Analisador paramétrico de Transisistores (Parametric BJT analyzer)
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Relatório de trabalho realizado no âmbito do projeto final. Deve ser lido em conjunto com os anexos - Testador paramétrico de transistores.
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Analisador paramétrico de Transisistores (Parametric BJT analyzer)
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- 11. 1 6./5&.&('134- /%8'6,1 -/'-2*+& /&51 2'656=134- )- 1./56?6,1)-% )& /-'O(,61 " "> )1)- +2/-%'1% 1+ '&(+I&+ & ,-%%&('&+ &(@-5@6)1+ (1 A5251 )& &5'6&% )& 1,-%)- ,-. 1+ +21+ ,1%1,'&%7+'6,1+ '656=1%1.*+& '&(+I&+ )& 156.&('134- +6.A'%6,1+ )& L ! )& .-)- 1 :2& &. ,1+- )& +1'2%134- )- 1./56?6,1)-% )& /-'O(,61 1 '&(+4- R &('%1)1 )1 ,A5251 (4- &B,&)1 - @15-% .8B6.- 1).6'6)- /-% &+'1 '&(+4- )& 156.&('134- .7(6.1 )&+'& 1./56?6,1)-% A )& ! (0&4('*+ 2% /0*'%&-.* 2, &?545, 2% %5'(%0 /%-'&,34- )1 ,A5251 A ,-(+'6'27)1 /-% )-6+ ,6%,26'-+ 6()&/&()&('&+K /%6.&6%- A 2. ,6%,26'- ,-./1%1)-% %&<&(&%1'6@- :2& )&'&,'1 +& 1 '&./&%1'2%1 .8B6.1 )1 ,A5251 DF N E> +& &(,-('%1 (- 56.61% 102+'1)- DF"N E . ,1+- )1 '&./&%1'2%1 +&% 1'6(<6)1> 1,'21%8 2. '%1(+7+'-% ,20- %&+25'1)- +&%8 1 )6.6(2634- )1 '&(+4- )& &('%1)1 )- ./56?6,1)-% /&%1,6-(15 )& -'O(,61 -.- ,-(+&:2O(,61 1 '&(+4- (1 &('%1)1 )1 ,A5251 )6.6(26%8 1;%2/'1.&('& /%-'&<&()- 1++6. 1 ,A5251 )& &@&('215 &.;151.&('- 'A%.6,- +&<2()- /%-'&<& 1 ,A5251 (1 &@&('2156)1)& )& &B6+'6%&. @15-%&+ )& '&(+4- &. .S)25- +2/&%6-%&+ 1 1/%-B6.1)1.&('& & ,-%%&('&+ +2/&%6-%&+ 1 >F /%-,&++- )& /%-'&,34- 2'656=1)- ,-(+6+'& (2.1 /-('& )& )7-)-+ &. +A%6& :2& ?2(,6-(1 /1%1 '&(+I&+ /-+6'6@1+ & (&<1'6@1+ )& .-)- 1 :2& &. ,-()63I&+ )& '&(+4- 6(?&%6-%&+ 1 > (4- &B6+'1 6(?52O(,61 (- ,6%,26'- %1+*0 % ,&*12(&(*1,2*0 2, '%7/%0,'40, ,7#(%1'% -(+'6'27)- /-% 2. +&(+-% )& '&./&%1'2%1 D ZE & 1,-()6,6-(1)-% )& +6(15 :2& &+'1;&5&,& -+ @15-%&+ )& '&(+4- /%-/-%,6-(16+ R '&./&%1'2%1 1.;6&('& )& ?-%.1 1 +&%&. .&)6)1+ /&5- ,-(@&%+-% G 1%1 &?&,'21% 1 6('&%?1,& 1- C -/'-2*+& /-% 2'656=1% 2. 1./56?6,1)-% -/&%1,6-(15 (1 .-('1<&. )& 1./56?6,1)-% (4- 6(@&%+-% & %&?&%6% :2& - +&(+-% 1/%&+&('1 2.1 ?2(34- ,1%1,'&%7+'6,1 " " − = ∆ ∆ ! ! /1%1 [ ]"∈ "
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≅(( ?6< * 61<%1.1 )& 5-,-+ ,-('%-51)-% &. .1591 ?&,91)1 +'& ,-('%-5- %&[(& 1+ 1,3I&+ /%-/-%,6-(15> 6('&<%15 & )&%6@1'6@1> - :2& +& '%1)2= R /1%'6)1 /&51 &B6+'O(,61 )& 2.1 ?2(34- )& '%1(+?&%O(,61 ,-. )-6+ =&%-+ 102+'8@&6+> 2. <1(9- ( 6<215.&('& 102+'8@&5 & 2. /S5- (1 -%6<&. ?2(34- )& '%1(+?&%O(,61 )- ,-('%-51)-% A )1)1 /-%K EDED ( ++= &()-K * -(+'1('& )& '&./- )- &?&6'- )&%6@1'6@- * -(+'1('& )& '&./- )- &?&6'- 6('&<%15
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−= _* ^ −= ^ −=
- 17. V *7,2*0 &*7 1)%0+*0 EEDEDEDDED FV! !!!! ++−= 1%1 -;'&% ,-&?6,6&('&+ 6<216+ )1+ @8%61+ '&(+I&+ )- +-.1)-%> FV! == K EEDEDEDDED ! !!!! ++−= & F () −= ED ED! −= * /&%134- &?&,'21)1 /&5- ('&<%1)-% EDED! −= * /&%134- &?&,'21)1 /&5- )6?&%&(,61)-% EDED! −= * /&%134- &?&,'21)1 /&5- -('- )& -.1 -(+6)&%1()- F () −= > +2;+'6'26()- ED! > ED! & ED! -;'A.*+&K EDEDED F (! ) ++ = ++= '%6;26()- = -;'A.*+&> ++ = (! )ED >5&45* 2*+ ,0G7%'0*+ 2* *1'0*5,2*0 /S+ 1 -;'&(34- )1 ?2(34- ,1%1,'&%7+'6,1 &. .1591 1;&%'1> )1+ ,1%1,'&%7+'6,1+ &5A,'%6,1+ )1 ,A5251 D1(&B- `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
- 18. F +'& ,-./-%'1.&('- /-)& +&% 1/%-B6.1)- 1 2. +6+'&.1 )& /%6.&6%1 -%)&. ,-. 1'%1+-> +&()- 1 ?2(34- )& '%1(+?&%O(,61 )1)1 /&51 &B/%&++4-K ED + = − , ?6< * 2%@1 )& %&1,34- )- /%-,&++- -.K ` ./56'2)& )- &+,154- ` &./- )& +2;6)1 '6%1)- )1 '1(<&('& R ,2%@1 )& +2;6)1 ` &./- )& 1'%1+- '6%1)- )1 '1(<&('& R ,2%@1 %&5134- A )1)1 /-%K τ = &()- &. ,-('1 1 %&+/-+'1 1- &+,154-> @4-*+& &B'%16% /1%1 +6+'&.1+ &. ,1)&61 1;&%'1 -+ /1%J.&'%-+ &<2()- * + -+ @15-%&+ 1,-(+&591)-+ /1%1 +6+'&.1+ &. ,1)&61 1;&%'1 ,-. &+'& '6/- )& %&+/-+'1 D,2%@1 %&1,34- /%-,&++-E +4- )1)-+ /-%K ( > = ^ = ^ >"= 1 ,2%@1 &B/&%6.&('15 :2& +&%@62 ,-.- %&?&%O(,61 1- .-)&5- D@&% ?6< VE> & 2'656=1()- 1 @1%6134- 1;+-52'1 )& 1./56'2)& 'A%.6,1> -;'A.*+&K V=−≈τ +
- 19. V ≅== N G+ & _ + ?6< * 2%@1 )& %&1,34- -;'6)1 /1%1 1 ,A5251 )& &5'6&% '656=1()- -+ @15-%&+ /1%1 1./56'2)& (4- (-%.156=1)1 1,-(+&591)-+ '&.*+&K > Va>> ===( ^ a === ^ a>">" === > & -+ /1%J.&'%-+ )&'&%.6(1)-+ /-% &+'& .A'-)- )& %&<%1+ -;'A.*+&K ++= ++= ( >">ED !>"ED + = & 1,-%)- ,-. 1 &B'%1,34- )1+ ,1%1,'&%7+'6,1+ &B/&%6.&('16+K E>"D > EGD Va!>"ED + = + = &.*+& &('4- 2. /S5- (1 -%6<&. & )-6+ =&%-+ &. `">
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b (-+ ,6%,26'-+ ,-. 1./56?6,1)-%&+ -/&%1,6-(16+ '%6;26()- - @15-% )& "" Qb 1 FV! == > %&+25'1 ""a"a F.1B .1B === ! b> & """"aa.6( .6( === ! Qb /'-2*+& /&51 ,-5-,134- &. +A%6& )& 2.1 %&+6+'O(,61 )& ""Qb &. +A%6& ,-. 2. /-'&(,6S.&'%- .25'6*@-5'1 )& b • *1'0*5* %0(),'()* & = ,-. = +,-59&()- 2. ,-()&(+1)-% )& _ >VY > '&.*+& _ b • *1'0*5* 1'%90,5 == %;6'%1()- _ >V2 )& ?-%.1 1 '&%.-+ @15-%&+ )& c_ b> ,15,251()- > ! "aV> − = ^ = Qb
- 21. • *7,2*0 &*7 1)%0+*0 % ,1H* 0*/*0&(*1,5K 1=&()- !_ V_ F_ _ ""Qb> 1'%6;262*+& - @15-% )& _ b> /1%1 102+'&+ )- <1(9- /%-/-%,6-(15 )1 .-('1<&.> +&()- - ,6%,26'- %&15 ,-(+'6'27)- /-% 2.1 %&+6+'O(,61 ?6B1 &. +A%6& ,-. 2. /-'&(,6S.&'%- + @15-%&+ ,15,251)-+ +4- @15-%&+ 1/%-B6.1)-+ /1%1 %&?&%O(,61> (- &('1('- (- ,6%,26'- %&15 '-%(1*+& (&,&++8%6- 1 6./5&.&('134- )& @1%613I&+ (-+ /1%J.&'%-+ )- ,-('%-51)-% /1%1 &@&('215 102+'& ,-. 1 (&,&++8%61 2'656=134- )& /-'&(,6S.&'%-+ (- ,-('%-5- )&%6@1'6@-> 6('&<%15 & /%-/-%,6-(15 ?6< * -('%-51)-% /%8'6,- /S+ - ,85,25- )-+ /1%J.&'%-+ )& 2. ,-('%-51)-% & /1%1 ,-(?6%.134- )&++&+ /1%J.&'%-+ -;'6)-+ /&5- .A'-)- )& * + > 6%8 +&% &+'6.1)- 2. .-)&5- /1%1 1 ,A5251 )& &5'6&%
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∞= −−− -- γβα & ' &B/%&++4- :2& ,1%1,'&%6=1 1 &B6+'O(,61 )& /S5-+ & =&%-+ %&16+ (2.1 /-++7@&5 ?2(34- )& '%1(+?&%O(,61 )- +6+'&.1 &. &+'2)- & +2;'%16%.-+ 1 D E - @15-% ?6(15 & 1++2.6()- :2& α− A - /S5- )& .16-% @15-% /-)&%&.-+ %&+,%&@&% 1 &B/%&++4- 1/%-B6.1)1K -- α− ≅∞− EDED D E -2 16()1K [ ]-- """ 5-<5-<ED5-< α−≅∞− D E &()- >"5-< " = > '&%&.-+ :2&K [ ]-- α>"5-<ED5-< "" −≅∞− D E ;'A.*+& 1++6. 1 &:2134- )& 2.1 %&,'1 ,20- )&,56@& )&'&%.6(1 - ,-&?6,6&('& α & 1 -%)&(1)1 (1 -%6<&. )&'&%.6(1 - ,-&?6,6&('& + ,-&?6,6&('&+ & β )1 &:2134- D E +4- -;'6)-+ )& ?-%.1 1(85-<1 & R &B/%&++4- D E +2;'%16%.-+ 1<-%1 - α− +∞ED /-)&%&.-+ &+,%&@&% 1 &B/%&++4- 1/%-B6.1)1K -- βα −− ≅+∞− EEDDED D E %-,&)&()- '15 ,-.- &. D E & D E /1%1 &+'1 (-@1 &B/%&++4-> -;'&%&.-+ -+ @15-%&+ )& β & +'& /%-,&)6.&('- A %&/&'6)-> /1%1 -+ -2'%-+ /1%J.&'%-+ E>>D δ > 1'A :2& 1 /%&,6+4- )-+ )1)-+ (4- /&%.6'1 (-@1 6'&%134-
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− = D!E &. :2&K + * &./&%1'2%1 (-%.156=1)1 )1 1.-+'%1 * &./&%1'2%1 )1 1.-+'%1 1 %&/%&+&('1% . * &./&%1'2%1 ?6(15 )1 &B/&%6O(,61 * &./&%1'2%1 6(6,615 )1 &B/&%6O(,61 '&./- &('%& 1.-+'%1<&(+ ?-6 )& 2. +&<2()- & ,-(+'1('& (1+ @8%61+ &B/&%6O(,61+ /S+ 1 +21 %&156=134-> +&5&,,6-(-2*+& 1:2&51 :2& +&%@6%61 )& %&?&%O(,61 ++6.> ?-6 &+,-596)1 )& ?-%.1 &./7%6,1 1 :2& 1/%&+&('1@1 2. ,-./-%'1.&('- .16+ /%SB6.- )1 .A)61 )1+ &B/&%6O(,61+ %&156=1)1+ D@&% ?6< !E
- 24. # $ % ! &' " ( 9 9! 9. 9: 9 9 9! ! ! ) * +" # - 0 .; " # 4 :! <; < !; &0 .; ' ?6< ! ` &+/-+'1 1- &+,154- &. .1591 1;&%'1 ,A5251 &5'6&% &156=-2*+& 1 ,-./1%134- )1 %&+/-+'1 '%1(+6'S%61 1- &+,154- -;'6)1 &B/&%6.&('15.&('& ,-. 1 )- .-)&5- &+'6.1)- D@&% ?6< VE & " , - $ 9 9 9! 9 9. 9 9: 9< 9 9= 9 ! ! ) * +" # - 2 # - ?6< V * -./1%134- &('%& .-)&5- &+'6.1)- & &B/&%6O(,61
- 25. '656=1()- )1)-+ &B'%17)-+ )& 2.1 )1+ &B/&%6O(,61+ %&156=1)1+ ?-6 &?&,'21)1 1 %&/%&+&('134- <%8?6,1 )1 ?2(34- )& .-)- 1 &+'6.1%*+& @15-% )- ,-&?6,6&('& & α ?6< F * ;'&(34- )- ?1,'-% (1 6)&('6?6,134- )- +6+'&.1 "V!!>""!V>"dEDd5-< +−=− += /- 0- >"dd "V!!>" == )1)- :2& EDED -- <∞ &('4- "dd < /&5- :2& >−= FV>" >"a dd"5-<dd"5-< = ∆ − = . α & +2;'%16% α + 1-+ )1)-+ & +& ?-% &?&,'21)1 1 %&/%&+&('134- )- %&+25'1)-> /-)&%*+&*8 &+'6.1% - @15-% )- ?1,'-% K dEEDEDDd"5-< - α− +− F>"" V F>" == − " 6./56?6,1()- -;'A.*+&K = ">
- 26. ! > % 0 .: :? 0 < ! ! % . 0 0 : 0 . 0 ! 0 0 0 : 0 . 0 ! ! ! . / ," @ & ' ?6< * ;'&(34- )- ?1,'-% (1 6)&('6?6,134- )- +6+'&.1 !!V> >"a dd"5-<dd"5-< >"a = ∆ − = ∆ ∆ = . β @15-% )& 5-< "d D"Ed A -;'6)- /&5- @15-% )1 -%)&(1)1 (1 -%6<&.> +&()- )& *">V F -.;6(1()- -+ %&+25'1)-+ -;'6)-+ /1%1 -+ ,-&?6,6&('&+ α> & β> ?-6 )&'&%.6(1)1 1 ?2(34- )& '%1(+?&%O(,61 )- .-)&5- &+'6.1)- - >!!!"> FV >">EDe −− +−= /56,1()- 1 '%1(+?-%.1)1 )& 1/51,& 1 EDe- K E!!!>D >" EFV>"D > ED f + + + −=' > 1 /1%'6% )- @15-% )& f ED' -;'A.*+& - @15-% &+/&%1)- )& ' , G ED ED f f = '%1@A+ )1 2'656=134- )& /%-<%1.134- &. 1' 1;# )6+/-(7@&5 (- 1(&B- %&?&%&('& R ,-./-(&('& )& /%-<%1.134- (-.&1)1.&('& 1'%1@A+ )-+ & ;'&% + -5-+ > -;'A.*+& 1+ ?2(3I&+ )& '%1(+?&%O(,61K V>"F> VV>"!>"> ED V ++ +− = − ,
- 27. V &+/%&=1()- - ?1,'-% )& +&<2()1 -%)&. )- (2.&%1)-% & +6./56?6,1()- -;'A.*+&K E!!V>EDFV>"D EVV>D!>" ED ++ −− =, DVE 1 ?6< " &+'8 %&/%&+&('1)1 1 %&+/-+'1 1- &+,154- )- .-)&5- &+'6.1)- %&/%&+&('1)- /-% ED f , K ?6< " * &+/-+'1 1- &+,154- )& .-)&5- & 1,-%)- ,-. 1 &+'6.134- )- .-)&5-> )1 ,A5251 &. .1591 1;&%'1 & 2'656=1()- - +6+'&.1 )& ,-('%-5- /-% %&156.&('134-> %&/%&+&('1)- (1 ?6< > +6.25-2*+& 1 %&+/-+'1 )- +6+'&.1 ,-. -+ +&<26('&+ /1%J.&'%-+ 6(6,616+K D E_ > # D E_ > E!!!>EDFV>"D EVV>D!>" ED ++ −− =, 16 2'656=1%*+& - % )& .-)- 1 -;'&%*+& 2.1 -%)&. )& <%1()&=1 )- <1(9- /%-/-%,6-(15> /1%1 - :215 - +6+'&.1 16()1 A &+'8@&5> (4- +& 2'656=1()- - '&%.- 6('&<%15 & )&%6@1'6@-
- 28. F 5 0 ED'ED ED ED, ED# ?6< * 61<%1.1 )& 5-,-+ )& 6+'&.1 )& -('%-5- /-% &156.&('134- ED * 6(15 )& %&?&%O(,61 D E* -('%-51)-% G -./&(+1)-% # D E ` 2(34- )& '%1(+?&%O(,61 &(+-% , D E ` 2(34- )& '%1(+?&%O(,61 )1 ,1)&61 )& 1,34-> 6+'&.1 1 ,-('%-51% ?&,'2-2*+& 1 +6.25134- ,-. 2. <1(9- /%-/-%,6-(15 :2& ,-5-,1++& - +6+'&.1 &. .-)- ,%6'6,1.&('& &+'8@&5 D/S5-+ (- &6B- 6.1<6(8%6-E & 1,-%)- ,-. - +6+'&.1> %&/%&+&('1)- D@&% ?6< E & /1%J.&'%-+ 6(6,616+ )&?6(6)-+ (- /-('- 1('&%6-% -;'&@&*+& - +&<26('& )61<%1.1 )& @1(+ D % EK % K ?6< * % +6+'&.1
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−− =, K 0. 0 0! 0 ! . 0 ! 0 ! 0 0 0 ! ! ! =. ==< ==. . ! === = . =.==. === === === ==< = . ? @ 8 & ' "A? ?6< * % +6+'&.1 &. .1591 ?&,91)1 /1%1 .-)&5- &+'6.1)- &
- 30. " ?&,'21()- 1 %&/%&+&('134- )1 %&+/&,'6@1 %&+/-+'1 1- &+,154-K & ' ! ! . . 0 ! ! . : ! ?6< * &+/-+'1 )- +6+'&.1 1- &+,154- & 1,-%)- ,-. 1 1(856+& )- % > - +6+'&.1> 1/S+ 1 6('%-)234- )& )-6+ =&%-+ %&16+ & 2. /S5- (1 -%6<&. D,-. 1 6('%-)234- )- ,-('%-51)-% E> ?6,-2 ,-. )-6+ /S5-+ )-.6(1('&+ ,-./5&B-+ ,-(02<1)-+ 6.251()- 1 15'&%134- )- <1(9- &+'8'6,- D.1('&()- - /S5- (1 -%6<&. & -+ )-6+ =&%-+ &. `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−= !V"V> - ,1+- &. :2& 1 '&./&%1'2%1 )& '&+'& A 6(?&%6-% R '&./&%1'2%1 )- )6++6/1)-% -;'&@&*+& 1 +&<26('& &B/%&++4-K 0- +−= V>"
- 32. 4 ( 5 *1&%/-.* 2* /0*90,7, & 1,-%)- ,-. 1 &+/&,6?6,134-> '-%(-2*+& (&,&++8%61 1 2'656=134- )& @8%61+ 56(<21<&(+ )& /%-<%1.134-> /1%1 6./5&.&('134- )- /%-0&,'- & :2& +& %&?&%&(,61. (- &+:2&.1 1/%&+&('1)- D@&% ?6< E " # B C D "D 2 7 3+ * E 3+ 7 3+ A 3+ + * D "D F " A 4 3+ 2 4 @ A A A 3+ 2 @ : ?6< * &/%&+&('134- &+:2&.8'6,1 )1+ 56(<21<&(+ )& /%-<%1.134- 2'656=1)1+
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
- 34. 7/5%7%1',-.* 1)1 1 (&,&++6)1)& )& ,-.2(6,134- &('%& 1 1/56,134- & - C > 6./5&.&('-2*+& - /%-'-,-5- )& ,-.2(6,13I&+ 1'%1@A+ )& 2. ,-(02('- )& ,-.1()-+ )& 1,-%)- ,-. - &+/&,6?6,1)- ,-. - D@&% 1;&51 E ,-(02('- )& ,-.1()-+ 6./5&.&('1)-+ ?-6 - +&<26('&K *7,12* ,0G7%'0*+ #$%&'()* K K K K -%%&('& &. . ] %-<%1.1% 1 ,-%%&('& )& 1+& K g &./&%1'2%1 N ] 6++6/1)-%> .;6&('&> A5251 ?&,'21% 5&6'2%1 '&./&%1'2%1+ K K g -%%&('& -5&,'-% ?&,'21% 5&6'2%1 ,-%%&('& ,-5&,'-% 15K155 &(92. 156;%134- )- &:26/1.&('- a g &(92. 1%1,'&%7+'6,1+ -.2(6,134- +A%6& a g &(92. &+'& )- 6(+'%2.&('- &(92. )&('6?6,134- )- 6(+'%2.&('- K &./&%1'2%1 &. N ] %-<%1.1% '&./&%1'2%1 )& '&+'& K K K K &(+4- &. ] %-<%1.1% '&(+4- )& ,-5&,'-% h K K X g &(92. ;'&(34- )& &%%- -,-%%6)- h K g &(92. 1'1 )1 (-%.1 6./5&.&('1)1 1;&51 * 6+'1 )& ,-.1()-+ 6./5&.&('1)-+ +6+'&.1 6./5&.&('1)- '&. +&(+-%&+ & 1,'21)-%&+ &> (- ,1+- )-+ 1,'21)-%&+> 1 ,-.2(6,134- &('%& - C & -+ ,6%,26'-+ )& 1,'2134- 1++-,61)-+ %&,-%%& R 2'656=134- )& ,-(@&%+-%&+ 6<6'15G (15S<6,- &+,-591 %&,162 +-;%& - ,-(@&%+-% .-)&5- V! F )& F ;6'+> :2& '&. ,-.- 2.1 )1+ +21+ ,1%1,'&%7+'6,1 1 /-++6;656)1)& )1 '&(+4- )& +17)1 /-)&% @1%61% &('%& M & ` %&51'6@1.&('& R '&(+4- )& " +'1 ,1%1,'&%7+'6,1 A (&,&++8%61 /1%1 1 6./5&.&('134- &?&,'21)1 (1 <&%134- )1 ,-%%&('& )& ;1+&> '&(+4- )& ,-5&,'-% & 16()1 )1 /%-<%1.134- )1 '&./&%1'2%1 )1 ,A5251
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• &(+4- ,-5&,'-% &.6++-%> • -%%&('& ,-5&,'-% .8B6.1> • -%%&('& )& ;1+& .8B6.1> • 1++- ,-%%&('& )& ;1+&> • 1++- '&(+4- ,-5&,'-% &.6++-%> • -'O(,61 .8B6.1> -. 1 6('&%?1,& ,-. - 2'656=1)-% D@&% 1(&B-+ +&,34- ('&%?1,& 1' 1;# * &(2 /%6(,6/15E ?6,1 )6+/-(7@&5 1 15'&%134- )-+ @15-%&+ /-% )&?&6'- )&?6(6)-+ /1%1 - '&+'& )- '%1(+7+'-%
- 36. ! " C $ G* 2 @ 4 8 * F # A 3+ " 4 F C 3+ E 3+ ( ( A 3+ F 3+ * * " ( 3+ 4 " # 3+ " & ' * 1 ( * ( 8 * ( A H 8? 7 * 3+ # 8 * 2 A I# * J 3+ E 3+ K " # ( ( ( ?6< * 52B-<%1.1 <&%15 )& 6./5&.&('134- ,S)6<- &. 1' 1;#
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i> i +'& )&56.6'1)-% /-)& +&% ?1,65.&('& 15'&%1)- D ` :26+6,1- 1)-+ . @1%68@&5 +&/1%1)-%E &+'%2'2%1 <&%15 )& ,-./-%'1.&('- )- /%-<%1.1> :21()- )1 %&,-591 )& 1.-+'%1+ %&156=1)1+ )2%1('& 1 &B/&%6O(,61 &(,-('%1*+&> )&?6(6)1 D@&% ?6< E
- 38. F " C E 8 * " L + 4 2 A G? I L 4 9 9 - 3+ 9 G? ( 8 * " & ' 2 & 4 M M 1 ' 2 7 * 3+ # 8 * % A ? G? A N ( ?6< * ./5&.&('134- 1:26+634- )& )1)-+
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
- 40. " K2(9* 2* (&0* *1'0*5,2*0 & ?-%.1 1 /-)&% 56<1%*+& - 1' 1;# 1- ,S)6<- )- C > 6./5&.&('-2*+& - ,S)6<- :2&> )& 1,-%)- ,-. - +'1()1%) > %&,&;& -+ ,-.1()-+ & 1,'21 (- &:26/1.&('-> '15 ,-.- %&/%&+&('1)- <&(&%6,1.&('& D@&% ?6< E " C $ G* * 4 2 @ 4 # D ND ( 3+ " 8 * 3+ + D D ( 4 * N 8 * 3+ + D D ( " " " & ' " " " & ' " " * F ?6< * S)6<- 2'656=1)- (- C
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j -%)&.> /-)& &B'%16%*+& )1 -;+&%@134- )1 ?6<2%1> 2'656=1()- -+ @15-%&+ )1 &B/&%6O(,61> &('%& -2'%-+> -+ +&<26('&+ /1%J.&'%-+K * &./- )& /6,- ≈ > * &./- )& ,%&+,6.&('- * !≈ >
- 42. * &./- )& &+'1;&5&,6.&('- !≈ > 15,251.*+& -+ %&+'1('&+ /1%J.&'%-+ 1'%1@A+ )1 1(856+& )1 ?6<2%1 %&+/&,'6@1 D@&% ?6< E> & -;'A.*+&K * -;%&*&5&@134-> ED EDED ""DkE +∞ +∞− = - -- > k > ""DkE = − = * &%7-)- )1+ -+,6513I&+ π = > &/%&+&('1()- - 6(+'1('& )& '&./- /1%1 - :215 -,-%%& - N .8B6.- )& -D E %&+25'1 &('4-K == π -2 a= a == * %&:2O(,61 )1+ -+,6513I&+ 1.-%'&,6)1+> ! F>"== π %1)G+ ;'6)-+ -+ @15-%&+ &B/&%6.&('16+ %&51'6@-+ R %&+/-+'1 )- +6+'&.1 1- &+,154- /1%1 @8%61+ '&./&%1'2%1+ )& '&+'& D&('%& N & !"N E> - @15-% )1 +-;%&*&5&@134- DkE +6'21* +& (- 6('&%@15- &('%& " & Fk> /1%1 2. '&./- )& &+'1;&5&,6.&('- D E &('%& V+ 1 "N & "+ 1 !"N D@&% ?6<2%1+ 1(&B- %&+25'1)-+ ?6(16+E + @15-%&+ -;'6)-+ /1%1 1 +-;%&*&5&@134- 2'656=1()- 2. ,-('%-51)-% > & (- ,1+- )& +&%&. 1/56,1)1+ 1+ %&<%1+ )& * + &+'1%4- (2. 6('&%@15- )& "k 1 !"k )- @15-% .8B6.- )1 %&+/-+'1 1- &+,154- ] &+'& ,1+- - @15-% )& +-;%&*&5&@134- @1%61 &('%& " & Fk> - :2& 6()6,61 :2& - /%-,&++- )- A'-)- )1 2%@1 )& %&1,34- )& * + 2'656=1)- /-)& +&% 1/56,1)- (&+'& .-)&5- )& A5251 )& &5'6&% . -2'%- /1%J.&'%- 6./-%'1('& A - @15-% .8B6.- )1 '&./&%1'2%1 +2/-%'1)1 /&51 ,A5251 )& &5'6&% :2&> )& 1,-%)- ,-. 1 &+/&,6?6,134- )- ?1;%6,1('&> A )& F N 1%1 - ,1+- &. :2& 1 '&./&%1'2%1 )& '&+'& A )& !"N & ,-(+6)&%1()- - @15-% )& +-;%&*&5&@134- -;'6)- &B/&%6.&('15.&('& '&.*+&K
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- 44. %7/%0,'40, 2, &?545, 1%1 ,-(9&,&% -+ &%%-+ %&51'6@-+ )1 '&./&%1'2%1> 2'656=1%1.*+& -+ ,-.1()-+ /%&@61.&('& 6./5&.&('1)-+ ++6.> /-% ,1)1 5&6'2%1 )& '&./&%1'2%1 )&('%- )1 <1.1 )& '&./&%1'2%1+ )& 6('&%&++&> ?-%1. &(@61)-+ )-6+ ,-.1()-+K /%6.&6%- ,-. 1 +6('1B& )&?6(6)1 )- '6/- K &. :2& %&/%&+&('1 1 '&./&%1'2%1 )& '&+'& ^ - +&<2()- :2& )&@&%8 +&% &(@61)- 1/S+ 1 ,&%'6?6,134- )& :2& 1 '&./&%1'2%1 (1 ,A5251 +& &(,-('%1 (2.1 @6=6(91(31 &+'1;&5&,6)1 DMG* kE )- @15-% ?6(15 )1 %&+/-+'1 A K g /S+ 1 /%-<%1.134- (- C )1+ ?2(3I&+ ,1%1,'&%7+'6,1+ D@&% ?6< E -;'6@&%1.*+& -+ +&<26('&+ &%%-+ &B/&%6.&('16+ /1%1 2. 6('&%@15- )& '&./&%1'2%1+ &('%& N & !"N & /1%1 2.1 '&./&%1'2%1 1.;6&('& )& "N K 8 * *G 0 9 0.9 0 9 0!9 0 9 9 9 !9 9 .9 9 ! ! . . : : " )3&* #6)7* 8 * ?6< * %%- %&51'6@- '&./&%1'2%1 D'&./ )6++6/1)-% &('%& F>VN & > N E O.*+&> 1++6.> &%%-+ %&51'6@-+ (- 6('&%@15- &('%& k & k )- @15-% ?6(15 /%-<%1.1)- +'&+ &%%-+ &(,-('%1.*+& )&('%- )-+ @15-%&+ &B/&,'8@&6+ /1%1 - +6+'&.1 /%-0&,'1)-
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