceRadio

67


Practical Applications of ResonanceTelevision

68


Practical Applications of ResonanceRadio Telephony

69


Practical Applications of ResonanceMobile Phone = Cellphone

70

ResonanceParallel RLC circuit

Complex Numbersin AC Network Analysis

71


AC Network Analysis — Complex Numbers

Re A

j A

A1

Im A

jA2

j A complex number can be viewed as a p o i n tor a p o s i t i o n vector in a two-dimensionalCartesian coordinate system — the complexplane or Argand diagram.

72


AC Network Analysis — Complex NumbersAlgebraic Form

21 jAAA

A1, A2 — projection of a vector on the real and imaginary axes

1j — imaginary unit

73


AC Network Analysis — Complex NumbersTrigonometric Form

Acos = A1 — vector projection on the real axisAsin = A2 — vector projection on the imaginary axis

sincos jAA

74


AC Network Analysis — Complex NumbersPolar Form

jAeA

— modulus (absolute value) of a complex number

— argument (phase, angle) of a complex number

22

21 AAA

12arctg AA

75


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)( mm

76



ICjI

CjU

ILjUIRU

1UCjI

ULj

I

UGI

1

UYIIZU

Voltage and Current Relationshipsin the Time and Frequency Domains

Cj

C

jCj

jCjC

j

221

1since,1

77



jZjXRC

LjRZ e1

— modulus of the complex impedance22 XRZ

RXarctg

cosRe ZRZ

sinIm ZXZ

— argument of the impedance (phase shift)

— resistance of a circuit

— reactance of a circuit


78



— modulus of the complex admittance22 BGY

GBarctg

cosRe YGY

sinIm YBY

— argument of the admittance (phase shift)

— conductance of a circuit

— susceptance of a circuit

jYjBG

LCjGY e1


79



1Y

Z

2222

22

1

BGBX

BGGR

BGjBG

jBGjXR

2222

22

1

XRXB

XRRG

XRjXR

jXRjBG


80



iui

uj

j

jj

IU

IU

IUZZ

e

eee

iuIU

IUZ

m

m


81



iu jj IIUU ee mmmm

iu jj IIUU ee

)(mmm

)(mmm

eeee)(

eeee)(ii

uu

tjjtjtj

tjjtjtj

IIIti

UUUtu


82



IZU

jXRC

LjRZ

1

Cj

C

jCj

jCjC

j

221

1since,1

Ohm’s Law

83



21 ZZIUZ

n

i i

n

ii YY

ZZ11

11

I

U U1 U2

Z1 Z2

Ohm’s Law — Series Circuit

84


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

n

ii

n

ii XXjRRXjRZ

12121

1

21

21

221

221

arctgRRXX

XXRRZZ

85


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

U Z2 Z1 I1 I2

21 YYUIY

n

i i

n

ii ZZ

YY11

11

86


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

n

ii

n

ii BBjGGBjGY

12121

1

21

21

221

221

arctgGGBB

BBGGYY

87


AC Network Analysis — Complex NumbersOhm’s Law

0jIeI

jXR UejUUjXIRIIjXRIZU

— modulus of voltage22 XR UUU

RX UUarctg — argument of voltage (phase shift)

88


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.

89


AC Network Analysis — Complex NumbersExample

I

U UL

UC

UR

L1

C1

R1

L2

C2

R2

I1 I2

90



2122221111 ;

jbc

jbc eIjIIIeIjIII

I I I II

I I I II

c bb

c

c bb

c

1 12

12

11

1

2 22

22

22

2

arctg

arctg

91



jbbcc

bcbc

IeIIjIIjIIjIIIII

2121

221121

cc

bb

bbcc

IIII

IIIII

21

21

221

221

arctg

92


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

93


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]

94


AC Network Analysis — Complex NumbersAC Power — Power Triangle

Re S

j S

P

Im S jQ

95


AC Network Analysis — Complex NumbersPower (Phase) Factor

SP

cos

2

U

IC I2L I1L

I1

IC I2

1

IR

constand

R

C

I

UCjI

96


AC Network Analysis — Complex NumbersPower (Phase) Factor

212121 tgtgtgtg UPIIIII RRLLC

21 tgtg UPCU

212 tgtg UPC

0.1cos9.0

Juliusz B. Gajewskifluid.itcmp.pwr.wroc.pl/elektra/L 06.pdfR e s o n a n c e is generated o n l y...

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Transcript of Juliusz B. Gajewskifluid.itcmp.pwr.wroc.pl/elektra/L 06.pdfR e s o n a n c e is generated o n l y...