Corrente Alternada CENFIM

7/28/2019 Corrente Alternada CENFIM

http://slidepdf.com/reader/full/corrente-alternada-cenfim 1/61



CENFIM 1

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CENFIM 2

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CENFIM 3

./0123456721042455428980:245;5<=65<:2>2?45@

.A><55145<285<2BC.45DA2:.<<51752375<1242EF

45170/0:2<.6;23.<56/D142A5172061DA2:.<<51752375<1242

.A><55145<.:.A>.<72A517.426<56067G1:026HI.I01265:.1451624.<56

JD214.30824.61DA:0<:D07.45:.<<51752375<1242

.A><55145<2:.<<5:BC.4./2:7.<45>.7G1:02

067018D0<.660675A267<0/=60:.64.6A.1./=60:.6

1DA5<2<26;217285164.660675A267<0/=60:.65A<532BC.2.D7<.70>.45

60675A26

56.3;5<:0<:D07.645:.<<51752375<1242

5<2BC.42:.<<51752375<1242601D6.0423

23.<56/D142A51720642:.<<51752375<1242

0<:D07.:.A<56067G1:02>D<25A 0<:D07.:.AI.I012>D<25A

52:7K1:02014D70;2

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0<:D07.6L<05

0<:D07.6L<05

.7G1:0265A

5<2BC.42:.<<51752375<12427<0/=60:2



CENFIM 4

516M5660A>3565:.A>.6726

082BC.45:2<8265A567<53255A7<0K18D3.

.A>5162BC.4./2:7.<45>.7G1:02

!"#

D<2175 . 45651;.3;0A517. 45672 104245 5428980:2 45;5<C. 65< 5N5:D724.6

5N5<:O:0.6 5 45A.167<2BM56H45 /.<A2 2 JD5 65>.662 /0:2< :.A DA2 ;06C.7217.

JD217.>.66O;53D10;5<6234.6516012A517.624JD0<04.6

$%&'(

)*+,,-

5;5<=@

5<:.1P5:0A517.645=385I<2

5<:.1P5:0A517.645<08.1.A57<02

5;5<=51:.17<2<@

:.A>21P2A517. 245JD24. 4D<2175 . 45:.<<5< 4.6 5N5<:O:0.6 5

45A.167<2BM56

JD0>2A517.6>2<25N5:DBC.4.65N5<:O:0.6

.3P264501/.<A2BC.75:1.3980:2

+D567M5655N5<:O:0.6



CENFIM 5

./(.0'

./(.0'12



CENFIM 6

3.(4

.

567D4. 42 515<802 53L:7<0:2 JD5 /0?5A.6 1.6 :2>O7D3.6 2175<0.<56 266517.D 126

:.<<51756 5 7516M56:.17O1D26H 067. LH12JD5326 JD5 1C. ;2<02A 2.3.18.4. 75A>.H

JD5A217GA.A56A.651704.ED1040<5::0.1206F5.A56A.651704.

N0675A1.517217.H1DA5<.6262>30:2BM565AJD56C.AD07.40;5<62626;21728516

5A/D1BC.4. 75A>.H4267516M56H:.<<517565.D7<268<2145?26660A268<2145?26

53L:7<0:26>.45A:32660/0:2<655A/D1BC.4.75A>.H:.A.@

5-5- #!,6,7,89, 5+:-6,+:-6, ,6,7 #,5+,, :-5!+ ,5+,,

;'& .8<=/0:.:.<<5175<5><56517242L:5<72A5175:.1672175H>.<1C.;2<022.3.18.4.75A>.

08.<<5175:.1672175



CENFIM 7

;$'&40

:.<<5175<5><56517242>.66D0;23.<5640/5<517564501672175201672175A2672AILA

.A56A.651704.

08.<<5175;2<0=;53D1040<5::0.123

;<0

A28<2145?240?65>5<0940:2HJD214.65;5<0/0:2DA2<5>570BC.426D2:.1/08D<2BC.

2.3.18.4.75A>..567D4.JD50<5A.65/5:7D2<H6D<80<1.6C.40;5<626/.<A2645

.1426 >5<0940:26 5><565172A.6 4.06 70>.6 45 .1426 >5<0940:26@ .14D3279<026 5

>D36279<0265262375<12426>D<26

082F.<<5175.14D3279<02QIF.<<5175D1040<5::0.1235A451754565<<2

;=0.

082F516C.2375<12427<0218D32<QIF516C.2375<1242JD24<242Q:F.<<5175

601D6.0423



CENFIM 8

DA2 .142 2375<1242H . :.1RD17. 4.6 ;23.<56 266DA04.6 5A:242 DA 4.6 651704.6

456081265 >.< :-5>57, .D ,5 660AH 75A.6 2375<1K1:026 >.6070;26 5

2375<1K1:026158270;26

:.1RD17.454D262375<1K1:026:.165:D70;264560812657,7:

;23.<266DA04.H5A:24201672175H>.<DA2:.<<5175.D7516C.L:P2A24.9:

,5-5->5HJD565<5><565172>.<DA2357<2A01S6:D32H,?+;

D382<A5175H56756601206;C.240:0.12<6527516M56.D:.<<51756:.17O1D26>2<2JD5

.6535A517.62:70;.6<5230?5A..IR5:70;.><5751404.

660AH27516C.<5><56517242L26.I<5>.60BC.42:.A>.15175:.17O1D2E1.7565

JD528<2145?2L<5><56517242>.<DA2357<2A01S6:D325.O140:5LA20S6:D3.F

+@A+

<5A.6 28.<2 7<272< 4. 567D4. 45 :.<<51756 5 7516M56 2375<12426 601D6.04206

0A>.<7K1:02 1235:7<910:2 <56D372 4. /2:7. 45JD23JD5< 60123 >5<0940:. 2375<124. 65

>.45< :.16045<2< :.A. 2 6.A2 45 601206 2375<124.6 601D6.04206 45 /<5JDG1:026

AS370>326 .1;LAH >.06H 45/010<A.6 26 8<2145?26 JD5 :2<2:75<0?2A 0A 60123

601D6.0423

;B..

(

08&5<2BC.45DA2/5A601D6.0423



CENFIM 9

.. ..

>.7G1:02><.4D?042>.<5/507.45.D35L><.>.<:0.1232.JD24<24.420175160424545:.<<5175H3.8.0145>51451754.65D651704.45:0<:D32BC.T/2:7.JD52><.4DBC.45 515<802 :23.<O/0:2 L ;2<0=;53 45 01672175 2 01672175H 21D3214.65 A56A. 4D26;5?562.3.18.45DA>5<O.4.Q1.517217.H545;04.=01L<:027L<A0:24.6:.<>.6H26;2<02BM564575A>5<27D<26C.AD07.A2064LI506

.4.6.62>2<53P.67L<A0:.665<C.D7030?=;506H7217.5A:.<<5175:.17O1D2H:.A.5A:.<<5175 2375<1242 /<5JDG1:02 72AILA 01/3D51:02<= . /D1:0.12A517. 4.62>2<53P.6H>.<5N5A>3.H4263KA>24264501:21456:G1:02

5>.<;517D<2H2/<5JDG1:02/.<45A26024.I20N2E01/5<0.<2#!?FH275A>5<27D<24263KA>2426;2<02<=35172A5175565<021.79<02DA2:5<72:017032BC.

2/08D<2'<5><56517265DA2:DI2535:7<.3O70:2:.A6D3/27.45:.I<5HDA2I.I012>3212.<0517242 1240<5:BC. :.A DA2 28D3P2A281570?242H DA2 I.I012:.ADA2>5B245/5<<.A2:0.5/0.645><27256751404.65>2<2353.630A517214. . :0<:D07. 2 :.<<5175 :.17O1D2H P2;5<= 7<216>.<75 45 :.I<5 >.<535:7<9306552:BC.42I.I0126.I<5228D3P2A281570?242HJD5;2<02A45651704.:.A.651704.42:.<<51751.:0<:D07.5/507.45.D351.6:.14D7.<5645><272H227<2:BC.42I2<<245/5<<.52<5>D36C.517<5.6/0.645><272/2?6565170<0145>5145175A51754.651704.42:.<<51751.:0<:D07.

08'/507.642:.<<517553L:7<0:2

.2>30:2<65:.<<51752375<1242H1C.P=7<216>.<7545:.I<5H228D3P21C.65456;02H2><56517214.DA230850<2;0I<2BC.126D2>.60BC..<.D7<.324.H723:.A.5A:.<<5175:.17O1D2H.6/0.62JD5:5A5<5>535A65H52I2<<245/5<<.L27<2O42 28D3P2 A281L70:2 5A :.<<5175 2375<1242 72AILA L 6.30:07242 >.< /.<B26535:7<.A281L70:262656626.30:072BC.AD4245651704.$$;5?56>.<658D14.H>53.JD5H45;04.=01L<:02H>5<A215:5><270:2A51755A<5>.D6.

CT.75A>.5AJD545:.<<54D262375<1K1:026:.165:D70;26H.D65R2HL.75A>.8267.

1DA:0:3.5><56517265>.<55N><5662655A658D14.6



CENFIM 10

*+D57,ET.1SA5<.45:0:3.65/5:7D24.61DA658D14.5><56517265>.<E526D2D104245L . !? EP5<7?F .7565 JD5 2 /<5JDG1:02 5 . >5<O.4. 567C. <532:0.124.6 1DA2

><.>.<BC. 01;5<62 A :0:3. 45A.<2 2 <5230?2<65 1DA >5<O.4. Q 3.8. E :0:3.645A.<2<C.658D14.E@%6/<5JDG1:026426.142645>5145A4266D26D7030?2BM56660AH2515<80253L:7<0:2L 4067<0IDO42 2 #$ !? 82A2 426 2D40./<5JDG1:026 ;20 45 $ !? 2 $ U!? 5:.A>.<72.JD5;D382<A5175654560812>.<535:7<.2:S670:2=40.H7535;06C.HD37<26.16H <242< 5 A0:<..1426 :.A>.<72A 82A26 45 /<5JDG1:026 JD5 D37<2>2662A .6!?EA582P5<7?F5H>.<;5?56H.6!?E8082P5<7?FH

08(5<O.4.458<2145?2601D6.0423

F!:575<A012<.>5<O.4.45DA2.1422375<124245$U!? @%E @%3?3BF3< @B3µµµµ !:,-+

T.;23.<0167217K15.A206535;24.2701804.>5328<2145?22AILA654560812

>.<9:8F,

!=2A>307D456>.6070;265158270;26

.;23.<A5404.517<5.6;23.<56A=N0A.6>.6070;.5158270;.:P2A265 9:

!,7!,7

2/08D<2*H75<5A.6@

!!@$F8F;



CENFIM 11

08*5><565172BC.422A>307D4545DA2:.<<5175601D6.0423

:),

5A.167<265JD5@ @$%ππππ8FV3;G<H8F

08$23.<AL40.45DA2:.<<5175601D6.0423

5:.16045<2<A.628.<27516M562375<12426601D6.04206H75<5A.6@)@$%ππππF@3?G<H8F

5<5A.62JD0JD5:.16045<2<2>5126A572454.:0:3.45DA2:.<<51752375<1242601D6.0423H>.06.;23.<AL40.45DA:0:3.L?5<.5><565172.;23.< JD5 DA2 :.<<5175 :.17O1D2 45;5 >.66D0< >2<2 7<216>.<72< 1.A56A.75A>.H2A56A2JD2170424545535:7<0:04245



CENFIM 12

:E,7I= <5/5<0A.6 JD5 . :23.< 45651;.3;04. 12 <56067G1:02 >.< 5/507. 45 .D35 L

0145>51451754.651704.45:0<:D32BC.42:.<<51755675A.4.H5N0670<=DA2:.<<5175:.17O1D2JD51.A56A.0175<;23.4575A>.H.D65R2HDA>5<O.4.H><.4D?0<=2A56A2JD2170424545:23.<JD52><.4D?042>532:.<<51752375<12H 5662 JD21704245 :P2A265 9: E,7I 5 <5><56517265 >.< 275A270:2A5175H:P58265=:.1:3D6C.45JD5.JD24<24.42017516042455/0:2?L08D232.;23.<AL40.4.JD24<24.45,H.D65R2@$@$8F%$@8F%√√√√$@3?H3H8F

55A;5?45:.<<51756<5><565172<A.67516M56H;0<=4508D23A.4.@$@$

8F%$@8F%√√√√$@3?H3H8F

.<7217.H.9:E,7I+75-:-5L.;23.<4201751604245

JD545;5<0275<DA2:.<<5175:.17O1D2>2<2H1DA2<56067G1:02><.;.:2<.A56A.

5/507.:23.<O/0:.H1.A56A.0175<;23.4575A>.

2<2<523B2<20A>.<7K1:024.;23.<5/0:2?H<5/0<265JD56C.;23.<565/0:2?56JD5.6;.37OA57<.652A>5<OA57<.61.60140:2A2.A540<5A8<2145?26601D6.04206F!:$

s µ 6

08516C.601D6.0423



CENFIM 13

:-,95-J-5#,5+,:!5-5,K;?-,5$;2/<5JDG1:025.>5<O.4.

$;$.;23.<AL40.45DA22375<1K1:02$;<.;23.<5/0:2?

$;=.>.17.4545A.<22270180<.><0A50<.>0:.

:+"#$;&µ6VH#V&H#N$&6VHµ6/VVHN$&V'U!?$;$A=NV# AL4V$H&'NA=NV$H&'N#VH(#

$;<VA=N√V#N$H'$'VH##$;=VHµ6><0A50<.>0:..:.<<52H.D65R2H2>96$H&µ6V&$$16

F!:<

06>.A.645DA2<56067G1:0245$UΩ5(WH575<A012<@

<;.;23.<5/0:2?42A=N0A20175160424545:.<<5175JD52>.45>5<:.<<5<Q

<;$22A>307D4542A=N0A27516C.2JD5>.4565<6DIA57042

:+"#

<;V V ⇔V√V√(N$N$V&H#N$

V$H&#N$V$H&#AV&#µQ

<;$V V√ V√$N$(V$$

A=NV√ A=NV(H*

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L(.651.4.K18D3.JD5.;5:7.</2?:.A.50N.4262I:06626L@5αααα@%514.@8F ,@8F;5αααα



CENFIM 14

@,H;0<=H>.<6DI6707D0BC.>2<2;23.<0167217K15.42:.<<5175@,@8F5αααα

0851.4.K18D3.αEF

.'..

;5:7.<456:<5;5DAK18D3.45$ππππ<24021.65A:242;.372H.D65R2H5A:242

>5<O.4.

5/01565 ;53.:04245218D32< :.A. 6514. . 1SA5<.ϖϖϖϖ 45 <24021.6 >5<:.<<04.6

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$ππππ

ϖϖϖϖ 75<5A.6ϖϖϖϖ@$ππππ%.Dϖϖϖϖ@$ππππE 5N><5662 5A <24021.6 658D14.

.45A.6H 28.<2H 8515<230?2< 2 /9<AD32 <5><56517270;2 42 :.<<5175 601D6.0423H

4575<A01214. . ;23.< 4. K18D3. α 660AH . ;5:7.< 45A.<2 DA 75A>. 7 2

>5<:.<<5<.K18D3.α5DA>5<O.4.2>5<:.<<5<π<24021.6.8.@

αααα@$ππππ;-% .D20142 αααα@ωωωω-0<=@ ,@8F5ωωωω-

672/9<AD3275<=H20142H4575<5A:.172.K18D3.JD5.;5:7.</.<A2:.A2

.<085A 42 :.17285A 4.6 K18D3.6H 1. 01672175 010:023 5675 K18D3. :P2A265

E;

/9<AD32<5><56517270;245DA2:.<<517565<=@@8F1ωωωω-ϕϕϕϕ2



CENFIM 15

;G.(.

A85<24.<535A5172<L:.16707DO4.>.<DA2153:.14D7.<:.3.:24.45723/.<A2JD5

>.662 65< A.;0A51724. 1. 650. 45 DA :2A>. A281L70:. /0N. .A. 65 ;5<0/0:2H .

:2A>.A281L70:.L:<024.>.<DAOA21>5<A215175215345/0.:.14D7.<4560812

65>.<!,.D,5+I,

26 5N7<5A042456 45672 56>0<2 567C. 242>724.6 4.06 21L06H :.16707DO4.6 20142 >.<

A275<023:.14D7.<5 JD5 654560812A >.<5),7:7-;3082BC.2.5N75<0.<L

/5072>.<A50.454D26>5B26>5</5072A5175242>7=;5062.621L06:.35:7.<5645/.<A22

5672I535:5<I.A :.172:7.H . JD5 65:P2A2A791.<A23A5175:.167<DO4265A

8<2/075

123065A.6./D1:0.12A517.45DA85<24.<

085<24.<535A5172<

. <.42< 1. 651704. 0140:24.H 2 56>0<2 ;20 :.<72< 26 301P26 45 /.<B2 4. :2A>.

A281L70:.H ><.4D?04.X 65 1532 DA2 /.<B2 535:7<.A.7<0? 014D?042 JD5 ><.;.:2 2

:0<:D32BC.45DA2:.<<517527<2;L64.:0<:D07.5N75<0.<

.4563.:2<6542>.60BC.>2<22>.60BC.H256>0<2:.<72DA1SA5<.A=N0A.45301P2645/.<B2

53.JD52/5A014D?042270185DA;23.<A=N0A.2>966575<4563.:24.*$Y.75

65JD57217.12>2<75I<21:2:.A.12><572H6D<80<C./5AJD5;C.240:0.12<65H>53.

JD5 >.45 :.1:3D0<65 JD5 2 :.<<5175 1. :0<:D07. 5N75<0.< 65<= 7217. A206 535;242H

JD217.A20.</.<.1SA5<.45:.14D7.<56JD5:.<72A.:2A>.A281L70:.

+D214.256>0<2:.1701D22<.42<27L.:D>2<2>.60BC.01;5<62Z>.60BC.45>2<7042H

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>5<>5140:D32<A5175Z6<5/5<0426301P26H450N214.4565<654545/5A014D?042



CENFIM 16

08<.4DBC.45/5A601D6.0423:.A4563.:2A517.45*$Y

08#5A601D6.0423><.4D?042:.A4563.:2A517.45*$Y2($Y

>2<70<45675A.A517.H.651704.42/5A014D?0421256>0<2L:.17<=<0.2.2175<0.<

.517217.H52A51.64.651704.H7D4.65>26624.A.4.456:<07./5A014D?042

270185DA;23.<A=N0A.2>962<.72BC.45'$YH:.A5B214.240A01D0<27L6521D32<

JD214..>321.4256>0<265:.3.:2>5<>5140:D32<A5175Z6301P2645/.<B24.:2A>.

P.A.8L15.H.JD56D:5452>96DA2<.72BC.:.A>3572H;.37214.Z>.60BC.

08&5A.I7042>532<.72BC.517<5($Y5&$Y



CENFIM 17

.< :242 <.72BC. 42 56>0<2 .I75<65= DA2 601D69045 2A>307D45 42/5A .I7042

65<= 7217. A20.<H JD217. A20.< /.< . 1SA5<. 45 56>0<26 JD5 :.16707D0 . 014D?04.

.80:2A5175H2/<5JDG1:0245>5145<=42;53.:0424545<.72BC.H

456:<0BC. JD5 2:2I2A.6 45 5/5:7D2< :.14D?0<= 2.6 A56A.6 <56D3724.6 65 28.<2

406>D65<A.6 45 DA <97.< E>2<75 A9;53F :.16707DO4. >.< DA OA21 E535:7<.OA21

230A51724.5A:H:HF5DA567=7.<E>2<75/0N2FH:.16707DO4.>.<I.I0126.145;C.6D<80<

26 /5A 014D?0426H >.< A.;0A5172BC. 4. <97.< . 406>.6070;. 456:<07.H :2>2? 45

><.4D?0<:.<<51752375<1242601D6.0423H:P2A265:-5;

08'<.4DBC.45DA28<2145?2601D6.0423

4.(;.

5 2123062<A.6 2 5N>5<0G1:02 45 ;5<0/0:2BC. 42 50 45 PA A26 2>30:214. 28.<28<2145?262375<12426H:P582<5A.6Z:.1:3D6C.JD565A217LA:.1672175.JD.:05175% 5675 JD.:05175 :P2A2<5A.6 ,!>57,4.:0<:D07.H2.JD232>30:2A.627516C.2375<12425JD565<5><565172>.<&6D2D104245L.I;02A5175.M

660AH2.,M266DA52/.<A2HJD5L456081242>.<5045PA8515<230?242

@&;

40/5<51B2 517<5&545;565 2. /2:7. 45& 45>5145< 42 /<5JDG1:02660AH 5A:.<<5175 2375<1242H 2 <532BC. 517<5 2 7516C. 5 2 :.<<5175 45>5145H >2<2 DA2 4242/<5JDG1:02H420A>54K1:02&54.K18D3.45456/262A517.ϕϕϕϕ

.<45/010BC.4560812<65=@

:.6ϕ>.<,-D57,

651ϕ>.<7->57,N



CENFIM 18

08(<0K18D3.4260A>54K1:026

58D042A5175 567D42<5A.6 .6 :0<:D07.6 5A JD5 6D<85A :.<<51756 2375<12426601D6.04206HJD56C./.<A24.6>.<<56067G1:026HI.I01265:.1451624.<56

5<5A.6H1.517217.H5A><0A50<.3D82<H.6:0<:D07.6045206H.D65R2H.6:.16707DO4.6

>.<<56067G1:026H>.<I.I0126>D<26E65A<56067G1:02F5>.<:.1451624.<56>D<.6E65A

<56067G1:02645>5<426F23:.A.6D:54512<52304245

. 517217.H 238DA26 456726 7<G6 8<2145?26H JD5 /.<A2A .6 535A517.6 <5206

E<56067G1:02H<52:7K1:02014D70;25<52:7K1:02:2>2:070;2FH266DA5A;23.<567C.I20N.6

JD5>.45A456><5?2<65/2:52.6<56721756

T.:26.H>.<5N5A>3.H4263KA>24264501:21456:G1:02HJD5>.45AH65A8<21455<<.H

65<:.16045<2426:.A.<56067G1:026>D<26

(

.2>30:2<A.6DA27516C.2375<1242601D6.0423Z<56067G1:02>D<2A51759PA0:2H

JD2365<=2/.<A245.14242:.<<51751.:0<:D07.[

>30:214.25045PA2.66D:5660;.60167217565DA2;5?JD5VHR=JD52

<52:7K1:02L1D32H/2:03A517565;5<0/0:2JD5@

\ A54042 JD5 2 7516C. 2DA5172H 2 :.<<5175 72AILA . /2<=H R= JD5 65

<532:0.12A>5325045PAHVQ

+D214. 2 7516C. 2>30:242 AD42 45 >.32<04245H 72AILA 2 01751604245 45

:.<<5175AD42455-,;



CENFIM 19

08*56067G1:02>D<2230A5172425A

.8.26:D<;26<5><56517270;26427516C.542:.<<5175567C.5A/265H.D65R2H2

DA A=N0A. 42 7516C. :.<<56>.145 2 DA A=N0A. 45 :.<<5175H . A56A.

6D:54514.>2<2.6?5<.6

K6M,7

.65<3082422DA2<54545:.<<51752375<12D12<56067G1:02E>5ND123KA>242

4501:21456:G1:02FH20175160424545:.<<5175JD5:0<:D3275A2A56A2/.<A2

601D6.0423JD527516C.42<545+D214.27516C.>2662>53.;23.<?5<.H15675

016721751C.:0<:D32:.<<5175E6D201751604245L?5<.F+D214.27516C.23:21B2

.;23.<A=N0A.H201751604245L08D23A5175A=N0A20?65517C.JD527516C.

5201751604245567C.5A/265E/08*F

. :26. 45 :2<826 9PA0:26H 2 7516C. 5 2 01751604245 45 :.<<5175 ;2<02A60AD372152A5175

08$516C.5017516042455A/265



CENFIM 20

((

6 I.I0126 567C. ><5651756 5A 7.4.6 2JD5356 <5:5>7.<56 1.6 JD206 65R2

15:566=<0.2><.4DBC.45DA:2A>.A281L70:..<5N5A>3.1.6535:7<.OA2156H:.172:7.<56H A.7.<56H <52:7K1:026 45 2<<21JD5 45 3KA>2426 45 456:2<82

E/3D.<56:51756H;2>.<45A5<:S<0.H;2>.<456940.H57:FH7<216/.<A24.<56H57:

2<2 . 567D4. 4. :.A>.<72A517. 42 I.I012 45;5A.6 75< 5A :.172 JD5 26

I.I01266C.45:.I<5H>.<066.2><565172<5ADA24575<A01242<56067G1:02

75-75-C5+53082<A.6D12I.I0122D127516C.:.17O1D2H:0<:D32<=

>.< 532 DA2 :.<<5175 53L:7<0:2 JD5 65<= D10:2A5175 30A07242 >532 <56067G1:02

58D14.235045PA@@% 24.JD55672<56067G1:0265<>5JD512H565

2>30:2A.6D127516C.535;242ZI.I012H:0<:D32<=>532A56A2DA2:.<<5175:.A

DA2 01751604245 535;242H . JD5 .<08012<= DA2 /.<75 >.7G1:02 40660>242H JD5

>.45<=4567<D0<2I.I01245;04.2.:23.<85<24.

75-:-553082<A.62A56A2I.I0122D127516C.2375<12H>.45

65 :.A><.;2< 5N>5<0A51723A5175 JD5 2 :.<<5175 JD5 :0<:D32 >532 A56A2

2><565172 DA ;23.< A51.< 5 3082<A.6 DA ]277OA57<. >.45<5A.6 :.A><.;2<

JD5.:.16DA.45>.7G1:02L><270:2A51751D3.H2>562<425N067G1:0245D12

:5<72 :.<<5175 660A >.45<5A.6 :P582< Z :.1:3D6C. JD5 2 I.I012 2><565172

DA2:5<72.>.60BC.=>266285A42:.<<517540/5<51754.:26.2175<0.<5AJD52

.>.60BC.5<245;042Z<56067G1:02.PA0:2

.4.656756/519A51.645;5A652.5/507.452D7.014DBC.42I.I012@

+D214.2I.I012L>5<:.<<042>.<DA2:.<<51752375<1242H2>2<5:5DA2:.<<5175

;2<0=;53H5>.<7217.DA:2A>.A281L70:.72AILA;2<0=;5324.JD526301P26

45 /.<B2 4. /3DN. A281L70:.H JD5 532 A56A2 85<2H :.<72A 2 6D6 ><9><0.6

:.14D7.<56H6D<85D12/5A452D7.014DBC.JD5H658D14.23504551?H65;20.>.<Z:2D62JD53P545D.<085A

08/5A45

2D7.014DBC.42

I.I012.>M565Z

:.<<5175



CENFIM 21

+D214.2:.<<5175H658D014.26;2<02BM5642/D1BC.601D6.0423H751452:<56:5<H

.:2A>.A281L70:.72AILA:<56:5

>2<5:5 517C. DA2 /5A JD5 65 .>M5 2 JD5 2 :.<<5175 65 5672I535B2H><.;.:214.DA27<26.12:.A517553L:7<0:25A<532BC.Z7516C.EA2I.I012

2.65<3082422DA27516C.2375<1242H27516C.2>2<5:5450A54027.2.665D6

75<A01206F

K,5+-,9

5 2 DA2 <545 45 :.<<5175 2375<12 /.< 30824. DA <5:5>7.< 014D70;.H ;20 42<

.<085A 2 DA 456/262A517. 517<5 2 7516C. 5 2 01751604245 672 S370A2H 5A

;0<7D45422D7.014DBC.H5672<=27<262425A<532BC.Z7516C.

675456/262A517.H:.A.7.4..K18D3.H>.4565<A5404.5A8<2D6.:26.45

D12:2<82014D70;2>D<2H5AJD5<56067G1:029PA0:2L1D32H.456/262A517.65<=

45*$^

.:26.45:2<82014D70;2>D<2H20175160424545:.<<5175517527<26242*$Ê

45>5<O.4.F5A<532BC.Z7516C.

08028<2A2;5:7.<0234551DA2I.I012>D<2



CENFIM 22

5675 :26. 40?65 JD5 2 01751604245 567= 456/26242 5A 27<26. 5A <532BC. Z

7516C.5ADAJD2<7.45:0:3.

O(

.A.2.>.60BC.2><56517242>532I.I012Z:.<<51752375<124275A2;5<:.A.6

/519A51.6452D7.014DBC.67265<=A20.<JD217.A20.<65R2.:.5/0:0517545

2D7.014DBC..?5A206<=>042665R2A26;2<02BM5642:.<<51752375<1242

P2A2A.6 <52:7K1:02 014D70;2 N. Z .>.60BC. 2><56517242 >532 I.I012 Z

:.<<5175H>53.JD5@

fL X L π 2=

-V52:7K1:02014D70;25A.PA6EΩ F

/V<5JDG1:025AP5<7?E!?F

V.5/0:05175452D7.014DBC.5AP51<06E!F

P(

23:.A.0140:=A.62.><01:O>0.4567575A2HA545652>.7G1:0245DA2I.I012

>D<2H >.4565 :.A><.;2< JD5 2 3507D<2 27<2;L6 45 DA ]277OA57<. 0140:2 DA2>.7G1:0208D232?5<.

.:.17<=<0.4.JD5.:.<<51DA2<56067G1:02H1DA2I.I012>D<21C.65><.4D?

151PDA:.16DA.45515<802:23.<O/0:2

:.<<5175 JD5 >5<:.<<5 2 I.I012 65<;5 D10:2A5175 >2<2 85<2< . :2A>.

A281L70:.

2 <52304245. JD5 .:.<<5L JD5H 2.:<56:5< 2 :.<<5175 >532 I.I012H 72AILAH

2:.175:5.A56A.2.:2A>..:2A>.A281L70:.H><.4D?014.65DA:.16DA.45

515<80253L:7<0:25675:26.2515<802/3S04.85<24.<27LZI.I0125LJD214.

40?5A.6JD55672567=2:.16DA0<515<802535:7<.A281L70:2A2;5?23:21B242

2 :.<<5175 A=N0A2 5 . /3DN. A=N0A.H 56756 75145A 2 40A01D0< 658D014. 2

7<2R5:79<02601D6.0423H45651<.3214.65DA2/5A452D7.014DBC.45723651704.

JD585<2DA2515<80253L:7<0:2JD5H28.<2H/3S0456452I.I01227L2.85<24.<

5675:26.H2I.I01245;.3;52515<8022.85<24.<5672A2150<240?5A.6JD52

I.I0121C.:.16.A5<523A51752515<802HA2660A>356A517527.A25A><567242

4D<2175 DA JD2<7. 45 :0:3. >2<2 85<2< . 65D :2A>. 535:7<.A281L70:.H >2<2 2

45;.3;5<1.JD2<7.45:0:3.658D0175



CENFIM 23

24. JD5 . ]277OA57<. A545 . ;23.< AL40. 42 >.7G1:02 5 5672 L6 >.6070;2

4D<2175DAJD2<7.45:0:3. 5158270;21. 658D0175 6751C.0140:2 151PDA2

>.7G1:02

660A 2 I.I012 1C. :.16.A5 515<802 6 :.16721756 :2<826 5 456:2<826 42A56A2.<08012A:.AJD55:0<:D35DA24575<A01242:.<<51751.6:.14D7.<56H

>.<7217.H72AILA2>2<5:5DA2>.7G1:02H2JD5:P2A2<5A.6>.751:02<52:70;2

E+2

,7+,-.A2I.I012<5232><565172<56067G1:029PA0:245;04.Z6D2:.16707D0BC.

08.I012

660A >.45A.6 .I75< DA :0<:D07. 04G170:. 2. 42 /08D<2 5675 :26. 45;5<5A.6:.16045<2<.;23.<42<56067G1:02 5.42<52:7K1:02014D70;2N.65>2<24.6

08

+,5L@N.$1219:-!7-,92

22 L X R Z +=



CENFIM 24

.:26.4.:0<:D07.H.456/262A517.;2<02517<5$Y5*$YP

S = U . I (VA)

P = UR . I = U . I COSϕ (W)

QL = XL . I = U . I SENϕ (VAr)

08#.7G1:0261DA2I.I012

'

((

6:.1451624.<561C.6C.<5:5>7.<56JD565R2AD7030?24.6P2I07D23A5175:.A.26 <56067G1:026 5 26 I.I0126 . 517217. 6C. AD07. S7506H >.< 5N5A>3.H >2<2:.17<2<02< .6 /519A51.6 158270;.6 JD5 ><.4D?5A 26 >.7G1:026 <52:70;26 426I.I0126

75-75-,5+H 69 :0<:D32<= :.<<5175 53L:7<0:2 >53. :0<:D07. 4D<2175 2:2<82 4. :.1451624.< 5672 /.<A2H >.4565 40?5< JD5 DA :.1451624.< 1C.>5<A0752>266285A42:.<<5175:.17O1D2 75- :-5H . :.1451624.< /0:2<= :2<<5824. 2375<1242A5175>5<A07014.2660A:.<<51751.:0<:D07.

.:0<:D07.014D70;.2:.<<517527<2624245DAK18D3.ϕ 5A<532BC.Z7516C.



CENFIM 25

08&028<2A2;5:7.<0234551DA:.1451624.<

:2<82 :2>2:070;2 E:.1451624.<FH JD214. 0165<04. 1DA :0<:D07. 45 :.<<5175

2375<1242 .<08012 DA 456/262A517. 517<5 7516C. 5 01751604245 :.<<5175

2>2<5:55A2;21B.5A<532BC.Z7516C.

.:26.42:2<82:2>2:070;2>D<2H20175160424545:.<<51752402172*$Ê45

>5<O.4.F5A<532BC.Z7516C.

O(

A :.1451624.<H 5A H >5<A075 JD5 /3D2 :.1672175A5175 DA2 :.<<5175

53L:7<0:2 >53. :0<:D07. 45;04. Z6 :.16721756H :2<826 5 456:2<826 4. A56A. T

0A>.<72175 1.72< JD5 5672 :.<<5175 5+57 -9 ,:)7-,7

755HA2660A>53.6:.14D7.<564.:0<:D07.

P2A2A.6 7->57, 7!7,-,9 755 N Z .>.60BC. JD5 .

:.1451624.<2><565172Z:.<<5175

-:V52:7K1:02:2>2:070;2

5A.PA6

/V<5JDG1:025AP5<7?

V2>2:042454.

:.1451624.<5A/2<246

08'52:7K1:02:2>2:070;2

P(

8D232.JD5.:.<<5:.A2I.I012H65/.<A54042:.ADA]277OA57<.2>.7G1:02

45DA:.1451624.<H>.4565:.A><.;2<JD52>.7G1:020140:242L08D232?5<.

DA :.1451624.< 1C. 5N0675 :.16DA. 45 515<802 2:70;2 D<2175 . ><0A50<.

JD2<7.45:0:3..:.1451624.<L7K75K,:)7-,7E

fC X C

π 2

1=



CENFIM 26

7K :7--8-,7H >53. JD5 2 515<802 /3S0 4. 85<24.< 45 >2<2 .

:.1451624.< . JD2<7. 45 :0:3. 658D0175 . :.1451624.< 456:2<<582 >2<2 .

:0<:D07.27L2.85<24.<H45;.3;514.2.A56A.2515<8022:DAD3242

2AILA2>2<5:5D12>.7G1:02<52:70;2L7><.4D?042>532515<802JD5L7<.:242

517<5.:.1451624.<5.85<24.<

08'0<:D07.

5675:26..456/262A517.;2<02517<5$^5*$Ê/08'F

R = Z COS ϕ

XC = Z SEN ϕ

Z2= R

2+ X

2C

UR = U COS ϕ

UC = U SEN ϕ U

2= UR

2+ U

2C

S = U . I (VA)

P = U .I . cosϕ (W)

QC = U .I . sesϕ (VAr)

S2

= P2

+ Q2

C

08(

DA:.1451624.<@+:V-:EFE;.372A>5<5<52:70;.6F



CENFIM 27

.

5 3082<A.6 DA2 <56067G1:02H DA2 I.I012 5 DA :.1451624.< .I75A.6 DA

7,7+,-.;

08*0<:D07.

5675:26.20A>54K1:024.:0<:D07.;5A@

Z2

= R2

+ X2

Z2

= R2

+ (XL – XC)2

fC X

fL X

C

L

π

π

2

1

2

−=

=

fC fL X

π π

2

12 −=

08$

+D214.-V$HE-V-FHP=<566.1K1:021.:0<:D07.5675:26.20A>54K1:024.:0<:D07.L08D232

V



CENFIM 28

;HP(.

P(QOA:.<<51752375<124227516C.52:.<<5175;2<02A2:24201672175>53.JD52

>.7G1:0240660>24272AILA65<=;2<0=;53

5/01565>.7G1:020167217K152:.A.6514..><.4D7.4.6;23.<5645 +5,H5A

:24201672175

0/(5<0/0:265JD52>.7G1:020167217K152L65A><5>.6070;2>.7G1:02LA=N0A2

JD214. /.<5A A=N0A.6 2 7516C. 5 2 :.<<5175H 21D3214.65 JD214. D 5 0 65

21D32A

08

6;23.<560167217K15.64556C.424.6>53265JD2BM56@

@(8F5R-@(8F5R-

>.7G1:02 LH >.<7217.H 65A><5 >.6070;2H JD23JD5< JD5 65R2 . 01672175

:.16045<24.5.65D8<=/0:.45651;.3;56565A><52:0A24.50N.4.675A>.6

6012365A><5>.6070;.42>.7G1:02266.:02422JD23JD5<<5:5>7.<9PA0:.60810

/0:2 JD5 535 <5:5I5 :.1701D2A5175 515<802 42 <545H 067. LH 515<802 JD5 L

:.16DA042 5 5/5:70;2A5175 7<216/.<A242 2O 2 6D2 P2I07D23 4560812BC. 45

>.7G1:022:70;2

T56728<2145?2JD5>.45A.635<5AJD23JD5<]277OA57<.N><0A5655A]277H



CENFIM 29

:.A. 62I5A.6 6 :.1724.<56 45 515<802 2:70;2 42<C.H >.< :.1658D0175H 2

515<802:.16DA0424D<2175DA:5<7.75A>.5A_WPE_03.]277P.<2F

08;.3DBC.:.1RD172426/D1BM56:.<<5175H7516C.5>.7G1:021DA:0<:D07.>D<2A5175<560670;.

221=306542/08:.1:3DOA.620142JD5@

• 2>.7G1:0275A/<5JDG1:024D>32427516C.54201751604245

1O2

08

2/08D<251:.17<2A6545651P242626:D<;26 45D 50 52014242>.7G1:02

JD53P56;5A266.:0242

0

90 270180 360

p

u

i

u

i

p

Quadrante

+ +

++ -

- -

-

+ + ++

1º 2º 3º 4º



CENFIM 30

6;23.<560167217K15.64556C.424.6>53265JD2BM56@

@(8F5R-

@(8F51R-S3T2

.A.1.:26.2175<0.<@• .7G1:0275A/<5JDG1:024D>32427516C.54201751604245.><0A50<.575<:50<.JD2<7.645>5<O.4.57GA601206:.17<=<0.6H>53.JD5

156656JD24<217562>.7G1:0265<=158270;2H.JD560810/0:25672<265</.<15:042

515<802Z<545

.658D14.5JD2<7.JD24<21756H57GA.A56A.6012352>.7G1:027.A260123 >.6070;.H . JD5 60810/0:2H :.A. 62I5A.6H JD5 156756 0175<;23.6 2 I.I012

<5:5I5 515<802 12 A56A2 JD21704245 JD5 P2;02 2175<0.<A5175 51;024. >2<2

<545

08;.3DBC.:.1RD172426/D1BM56:.<<5175H7516C.5>.7G1:021DA:0<:D07.

>D<2A5175014D70;.

5A.6H 2660AH JD5 P= DA2 .6:032BC. :.17O1D2 45 515<802H .<2 42 <545 5A

40<5:BC.Z:2<82H.<245672>2<22JD532

u

i

p

Quadrante

+ +

+- +

- -

-

- - ++

1º 2º 3º 4º

90

0

180 270360

u i

p



CENFIM 31

662515<8021D1:2:P582265<7<216/.<A2425H>.<7217.H2>.7G1:02AL402JD5

3P5567= 266.:0242 L1D3267.L >.67.5A 5;04G1:02 >532 08D234245560A57<02

426=<5264530A072426>532/D1BC.>.7G1:02H:DR26.A2238LI<0:2<53270;22DA

>5<O.4.L1D32.45A.6517C.2/0<A2<JD5DA2014D7K1:02>D<21C.:.16.A5515<8022:70;2.<LAH5662515<802JD5:0<:D322375<1242A5175517<52<54552

:2<82H5AI.<21C.65R2:.16DA042H75ADA4575<A0124.;23.<5L456081242>.<

'H456081214.6508D23A5175>.<<52:70;22>.7G1:02JD53P5

;5A266.:0242

.7265>532357<2L++5,5;;).(

125>.4565<A54042>53.652515<802:.<<56>.145175>53.6

H A54042 5A P 662 515<802

:.<<56>.145 2. :2A>. A281L70:. :<024. 12 014D7K1:02 >532 >266285A 42

:.<<5175

.A.>.45A.6:.1:3D0<H2I.I012:.A5B2>.</.<15:5<515<802Z<545.45A.6

72AILA2/0<A2<@

12

08#

2 /08D<2 # 567= <5><5651724. DA :.1451624.< 517<5 :DR.6 75<A01206 567=

2>30:242DA27516C.2375<1242601D6.0423

.A.62I5A.6H2:.<<5175567=5A2;21B.45*$^H:.1/.<A51.6A.67<2A26

5JD2BM56@

@(8F5R-

A2014D7K1:02LDA/.<15:54.<45515<802<52:70;2+`$



CENFIM 32

@(8F51R-AS3T2

08&;.3DBC.42:.<<5175H7516C.5>.7G1:021DA:0<:D07.:2>2:070;.>D<.

2/08D<2&H+5,567C.<5><5651724268<2/0:2A5175RD172A5175:.A2:D<;242

>.7G1:02

.6Y5YJD24<217562>.7G1:02L>.6070;2H5158270;21.6Y5YJD24<21756

607D2BC.L.>.672Z;5<0/0:2422175<0.<A5175H>.06.:.1451624.<:.A5B2>.<

:.16DA0<515<80242<545E>.7G1:02>.6070;2F>2<212.6:032BC.658D0175;.372<2

/.<15:G3212A56A2JD21704245

53208D234245560A57<02426=<526>.6070;265158270;264530A072426>532:D<;25 658D014. DA <2:0.:O10. 04G170:. 2. R= D7030?24. 1.6 4.06 >.17.6 2175<0.<56H

>.45A.670<2<26658D01756:.1:3D6M56@

• >.7G1:0275A/<5JDG1:024D>32427516C.54201751604245• A2:2>2:04245>D<21C.:.16.A5515<8022:70;2V$• .17<2<02A5175ZI.I012HDA:.1451624.<:.A5B2>.<<5:5I5<515<802

42<545• A:.1451624.<LDA:.16DA04.<45515<802<52:70;2+a$• 2AILA2.:.1451624.<567=266.:0242DA2515<802.6:032175JD51C.

0

p

u

i

90 180270

360

u

i

p

Quadrante

+ +

-+ -

- -

+

+ + --

1º 2º 3º 4º



CENFIM 33

L7<216/.<A242 >.7G1:02 266.:0242 456081265 >.< P 672 75A >.<

D104245.EF5LA54042>53.6OH:.A.

62I5A.6

2<2A540<2515<802<52:70;2D7030?2A65H

:DR20140:2BC.L42425APE!F

'((.16045<5A.628.<2DA2:2<8270>.?.:.A.12/08D<2'

08'

+D214.2>30:2A.6DA27516C.1.665D675<A01206H2:.<<5175<56D372175;20H

127D<23A5175H5672<5A6514.45456/262A517./D1BC.4.6;23.<56455

/08D<2 ( JD5 70>0:2A5175 03D67<2 5672 607D2BC.H <5/5<565 5A >2<70:D32< Z5;.3DBC.42>.7G1:021DA2I.I012.1457516C.5:.<<5175567C.456/26242645

DAK18D3.40/5<517545*$YE14D7K1:02>D<2F

08(;.3DBC.7O>0:242:.<<5175H7516C.5>.7G1:021DA:0<:D07.><54.A012175A5175014D70;.

u

i

p

Quadrante

+

+

-

-

- - ++

1º 2º 3º 4º

-

0180

36090

ui

p

u



CENFIM 34

221=306542/D1BC.>.7G1:02;5A.6JD526=<526>.6070;26>.<5324530A0724266C.6D>5<0.<56Z6158270;26 6D2 40/5<51B2 :.<<56>.145 Z 515<802 5/5:70;2A5175 :.16DA042 4D<2175 DA

>5<O.4.H067.LHZ515<8022:70;22AILAP=226601232<DA2515<802<52:70;2JD5:.<<56>.145Z6=<526JD565:.A>5162A672515<802:0<:D3212<545H:.A.62I5A.6H65A65<7<216/.<A242

(P(.

08*.7G1:0265A:.<<51752375<1242

P

U1R--2

LP5/.<A221=3.82:P582<O2A.6Z5N><566C.@

19:-!7-,92P

19:-!2

@.7ϕ

Q = U IL sen ϕ

@.



CENFIM 35

3$.4

(4

D7030?2BC. 4.6 60675A26 7<0/=60:.6 5A 7.42 2 :24502 45 515<802 75A DA :2<=:75<

><270:2A5175 5N:3D60;. .A5175 2 1O;53 42 D7030?2BC. ;2A.6 51:.17<2< DA

60810/0:270;. 5 ;2<024. 1SA5<. 45 2>2<53P.6H 2660A :.A. 0167232BM56 45 >5JD512

>.7G1:02230A5172426:.A7516M56A.1./=60:26

'(4

• 2<2 2 A56A2 >.7G1:02 2 /.<15:5<H DA 2375<124.< 7<0/=60:. 75A A51.<

;.3DA5 5 ><5B. JD5 2 :.<<56>.145175 D104245 A.1./=60:2 5 A20.<

/02I03042454565<;0B.

• 6<5456457<216>.<755454067<0ID0BC.<56D372AA20660A>35655:.19A0:26

7030?2A7<G6:.14D7.<5645/2655H5;517D23A5175HDAJD2<7.:.14D7.<15D7<.H

406>516214. 6506 :.14D7.<56 JD5 <5JD5< DA2 <545 A.1./=60:2 5JD0;235175 \

5:.1.A02 4. :.I<5 5 A51.<56 >5<426 5A 301P2 2302A65 .6 A51.<56 :D67.6 5

A20.< 60A>30:04245 45 :.1:5>BC. 5 0A>32172BC. 426 567<D7D<26 45 2>.0. 426

301P26

• 60A>30:04245 45 :.167<DBC.H A51.<56 :D67.6 5 8<2145 /02I0304245 45

/D1:0.12A517. 4.6 7<216/.<A24.<56 7<0/=60:.6 5 20142 4.6 A.7.<56

266O1:<.1.6 45 :2A>. 80<2175H 45 5A><58. 8515<230?24. 5 JD5 1C. 7GA

5JD0;235175 5A A.1./=60:.H RD670/0:2A 69 >.< 60 2 5N067G1:02 45 60675A26

7<0/=60:.6

V.

12 1$2

085><565172BC.:2<7560212EF5;5:7.<023EF45DA60675A27<0/=60:.45

7516M56.72<JD5567C.456/262426517<56045$YH.D65R2H45>5<O.4.

120º

120º

120º

u1

u2u3

-400

-300

-200

-100

0

100

200

300

400

180 270 36090

u1 u2 u3



CENFIM 36

2/08H>.45A.6;5<5A !5-"#7-,525;.3DBC.4267516M56

45DA60675A27<0/=60:.2>2<70<4.6<56>5:70;.6;23.<560167217K15.6

.4520142/2?5<65DA2<5><565172BC.;5:7.<0234.A56A.60675A2E/08&$FH65275145<A.62JD5DA456/262A517.1.75A>.4545>5<O.4.5JD0;2352

DA2 40/5<51B2218D32< 45 $Y517<5 .6 ;5:7.<56 <5><56517270;.6 4267516M56

D>M565 7.4. . 60675A2 <.4214. 2 DA2 ;53.:04245 218D32< ω 1. 651704.

0140:24.HJD52<I07<2A.6:.A.>.6070;.

,3,2,1 U U U :.16707D5AH 2660AH DA :.1RD17. 45 7<G6 97- K,5- :DR2

8<2145?2A5455A9:E,7I26<5/5<04267516M56

LP

221=306542/08;5A.6JD5567=5A2;21B.<53270;2A517525

/2:7.H 65 56:.3P5<A.6H >.< 5N5A>3.H 2 .<085A 4.6 75A>.6 :.A. <5/5<G1:02H

;5<0/0:2A.6JD575AR=DA:5<7.;23.<>.6070;.1DA2/26545:<56:5175426D2

2375<1K1:02H51JD217.JD5L1D3.56928.<20<=010:02<22375<1K1:02>.6070;2

>.<;23.<566D:5660;2A5175:<56:51756

5<0/0:2A.608D23A5175JD5567=5A27<26.<53270;2A5175267.60810/0:2

JD5 65 24A070<A.6 7.4. . 60675A2 <.4214. 1. 651704. 0140:24. 12 /08 5

7.A2<A.6:.A.<5/5<G1:022>.60BC..:D>2422424.A.A517.>.<DA456656

;5:7.<56H>.<5N5A>3. 1U HJD5567265<=6D:5660;2A5175.:D>242>53.6;5:7.<56

2U 5 3U

65JDG1:02 45 /2656 0140:242 L H H 5 :P2A265 +)

.<7217.H 2 65JDG1:02 45/2656L 2 .<45A>532JD23 656D:545A 26

/26561DA60675A27<0/=60:.

.''4

N0675A4D26/.<A267O>0:2645266.:02BC.H26:P2A2426@

• .,K"#.;

• .,K"#O'.+.



CENFIM 37

08082BC.5A567<53245:2<826A.1./=60:26D7030?214.DAS10:.:.14D7.<

:.ADA:517<23E:.14D7.<15D7<.F5675:.14D7.<1C.:0<:D32:.<<5175HV

>.45<02A56A.6D><0A0<65

67<G6:2<826<5><565172426:2<2:75<0?2A65>53.A56A.;23.<45 0A>54K1:02H

067.LH&@&$@&<6<56>5:70;265N7<5A042456567C.30824262.675<A0120645

:242 DA 4.6 51<.32A517.6 4. 2375<124.< 5 6C. <5/5<51:02426 >5326 357<26N?

W?U&

.6 :.14D7.<56 45 230A5172BC. 5672I535:5A65 2660A 7<G6 :.<<51756 :.A .

A56A.;23.<5/0:2?A26E$3T

6;23.<560167217K15.6456626:.<<5175640/5<5AH:.17D4.H5A:242A.A517.5

5A:242DA2426/2656

.45<O2A.6<570<2<.4.:0<:D07.65A2375<2BC..D><5RDO?.

426 :.140BM56 45 /D1:0.12A517. 69 1C. . /2?5A.6 >.< DA2 A54042

><5;5170;2H 623;28D2<4214. 2660A 2 P0>97565 426 7<G6 :2<826 >.45<5A 6./<5<

JD23JD5<2375<2BC.



CENFIM 38

67263082BM56:.1/08D<2ADA2 -:H7217.12/.175:.A.12:2<82>.17.

:.ADA456081265>.<

.A.5N5A>3.H65:.16045<2<A.6.6;23.<5645HHE/08F:.<<56>.14514.2DA2:.<<51755/0:2?45(>.</2655<53270;.62.A.A517.5AJD5ω 7V*$YH

75A.6@

090=t ω

VA=N651ω 7VH651*$YVH

VA=N651Eω 7$YFVH651E$YFV#H&&

VA=N651ω 7VH651$YV#H&&

5675H:.A.5AJD23JD5<.D7<.01672175H26.A2238LI<0:2426:.<<51756L?5<.@

$VbbVH#H&&#H&&V

5AJD5$<5><5651722:.<<51751.15D7<.

/08 A.67<2 5675 60675A2 45 :.<<51756 5A <5><565172BC. :2<7560212H .145H

>2<2 :242 01672175H 2 6.A2 426 .<4512426 :.<<56>.1451756 Z 0175<65:BC. 42

;5<70:23:.A26<56>5:70;26601D690456L65A><508D232?5<.

08



CENFIM 39

(L.

+D214. 7.426 26 :2<826 7GA . A56A. ;23.<H 067. LH 2 A56A2 0A>54K1:02H .

60675A240?655JD030I<24.526:.<<517565A:242DA2426/26566C.08D206

.0.:26.2I.<424.1.>.17.2175<0.<

DA60675A27<0/=60:.5JD030I<24.26.A2;5:7.<023426:.<<51756L08D232?5<.

0 I :.<<51751.15D7<.0 ;5:7.<1D3.

V(.(

• (.

.16045<5A.6DA60675A27<0/=60:.:.A15D7<.HE:DR26301P2645230A5172BC.6C.

:.16707DO426>.<7<G6:.14D7.<5645/2655>53.:.14D7.<15D7<.H<5/5<51:024.6H

<56>5:70;2A5175H!.12?.$12?.<12;

.16045<5A.6 20142 .6 >.17.6?$?< :.<<56>.1451756 2.6 75<A01206 45

3082BC.

08516M5660A>356.D7516M5645/265

DA,--,E8,775+-75A.6H>.<7217.H7<G67516M5660A>356HJD5

4560812A.6 >.< ? $ < .A. 62I5A.6H 6C. 08D206 5A 8<2145?2 5

;5:7.<023A5175/.<A2ADA2567<5327<0/=60:2457516M56E/08&F

56081265 >.< 2 7516C. 5N0675175517<5 JD23JD5<

:.14D7.<45/2655.:.14D7.<15D7<.

03210 =++= I I I I



CENFIM 40

08#67<5327<0/=60:2457516M56

• (

08&516M56:.A>.6726

2 /08& 26601232A.6 26 <5/5<0426 7516M56 :.A 60AI.3.802 R= 1.662:.1P5:042H

067.L@

2112 U U U −=

1331 U U U −=

JD5<5><565172AH<56>5:70;2A5175H267516M56517<526/26565H5H5

6726 5N><566M56 >.660I03072A1.6 45651P2< .6 ;5:7.<56 <5><56517270;.6 426

7516M56 :.A>.6726 45 DA 60675A2 7<0/=60:. 2 >2<70< 426 <56>5:70;26 7516M56

60A>356

5/01565:.A.27516C.5N0675175517<54D26JD206JD5</26564.60675A27<0/=60:.

3223 U U U −=

U1

U2U3

120º120º

120º

0

3 2

1

U1

U2U3

U23

U12U31



CENFIM 41

.'&V(.(

.<56>5:70;.4028<2A2;5:7.<023ID6JD5A.62<532BC.517<527516C.:.A>.672

527516C.60A>[email protected]<G6;5:7.<56/.<A2ADA7<0K18D3.0696:5356H:DR.K18D3..I7D6.:.<<56>.1452.456/262A517.5N0675175517<5 1U 5 2U 067.LH$Y

6<567217564.06K18D3.6;235A$Y:242DA

V:.6$Y:.A.:.6$YV2

3

17C.V 3

514.VVV

5VVV

75A.6JD5

.145!5--5#,!:7-5#7!-

+D214.1.660675A267<0/=60:.6650140:2DA4575<A0124.;23.<427516C.65A

JD23JD5< 24R5:70;2BC.H 45;5 6DI5175145<65 JD5 535 65 <5/5<5 2 DA2 7516C.

:.A>.672.<5N5A>3.H6540665<A.6JD5DA24575<A01242301P245L45#

_H45;5A.6517514G32:.A.27516C.:.A>.672

D7<26;5?562>2<5:5A1.64D267516M5656:<0726658D042A51752>5126:.ADA

7<2B. .I3OJD. 2 65>2<=326 .< 5N5A>3.H DA2 <545 $$$ ><0A50<2

4560812517C.27516C.60A>35652658D14227516C.:.A>.672

(4L.

0?65JD5DA60675A27<0/=60:.L4565JD030I<24..D45:2<8264565JD030I<242665

260A>54K1:026>.</2651C./.<5A7.42608D206

5672 607D2BC.H . >2>53 4565A>51P24. >53. :.14D7.< L /D142A51723H

:.A.;5<5A.6

<5A.6 517C. 567D42< . /D1:0.12A517. 45 DA2 :2<82 7<0/=60:2 4565JD030I<242

308242 5A 567<532 :.A 15D7<.H 5 658D042A5175 65A 15D7<.H 5 2;2302< .6

<56D3724.6

UC = 3 US



CENFIM 42

.(

08'575<A012BC.;5:7.<02342:.<<51751.

:.14D7.<15D7<.5A

60675A267<0/=60:.6

4565JD030I<24.6

0<:D32<= 2660A DA2 :.<<5175 1. 15D7<. N I :.<<56>.145175 Z <56D372175 42

6.A2;5:7.<023426:.<<517561267<G6/2656

.1:3DOA.6JD5@

662:.<<5175L08D23Z6.A2;5:7.<023426:.<<51756426/2656

.(

6D><566C. 4. :.14D7.< 15D7<. 1DA 60675A2 4565JD030I<24. .<08012 DA

4565JD03OI<0. 426 7516M56 60A>356 6DR507214. .6 40;5<6.6 <5:5>7.<56 2 6D>.<72<

1D16:26.66.I<57516M56H1.D7<.67516M5601/5<0.<562.<56>5:70;.;23.<1.A0123

.'(O'.

DA60675A25A567<5324565JD030I<24.:0<:D3265A><5DA2:.<<51751.15D7<.

321 I I I I N ++=

T0A><56:014O;53.:.14D7.<15D7<.>2<242<>266285AZ:.<<51754545/507.



CENFIM 43

08(=<026>5<6>5:70;2645DA2A56A23082BC.5A7<0K18D3.

DA2 3082BC. 5A 7<0K18D3.H 26 7<G6 :2<826 3082A65 65JD51:023A5175

:.1/08D<214. DA2 A23P2 /5:P242 7<0218D32< E/08(FH 6514. :242 >.17. :.ADA

30824.2DA2/265

.45A.6;5<0/0:2<JD5@

• C.5N0675:.14D7.<15D7<.>.<1C.P2;5<>.17.:.ADAZ67<G6/2656

• 7516C.2>30:2422:242DA2426:2<826L27516C.:.A>.672

./

6:.<<5175645301P2H:.A.2><9><024560812BC.0140:2H6C.26:.<<51756JD5:0<:D32A1.6:.14D7.<5645230A5172BC.5JD512/08D<2/.<2A1.72426>.<H5

P2A2A65 :.<<51756 45 /265 / Z6 :.<<51756 JD5 :0<:D32A 1.6 <2A.6 4.

7<0K18D3.C.26:.<<51756H542/08&.4267GA.A56A.651704.

45:0<:D32BC.c2<I07<24.d

O'.L.

A60675A27<0/=60:.40?655A7<0K18D3.5JD030I<24.JD214.7.42626:2<8264.

7<0K18D3.6C.04G170:265>.<7217.7GA2A56A20A>54K1:02

5672607D2BC.26:.<<51756126301P266C.7.42608D206H067.LHVV2660A

:.A.26:.<<51756126/2656H.D65R2HVV5AJD23JD5<:26.456/262426

$Y517<560

../

IL= 3 If



CENFIM 44

672<532BC.>5<A0751.651D1:02<JD52:.<<517512301P2L 3 ;5?56A20.<JD5

2:.<<517545/265

O'.L.

+D214.26:2<8261C.6C.7.42608D206H.7<0K18D3.L4565JD030I<24.52<532BC.

4.>.17.2175<0.<1C.L;=30426:.<<5175645301P2450N2A4565<08D206H2660A

:.A. 26 :.<<51756 45 /265 217GA65H :.17D4.H 26 7516M56 45 /265 1.6

75<A0120645:242DA2426:2<826

P(4

(.((4

:=3:D3.45>.7G1:0265A:27<0/=60:2H1.A5242A5175426!-D57,7-,9?

7-,9 !5-H 6017570?2 5 658D5 DA2 /.<AD32BC. 04G170:2 Z 4.6

:.16DA.6>.</265

.16045<214. . :26. 85<23 JD5 :.175A>32 7.426 26 607D2BM56 45 :2<82 2 JD5

75A.6;014.2/2?5<<5/5<G1:02:2<8265JD030I<2426.D1C.H3082BC.5A567<532

.D5A7<0K18D3.H>.4565<2660A5JD2:0.124.@

'.

) Vbb

) +V+b+b+

) V 22QP +

242 DA2 426 >2<:5326 0145N2426 <5><565172 DA2 >.7G1:02 >.< /265H 5 26

<56>5:70;265JD2BM566C.@

/ V:.6 F ϕ

+/ V651 F ϕ

>.7G1:022:70;2H6514.:.<<56>.1451752DA2>.7G1:02:.16DA042HL65A><5

>.6070;252<5/5<0426.A2L2<07AL70:2



CENFIM 45

>.7G1:02 <52:70;2 7.723 L . I2321B. 42 >.7G1:02 JD5 :0<:D32 517<5 .6

:.A>.151756<52:70;.652<545H6514.158270;2JD214.:.16DA042H5>.6070;2

JD214./.<15:0426D26.A2LH>.<7217.H238LI<0:2

'L.

+D214. 26 :2<826 6C. 08D206 126 7<G6 /2656H 26 5N><566M56 >2<2 26 >.7G1:026

<56D372AA20660A>356123065A.6H517C.H263082BM565A567<53255A7<0K18D3.

.'(.

5672A.17285A5126:.140BM5651D1:02426H75A.6JD5@

• 6:.<<5175645/2656C.08D2065A8<2145?2508D206Z6:.<<5175645

301P2

• .675<A0120645:242:2<82567=2>30:242DA27516C.60A>356

56726:.16045<2BM56I26502A6526454DBM56>2<22640/5<51756>.7G1:026@

P

VVV:.6ϕ

VbbV:.6ϕ

.A.3C

S

U U =

V3C U

:.6ϕ

5>.<7217.

U1U--2

P:

5/.<A221=3.82:P582<O2A.6Z5N><566C.@

P = 3 UC IL cos ϕ



CENFIM 46

1:-!7-,92

P

1:-!2

.'(O'.

5672A.17285A5126:.140BM5651D1:02426H75A.6JD5@

• 6:.<<51756126301P266C. 3 6D>5<0.<56Z6:.<<51756126/265657GA.

A56A.;23.<5A7.4265326

• 7516C. 2>30:242 2 :242 DA 4.6 535A517.6 4. 7<0K18D3. L 2 7516C.

:.A>.672

514.2660AH454D?2A.6265N><566M56>2<22640/5<51756>.7G1:026@

P

VVV/:.6ϕ

V/:.6ϕ V3 L I :.6ϕ

5>.<7217.

U1R--2

P

5/.<A221=3.82:P582<O2A.6Z5N><566C.@

19:-!7-,92

P

19:-!2

Q = 3 UC IL sen ϕ

S = 3 UC IL

Q = 3 UC IL sen ϕ

S = 3 UC IL

P = 3 UC IL cosϕ



CENFIM 47

N '. P ( ( 4

L.

.A>2<214.265N><566M56JD5454D?0A.61.:26.423082BC.5A7<0K18D3.55A567<532 ;5<0/0:2A.6 JD5 6C. 08D206H >53. JD5 >.45A.6 56:<5;5<>2<2 2AI.6.6

:26.6@

P = 3 UC I cos ϕ Q = 3 UC I sen ϕ S = 3 UC I



CENFIM 48

3<(P

P

,5-5, 9, ! + 7!-? 6 + E7"# -? ,

!!,5-+7!55-7-,9?)!9,-EX-,:

A24242:.<<51752I6.<;04265<=7217.A53P.<2><.;507242JD217.A20.</.<26D2

:.A>.151752:70;25A206><9N0A.42D104245H>.<7217.H/.<./2:7.<45>.7G1:02

>2<75:.<<56>.145175Z6D2:.A>.15175<52:70;2:0<:D3260A>356A517512<545H65A

65<7<216/.<A242

2I6.<BC.H>.<>2<754.6<5:5>7.<56014D70;.6H45:.<<51756:DR.;23.<L6D>5<0.<2.

15:566=<0.>2<2.<56>5:70;./D1:0.12A517.75A0A>.<7217560A>30:2BM5627.4.6.6

1O;5064.0675A23L:7<0:.4515<802H:DR.601:.1;510517566C.658D042A51752>.1

724.6

(.VQ

• 5:56604245 45 2DA5172< 2 >.7G1:02 016723242 126 :517<206 ><.4D7.<26 68<D>.675<C.H2660AH4565<6.I<540A5160.124.6>2<2/2?5<5A/2:52DA2:2<82<52:70;224;5<62

• 6 301P26 75<C. 08D23A5175 45 65< 6.I<540A5160.12426 >2<2 JD5 >.662A

;50:D32<2:.<<5175>.<5662<2?C.5N:54517=<02

• 20.<56JD5426457516C.5A301P2@&5A20.<5626>5<426@&$

• .I<540A5160.12A517.457.4222>2<53P285A45:.<755><.75:BC.H:.A.H>.<5N5A>3.H65::0.124.<565406RD17.<56

• DA517. 42 >.7G1:02 4.6 7<216/.<A24.<56 567=70:.6 126 6DI5672BM56

7<216/.<A24.<26

(.V0(6:D67.66C.><.>.<:0.1206Z>.7G1:02D107=<024.6

8<D>.6

• 01P26 45 A20.< 65:BC.H 127D<23A5175 A206 :2<26 01;5670A517. 12 6D2

:.167<DBC.1C.75AH2660AH2:.17<2>2<704245A20.<1SA5<.45D7030?24.<56H.

JD5657<24D?1DA2>5<4242>.751:023<5:507245/2:7D<2BC.



CENFIM 49

• .172I030?2BC.426>5<426>.<5/507..D35126301P26

• .4222>2<53P285A42<545L45:D67.A206535;24.

• 27D<23A5175H.28<2;2A517.45:D67.665<=6D>.<724.>53.D7030?24.<5.D7030?24.<L45I20N27516C.H70>.4.AL670:.H./2:7.<45>.7G1:02LAD07.2><.

N0A242A5175D107=<0.5 .4067<0ID04.<2>5126/2:7D<22515<8022:70;2:.16DA0426

:D67.6501:.1;5105175645<0;24.642D7030?2BC.45<5:5>7.<564.70>.014D70;.H:.A.

3KA>2426/3D.<56:51756 556>.<240:2A5175 A.7.<56H 6C.85<23A51756D>.<724.6 >532

517042454067<0ID04.<2

5.D7030?24.<L45I20N27516C.A26>.66D00167232BM56:.AI20N./2:7.<45>.7G1:02H

517C..4067<0ID04.<.I<08=3.=20167232<DA:.1724.<45515<802<52:70;2H.I<08214..26D>.<72<.6:D67.6426D2><9><020167232BC.1C.:.<<08042

2<2/2:7.<5645>.7G1:02:DR.;23.<L08D23.D6D>5<0.<2$H(H.4067<0ID04.<1C..I<082

Z0167232BC.45:.1724.<5645515<802<52:70;2

$(P

53.JD5/0:.D407.<566237220A>.<7K1:0242:.<<5:BC.4./2:7.<45>.7G1:02H067.LH2

>.660I030424545.7.<12<.A206><9N0A.>.66O;5342D104245

.1658D565 723 :.<<5:BC. /2?514. 40A01D0< 2 :.A>.15175 <52:70;2 42 :.<<5175H

<5:.<<514.2.5/507.:.1RD824.45:2>2:042451.:0<:D07.

.A.62I5A.6H2:.A>.15175<52:70;242:.<<5175567=5AJD24<27D<2527<26.45

*$^<53270;2A5175Z7516C.

082<82014D70;22175642:.<<5:BC.H

20N./2:7.<45>.7G1:02

S

S’

Q’

QC

QC

Q

ϕ ´,

ϕ



CENFIM 50

:.16DA.45515<802<52:70;2567=30824.2DAA2DE7-!-D57,1+7ϕ 25Z0167232BC.1.BC.45/2:7.<45>.7G1:0245;55672<266.:02422.640/5<51756<5:5>7.<56D7030?24.612014S67<0251.75<:0=<0.5<A0751.6 045170/0:2< .6 2>2<53P.6 :DR. :.16DA. 5A 515<802 <52:70;2 L A206 .DA51.60A>.<72175

!:M 7ϕ -Kϕ

• .7.<266O1:<.1. $H'2$H(# #H(2$H&

• KA>242601:21456:51756 $

• KA>2426/3D.<56:51756 $H# H'

• KA>242645456:2<82 $H2$H& H*2H• .<1.645<56067G1:02 $

• .<1.645014DBC. $H(# $H&

• .<1.6452JD5:0A517.4053L:7<0:. $H(# $H&

• =JD0126456.342<45<56067G1:02 $H(2$H* $H'#2$H(

• .67.6 567=70:.6 A.1./=60:.6 45 6.3424D<2 >.<2<:.

$H# H'

• <D>.6<.7270;.6456.3424D<2>.<2<:. $H'2$H* H$2$H(

• <216/.<A24.<56<5:70/0:24.<56 45 6.3424D<2 >.<2<:. $H'2$H( H$2$H'#

• .<1.6452<:. $H( $H'#

<('

LP.(PY

0167232BC.4DA:.1451624.<5AJD23JD5<<54553L:7<0:2E!.DFRD670/0:265>532@

• D><566C.426>5123042456>.<:.16DA.45515<802<52:70;2

• 0A01D0BC.42>.7G1:022>2<51756DI6:<072

• 0A072BC.426>5<42645515<8022:70;2E>5<42645R.D35F1.6:2I.6

• .660I0304245 45 75< DA 2DA517. 42 >.7G1:02 2:70;2 EUWF 406>.1O;53 1.

65:D14=<0.4.7<216/.<A24.<!

• 53P.<024.1O;53457516C.1./0A42301P2

755!-<5C9,,E5-5C1L'2



CENFIM 51

('..

08.A>5162BC.83.I23

• 6D><0A526>5123042456>.<:.16DA.5N:5660;.45515<802<52:70;2

• 2RD672 2 15:56604245 <523 42 0167232BC. 5A _W Z 6DI6:<0BC. 42 >.7G1:02

2>2<5175E(5A_F

• 230;02.>.67.457<216/.<A2BC.E>.7G1:02406>.1O;535A_WF

• >5<A075D7030?2<DA406RD17.<A2065:.19A0:.

9"Z• 2:.<<5175<52:70;2E<F567=><565175120167232BC.45645.1O;5327L2.6

<5:5>7.<56

• 26>5<426>.<5/507.45.D351.6:2I.61C.6C.40A01DO426E_WPF

(.

08.A>5162BC.2<:023



CENFIM 52

• 6D><0A526>5123042456>.<:.16DA.5N:5660;.45515<802<52:70;2

• .>70A0?2DA2>2<75420167232BC.H2:.<<5175<52:70;21C.L;50:D3242517<5.6

1O;5065

• 230;02.>.67.457<216/.<A2BC.E>.7G1:02406>.1O;535A_WF

• >5<A075D7030?2<DA406RD17.<A2065:.19A0:.2A.1721754.:.1451624.<

9"Z• 2 :.<<5175 <52:70;2 < 567= ><565175 12 0167232BC. 45645 . 1O;53 27L 2.6

<5:5>7.<56

• 26>5<426>.<5/507.45.D351.6:2I.66C.40A01DO426E_WPF

(..11.6I.<15645:242<5:5>7.<4.70>.014D:70;.F

08.A>5162BC.3.:23

• 6D><0A526>5123042456>.<:.16DA.5N:5660;.45515<802<52:70;2

• .>70A0?27.422<54553L:7<0:2:.<<5175<52:70;2<L/.<15:0421.3.:234.

65D:.16DA.• 230;02.>.67.457<216/.<A2BC.E>.7G1:02406>.1O;535A_WF

• >5<A075D7030?2<DA406RD17.<A2065:.19A0:.2A.1721754.:.1451624.<

9"Z• 2:.<<5175<52:70;2R=1C.:0<:D32>53.6:2I.6420167232BC.

• 26>5<426>.<5/507.45.D351.6:2I.66C.7.723A51756D><0A0426E_WPF



CENFIM 53

F7C7,:9,

DA20167232BC./2I<0345$$$#$!?;5<0/0:.D65JD526515<8026

2:70;2 5 <52:70;2 A540426 >53.6 :.1724.<56 2. /0A 45 $$ P.<26 45/D1:0.12A517. 5<2A <56>5:70;2A5175 45 #$$$ _WP 5 &$$ _P

23:D35@

2F6>.7G1:0262:70;25<52:70;2

IF/2:7.<45>.7G1:02420167232BC.

:F :2>2:04245 45:242 :.1451624.<H 2 3082< 5A7<0K18D3.H>2<2 535;2< .

/2:7.<45>.7G1:02420167232BC.>2<2$H(

.2F VW 7V#$$$$$V('H#U

+VW+ 7V&$$$$V(U

IF 78ϕ 0V+ V$H*&V`:.6ϕ 0V$H'

:F :.6ϕ /V$H(V`78ϕ /V$H'#

+V78ϕ /V('H#N$H'#V&#H&U

+VE78ϕ 078ϕ / FV+0+/V(&#H&V(H'U

+V

+

V(H'V&HUV&E:242:.1451624.<F

V+ E f .π FV&E$$NN .π N#$FVHN$V

µ



CENFIM 54

LVN

LVN

[23:D35.>5<O.4.:.<<56>.1451752DA2/<5JDG1:0245$_!?

$[23:D352/<5JDG1:0245DA2:2601D6.0423:DR.>5<O.4.L$H6

<[+D217.6:0:3.6/2?1DA658D14.DA2:.<<5175:DR.>5<O.4.L$H$$6[

=[+D217.75A>.45A.<2DA2:.<<517525/5:7D2<$$:0:3.662I514.JD526D2/<5JDG1:02L45#$$$!?[

B[+D23L.>5<O.4.45DA2.14545<=40.:DR2/<5JDG1:02L#$$_!?[ µ

G[A2 .145 45 <242< 75A DA >5<O.4. 45 121.6658D14. 23:D35 2 6D2

/<5JDG1:02

H[A;.37OA57<.45:20140:21.6$$23:D35@

2F;23.<5/0:2?427516C.

IF;23.<A=N0A.427516C.

\[A:23.<O/5<.45#$$W:.16.A55A:.<<5175:.17O1D2H5.3082<A.62

:.<<5175 2375<1242 42 A56A2 7516C.H JD23 65<= . ;23.< 5/0:2? 42 :.<<5175

:.16DA042[

S[;23.<5/0:2?45DA27516C.2375<1242601D6.0423L45#$$+D2365<=.;23.<

2<07AL70:.AL40.[

3[+D23426/<5JDG1:026E#$5$$!?F45A.<2A20675A>.25/5:7D2<:0:3.[

[5JD2BC.0V651Eπ /7F<5><565172DA2:.<<51752375<1242601D6.0423

2F140JD5.;23.<422A>307D4542:.<<5175

IF23:D35.;23.<5/0:2?

:F140JD5.;23.<42;53.:04245218D32<

4F23:D35.;23.<42/<5JDG1:02

5F23:D35.;23.<4.>5<O.4.

/F23:D35.;23.<0167217K15.42:.<<5175H1.016721757V$H$$#6

8F2B22<5><565172BC.75A>.<2345672:.<<5175

$[A2I.I01275ADA2<56067G1:0245$Ω 5DA2<52:7K1:0245$Ω 23:D35@

2F0A>54K1:0242I.I012Ω

IF01751604245JD5>5<:.<<52I.I012JD214.653P52>30:2$

LVN



CENFIM 55

<[ DA I23267<.H :DR2 <56067G1:02 L 45 $Ω 52<52:7K1:02$Ω HL2>30:242

DA27516C.45($23:D35@

2F6D20A>54K1:02Ω

IF01751604245JD5.>5<:.<<5=[A:0<:D07.53L:7<0:.H:DR27516C.L$HL:.16707DO4.>.<DA2<56067G1:0245

$$Ω 5DA2I.I012>D<245#$$Ω 23:D35@

2F0A>54K1:0252<52:7K1:0242I.I012Ω Ω

IF0A>54K1:024.:0<:D07.Ω

:F017516042452I6.<;042

B[A2 I.I0121C.>D<2H:DR2 <52:7K1:02L$$Ω H2I6.<;5DA20175160424545

$H(JD214.3082422$23:D35@

2F0A>54K1:0242I.I012 Ω IF6D2<56067G1:02Ω

G[23:D352<52:7K1:0245DA2I.I012JD52I6.<;5$H#JD214.6DIA570422

#$6D2<56067G1:02L45$$Ω Ω

H[23:D3520A>54K1:0245DA2I.I012:DR.:.5/0:05175452D7.014DBC.LH#!5

:DR2<56067G1:02L$$Ω E/V#$!?F Ω

\[A2I.I012L>5<:.<<042>.<H#JD214.653P52>30:2DA27516C.:.17O1D2

45#$.2>30:2<3P5#$5A:.<<51752375<1242201751604245I20N2>2<2$H

23:D35@2F<56067G1:0242I.I012Ω

IF<52:7K1:0242I.I012Ω

:F65D:.5/0:05175452D7.014DBC.H65/V#$!?

S[A2JD5:54.<:DR2<56067G1:02L08D232'$ Ω 567=30824.2$E:2F

23:D35@

2F017516042452I6.<;042

IF>.7G1:022>2<5175

:F>.7G1:022:70;24F/2:7.<45>.7G1:02

$3[A2 I.I012 75A DA2 <56067G1:02 45 # Ω 5 DA2 <52:7K1:02 45 #$ Ω

23:D35H2.65<3P52>30:242DA27516C.45#$H26658D017568<2145?26@

2F017516042452I6.<;042

IF>.7G1:022:70;21.:0<:D07.

:F>.7G1:022>2<5175

4F>.7G1:02<52:70;2

5F/2:7.<45>.7G1:02



CENFIM 56

$[53270;2A5175 2. ><.I35A2 2175<0.<H 6D>.1P2 JD5 2>30:2;2 DA2 7516C.

:.17O1D245#$23:D35@

2F017516042452I6.<;042

IF>.7G1:0240660>242>.<5/507.45.D35$$[A:.1451624.<45$ µ L30824.Z<545454067<0ID0BC.45$H#$!?

23:D35@

2F6D2<52:7K1:02Ω

IF6D20A>54K1:02E65456><5?2<A.62<56067G1:02><9><02FΩ

:F017516042452I6.<;042>53.:.1451624.<

$<[23:D352:2>2:0424545DA:.1451624.<>D<.EV$FJD52I6.<;5JD214.

30824.2$E#$!?F µ

$=[A:.1451624.<>D<.45# µ 2I6.<;5$HE#$!?F23:D35@

2F6D2<52:7K1:02Ω

IF7516C.JD53P5/.02>30:242

$B[A:0<:D07.6L<05H:.16707DO4.>.<DA:.1451624.<>D<.45'H' µ 5DA2

<56067G1:02 45 #$ Ω HL230A51724.>.<DA2/.17545:.<<51752375<124245

$H#$!?23:D35@

2F0A>54K1:024.:0<:D07.Ω

IF017516042452I6.<;042

:F7516C.5A:242535A517.Q

4F6>.7G1:0262:70;25<52:70;2

5F2B2.4028<2A2;5:7.<0234.:0<:D07.

$G[A:0<:D07.6L<05L:.16707DO4.>.<DA2<56067G1:0245$ Ω H:.5/0:05175

452D7.014DBC.08D232$H!5:2>2:0424545$ µ 7516C.2>30:242L45

$$E:2F23:D35@

2F<52:7K1:0242I.I01254.:.1451624.< Ω Ω

IF0A>54K1:024.:0<:D07.Ω

:F017516042452I6.<;042

4F7516C.5A:242535A517.

5F6>.7G1:0262:70;2H<52:70;252>2<5175

/F2B2.4028<2A2;5:7.<0234.:0<:D07.

$H[<G6<56067G1:02630824265A567<532:.A15D7<.2I6.<;5A<56>5:70;2A5175H

5&7516C.60A>356L45$23:D35.6;23.<5645@

2FHH Ω Ω Ω

IF

:FHH



CENFIM 57

$\[<G63KA>242601:21456:51756H30824265A567<532:.A15D7<.2DA2<54545

$$$H7GA.6;23.<5645Ω HH#Ω 5(Ω 23:D35@

2FH5

IF:FHH

$S[<G6 <5:5>7.<56H 30824.6 5A 567<532 :.A 15D7<.H 7GA <56>5:70;2A5175 26

>.7G1:02645#$$WH'#$W5$$$W<545L45$$$23:D35@

2FH5

IFHHΩ Ω Ω

:F

<3[<G6:2<826014D70;2630824265A567<532:.A15D7<.Z<54545$$$7GA

<56>5:70;2A5175.6658D01756;23.<56@V$$Ω H-V$$Ω HV$$Ω H-V$$Ω HV$$Ω H

-V$$Ω

2F23:D3520A>54K1:0245:242:2<82Ω Ω Ω

IF23:D35.;23.<42017516042455A07.

4F23:D35.;23.<42017516042451.15D7<.

5F23:D35./2:7.<45>.7G1:0245:242:2<82

<[<G6:2<826<560670;2608D206H30824265A7<0K18D3.2$$H2I6.<;5A42<545

DA2:.<<5175E45301P2F:DR201751604245L23:D35@

2F017516042455A:242:2<82

IF<56067G1:0245:242:2<82:242:2<82

:F2B2.4028<2A2;5:7.<0234.:0<:D07.

4F>.7G1:0245:242:2<82

<$[ DA2 <545 45 $$ L 30824. DA A.7.< 7<0/=60:. :DR.6 51<.32A517.6 6C.

30824.6 5A 7<0K18D3. 01751604245 12 301P2 L &Q . /2:7.< 45 >.7G1:02 4.

A.7.<L$H(23:D35@

2F017516042455A:24251<.32A517.4.A.7.<

IF0A>54K1:025A:242/2654.A.7.<Ω

:F>.7G1:022:70;22I6.<;042>.</265

<<[67<G6<2A.645DA23082BC.5A7<0K18D3.6C.:.16707DO4.6>.<7<G6:0<:D07.6

6L<05H <56>5:70;2A5175 :.A@ V $$ Ω H 1ϕ V$HQ V#$ Ω H

:.6 2ϕ VH#HV$$Ω H:.6 3ϕ V$H&<545L45$$$23:D35@

2F<56067G1:0245:242<2A.Ω Ω Ω

IF01751604245JD5>5<:.<<5:242<2A.



CENFIM 58

:F>.7G1:022:70;245:242<2A.

4F>.7G1:02<52:70;245:242<2A.

<=[A:0<:D07.4503DA012BC.L:.16707DO4.>.<&$3KA>2426/3D.<56:5175645#(

W$ /2:7.< 45 >.7G1:02 45 :242 3KA>242 L $H# 6 3KA>2426 567C.08D23A5175 4067<0IDO426 >5326 7<G6 /2656 45 DA 60675A2 7<0/=60:.H :DR2 7516C.

:.A>.672L45$$#$!?23:D35@

2F>.7G1:022:70;2120167232BC.

IF0175160424512301P2JD5230A5172.:0<:D07.

:F>.7G1:022>2<5175420167232BC.

4F :2>2:04245 45 :242 :.1451624.<H 2 3082< 5A 7<0K18D3.H >2<2 5/5:7D2< 2

:.A>5162BC.7.7234./2:7.<45>.7G1:02420167232BC.E:.6ϕ VF µ

<B[ DA2 <545 7<0/=60:2 45 $$$ #$ !? 567= 30824. DA <5:5>7.< 7<0/=60:.:.16707DO4. >.< 7<G6 I.I0126 08D206 45 $ Ω 45 <52:7K1:02 5 $ Ω 45

<56067G1:0230824265A7<0K18D3.23:D35@

2F01751604245JD5>26625A:242I.I012

IF0175160424512301P2JD5230A5172.<5:5>7.<

:F/2:7.<45>.7G1:0242:2<82

4F:2>2:0424545:242DA4.6:.1451624.<5623082<5A7<0K18D3.45A.4.2

/2?5<2:.A>5162BC.7.7234./2:7.<45>.7G1:02E:.6ϕ VF µ

<G[

I65<;52/08D<25:23:D35@

2F017516042450140:242>53.2A>5<OA57<.

IF/2:7.<45>.7G1:0242:2<82

:F>.7G1:022:70;2

4F :2>2:04245 45 :242 DA 4.6 :.1451624.<56H 5A 7<0K18D3.H 45 A.4. 2

:.A>5162<7.723A51752:2<82E:.6ϕ VF µ



CENFIM 59

<H[AA.7.<7<0/=60:.75ADA2>.7G1:02S70345#Hη V($eH:.6ϕ V$H(#H:V$$

23:D35@

2F017516042452I6.<;042>53.A.7.<

IF :2>2:04245 45 :242 :.1451624.<H 30824.6 5A 7<0K18D3.H 45 A.4. JD5:.6ϕ V µ

:F017516042457.7232I6.<;04245>.0645:.A>51624..A.7.<

<\[

7516C.42<54542/08D<2L45$$

.A.0175<<D>7.<45630824..62>2<53P.6A2<:2;2A@#5(_W

.3082<HA2<:2<2A@5(_W23:D35@

2F/2:7.<45>.7G1:0242:2<82014D70;2

IF>.7G1:02<52:70;27<0/=60:242:2<82014D70;2E+010:023F

:F/2:7.<45>.7G1:02/01234.:0<:D07.

4F>.7G1:02<52:70;27.7234.:0<:D07.H:.A30824.E+/0123F

5F:2>2:0424545:242:.1451624.< µ



(%.' ;(

672 D104245 >5428980:2 ><575145D /.:2< .6 26>5:7.6 A206 <535;21756 <532:0.124.6

:.A2:.<<51752375<1242H7514.5A:.16045<2BC..8<2D452><5140?285A2JD565

4567012

;.'

ELECTROTECNIA-TEÓRICA E APLICADA, (ÁLVARO BADONI - PORTO EDITORA)

ELEMENTOS DE ELECTRICIDADE, (SIMÕES MORAIS - PORTO EDITORA)

ELECTROTECNIA – PROBLEMAS E ITENS (DIDÁTICA EDITORA-JOSÉ VAGOS

MATIAS)

ELECTROTECNIA – PROBLEMAS E ITENS – 2 (DIDÁTICA EDITORA-JOSÉ VAGOSMATIAS)

CATÁLOGO GERAL-91 (MERLIN GERIN)

ELECTROTECNIA (PARANINFO-PABLO ALCADE SAN MIGUEL)

(%.'

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