!
"
$
%
#
&
&
&
%
"
'
"
'
"
'
#
(
&
#
'
)
(
$
%
#
%
#
$
)
)
CENFIM
(
(
1
)
*
#$
+
+
,
,
-
-
#
#
&$
&$
&$
CENFIM
2
. /012345672
104245 5428980:2 45;5<= 65< :2>2?45@
.A><55145< 2 85<2BC. 45 DA2 :.<<5175 23
75<1242 E
F
45170/0:2< .6;23
.<56/D142A5172061DA2 :.<<5175 23
75<1242
.A><55145< . :.A>.<72A517. 426<56067G1:026H I.I01265 :.1451624.<56
JD214. 3
0824.61DA :0<:D07. 45 :.<<5175 23
75<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 <53
2BC. 2 .D7<. 70>. 45
60675A26
56.3
;5< :0<:D07.645 :.<<5175 23
75<1242
5<2BC. 42 :.<<5175 23
75<1242 601D6.0423
23
.<56/D142A51720642 :.<<5175 23
75<1242
0<:D07. :.A <56067G1:02 >D<2 5A
0<:D07. :.A I.I012 >D<2 5A
52:7K1:02 014D70;2
0<:D07. :.A :.1451624.< >D<. 5A
52:7K1:02 :2>2:070;2
0<:D07. 6L<05
0<:D07. 6L<05
0<:D07. 6L<05
.7G1:0265A
5<2BC. 42 :.<<5175 23
75<1242 7<0/=60:2
CENFIM
3
516M5660A>3
565 :.A>.6726
082BC. 45 :2<8265A 567<53
2 5 5A 7<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<2BM56H 45 /.<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 =3
85I<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<2 5N5:DBC. 4.65N5<:O
:0.6
.3
P2645 01/.<A2BC. 75:1.3
980:2
+D567M565 5N5<:O
:0.6
CENFIM
4
./
./
CENFIM
(
(
.0'
1
2
.0'
5
3
.
(
4
.
567D4. 42 515<802 53
L:7<0:2 JD5 /0?5A.6 1.6 :2>O
7D3
.6 2175<0.<56 266517.D 126
:.<<517565 7516M56:.17O
1D26H 067. LH 12JD53
26JD5 1C. ;2<02A 2. 3
.18. 4. 75A>.H
JD5 A217GA . A56A. 651704. E
D1040<5::0.1206F 5 . A56A. 651704.
N0675A 1. 517217.H 1DA5<.6262>3
0:2BM565A JD5 6C. AD07. 40;5<62626;21728516
5A /D1BC. 4. 75A>.H 4267516M56H :.<<517565 .D7<268<2145?26
660A 268<2145?26
53
L:7<0:26>.45A :3
2660/0:2< 65 5A /D1BC. 4. 75A>.H :.A.@
5 - 5# ! ,6 ,7
,89 ,
5 +:-6 ,
+: -6 ,
# ,5+ , ,
:
- 5
!+
,5+
,6 ,7
; '
. 8<=/0:.
75A>.
,
&
:.<<5175 <5><56517242 L:5<72A5175 :.1672175H >.< 1C. ;2<02 2. 3
.18. 4.
08
CENFIM
,
.<<5175 :.1672175
6
;
$'
&
4
0
:.<<5175 <5><56517242 >.66D0 ;23
.<5640/5<5175645 01672175 2 01672175 A2672AILA
. A56A. 651704.
08
;
<
.<<5175 ;2<0=;53D1040<5::0.123
0
A2 8<2145?2 40?65 >5<0940:2H JD214. 65 ;5<0/0:2 DA2 <5>570BC. 42 6D2 :.1/08D<2BC.
2. 3
.18. 4. 75A>.
.1426 >5<0940:26
. 567D4. JD5 0<5A.65/5:7D2<H 6D<80< 1.6C. 40;5<626/.<A2645
5><565172A.6 4.06 70>.6 45 .1426 >5<0940:26@ .14D3
279<026 5
>D3
6279<0265 2623
75<12426>D<26
08
;
=
0
08
CENFIM
2F
.<<5175 .14D3
279<02QIF
.<<5175 D1040<5::0.1235A 45175 45 65<<2
.
2F 516C. 23
75<1242 7<0218D3
2<QIF 516C. 23
75<1242 JD24<242Q:F
601D6.0423
.<<5175
7
DA2 .142 23
75<1242H . :.1R
D17. 4.6 ;23
.<56 266DA04.6 5A :242 DA 4.6 651704.6
4560812 65 >.<
:
- 5>57,
.D
,
5
660AH 75A.6 23
75<1K1:026 >.6070;26 5
23
75<1K1:026158270;26
:.1R
D17. 45 4D2623
75<1K1:026:.165:D70;264560812 65 7,7:
;23
.< 266DA04.H 5A :242 01672175H >.< DA2 :.<<5175 .D 7516C. L :P2A24. 9 :
,5 - 5->5
H JD5 65 <5><565172 >.< DA2 3
57<2 A01S6:D3
2H ,?+;
D3
82<A5175H 56756601206;C. 240:0.12< 65 2 7516M56.D :.<<51756:.17O
1D26>2<2 JD5
.653
5A517.62:70;.6<523
0?5A . .IR
5:70;. ><5751404.
660AH 2 7516C. <5><56517242 L 2 6.I<5>.60BC. 42 :.A>.15175 :.17O
1D2
E
1.75 65
JD5 2 8<2145?2 L<5><56517242 >.< DA2 3
57<2 A01S6:D3
25.O
140:5 LA20S6:D3
.F
+
@
A+
<5A.6 28.<2 7<272< 4. 567D4. 45 :.<<51756 5 7516M56 23
75<12426 601D6.04206
0A>.<7K1:02 12
3
5:7<910:2 <56D3
72 4. /2:7. 45 JD23
JD5< 60123>5<0940:. 23
75<124. 65
>.45< :.16045<2< :.A. 2 6.A2 45 601206 23
75<124.6 601D6.04206 45 /<5JDG1:026
AS3
70>3
26
.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. 45
DA2 / 5 A
601D6.0423
CENFIM
8
.
.
.
.
>.7G1:02 ><.4D?042 >.< 5/507. 45 .D3
5 L><.>.<:0.1232. JD24<24. 42 01751604245
45 :.<<5175H 3
.8. 0145>5145175 4. 65D651704. 45 :0<:D3
2BC. T/2:7. JD5 2 ><.4DBC.
45 515<802 :23
.<O
/0:2 L ;2<0=;5345 01672175 2 01672175H 21D3
214. 65 A56A. 4D26
;5?562. 3
.18. 45 DA >5<O
.4.Q1. 517217.H 5 45;04. = 01L<:02 7L<A0:2 4.6:.<>.6H
26;2<02BM5645 75A>5<27D<2 6C. AD07. A2064LI506
.4.6.62>2<53
P.67L<A0:.665<C. D703
0?=;506H 7217. 5A :.<<5175 :.17O
1D2H :.A. 5A
:.<<5175 23
75<1242
/<5JDG1:02 72AILA 01/3
D51:02<= . /D1:0.12A517. 4.6
2>2<53
P.6H >.< 5N5A>3
.H 4263
KA>242645 01:21456:G1:02
5 >.< ;517D<2H 2 /<5JDG1:02 /.< 45A26024. I20N2 E
01/5<0.< 2 # !?FH 2 75A>5<27D<2
4263
KA>2426;2<02<= 3
5172A5175 5 65<02 1.79<02 DA2 :5<72 :01703
2BC.
2 /08D<2 ' <5><565172 65 DA2 :DI2 53
5:7<.3
O
70:2 :.A 6D3
/27. 45 :.I<5H DA2 I.I012
>3
212 .<0517242 12 40<5:BC.
:.A DA2 28D3
P2 A281570?242H DA2 I.I012 :.A
DA2 >5B2 45 /5<<. A2:0. 5 /0.645 ><272 56751404.65 >2<23
53
.6
3
0A517214. . :0<:D07. 2 :.<<5175 :.17O
1D2H P2;5<= 7<216>.<75 45 :.I<5 >.<
53
5:7<93
065 5 2:BC. 42 I.I012 6.I<5 2 28D3
P2 A281570?242H JD5 ;2<02A 45 651704.
:.A . 651704. 42 :.<<5175 1. :0<:D07.
5/507. 45 .D3
5 1.6:.14D7.<5645 ><272H 2 27<2:BC. 42 I2<<2 45 /5<<. 5 2 <5>D3
6C.
517<5 .6/0.645 ><272 /2?65 65170< 0145>5145175A5175 4. 651704. 42 :.<<5175 1.
:0<:D07.
08' /507.642
:.<<5175 53
L:7<0:2
. 2>3
0:2< 65 :.<<5175 23
75<1242H 1C. P= 7<216>.<75 45 :.I<5H 2 28D3
P2 1C. 65 456;02H
2><56517214. DA2 3
0850<2 ;0I<2BC. 12 6D2 >.60BC.
.< .D7<. 3
24.H 723:.A. 5A
:.<<5175 :.17O
1D2H .6/0.62JD5:5A 5 <5>53
5A 65H 5 2 I2<<2 45 /5<<. L27<2O
42
28D3
P2 A281L70:2 5A :.<<5175 23
75<1242 72AILA L 6.3
0:07242 >.< /.<B26
53
5:7<.A281L70:26 265662 6.3
0:072BC. AD42 45 651704. $$ ;5?56>.< 658D14.H >53
.
JD5H 45;04. = 01L<:02H >5<A215:5 ><270:2A5175 5A <5>.D6.
C
T. 75A>. 5A JD5 45:.<<5 4D2623
75<1K1:026:.165:D70;26H .D65R
2H L. 75A>. 8267.
1DA :0:3
. 5><565172 65 >.< 5 5N><5662 65 5A 658D14.6
CENFIM
9
*+D57, E
T. 1SA5<. 45 :0:3
.65/5:7D24.61DA 658D14. 5><565172 65 >.< E 5 2 6D2 D104245
L . !? E
P5<7?F
.75 65 JD5 2 /<5JDG1:02 5 . >5<O
.4. 567C. <53
2:0.124.6 1DA2
><.>.<BC. 01;5<62
A :0:3
. 45A.<2 2 <523
0?2< 65 1DA >5<O
.4. Q 3
.8. E :0:3
.6
45A.<2<C. 658D14.
E@ %
6/<5JDG1:026426.142645>5145A 4266D26D703
0?2BM56 660AH 2 515<802 53
L:7<0:2
L 4067<0IDO
42 2 #$ !?
82A2 426 2D40./<5JDG1:026 ;20 45 $ !? 2 $ U!? 5
:.A>.<72 . JD5 ;D3
82<A5175 65 4560812 >.< 53
5:7<.2:S670:2 =40.H 753
5;06C.H D3
7<2
6.16H <242< 5 A0:<..1426 :.A>.<72A 82A26 45 /<5JDG1:026 JD5 D3
7<2>2662A .6
!?E
A582 P5<7?F 5H >.< ;5?56H .6 !?E
8082 P5<7?FH
08(
F
5<O
.4. 45 8<2145?2 601D6.0423
!:
575<A012< . >5<O
.4. 45 DA2 .142 23
75<1242 45 $ U!?
@
%E
@
%3?
3B F 3 <
@ B3 µ
!:
,-+
T . ;23
.< 0167217K15. A20653
5;24. 2701804. >53
2 8<2145?2
>.< 9 :
2AILA 65 4560812
8F,
!= 2A>3
07D456>.6070;265 158270;26
. ;23
.< A5404. 517<5 .6;23
.<56A=N0A.6>.6070;. 5 158270;. :P2A2 65 9 :
!,7
!,7
2 /08D<2 *H 75<5A.6@
!! @
CENFIM
$F
8F;
10
08*
:
5><565172BC. 42 2A>3
07D45 45 DA2 :.<<5175 601D6.0423
),
5<5A.62JD0 JD5 :.16045<2< 2>5126A57245 4. :0:3
. 45 DA2 :.<<5175
23
75<1242 601D6.0423
H >.06. ;23
.< AL40. 45 DA :0:3
. L?5<. 5><565172 .
;23
.< JD5 DA2 :.<<5175 :.17O
1D2 45;5 >.66D0< >2<2 7<216>.<72< 1.
A56A. 75A>.H 2 A56A2 JD21704245 45 53
5:7<0:04245
@ $ %π
5A.167<2 65 JD5@
08 $
8FV
3;
G<H
8F
23
.< AL40. 45 DA2 :.<<5175 601D6.0423
5 :.16045<2<A.628.<2 7516M5623
75<12426601D6.04206H 75<5A.6@
)
CENFIM
@ $ %π
F@
3?
G<H
8F
11
:
E,7 I
= <5/5<0A.6 JD5 . :23
.< 45651;.3
;04. 12 <56067G1:02 >.< 5/507. 45
0145>5145175 4. 651704. 45 :0<:D3
2BC. 42 :.<<5175
.D3
5 L
5675 A.4.H 5N0670<= DA2 :.<<5175 :.17O
1D2 JD5 1. A56A. 0175<;23
. 45 75A>.
H .D 65R
2H DA >5<O
.4.H ><.4D?0<= 2 A56A2 JD21704245 45 :23
.< JD5 2 ><.4D?042
>53
2 :.<<5175 23
75<12H
5662 JD21704245 :P2A2 65 9 :
E,7 I 5 <5><565172 65 >.<
275A270:2A5175H :P582 65 = :.1:3
D6C. 45 JD5 . JD24<24. 42 01751604245 5/0:2?
L08D232. ;23
.< AL40. 4. JD24<24. 45 ,H .D65R
2@
$
@
$
8F % $
@
8F % √$
@ 3?
H3H
8F
5 5A ;5?45 :.<<51756<5><565172<A.67516M56H ;0<= 45 08D23A.4.@
$
@
$
8F % $
.< 7217.H . 9 :
@
E,7 I
8F % √$
+
@ 3?
H3H
7
5-
8F
:
- 5
L. ;23
.< 42 01751604245
JD5 45;5<02 75< DA2 :.<<5175 :.17O
1D2 >2<2H 1DA2 <56067G1:02 ><.;.:2< . A56A.
5/507. :23
.<O
/0:.H 1. A56A. 0175<;23
. 45 75A>.
2<2 <523
B2< 2 0A>.<7K1:02 4. ;23
.< 5/0:2?H <5/0<2 65 JD5 6C. ;23
.<565/0:2?56JD5
.6;.3
7O
A57<.65 2A>5<O
A57<.61.60140:2A 2. A540<5A 8<2145?26601D6.04206
F
!: $
6 µs
08
CENFIM
516C. 601D6.0423
12
:-,9
5-
J- 5 #
,5+
,
: !
5-
5
$;
2 /<5JDG1:02 5 . >5<O
.4.
$;
$
. ;23
.< AL40. 45 DA2 23
75<1K1:02
$;
<
. ;23
.< 5/0:2?
$;
=
. >.17. 45 45A.<2 2 270180< . ><0A50<. >0:.
,K;
?
-
,5
:
+"#
$; & µ6V
U!?
$;
$
A=N V
H#
V&
#
AL4 V
V
$;
=
V H µ6
A=N
&
H µ6 / V
6V
$H& ' N
A=N V
V
H N $
&
V
'
$H& ' N# V H (#
√ V # N$H'$' V H# #
$;
<
><0A50<. >0:. .:.<<5 2
F
H# N $
H .D65R
2H 2>96$H& µ 6V &$$ 16
!: <
06>.A.645 DA2 <56067G1:02 45
$ UΩ 5
( WH
575<A012<@
<;
. ;23
.< 5/0:2?42 A=N0A2 01751604245 45 :.<<5175 JD5 2 >.45 >5<:.<<5<Q
<;
$
2 2A>3
07D45 42 A=N0A2 7516C. 2 JD5 >.45 65< 6DIA57042
:
+"#
<;
⇔
V
V
V $H& # N $
<;
$
V $H& # A
√
V& #µ
V√
A=N V
' 4
L
(
5α@
(N $N $ V &H #N $
Q
$N $
( V $$
( H*
'
(
&
.
.
651. 4. K18D3
. JD5 . ;5:7.<
CENFIM
V√
V√
V
A=N V
514.
V√
/2?:.A . 50N. 4262I:06626L@
%
@
8F
,@
8F ;
5α
13
@ , H ;0<=H >.< 6DI6707D0BC. >2<2 ;23
.< 0167217K15. 42 :.<<5175@
,@
5α
8F
51. 4. K18D3
.αE
08
.
F
' .
;5:7.<
.
456:<5;5 DA K18D3
. 45 $π
π <24021.65A :242 ;.3
72H .D65R
2H 5A :242
>5<O
.4.
5/015 65 ;53
.:04245 218D3
2< :.A. 6514. . 1SA5<. ϖ 45 <24021.6 >5<:.<<04.6
>.< 658D14.
660A@
$π
π
ϖ
75<5A.6
ϖ @ $π
π%
.D ϖ @ $π
πE
5N><5662
5A
<24021.6
658D14.
.45A.6H 28.<2H 8515<23
0?2< 2 /9<AD3
2 <5><56517270;2 42 :.<<5175 601D6.0423
H
4575<A01214. . ;23
.< 4. K18D3
. α
>5<:.<<5< . K18D3
. α 5 DA >5<O
.4.
α @ $π
π ;- %
0<= @
,@
660AH . ;5:7.< 45A.<2 DA 75A>. 7 2
2 >5<:.<<5< π <24021.6 .8.@
.D20142
8F
α@ω-
5 ω-
672 /9<AD3
2 75<=H 20142H 45 75< 5A :.172 . K18D3
. JD5 . ;5:7.< /.<A2 :.A 2
.<085A 42 :.17285A 4.6 K18D3
.6H 1. 01672175 010:023
E
5675 K18D3
. :P2A2 65
;
/9<AD3
2 <5><56517270;2 45 DA2 :.<<5175 65<=@
CENFIM
@
8F
1ω-
ϕ2
14
;
G
.
(
.
A 85<24.< 53
5A5172< L :.16707DO
4. >.< DA 2153:.14D7.< :.3
.:24. 45 723/.<A2 JD5
>.662 65< A.;0A51724. 1. 650. 45 DA :2A>. A281L70:. /0N.
:2A>. A281L70:. L:<024. >.< DA O
A21 >5<A215175
65 >.<
!,
.A. 65 ;5<0/0:2H .
215345 /0. :.14D7.< 4560812
.D,5 +I,
26 5N7<5A042456 45672 56>0<2 567C. 242>724.6 4.06 21L06H :.16707DO
4.6 20142 >.<
A275<023:.14D7.< 5 JD5 65 4560812A >.<
5),
7 :7-
;
3
082BC. 2. 5N75<0.< L
/5072 >.< A50. 45 4D26>5B26>5</5072A5175 242>7=;5062.621L06:.3
5:7.<5645 /.<A2 2
5672I53
5:5< I.A :.172:7.H . JD5 65 :P2A2A
79
1.<A23
A5175 :.167<DO
4265A
8<2/075
123
065A.6. /D1:0.12A517. 45 DA 85<24.<
08
5<24.< 53
5A5172<
. <.42< 1. 651704. 0140:24.H 2 56>0<2 ;20 :.<72< 26 3
01P26 45 /.<B2 4. :2A>.
A281L70:.H ><.4D?04.X 65 153
2 DA2 /.<B2 53
5:7<.A.7<0? 014D?042 JD5 ><.;.:2 2
:0<:D3
2BC. 45 DA2 :.<<5175 27<2;L64. :0<:D07. 5N75<0.<
. 4563
.:2< 65 42 >.60BC.
>2<2 2 >.60BC.
H 2 56>0<2 :.<72 DA 1SA5<. A=N0A. 45
3
01P2645 /.<B2
53
. JD5 2 / 5 A
014D?042 270185 DA ;23
.< A=N0A. 2>9665 75< 4563
.:24. *$Y
.75
65 JD5 7217. 12 >2<75 I<21:2 :.A. 12 ><572H 6D<80<C. / 5 A JD5 ;C. 240:0.12< 65H >53
.
JD5 >.45 :.1:3
D0< 65 JD5 2 :.<<5175 1. :0<:D07. 5N75<0.< 65<= 7217. A206 53
5;242H
JD217. A20.< /.< . 1SA5<. 45 :.14D7.<56JD5 :.<72A . :2A>. A281L70:.
+D214. 2 56>0<2 :.1701D2 2 <.42< 27L .:D>2< 2 >.60BC. 01;5<62 Z >.60BC. 45 >2<7042H
;20 :.<7214. DA A51.< 1SA5<. 45 3
01P26 45 /.<B2 27L . 65D >3
21. 65 406>.<
>5<>5140:D3
2<A5175 Z6<5/5<04263
01P26H 450N214. 45 65< 6545 45 / 5 A 014D?042
CENFIM
15
08
08 #
<.4DBC. 45 / 5 A 601D6.0423:.A 4563
.:2A517. 45 *$Y
5 A 601D6.0423><.4D?042 :.A 4563
.:2A517. 45 *$Y2 ($Y
>2<70< 45675 A.A517.H . 651704. 42 / 5 A 014D?042 12 56>0<2 L:.17<=<0. 2. 2175<0.<
. 517217.H 5 2 A51.64. 651704.H 7D4. 65 >2662 4. A.4. 456:<07.
270185 DA ;23
.< A=N0A. 2>962 <.72BC. 45
/ 5 A 014D?042
'$YH :.A5B214. 2 40A01D0< 27L65 21D3
2<
JD214. . >3
21. 42 56>0<2 65 :.3
.:2 >5<>5140:D3
2<A5175 Z63
01P2645 /.<B2 4. :2A>.
P.A.8L15.H . JD5 6D:545 2>96DA2 <.72BC. :.A>3
572H ;.3
7214. Z>.60BC.
08 &
CENFIM
5 A .I7042 >53
2 <.72BC. 517<5 ($Y5 &$Y
16
.< :242 <.72BC. 42 56>0<2 .I75< 65 = DA2 601D69045
2A>3
07D45 42 / 5 A .I7042
65<= 7217. A20.<H JD217. A20.< /.< . 1SA5<. 45 56>0<26 JD5 :.16707D0 . 014D?04.
.80:2A5175H 2 /<5JDG1:02 45>5145<= 42 ;53
.:04245 45 <.72BC.H
456:<0BC. JD5 2:2I2A.6 45 5/5:7D2< :.14D?0<= 2.6 A56A.6 <56D3
724.6 65 28.<2
406>D65<A.6 45 DA <97.< E
>2<75 A9;53
F :.16707DO
4. >.< DA O
A21 E
53
5:7<.O
A21
23
0A51724. 5A :H:HF 5 DA 567=7.< E
>2<75 /0N2FH :.16707DO
4. >.< I.I0126.145 ;C. 6D<80<
26 / 5 A 014D?0426H >.< A.;0A5172BC. 4. <97.<
><.4D?0< :.<<5175 23
75<1242 601D6.0423
H :P2A2 65
08 '
4.
. 406>.6070;. 456:<07.H :2>2? 45
:
- 5
;
<.4DBC. 45 DA2 8<2145?2 601D6.0423
(
;
.
5 2123
062<A.6 2 5N>5<0G1:02 45 ;5<0/0:2BC. 42 50 45 PA A26 2>3
0:214. 28.<2
8<2145?2623
75<12426H :P582<5A.6Z:.1:3
D6C. JD5 65 A217LA :.1672175 . JD.:05175
%
5675 JD.:05175 :P2A2<5A.6 , ! >57, 4. :0<:D07.H 2. JD232>3
0:2A.6 2
7516C. 23
75<1242 5 JD5 65 <5><565172 >.< &
6D2 D104245 L.I;02A5175 . M
660AH 2 . ,
M
266DA5 2 /.<A2H JD5 L456081242 >.< 50 45
PA 8515<23
0?242
@ &;
40/5<51B2 517<5 & 5
45;5 65 2. /2:7. 45 & 45>5145< 42 /<5JDG1:02
660AH 5A
:.<<5175 23
75<1242H 2 <53
2BC. 517<5 2 7516C. 5 2 :.<<5175 45>5145H >2<2 DA2 4242
/<5JDG1:02H 42 0A>54K1:02 & 5 4. K18D3
. 45 456/262A517. ϕ
.< 45/010BC. 4560812< 65 =@
:.6ϕ
>.<
, -D57,
651ϕ
>.<
7->57, N
CENFIM
17
08 (
<0K18D3
. 4260A>54K1:026
58D042A5175 567D42<5A.6 .6 :0<:D07.6 5A JD5 6D<85A :.<<51756 23
75<12426
601D6.04206H JD5 6C. /.<A24.6>.< <56067G1:026H I.I01265 :.1451624.<56
5<5A.6H 1. 517217.H 5A ><0A50<. 3
D82<H .6:0<:D07.6045206H .D 65R
2H .6:.16707DO
4.6
>.< <56067G1:026H >.< I.I0126>D<26E
65A <56067G1:02F 5 >.< :.1451624.<56>D<.6E
65A
<56067G1:02645 >5<426F
23:.A. 6D:545 12 <523
04245
. 517217.H 23
8DA26 456726 7<G6 8<2145?26H JD5 /.<A2A .6 53
5A517.6 <5206
E
<56067G1:02H <52:7K1:02 014D70;2 5 <52:7K1:02 :2>2:070;2FH 266DA5A ;23
.<567C. I20N.6
JD5 >.45A 456><5?2< 65 /2:5 2.6<56721756
T. :26.H >.< 5N5A>3
.H 4263
KA>242645 01:21456:G1:02H JD5 >.45AH 65A 8<2145 5<<.H
65< :.16045<2426:.A. <56067G1:026>D<26
(
. 2>3
0:2<A.6DA2 7516C. 23
75<1242 601D6.0423Z<56067G1:02 >D<2A5175 9PA0:2H
JD2365<= 2 /.<A2 45 .142 42 :.<<5175 1. :0<:D07.[
>3
0:214. 2 50 45
PA 2.66D:5660;.60167217565 DA2 ;5?JD5
V
HR
= JD5 2
<52:7K1:02 L1D3
2H /2:03
A5175 65 ;5<0/0:2 JD5@
\ A54042 JD5 2 7516C. 2DA5172H 2 :.<<5175 72AILA . /2<=H R
= JD5 65
<53
2:0.12A >53
2 50 45
PAH
V
Q
+D214. 2 7516C. 2>3
0:242 AD42 45 >.3
2<04245H 72AILA 2 01751604245 45
:.<<5175 AD42 45
CENFIM
5-,
;
18
08 *
56067G1:02 >D<2 23
0A517242 5A
.8. 26:D<;26<5><56517270;2642 7516C. 5 42 :.<<5175 567C. 5A /265H .D65R
2H 2
DA A=N0A. 42 7516C. :.<<56>.145 2 DA A=N0A. 45 :.<<5175H . A56A.
6D:54514. >2<2 .6?5<.6
K 6M ,7
. 65< 3
08242 2 DA2 <545 45 :.<<5175 23
75<12 D12 <56067G1:02 E
> 5N D12 3
KA>242
45 01:21456:G1:02FH 2 01751604245 45 :.<<5175 JD5 :0<:D3
2 75A 2 A56A2 /.<A2
601D6.0423JD5 2 7516C. 42 <545 +D214. 2 7516C. >2662 >53
. ;23
.< ?5<.H 15675
01672175 1C. :0<:D3
2 :.<<5175 E
6D2 01751604245 L?5<.F +D214. 2 7516C. 23
:21B2
. ;23
.< A=N0A.H 2 01751604245 L08D23
A5175 A=N0A2
0?65 517C. JD5 2 7516C.
5 2 01751604245 567C. 5A /265 E
/08 *F
. :26. 45 :2<826 9PA0:26H 2 7516C. 5 2 01751604245 45 :.<<5175 ;2<02A
60AD3
72152A5175
08 $
CENFIM
516C. 5 01751604245 5A /265
19
(
(
6 I.I0126 567C. ><5651756 5A 7.4.6 2JD53
56 <5:5>7.<56 1.6 JD206 65R
2
15:566=<0. 2 ><.4DBC. 45 DA :2A>. A281L70:.
.< 5N5A>3
. 1.653
5:7<.O
A2156H
:.172:7.<56H A.7.<56H <52:7K1:026 45 2<<21JD5 45 3
KA>2426 45 456:2<82
E
/3
D.<56:51756H ;2>.< 45 A5<:S<0.H ;2>.< 45 6940.H 57: FH 7<216/.<A24.<56H 57:
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<5A DA2 4575<A01242 <56067G1:02
7
5-
7 5-C
5+
53
082<A.6D12 I.I012 2 D12 7516C. :.17O
1D2H :0<:D3
2<=
>.< 53
2 DA2 :.<<5175 53
L:7<0:2 JD5 65<= D10:2A5175 3
0A07242 >53
2 <56067G1:02
58D14. 2 3
50 45
PA@
@
%
24. JD5 5672 <56067G1:02 65< >5JD512H 5 65
2>3
0:2A.6D12 7516C. 53
5;242 ZI.I012H :0<:D3
2<= >53
2 A56A2 DA2 :.<<5175 :.A
DA2 01751604245 53
5;242H . JD5 .<08012<= DA2 /.<75 >.7G1:02 40660>242H JD5
>.45<= 4567<D0< 2 I.I012 45;04. 2. :23
.< 85<24.
7
5-
:
- 5
53
082<A.62 A56A2 I.I012 2 D12 7516C. 23
75<12H >.45
65 :.A><.;2< 5N>5<0A51723
A5175 JD5 2 :.<<5175 JD5 :0<:D3
2 >53
2 A56A2
2><565172 DA ;23
.< A51.<
5 3
082<A.6 DA ]277O
A57<. >.45<5A.6 :.A><.;2<
JD5 . :.16DA. 45 >.7G1:02 L ><270:2A5175 1D3
.H 2>562< 42 5N067G1:02 45 D12
:5<72 :.<<5175
660A >.45<5A.6 :P582< Z :.1:3
D6C. JD5 2 I.I012 2><565172
DA2 :5<72 .>.60BC. = >266285A 42 :.<<5175 40/5<5175 4. :26. 2175<0.< 5A JD5 2
.>.60BC. 5<2 45;042 Z<56067G1:02 .PA0:2
.4.656756/519A51.645;5A 65 2. 5/507. 45 2D7. 014DBC. 42 I.I012@
+D214. 2 I.I012 L>5<:.<<042 >.< DA2 :.<<5175 23
75<1242H 2>2<5:5 DA2 :.<<5175
;2<0=;53
H 5 >.< 7217. DA :2A>. A281L70:. 72AILA ;2<0=;53 24. JD5 263
01P26
45 /.<B2 4. /3
DN. A281L70:.H JD5 53
2 A56A2 85<2H :.<72A 2 6D6 ><9><0.6
:.14D7.<56H 6D<85 D12 / 5 A 45 2D7. 014DBC. JD5H 658D14. 2 3
50 45 51?H 65 ;20
.>.< Z:2D62 JD5 3
P5 45D.<085A
08
/ 5 A 45
2D7. 014DBC. 42
I.I012 .>M5 65 Z
:.<<5175
CENFIM
20
+D214. 2 :.<<5175H 658D014. 26;2<02BM5642 /D1BC. 601D6.0423
H 75145 2 :<56:5<H
. :2A>. A281L70:. 72AILA :<56:5
>2<5:5 517C. DA2 / 5 A
JD5 65 .>M5 2 JD5 2 :.<<5175 65 5672I53
5B2H
><.;.:214. DA 27<26. 12 :.A5175 53
L:7<0:2 5A <53
2BC. Z 7516C. E A2 I.I012
2. 65< 3
08242 2 DA2 7516C. 23
75<1242H 2 7516C. 2>2<5:5 45 0A54027. 2.665D6
75<A01206F
K ,5 +-,9
5 2 DA2 <545 45 :.<<5175 23
75<12 /.< 3
0824. DA <5:5>7.< 014D70;.H ;20 42<
.<085A 2 DA 456/262A517. 517<5 2 7516C. 5 2 01751604245
672 S3
70A2H 5A
;0<7D45 42 2D7. 014DBC.H 5672<= 27<26242 5A <53
2BC. Z7516C.
675 456/262A517.H :.A. 7.4. . K18D3
.H >.45 65< A5404. 5A 8<2D6
. :26. 45
D12 :2<82 014D70;2 >D<2H 5A JD5 <56067G1:02 9PA0:2 L1D3
2H . 456/262A517. 65<=
45 *$^
. :26. 45 :2<82 014D70;2 >D<2H 2 01751604245 45 :.<<5175 5175 27<26242 *$^E
45 >5<O
.4.F 5A <53
2BC. Z7516C.
08
CENFIM
028<2A2 ;5:7.<02345
5
1DA2 I.I012 >D<2
21
5675 :26. 40?65 JD5 2 01751604245 567= 456/26242 5A 27<26. 5A <53
2BC. Z
7516C. 5A DA JD2<7. 45 :0:3
.
O
(
.A. 2 .>.60BC. 2><56517242 >53
2 I.I012 Z:.<<5175 23
75<1242 75A 2 ;5< :.A .6
/519A51.645 2D7. 014DBC.
672 65<= A20.< JD217. A20.< 65R
2 . :.5/0:05175 45
2D7. 014DBC. .?5 A206<=>042665R
2A 26;2<02BM5642 :.<<5175 23
75<1242
2 I.I012 Z
P2A2A.6 <52:7K1:02 014D70;2 N. Z .>.60BC. 2><56517242 >53
:.<<5175H >53
. JD5@
X L = 2πfL
- V
52:7K1:02 014D70;2 5A .PA6EΩ F
/ V <5JDG1:02 5A P5<7?E
!?F
V
.5/0:05175 45 2D7. 014DBC. 5A P51<06E
!F
P
(
23:.A. 0140:=A.62. ><01:O
>0. 45675 75A2H A545 65 2 >.7G1:02 45 DA2 I.I012
>D<2H >.45 65 :.A><.;2< JD5 2 3
507D<2 27<2;L6 45 DA ]277O
A57<. 0140:2 DA2
>.7G1:02 08D232 ?5<.
. :.17<=<0. 4. JD5 .:.<<5 1DA2 <56067G1:02H 1DA2 I.I012 >D<2 1C. 65 ><.4D?
151PDA :.16DA. 45 515<802 :23
.<O
/0:2
:.<<5175 JD5 >5<:.<<5 2 I.I012 65<;5 D10:2A5175 >2<2 85<2< . :2A>.
A281L70:.
2 <523
04245 . JD5 .:.<<5 L JD5H 2. :<56:5< 2 :.<<5175 >53
2 I.I012H 72AILAH
2:.175:5 . A56A. 2. :2A>. . :2A>. A281L70:.H ><.4D?014. 65 DA :.16DA. 45
515<802 53
L:7<0:2
5675 :26. 2 515<802 /3
S0 4. 85<24.< 27L Z I.I012 5 L JD214.
40?5A.6JD5 5672 567= 2 :.16DA0< 515<802 53
5:7<.A281L70:2
A2 ;5?23
:21B242
2 :.<<5175 A=N0A2 5 . /3
DN. A=N0A.H 56756 75145A 2 40A01D0< 658D014. 2
7<2R
5:79<02 601D6.0423
H 45651<.3
214. 65 DA2 / 5 A 45 2D7. 014DBC. 45 723651704.
JD5 85<2 DA2 515<802 53
L:7<0:2 JD5H 28.<2H /3
S0 45645 2 I.I012 27L 2. 85<24.<
5675 :26.H 2 I.I012 45;.3
;5 2 515<802 2. 85<24.<
5672 A2150<2 40?5A.6JD5 2
I.I012 1C. :.16.A5 <523
A5175 2 515<802H A2660A>3
56A5175 2 7.A2 5A><567242
4D<2175 DA JD2<7. 45 :0:3
. >2<2 85<2< . 65D :2A>. 53
5:7<.A281L70:.H >2<2 2
45;.3
;5< 1. JD2<7. 45 :0:3
. 658D0175
CENFIM
22
24. JD5 . ]277O
A57<. A545 . ;23
.< AL40. 42 >.7G1:02 5 5672 L6 >.6070;2
4D<2175 DA JD2<7. 45 :0:3
. 5 158270;2 1. 658D0 175
675 1C. 0140:2 151PDA2
>.7G1:02
660A 2 I.I012 1C. :.16.A5 515<802
6 :.16721756 :2<826 5 456:2<826 42
A56A2 .<08012A :.A JD5 5 :0<:D3
5 DA2 4575<A01242 :.<<5175 1.6:.14D7.<56H
>.< 7217.H 72AILA 2>2<5:5 DA2 >.7G1:02H 2 JD5 :P2A2<5A.6>.751:02 <52:70;2
E
+2
+
, 7+,-
,5
L @ N.
$
1
2 19 :
-
!
7-,9 2
.
A2 I.I012 <5232><565172 <56067G1:02 9PA0:2 45;04. Z6D2 :.16707D0BC.
Z = R 2 + X L2
08
.I012
660A >.45A.6 .I75< DA :0<:D07. 04G170:. 2. 42 /08D<2
:.16045<2< . ;23
.< 42 <56067G1:02
5675 :26. 45;5<5A.6
5 . 42 <52:7K1:02 014D70;2 N. 65>2<24.6
08
CENFIM
23
. :0<:D07. 014D70;. 2 :.<<5175 27<26242 45 DA K18D3
.
. :26. 4. :0<:D07.
ϕ
5A <53
2BC. Z7516C.
H . 456/262A517. ;2<02 517<5 $Y5 *$Y
P
08 #
S=U.I
(VA)
P = UR . I = U . I COS ϕ
(W)
QL = XL . I = U . I SEN ϕ
(VAr)
.7G1:0261DA2 I.I012
'
(
(
6:.1451624.<561C. 6C. <5:5>7.<56JD5 65R
2A D703
0?24.6P2I07D23
A5175 :.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;26 426
I.I0126
7
5- 7 5-,5+ H 69 :0<:D3
2<= :.<<5175 53
L:7<0:2 >53
. :0<:D07. 4D<2175 2
:2<82 4. :.1451624.<
5672 /.<A2H >.45 65 40?5< JD5 DA :.1451624.< 1C.
>5<A075 2 >266285A 42 :.<<5175 :.17O
1D2
7
5:
- 5
H . :.1451624.< /0:2<= :2<<5824. 23
75<1242A5175
>5<A07014. 2660A :.<<5175 1. :0<:D07.
CENFIM
24
08 &
028<2A2 ;5:7.<02345
5
1DA :.1451624.<
:2<82 :2>2:070;2 E
:.1451624.<FH JD214. 0165<04. 1DA :0<:D07. 45 :.<<5175
23
75<1242 .<08012 DA 456/262A517. 517<5 7516C. 5 01751604245
:.<<5175
2>2<5:5 5A 2;21B. 5A <53
2BC. Z7516C.
. :26. 42 :2<82 :2>2:070;2 >D<2H 2 01751604245 45 :.<<5175 2402172 *$^E
45
>5<O
.4.F 5A <53
2BC. Z7516C.
O
(
A :.1451624.<H 5A
H >5<A075 JD5 /3
D2 :.1672175A5175 DA2 :.<<5175
53
L: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
75
5
P2A2A.6
-
9
, :
)7- ,7
H A2660A >53
.6:.14D7.<564. :0<:D07.
7->57,
7 ! 7,-,9
75
5
N
Z .>.60BC. JD5 .
:.1451624.< 2><565172 Z:.<<5175
-: V
52:7K1:02 :2>2:070;2
5A .PA6
1
XC =
2πfC
/ V <5JDG1:02 5A P5<7?
V
2>2:04245 4.
:.1451624.< 5A /2<246
08 '
P
52:7K1:02 :2>2:070;2
(
8D232. JD5 .:.<<5 :.A 2 I.I012H 65 /.< A54042 :.A DA ]277O
A57<. 2 >.7G1:02
45 DA :.1451624.<H >.45 65 :.A><.;2< JD5 2 >.7G1:02 0140:242 L08D232 ?5<.
DA :.1451624.< 1C. 5N0675 :.16DA. 45 515<802 2:70;2
JD2<7. 45 :0:3
. . :.1451624.< L 7
CENFIM
K
7
5
K,
D<2175 . ><0A50<.
:
)7- ,7
E
25
7 K
:7-
:.1451624.<
-8-,7 H >53
. JD5 2 515<802 /3
S0 4. 85<24.< 45
>2<2 .
. JD2<7. 45 :0:3
. 658D0175 . :.1451624.< 456:2<<582 >2<2 .
:0<:D07. 27L2. 85<24.<H 45;.3
;514. 2. A56A. 2 515<802 2:DAD3
242
2AILA 2>2<5:5 D12 >.7G1:02 <52:70;2 L7><.4D?042 >53
2 515<802 JD5 L7<.:242
517<5 . :.1451624.< 5 . 85<24.<
DA :.1451624.<@ +: V -:
08 '
E
F
E;.3
7 2A>5<5 <52:70;.6F
0<:D07.
5675 :26. . 456/262A517. ;2<02 517<5 $^5 *$^E
/08 'F
R = Z COS ϕ
XC = Z SEN ϕ
Z2 = R2 + X2C
UR = U COS ϕ
UC = U SEN ϕ
U 2 = U R 2 + U 2C
S = U . I (VA)
P = U .I . cos ϕ (W)
QC = U .I . ses ϕ (VAr)
S2 = P2 + Q2C
08 (
CENFIM
26
.
5 3
082<A.6 DA2 <56067G1:02H DA2 I.I012 5 DA :.1451624.< .I75A.6 DA
7, 7+,-
. ;
08 *
0<:D07.
5675 :26. 2 0A>54K1:02 4. :0<:D07. ;5A@
Z2 = R2 + X2
Z2 = R2 + (XL – XC)2
X L = 2πfL
XC = −
1
2πfC
X = 2πfL −
1
2πfC
+D214. - V $H E
- V - FH P= <566.1K1:02 1. :0<:D07.
5675 :26. 2 0A>54K1:02 4. :0<:D07. L08D232
V
08 $
CENFIM
27
;
H
P
(
P
.
(Q
O
A :.<<5175 23
75<1242 2 7516C. 5 2 :.<<5175 ;2<02A 2 :242 01672175 >53
. JD5 2
>.7G1:02 40660>242 72AILA 65<= ;2<0=;53
5/015 65 >.7G1:02 0167217K152 :.A. 6514. . ><.4D7. 4.6;23
.<5645 + 5 ,H 5A
:242 01672175
0/(
5<0/0:2 65 JD5 2 >.7G1:02 0167217K152 L65A><5 >.6070;2
>.7G1:02 LA=N0A2
JD214. /.<5A A=N0A.6 2 7516C. 5 2 :.<<5175H 21D3
214. 65 JD214. D 5 0 65
21D3
2A
08
6;23
.<560167217K15.645
@
@
(8F
(8F
5
6C. 424.6>53
265JD2BM56@
5 R5 R-
>.7G1:02 LH >.<7217.H 65A><5 >.6070;2H JD23
JD5< JD5 65R
2 . 01672175
:.16045<24. 5 . 65D8<=/0:. 45651;.3
;5 65 65A><5 2:0A2 4. 50N. 4.675A>.6
6012365A><5 >.6070;. 42 >.7G1:02 266.:0242 2 JD23
JD5< <5:5>7.< 9PA0:. 60810
/0:2 JD5 53
5 <5:5I5 :.1701D2A5175 515<802 42 <545H 067. LH 515<802 JD5 L
:.16DA042 5 5/5:70;2A5175 7<216/.<A242
2O2 6D2 P2I07D234560812BC. 45
>.7G1:02 2:70;2
T 5672 8<2145?2 JD5 >.45A.63
5< 5A JD23
JD5< ]277O
A57<.
CENFIM
N><0A5 65 5A ]277H
28
:.A. 62I5A.6
6 :.1724.<56 45 515<802 2:70;2 42<C.H >.< :.1658D0175H 2
515<802 :.16DA042 4D<2175 DA :5<7. 75A>. 5A _WPE
_03
.]277 P.<2F
p
u
i
0
180
90
08
360
270
u
+
+
-
-
i
+
+
-
-
p
+
+
+
+
Quadrante
1º
2º
3º
4º
;.3
DBC. :.1R
D172 426/D1BM56:.<<5175H 7516C. 5 >.7G1:02 1DA :0<:D07.
>D<2A5175 <560670;.
2 21=3
065 42 /08
•
:.1:3
DO
A.620142 JD5@
2 >.7G1:02 75A /<5JDG1:02 4D>3
2 42 7516C. 5 42 01751604245
1
O
2
08
2 /08D<2
51:.17<2A 65 45651P242626:D<;2645 D 5 0 5 20142 42 >.7G1:02
JD5 3
P56;5A 266.:0242
CENFIM
29
6;23
.<560167217K15.645
@
@
(8F
(8F
5
6C. 424.6>53
265JD2BM56@
5 R5 1R-
S3T2
.A. 1. :26. 2175<0.<@
•
.7G1:02 75A /<5JDG1:02 4D>3
2 42 7516C. 5 42 01751604245
. ><0A50<. 5 75<:50<. JD2<7.645 >5<O
.4.
5
7GA 601206:.17<=<0.6H >53
. JD5
156656JD24<217562 >.7G1:02 65<= 158270;2H . JD5 60810/0:2 5672< 2 65< /.<15:042
515<802 Z<545
. 658D14. 5 JD2<7. JD24<21756H
5
7GA . A56A. 601235 2 >.7G1:02 7.A2
60123>.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
p
u
i
0
08
360
90
180
270
u
+
+
-
-
i
-
+
+
-
p
-
+
-
+
Quadrante
1º
2º
3º
4º
;.3
DBC. :.1R
D172 426/D1BM56:.<<5175H 7516C. 5 >.7G1:02 1DA :0<:D07.
>D<2A5175 014D70;.
5A.6H 2660AH JD5 P= DA2 .6:03
2BC. :.17O
1D2 45 515<802H .<2 42 <545 5A
40<5:BC. Z:2<82H .<2 45672 >2<2 2JD53
2
CENFIM
30
662 515<802 1D1:2 :P582 2 65< 7<216/.<A242 5H >.<7217.H 2 >.7G1:02 AL402 JD5
3
P5 567= 266.:0242 L 1D3
2
67. L >.67. 5A 5;04G1:02 >53
2 08D23
4245 5 60A57<02
426=<526453
0A072426>53
2 /D1BC. >.7G1:02H :DR
2 6.A2 23
8LI<0:2 <53
270;2 2 DA
>5<O
.4. L 1D3
2
.45A.6517C. 2/0<A2< JD5 DA2 014D7K1:02 >D<2 1C. :.16.A5
515<802 2:70;2
.<LAH 5662 515<802 JD5 :0<:D3
2 23
75<1242A5175 517<5 2 <545 5 2
:2<82H 5AI.<2 1C. 65R
2 :.16DA042H 75A DA 4575<A0124. ;23
.< 5 L456081242 >.<
'
H 456081214. 65 08D23
A5175 >.< <52:70;2 2 >.7G1:02 JD5 3
P5
;5A 266.:0242
.72 65 >53
2 3
57<2 L
1
+
+5,
5
;;)
2 5 >.45 65< A54042 >53
.6
.
(
5 2 515<802 :.<<56>.145175 >53
.6
H
A54042
5A
P
662
515<802
:.<<56>.145 2. :2A>. A281L70:. :<024. 12 014D7K1:02 >53
2 >266285A 42
:.<<5175
.A. >.45A.6:.1:3
D0<H 2 I.I012 :.A5B2 >.< /.<15:5< 515<802 Z<545
.45A.6
72AILA 2/0<A2<@
A2 014D7K1:02 LDA /.<15:54.< 45 515<802 <52:70;2
+`$
1
2
08 #
2 /08D<2
# 567= <5><5651724. DA :.1451624.<
517<5 :DR
.6 75<A01206 567=
2>3
0:242 DA2 7516C. 23
75<1242 601D6.0423
.A. 62I5A.6H 2 :.<<5175
567= 5A 2;21B. 45 *$^H :.1/.<A5 1.6A.67<2A 26
5JD2BM56@
@
CENFIM
(8F
5 R-
31
@
5 1R- A S3T2
(8F
p
u
i
0
90
180
u
+
+
-
-
i
+
-
-
+
p
+
-
+
-
1º
2º
3º
4º
Quadrante
08 &
2 /08D<2
360
270
;.3
DBC. 42 :.<<5175H 7516C. 5 >.7G1:02 1DA :0<:D07. :2>2:070;. >D<.
&H + 5 , 567C. <5><5651724268<2/0:2A5175 R
D172A5175 :.A 2 :D<;2 42
>.7G1:02
.6 Y5 YJD24<217562 >.7G1:02 L>.6070;2H 5 158270;2 1.6 Y5 YJD24<21756
607D2BC. L .>.672 Z ;5<0/0:242 2175<0.<A5175H >.06. :.1451624.< :.A5B2 >.<
:.16DA0< 515<802 42 <545 E
>.7G1:02 >.6070;2F >2<2 12 .6:03
2BC. 658D0175 ;.3
72< 2
/.<15:G 3
2 12 A56A2 JD21704245
53
2 08D23
4245 5 60A57<02 426=<526>.6070;265 158270;26453
0A072426>53
2 :D<;2
5 658D014. DA <2:0.:O
10. 04G170:. 2. R
= D703
0?24. 1.6 4.06 >.17.6 2175<0.<56H
>.45A.670<2< 26658D01756:.1:3
D6M56@
•
•
•
CENFIM
>.7G1:02 75A /<5JDG1:02 4D>3
2 42 7516C. 5 42 01751604245
A2 :2>2:04245 >D<2 1C. :.16.A5 515<802 2:70;2
V$
•
.17<2<02A5175 Z I.I012H DA :.1451624.< :.A5B2 >.< <5:5I5< 515<802
42 <545
A :.1451624.< LDA :.16DA04.< 45 515<802 <52:70;2 +a$
•
2AILA 2. :.1451624.< 567= 266.:0242 DA2 515<802 .6:03
2175 JD5 1C.
32
L7<216/.<A242
>.7G1:02 266.:0242 4560812 65 >.<
D104245 .
P
E
672 75A >.<
F 5 LA54042 >53
.6
O
H :.A.
62I5A.6
2<2 A540< 2 515<802 <52:70;2 D703
0?2A 65
:DR
2 0140:2BC. L4242 5A
'
H
PE
(
!
F
(
.16045<5A.628.<2 DA2 :2<82 70>.
?. :.A. 12 /08D<2 '
08 '
+D214. 2>3
0:2A.6DA2 7516C.
1.665D675<A01206H 2 :.<<5175
<56D3
72175 ;20H
127D<23
A5175H 5672< 5A 6514. 45 456/262A517. /D1BC. 4.6;23
.<5645
/08D<2
5
( JD5 70>0:2A5175 03
D67<2 5672 607D2BC.H <5/5<5 65 5A >2<70:D3
2< Z
5;.3
DBC. 42 >.7G1:02 1DA2 I.I012 .145 7516C. 5 :.<<5175 567C. 456/26242645
DA K18D3
. 40/5<5175 45 *$YE14D7K1:02 >D<2F
p
u
i
0
90
u
u
08 ( ;.3
DBC. 7O
>0:2 42
:.<<5175H 7516C. 5 >.7G1:02
1DA :0<:D07.
><54.A012175A5175
014D70;.
-
+
i
-
p
-
+
1º
2º
Quadrante
CENFIM
360
180
+
-
+
3º
4º
33
2 21=3
065 42 /D1BC. >.7G1:02 ;5A.6JD5 26=<526>.6070;26>.< 53
2 453
0A072426
6C. 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.H 067. LH Z515<802 2:70;2
2AILA P= 2 2660123
2< DA2 515<802 <52:70;2 JD5 :.<<56>.145 Z6=<526JD5 65
:.A>5162A 672 515<802 :0<:D3
2 12 <545H :.A. 62I5A.6H 65A 65< 7<216/.<A242
(
P
(
.
08 *
.7G1:0265A :.<<5175 23
75<1242
P
@
L
.
7
ϕ
U 1R --2
P
5 /.<A2 21=3
.82 :P582<O
2A.6Z5N><566C.@
Q = U IL sen ϕ
19 :
-
!
7-,9 2
P
@
CENFIM
.
19 :
-
!
2
34
3$
.
(
4
4
D703
0?2BC. 4.6 60675A26 7<0/=60:.6 5A 7.42 2 :24502 45 515<802 75A DA :2<=:75<
><270:2A5175
5N:3
D60;.
.A5175
2
1O
;53 42
D703
0?2BC.
;2A.6 51:.17<2<
DA
60810/0:270;. 5 ;2<024. 1SA5<. 45 2>2<53
P.6H 2660A :.A. 016723
2BM56 45 >5JD512
>.7G1:02 23
0A5172426:.A 7516M56A.1./=60:26
'
•
(
4
2<2 2 A56A2 >.7G1:02 2 /.<15:5<H DA 23
75<124.< 7<0/=60:. 75A A51.<
;.3
DA5 5 ><5B. JD5 2 :.<<56>.145175 D104245 A.1./=60:2 5 A20.<
/02I03
04245 45 65<;0B.
72A A20660A>3
565 5:.19A0:26
• 6<545645 7<216>.<75 5 45 4067<0ID0BC. <56D3
703
0?2A 7<G6:.14D7.<5645 /265 5H 5;517D23
A5175H DA JD2<7. :.14D7.< 15D7<.H
406>516214. 6506 :.14D7.<56 JD5 <5JD5< DA2 <545 A.1./=60:2 5JD0;23
5175 \
5:.1.A02 4. :.I<5 5 A51.<56 >5<426 5A 3
01P2 23
02A 65 .6 A51.<56 :D67.6 5
A20.< 60A>3
0:04245 45 :.1:5>BC. 5 0A>3
2172BC. 426 567<D7D<26 45 2>.0. 426
3
01P26
•
60A>3
0:04245 45 :.167<DBC.H A51.<56 :D67.6 5 8<2145 /02I03
04245 45
/D1:0.12A517.
4.6 7<216/.<A24.<56 7<0/=60:.6 5
20142
4.6 A.7.<56
266O
1:<.1.6 45 :2A>. 80<2175H 45 5A><58. 8515<23
0?24. 5 JD5 1C. 7GA
5JD0;23
5175 5A A.1./=60:.H R
D670/0:2A 69 >.< 60 2 5N067G1:02 45 60675A26
7<0/=60:.6
V
400
.
u1
u2
u1
u3
300
200
100
0
-100
90
180
270
-200
12
-300
1$2
-400
08
u3
120º
u2
5><565172BC. :2<7560212 E F 5 ;5:7.<023E F 45 DA 60675A2 7<0/=60:. 45
7516M56 .72< JD5 567C. 456/262426517<5 60 45
CENFIM
120º
120º
360
$YH .D65R
2H
45 >5<O
.4.
35
2 /08
H >.45A.6;5< 5A
!
5- "# 7 -
, 5
2 5;.3
DBC. 4267516M56
45 DA 60675A2 7<0/=60:. 2 >2<70< 4.6<56>5:70;.6;23
.<560167217K15.6
.45 20142 /2?5< 65 DA2 <5><565172BC. ;5:7.<0234. A56A. 60675A2 E
/08&$
65 275145<A.62 JD5 DA 456/262A517. 1. 75A>. 45
DA2 40/5<51B2 218D3
2< 45
FH
45 >5<O
.4. 5JD0;23
5 2
$Y 517<5 .6 ;5:7.<56 <5><56517270;.6 426 7516M56
D>M5 65 7.4. . 60675A2 <.4214. 2 DA2 ;53
.:04245 218D3
2<
ω
1. 651704.
0140:24.H JD5 2<I07<2A.6:.A. >.6070;.
U 1,U 2,U 3, :.16707D5AH 2660AH DA :.1R
D17. 45 7<G6 9 78<2145?2 A545 5A 9 :
K,
5-
:DR
2
567= 5A 2;21B. <53
270;2A5175 2
5
E,7 I26<5/5<04267516M56
L P
2 21=3
065 42 /08
;5A.6JD5
/2:7.H 65 56:.3
P5<A.6H >.< 5N5A>3
.H 2 .<085A 4.6 75A>.6 :.A. <5/5<G1:02H
;5<0/0:2A.6JD5
75A R
= DA :5<7. ;23
.< >.6070;. 1DA2 /265 45:<56:5175 42 6D2
23
75<1K1:02H 51JD217. JD5
L 1D3
. 5 69 28.<2 0<= 010:02< 2 23
75<1K1:02 >.6070;2
>.< ;23
.<566D:5660;2A5175 :<56:51756
5<0/0:2A.608D23
A5175 JD5
567= 5A 27<26. <53
270;2A5175 2
67. 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<G1:02 2 >.60BC. .:D>242 2 424. A.A517. >.< DA 456656
;5:7.<56H >.< 5N5A>3
.
U 1 H JD5 5672 65<= 6D:5660;2A5175 .:D>242 >53
.6;5:7.<56
U2 5 U3
65JDG1:02 45 /2656 0140:242 L H
H
5 :P2A2 65
+ )
.<7217.H 2 65JDG1:02 45 /2656L 2 .<45A >53
2 JD2365 6D:545A 26
/26561DA 60675A2 7<0/=60:.
. '
'
4
N0675A 4D26/.<A267O
>0:2645 266.:02BC.H 26:P2A2426@
CENFIM
• .,K "#
. ;
• .,K "#
O ' .
+
.
36
08
082BC. 5A 567<53
2 45
:2<826A.1./=60:26D703
0?214. DA S10:. :.14D7.<
:.ADA :517<23E
:.14D7.< 15D7<.F
5675 :.14D7.< 1C. :0<:D3
2 :.<<5175H
V
>.45<02 A56A. 6D><0A0< 65
67<G6:2<826<5><565172426:2<2:75<0?2A 65 >53
. A56A. ;23
.< 45 0A>54K1:02H
067. LH & @ &$ @ &<
6<56>5:70;265N7<5A042456567C. 3
0824262.675<A0120645
:242 DA 4.6 51<.3
2A517.6 4. 23
75<124.< 5 6C. <5/5<51:02426 >53
26 3
57<26 N?
W?U &
.6 :.14D7.<56 45 23
0A5172BC. 5672I53
5:5A 65 2660A 7<G6 :.<<51756 :.A .
A56A. ;23
.< 5/0:2?A26
E
$3T
6;23
.<560167217K15.6456626:.<<5175640/5<5AH :.17D4.H 5A :242 A.A517. 5
5A :242 DA2 426/2656
.45<O
2A.6<570<2< .
426 :.140BM56 45 /D1:0.12A517.
4. :0<:D07. 65A 23
75<2BC. .D><5R
DO
?.
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<
JD23
JD5< 23
75<2BC.
CENFIM
37
67263
082BM56:.1/08D<2A DA2
-
:H 7217. 12 /.175 :.A. 12 :2<82
>.17.
:.ADA 4560812 65 >.<
.A. 5N5A>3
.H 65 :.16045<2<A.6.6;23
.<5645
DA2 :.<<5175 5/0:2?45 (
H
H
E
/08 F :.<<56>.14514. 2
>.< /265 5 <53
270;.62. A.A517. 5A JD5
ω7
V *$YH
75A.6@
ωt = 90 0
ω7
V
51
A=N 6
V
51E
A=N 6
V
51
A=N 6
V
ω7
$YF
ω7
H
V
V
651*$YV
H
H
H
651E $YF V #H&&
651
$YV #H&&
5675H :.A. 5A JD23
JD5< .D7<. 01672175H 2 6.A2 23
8LI<0:2 426:.<<51756L?5<.@
$
V
b
5A JD5
/08
b
$
V
H
#H&&
#H&& V
<5><565172 2 :.<<5175 1. 15D7<.
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:.A 26<56>5:70;26601D690456L65A><5 08D232 ?5<.
08
CENFIM
38
(
L
.
+D214. 7.426 26 :2<826 7GA . A56A. ;23
.<H 067. LH 2 A56A2 0A>54K1:02H .
60675A2 40?65 5JD03
0I<24. 5 26:.<<517565A :242 DA2 426/26566C. 08D206
.0 . :26. 2I.<424. 1. >.17. 2175<0.<
DA 60675A2 7<0/=60:. 5JD03
0I<24. 2 6.A2 ;5:7.<023426:.<<51756L08D232 ?5<.
I 0 = I1 + I 2 + I 3 = 0
I 0 :.<<5175 1. 15D7<. 0 ;5:7.< 1D3
.
V
•
( .
(
( .
.16045<5A.6DA 60675A2 7<0/=60:. :.A 15D7<.H E
:DR
263
01P2645 23
0A5172BC. 6C.
:.16707DO
426>.< 7<G6:.14D7.<5645 /265 5 >53
. :.14D7.< 15D7<.H <5/5<51:024.6H
<56>5:70;2A5175H !
. 1 2? .$ 1 2?
.< 1 2
.16045<5A.6 20142 .6 >.17.6 ?$?<
;
:.<<56>.1451756 2.6 75<A01206 45
3
082BC.
08
516M5660A>3
56.D7516M5645 /265
560812 65 >.<
2 7516C. 5N0675175 517<5 JD23
JD5<
:.14D7.< 45 /265 5 . :.14D7.< 15D7<.
DA
, -
- ,E8 ,7 7
4560812A.6 >.<
?
$
5 +<
75A.6H >.<7217.H 7<G67516M5660A>3
56H JD5
.A. 62I5A.6H 6C. 08D206 5A 8<2145?2 5
;5:7.<023
A5175 /.<A2A DA2 567<53
2 7<0/=60:2 45 7516M56E
/08& F
CENFIM
39
U1
120º
120º
08#
67<53
2 7<0/=60:2 45 7516M56
•
120º
U3
U2
(
1
U1
U31
U3
3
0
U12
U2
U23
2
08&
5/015 65
/26564. 60675A2 7<0/=60:.
516M56:.A>.6726
:.A. 2 7516C. 5N0675175 517<5 4D26JD206JD5<
2 /08& 2660123
2A.6 26 <5/5<0426 7516M56 :.A 60AI.3
.802 R
= 1.662 :.1P5:042H
067. L@
U 12 = U 1 − U 2
U 23 = U 2 − U 3
U 31 = U 3 − U 1
JD5 <5><565172AH <56>5:70;2A5175H 267516M56517<5 26/2656 5
H
5 H
5
6726 5N><566M56 >.660I03
072A 1.6 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>3
56
CENFIM
40
.
'
&
V
( .
(
. <56>5:70;. 4028<2A2 ;5:7.<023ID6JD5A.62 <53
2BC. 517<5 2 7516C. :.A>.672
5 2 7516C. 60A>3
56@ .67<G6;5:7.<56/.<A2A DA 7<0K18D3
. 0696:53
56H :DR
. K18D3
.
.I7D6. :.<<56>.145 2. 456/262A517. 5N0675175 517<5
U 1 5 U 2 067. LH
$Y
6<567217564.06K18D3
.6;23
5A $Y:242 DA
V
:.6 $Y:.A. :.6 $YV
17C.
V
514.
V
5
V
3
V
V
V
V
UC = 3 US
75A.6JD5
.145
3
2
!
5-
- 5 #
,
!:
7 - 5 # 7
! -
+D214. 1.660675A267<0/=60:.665 0140:2 DA 4575<A0124. ;23
.< 42 7516C. 65A
JD23
JD5< 24R
5:70;2BC.H 45;5 6DI5175145< 65 JD5 53
5 65 <5/5<5 2 DA2 7516C.
:.A>.672
.< 5N5A>3
.H 65 40665<A.6JD5 DA2 4575<A01242 3
01P2 45
L45
#
_ H 45;5A.6517514G 3
2 :.A. 2 7516C. :.A>.672
D7<26;5?562>2<5:5A 1.64D267516M5656:<0726658D042A5175 2>5126:.A DA
7<2B. .I3
O
JD. 2 65>2<= 3
26
.< 5N5A>3
.H DA2 <545
$
$$
><0A50<2
4560812 517C. 2 7516C. 60A>3
565 2 658D142 2 7516C. :.A>.672
(
4
L
.
0?65 JD5 DA 60675A2 7<0/=60:. L4565JD03
0I<24. .D45 :2<8264565JD03
0I<242665
260A>54K1:026>.< /265 1C. /.<5A 7.42608D206
5672 607D2BC.H . >2>534565A>51P24. >53
. :.14D7.<
L /D142A51723
H
:.A. ;5<5A.6
<5A.6 517C. 567D42< . /D1:0.12A517. 45 DA2 :2<82 7<0/=60:2 4565JD03
0I<242
3
08242 5A 567<53
2 :.A 15D7<.H 5 658D042A5175 65A 15D7<.H 5 2;23
02< .6
<56D3
724.6
CENFIM
41
.
(
08'
575<A012BC.
;5:7.<02342 :.<<5175 1.
:.14D7.< 15D7<. 5A
60675A267<0/=60:.6
4565JD03
0I<24.6
0<:D3
2<= 2660A DA2 :.<<5175 1. 15D7<.
I N :.<<56>.145175 Z <56D3
72175 42
6.A2 ;5:7.<023426:.<<517561267<G6/2656
.1:3
DO
A.6JD5@
DA 60675A2 5A 567<53
2 4565JD03
0I<24. :0<:D3
2 65A><5 DA2 :.<<5175 1.
15D7<.
662 :.<<5175 L08D23Z6.A2 ;5:7.<023426:.<<51756426/2656
I N = I1 + I 2 + I 3
T0A><56:014O
;53. :.14D7.< 15D7<. >2<2 42< >266285A Z:.<<5175 45
45/507.
.
(
6D><566C. 4. :.14D7.< 15D7<. 1DA 60675A2 4565JD03
0I<24. .<08012 DA
4565JD03
O
I<0. 426 7516M56 60A>3
56 6DR
507214. .6 40;5<6.6 <5:5>7.<56 2 6D>.<72<
1D16:26.66.I<57516M56H 1.D7<.67516M5601/5<0.<562. <56>5:70;. ;23
.< 1.A0123
. '
CENFIM
(
O ' .
42
08(
DA2
3
082BC.
=<026>5<6>5:70;2645 DA2 A56A2 3
082BC. 5A 7<0K18D3
.
5A
7<0K18D3
.H
26 7<G6 :2<826 3
082A 65
65JD51:023
A5175
:.1/08D<214. DA2 A23
P2 /5:P242 7<0218D3
2< E
/08(FH 6514. :242 >.17. :.ADA
3
0824. 2 DA2 /265
.45A.6;5<0/0:2< JD5@
• C. 5N0675 :.14D7.< 15D7<. >.< 1C. P2;5< >.17. :.ADA Z67<G6/2656
•
7516C. 2>3
0:242 2 :242 DA2 426:2<826L2 7516C. :.A>.672
.
/
6:.<<5175645 3
01P2
H :.A. 2 ><9><02 4560812BC. 0140:2H 6C. 26:.<<51756JD5
:0<:D3
2A 1.6:.14D7.<5645 23
0A5172BC. 5 JD5 12 /08D<2 /.<2A 1.72426>.<
P2A2A 65 :.<<51756 45 /265
7<0K18D3
.
C. 26:.<<51756
H
/
H
5
Z6 :.<<51756 JD5 :0<:D3
2A 1.6 <2A.6 4.
5
42 /08
&
.4267GA . A56A. 651704.
45 :0<:D3
2BC. c2<I07<24.d
O ' .
L
.
A 60675A2 7<0/=60:. 40?65 5A 7<0K18D3
. 5JD03
0I<24. JD214. 7.42626:2<8264.
7<0K18D3
. 6C. 04G170:265 >.<7217. 7GA 2 A56A2 0A>54K1:02
5672 607D2BC. 26:.<<517561263
01P266C. 7.42608D206H 067. LH
:.A. 26:.<<51756126/2656H .D65R
2H
V
V
V
V
2660A
5A JD23
JD5< :26. 456/262426
$Y517<5 60
.
.
/
I L= 3 I f
CENFIM
43
672 <53
2BC. >5<A075 1.651D1:02< JD5 2 :.<<5175 12 3
01P2 L
3 ;5?56A20.< JD5
2 :.<<5175 45 /265
O ' .
L
.
+D214. 26:2<8261C. 6C. 7.42608D206H . 7<0K18D3
. L4565JD03
0I<24. 5 2 <53
2BC.
4. >.17. 2175<0.< 1C. L;=3
042
6:.<<5175645 3
01P2 450N2A 45 65< 08D206H 2660A
:.A. 26 :.<<51756 45 /265
217GA 65H :.17D4.H 26 7516M56 45 /265 1.6
75<A0120645 :242 DA2 426:2<826
P
(
4
( .
(
(4
:=3
:D3
. 45 >.7G1:0265A : 2 7<0/=60:2H 1.A5242A5175 426! -D57,
7-,9
!
7-,9 ?
5- H 6017570?2 5 658D5 DA2 /.<AD3
2BC. 04G170:2 Z 4.6
:.16DA.6>.< /265
.16045<214. . :26. 85<23JD5 :.175A>3
2 7.426 26 607D2BM56 45 :2<82 2 JD5
75A.6;014. 2 /2?5< <5/5<G1:02
:2<8265JD03
0I<2426.D 1C.H 3
082BC. 5A 567<53
2
.D5A 7<0K18D3
. H >.45 65< 2660A 5JD2:0.124.@
'
.
)
V
)
b
b
+ V + b+ b+
)
V
P2 + Q2
242 DA2 426 >2<:53
26 0145N2426 <5><565172 DA2 >.7G1:02 >.< /265H 5 26
<56>5:70;265JD2BM566C.@
V
:.6ϕ F
+/ V
651 ϕ F
/
>.7G1:02 2:70;2H 6514. :.<<56>.145175 2 DA2 >.7G1:02 :.16DA042H L 65A><5
>.6070;2 5 2 <5/5<042 6.A2 L2<07AL70:2
CENFIM
44
>.7G1:02 <52:70;2 7.723 L . I23
21B. 42 >.7G1:02 JD5 :0<:D3
2 517<5 .6
:.A>.151756<52:70;.65 2 <545H 6514. 158270;2 JD214. :.16DA042H 5 >.6070;2
JD214. /.<15:042
'
L
6D2 6.A2 LH >.<7217.H 23
8LI<0:2
.
+D214. 26 :2<826 6C. 08D206 126 7<G6 /2656H 26 5N><566M56 >2<2 26 >.7G1:026
<56D3
72A A20660A>3
56 123
065A.6H 517C.H 263
082BM565A 567<53
2 5 5A 7<0K18D3
.
. '
(
.
5672 A.17285A 5 126:.140BM5651D1:02426H 75A.6JD5@
•
6 :.<<51756 45 /265 6C. 08D206 5A 8<2145?2 5 08D206 Z6 :.<<51756 45
3
01P2
•
.675<A0120645 :242 :2<82 567= 2>3
0:242 DA2 7516C. 60A>3
56
56726:.16045<2BM56I26502A 65 26454DBM56>2<2 2640/5<51756>.7G1:026@
P
V
V
V
V
b
b
.A.
US =
V
UC
3
:.6ϕ
V
:.6ϕ
UC
3
:.6ϕ
5 >.<7217.
P=
P
3 UC IL cos ϕ
U 1U --2
:
5 /.<A2 21=3
.82 :P582<O
2A.6Z5N><566C.@
CENFIM
45
3 UC IL sen ϕ
Q=
1 :
-
!
7-,9 2
P
3 UC IL
S=
. '
1 :
-
(
!
2
O ' .
5672 A.17285A 5 126:.140BM5651D1:02426H 75A.6JD5@
•
3 6D>5<0.<56Z6:.<<51756126/26565 7GA .
6:.<<517561263
01P266C.
A56A. ;23
.< 5A 7.42653
26
•
7516C. 2>3
0:242 2 :242 DA 4.6 53
5A517.6 4. 7<0K18D3
. L 2 7516C.
:.A>.672
514. 2660AH 454D?2A.6265N><566M56>2<2 2640/5<51756>.7G1:026@
P
V
V
V
V
/
/
:.6ϕ
:.6ϕ V
IL
3
:.6ϕ
5 >.<7217.
P=
3 UC IL cos ϕ
U 1R --2
P
5 /.<A2 21=3
.82 :P582<O
2A.6Z5N><566C.@
Q=
3 UC IL sen ϕ
19 :
-
!
7-,9 2
P
S=
CENFIM
3 UC IL
19 :
-
!
2
46
N
L
'
.
P
(
(
4
.
.A>2<214. 265N><566M56JD5 454D?0A.61. :26. 42 3
082BC. 5A 7<0K18D3
. 5 5A
567<53
2 ;5<0/0:2A.6 JD5 6C. 08D206H >53
. JD5 >.45A.6 56:<5;5< >2<2 2AI.6 .6
:26.6@
P=
CENFIM
3 UC I cos ϕ
Q=
3 UC I sen ϕ
S=
3 UC I
47
3<
(
P
P
,5- 5 ,
! !,
9,
5-
+ 7
!
+
! 5 5-
7 !-
? 6 +
7-,9 ?) ! 9 ,-
E
7"#
E
- ?
,
X-,:
A2 4242 :.<<5175 2I6.<;042 65<= 7217. A53
P.< 2><.;507242 JD217. A20.< /.< 2 6D2
:.A>.15175 2:70;2 5 A206><9N0A. 42 D104245H >.<7217.H /.< . /2:7.< 45 >.7G1:02
>2<75 :.<<56>.145175 Z 6D2 :.A>.15175 <52:70;2 :0<:D3
2 60A>3
56A5175 12 <545H 65A
65< 7<216/.<A242
2I6.<BC.H >.< >2<75 4.6<5:5>7.<56014D70;.6H 45 :.<<51756:DR
. ;23
.< L 6D>5<0.< 2.
15:566=<0. >2<2 . <56>5:70;. /D1:0.12A517. 75A 0A>.<7217560A>3
0:2BM562 7.4.6.6
1O
;5064.
0675A2
3
L:7<0:. 45
15<802H :DR
.601:.1;510517566C. 658D042A5175 2>.1
724.6
( .
V
Q
•
5:56604245 45 2DA5172< 2 >.7G1:02 016723
242 126 :517<206 ><.4D7.<26 6
8<D>.675<C.H 2660AH 45 65< 6.I<540A5160.124.6>2<2 /2?5<5A /2:5 2 DA2 :2<82
<52:70;2 24;5<62
•
63
01P26 75<C. 08D23
A5175 45 65< 6.I<540A5160.12426 >2<2 JD5 >.662A
;50:D3
2< 2 :.<<5175 >.< 5662 <2?C. 5N:54517=<02
•
20.<56JD542645 7516C. 5A 3
01P2
@&
5 A20.<5626>5<426 @ &
$
•
.I<540A5160.12A517. 45 7.42 2 2>2<53
P285A 45 :.<75 5 ><.75:BC.H :.A.H >.<
5N5A>3
.H 65::0.124.<565 406R
D17.<56
•
DA517. 42 >.7G1:02
7<216/.<A24.<26
( .
V
0(
4.6 7<216/.<A24.<56 567=70:.6 126 6DI5672BM56
6:D67.66C. ><.>.<:0.1206Z>.7G1:02 D107=<02 4.6
8<D>.6
•
01P26 45 A20.< 65:BC.H 127D<23
A5175 A206 :2<26
01;5670A517. 12 6D2
:.167<DBC. 1C. 75AH 2660AH 2 :.17<2>2<7042 45 A20.< 1SA5<. 45 D703
0?24.<56H .
JD5 65 7<24D?1DA2 >5<42 42 >.751:023<5:5072 45 /2:7D<2BC.
CENFIM
48
•
.172I03
0?2BC. 426>5<426>.< 5/507. .D3
5 1263
01P26
•
.42 2 2>2<53
P285A 42 <545 L45 :D67. A20653
5;24.
•
27D<23
A5175H . 28<2;2A517. 45 :D67.665<= 6D>.<724. >53
. D703
0?24.<
5 . D703
0?24.< L 45 I20N2 7516C.H 70>. 4.AL670:.H . /2:7.< 45 >.7G1:02 L AD07. 2><.
N0A242A5175 D107=<0. 5 . 4067<0ID04.< 2>5126/2:7D<2 2 515<802 2:70;2 :.16DA042
6
:D67.65 01:.1;5105175645<0;24.642 D703
0?2BC. 45 <5:5>7.<564. 70>. 014D70;.H :.A.
3
KA>2426/3
D.<56:517565 56>.<240:2A5175 A.7.<56H 6C. 85<23
A5175 6D>.<724.6>53
2
51704245 4067<0ID04.<2
5 . D703
0?24.< L45 I20N2 7516C. A26>.66D0 016723
2BM56:.A I20N. /2:7.< 45 >.7G1:02H
517C. . 4067<0ID04.< .I<08= 3
. = 2 016723
2< DA :.1724.< 45 515<802 <52:70;2H .I<08214.
. 2 6D>.<72< .6:D67.642 6D2 ><9><02 016723
2BC. 1C. :.<<08042
2<2 /2:7.<5645 >.7G1:02 :DR
. ;23
.< L08D23
.D6D>5<0.< 2 $H(H . 4067<0ID04.< 1C. .I<082
Z016723
2BC. 45 :.1724.<5645 515<802 <52:70;2
$
(
P
53
. JD5 /0:.D 407. <56623
72 2 0A>.<7K1:02 42 :.<<5:BC. 4. /2:7.< 45 >.7G1:02H 067. LH 2
>.660I03
04245 45 . 7.<12< . A206><9N0A. >.66O
;5342 D104245
.1658D5 65 723 :.<<5:BC. /2?514. 40A01D0< 2 :.A>.15175 <52:70;2 42 :.<<5175H
<5:.<<514. 2. 5/507. :.1R
D824. 45 :2>2:04245 1. :0<:D07.
.A. 62I5A.6H 2 :.A>.15175 <52:70;2 42 :.<<5175 567= 5A JD24<27D<2 5 27<26. 45
*$^<53
270;2A5175 Z7516C.
QC
S
Q
S’
ϕ
ϕ,
08
CENFIM
2<82 014D70;2 2175642 :.<<5:BC.H
20N. /2:7.< 45 >.7G1:02
Q’
´
QC
49
:.16DA. 45 515<802 <52:70;2 567= 3
0824. 2 DA A2DE 7-
! -D57, 1 + 7
ϕ2
5 Z016723
2BC.
1.BC. 45 /2:7.< 45 >.7G1:02 45;5 5672< 266.:0242 2.640/5<51756<5:5>7.<56D703
0?24.6
12 014S67<02 5 1. 75<:0=<0.
5<A075 1.6 045170/0:2< .6 2>2<53
P.6 :DR
. :.16DA. 5A 515<802 <52:70;2 L A206 .D
A51.60A>.<72175
!
-Kϕ
•
.7.< 266O
1:<.1.
•
KA>242601:21456:51756
•
KA>2426/3
D.<56:51756
•
KA>242645 456:2<82
•
.<1.645 <56067G1:02
•
.<1.645 014DBC.
$H(#
$H&
•
.<1.645 2JD5:0A517. 4053
L:7<0:.
$H(#
$H&
•
=JD012645 6.3
42< 45 <56067G1:02
$H( 2 $H*
$H'# 2 $H (
$H#
H'
•
$H ' 2 $H(#
$H#
H *2 H
$
<216/.<A24.<56<5:70/0:24.<56 45 6.3
424D<2 >.<
2<:.
•
H'
$H 2 $H&
<D>.6<.7270;.645 6.3
424D<2 >.< 2<:.
•
#H( 2 $H&
$
.67.6 567=70:.6 A.1./=60:.6 45 6.3
424D<2 >.<
2<:.
•
<
ϕ
7
:
M
.<1.645 2<:.
$H' 2 $H*
H$ 2 $H (
$H' 2 $H(
H$ 2 $H'#
$H(
(
$H'#
'
L P
.
(
P
Y
016723
2BC. 4DA :.1451624.< 5A JD23
JD5< <545 53
L:7<0:2 E
! .D
FR
D670/0:2 65 >53
2@
•
D><566C. 426>5123
042456>.< :.16DA. 45 515<802 <52:70;2
•
0A01D0BC. 42 >.7G1:02 2>2<5175 6DI6:<072
•
0A072BC. 426>5<42645 515<802 2:70;2 E
>5<42645 R
.D3
5F 1.6:2I.6
•
.660I03
04245 45 75< DA 2DA517. 42 >.7G1:02 2:70;2 E
UWF 406>.1O
;53 1.
65:D14=<0. 4. 7<216/.<A24.< !
•
75
CENFIM
53
P.<02 4. 1O
;5345 7516C. 1. /0A 42 3
01P2
5
!
-
< 5C
9,
,E
5-
5
C
1L' 2
50
(
'.
.
08
.A>5162BC. 83
.I23
•
6D><0A5 26>5123
042456>.< :.16DA. 5N:5660;. 45 515<802 <52:70;2
•
2R
D672 2 15:56604245 <52342 016723
2BC. 5A _W Z 6DI6:<0BC. 42 >.7G1:02
2>2<5175 E
( 5A _ F
•
23
0;02 . >.67. 45 7<216/.<A2BC. E
>.7G1:02 406>.1O
;535A _WF
•
>5<A075 D703
0?2< DA 406R
D17.< A2065:.19A0:.
9 "Z
•
2 :.<<5175 <52:70;2 E<F 567= ><565175 12 016723
2BC. 45645 . 1O
;53 27L 2.6
<5:5>7.<56
•
26>5<426>.< 5/507. 45 .D3
5 1.6:2I.61C. 6C. 40A01DO
426E
_WPF
(
.
08
CENFIM
.A>5162BC. 2<:023
51
•
6D><0A5 26>5123
042456>.< :.16DA. 5N:5660;. 45 515<802 <52:70;2
•
.>70A0?2 DA2 >2<75 42 016723
2BC.H 2 :.<<5175 <52:70;2 1C. L;50:D3
242 517<5 .6
1O
;506 5
•
23
0;02 . >.67. 45 7<216/.<A2BC. E
>.7G1:02 406>.1O
;535A _WF
•
>5<A075 D703
0?2< DA 406R
D17.< A2065:.19A0:. 2 A.172175 4. :.1451624.<
9 "Z
• 2 :.<<5175 <52:70;2
< 567= ><565175 12 016723
2BC. 45645 . 1O
;53
27L 2.6
<5:5>7.<56
• 26>5<426>.< 5/507. 45 .D3
5 1.6:2I.66C. 40A01DO
426E
_WPF
(
.
. 11.6I.<15645 :242 <5:5>7.< 4. 70>. 014D:70;.F
08
.A>5162BC. 3
.:23
• 6D><0A5 26>5123
042456>.< :.16DA. 5N:5660;. 45 515<802 <52:70;2
• .>70A0?2 7.42 2 <545 53
L:7<0:2
:.<<5175 <52:70;2 < L/.<15:042 1. 3
.:234.
65D:.16DA.
• 23
0;02 . >.67. 45 7<216/.<A2BC. E
>.7G1:02 406>.1O
;535A _WF
• >5<A075 D703
0?2< DA 406R
D17.< A2065:.19A0:. 2 A.172175 4. :.1451624.<
9 "Z
• 2 :.<<5175 <52:70;2 R
= 1C. :0<:D3
2 >53
.6:2I.642 016723
2BC.
• 26>5<426>.< 5/507. 45 .D3
5 1.6:2I.66C. 7.723
A5175 6D><0A0426E
_WPF
CENFIM
52
F 7C
7,
:
9,
DA2 016723
2BC. /2I<0345
$
$$
#$ !?;5<0/0:.D 65 JD5 26515<8026
2:70;2 5 <52:70;2 A540426 >53
.6 :.1724.<56 2. /0A 45
/D1:0.12A517. 5<2A <56>5:70;2A5175 45
#$$$ _WP 5
$$ P.<26 45
&$$ _
P
23
:D3
5@
2F
6>.7G1:0262:70;2 5 <52:70;2
IF
/2:7.< 45 >.7G1:02 42 016723
2BC.
:F
:2>2:04245 45 :242 :.1451624.<H 2 3
082< 5A 7<0K18D3
.H >2<2 53
5;2< .
/2:7.< 45 >.7G1:02 42 016723
2BC. >2<2 $H(
.
2F
VW
+ V W+
7 V #$$$
$$ V ('H# U
7V
$$ V ( U
IF 78 ϕ 0 V +
&$$
V $H*& V` :.6ϕ 0 V $H'
:F :.6ϕ / V $H( V` 78 ϕ / V $H'#
+ V
+
+
78 ϕ / V ('H# N $H'# V &#H& U
V E
78 ϕ 0
V
+
V+
78 ϕ /F V +0 +/ V (
V (H '
E
π. f
V &H
FV&
U
&#H& V (H ' U
V&
E
:242 :.1451624.<F
E $$ N N π . N#$F V
H
N $
V
µ
CENFIM
53
L
L
L
V
V
V
N
N
N
[ 23
:D3
5 . >5<O
.4. :.<<56>.145175 2 DA2 /<5JDG1:02 45 $ _!?
$[ 23
:D3
5 2 /<5JDG1:02 45 DA2 : 2 601D6.0423:DR
. >5<O
.4. L$H 6
<[ +D217.6:0:3
.6/2?1DA 658D14. DA2 :.<<5175 :DR
. >5<O
.4. L$H$$ 6[
=[ +D217. 75A>. 45A.<2 DA2 :.<<5175 2 5/5:7D2<
$$ :0:3
.662I514. JD5 2 6D2
/<5JDG1:02 L45 #$$$ !?[
B[ +D23L. >5<O
.4. 45 DA2 .145 45 <=40. :DR
2 /<5JDG1:02 L#$$ _!?[
µ
G[ A2 .145 45 <242< 75A DA >5<O
.4. 45
121.6658D14.
23
:D3
5 2 6D2
/<5JDG1:02
H[ A ;.3
7O
A57<. 45 : 2 0140:2 1.6 $$
2F
;23
.< 5/0:2?42 7516C.
IF
;23
.< A=N0A. 42 7516C.
23
:D3
5@
\[ A :23
.<O
/5<. 45 #$$ W :.16.A5 5A :.<<5175 :.17O
1D2
H
5 . 3
082<A.62
:.<<5175 23
75<1242 42 A56A2 7516C.H JD2365<= . ;23
.< 5/0:2? 42 :.<<5175
:.16DA042[
S[
;23
.< 5/0:2?45 DA2 7516C. 23
75<1242 601D6.0423L45 #$$
+D2365<= . ;23
.<
2<07AL70:. AL40.[
3[+D23426/<5JDG1:026E
#$ 5
$$ !?F 45A.<2 A20675A>. 2 5/5:7D2<
:0:3
.[
[ 5JD2BC. 0V 651E π / 7F <5><565172 DA2 :.<<5175 23
75<1242 601D6.0423
2F 140JD5 . ;23
.< 42 2A>3
07D45 42 :.<<5175
IF
23
:D3
5 . ;23
.< 5/0:2?
:F 140JD5 . ;23
.< 42 ;53
.:04245 218D3
2<
4F
23
:D3
5 . ;23
.< 42 /<5JDG1:02
5F
23
:D3
5 . ;23
.< 4. >5<O
.4.
/F
23
:D3
5 . ;23
.< 0167217K15. 42 :.<<5175H 1. 01672175 7 V $H$$# 6
8F 2B2 2 <5><565172BC. 75A>.<2345672 :.<<5175
$[ A2 I.I012 75A DA2 <56067G1:02 45 $ Ω 5 DA2 <52:7K1:02 45 $ Ω
Ω
2F
0A>54K1:02 42 I.I012
IF
01751604245 JD5 >5<:.<<5 2 I.I012 JD214. 65 3
P5 2>3
0:2 $
CENFIM
23
:D3
5@
54
$ Ω 5 2 <52:7K1:02
<[ DA I23
267<.H :DR
2 <56067G1:02 L 45
DA2 7516C. 45 ($
$ Ω H L 2>3
0:242
23
:D3
5@
Ω
2F
6D2 0A>54K1:02
IF
01751604245 JD5 . >5<:.<<5
=[ A :0<:D07. 53
L:7<0:.H :DR
2 7516C. L
$
$$ Ω 5 DA2 I.I012 >D<2 45 #$$ Ω
H L:.16707DO
4. >.< DA2 <56067G1:02 45
23
:D3
5@
2F
0A>54K1:02 5 2 <52:7K1:02 42 I.I012
IF
0A>54K1:02 4. :0<:D07.
:F
01751604245 2I6.<;042
Ω
Ω
Ω
B[ A2 I.I012 1C. >D<2H :DR
2 <52:7K1:02 L $$ Ω H 2I6.<;5 DA2 01751604245 45
$H(
JD214. 3
08242 2
$
2F
0A>54K1:02 42 I.I012
IF
6D2 <56067G1:02
23
:D3
5@
Ω
Ω
G[ 23
:D3
5 2 <52:7K1:02 45 DA2 I.I012 JD5 2I6.<;5 $H#
6D2 <56067G1:02 L45 $$ Ω
#$
JD214. 6DIA57042 2
Ω
H[ 23
:D3
5 2 0A>54K1:02 45 DA2 I.I012 :DR
. :.5/0:05175 45 2D7. 014DBC. L H# ! 5
:DR
2 <56067G1:02 L $$
Ω E/V#$ !?F
\[ A2 I.I012 L >5<:.<<042 >.<
45 #$
. 2>3
0:2< 3
P5 #$
H#
Ω
JD214. 65 3
P5 2>3
0:2 DA2 7516C. :.17O
1D2
5A :.<<5175 23
75<1242 2 01751604245 I20N2 >2<2 $H
23
:D3
5@
2F
<56067G1:02 42 I.I012
Ω
IF
<52:7K1:02 42 I.I012
Ω
:F
65D:.5/0:05175 45 2D7. 014DBC.H 65 /V#$ !?
S[ A 2JD5:54.< :DR
2 <56067G1:02 L 08D232 '$
Ω 567= 3
0824. 2
$
E
: 2F
23
:D3
5@
2F
01751604245 2I6.<;042
IF
>.7G1:02 2>2<5175
:F
>.7G1:02 2:70;2
4F
/2:7.< 45 >.7G1:02
$3[ A2 I.I012 75A DA2 <56067G1:02 45
#
Ω 5 DA2 <52:7K1:02 45
23
:D3
5H 2. 65< 3
P5 2>3
0:242 DA2 7516C. 45 #$
2F
01751604245 2I6.<;042
IF
>.7G1:02 2:70;2 1. :0<:D07.
:F
>.7G1:02 2>2<5175
4F
>.7G1:02 <52:70;2
5F
/2:7.< 45 >.7G1:02
CENFIM
#$
Ω
H 26658D017568<2145?26@
55
$ [ 53
270;2A5175 2. ><.I3
5A2 2175<0.<H 6D>.1P2 JD5 2>3
0:2;2 DA2 7516C.
:.17O
1D2 45 #$
23
:D3
5@
2F
01751604245 2I6.<;042
IF
>.7G1:02 40660>242 >.< 5/507. 45 .D3
5
$$[ A :.1451624.< 45
$
µ
L3
0824. Z <545 45 4067<0ID0BC. 45
$
H #$ !?
23
:D3
5@
Ω
2F
6D2 <52:7K1:02
IF
6D2 0A>54K1:02 E
65 456><5?2<A.62 <56067G1:02 ><9><02F
:F
01751604245 2I6.<;042 >53
. :.1451624.<
Ω
$<[ 23
:D3
5 2 :2>2:04245 45 DA :.1451624.< >D<. E V$F JD5 2I6.<;5
3
0824. 2
$
µ
E
#$ !?F
$=[ A :.1451624.< >D<. 45 #
µ
2I6.<;5 $H
6D2 <52:7K1:02
IF
7516C. JD5 3
P5 /.0 2>3
0:242
<56067G1:02 45
$
E
#$ !?F
23
:D3
5@
Ω
2F
$B[ A :0<:D07.
6L<05H :.16707DO
4. >.< DA :.1451624.< >D<. 45
#$
JD214.
'H'
µ
5 DA2
Ω H L 23
0A51724. >.< DA2 /.175 45 :.<<5175 23
75<1242 45
H #$ !? 23
:D3
5@
Ω
2F
0A>54K1:02 4. :0<:D07.
IF
01751604245 2I6.<;042
:F
7516C. 5A :242 53
5A517.
4F
6>.7G1:0262:70;2 5 <52:70;2
Q
5F 2B2 . 4028<2A2 ;5:7.<0234. :0<:D07.
$G[ A :0<:D07.
6L<05 L :.16707DO
4. >.< DA2 <56067G1:02 45
45 2D7. 014DBC. 08D232 $H
$$
E
:2F
! 5 :2>2:04245 45
µ
Ω H :.5/0:05175
7516C. 2>3
0:242 L45
23
:D3
5@
2F
<52:7K1:02 42 I.I012 5 4. :.1451624.<
IF
0A>54K1:02 4. :0<:D07.
:F
01751604245 2I6.<;042
4F
7516C. 5A :242 53
5A517.
5F
$
$
Ω
Ω
Ω
6>.7G1:0262:70;2H <52:70;2 5 2>2<5175
/F 2B2 . 4028<2A2 ;5:7.<0234. :0<:D07.
$H[ <G6<56067G1:0263
0824265A 567<53
2 :.A 15D7<. 2I6.<;5A <56>5:70;2A5175
5&
2F
7516C. 60A>3
56L45
H
H
Ω
Ω
$
H
23
:D3
5 .6;23
.<5645@
Ω
IF
:F
CENFIM
H
H
56
$\[ <G63
KA>242601:21456:51756H 3
0824265A 567<53
2 :.A 15D7<. 2 DA2 <545 45
$
2F
$$ H 7GA .6;23
.<5645
H
5
H
H
Ω H
Ω 5 ( Ω
H#
23
:D3
5@
IF
:F
$S[ <G6 <5:5>7.<56H 3
0824.6 5A 567<53
2 :.A 15D7<.H 7GA <56>5:70;2A5175 26
>.7G1:02645 #$$ WH '#$ W 5 $$$ W
2F
IF
H
<545 L45
$
$$
23
:D3
5@
5
H
Ω
H
Ω
Ω
:F
<3[ <G6:2<826014D70;263
0824265A 567<53
2 :.A 15D7<. Z<545 45
V
<56>5:70;2A5175 .6658D01756;23
.<56@
-
V $$
ΩH
-
V $$
Ω
V $$
$
Ω H - V $$ Ω H
$$
$$
V $$
7GA
ΩH
ΩH
Ω
2F
23
:D3
5 2 0A>54K1:02 45 :242 :2<82
IF
23
:D3
5 . ;23
.< 42 01751604245 5A 07.
4F
23
:D3
5 . ;23
.< 42 01751604245 1. 15D7<.
5F
23
:D3
5 . /2:7.< 45 >.7G1:02 45 :242 :2<82
Ω
< [ <G6:2<826<560670;2608D206H 3
0824265A 7<0K18D3
. 2
DA2 :.<<5175 E
45 3
01P2F :DR
2 01751604245 L
2F
01751604245 5A :242 :2<82
IF
<56067G1:02 45 :242 :2<82
$$
Ω
H 2I6.<;5A 42 <545
23
:D3
5@
:242 :2<82
:F 2B2 . 4028<2A2 ;5:7.<0234. :0<:D07.
4F
>.7G1:02 45 :242 :2<82
<$[ DA2 <545 45
$$
L3
0824. DA A.7.< 7<0/=60:. :DR
.6 51<.3
2A517.6 6C.
3
0824.6 5A 7<0K18D3
.
01751604245 12 3
01P2 L & Q . /2:7.< 45 >.7G1:02 4.
A.7.< L$H(
23
:D3
5@
2F
01751604245 5A :242 51<.3
2A517. 4. A.7.<
IF
0A>54K1:02 5A :242 /265 4. A.7.<
:F
>.7G1:02 2:70;2 2I6.<;042 >.< /265
Ω
<<[ 67<G6<2A.645 DA2 3
082BC. 5A 7<0K18D3
. 6C. :.16707DO
4.6>.< 7<G6:0<:D07.6
6L<05H <56>5:70;2A5175 :.A@
:.6ϕ 2 V H#H
V $$
V
Ω H :.6ϕ 3 V$H&
<545 L45
Ω
2F
<56067G1:02 45 :242 <2A.
IF
01751604245 JD5 >5<:.<<5 :242 <2A.
CENFIM
$$
Ω
ϕ1
ΩH
$
$$
V$H Q
V #$
ΩH
23
:D3
5@
Ω
57
:F
>.7G1:02 2:70;2 45 :242 <2A.
4F
>.7G1:02 <52:70;2 45 :242 <2A.
<=[ A :0<:D07. 45 03
DA012BC. L :.16707DO
4. >.< &$ 3
KA>2426/3
D.<56:5175645 #(
W
$
/2:7.< 45 >.7G1:02 45 :242 3
KA>242 L $H#
63
KA>2426 567C.
08D23
A5175 4067<0IDO
426 >53
26 7<G6 /2656 45 DA 60675A2 7<0/=60:.H :DR
2 7516C.
:.A>.672 L45 $$
#$ !? 23
:D3
5@
2F
>.7G1:02 2:70;2 12 016723
2BC.
IF
01751604245 12 3
01P2 JD5 23
0A5172 . :0<:D07.
:F
>.7G1:02 2>2<5175 42 016723
2BC.
4F
:2>2:04245 45 :242 :.1451624.<H 2 3
082< 5A 7<0K18D3
.H >2<2 5/5:7D2< 2
:.A>5162BC. 7.7234. /2:7.< 45 >.7G1:02 42 016723
2BC. E
:.6ϕ V F
<B[ DA2 <545 7<0/=60:2 45
$
$$
#$ !? 567= 3
0824. DA <5:5>7.< 7<0/=60:.
:.16707DO
4. >.< 7<G6 I.I0126 08D206 45
<56067G1:02 3
0824265A 7<0K18D3
.
µ
$
Ω 45 <52:7K1:02 5
$
Ω 45
23
:D3
5@
2F
01751604245 JD5 >2662 5A :242 I.I012
IF
01751604245 12 3
01P2 JD5 23
0A5172 . <5:5>7.<
:F
/2:7.< 45 >.7G1:02 42 :2<82
4F
:2>2:04245 45 :242 DA 4.6:.1451624.<562 3
082< 5A 7<0K18D3
. 45 A.4. 2
/2?5< 2 :.A>5162BC. 7.7234. /2:7.< 45 >.7G1:02 E
:.6ϕ V F
µ
<G[
I65<;5 2 /08D<2 5 :23
:D3
5@
2F
01751604245 0140:242 >53
. 2A>5<O
A57<.
IF
/2:7.< 45 >.7G1:02 42 :2<82
:F
>.7G1:02 2:70;2
4F
:2>2:04245 45 :242 DA 4.6 :.1451624.<56H 5A 7<0K18D3
.H 45 A.4. 2
:.A>5162< 7.723
A5175 2 :2<82 E
:.6ϕ V F
CENFIM
µ
58
<H[ A A.7.< 7<0/=60:. 75A DA2 >.7G1:02 S70345 #
H
η V($eH :.6ϕ V$H(#H
:V $$
23
:D3
5@
2F
01751604245 2I6.<;042 >53
. A.7.<
IF
:2>2:04245 45 :242 :.1451624.<H 3
0824.6 5A 7<0K18D3
.H 45 A.4. JD5
:.6ϕ V
:F
µ
01751604245 7.7232I6.<;042 45>.0645 :.A>51624. . A.7.<
<\[
7516C. 42 <545 42 /08D<2 L45 $$
.A . 0175<<D>7.<
.3
082<
4563
0824. .62>2<53
P.6A2<:2;2A@ #
H A2<:2<2A@
5 ( _W
5 ( _W
23
:D3
5@
2F
/2:7.< 45 >.7G1:02 42 :2<82 014D70;2
IF
>.7G1:02 <52:70;2 7<0/=60:2 42 :2<82 014D70;2 E
+ 010:023
F
:F
/2:7.< 45 >.7G1:02 /01234. :0<:D07.
4F
>.7G1:02 <52:70;2 7.7234. :0<:D07.H :.A
3
0824. E
+ /0123
F
5F
:2>2:04245 45 :242 :.1451624.<
µ
CENFIM
59
( %
( %
;
.
.
'
'
(
672 D104245 >5428980:2 ><575145D /.:2< .6 26>5:7.6 A206 <53
5;21756 <53
2:0.124.6
:.A 2 :.<<5175 23
75<1242H 7514. 5A :.16045<2BC. . 8<2D 45 2><5140?285A 2 JD5 65
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 VAGOS
MATIAS)
CATÁLOGO GERAL-91 (MERLIN GERIN)
ELECTROTECNIA (PARANINFO-PABLO ALCADE SAN MIGUEL)
CENFIM
60