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8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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UNIVERSIDAD NACIONAL DE ROSARIOFACULTAD DE CIENCIAS EXACTAS, INGENIERIA Y AGRIMENSURA
FISICA IIIDEPARTAMENTO DE FÍSICA Y QUÍMICA (E.F.B.)
ACTIVIDAD DE LABORATORIO Nº5
TITULO “CIRCUITOS EN CORRIENTE ALTERNA”
RESUMEN: En el !"#!en$e $%&'& *%+$!, !n$en$&%e- '$ene% l& &%!&!ne/e l& n/!!ne /e 0#n!n&-!en$ /e !%#!$ &*&!$! e!n/#$!, #&n/ l& 0%e#en!& /e en$%&/& &%1&, 2 l& e3&l /e en$%&/&e #n& n/& &l$e%n& !n#!/&l.
0
PROFESOR:
• Te%1&4 PROF. CARLOS ABECASIS• P%+$!& 4 PROF. MARICEL MATAR• L&'%&$%!4 PROF. CARMELO CELAURO
ALUMNOS:• CAPELLO, N!l+ C567789:• GIO;ANON, B%#n G56:669<• MARCU==I, F&'! M567>9:• ;ENTICINQUE, P&'l ;5@67@97
HORARIO DE LABORATORIO:
;!e%ne /e @@4?? @64??
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OBJETIVOS
An&l!&%e- el -*%$&-!en$ /e l !%#!$ RL, RC, LRC, en %"!-en /e%%!en$e &l$e%n&.
INTRODUCCIÓN TEÓRICA
CIRCUITO SERIE EN CORRIENTE ALTERNA
L& !"#!en$e 0!"#%&, n -#e$%& l& e$%#$#%& l+!& /e #n "ene%&/% !n/#$%!&len l& #&l e &*%e!& l& e$%#$#%& /el !n/#$% 2 /el !n/#!/. Al%e/e/% /e l&!%#n0e%en!& !n$e%n& /el e$&$% e$+n /!$%!'#!/ !e%$ n-e% /e *&%e /e*l. C- &/& n/#$% !$#&/ '%e l& #*e%0!!e
/el !n/#!/ %$% %$& el &-* -&"n$!, e*%/#e en l #n& 0e- !n/#!/&, en /e$e%-!n&/ en$!/#&n/ el n/#$% *&& *% #n *l n%$e 2 en en$!/*#e$ #&n/ *&& *% #n *l #%. L& 0e- !n/#!/&e, *% n!"#!en$e, &l$e%n&, 2 el n-e% /e !l-*le$ en &/& #el$& e !"#&l &l n-e% /e *&%e /e*l. E$& /!*!!n -#l$!*l&% *e%-!$e &l&n&% #n&0%e#en!& #0!!en$e-en$e "%&n/e !n #e e& nee&%!#$!l!&% #n& el!/&/ &n"#l&% *el!"%&-en$e ele&/&.L& 0e- !n/#!/& /e #n &l$e%n&/% !n/#$%!&l *#e/e /!0e%!% /e l& 0%-& !n#!/&l*e%0e$&, *e%, & -en #e n e /!"& e*%e&-en$e $%& &, #*n/%e-#e #$!l!&- #n &l$e%n&/% #e -&n$!ene en$%e # '%ne #n& /!0e%en!& /e*$en!&l !n#!/&l /&/& *%4
)( t senV v m ω = (@)
!en/ l& /!0e%en!& /e *$en!&l !n$&n$+ne&, V m l& /!0e%en!& /e *$en!&l-+!-& 2 ω l& 0%e#en!& &n"#l&%, !"#&l &l n-e% /e %el#!ne *% e"#n//el %$% -#l$!*l!&/ *% el n-e% /e *&%e /e *l. En N%$e&-%!&, *&%&-#H /e l &l$e%n&/%e, 0
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!en/ !, /! 9 /$ 2 l& !n$en!/&/ !n$&n$+ne& /e l& %%!en$e, # /e%!&/& %e*e$&l $!e-* 2 l& &%"& /el n/en&/%, %e*e$!&-en$e. De%!&n/ e$& e#&!n%e*e$ & $, 2 #$!$#2en/ / 9 /$ *% !, e '$!ene4
iC dt
id L
dt
di Rt V m .
1..)cos(
2
2
++=ω ω
E$& e #n& e#&!n /!0e%en!&l /e e"#n/ %/en, #2& l#!n e4
bt m At sen
C L R
V i −+−
−+
= ε ϕ ω
ω ω
.)(.
)1
( 22
/n/e
R
C L
Tg ω ω
ϕ
1−
=
E$& e#&!n *&%ee -*l!&/& *e% # !n$e%*%e$&!n n e /!01!l.
Cn!/e%e- en *%!-e% l#"&% el $%-!n A.ε5'$. L& -&"n!$#/e A 2 ' /e*en/en /el& n$&n$e /el !%#!$ 2 /e l& n/!!ne !n!!&le, e$ e, /e l &l%e /e;, !, /!9/$ 2 en el !n$&n$e /e e%%&% el !%#!$. El 0&$% ε5'$ /e%eee*nen!&l-en$e n el $!e-* 2 e H&e /e*%e!&'le &l &' /e #n $!e-*#0!!en$e (/e %/!n&%! -#2 *e#e3). E$e 0&$% e $%&n!$%! 2 n#e en l&*%+$!& l& /!0e%en!& /e *$en!&l 2 l& %%!en$e $%&n!$%!& *#e/en &/#!%!% &l%e *el!"%&-en$e "%&n/e, n l $en/%e- en #en$& 2 ln!/e%&%e- el *%!-e% $%-!n, #e e /en-!n& l#!n /el e$&/e$&!n&%!.
Se e #e l& %%!en$e %%e*n/!en$e &l e$&/ e$&!n&%! &%1&!n#!/&l-en$e n el $!e-*, l -!- #e el l$&e en$%e l '%ne. S# &l% -+!- e4
22)
1(C
L R
V I mm
ω ω −+
=
L& !n$en!/&/ /e l& %%!en$e en el e$&/ e$&!n&%! *#e/e e%!'!%e, *% $&n$,
)(. ϕ ω −= t sen I i m (:)
2
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L& 0%e#en!& /e l& !n$en!/&/ /e l& %%!en$e e, *% $&n$, l& -!-& #e l& /ell$&e, *e% &-'& /!0!e%en en l& 0&e, e&, e$+n /e0&&/& #n +n"#l ϕ.
In$%/#&- l& !"#!en$e !-*l!0!&!ne4
Z X R
X X X C
L
X C
X L
C L
C
L
=+
=−=−
=
=
22
1
1
ω ω
ω
ω
En$ne,
Z
V
X R
V
X X R
V I mm
C L
mm =
+
=
−+
=2222 )(
(6)
R
X Tg =ϕ (8)
L& -&"n!$#/ = e /en-!n& !-*e/&n!& /el !%#!$K X e l& %e&$&n!&, 2 X L 2 XCn l& %e&$&n!& !n/#$!& 2 l& %e&$&n!& &*&!$!&, %e*e$!&-en$e. L&!-*e/&n!& 2 l& %e&$&n!& e e*%e&n en OH-.
L& !n$en!/&/ -+!-& /e %%!en$e e$+ %el&!n&/& n l& -+!-& /!0e%en!& /e*$en!&l *% #n& e#&!n #e $!ene l& -!-& 0%-& #e l& le2 /e OH-, *&%& l&
%%!en$e n$!n#&, %%e*n/!en/ l& !-*e/&n!& = & l& %e!$en!& R.
MEDIAS CUADRÁTICAS O VALORES EFICACES
L &l%e !n$&n$+ne /e #n& !n$en!/&/ /e %%!en$e, 0e- /!0e%en!& /e*$en!&l &l$e%n&, &%1&n /e #n -/ n$!n# /e/e #n &l% -+!- en #nen$!/, *&&n/ *% e%, H&$& #n -+!- en en$!/ *#e$, 2 &1#e!&-en$e. E -+ -/ e$#/!&% l& %%!en$e &l$e%n& n!/e%&n/ #valores cuadráticos medios en l#"&% /e # &l%e -+!-.
El &l% #&/%+$! -e/! /e l& !n$en!/&/ /e #n& %%!en$e &%!&'le e /e0!ne- el &l% /e l& !n$en!/&/ /e #n& %%!en$e n$&n$e #e /e&%%ll&e l&-!-& &n$!/&/ /e &l% en !"#&l $!e-* 2 en l& -!-& %e!$en!&K 2 *% e$&%&n e /en-!n& &l &l% #&/%+$! -e/!, valor eficaz /e l& %%!en$e &%!&'le.L& /e%!&/& %e*e$ &l $!e-* /e l& &n$!/&/ /e &l% /e&%%ll&/& en #n&%e!$en!& R, #e $%&n*%$& #n& %%!en$e &l$e%n& !n#!/&l )(. t sen I i m ω = , e4
)(.. 222 t sen R I Ri m ω =
L& &n$!/&/ $$&l /e &l% /e&%%ll&/& en #n $!e-* T, !"#&l & #n *e%1/, e4
3
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RT I t sen R I Rdt i H m
T
m
T
2
0
22
0
2
2
1)(.. === ∫ ∫ ω
L& &n$!/&/ /e &l% /e&%%ll&/& *% #n& %%!en$e /e !n$en!/&/ n$&n$e I e enel -!- $!e-* e4
RT I H e .2
=
P#e$ #e *% /e0!n!!n /e I e l& &n$!/&/e /e &l% n !"#&le,
2.
2
1.
22 meme
I I RT I RT I =⇒=
P% n!"#!en$e, ! l& !n$en!/&/ /e #n& %%!en$e &%1& !n#!/&l-en$e, # &l% e0!& e 2/1 ≈ ?,? ee # &l% -+!-. A!-!-, el &l% e0!& /e #n&/!0e%en!& /e *$en!&l #e &%1e !n#!/&l-en$e e 2/1 ee # &l% -+!-. S! /!!/!en/ *% 2/1 el *%!-e% 2 el l$!- -!e-'% /e l& e#&!n(3), '$ene-4
Z
V I mm 2/
2=
e&,
Z
V I ee = (>)
De &H%& en &/el&n$e, e !n$e%*%e$&%+ #e l& le$%& I , ε V !n #'1n/!e, e%e0!e%en & l &l%e e0!&e /e l& -&"n!$#/e %%e*n/!en$e. L& e#&!n(5) e e%!'!%+, *% $&n$,
Z
V I =
Cn!/e%e- &H%& l 0&$%e #e /e$e%-!n&n l& %e!$en!&, %e&$&n!& e!-*e/&n!& /e #n !%#!$. En *%!-e% l#"&%, l& %e!$en!& /e #n n/#$% #e$%&n*%$& #n& %%!en$e &l$e%n& *#e/e n e% l& -!-& #e l& %e!$en!& /el
-!- n/#$% #&n/ $%&n*%$& #n& %%!en$e n$&n$e. L& /!0e%en!& e/e'e &l HeH /e #e l& /en!/&/ /e %%!en$e en #n H!l #e $%&n*%$& #n&%%!en$e &l$e%n& n e #n!0%-e en $/& l& e!n /el -!-, !n -&2% #n$ &l& #*e%0!!e. E$e 0en-en e ne n el n-'%e /e e0e$ %$!&l “e0e$!n”. L& e!n e0e$!& /el n/#$% e %e/#e 2 # %e!$en!& -en$&. Ele0e$ %$!&l e %!"!n&/ *% l& 0e- $!n/#!/& %e&/& *% l& &%!&!ne /el0l# !n$e%n en #n n/#$%, 2 e -&2% #&n$ -+ ele&/& e l& 0%e#en!&. A#n#e e$e e0e$ e #n 0&$% !-*%$&n$e *&%& l& 0%e#en!& #$!l!&/& en%&/!, *#e/e /e*%e!&%e /e %/!n&%! *&%& l& 0%e#en!& /e
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#n& $!n/#!n, X L :π0.L, e *%*%!n&l & l& e &l e0!!en$e /e$!n/#!n 2 & l& 0%e#en!&. S! H&2 #n nle /e H!e%% &!&/ &l !n/#$%,en$ne el e0!!en$e /e $!n/#!n n e n$&n$e, *e% /e n#e *&%&-&2% en!lle n $en/%e- en #en$& e$& &%!&!n.
L& %e&$&n!& &*&!$!&, X C @ 9 :π0.C, e !ne%&-en$e *%*%!n&l & l&&*&!/&/ 2 & l& 0%e#en!&.
L& %el&!n en$%e l& !-*e/&n!& = /e #n !%#!$ en e%!e 2 l &l%e /e R, X L 2XC *#e/e %e*%een$&%e "%+0!&-en$e n!/e%&n/ $/& e$& -&"n!$#/e -e$%e. L& %e!$en!& R e %e*%een$& *% #ne$% !$#&/ '%e el ee X, 2 #2 en$!/!n!/e n el en$!/ *!$! /el -!-K 2 l&%e&$&n!& XL 2 XC *% e$%e !$#&/ '%eel ee Y 2 en l en$!/ *!$! 2 ne"&$!,%e*e$!&-en$e. L& !-*e/&n!& e el e$% #-& "e-$%!& %e#l$&n$e /e e$ $%ee$%e, - e e en l& !"#!en$e 0!"#%&. E$&0!"#%& e H& /!'#&/ *&%& el & X L XC 2, *% $&n$, X e *!$!&. S! XL XC, %e#l$& X ne"&$!&2 e /!%!"e H&!& &'& en e /e H&!& &%%!'&.
RELACIÓN ENTRE LAS FASES DEL VOLTAJE Y DE LA INTENSIDAD DECORRIENTE
L& !n$en!/&/ e0!& en #n !%#!$ en e%!e e !"#&l & l& %&n /el l$&ee0!& & l& !-*e/&n!& /el !%#!$. Cn!/e%e- &H%& l& 0&e /e l& !n$en!/&//e %%!en$e %e*e$ & l& /el l$&e. L& e#&!ne (1), (2) 2 (4) *#e/ene%!'!%e #n$& *&%& -&2% -/!/&/,
R
X Tg
t sen I i
t senV v
m
m
=
−=
=
ϕ
ϕ ω
ω
)(.
)(.
El &l% ω %e*%een$& #n +n"#l (en %&/!&ne), 2 # &l% en #&l#!e% !n$&n$e $ e/en-!n& 0&e /el ;l$&e. An+l"&-en$e, l& -&"n!$#/ (ω$5ϕ), e /en-!n& 0&e/e l& !n$en!/&/ /e l& %%!en$e. Se /!e #e l& %%!en$e *%een$& #n& /!0e%en!&/e 0&e n el l$&e, #e e$+ /e0&&/& #n +n"#l ϕ %e*e$ &l l$&e.P#e$ #e l& %e&$&n!& X *#e/e e% *!$!& ne"&$!&, l -!- le #e/e &l+n"#l ϕ. S! XL XC, X e en$ne *!$!&, ϕ e *!$!, 2 l -+!-,-1n!-, e$., /e l& !n$en!/&/ /e l& %%!en$e &*&%een en !n$&n$e *$e%!%e#e l #e %%e*n/en &l l$&e. Se /!e #e l& !n$en!/&/ e$+ %e$%&&/&
%e*e$ &l l$&e. P% el n$%&%!K ! XL XC, X e ne"&$!&, ϕ $&-'!n ene"&$! 2 l& !n$en!/&/ e$+ &&n&/& %e*e$ &l l$&e.
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El +n"#l ϕ *#e/e /e$e%-!n&%e !n-e/!&$&-en$e -e/!&n$e el /!&"%&-& /el e$% !-*e/&n!&, *#e$ #e R X Tg /=ϕ . C- & e*e!&l, e e!/en$e #e ! #n!%#!$ e$+ 0%-&/ *% #n& %e!$en!& *#%& ne$&/& & #n& /!0e%en!& /e*$en!&l &l$e%n&, X ?, = R, ϕ ?, 2 l& !n$en!/&/ 2 el l$&e e$+n en 0&e,
- !n/!& l& !"#!en$e 0!"#%&.
S! el !%#!$ e -*ne /e #n& $!n/#!n *#%&, R ?, = X L, 2/π ϕ += , 2 l&!n$en!/&/ e$+ %e$%&&/& %e*e$ &l l$&e #n +n"#l /e π 9 : %&/!&ne, 7?,e"n e e en l& !"#!en$e 0!"#%&.
S! el !%#!$ n$!ene l &*&!/&/, R ?, = X C, 2/π ϕ −= 2 l& !n$en!/&/ e$+&&n&/& %e*e$ &l l$&e #n +n"#l /e 7?, - !n/!& l& 0!"#%&
L& !n$en!/&/ /e l& %%!en$e $!ene l& -!-& 0&e en $/& l& *&%$e /e #n !%#!$
en e%!eK e&, e -+!-& en l& %e!$en!&, l& $!n/#!n 2 el n/en&/% &l
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-!- $!e-*K n#l& en l $%e #n !n$&n$e /e*#K -+!-&, *e% /e en$!/*#e$, $% !n$&n$e $/&1& *$e%!%, 2 &1 #e!&-en$e.
DIFERENCIA DE POTENCIAL ENTRE LOS PUNTOS DE UN CIRCUITORECORRIDO POR UNA CORRIENTE AL TERNA
L& /!0e%en!& /e *$en!&l e0!& en$%e / *#n$ #&le#!e%& /e #n!%#!$ en e%!e e !"#&l &l *%/#$ /e l& !n$en!/&/ e0!& *% l& !-*e/&n!& /el!%#!$ en$%e l / *#n$ n!/e%&/, !e-*%e #e n e!$& n!n"#n& 0e--*%en/!/& en$%e ell4
abab Z I V .=
L& /!0e%en!& /e 0&e ϕ en$%e V ab e I e4
ab
ab
R
X Tg =ϕ
En l& 0!"#%&, l& !-*e/&n!& Z ab en$%e & 2 ' e !"#&l & R, *#e$ #e n H&2 $%ele-en$ /e !%#!$ en$%e e$ *#n$. P% n!"#!en$e, V ab I.R 2 ϕ &% $"? ?. E$ e, l& /!0e%en!& /e *$en!&l en$%e l $e%-1n&le /e #n& %e!$en!&*#%& e$+ en 0 &e n l& !n$en!/&/ /e l& %%!en$e en l& %e!$en!&. En$%e l
*#n$ ' 2 , Z bc X L, V bc = I.X L, ϕ &% T" ∞ π 9 :. L& /!0e%en!& /e *$en!&len$%e l e$%e- /e #n& $!n/#!n *#%& e$+ &&n&/& 7? %e*e$ & l&!n$en!/&/. En$%e l *#n$ 2 /, Z cd X C , V cd I.X C , ϕ &% $" (5∞) 5 π9:. L&/!0e%en!& /e *$en!&l en$%e l '%ne /e #n n/en&/% e$+ %e$%&&/& 7?%e*e$ & l& %%!en$e en el n/en&/%.
L& #-& /e l& /!0e%en!& /e *$en!&l e0!&e en$%e l e$%e- /e!e%$ n-e% /e ele-en$ /e #n !%#!$ en, e%!e, n e !"#&l & l& /!0e%en!& /e*$en!&l e0!& en$%e l e$%e- /el !%#!$ en n#n$. E$& &n-&l1& e
e*l!&, n '$&n$e, en!ll&-en$e, ! e $!enen en #en$& l& %el&!ne /e 0&een$%e l& /!0e%en!& /e *$en!&l /e l& /!$!n$& *&%$e.
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L& l1ne& llen& %e*%een$& l& !n$en!/&/ !n$&n$+ne& /e l& %%!en$e, l& -!-& en$/& l& *&%$e /el !%#!$, 2 #e $!ene #n &l% -+!- /e 2.e I A. L& $%&$%e #%& %e*%een$&n l& /!0e%en!& /e *$en!&l !n$&n$+ne& en$%e & 2 'K ' 2, 2 2 /, #2 &l%e -+!- n 2.abV , 2.bcV 2 2.cd V %e*e$!&-en$e, 2 #2& /!0e%en!& /e 0&e n %el&!n & i n l& %e5*%een$&/&.
L& /!0e%en!& /e *$en!&l !n$&n$+ne& v ad e !"#&l & l& #-& /e l& /!0e%en!& /e*$en!&l !n$&n$+ne& v ab, v bc 2 v cd . S! e #-&n l& $%e #%& /e $%&, l& #%&'$en!/& %e*%een$&, *% $&n$, l& /!0e%en!& /e *$en!&l !n$&n$+ne& en$%e & 2 /.
DIARAMAS DE VECTOR ROTATORIO
L& /!0!#l$&/ e!/en$e /e /!'#&% e !n$e%*%e$&% /!&"%&-& $e-*%&le -el &n$e%!%, H&e /ee&'le !n$%/#!% &l"n -$/ "%+0! -+ en!ll *&%&%e*%een$&% l& %el&!ne /e 0&e en$%e !n$en!/&/e 2 /!0e%en!& /e *$en!&l enl !%#!$ /e %%!en$e &l$e%n&, l #e *#e/e ne"#!%e - !"#e.
S#*n"&- #e e /ee& %e*%een$&% #n& %%!en$e #e &%1e !n#!/&l-en$e,/e 0%e#en!& 0 2 &l% -+!- I m. Cn$%#2&- #n e$% I m, - el /e l& 0!"#%&& !e%$& e&l& nen!en$e, e !-&"!ne- #e e$e e$% e$ "!%&n/ &l%e/e/% /e O en en$!/ n$%&%! &l /e l& &"#& /e #n %el n #n& el!/&/ &n"#l&% #n!0%-e ω :π.0 %&/9e".
S! el e$% #*& l& *!!n H%!n$&l en el !n$&n$e $ ?, # *%2e!n -*nen$e '%e #n ee e%$!&l, en #&l#!e% !n$&n$e $, e%+ !"#&l &l &l% /e l&%%!en$e en /!H !n$&n$e, *#e$ #e e"n l& 0!"#%& l& -*nen$e e%$!&l e4
i ft sen I sen I mm == π θ 2
8
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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N&$#%&l-en$e, el /!&"%&-& l %e*%een$& el &l% /e i en #n !n$&n$e/e$e%-!n&/, *e% el le$% *#e/e #*l!% l& %$&!n -en$&l-en$e, 2 e"#!% l&0l#$#&!ne /e i #&n/ I m "!%&. E$e -$/ e e#!&len$e &l #e e #$!l!&*&%& %e*%een$&% #n -!-!en$ &%-n! !-*le *% l& *%2e!n '%e #n
/!+-e$% /e #n *#n$ #e e -#ee en #n& !%#n0e5
%en!& n el!/&/ &n"#l&% n$&n$e. S! e /ee&%e*%een$&% en el -!- /!&"%&-& l &l%e!n$&n$+ne /e #n& /!0e%en!& /e *$en!&l /&/&
)2( ϕ π −= ft senV v m #e $en"& l& -!-& 0%e#en!&
#e l& %%!en$e, *e% %e$%&&/& %e*e$ & $& #n+n"#l ϕ, e n$%#2e #n e"#n/ e$% /e ln"!5
$#/ V m & !e%$& e&l& nen!en$e, *e% /e*l&&/ %e*e$ & I m #n +n"#l ϕ enel en$!/ /e l& &"#& /el %el. C#&n/ &-' e$%e "!%&n en en$!/n$%&%! &l /e l& &"#& /el %el, e e!/en$e #e l &l%e !n$&n$+ne /e v *%een$&%+n !e-*%e #n %e$&%/ /e 0&e ϕ %e*e$ & l /e i .
;l!en/ &l !%#!$ LRC &n$e%!%, %e*%een$&- *% e$e -$/ /el e$% %$&$%! l& !n$en!/&/ 2 l& /!0e%en!& /e *$en!&l en$%e l e$%e- /e l&/!$!n$& *&%$e /el !%#!$.
P&%& %e*%een$&% l& !n$en!/&/ /e l& %%!en$e '&$&%+ #n l e$% I m, 2& #e$e $!ene el -!- &l% 2 l& -!-& 0&e en $/& l& *&%$e /el !%#!$. L&/!0e%en!& /e *$en!&l en$%e l '%ne /e l& %e!$en!& e$+ %e*%een$&/& *% ele$%
abmV #e $!ene l& -!-& /!%e!n #e I m, *#e$ #e v ab e i e$+n en 0&e,
2 #2& ln"!$#/ e V V abe 2. . An+l"&-en$e, l e$%e
bcmV /e ln"!$#/
V V bce
2. 2cd m
V /e ln"!$#/ V V cd e
2. %e*%een$&n l& /!0e%en!& /e *$en!&l&&n&/& en$%e l '%ne /e l& $!n/#!n 2 l& /!0e%en!& /e *$en!&l%e$&%/&/& en$%e l '%ne /el n/en&/%. C#&n/ l e$%e "!%&n, #-*nen$e e%$!&le e*e%!-en$&n l& -!-& &%!&!ne #e l& %/en&/&/e l& #&$% #%& /el /!&"%&-& $e-*%&l &n$e%!%.
9
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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Q# #e/e n l& /!0e%en!& /e *$en!&l en$%e & 2/. U$!l!&n/ el /!&"%&-& /el e$% %$&$%!, e$ e'$!ene #-&n/ l& -*nen$e e%$!&le /e
abmV ,
bcmV
cd mV . Pe% l& #-& /e l& -*nen$e
e%$!&le /e e$ e$%e e !"#&l & l& -*nen$ee%$!&l /e # #-& "e-$%!& %e#l$&n$e. P% $&n$,! e$& %e#l$&n$e e '$!ene *% #&l#!e% -$/&/e#&/, # -*nen$e e%$!&l %e*%een$& l&/!0e%en!& /e *$en!&l !n$&n$+ne& en$%e l e$%e-/el !%#!$, 2 el +n"#l ϕ, #e 0%-& n I m, !n/!& l&/!0e%en!& /e 0&e /el !%#!$ en n#n$. L& *%*!&
%e#l$&n$e %e*%een$&, *% $&n$, l& /!0e%en!& /e *$en!&l -+!-& en$%e & 2 /, e&
abmV . A#n#e el /!&"%&-& /el e$% %$&$%! e een!&l-en$e #n -$/
*&%& %e*%een$&% &l%e !n$&n$+ne, en l& *%+$!&, e #$!l!& &! !e-*%eel#!&-en$e n l &l%e e0!&e 2 l& /!0e%en!& /e 0&e. S! !-&"!n&-#e e -/!0!& l& e&l& /e l /!&"%&-& 0&%!&le en el 0&$% 2 , l -!-e$%e #e %e*%een$&n &l%e -+!-, %%e*n/e%+n en l& n#e& e&l& &&l%e e0!&e, *e%-&ne!en/ !n&%!&'le l +n"#l # %e*%een$&n/!0e%en!& /e 0&e. P% ell e *%+$!& n%-&l n$%#!% e$ /!&"%&-& n le$%e #e %e*%een$&n l &l%e e0!&e en l#"&% /e l &l%e -+!-. S!e /ee&n l &l%e !n$&n$+ne l e nee&%! !-&"!n&% -#l$!*l!&/& *%
2 l& ln"!$#/ /e $/ l e$%e 2 #*ne% el /!&"%&-& en %$&!n.
E -*le$&-en$e &%'!$%&%!& l& %!en$&!n #e e / &l /!&"%&-& e$%!&l. En #n!%#!$ en e%!e e -!en& /e %/!n&%! *% /!'#&% el e$% !n$en!/&/0%-&n/ #n +n"#l #&l#!e%& 2 e n$%#2en l $% e$%e n l&%!en$&!n %el&$!& &/e#&/&. Se -!$en "ene%&l-en$e l ee X e Y. A-'&0!"#%& n !"#&le, &l #e el e$% !n$en!/&/ e H%!n$&l 2 #e l& e&l& eH& -/!0!&/ *&%& #e l e$%e %e*%een$en &l%e e0!&e. P#e$ #e V ab IR , V bc = IX L, V cd = IX C 2 V ad = IZ , l e$%e l$&e e$+n %el&!n&/e&$&-en$e /el -!- -/ #e l e$%e en #n /!&"%&-& /e e$% !-*e/&n!& $&l - el /!&"%&-& /e !-*e/&n!&. En e0e$, el /!&"%&-& /ele$% l$&e /e #n !%#!$ en e%!e *#e/e '$ene%e & *&%$!% /e # /!&"%&-& /ee$% !-*e/&n!&, -#l$!*l!&n/ &/& e$% %e!$en!&, %e&$&n!& !-*e/&n!&
*% l& !n$en!/&/.
10
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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CIRCUITOS EN PARALELO
Cn!/e%e- &H%&, el &*l&-!en$ en *&%&lel /e l ele-en$ /el!%#!$ /e l& 0!"#%&. L& /!0e%en!& /e *$en!&l !n$&n$+ne& v ab en l e$%e- /el& / /e%!&!ne e l& -!-&. L& !n$en!/&/ i 1 en l& /e%!&!n #*e%!% (&*&!$!&) e$&%+ &&n&/& %e*e$ & v abK l& /e l& /e%!&!n !n0e%!% (!n/#$!&)e$&%+ %e$&%/&/&. Se"n l& le2 /e l n#/ /e !%HH00, l& !n$en!/&/ !n$&n$+ne&I l /e l& l1ne& e !"#&l & l& #-& /e l& !n$en!/&/e !n$&n$+ne& i 1 e i 2 . El/!&"%&-& e$%!&l /el !%#!$ e$+ !n/!&/ & #n $&/. El e$% V ab %e*%een$&
l& /!0e%en!& /e *$en!&l -n en$%e l e$%e- /e &/& /e%!&!n, 2 le$%e I 1 e I 2 , l& !n$en!/&/e. I 1 e$+ &&n&/& %e*e$ & V , #n +n"#l ϕ@ &%$" X l /R l K I 2 e$+ %e$%&&/& %e*e$ & ; #n +n"#l ϕ: &% $" X 2 /R 2 . L& !n$en!/&/en l& l1ne& e$+ %e*%een$&/& *% I l , #-& "e-$%!& /e I 1 e I 2 , 2 ϕ e l&/!0e%en!& /e 0&e en$%e l& !n$en!/&/ en l& l1ne& 2 V .
Se /e0!ne l& !-*e/&n!& e#!&len$e /el !%#!$ - l& %&n /e V ab & l&!n$en!/&/ en l& l1ne& I l . E e!/en$e #e ! V ab 2 l& n$&n$e /el !%#!$ nn!/& (/&$), el +l#l /e I l lle& n!" el +l#l /e l& !n$en!/&/e
!n/!!/#&le en &-'& /e%!&!ne, 2 # /!0e%en!& /e 0&e. P% n!"#!en$e,l& !-*e/&n!& e#!&len$e *#e/e e*%e&%e en 0#n!n /e l& %e!$en!& 2%e&$&n!& /e l& /e%!&!ne. L& e*%e!n #e %e#l$& e &l" -*l!&/&,n#e *#e/e /e/#!%e 0+!l-en$e -e/!&n$e l& "%&0!& &n$e%!%e.
RESONANCIA
Un & *&%$!#l&% /e !n$e% e *%een$& en #n !%#!$ en e%!e #e n$!ene$!n/#!n 2 &*&!/&/, #&n/ L, C 2 f $!enen &l%e $&le #e
11
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fC fL π π 2/12 = , e$ e, #&n/ X L X C . En$ne X ?, Z = R 2 ϕ &% $" X 9 R ?. L& !-*e/&n!& /e !%#!$ e !-*le-en$e !"#&l & # %e!$en!&, 2 l&!n$en!/&/ e$+ en 0&e n l& /!0e%en!& /e *$en!&l en$%e l '%ne /el !%#!$.L& %%!en$e #e &$%&!e& #n !%#!$ %en&n$e en e%!e, ! # %e!$en!& e*e#e3&, e%+ "%&n/e, 2 l& /!0e%en!& /e *$en!&l en$%e l '%ne /e l&
$!n/#!n 2 /e l& &*&!/&/ *#e/e e% -#H -&2% #e l& #e e!$e &$%& /e $/ el !%#!$.
;e&- &H%&, el 0en-en n #n !%#!$ RLC e%!e /e ee-*l.
Cn!/e%e- #n !%#!$ en e%!e en el #&l $ene- R , X L X C , ne$&/ & #n&
/!0e%en!& /e *$en!&l &l$e%n& #2 &l% e0!& e ε. L& !n$en!/&/ /e l& %%!en$een el !%#!$ e4
R
V
Z
V I ==
L& /!0e%en!& /e *$en!&l en$%e l '%ne /e l& %e!$en!&,
IRV R =
L& /!0e%en!& /e *$en!&l en$%e l '%ne /e l& $!n/#!n,
L L IX V =
12
I
ε
V R
V L
V C
V X
R XLX
C
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L& /!0e%en!& /e *$en!&l en$%e l& &%-&/#%& /el n/en&/%,
C C IX V =
Y l& /!0e%en!& /e *$en!&l en$%e l e$%e- /e @& -'!n&!n $!n/#!n5
&*&!/&/ (V X ) e4 IX V X =
Se 'e%& #e l& /!0e%en!& /e *$en!&l !n$&n$+ne& en$%e l e$%e- /e l&$!n/#!n 2 /el n/en&/% e$+n /e0&&/& @?, 2 n#e l &l%ee0!&e /e &/& #n& *#e/en e% "%&n/e, # %e#l$&n$e en $/ !n$&n$e e n#l&.En ne#en!& #n l$1-e$% &*l!&/ en$%e ' 2 /, !n/!&%+ e%.L& 0!"#%& %e*%een$& #n !%#!$ en *&%&lel en el #&l X L X C , !en/ R l& -!-&en &/& /e%!&!n. L& !n$en!/&/ e0!& e, *% n!"#!en$eK !"#&l en &/&/e%!&!n, e$&n/ #n& /e e$& %%!en$e %e$&%/&/& 2 l& $%& &/el&n$&/& el
-!- +n"#l %e*e$ & l& /!0e%en!& /e *$en!&l. S! X L 2 X C n -#H-&2%e #e R , l& /!0e%en!& /e 0&e n &*%!-&/&-en$e 7? 2, - %e#l$&e!/en$e, l& !n$en!/&/ I l en l& l1ne& e -#H -en% #e l& !n$en!/&/ en &/&/e%!&!n. De $%& 0%-&, l& !-*e/&n!& e#!&len$e, V 9 I l , /el !%#!$ en n#n$e -#2 ele&/&, -#H -&2% #e l& /e #n& #&l#!e%& /e l& /e%!&!ne. De-/ '&$&n$e *&%&/!, #&n$ -en% e l& %e!$en!&, -&2% e l&!-*e/&n!&, 2& #e #&n/ R $!en/e & e%, l& /!0e%en!& /e 0&e $!en/en & 7?,2 l& %%!en$e /e l& l1ne& $!en/e & e%. Se /!e #e el !%#!$ e$+ en %en&n!&*&%&lel&.
De'e e!$&%e l& %en&n!& en l& l1ne& /e $%&n*%$e /e ene%"1&, *#e$ #e,!n/#/&'le-en$e, %e#l$&%+n !n$en!/&/e /e %%!en$e 2 /!0e%en!& /e *$en!&l
ele&/& en !e%$& *&%$e /el !%#!$. P% el n$%&%!, en l !%#!$ /e %&/! e&*%eH& l& en$&& /e l& %en&n!& en el *%e /e !n$n!&!n. El !%#!$/e &n$en& /e #n %&/!%%ee*$% n$!ene #n& $!n/#!n 2 #n n/en&/% ene%!e. L& e$&!ne e-!%& -*%en/!/& /en$% /el !n$e%&l /e 0%e#en!&/el %ee*$% !n/#en en e$e !%#!$ 0#e%& ele$%-$%!e /e 0%e#en!&!"#&le & l& 0%e#en!& *%$&/%& /e /!H& e$&!ne. C#&n/ eln/en&/% /e !n$n1& e &#$& /e -/ #e el !%#!$ e$ en %en&n!& nl& 0%e#en!& /e #n& e$&!n /ee&/&, l& !n$en!/&/ /e %%!en$e #e %%e5*n/e & &#ell& 0%e#en!& *&%$!#l&% e "%&n/e, 2 e '$!ene #n& /!0e%en!& /e*$en!&l $&-'!n "%&n/e en$%e l& &%-&/#%& /el n/en&/%K *e%, - #nn/en&/% l *#e/e e$&% en %en&n!& *&%& #n& 0%e#en!&, l& $%&e$&!ne n *%/#!%1&n -+ #e !n$en!/&/e 2 /!0e%en!& /e *$en!&l/e*%e!&'le, 2, *% n!"#!en$e, n e%+n 1/&.
13
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L& n/!!n nee&%!& *&%& l& %en&n!& e4
C L X X
LC ω
ω 1
=⇒=
/e l& #&l /e/#!-4
L
C
LC f
/1.
2
11.
2
1
π π == (
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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ELEMENTOS UTILI"ADOS
O!l*!K Gene%&/% /e n/&K C!%#!$ /e P%#e'& (e"n 0!"#%&), #2&l%e n4
R#$%$'%$ C*%&+,#$ I'-.&'%$R@ @>? Ω C@ ?,?8 µF L@ 8, -J2R: 6,6 Ω C: ?,@ µF L: >?? µJ2 99 @
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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CIRCUITO R3L1FRECUENCIA: 3/0H
seg rad Hz f
seg T Hz f
T
5
5
10.45,239000..2..2
10.5,239000
11
=⇒==
=⇒== −
ω π π ω
( )mA
V
Z
V I teorica 53,0
)107,41045,2(680
5,02
2352=
⋅⋅⋅+Ω
⋅==
−
t ! $!e-* /e /e0&&e ⇒ Se '$!ene /e l& "%+0!& /e l& e3&le en el!l*!
E*e%!-en$&l-en$e, -e/!-4 t = 8 µe".
ϕ 4 +n"#l /e /e0&&e
º2,5698,010.4.10.45,2. 65 =⇒=== − ϕ ω ϕ rad seg seg
rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º6003,1680
7,4.10.45,2 5
=⇒=Ω
=== ϕ ω
ϕ rad mHy seg
rad
arcTg R
LarcTg
R
X acrTg L
mAdiv
V div
I practico 47,0680
2,06,1
=Ω
⋅
=
RAFICA Y V R (I ) VS TIEMPO
16
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DIARAMA FASORIAL
FRECUENCIA: 130H
seg rad Hz f
seg T
Hz f
T
4
5
10.2,813000..2..2
10.7,7
13000
11
=⇒==
=⇒== −
ω π π ω
( )mA
V
Z
V I teorica 90,0
)107,4102,8(680
5,02
2342=
⋅⋅⋅+Ω
⋅==
−
E*e%!-en$&l-en$e, -e/!- el /e0&&e4 t = V9/!
V: 6,8 /!V
R: @,< /!
D#$: ?, /!
ε
i
VR
VL
φ
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º2,285,010.6.10.2,8. 64 =⇒=== − ϕ ω ϕ rad seg seg
rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º5,2952,0680
7,4.10.2,8 4
=⇒=Ω
=== ϕ ω
ϕ rad mHy seg
rad
arcTg R
LarcTg
R
X acrTg L
mAdiv
V div
I practico 79,0680
2,07,2
=Ω
⋅
=
RAFICA Y V R (I ) VS TIEMPO
DIARAMA FASORIAL
FRECUENCIA: !0H
18
13KHz
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: @?V9/!
V: 6,: /!
VR: :, /!D#$: ?,< /!
ε
i
VR
VL
φ
8/17/2019 Fisica3 - TP5 “CIRCUITOS EN CORRIENTE ALTERNA”
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seg rad Hz f
seg T Hz f
T
4
4
10.8,36000..2..2
10.7,16000
11
=⇒==
=⇒== −
ω π π ω
( )mA
V
Z
V I teorica 01,1
)107,4108,3(680
5,02
2342=
⋅⋅⋅+Ω
⋅==
−
E*e%!-en$&l-en$e, -e/!- el /e0&&e4 t = µe".
ϕ 4 +n"#l /e /e0&&e
º3,1527,010.7.10.8,3. 64 =⇒=== − ϕ ω ϕ rad seg seg
rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º1,1526,0680
7,4.10.8,3 4
=⇒=Ω
=== ϕ ω
ϕ rad mHy seg
rad
arcTg R
LarcTg
R
X acrTg L
mAdiv
V div
I practico 88,0
680
2,03
=
Ω
⋅
=
RAFICA Y V R (I ) VS TIEMPO
DIARAMA FASORIAL
19
6KHz
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: :?V9/!
V: 6,: /!V
R: 6 /!
D#$: ?,6> /!
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CIRCUITO R1C1FRECUENCIA: 460H
seg rad Hz f
seg T Hz f
T
5
5
10.52,240000..2..2
10.5,240000
11
=⇒==
=⇒== −
ω π π ω
( )
mAV
Z
V I teorica 11,4
)10047,01052,2
1(150
5,02
2
65
2
=
⋅⋅⋅+Ω
⋅==
−
t ! $!e-* /e /e0&&e ⇒ Se '$!ene /e l& "%+0!& /e l& e3&le en el!l*!
E*e%!-en$&l-en$e, -e/!-4 t = : µe".
ϕ 4 +n"#l /e /e0&&e
º92,2851,010.2.10.52,2. 65 =⇒=== − ϕ ω ϕ rad seg seg rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º3,2952,0047,0.150.10.52,2
11
5=⇒=
Ω
=== ϕ µ
ω ϕ rad
F seg
rad arcTg
R
C arcTg R
X acrTg C
mAdiv
V div
I practico 10,3150
2,03,2
=Ω
⋅
=
20
ε
i
VR
VL
φ
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RAFICA Y V R (I ) VS TIEMPO
DIARAMA FASORIAL
FRECUENCIA: 130H
seg rad
Hz f
seg T Hz f
T
4
5
10.2,813000..2..2
10.7,713000
11
=⇒==
=⇒== −
ω π π ω
21
40KHz
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: >V9/!
V: :, /!V
R: :,6 /!
D#$: ?,8 /!
ε
i
VR
VC
φ
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( )
mAV
Z
V I teorica 36,2
)10047,0102,8
1(150
5,02
2
64
2
=
⋅⋅⋅+Ω
⋅==
−
E*e%!-en$&l-en$e, -e/!- el $!e-* /e /e0&&e4 t = @6 µe".
ϕ 4 +n"#l /e /e0&&e
º3,6107,110.13.10.2,8. 64 =⇒=== − ϕ ω ϕ rad seg seg
rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º6005,1047,0.150.10.2,8
11
4=⇒=
Ω
=== ϕ µ
ω ϕ rad F
seg rad
arcTg RC arcTg
R X acrTg C
mAdiv
V div
I practico 00,2150
2,05,1
=Ω
⋅
=
RAFICA Y V R (I ) VS TIEMPO
DIARAMA FASORIAL
22
13KHz
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: @?V9/!
V: 6,@ /!V
R: @,> /!
D#$: @,6 /!
ε
i
VR
VC
φ
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FRECUENCIA: 70H
seg rad Hz f
seg T Hz f
T
4
4
10.1,58000..2..2
10.25,18000
11
=⇒==
=⇒== −
ω π π ω
( )
mAV
Z
V I teorica 6,1
)10047,0101,5
1(150
5,02
2
64
2
=
⋅⋅⋅+Ω
⋅==
−
E*e%!-en$&l-en$e, -e/!- el $!e-* /e /e0&&e4 t = :8 µe".
ϕ 4 +n"#l /e /e0&&e
º2,7023,110.24.10.1,5. 64 =⇒=== − ϕ ω ϕ rad seg seg
rad t
C-*&%+n/l n el &l% $e%!, $ene-4
º3,7023,1047,0.150.10.1,5
11
4=⇒=
Ω
=== ϕ µ
ω ϕ rad
F seg
rad arcTg
R
C arcTg R
X acrTg C
mAdiv
V div
I practico 33,1150
2,01
=Ω
⋅
=
RAFICA Y V R (I ) VS TIEMPO
23
8KHz
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: :?V9/!
V: 6,: /!
VR: @ /!D#$: @,: /!
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DIARAMA FASORIAL
OBSERVACIONES
N$&- #e, en el !%#!$ R L1, #&n/ l& 0%e#en!& /!-!n#2e, el +n"#l/e /e0&&e e &H!&, /e'!/ & #e /!-!n#2e l& %e&$&n!& !n/#$!& X L, 2& #el& -!-& e /!%e$&-en$e *%*%!n& & l& 0%e#en!&
P% n$%&*&%$e, e- #e #&n/ /!-!n#!- l& 0%e#en!&, el +n"#l/e /e0&&e -en$&, /e'!n/e e$, & #e X C e !ne%&-en$e *%*%!n&l &l& 0%e#en!& /e $%&'&.
C&'e &l&%&% #e l& /!0e%en!& '$en!/& en$%e l &l%e $e%!&l#l&/ 2 l &l%e *%+$!, n /e'!/& & e%%%e /e -e/!!n, /e&*%e!&!n, -&l n$&$, *&&e /e l& "%+0!& /el !l*! & l&#&/%!#l&, e$.
24
ε
i
VR
VC
φ
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CIRCUITO R1R3L1C1
P&%& */e% /e$e%-!n&% l& 0%e#en!& /e %en&n!&, $ene- / 0%-&4
- Enn$%&% l& I I -&- J&ll&% #n /e0&&e n#l (ϕ ?)
U&- l& l$!-& /e l& *!ne *#e$ #e e -& *%e!& 2 -+ en!ll&,2& #e e -+ !-*le &*%e!&% #n /e0&&e en el !l*!, #e /e$e%-!n&% en#e --en$ l& %%!en$e I , e H&ll& en # &l% -+!-.
E$e -$/ n!$e en &%!&% l& 0%e#en!& /el "ene%&/% /e e3&le, H&$&#e l& "%+0!& (/e ε 2 V R ) en el !l*! !n!/&n. E$& 0%e#en!&enn$%&/& e, en$ne, l& ll&-&/& 0%e#en!& /e %en&n!&.
E$e 0en-en e /& #&n/ X L X C , 2& #e4
? /e0&&(e =⇒=−
= ϕ ϕ R
arcTg R
X X arcTg C L
0
e- #e #&n/ e$ #e/e4 V L V C ⇒ ε R 1..I R 2 .I X L.I 5 X C .I
⇒ el !%#!$ e *#%&-en$e %e!$!
CÁLCULO TEÓRICO DE f 0
sT kHz f
T
kHz f kHz f kHz f
seg rad
f f seg
rad
F mHy
LC LC C L X X X
T!RIC!T!RIC!
T!RICA
C L
T!RICA
5
0
000
4
4
2
10.34,971,10
11
07,165,135,55,071,10
.2
10.8,6
2210.8,6
047,0.7,4
1
1110
−=⇒==
=⇒=⇒=
⇒==⇒=⇒==
⇒=⇒=⇒=⇒=⇒=
π π
ω π ω
µ ω
ω ω ω
ω
CALCULO PRÁCTICO DE f 0
C#&n/ l& "%+0!& /e ε 2 V R !n!/en, el *e%1/ -e/!/ /el !l*! e4
" #R$C"IC% 78 µe".
kHz f seg T
f "RACTIC!
"RACTIC!
"RACTIC! 64,10
9411 =⇒== µ
25
X ?
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RAFICA Y V R VS TIEMPO
C- *#/!- &*%e!&% en l& "%+0!&, e- #e #&n/ &%!&- l&0%e#en!& /e %en&n!& f & & ?,>f & , e- #e l& %%!en$e &/el&n$& & l& $en!n,
26
f 0
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: @?V9/!
V: 6,: /!V
R: :,< /!
D#$: ? /!
0,5 · f 0
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: :?V9/!
V: 6,: /!V
R: :,: /!
D#$: ?, /!
1,5 · f 0
V ( ): ?,: ;9/!V
R (): ?,: ;9/!
&: @?V9/!
V: 6,: /!V
R: :,> /!
D#$: ?,6 /!
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*%e/-!n&n/ #n -*%$&-!en$ &*&!$!K -!en$%& #e #&n/ l& 0%e#en!&/e %en&n!& f & l& &%!&- & @,>f & , e 'e%& #e l& %%!en$e &$%&&%+ & l&$en!n, *%e/-!n&n/ #n -*%$&-!en$ !n/#$!.
El -*%$&-!en$ /e l& %%!en$e en #n !%#!$ en e%!e /e %%!en$e&l$e%n&, #&n/ &%1& l& 0%e#en!& /e l& e3&l /e en$%&/&, e e&$&-en$e
&n+l" & l& %e*#e$& /el !$e-& -&&5%e%$e, #e $!ene 0#e%& /e&-%$!"#&-!en$ !, &l &%!&% l& 0%e#en!& /e l& 0#e%& !-*#l%&.
OBSERVACIONES
P&%& l& 0%e#en!&, !"#&l & l& 0%e#en!& /e %en&n!&, X L X C , *% l #eV L V C , en$ne el !%#!$ e%+ *#%&-en$e %e!$! ε R 1.i R 2 .i
Al /!-!n#!% l& 0%e#en!& &*&%ee #n +n"#l /e /e0&&e, &/el&n$&n/ l&%%!en$e & l& $en!n, 2 el !%#!$ $en/%+ #n -*%$&-!en$ &*&!$!. M!en$%&#e &l -en$&% l& 0%e#en!&, n %e*e$ & l& 0%e#en!& /e %en&n!&,
$&-'!n &*&%ee%+ el +n"#l /e /e0&&e, -!en$%& #e l& %%!en$e %e$%&&%+ & l&$en!n, 'e%+n/e #n -*%$&-!en$ !n/#$!.
CONCLUSIONES
Cn l !$ &n$e%!%-en$e */e- /e/#!%, #e n e nee&%! &-'!&% l -*nen$e /e #n !%#!$, *&%& &-'!&% # 0#n!n, e /e!%, *&%& &%!&% #-*%$&-!en$. P% ee-*l /e &*&!$! & !n/#$! e *#e/e l"%&% -e/!&n$el& &%!&!n /e l& 0%e#en!& /e l& e3&l /e en$%&/&. W$e -*%$&-!en$ e -#2$!l, *% ee-*l, en l &*&%&$ /e %&/! %ee*!n !n nee!/&/ /e $ene% #e&-'!&% l ele-en$ n$!$#$! /e l !%#!$ /e l &*&%&$. L -!-#e/e, ! #e%e- &-'!&% # -*%$&-!en$ /e !n/#$! & &*&!$!.
T&-'!n !- #e e *#e/e %e"#l&% el +n"#l /e /e0&&e en$%e l&e3&le /e en$%&/& 2 /e l& %%!en$e #e !%#l& & $%& /e #n !%#!$, !n-/!0!&% l -*nen$e /el -!-. E$ e %e&l!& -e/!&n$e el &-'! /e0%e#en!&.
BIBLORAFIA
• '()IC* +IV-R)I"*RI*, /e SEARS5=EMANSY5YOUNG, Se$& e/!!nEDIT. A//!n5ele2 I'e%&-&%!&n& S.A., !l-!n"$n, Del&&%e, E.U.A. @7C&*1$#l 6