Electrotecnia Apuntes y Problemas 08-09
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&& '4* *7-*?%- 9'"45=O "• G $-:.-% ' ;%-%#E'
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(=O =
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11
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4))$
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- ))
6
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4* &.07/$
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# $ A
A % T4))@E @E """"">l-!?%5#@*"*" >l-!?%5#@*"%".2/E" E".#;/
= = =
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JRgRC,h#E
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=O =
=O −
=
II&&&3C/0-E'
=O 6 + Ω = Ω
/8G % 119@@T$ 11$$ ,>-L@-H5' 1G$'
=O=O
=O
(=,H
11
$
−
+
= = =Ω
;)2 1$G$;$G -GH)$ ) ' 8
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=
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EII'4)) )) ''' & & J"4))$
)) [
- ))
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4*@E &.07/$
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6
6 9
9
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= = =
>l-!?%5#@*"%".2/E"#;"
. / . /. /
9 6 9 6
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+
11!&=;$$1 $$$$1$:1$=$1$ $$$$17 7)$$119'
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A70""5W I/!H)$#)9$#$:$$1$ $''$$ $9 $#$$ $;$G$1 '/7 7$ $91 $#$ 11$;$ /0-'
=O =
c
O
=O = O
ϕ
ϕ−
=
< < −
II&&3 K &C-EC/0-E'( )=O =O 6 ++ Ω
&''' !" -./-0
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/8G % 7 71$$1-T:1@@T$ 11$$ ,>-L@-H5' 1G$'
( ) ( )=O =O
HH11OHH11O
(= (=,,
= 11 11
11 11
)) [
- ))
7
$
6 6
2% 9
9
−= = = =
+ + Ω Ω
= Ω = Ω== Ω
= Ω
;)2 1$G$;$G -GH)$ ) '
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HH11O,, $−
=
$ 7$9$1 7 77 7 7G$'
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=
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=
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E&&'4#*E@8#@5%8E-78#% "#9 *%- %G8#5*%- % "" P*"8))+ "# + "
4*E@8#@5%8E-$
- 6 = − = Ω
4*E
# $ A
*A*>4#*@E @E """"">l-!?%5#@*"*" >l-!?%5#@*"%".2/E" E".#;/
= = =
= = =
$ F$1'':; $111952 G$ 1' G19'$1 71 '>E-!%8#E@8%8#%6#@57l@G%*E--7#@6##@#*E@8#@5%8E-"4 8 =O "* " )))+"• ;?@6ER%S$* 0 "• ;?@6ERFSG& 0"
A70""5W I
=× × × ×C, E
ω πω πω πω π
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=O +=
=O =
&&&&3CQ0-E'
=O 6 −
Ω = − Ω
/8G % 1G19@@T$ 11$$ ,>-L@-H5' 1G$'
=O=O
=O
(=,H
11
$
+
−
= = =Ω
;)2 1$G$;$G -GH)$ ) ' 8
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=O(= =
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" ' 0 +"
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=
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;A20705r
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r4; 6; "
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48 ' ) < )) *r'"""/!& I&• $ "5
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c
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4 !E6E-#5$
8 "krq4=O"*4=OA=" $c]-:$ $c97=;$G$ $c '
; % - "#+$? 1*2 (=2*rA=H"*7;; "
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ϕ ϕ ϕ
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< <
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8G %? (1==f (=2"* "
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L! >' ''&'LL'& 3'>''!& '&'>',' & &''&'>'>'''&'& 3'4# "" ' & "4#)+ 6)5?!%-6#@57E@#5#@*"%" "" ))"4; ""9+";>E-!%F7@l!7*%"4-6 ))"
=O =O =O- $ F)$ *$ + −Ω Ω Ω
4T &)F)* "#$6"""7$7 "$F)"*$*"#4474*4##447$#))#$ =O7$ "74*4#$#*7$ =O#$
&''' !" -./-0
>/!)2.?
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RQ,Q>Q'''Q/''!&1 $ 1 $$!&VG $P $ G8 $G$H:;H I3'
+R+Q+,Q+>Q'''Q+
/8G % 1 $$1-@T=7 7JRB@T: $ JR.0T
=O
=O
=O
((O
- $ =1 .=1 =/
F)$ N1 .= N1/
*$ H .= H/
.=1 H/ (1N
6
6
6
9 6
+
+
+
= Ω = + Ω= Ω = + Ω= Ω= − Ω
= + Ω = Ω
4# + "/8G % 1 $$1.-T=7 7>--M: $ ,-i$ 1@-H5'+
(
=O
=O
=O
F)$ 1= (== = ,,
*$ 11
1= = =
- $ H= .H= =/
F)$ ,, .= ,,/
*$ 11 .= 11/
.H= ,/
' 8
' 4
6
6
6
9 6
ω π
ω π
−
−
+
+
= = × × = Ω
= = = Ω× ×
= Ω = + Ω= Ω = + Ω= Ω= − Ω
= − Ω = H(O(, − Ω
&''' !" -./-0
>/!)2.@
>',' & &''&'4>""5?!%6#@57E@#5;%-*7%#56#@57l@6E6%"4*5?!%@\!#-E5*E!;#:E5"/8G ''%6 =1H[ 5#-7#((2"* "O- 6 "-6A.=L1LH/[A(([O E"%5 "
((
((
$
= = =
Ω
(O E"2A-7A=[%A=22A-7A1[%A122(A-(7AH[%AH2,O) "26A((2/8G ''% 1 $$1.-T=7 7>--M: $ ,-i$ 10=>?L@-H5'+O76 "
(
=O
=O
=O
F)$ 1= (== = ,,
*$ 11
1= = =
- $ H= .H= =/
F)$ ,, .= ,,/
*$ 11 .= 11/
.H= ,/
' 8
' 4
6
6
6
9 6
ω π
ω π
−
−
+
+
= = × × = Ω
= = = Ω× ×
= Ω = + Ω= Ω = + Ω= Ω= − Ω
= − Ω = H(O(, − Ω
&''' !" -./-0
>/!)2.A
O E"%5 ">;"
=OH(O
H(
(,
(,
$
+
−
= = =Ω
(O E"
=O LH(O LH(O
L=O LH(O =H(O
( *( =O LH(O 4N=NO
2 A- 7AH= %AH= 2
2 AP 7A,, %A,, 2
2 AP 7 A11 %A11 2−
Ω
Ω
,O) "
=O LH(O LH(O
L=O LH(O =H(O
( *( =O LH(O
2 A- 7AH= %AH= 2 A .(L+=/2
2 AP 7A,, %A,, 2 A .4(L+HH1/2
2 AP 7 A11 %A−
Ω
Ω
4N=NO
6 =O
11 2A.1(HH4+,N/2
2 .(,L + =/2A(, =
H(OH= +
=H(O,, +
N=NO11 −
26A =O(,
&''' !" -./-0
>/!)2.B
>'>'''&'4G .))/ ./4'"4@ )+1= )+ ) # "45 -#5E@%@*7%)) l"PAP*
ω ω ω πω
π
= = = = = =
=
• $T "• $* % ))T"• *$**>"
4# 5#-7#6#@57E@#5)) "
L=O * 4=OP AP .= / .= / 6 6
+1
→ + = −
4%PP*"5))9 ))"/8G $ '?))H1T1`> 1=[ ""1=2)"#& "T ./" TRSR*S"
(((
(
H1T 0T H1T TH1 =
= T= T T
1 > 0> 1 > 1 =
= >= > >
=N
H1 = 1 =
# 8
4# 4
8 4
µ µµµ
π π
−
−
− −
→ ×→ = = ×→
→ ×→ = = ×→
= = =× × ×
&''' !" -./-0
>/!)2..
(=O
=O
=O =O =O
=O
=N H1 = 1H(=
1H(=
=N 1 =
1= 1H(= 1H(= .1= =/
.= 1H(=/
.= 1H(=/
.1= =/ 1=
' 8
' 4
9 6
6
6
6
ω π
ω π
−+
−−
+ −
= = × = Ω
= = = Ω×
= Ω+ Ω+ Ω = + Ω+ Ω− Ω+ Ω = Ω
4*F "4; E"
=O=O
=O
1=
1=
$
= = =
Ω
4; E"
=O =O =O
=O =O =O
=O =O =O
1= 1=
1H(= 1H(=
1H(= 1H(=
$
$
$
+ +
− −
= × = Ω × =
= × = Ω × =
= × = Ω × =
?0 -#5E@%@*7%5#-7# "# "* ))"
L=O2 A1H( 2
=O7A $
- =O2 A1= 2
* 4=O2 A1H( 2
&''' !" -./-0
?/!)2.0
L! ?' !''&'LL!'& 3'?''!& !'&'?',' &&& !&''&'?'>'!''&'?'?'!& '?'@'&!c'(&!c!& '?'A'K &3&& ''K' ?'A'') G 41$ $ 1 $'
?'A',') G 1)F'11 $ 1 $'?'A'>'1 $ $ 1$G1 $''
& 3'4# "" ' & "4#)+ 6)5?!%-7@6#@578%8#5#@*"%" "" ))"4;7 ""9+";>E-!%F7@l!7*%"4-6 ))"
=O =O =O- $ F)$ *$ + −Ω Ω Ω 4T &)F)* "#$6"""7$7 "$F)"*$*"#4474*4##447$#))#$ =O7$ "74*4#$#*7$ =O#$
&''' !" -./-0
?/!)20-
?''!& !'&'/$9 ;''1 !$9$ $9$$1 $$'
(
"""
,
=+ + + +
/''!&1 $ 1 $$!&VG $P $ G8 $G$H:;H I3'
(
"""
,
9
9 9 9 9
=+ + + +
/ $G 7$ $915$1#G ;G $ ) $:2 ' ) ;$$1$=$9H$ 1$:2 $$G 71'/8G ''%6 =1H[ ;%-%#E((2"* 7 "O7 "
((((
=
$
= = =
Ω
((
1
$
= = =
Ω (
(
((,1
H
$
= = =
Ω
O5 "76A7L7L7(A((%L%L,1%A1%(O5- E"
(((,H
1
$= = = Ω
J)- ;"
(
(,H
== 1 H
= = = = Ω+ + + +
Ω Ω Ω
&''' !" -./-0
?/!)20
8G ''#+$? ;%-%#E =1[))PAN1[P*AH[ (=2M1="* 76 "
j4*E )"
= + +
=O
=O=O
=O
4=O=O
=O
L=O=O
(
(=- $ = .= =/
=1
(=F)$ (, .= (,/
N1
(=*$ 1H, .= 1H,/
H
.= N=/ 1(
$ 6 $
$ 6 $
$ 6 $
6 $
+
−
= = = = +Ω
= = = = −Ω
= = = = +Ω
= + =
=O
=O
(=O(=O
(==H1
1(
$
9
$ −= = = Ω
4# + "
P
-
P*
&''' !" -./-0
?/!)20,
/8G % 1 !$$1.-T=7 7>--M: $ ,-i$ 1! ,>-/@-H5'+
(
=O
=O=O
=O
4=O=O
F)$ 1= (== = ,,
*$ 11
1= = =
(=- $ NN .NN =/
H=
(=F)$ ,,= .= ,,=/
,,
*
' 8
' 4
$ 6 $
$ 6 $
ω π
ω π
−
−
+
= = × × = Ω
= = = Ω× ×
= = = = +Ω
= = = = −Ω
=O
L=O=O
(NO
=O
(NO(NO
(=$ ,,1 .= ,,1/
11
.NN =1/ H
(=,=N
H
$ 6 $
6 $ $
9
$
+
−
+
−
= = = = +Ω
= − =
= = = Ω
=OA,, $−
=ONN $
6 =O2 A(=
6 (NO7 AH $−
* =O7 A,,1 $+
&''' !" -./-0
?/!)20>
?',' &&& !&''&'4>""5?!%7@6#@578%8#5;%-*7%#57@6#@578%86E6%"4*5?!%@\!#-E5*E!;#:E5"4# 7!;#8%@*7%6E6%"
?'>'!''&'4G .))/ ./4'"4@ )+1= )+ ) # "45 -#5E@%@*7%)) l"PAP*
ω ω ω πω
π
= = = = = =
=
• $T "• $* % ))T"• *$**>"
4# ;%-%#E7@6#@578%8#5)) "E5#%@?%@"
L=O * 4=OP AP .= / .= / 6 6
+1
→ + = −
4%PP*;%-%#Ex"5))9 ))"!3&c4#*E@8#@5%8E- $ @-H5 7@8?*672E*r"
&''' !" -./-0
?/!)20?
/8G $ !'?))H1T1`> 1=[ ; ""1=2)"#& "T ./" TRSR*S"
(((
(
H1T 0T H1T TH1 =
= T= T T
1 > 0> 1 > 1 =
= >= > >
=N
H1 = 1 =
# 8
4# 4
8 4
µ µµµ
π π
−
−
− −
→ ×→ = = ×→
→ ×→ = = ×→
= = =× × ×
(
=O
=O
=N H1 = 1H(=
1H(=
=N 1 =
' 8
' 4
ω π
ω π
−+
−−
= = × = Ω
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Ω× Ω×= = + + = Ω+ Ω+ = ΩΩ
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