Izvanredni-drugi dio zadatci za kolokvij s... · Zadaci prije kolokvija 1. Dvije nabijene kugle u...

13 Odredi kolika se snaga troši na otporu R4 ako su zadani elementi mreže spojeni prema slici U1=U2=12VR1=02Ω R2=03Ω R3=01Ω R4=02Ω

14 Zadana je mreža prema slici Odredite struju I3 kroz R3 ako su vrijednosti U1=64V U2=24V U3=30V R1=2Ω R2=3Ω R3=4Ω R4=7Ω i R5=3Ω

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Nastavak zadatka 14 15 U mreži prema slici odredite struju IR kroz otpornik R i snagu koja se na njemu troši Zadani su elementi u mreži U1=12V U2=12VIk=14A R1=5Ω R2=4Ω R3=1Ω R4=2Ω i R=10Ω

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16 U mreži prema slici potrebno je izračunati snagu P na otporniku R9 te ukupnu struju I i ukupan napon U spoja Zadana je struja I7 kroz otpornik R7 od 4A te sljedeće vrijednosti otpornika R1=R2=R3=1Ω R5=R6=2Ω R4=3Ω R7=4Ω R8=R9=5Ω 16 Dva otpornika R1 i R2 od 25Ω i jedan otpornik R3=50Ω te kondenzator kapaciteta C=5μF spojeni su na izvor elektromotorne sile E prema slici Odredite EMS izvora ako je naboj na kapacitetu Q=200μC

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17 Zadane su vrijednosti za napone izvora i otpore spojene prema shemi na slici (U1=150V U2=120V R1=4Ω R2=12Ω R3=6Ω R4=3Ω) Treba odrediti a) struje grane (I1I2I3) i b) razvijene snage na radnim otporima Kako bi odredili struje grana treba naći napon između a i b Nadomjesna shema za paralelni spoj izvora

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Smjer obilaska

I1

I2

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18 Za spoj prema slici odredite napone na svim otporima Zadano je R1=R7=4Ω R2=12Ω R3=7Ω R4=R5=20Ω R6=80Ω i U=200V

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Zadaci prije kolokvija

1 Dvije nabijene kugle u vakuumu se privlače silom F Iznosi naboja na kuglama su jednaki Ako s jedne kugle uzmemo 23 njezinog naboja i premjestimo ga na drugu kuglu kolika je sada privlačna sila

2

2 Točkasti naboj Q=6nC nalazi se na graničnoj površini zrak-staklo čija je relativna permitivnost εr= 7 Kolika je jakost električnog polja i električnog potencijala u točki A u zraku na udaljenosti d=12 cm od točkastog naboja a koliko u točki B u staklu (Obavezno nacrtati sliku)

122

2

212

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εεπ

3 Kondenzator kapaciteta C1=6μF nabija se na napon 220V Isključivši C1 s izvora na njega je priključen prazan kondenzator nepoznatog kapaciteta koji se pri tome nabio na 22V Koliki je iznos C2 Q=UC1=220V6μF=1320μC U=220V Dio naboja će preći sa C1 na C2 sve dok se naponi ne izjednače Q1+Q2=Q ndash jer nismo doveli nove naboje Ursquo=U1=U2=22V Q1=U1C1=22V3μF=66μC Q2=Q-Q1=1320 μC -66 μC =1254μC

FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2

4 Odredite koliki je kapacitet kondenzatora C3 u spoju na slici Napon izvora je U=15V napon na kondenzatoru C4=5 middot10-6F je UC4=5V a naboj na kondenzatoru C2= 7 middot10-6F iznosi Q2=25 middot10-6C Kapacitet kondenzatora C1=5 middot10-6F 5 Zadan je spoj prema slici uz R1 = 2 Ω R2 = 3 Ω R3 = 9 Ω R4 = 18 Ω i R5 = 3 Ω te pad napona na otporniku R3 od 39 V Odrediti napon U2

R3

R4

+

R1 R5

R2

U2=

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VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

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431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==

6 Kad kroz akumulator teče struja jakosti 10 A napon na stezaljkama je 12V dok pri struji od 25A napon je 114V Koliki je unutarnji otpor i elektromotorna sila akumulatora

Magnetizam

1 Kroz dva paralelna vodiča duljine 15m teče struja I=60A Kolika mora biti njihova međusobna udaljenost da oni jedan na drugi djeluju silom od 05N

mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ

2 Ravan vodič duljine 2m mase 15g smješten je u zraku okomito na magnetsko polje jakosti 5middot103 Am Nađite jakost struje koju treba propustiti vodič da on u magnetskom polju miruje Uvjet mirovanja F=G

AlH

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711

1sinsin

0

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μ

μμα

α

3 Kroz dva dugačka paralelna pravocrtna vodiča udaljena 20cm teku struje I1=4A i I2=6A na kojoj udaljenost od prvog vodiča iščezava magnetsko polje

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4 Tri vodiča prema slici protjecana su strujama Odrediti jakost magnetskog polja u točki A ako je zadano I1=10A I2=5A I3=5A d1=225cm d2=225cm d3=525cm

5 Kroz horizontalan štap duljine 02m prolazi el struja Štap se nalazi na horizontalnom magnetnom polju od 008T koje sa smjerom struje zatvara kut od 30˚ Sila kojom polje djeluje na štap iznosi 024N

bull koliki je iznos el struje koji prolazi kroz štap bull kolika bi sila bila da el struja ima smjer magnetskog polja

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sin)

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blBFI

lIBFa

6Zadan je raspored vodiča prema slici Razmak vodiča je l=50cm a struje kroz vodiče su I1=50A i I2=100A treba odrediti jakost polja H u točkama A i B

7Kroz tri paralelna vodiča protječu struje I1=10A I2=5A I3=15A Odredite smjer i iznos (po jedinici duljine) sila na vodiče ako presjeci vodiča čine jednakostraničan trokut stranice 1m

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=

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Nastavak 7 zadatka Po jedinici duljine

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8 Zavojnica induktiviteta L=2H duljine l=10 cm s brojem zavoja N=500 protjecana je strujom od 2A

a) Koliki se napon inducira u zavojnici ako se u vremenu od 001 s struja poveća na 7 A b) Kolika je jakost magnetskog polja H u jezgri zavojnice

Zavojnica duljine 1m sa 1000 zavoja sa jezgrom presjeka 5cm2 od transformatorskog lima kod struje 1A ima indukciju 08T Odrediti relativnu permabilnost jezgre

10 U homogenom magntskom polju indukcije B=300T u vertikalnoj ravnini su postavljene tračnice od kovine na rastojanju l= 500 cm Tračnice su na vrhu spojene Pravac vektora magnetske indukcije je normalan na ravan koju obrazuju tračnice Niz tračnice klizi bez trenja (ali u stalnom kontaktu sa tračnicama) prav vodič duljine ab= 500 cm stalnom brzinomm v=5ms Masa vodiča ab je m= 1 g Odrediti otpor pokretnog vodiča ako je otpor ostalih vodiča zanemariv

B

v

b

l

a

Izmjenična struja 1 Serijski spoj otpornika otpora R=2Ω zavojnice induktiviteta L=16mH i kondenzator kapaciteta C=20microF priključen je na izvor izmjeničnog napona efektivne vrijednostiu U=6V kružne frekvencije ω=5000 rads Odrediti struju u krugu te padove napona na pojedinim elementima kruga

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1010205000

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Impedancija kruga

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22 arctgarctgϕ

Fazor struje

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Padovi napona

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Vektorski dijagram

IUR

UL

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2 Na izmjenični izvor efektivnog napona 10V i kružne frekvencije 1000Hz paralelno priključimo otpor R=4Ω kondenzator C=0025mF i induktivitet L=8mH

a) Izračunaj impedanciju priključenu na izvor b) Kolika je efektivna struja i kolike su njene komponente

a) b)

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3 U serijski strujni krug spojeni su otpor R=4 Ω zavojnica L=3mH i izmjenični naponski izvor u=10radic2sin(1000t)

a) Napiši izraz za struju b) Koliki su padovi napona na otporu i induktivitetu c) Skiciraj vektorski prikaz napona i struje

a)

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Vektorski dijagram

4 Efektivni pad napona na kombinaciji elemenata L R1 i C (između točaka A i C) je U1=10V Ako je impedancija zavojnice jednaka XL=10Ω a impedancija kondenzatora XC=5Ω koliki je efektivni pad napona između točaka B i D Vrijednosti otpornika su R1=3Ωi R2=4Ω

A B C D

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5 Na serijski spoj omskog i jalovog otpora priključena je napon tu π100sin2220= V

Krugom teče struja ti ⎟⎠⎞

⎜⎝⎛ +=

6100sin2 ππ A

bull Kolika je efektivna vrijednost struje bull Odredite faktor snage bull Kolike su vrijednosti elemenata bull Kolika se energija pretvara u toplinu za 1 minutu

a) b) c)

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===

ϕ

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ρρ

gt

6 Žarulja 220V60W spojena je na napon tu π100sin2220= Na raspolaganju su nam kondenzator C=1microF i zavojnica L=1H

bull Da li je potrebno spojiti neki od elemenata sa žaruljom da bi ona svjetlila punim sjajem bull Kolika je i=f(t) ako u seriju sa žaruljom spojimo zavojnicu bull Koliku struju mjeri ampermetar u slučaju b bull Koliki mora biti iznos kapaciteta kondenzatora spojenog serijski sa zavojnicom u b

slučaju da bi žarulja svijetlila punim sjajem a) b) c) d)

potrebno 60

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7 Jednofazno trošilo priključeno na 220V50Hz ima radnu snagu 400W a jalovu 200Var induktivnog karaktera

bull Koliki je iznos struje spoja bull Koja je vrijednost faktora snage cosφ bull Ako je trošilo izvedeno kao serijski spoj otpora i idealne zavojnice koju vrijednost ima

otpor a) b) c)

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8 U Europi je frekvencija gradske mreže 50 Hz a u Americi 60Hz Za koliko je posto induktivni otpor neke zavojnice induktiviteta L veći u Americi nego u EU 9 Zavojnica nekog omskog otpora 10Ω i induktiviteta L priključena je na izmjenični napon U=127V i f=50Hz Zavojnica troši snagu P=400W a fazni pomak između struje i napona je 60deg

bull Koliki je induktivitet zavojnice bull Kolika je struja kroz zavojnicu bull Kolika je jalova snaga

Prividna snaga Radna (Djelatna) snaga ndash snaga na otporima Reaktivan (Jalova) snaga

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a) b) c) 10 Serijski su spojeni R=4Ω L=3mH i C=166μF priključeni su na napon u=100 2 sin1000t Odredite a) ukupnu impedanciju i admitanciju spoja b) struju spoja c) napone na pojedinim elementima

a)

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153120912660

80

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Nastavak zadatka 14 15 U mreži prema slici odredite struju IR kroz otpornik R i snagu koja se na njemu troši Zadani su elementi u mreži U1=12V U2=12VIk=14A R1=5Ω R2=4Ω R3=1Ω R4=2Ω i R=10Ω

( )AI

UukRRukIVRukIukUuk

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5

6336

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4512

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846

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231

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3

2

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2

54

213

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2

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11

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AB

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812

84416828828

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92

899

123

123123

89456

321123

456456

564456

6556

897789

8989

9889

777

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=sdot=

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VUUEVRIU

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2001604016023

612540

0

40

0

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21

21

221121

21

33

21

=+=+==sdot=

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=====

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R1

R2

R3

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C

I1

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BR1 R2 R3

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R5

R6

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34

0

28

0

18

6

3

30215

15

2

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25

4

12322

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3

12311

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123

34123

21

2112

21

43

4334

4

22

3

11

=sdot+

=

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I2

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8

16040

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20416

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3

12

76

77

6

34512

54

5445

21

2112

=sdot==sdot==sdot=

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2


122

2

212

2

2

221

12

91

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913

131

41

41

41

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A

B

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12 cm

Vrq

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A

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1

64204

1

4504

1

544

1

2

2

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=

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2

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10741

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573

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α B

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5212

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3

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==

==

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π

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sin)

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B

B

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A

A

A

A

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57792

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69632

89312

8931

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11

21

22

11

21

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==

minus=minus=

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π

π

π

laIIoFF

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sdot==

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2

2

2

323223

313113

212112


mN

aIIo

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mN

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mN

aIIo

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μ

μπ

μ

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152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

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6839120cos2

291360cos2

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12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

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B

v

b

l

a


Ω=sdotsdot

=sdot

=

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minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

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vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

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C

XcXcX

AIef

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fLX

L

L

L

μπ

ϕϕ

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1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

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deg=equiv===

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=sdot

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4108032

220

548

598

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LLLLL

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QL

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R

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gt

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2021100120

100120

1201002

1

2

2

1

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XX

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L

L

L

L

L

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2

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ϕ

ϕ


a)

( )

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2015

02613

22

ϕϕ

ω

ω

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CX

LX

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C

L

b)

( )deg+=

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==

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20

1002

tti

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c)

VXIUVXIU

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CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


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RItg m

VAr691sin

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cos

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QIUQ

UPIIUP

ϕ

ϕϕ


WRIP

VUUUukVRIukU

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ARU

I

RRRRRR

AIII

AR

UI

RRRVRIU

UU

AB

AB

AB

AB

812

84416828828

693

4

4

111165

61

1016

92

899

123

123123

89456

321123

456456

564456

6556

897789

8989

9889

777

7

=sdot=

=+=+==sdot==+=

Ω=++=

==

Ω=+=

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==

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=

VUUEVRIU

AIII

AII

RIRIUU


VCQUUUU


BCAB

BC

RR

RBC

RRCAB

BCAB

2001604016023

612540

0

40

0

3

21

21

221121

21

33

21

=+=+==sdot=

=+=

===

sdot=sdot==


=====

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R1

R2

R3

U+

C

I1

I2I

BR1 R2 R3

R4

R5

R6

R7

R8 R9

U+

I7

A


R1

E2

R4

+

E1 +

R2 R3

I3

I1

I2

a

b

AR

RIUI

RIRIU

AR

RIUI

RIURIRIRIU

VRIUab

AR

UukI

RRRRR

AARukIukUuk

AIIIuk

RRRRR

ARUI

ARUI

R

34

0

28

0

18

6

3

30215

15

2

40

25

4

12322

123422

3

12311

123111

123111

123

34123

21

2112

21

43

4334

4

22

3

11

=sdot+

=

=sdotminussdotminus

=sdot+

=


minus=sdotminus=

=+

=

Ω=+sdot

=

=Ωsdot=sdot=

=primeminusprime=

Ω=+sdot

=

==prime

==prime

I2

R12

Uab

R4

U2

I3

Smjer obilaska

+

-

R12

a

b

Uab

Euk

I3

R3

Smjer obilaska

I1

I2

+

-


VUVUVU

ARUI

VUUUUVRIukU

ARukUukIuk

Ruk

R

RRRRRRRRR

RRRRR

AB

ABAB

AB

AB

AB

8010856782438

8

16040

10

20416

1680208020

20

1020202020

3124124

45

3

12

76

77

6

34512

54

5445

21

2112

=sdot==sdot==sdot=

==

=minus===sdot=

==

Ω=+=

Ω=+sdot

=

Ω=++=

Ω=+sdot

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=

Ω=+sdot

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=



2


122

2

212

2

2

221

12

91

41

913

131

41

41

41

Frq

r

qqF

rq

rQQF

roro

roro

=sdotsdotsdot

sdot=⎟⎠⎞

⎜⎝⎛ minussdot

sdotsdotsdot

=prime

sdotsdotsdot

=sdot

sdotsdotsdot

=

εεπεεπ

εεπεεπ

zrak

stakloQ

A

B

T12 cm

12 cm

Vrq

mVrqE

Vrq

mVrqE

Ro

B

Ro

B

Ro

A

Ro

A

66504

1

64204

1

4504

1

544

1

2

2

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

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=

εεπϕ

εεπ

εεπϕ

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FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

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AlH

gmI

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711

1sinsin

0

0

0

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dd-r

r

I1

I2

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cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

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α

α

sin0sin

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sin)

lIBF

blBFI

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rH

π21

=

( )

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IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

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=

==

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π

π

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laIIoFF

la

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2

2

2

323223

313113

212112


mN

aIIo

lF

mN

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lF

mN

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μ

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152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

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6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

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v

b

l

a


Ω=sdotsdot

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minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

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vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

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5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

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I

AR

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AZU

I

ef

C

L

ef

L

ef

R

ef

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ef

degang==

degminusang==

==

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9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


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LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

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AC

AC

BC

CBC

AC

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7514

711

608

83522

12

2

222

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4310510

2

1

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AC

C

L



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a) b) c)

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22

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C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

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===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

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L

L

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1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

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598

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2021100120

100120

1201002

1

2

2

1

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XX

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2

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ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


R1

E2

R4

+

E1 +

R2 R3

I3

I1

I2

a

b

AR

RIUI

RIRIU

AR

RIUI

RIURIRIRIU

VRIUab

AR

UukI

RRRRR

AARukIukUuk

AIIIuk

RRRRR

ARUI

ARUI

R

34

0

28

0

18

6

3

30215

15

2

40

25

4

12322

123422

3

12311

123111

123111

123

34123

21

2112

21

43

4334

4

22

3

11

=sdot+

=

=sdotminussdotminus

=sdot+

=


minus=sdotminus=

=+

=

Ω=+sdot

=

=Ωsdot=sdot=

=primeminusprime=

Ω=+sdot

=

==prime

==prime

I2

R12

Uab

R4

U2

I3

Smjer obilaska

+

-

R12

a

b

Uab

Euk

I3

R3

Smjer obilaska

I1

I2

+

-


VUVUVU

ARUI

VUUUUVRIukU

ARukUukIuk

Ruk

R

RRRRRRRRR

RRRRR

AB

ABAB

AB

AB

AB

8010856782438

8

16040

10

20416

1680208020

20

1020202020

3124124

45

3

12

76

77

6

34512

54

5445

21

2112

=sdot==sdot==sdot=

==

=minus===sdot=

==

Ω=+=

Ω=+sdot

=

Ω=++=

Ω=+sdot

=+sdot

=

Ω=+sdot

=+sdot

=



2


122

2

212

2

2

221

12

91

41

913

131

41

41

41

Frq

r

qqF

rq

rQQF

roro

roro

=sdotsdotsdot

sdot=⎟⎠⎞

⎜⎝⎛ minussdot

sdotsdotsdot

=prime

sdotsdotsdot

=sdot

sdotsdotsdot

=

εεπεεπ

εεπεεπ

zrak

stakloQ

A

B

T12 cm

12 cm

Vrq

mVrqE

Vrq

mVrqE

Ro

B

Ro

B

Ro

A

Ro

A

66504

1

64204

1

4504

1

544

1

2

2

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

εεπϕ

εεπ

εεπϕ

εεπ


FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


VUVUVU

ARUI

VUUUUVRIukU

ARukUukIuk

Ruk

R

RRRRRRRRR

RRRRR

AB

ABAB

AB

AB

AB

8010856782438

8

16040

10

20416

1680208020

20

1020202020

3124124

45

3

12

76

77

6

34512

54

5445

21

2112

=sdot==sdot==sdot=

==

=minus===sdot=

==

Ω=+=

Ω=+sdot

=

Ω=++=

Ω=+sdot

=+sdot

=

Ω=+sdot

=+sdot

=



2


122

2

212

2

2

221

12

91

41

913

131

41

41

41

Frq

r

qqF

rq

rQQF

roro

roro

=sdotsdotsdot

sdot=⎟⎠⎞

⎜⎝⎛ minussdot

sdotsdotsdot

=prime

sdotsdotsdot

=sdot

sdotsdotsdot

=

εεπεεπ

εεπεεπ

zrak

stakloQ

A

B

T12 cm

12 cm

Vrq

mVrqE

Vrq

mVrqE

Ro

B

Ro

B

Ro

A

Ro

A

66504

1

64204

1

4504

1

544

1

2

2

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

εεπϕ

εεπ

εεπϕ

εεπ


FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

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sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

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==equivsdot=

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deg=equiv===

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=sdot

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996

4108032

220

548

598

6446032220

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572621

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PP

tg

AU

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L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

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ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

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2

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RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

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rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ



2


122

2

212

2

2

221

12

91

41

913

131

41

41

41

Frq

r

qqF

rq

rQQF

roro

roro

=sdotsdotsdot

sdot=⎟⎠⎞

⎜⎝⎛ minussdot

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=sdot

sdotsdotsdot

=

εεπεεπ

εεπεεπ

zrak

stakloQ

A

B

T12 cm

12 cm

Vrq

mVrqE

Vrq

mVrqE

Ro

B

Ro

B

Ro

A

Ro

A

66504

1

64204

1

4504

1

544

1

2

2

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

=sdotsdotsdotsdot

=

εεπϕ

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εεπϕ

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FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

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==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

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μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


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μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

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=

sdotsdotsdot=

αα

α

α

sin0sin

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sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

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π

π

π

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la

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la

IIoFF

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2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


FUQC μ57

2

22 ==

++

Q1C1

++

Q2C2


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


R3

R4

+

R1 R5

R2

U2=

FUQC

VUUUU

VCQU

CQQ

5

3

33

3432

2

22

6

32

10741

431

573

1025

minus

minus

sdot==

=rArr=+

==

sdot==


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


Magnetizam


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


mF

lIrrlI

rlIIF o

oo 02160222

2221 =sdot

=rArrsdot

=sdotsdot

sdot=π

μπ

μπ

μ


AlH

gmI

gmlIHHB

gmlIB

711

1sinsin

0

0

0

=sdotsdot

sdot=


=sdot=sdotsdotsdot

μ

μμα

α


dd-r

r

I1

I2

( )

cmrrr

rdI

rI

rIH

820

6422

2

21

==minus

=

minus=

=

ππ

π




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ




α B

I

mAd

IH

mAd

IH

mAd

IH

mAHHHHH

5212

5332

0772

089 vektor-

3

33

2

22

1

11

321

==

==

==

=minus+=

π

π

π

r

r

r

rrrr

r

=sdotsdotsdot=deg=

=sdotsdot

=

sdotsdotsdot=

αα

α

α

sin0sin

)sin

sin)

lIBF

blBFI

lIBFa



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ



rH

π21

=

( )

( ) mAll

IH

mAl

IH

mAHHH

mAll

IH

mAl

IH

mAHHH

B

B

B

B

BBB

A

A

A

A

AAA

78392

57792

7939

69632

89312

8931

22

11

21

22

11

21

=minus

=

==

=minus=

=minus

=

==

minus=minus=

π

π

π

π

laIIoFF

la

IIoFF

la

IIoFF

sdot==

sdot==

sdot==

πμ

πμ

πμ

2

2

2

323223

313113

212112


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


mN

aIIo

lF

mN

aIIo

lF

mN

aIIo

lF

μπ

μ

μπ

μ

μπ

μ

152

302

102

3223

3113

2112

==

==

==

mNlFFFFFF

mNlFFFFFF

mNlFFFFFF

6839120cos2

291360cos2

452660cos2

12313

2

23

2

13

2

3

22312

2

23

2

12

2

2

11312

2

13

2

12

2

1

μ

μ

μ

==degminus+=

==degminus+=

==degminus+=





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ





B

v

b

l

a


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


Ω=sdotsdot

=sdot

=

Ω=sdotsdot=sdot=

minus

minus

1010205000

11810615000

6

3

CX

LX

C

L

ω

ω

Impedancija kruga



⎜⎝⎛ minus= 451

22 arctgarctgϕ

Fazor struje

A452514522

06degang=

degminusangdegang

==ZUI v

vv

Padovi napona


vv



Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ

Vektorski dijagram

IUR

UL

UC

α



a) b)


Ωminus=

Ω==

Ω=

5726573612311114

84

ZjZXXRZ

jXjjXX

R

CL

C

LL

vvvvv

v

v

AAcX

UI

AXU

I

AR

UI

AZU

I

ef

C

L

ef

L

ef

R

ef

ef

ef

degang==

degminusang==

==

degang=degminusang

degang==

9052

90251

52

57267925726573

010

vv

vv

v

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ

Vektorski dijagram

UIR

IL

IC

α



a)

Ω=sdot= 3LX L ω


Ω=+= 522

LXRZ

AZUI 2

510

=== v

vv

b)


vv


Vektorski dijagram


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


A B C D

U1

U2

L R1 R2C

( )

VZIU

AZUI

ZXRZ

ZXXRZ

BCBC

AC

AC

BC

CBC

AC

CLAC

7514

711

608

83522

12

2

222

=sdot=

==

Ω=+=

Ω=minus+=

Ω=Ω=

=Ω=Ω=

4310510

2

1

RR

VUXX

AC

C

L



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ



⎜⎝⎛ +=

6100sin2 ππ A


a) b) c)

AIef 4112

22

Im===

866023cos

6==equiv= ϕπϕ gt

JtIUW

FXcf

CI

UcXc

IUR

VUUVUU

VUmU

R

C

R

166166086602220cos

9402

18772

110

71342

519011050220sin

51908660220cos

2202

22202


=sdot

=equivΩ===

Ω===


===

ϕ

μπ

ρρ

gt




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ




potrebno 60

273067806

220

6780660

220

2202220

2

22

nijeWRžIP

ARUI

PUnRž

RUI

VUefUm

minus=sdot=

===

Ω====

=equiv= gt

( ) ( )

FXcf

C

XcXcX

AIef

ttiZRž

XRZ

fLX

L

L

L

μπ

ϕϕ

ππ

1102

1314

25302

Im

5621314sin3580

56219306865

67806cos

6865

314100222

=sdot

=

Ω=equiv=

==

degminussdot=

deg=equiv===

Ω=+=

Ω===

gt

gt



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ



otpor a) b) c)

Ω=minus=

Ω===

Ω==equivsdot=

==equivsdot=

=sdot=sdot=

=deg=

deg=equiv===

=sdot

=sdot

=equivsdotsdot=

996

4108032

220

548

598

6446032220

8905726coscos

572621

400200

0325726cos220

400cos

cos

22 XLZRI

UZ

IUXXIU

VIP

UUIP

VAIUPs

PP

tg

AU

PIIUP

L

LLLLL

L

QL

LLLQL

R

Q

gt

gt

gt

gt

ϕ

ϕϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ




2021100120

100120

1201002

1

2

2

1

gtequiv===

sdot==sdot==

sdot==

LL

XX

LLXLLX

LfLX

L

L

L

L

L

ππ

πωπω

πω

reaktivni sin

cos2

2

AmperVoltVArIUQ

RURIP

WIUP

VAIUS

minusrArrsdotsdot=

=sdot=

rArrsdotsdot=

rArrsdot=

ϕ

ϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ


a)

( )

degminus=rArr

Ω==

Ω=minus+=

Ω==

Ω=sdot=

936cos

2015

02613

22

ϕϕ

ω

ω

ZY

XXRZ

CX

LX

CL

C

L

b)

( )deg+=

==

==

936sin220)(

20

1002

tti

AZ

UefI

VUmUef

ω

c)

VXIUVXIU

VRIU

CC

LL

R

153120912660

80


=sdot=

H055060 =sdot


RL

RItg m

VAr691sin

A296cos

cos

==sdotsdot=

==rArrsdotsdot=

QIUQ

UPIIUP

ϕ

ϕϕ

Izvanredni-drugi dio zadatci za kolokvij s... · Zadaci prije kolokvija 1. Dvije nabijene kugle u...

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Transcript of Izvanredni-drugi dio zadatci za kolokvij s... · Zadaci prije kolokvija 1. Dvije nabijene kugle u...