relatório - dtr - distribuição de tempos de residência - feq eq801 - unicamp - grupo 2 - final

7/23/2019 Relatório - DTR - Distribuição de Tempos de Residência - FEQ EQ801 - UNICAMP - Grupo 2 - Final

http://slidepdf.com/reader/full/relatorio-dtr-distribuicao-de-tempos-de-residencia-feq-eq801-unicamp 1/35

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AHOGPM\AS

AHOGPM\AS......................................................................................................................63

MKPXA@WÎÃA..................................................................................................................63

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DGDÓXMJ @G CÊICWIA..................................................................................................6=

XGSWIPJ@AS G @MSCWSSÃA.............................................................................................:8

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3

AHOGPM\AS

Kgstg gxpgrmdgkta ahogtmvau-sg jkjimsjr g vgrmlmcjr a cadpartjdgkta `g rgjtargs kãa-

m`gjms. Vjrj mssa `gtgrdmkau-sg j `mstrmhumîãa à tgdpa `g rgsm`ïkcmj (@PX) gd rgjtargs à

tmpa VLX g CSPX, cjicuiau-sg a tgdpa dí`ma `g rgsm`ïkcmj, cadpjrau-sg js `mstrmhumî÷gs

gxpgrmdgktjms cad da`gias tgórmcas g, par lmd, ahsgrvau-sg a cadpartjdgkta `g

gscajdgktas kãa m`gjms.

MKPXA@WÎÃA

Xgjtargs sãa as gqumpjdgktas qug rgngd a pracgssa `g praùîãa `g ud `gtgrdmkjà

praùta quädmca. Sgkà jssmd, a praogta g gscaibj `g autras gqumpjdgktas sãa mkliugkcmjàs

pgia tmpa `g rgjtar g suj glmcmïkcmj. Kgssg caktgxta, gstuàs j rgspgmta à lukcmakjdgkta

`gssgs gqumpjdgktas sãa gxtrgdjdgktg mdpartjktgs pjrj j dgibarmj às pracgssas. Gssj

dgibarmj í kgcgssêrmj pams, gd ud pracgssa rgji, bê `gsvmas `j m`gjim`j`g qug ljzgd cad qug

a rgjtar kãa smnj gxjtjdgktg j cmkítmcj g j liumà`mkèdmcj m`gjidgktg gspgrj`js.

Hjsmcjdgktg bê àms tmpas `g gscajdgktas m`gjms? tuhuijr gdpmstakjà g dmsturj

pgrlgmtj. Gd ud rgjtar pmstãa ta`js js daiícuijs pgrdjkgcgd ka smstgdj a dgsda mktgrvjia

`g tgdpa(dgsda tgdpa `g rgsm`ïkcmj) g gd ud rgjtar cad dmsturj pgrlgmtj a tgdpa `g

rgsm`ïkcmj `js daiícuijs pa`g vjrmjr `g zgra jtí mklmkmta.0

Gd ud gscajdgkta rgji, ta`jvmj, as gigdgktas `g liumà pgrcarrgd cjdmkbas

`mlgrgktgs uks às autras g, `gssj lardj, passugd tgdpas `mlgrgktgs `g sjä`j. Gssg `gsvma

`j m`gjim`j`g í jnrjvjà par lgkðdgkas cada zakjs `g gscajdgktas prglgrgkcmjms (curta-

cmrcumta, suhpjssjngd g cjkjimzjîãa), zakjs dartjs (gstjnkjîãa), `mspgrsãa jxmji gd rgjtargstuhuijrgs g sgnrgnjîãa rgsuitjktg `js cak`mî÷gs `g dmsturj.7

Lmnurj : – Gxgdpias `g caklardjî÷gs ngadítrmcjs qug jlgtjd gqumpjdgktas kãa m`gjms.

Vjrj `mjnkastmcjr gssgs `gsvmas `j m`gjim`j`g í utmimzjà a cakcgmta `g @mstrmhumîãa `g

Pgdpas `g Xgsm`ïkcmj (@PX), qug caksmstg gd `gtgrdmkjr a tgdpa qug js daiícuijspgrdjkgcgd mk`mvmùjidgktg ka rgjtar.



0

J @PX í ahtm`j gxpgrmdgktjidgktg g js tíckmcjs djms caduks sãa j pgrturhjîãa gd

puisa g gd `gnrju. Gstjs caksmstgd gd jpimcjr ka smstgdj ud djrcjàr, qug kãa `gvg

mktgrlgrmr kj dmsturj kgd kj `mkèdmcj à gscajdgkta0, g jkjimsjr j cadpasmîãa `j carrgktg `g

sjä`j ja iakna à tgdpa.

J jkêimsg `j `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj larkgcg mklardjî÷gs qug pgrdmtgd

mktgrprgtjr g prgvgr a cadpartjdgkta `g smstgdjs caktäkuas rgjms. Sg j `mstrmhumîãa jprgsgktj

udj dí`mj qug `mlgrg smnkmlmcjtmvjdgktg à tgdpa `g rgsm`ïkcmj à rgjtar, tïd-sg ud

mk`mcjtmva `g qug jinud `gsvma gstê sgkà ahsgrvjà ka pracgssa. @gssj lardj pa`g-sg

`mjnkastmcjr prahigdjs gd rgjtargs gd apgrjîãa au prgvgr j `mkèdmcj `g udj rgjîãa cjsa

gij vgkbj j sgr rgjimzj`j gd ud rgjtar `mspakävgi.

XG\MSÃA PGÓXMCJ

Parkj-sg kgcgssêrmj udj rgvmsãa à da`gia tgórmca `g rgjtargs `gvmà ja `gsvma `jm`gjim`j`g às qug larjd utmimzjàs ka gxpgrmdgkta. Gd ud rgjtar CSPX pgrlgmta, a tgdpa `g

rgsm`ïkcmj `js daiícuijs kgig prgsgktgs pa`g vjrmjr `g zgra j mklmkmta, sgkà zgra a cjsa gd

qug bê curta-cmrcumta g mklmkmta a cjsa qug bê zakj dartj. Vjrj rgjtargs m`gjms, a tgdpa `g

rgsm`ïkcmj í `g ud vjiar lmxa ùrjktg taà a pracgssa, kãa bjvgkà gstjnkjîãa au curta-

cmrcumtas. Oê pjrj a VLX, tgdas gd ngrji ud gscajdgkta turhuigkta, kãa bjvgkà rgnmãa `g

zakj dartj. Varíd, bê a cjsa `g igmtas lmxas (VLX”s cad j prgsgkîj `g cjtjimzjàrgs sóimàs ja

iakna à rgjtar) j lardjîãa `g cjdmkbas prglgrgkcmjms. Vjrj rgjtargs VLX m`gjms, mssa kãa

jcaktgcg. Gssjs acarrïkcmjs sãa `gscrmtjs kjs Lmnurjs 8 g 3.

Lmnurj 8 – Lardjîãa `g curta-cmrcumtas (sbart-cmrcumtmkn) g zakjs-dartjs (stjnkjkt

rgnmaks)



7

Lmnurj 3 – Lardjîãa `g cjdmkbas prglgrgkcmjms (cbjkkgimkn) gd igmtas gdpjcatjàs

( pjceg` hg`).

Gstu`j-sg j `gtgrdmkjîãa à tgdpa `g rgsm`ïkcmj (@PX) pjrj jkjimsjr cada a rgjtar rgji

gstê `gsvmjkà `j m`gjim`j`g. Caklardg as rgsuitjàs, du`jkîjs pa`gd sgr lgmtjs pjrj gvmtjr

as `gsvmas.

Vjrj ahtgrdas a @PX, kardjimzj-sg j curvj `g tgdpa `g rgsm`ïkcmj, pjrj qug j êrgj sah

gssj curvj gqumvjibj j :?

Jssmd, j lrjîãa `j carrgktg `g sjä`j cad tgdpa `g rgsm`ïkcmj gktrg t g t+`t í?

Vjrj glgmtas `g cadpjrjîãa, g par caksgnumktg, `gcmsãa `g cada jnmr, as pjrèdgtras ̇g sg dastrjd `g gxtrgdj mdpartèkcmj gd @PX. ̇ gqumvjig ja tgdpa dí`ma gd qug js

daiícuijs rgsm`mrjd ka rgjtar, gkqujkta í `jà par pjrèdgtras prgvmjdgktg `gtgrdmkjàs,

qug í a vaiudg à rgjtar (\) g j vjzãa vaiudítrmcj `g jimdgktjîãa (Z). Pjms pjrèdgtras sãa

cjicuijàs `js sgnumktgs djkgmrjs?

̇



<

Cadpjrj-sg gktãa tjms vjiargs `g ̇g . Cjsa?

̇ 4 ? A rgjtar í m`gji. Wd vjiar m`ïktmca kãa jcaktgcgrê kj rgjim`j`g, paríd

qujkta djms próxmda, dgibar.

̇ 9 ? Gxmstgd zakjs dartjs.

̇; ? Gxmstgd curtas-cmrcumtas.

G jssmd tadj-sg j `gvm`j `gcmsãa pjrj dgibarmj à rgjtar.

DJPGXMJMS G DÍPA@AS

M. Djtgrmjms

A pracg`mdgkta gxpgrmdgktji sg rgjimzau cad a usa às sgnumktgs djtgrmjms?

Xgjtar Puhuijr

Pjkqug Jnmtjà

Carjktg `g jimdgktas (trjîjàr)

Vravgtj (:66di)

Crakðdgtra

Gspgctralatðdgtra

Cuhgtjs

MM.

Dítaàs Gxpgrmdgktjms

Cakùzmu-sg a gxpgrmdgkta gd 3 smtujî÷gs?

Pjkqug jnmtjà `g liuxa caktäkua cad j pê `g jnmtjîãa j :cd à lukà à vaiudg

rgjcmakji, CSPX sah j igngk`j (Hjsg – Lukà) g jtrmhuä`j j jkêimsg `g @gsvma5

Pjkqug jnmtjà `g liuxa caktäkua cad j pê `g jnmtjîãa j :cd `j supgrläcmg à

iäqumà, CSPX sah j igngk`j (Papa – Supgrläcmg) g jtrmhuä`j j jkêimsg `g _akj

Dartj5

Xgjtar tuhuijr, VLX.



>

MMM. Xgjtar tjkqug jnmtjà `g liuxa caktäkua (CSPX)

Lmnurj 0 – Daktjngd gxpgrmdgktji à rgjtar tjkqug jnmtjà

m.

Gtjpj `g prgpjrjîãa

Joustj-sg j pê à jnmtjàr j, jpraxmdj`jdgktg, :cd à lukà à vjsa g imnj-sg j

jnmtjîãa5

Joustj-sg js vjz÷gs `g ênuj `g gktrj`j pjrj :36 di/dmk (`gmxjkà j tarkgmrj

cadpigtjdgktg jhgrtj g joustjkà j vjzãa par dgma `j vêivuij \6)5

Ka dadgkta gd qug a rgjtar gkcaktrj-sg tatjidgktg cbgma, cad ênuj liumkà

pgij sjä`j rg`mrgcmakj-sg j gktrj`j `g ênuj pjrj ud iacji `g `gscjrtg, jlmd `gdjktgr j vjzãa cakstjktg. Kgstj gtjpj, kãa sg djkmpuij vêivuijs `g caktraig `g

vjzãa5

Ljz-sg j `gtgrdmkjîãa à vaiudg à rgjtar jtrjvís à vaiudg `g iäqumà prgsgktg

kgstg5

Gsvjzmj-sg a tjkqug, cadpigtjkà-a, pastgrmardgktg, cad a carjktg (trjîjàr)5

Imnj-sg a smstgdj `g jnmtjîãa.

mm. Gtjpj gxpgrmdgktji

Jhrg-sg cadpigtjdgktg j tarkgmrj pjrj gktrj`j `g ênuj g jcmakj-sg a

crakðdgtra5

Sãa rgtmrj`js jdastrjs gd tuhas `g gksjma kas tgdpas 6, :6””, 86””, 36””, 06””, 76””,

:”66—, :”:6—, :”36—, :”76—, 8”:6—, 8”06—, 3”:6—, 0”66—, 7”66—, <”66—, >”66—, :6”66—,

:8”36—, :7”66—, 86”66—, 87”66—, 37”66—, 07”66—5

Jkjimsj- d)5

J prêtmcj í rgpgtm`j cad j pê `g jnmtjîãa j, jpraxmdj`jdgktg, :cd `j supgrläcmg

à iäqumà.



2

M\. Gscajdgkta tuhuijr (VLX)

Lmnurj 7 – Daktjngd gxpgrmdgktji à rgjtar tuhuijr

M. Gtjpj `g prgpjrjîãa

Jhrg-sg j tarkgmrj g j vêivuij \6 pjrj gkcbgr a rgjtar (vêivuij \8 g prgsmibj V8,

lgcbj`js)5

Cad rgjtar cbgma, joustj-sg j vjzãa `g ênuj gd :66 di/dmk par dgma `j vêivuij

\35

Lgcbj-sg j prgsmibj V: g jhrg-sg V8 pjrj gsvjzmjr a tuha. Caigtj-sg a iäqumà pjrj

`gtgrdmkjîãa à vaiudg à rgjtar5

Lgcbj-sg V8 kavjdgktg g jhrg-sg \8 pjrj qug acarrj a trjîjàr gkcbj a tuha5

Jpós a gkcbmdgkta, lgcbj-sg \8.

MM. Gtjpj gxpgrmdgktji

Ka dadgkta gd qug bê trjîjàr gd taà a tuha, jhrg-sg V: g jcmakj-sg a

crakðdgtra5

Sãa rgtmrj`js jdastrjs gd tuhas `g gksjma kas tgdpas 6, :”, 3”, 7”, >”, =”, ::”, :3”,

:0”, :7”, :<”, :>”, :>” 36—, :2”, :2” 36—, :=”, :=” 36—, 86”, 86” 36—, 8:”, 8:” 36—, 88”,

88”36—, 83”36—, 80”, 80”36—,87”, 87”36—, 8<”, 8<”36—, 8>”,82”, 8=”, 36”,38”, 37”, 32”,

06”5

Jkjimsjd-sg, gktãa, js jdastrjs cad juxäima ̀ a gspgctra



=

DGDÓXMJ @G CÊICWIA

:. Cêicuia à vaiudg às rgjtargs g `g ?

As vaiudgs às rgjtargs VLX g CSPX larjd dg`màs cad udj pravgtj nrjùj`j? g 4 :326 dI. Cad gstg vjiar g cad j vjzãa `g ênuj utmimzj`j, cjicuiau-sg a

tgdpa `g rgsm`ïkcmj ()?

8. Jkêimsg `j jhsarhèkcmj

Vgij Igm `g Hggr,

@gtgrdmkj-sg (Gqujîãa :6)?

Gxgdpimlmcjkà cad j dg`m`j pjrj a VLX cad t 4 : dmk?

3. Cêicuia `js lukî÷gs L(t) g G(t)

J lukîãa L(t) í `glmkm`j cada?

J lukîãa G(t) í `glmkm`j par?

G(t)lam gktãa ahtmà pgij vjrmjîãa `g C/C6 `mvm`mà pgij vjrmjîãa `g tgdpa, cada udj

jpraxmdjîãa?

(:0)

(:7)



:6

0. Cêicuia à Pgdpa `g Xgsm`ïkcmj Dí`ma?

A tgdpa `g rgsm`ïkcmj dí`ma lam ahtmà jtrjvís à pranrjdj Armnmk par mktgnrjîãa

kudírmcj.

7. \jrmèkcmj

J vjrèkcmj pa`g sgr cêicuij`j jtrjvís `j gqujîãa j sgnumr?

̇

Cada jpraxmdjîãa, j vjrmèkcmj lam cjicuij`j `j sgnumktg djkgmrj?

√ ̇

<. CSPX m`gji

J lukîãa G(t) pjrj a CSPX m`gji í `jà par?

Vjrj a tgdpa t 4 7 dmk, tgd-sg?

( )

J lukîãa L(t) í `j`j par?

Vjrj j dgsdj dg`m`j, tgd-sg qug?

6,=20

Da`gia cad curta-cmrcumta g tgdpa darta?

J lukîãa G(t) g L(t), pjrj gstg a cjsa, sãa `j`js par?



::

Vgij imkgjrmzjîãa `j gqujîãa í passävgi ahtgr jilj g hgtj jtrjvís à nrêlmca `g ?

(

) (

) (

)

>. Da`gia pjrj K tjkqugs CSPX g VLX gd sírmg?

Cad hjsg ka tgdpa gspjcmji cjicuij-sg a kÿdgra `g rgjtargs CSPX gd sírmg?

Jssmd, j `mstrmhumîãa às tgdpas `g rgsm`ïkcmj í `j`j par?

( )

Ak`g

Da`gia m`gji pjrj a VLX?

Vjrj a VLX m`gji, caksm`grj-sg qug G(t) g L(t) gqumvjigd j zgra jtí t48<,7 dmk, qujkà

pjssjd j vjigr : g mklmkmta, rgspgctmvjdgktg. Jpós t48<,7 dmk, L(t) vaitj j vjigr zgra,

ianmcjdgktg.

2.

VLX cad gscajdgkta ijdmkjr sgnrgnjà

Kgstg cjsa, tgdas qug?

Zujkà t 9 π/8 - L(t) 4 G(t) 4 6

Zujkà t ; π/8 - g

Vjrj a tgdpa t 4 > dmk, tgdas?

L(t) g G(t) 4 6

Vjrj tgdpa t 4 8: dmk, tgdas?



:8

XGSWIPJ@AS G @MSCWSSÃA

@jà a pracg`mdgkta gxpgrmdgktji jàtjà, trjtj-sg `g gxpgrmdgkta à tmpa ’wjsb-

aut— au `gnrju kgnjtmva. @gstj lardj, j cakcgktrjîãa `g trjîjàr gd lukîãa à tgdpa

kardjimzj`j pgij cakcgktrjîãa dêxmdj `g trjîjàr, au sgoj, j dg`m`j `g cakcgktrjîãa

rgijtmvj – C (t)/C6 – tgd cadpartjdgkta mkvgrtmà. Caksgqugktgdgktg, kãa sg ahtïd a nrêlmca

`j `mstrmhumîãa cuduijtmvj às tgdpas `g rgsm`ïkcmj (L(t)) `mrgtjdgktg g smd j lukîãa U: –

L(t)T.

Jprgsgktjd-sg j sgnumr kjs Pjhgijs : g 8 a cakoukta `g `jàs rglgrgktg jas àms

gksjmas rgjimzjàs? (m) smduijkà ud rgjtar VLX5 (mm) smduijkà ud rgjtar CSPX, cad js pês à

mdpgimàr à smstgdj `g jnmtjîãa próxmdj ja lukà (Hjsg – @gsvma) g próxmdj æ supgrläcmg

(_akj Dartj – Papa). As `jàs sg rglgrgd æs dg`m`js `g jhsarhèkcmj lgmtjs cad js jiäquatjs

rgtmrj`js ja iakna à gxpgrmdgkta, g jas `jàs `g cakcgktrjîãa kardjimzjàs j pjrtmr às

rgsuitjàs ̀ a gspgctalatðdgtra.

Pjhgij : – @jàs gxpgrmdgktjms rglgrgktgs æs jhsarhèkcmjs dg`m`js g rjz÷gs `g

cakcgktrjî÷gs gd rgijîãa æ cakcgktrjîãa mkmcmji pjrj as rgjtargs CSPX

Jhsarhèkcmj (7:8 kd) C/C6 U:-L(t)T : - C/C6 UL(t)T

Pgdpa(s)

@gsvma(Hjsg)

_akj Dartj(Papa)

@gsvma(Hjsg)

_akj Dartj(Papa)

@gsvma(Hjsg)

_akj Dartj(Papa)

6 6,>>> 6,>23 :,666 :,666 6,666 6,666

:6 6,<<< 6,<20 6,27> 6,2>0 6,:03 6,:8<

86 6,<02 6,<>8 6,230 6,272 6,:<< 6,:08

36 6,<0 6,<<0 6,280 6,202 6,:>< 6,:78

06 6,<08 6,<77 6,28< 6,23> 6,:>0 6,:<3

76 6,<3 6,<70 6,2:: 6,237 6,:2= 6,:<7

<6 6,<88 6,<00 6,26: 6,288 6,:== 6,:>2

>6 6,<:< 6,<33 6,>=3 6,262 6,86> 6,:=8

=6 6,<60 6,<:0 6,>>> 6,>20 6,883 6,8:<

::6 6,722 6,<6> 6,>7> 6,>>7 6,803 6,887

:36 6,7>7 6,728 6,>06 6,>03 6,8<6 6,87>

:<6 6,770 6,7<= 6,>:3 6,>8> 6,82> 6,8>3

:=6 6,702 6,77: 6,>67 6,>60 6,8=7 6,8=<

806 6,76< 6,7:> 6,<7: 6,<<6 6,30= 6,306

366 6,0<> 6,02> 6,<6: 6,<88 6,3== 6,3>2

3<6 6,03> 6,07> 6,7<8 6,720 6,032 6,0:<

086 6,06> 6,08: 6,780 6,732 6,0>< 6,0<8

<66 6,33: 6,377 6,08< 6,073 6,7>0 6,70>

>76 6,8>> 6,363 6,37< 6,32> 6,<00 6,<:3

=66 6,83: 6,87> 6,8=> 6,382 6,>63 6,<>8

:866 6,:<< 6,:20 6,8:0 6,837 6,>2< 6,><7

:766 6,::> 6,:0: 6,:7: 6,:26 6,20= 6,286

8:66 6,6=8 6,6>3 6,::2 6,6=3 6,228 6,=6>

8>66 6,682 6,608 6,63< 6,670 6,=<0 6,=0<



:3

Pjhgij 8 – @j`as gxpgrmdgktjms rglgrgktgs æs jhsarhèkcmjs dg`m`js g rjz÷gs `g

cakcgktrjî÷gs gd rgijîãa æ cakcgktrjîãa mkmcmji pjrj a rgjtar VLX

Pgdpa (dmk)Jhsarhèkcmj

(7:8kd)C/C6 (:-L(t)) : - C/C6 (L(t)) G(t) (`L/`t)

6 6,>26 :,666 6,666 6,6<0

: 6,>36 6,=3< 6,6<0 6,680

3 6,>80 6,=82 6,6>8 6,66:

7 6,>82 6,=33 6,6<> -6,66<

> 6,>08 6,=7: 6,60= -6,668

= 6,>37 6,=08 6,672 6,66:

:: 6,>32 6,=0< 6,670 -6,667

:3 6,>76 6,=<8 6,632 -6,663

:0 6,>07 6,=77 6,607 6,663

:7 6,>07 6,=77 6,607 6,66<

:< 6,>37 6,=08 6,672 6,672

:> 6,<70 6,232 6,:<8 6,::7

:>,7 6,<66 6,><= 6,83: 6,6=8

:2 6,728 6,>0< 6,870 6,:>8

:2,7 6,0<< 6,7=> 6,063 6,88:

:= 6,0:6 6,78< 6,0>0 -6,66:

:=,7 6,0<> 6,7== 6,06: 6,680

86 6,3=: 6,76: 6,0== 6,:67

86,7 6,327 6,0=0 6,76< -6,6:8

8: 6,066 6,7:3 6,02> 6,6<=

8:,7 6,33: 6,080 6,7>< 6,676

88 6,3<: 6,0<3 6,73> 6,60<

88,7 6,8=7 6,3>2 6,<88 6,62:

83 6,8=2 6,328 6,<:2 6,62<

83,7 6,882 6,8=8 6,>62 6,688

80 6,82: 6,3<6 6,<06 -6,633

80,7 6,870 6,38< 6,<>0 6,67:

87 6,80: 6,36= 6,<=: 6,68<

87,7 6,830 6,366 6,>66 -6,6><

8< 6,366 6,327 6,<:7 -6,6>:

8<,7 6,82= 6,3>: 6,<8= 6,670

8> 6,872 6,33: 6,<<= 6,6<3

82 6,8:7 6,8>< 6,>80 6,6:0

8= 6,83< 6,363 6,<=> -6,6:2

36 6,803 6,3:8 6,<22 -6,66=

38 6,87> 6,38= 6,<>: 6,638

37 6,:86 6,:70 6,20< 6,632

32 6,6>> 6,6== 6,=6: 6,6:0

06 6,6<< 6,627 6,=:7 6,66>



:0

@a cêicuia `j vjzãa pjrj a gxpgrmdgkta cad a rgjtar CSPX rgsuitjd as `jàs

prgsgktgs kj Pjhgij 3. As cêicuias larjd lgmtas j pjrtmr `j dg`mîãa `mrgtj `g ud vaiudg gd

ud mktgrvjia `g tgdpa, jàtj-sg kavjdgktg j igngk`j ’@gsvma – Hjsg— pjrj a gksjma lgmta

cad a mdpgimàr próxmda ja lukà à híqugr gxpgrmdgktji g ’_akj Dartj – Papa— próxmda æ

supgrläcmg à vaiudg rgjcmakji, cada sg vgrê pastgrmardgktg gssj kadgkcijturj cak`mz cad

as rgsuitjàs ahsgrvjàs. Pjdhíd gstãa prgsgktgs as `jàs à tgdpa gspjcmji pjrj cj`j

smstgdj.

Pjhgij 3 – \aiudg g vjz÷gs dg`m`js, jssmd cada tgdpas gspjcmjms cjicuijàs, pjrj jdhas as

rgjtargs CSPX, cad a jnmtjàr próxmda ja tapa g à lukà.

Xgjtar @gsvma (Hjsg) _akj Dartj (Papa)

\aiudg (dI) :326

\jzãa Dí`mj (dI/dmk) ::3,0 =6,6

Pgdpa Gspjcmji – π (s) >36,0 =86,6

As rgsuitjàs gqumvjigktgs pjrj a rgjtar VLX sg gkcaktrjd kj Pjhgij 0?

Pjhgij 0 – \aiudg g vjz÷gs dg`m`js, jssmd cada tgdpas gspjcmjms cjicuijàs, pjrj jdhas a

rgjtar VLX, cad a jnmtjàr próxmda ja tapa g à lukà.

Xgjtar VLX

\aiudg (dI) :=26

\jzãa Dí`mj (dI/dmk) >0,67

Pgdpa Gspjcmji - π (dmk) 8<,>0

CSPX kãa M`gji

\. @mstrmhumîãa `g Pgdpas `g Xgsm`ïkcmj pjrj Xgjtargs CSPX M`gjms

Vgia hjijkîa `g djssj `g ud trjîjàr armnmkjà `g ud `mstÿrhma `gnrju, tgd-sg j

lukîãa `g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj cuduijtmvj, L, qug í `j`j par?

( )

Kavjdgktg rgssjitj-sg qug a gxpgrmdgkta jàtj ud `gnrju kgnjtmva ’wjsb-aut— a

quji sg cjrjctgrmzj par udj lukîãa cuduijtmvj cad cadpartjdgkta mkvgrsa. @gssj djkgmrj bê

udj `mlgrgkîj gd tgrdas à trjtjdgkta `g `jàs qujkà cadpjrjà ja `gnrju pasmtmva.

Varíd, `g qujiqugr lardj, gd tgrda tgórmca sg djktíd a cadpartjdgkta gspgrjà.

Var datmvas rgijcmakjàs æ `mkèdmcj às gxpgrmdgktas, j vjzãa `g jimdgktjîãa g,

caksgqugktgdgktg, j `g sjä`j às rgjtargs vjrmjrjd caklardg as gxpgrmdgktas larjd

gxgcutjàs. @gssj lardj, a sgnukà gxpgrmdgkta, par tgr dgkar vjzãa `g jimdgktjîãa qug a

prmdgmra, passum djmar tgdpa `g gspjcmji.



:7

J sgnumr, tïd-sg as nrêlmcas cad a cadpartjdgkta m`gji `g L(t) gd lukîãa à tgdpa

da`gijà caklardg j gqujîãa prgsgktg jcmdj, jpimcj`j j pjrtmr às `jàs gxpgrmdgktjms `g

tgdpas `g rgcaibmdgkta `js jiäquatjs g à tgdpa gspjcmji. Sgnug-sg j igngk`j `js curvjs

pjrgj`j cad as tgdpas gspjcmjms, jprgsgktjàs kj Pjhgij 3?

Lmnurj < – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmva L(t) gd lukîãa

à tgdpa, tgórmcj, pjrj a tgdpa gspjcmji `g cj`j gxpgrmdgkta.

Jdhjs js curvjs jprgsgktjd ud cadpartjdgkta smdmijr sgnumkà ud `gcríscmda

gxpakgkcmji `j lukîãa tgórmcj. J curvj gqumvjigktg ja rgjtar tgdpa gspjcmji dgkar jprgsgktj

j lukîãa cuduijtmvj cbgnjkà æ ukm`j`g djms rjpm`jdgktg, carrgspak`gktg æs `mlgrgktgs

vjz÷gs gxpgrmdgktjms dg`m`js.

J lukîãa `g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj, G, í `j`j, pgia hjijkîa `g djssj `g

ud trjîjàr armukà `g ud `mstÿrhma puisa.

( )

Wd `mstÿrhma `gnrju pa`g sgr caksm`grjà cada udj sírmg mklmkmtj `g `mstÿrhmas

puisas. Caksgqugktgdgktg, j sadjtórmj `g mklmkmtas puisas igvjrmj j ud `mstÿrhma `gnrju.

Djtgdjtmcjdgktg, j lukîãa `g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj cuduijtmvj, L, pa`g sgr`j`j pgij sadjtórmj `g k, tgk`gkà j mklmkmta, lukî÷gs `g `mstrmhumîãa `g tgdpas `g

rgsm`ïkcmj, G.

√

G pgia tgargdj luk`jdgktji à cêicuia,

6%

86%

06%

<6%

26%

:66%

:86%

6 :666 8666 3666 0666 7666 <666

L ( t )

Pgdpa (s)

π (s) 4 >36

π (s) 4 =86



:<

Ahtíd-sg `gssj djkgmrj j curvj `g G(t) gd lukîãa à tgdpa pjrj ud joustg m`gji às

`jàs gxpgrmdgktjms pjrj as àms gksjmas rgjimzjàs cad a rgjtar CSPX?

Lmnurj > – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj G(t) gd lukîãa à tgdpa,

tgórmcj, pjrj a tgdpa gspjcmji `g cj`j gxpgrmdgkta.

\M. CSPX”s kãa M`gjms – Xgsuitjàs Gxpgrmdgktjms

As àms prmkcmpjms prahigdjs cjrjctgrästmcas `g rgjtargs caktäkuas `g tjkqugs jnmtjàs

(CSPX”s) qug `gsvmjd a cadpartjdgkta m`gji sãa? `gsvmas au ’curta-cmrcumta— g j prgsgkîj `g

rgnm÷gs `g gstjnkjîãa au vaiudgs dartas.

Jprgsgktjd-sg j sgnumr js curvjs `g L(t) g (:-L(t)) pgia tgdpa cad hjsg kas `jàs

gxpgrmdgktjms `g cakcgktrjîãa(au jhsarhèkcmj) jprgsgktjàs ka mkäcma `gssj sgîãa.

Lmnurj 2 – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmvj cadpigdgktjr

U:-L(t)T gd lukîãa à tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd à lukà à rgcmpmgktg.

6.66%

6.68%

6.60%

6.6<%

6.62%

6.:6%

6.:8%

6.:0%

6.:<%

6 :666 8666 3666 0666 7666 <666

G ( t )

Pgdpa (s)

π (s) 4 >36

π (s) 4 =86

6.6

6.8

6.0

6.<

6.2

:.6

:.8

6 766 :666 :766 8666 8766 3666

: -

L (

t ) a u U

C / C 6

T

Pgdpa (s)



:>

Lmnurj = – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmvj L(t) gd lukîãa

à tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd à lukà à rgcmpmgktg.

Lmnurj :6 – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmvj

cadpigdgktjr U:-L(t)T gd lukîãa à tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j

supgrläcmg `j dmsturj.

6.6

6.8

6.0

6.<

6.2

:.6

:.8

6 766 :666 :766 8666 8766 3666

L ( t )

Pgdpa (s)

6.6

6.8

6.0

6.<

6.2

:.6

:.8

6 766 :666 :766 8666 8766 3666

: -

L ( t )

Pgdpa (s)



:2

Lmnurj :: – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmvj L(t) gd

lukîãa à tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j supgrläcmg `j dmsturj.

As prmdgmras paktas `js Lmnurjs 2 g :6 `gdakstrjd cijrjdgktg j prgsgkîj `g paktas

`g curta-cmrcumta ka gscajdgkta à rgjtar gxpgrmdgktji. Gssjs dgsdjs curvjs sãa

cadpjtävgms cad a dítaà gdprgnjà, `g ud `gnrju kgnjtmva, ak`g sg vï qug mkmcmjidgktg

js jiäquatjs rgtmrj`js djktmkbjd prjtmcjdgktg j dgsdj jhsarhèkcmj(cakcgktrjîãa) à

trjîjàr(carjktg) utmimzjà, caklardg sg pracg`g a gscajdgkta pgia rgjtar j cakcgktrjîãa

`gcjm jtí sg gstjhmimzjr kas ÿitmdas paktas gxpgrmdgktjms.

Js Lmnurjs = g :: gvm`gkcmjd a cadpartjdgkta cuduijtmva `j cakcgktrjîãa `g sjä`j

à rgjtar, ka mkäcma bê udj nrjk`g vjrmjîãa ka tatji `g trjîjàr qug `gmxj a smstgdj,

caklardg a tgdpa judgktj a jcÿduia crgscg cj`j vgz dgkas, `gdakstrjkà udj tgk`ïkcmj

`j curvj gd sg gstjhmimzjr pjrj tgdpas supgrmargs j 8766s.

\MM. Pgdpa Dí`ma `g Xgsm`ïkcmj ( )̇ g \jrmèkcmj (S¸)

A tgdpa dí`ma `g rgsm`ïkcmj `g `gtgrdmkj`j gspícmg `gktra `g ud rgjtar pa`g sgr

`gtgrdmkj`j j pjrtmr `j mksgrîãa `g ud trjîjàr cada `mstÿrhma `gnrju kj jimdgktjîãa. Cada

Vgia tgargdj à vjiar dí`ma?

̇ ∥ ∥

Vgij rgijîãa jktgrmar gktrg j lukîãa G g L , puisa g `gnrju, rgspgctmvjdgktg, g jinudjsdjkmpuijî÷gs jiníhrmcjs?

6.6

6.:

6.8

6.3

6.0

6.7

6.<

6.>

6.2

6.=

:.6

6 766 :666 :766 8666 8766 3666

L ( t )

Pgdpa (s)



:=

Cada

Suhstmtumkà

̇

Vartjkta, utmimzjkà `g mktgnrjîãa kudírmcj, j pjrtmr saltwjrg djtgdêtmca ArnmkVra

2.6, g igvjkà gd caksm`grjîãa qug, pjrj as `jàs gdpärmcas, armukàs `g `mstÿrhma `gnrju, j

prgcmsãa `j mktgnrjîãa pjrj ahtgkîãa à tgdpa `g rgsm`ïkcmj dí`ma gd L í djmar, ahtmvgrjd-sg sgus vjiargs.

Vjrj dgksurjr j `mspgrsãa `j `mstrmhumîãa às tgdpas `g rgsm`ïkcmj, gstmdau-sg a

pjrèdgtra gstjtästmca, vjrmèkcmj (S¸ ), kudgrmcjdgktg, ljzgkà usa à dgsda saltwjrg

djtgdêtmca.

̇

Vjrj as gksjmas cad a mdpgimàr kj hjsg g ka tapa à vaiudg rgjcmakji, tïd-sg as

sgnumktgs rgsuitjàs?

Pjhgij 7 – Pgdpa dí`ma `g rgsm`ïkcmj gxpgrmdgktji pjrj as rgjtargs CSPX cad jnmtjîãa

próxmdj j supgrläcmg (tapa) g próxmdj ja lukà (hjsg).

Xgjtar @gsvma (Hjsg) _akj Dartj (Papa) ̇(s) <06,> <8<,8

π (s) >36,0 =86,6

π (s) - ̇ (s) 2=,> 8=3,2

0826>2,< 780:62,2

Cada as vjiargs `j vjrmèkcmj pjrj jdhas as cjsas sãa próxmdas ja quj`rjà às

rgspgctmvas tgdpas dí`mas, cakcium-sg qug bê nrjk`g `mspgrsãa às `jàs gd tarka `j dí`mj.

Ljta qug gstj rgijcmakjà js cak`mî÷gs g j lardj cada a gdpmrmsda lam gdprgnjà.

Cada gspgrjà, as vjiargs `g tgdpa `g rgsm`ïkcmj ahtmàs sãa dgkargs à qug as

tgdpas gspjcmjms, pjrj jdhas as gksjmas. Gssg `gsvma gktrg ud cadpartjdgkta m`gji, tgdpa

gspjcmji, g a rgji, tgdpa dí`ma `g rgsm`ïkcmj, carrgspak`g j ud smstgdj `g jnmtjîãa kãa-m`gji

qug jcjhj par armnmkjr cjdmkba prglgrgkcmjms `g gscajdgkta `g daà qug a liumà

pgrdjkgcg dgkas tgdpa mktgrkjdgktg ja rgjtar à qug a gspgrjà.



86

Wdj jkêimsg djms crmtgrmasj `gssj mkliuïkcmj gxmng qug `gscrgvj cad djmar `gtjibgs a

smstgdj `g jnmtjîãa mdpigdgktjà, gspgcmlmcjkà a da`gia `g mdpgimàr, g j `mkèdmcj à

gscajdgkta ngrjà par sgu davmdgkta. @gvg-sg cakbgcgr hgd sg bê ngrjîãa `g liuxa tjkta

jxmji qujkta iaknmtu`mkji g rj`mji, `g daà j sg pgrdmtmr j dmsturj `js `mlgrgktgs cjdj`js à

vaiudg rgjcmakji. Mdpartjktg tjdhíd vgrmlmcjr j mkliuïkcmj `g jinudjs cjrjctgrästmcjs à

liumà, cada vmscasm`j`g g `gksm`j`g, gd cadpjtmhmim`j`g cad js à jnmtjàr, cada `mèdgtra

g jiturj.

@g qujiqugr lardj, jkjimsjkà as rgsuitjàs pjrj tgdpa gspjcmji g `g rgsm`ïkcmj

dí`ma, pa`g-sg vgrmlmcjr qug bê djmar `mlgrgkîj gktrg gspgrjà g ahtmà pjrj a cjsa à ’Papa

– _akj Dartj— à vaiudg rgjcmakji. Kgssg cjsa, js pês à mdpgimàr `mrgcmakjd a liumà par

cjdmkbas prglgrgkcmjms `gvmà j praxmdm`j`g gktrg j jimdgktjîãa g j sjä`j à liumà kj pjrtg

supgrmar à rgjtar. @gssj lardj, pa`g-sg jlmrdjr qug j jnmtjîãa à rgspgctmva liumà

gxpgrmdgktji pjrj a gscajdgkta mdpigdgktjà í dgibar cad a smstgdj `g jnmtjîãa

iacjimzjà próxmda ja lukà à rgjtar à qug próxmda æ supgrläcmg.

\MMM. Curvj `g @mstrmhumîãa às Pgdpas `g Pgdpas `g Xgsm`ïkcmj - G

J bmpótgsg `g djmar prgcmsãa qujkà utmimzj`j j mktgnrjîãa gd L í oustmlmcj`j par

bjvgr j utmimzjîãa `g ud dítaà kudírmca j dgkas. Cjsa sg utmimzjssg a dítaà gd G , as

`jàs gd L tgrmjd qug sgr `mlgrgkcmjàs kudgrmcjdgktg pjrj pastgrmar mktgnrjîãa. As

nrêlmcas gqumvjigktgs j cj`j ud às gksjmas sg sgnugd.

Lmnurj :8 – Curvj `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj G(t) gd lukîãa à

tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j supgrläcmg `j dmsturj.

@gvmà æ prgsgkîj `g curtas-cmrcumtas gd jdhas as cjsas, cj`j ud cad suj

mktgksm`j`g, `g jcarà cad a smstgdj `g jnmtjîãa mdpigdgktjà, as prmdgmras paktas, cuoas

vjiargs `gstajvjd às `gdjms, kãa pgrdmtmkà a joustg `j curvj gxpakgkcmji, larjd

suprmdmàs.

y 4 =.267G-0g-:.6>=G-3x

X¸ 4 =.3=:G-:

6G+6

8G-0

0G-0

<G-0

2G-0

:G-3

:G-3

:G-3

8G-3

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)



8:

Vartjkta, a vjiar `j êrgj jhjmxa `g jdhjs js curvjs `js Lmnurjs :8 g :3 rgprgsgktj j

lrjîãa `g cakcgktrjîãa `g trjîjàr qug rgjidgktg pgrdgau a dgma jnmtjà. Cada j mktgnrji

`j curvj `g G gd ta`j suj gxtgksãa, par `glmkmîãa, tgd vjiar ukmtêrma, js curvjs gd qugstãa

jprgsgktjd vjiargs mklgrmargs j ukm`j`g, `gdakstrjkà, kgstj `mlgrgkîj, j prgsgkîj à curta-

cmrcumta. J lrjîãa `g trjîjàr qug kãa pgrdgau a dgma, par caksgquïkcmj, í j `mlgrgkîj gktrg

as vjiargs `js êrgjs `js mktgnrjms `js rgspgctmvjs curvjs.


tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j supgrläcmg `j dmsturj.

Js curvjs tjdhíd mkciugd a joustg `g `jàs par udj lukîãa gxpakgkcmji j lmd `g

cadpjrjr a rgsuitjà gxpgrmdgktji cad a da`gia tgórmca `g ud rgjtar m`gji. As caglmcmgktgs

`g `gtgrdmkjîãa, X8, tjdhíd sg gkcaktrjd `gtgrdmkjàs g `gdakstrjd qug jpós j gxciusãa

a joustg gxpgrmdgktji `gdakstrau haj cagrïkcmj.


tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j supgrläcmg `j dmsturj, oukta jas paktas

mkmcmjms, mk`mcjàrgs `j prgsgkîj `g curta-cmrcumta.

y 4 2.30=G-0g-=.60:G-0x

X¸ 4 2.682G-:

6G+6

8G-0

0G-0

<G-0

2G-0

:G-3

:G-3

:G-3

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)

y 4 :.36<G-3g-:.8>3G-3x

X¸ 4 <.736G-:

6G+6

8G-3

0G-3

<G-3

2G-3

:G-8

:G-8

:G-8

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)



88


tgdpa, pjrj pês à mdpgimàr pasmcmakj`j j 8 cd `j supgrläcmg `j dmsturj, oukta jas paktas

mkmcmjms, mk`mcjàrgs `j prgsgkîj `g curta-cmrcumta.

Js Lmnurjs :0 g :7 mkciugd as paktas mk`mcjàrgs `g curta-cmrcumta. Jiíd `msta, as

joustgs `js curvjs gxpakgkcmjms cjrjctgrästmcjs `j curvj G , g sgus rgspgctmvas caglmcmgktgs `g

`gtgrdmkjîãa, X¸ , cadpravjkà qug as paktas mkmcmjs kãa rgprgsgktjd a dgsda lgkðdgka

qug as `gdjms.

Dítaàs gstmdjtmvas, j pjrtmr `g da`gias, `j djnkmtu`g às `gsvmas gd rgijîãa æ

vjzãa `g jimdgktjîãa à rgjtar sgrãa jhar`jàs djms j`mjktg.

MR. Da`gia `g CSPX cad Gspjîa Darta g @gsvma

Kj prgsgkîj `g `gsvma, a gigdgkta `g liumà qug gktrj ka rgjtar tgd tgdpa `g

rgsm`ïkcmj djmar qug a tgdpa gspjcmji, cjicuijà caksm`grjkà a rgjtar cada m`gji. Msta sg

`gvg ja ljta `g qug j vjzãa jimdgktj`j qug j`gktrj a rgjtar í dgkar qug j vjzãa jimdgktj`j

`gvmà ja `gsvma `g pjrtg `gstj.

Wd da`gia `g rgjtar kãa m`gji cuoas pjrèdgtras gstmdjd j prgsgkîj g j mktgksm`j`g

`g cj`j `gsvma `j m`gjim`j`g í a da`gia `g CSPX cad Gspjîa Darta g @gsvma. J pjrtmr `g ud

pracgssa tgórmca gd qug bê ud rgjtar cuoa vaiudg í udj lrjîãa à vaiudg qug sg sup÷g tgr gpjrtg `j jimdgktjîãa í `gsvmj`j par ud hy-pjss, pgias hjijkîas `g djssj `g ud trjîjàr

armnmkjà `g ud `mstÿrhma kjs oukî÷gs, pa`g-sg ahtgr j sgnumktg rgijîãa pjrj cakcgktrjîãa `g

trjîjàr kj sjä`j, CP, g cakcgktrjîãa à `mstÿrhma kj gktrj`j, CP6?

U ()T

Gd qug a pjrèdgtra í j lrjîãa `g vjzãa `g gktrj`j qug sg sup÷gs qug í `gsvmj`j,

í j rjzãa gktrg a vaiudg rgji à rgjtar g a qug sg prgsudg tgr g

a tgdpa gspjcmji.

y 4 :.86>G-3g-:.:73G-3x

X¸ 4 7.8<=G-:

6G+6

8G-3

0G-3

<G-3

2G-3

:G-8

:G-8

:G-8

8G-8

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)



83

@g lardj j sg ahtgr djms ljcmidgktg as vjiargs `g g , imkgjrmzj-sg j gqujîãa

jktgrmar?

(

) (

) (

)

Xgjrrjkojkà a prmdgmra tgrda?

Padjkà-sg par hjsg as `jàs gxpgrmdgktjms, gijharjrjd-sg js curvjs j sgnumr?

Lmnurj :< – Curvj `j gqujîãa à da`gia `g rgjtar CSPX cad `gsvma g vaiudg darta pjrj a

jnmtjàr j 8 cd à lukà à rgjtar.

Lmnurj :> – Curvj `j gqujîãa à da`gia `g rgjtar CSPX cad `gsvma g vaiudg darta pjrj a

jnmtjàr j 8 cd `j supgrläcmg à rgjtar.

y 4 :.:>G-3x + :.0=G-:

X¸ 4 :.66G+6

6.6

6.7

:.6

:.7

8.6

8.7

3.6

3.7

6 766 :666 :766 8666 8766 3666

i k ( : / ( : - L ) )

tgdpa (s)

Curta Cmrcumta

y 4 :.67G-3x + :.76G-:

X¸ 4 =.==G-:

6.6

6.7

:.6

:.7

8.6

8.7

3.6

3.7

6 766 :666 :766 8666 8766 3666

i k ( : / ( :

- L ) )

tgdpa (s)

_akj Dartj



80

J pjrtmr às caglmcmgktgs jknuijr g imkgjr `js rgtjs joustj`js, pa`g-sg ahtgr as

pjrèdgtras g .

( )

As vjiargs `g g gstãa `mspastas kj tjhgij j sgnumr?

Pjhgij < – Vjrèdgtras g à da`gia `g rgjtargs CSPX cad zakj dartj g `gsvma pjrj as

`jàs gxpgrmdgktjms `g cj`j gxpgrmdgkta.

Xgjtar @gsvma (Hjsg) _akj Dartj (Papa)

Pgdpa Gspjcmji – π (s) >36,0 =86,6

Caglmcmgktg Jknuijr (:6³) :,:>83 :,60>>Caglmcmgktg Imkgjr 6,:0=: 6,:0=2

κ (%) :66 2=,8

θ (%) :3,20 :3,=8

Sgkà a pjrèdgtra jpraxmdj`jdgktg mnuji pjrj jdhjs js pasmî÷gs à jnmtjàr, lmcj

gvm`gktg qug gstj kãa gxgrcg mkliugkcmj glgtmvj ka qug `mz rgspgmta j prgsgkîj `g `gsvmas kjs

cak`mî÷gs à gxpgrmdgkta. Vartjkta, gstjkà js pês à mdpgimàr próxmdjs au jljstj`js `j

supgrläcmg `j dmsturj, kjs cak`mî÷gs ngadítrmcjs, cmkítmcjs g `mkèdmcjs à gxpgrmdgkta, j

dgsdj lrjîãa `g rgjngktg jimdgktjà `gsvmjrê à sgma à rgjtar `mrgtjdgktg pjrj j sjä`j.

Gd caktrjpjrtm`j, jkjimsjkà a pjrèdgtra , pgrcghg-sg qug, cada gspgrjà, a rgjtar

cad js pês djms próxmdjs `j supgrläcmg, `gvmà æ jusïkcmj `g jnmtjîãa kjs rgnm÷gs mklgrmargs

qug lmcjd gstjnkj`js, passum ud vaiudg rgji `g dmsturj mklgrmar ja cjicuijà, kj ar`gd `g

:6% j dgkas. Caksgqugktgdgktg, cad dgkar vaiudg rgjcmakji, pjrj udj dgsdj vjzãa, as

rgjngktgs jimdgktjàs gd ud rgjtar cuoj jnmtjîãa acarrj sadgktg próxmdj `j supgrläcmg tïd

ud dgkar tgdpa `g rgsm`ïkcmj, mkliugkcmjkà sahrgdjkgmrj gd qujiqugr rgjîãa quädmcj qug,

parvgkturj, passj acarrgr kgstg rgjtar CSPX.



87

R. Cadpjrjîãa gktrg da`gias `g CSPX – Gspjîa Darta g @gsvma, M`gji g Gxpgrmdgktji

J sgnumr jprgsgktjdas rgukmàs as rgsuitjàs gxpgrmdgktjms ahtmàs g as da`gias

utmimzjàs ka joustg `g `jàs pjrj da`gijr a cadpartjdgkta à rgjtar, j lmd `g cadpjrjr js

`mlgrgkîjs da`gia-gxpgrmdgktjms g m`gktmlmcjr ka rgjtar gxpgrmdgktji j armngd `gssjs

`mvgrnïkcmjs. As nrêlmcas `g G g L j sgnumr jprgsgktjd j `mspgrsãa às `jàs gxpgrmdgktji, g

as da`gias `g cadpjrjîãa m`gji, g kãa-m`gji cad vaiudg darta g cjkjimzjîãa, pjrj as gksjmas

cad pê à jnmtjàr próxmdj ja lukà g próxmdj æ supgrläcmg à vaiudg rgjcmakji,

rgspgctmvjdgktg.

Lmnurj :2 – Nrêlmca cadpjrjtmva `js curvjs G j`vmk`js às `mlgrgktgs `jàs `g da`gias g

gxpgrmdgktji pjrj a rgjtar CSPX cad tgdpa gspjcmji tgórmca `g >36s (Hjsg – Lukà).

Lmnurj := – Nrêlmca cadpjrjtmva `js curvjs L j`vmk`js às `mlgrgktgs `jàs `g da`gias g

gxpgrmdgktji pjrj a rgjtar CSPX cad tgdpa gspjcmji tgórmca `g >36s (Hjsg – Lukà).

6.66%

6.68%

6.60%

6.6<%

6.62%

6.:6%

6.:8%

6.:0%

6.:<%

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)

π (s) 4 >36 (m`gji)

π (s) 4 >36 (Da`gia)

π (s) 4 >36 (gxpgrmdgktji)

6%

:6%

86%

36%

06%

76%

<6%

>6%

26%

=6%

:66%

6 :666 8666 3666

L ( t )

Pgdpa (s)

π (s) 4 >36 (m`gji)

π (s) 4 >36 (Da`gia)

π (s) 4 >36 (gxpgrmdgktji)



8<

Lmnurj 86 – Nrêlmca cadpjrjtmva `js curvjs G j`vmk`js às `mlgrgktgs `jàs `g da`gias g

gxpgrmdgktji pjrj a rgjtar CSPX cad tgdpa gspjcmji tgórmca `g =86s (Papa – Supgrläcmg).

Lmnurj 8: – Nrêlmca cadpjrjtmva `js curvjs L j`vmk`js às `mlgrgktgs `jàs `g da`gias g

gxpgrmdgktji pjrj a rgjtar CSPX cad tgdpa gspjcmji tgórmca `g =86s (Papa – Supgrläcmg).

Js Lmnurjs :2 g 86 dastrjd qug pjrj a cadpartjdgkta gxpgrmdgktji g às da`gias

`g cadpjrjîãa G(t) `mdmkum caklardg a tgdpa `g gxpgrmdgkta judgktj, æ dg`m`j qug

`mdmkum j cakcgktrjîãa `g trjîjàr kj sjä`j à smstgdj. J djmar prgsgkîj às paktas

gxpgrmdgktjms pjrj hjmxas tgdpas mkliugkcmj ka cêicuia `g ud tgdpa dí`ma `g rgsm`ïkcmj

dgkar à qug a tgdpa gspjcmji pjrj jdhas as gksjmas. Katj-sg hgd jmk`j qug gssgs

prmdgmras paktas sg `mstjkcmjd djms `js lukî÷gs `g cadpjrjîãa, qug sgrvgd `g rglgrïkcmj æ

jkêimsg, rgvgijkà qug, ka mkäcma à pracg`mdgkta, j lardjîãa `g cjkjms prglgrgkcmjms `g

gscajdgkta g j mksulmcmïkcmj `j dmsturj sãa gxpimcjî÷gs sahrg a lgkðdgka gxpgrmdgktji

cak`mzgktgs. Caklardg a tgdpa judgktj, bê djmar cadpjtmhmim`j`g gktrg as rgsuitjàs

gxpgrmdgktjms g as da`gias

6.66%

6.68%

6.60%

6.6<%

6.62%

6.:6%

6.:8%

6.:0%

6.:<%

6.:2%

6 766 :666 :766 8666 8766 3666

G ( t )

Pgdpa (s)

π (s) 4 =86 (m`gji)

π (s) 4 =86 (Da`gia)

π (s) 4 =86 (gxpgrmdgktji)

6%

:6%

86%

36%

06%

76%

<6%

>6%

26%

=6%

:66%

6 :666 8666 3666

L ( t )

Pgdpa (s)

π (s) 4 =86 (m`gji)

π (s) 4 =86 (Da`gia)

π (s) 4 =86 (gxpgrmdgktji)



8>

@g qujiqugr lardj, js curvjs gvm`gkcmjd pjrj j lukîãa `g `mstrmhumîãa às tgdpas `g

rgsm`ïkcmj G(t) qug j `mspgrsãa às paktas gxpgrmdgktjms sg j`gquj djms ja da`gia

mdpigdgktjà à qug j caklmnurjîãa m`gji. Mssa smnkmlmcj qug ja caksm`grjr kj da`gijngd à

smstgdj as glgmtas `g ud vaiudg darta rglgrgktg j udj zakj `g gstjnkjîãa g `j cjkjimzjîãa

à gscajdgkta sg jtmkng udj djmar praxmdm`j`g cad as rgsuitjàs rgjms.

Gvm`gktgdgktg gssj dgibar joustg grj gspgrj`j, a da`gia m`gji sg hjsgmj gd udj

jnmtjîãa pgrlgmtj g mkstjktèkgj à liumà ka rgjtar. Kj smtujîãa rgji, j gxpgrmdgktji, j dmsturj

í mksulmcmgktg? bê gigdgktas `g liumàs qug `gmxjd a smstgdj rgjcmakji jhruptjdgktg iana

jpós æ jimdgktjîãa, g sg lardj rgnm÷gs ak`g a liumà, g a trjîjàr, gstjnkj.

Js Lmnurjs := g 8: dastrjd j `mstrmhumîãa cuduijtmvj L(t) kavjdgktg j djmar

cadpjtmhmim`j`g sg `ê cad a da`gia `g cadpjrjîãa utmimzjà à qug cad j curvj `g ud

rgjtar m`gji. Cada j lukîãa í cuduijtmvj g cjrrgnj caksmna j `mspgrsãa caktmkujdgktg ja

iakna à tgdpa, as paktas gxpgrmdgktjms mkmcmjms qug `gstajvjd às da`gias `g cadpjrjîãa

gd tgdpas hjmxas `gssj vgz cakcar`jd djms cad j curvj à da`gia.

J cadpjrjîãa gktrg js ùjs caklmnurjî÷gs `g rgjtargs CSPX, (Hjsg – Lukà) g (Papa –

Supgrläcmg) rgvgij qug a smstgdj `g jnmtjîãa cad a mdpgimàr próxmda ja lukà à rgjtar ngrj

ud dgibar smstgdj `g jnmtjîãa g jpraxmdj a rgjtar ja smstgdj m`gji. Gssg ljta lmcj gvm`gktg

pgij dgkar `mspgrsãa às prmdgmras paktas gxpgrmdgktjms, j hjmxas vjiargs `g tgdpa, gd

tarka à da`gia caksm`grjà pjrj j Lmnurj :2.

VLX kãa M`gji

RM.

@mstrmhumîãa `g Pgdpas `g Xgsm`ïkcmj pjrj Xgjtargs VLX M`gjms

Js lukî÷gs `g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj, G , g j cuduijtmvj, L , sãa

cjrjctgrästmcjs à `mstÿrhma `g trjîjàr jpimcjà g à tmpa `g rgjtar g gscajdgkta. Ka cjsa

m`gji, katj-sg a imdmtg hgd `glmkmà gktrg a dadgkta gd qug acarrg a `mstÿrhma g a dadgkta

gd qug sg dg`g a trjîjàr kj sjä`j. A mktgrvjia `g tgdpa gktrg gssgs àms dadgktas í a

tgdpa gspjcmji à rgjtar.

Pjhgij > – \aiudg g vjzãa dg`m`j g tgdpa gspjcmji pjrj a rgjtar VLX.

Xgjtar VLX

\aiudg (dI) :=26

\jzãa Dí`mj (dI/s) :,83

Pgdpa Gspjcmji – π (s) :<60,3

Js lmnurjs j sgnumr carrgspak`gd æs `mstrmhumî÷gs `g tgdpas `g rgsm`ïkcmj pjrj

rgjtargs VLX m`gjms kjs cak`mî÷gs à gxpgrmdgkta.



82

Lmnurj 88 – Lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj tgórmcj pjrj j vjzãa g vaiudg

dg`màs ka gxpgrmdgkta cad gscajdgkta.

Lmnurj 83 – Lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj cuduijtmvj tgórmcj pjrj j vjzãa g

vaiudg dg`màs ka gxpgrmdgkta.

6.666 :6.666 86.666 36.666 06.666 76.666

G ( t )

Pgdpa (dmk)

π

∛

6%

:6%

86%

36%

06%

76%

<6%

>6%

26%

=6%

:66%

6.666 :6.666 86.666 36.666 06.666 76.666

L ( t )

Pgdpa (dmk)

π



8=

RMM. Xgsuitjàs Gxpgrmdgktjms – Xgjtar VLX kãa m`gji

J pjrtmr às `jàs gxpgrmdgktjms pjrj a gscajdgkta gd ud rgjtar VLX prgsgktgs kj

Pjhgij > sg ahtïd js curvjs `js lukî÷gs `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj, G , g

cuduijtmvj, L , g `j vjrmjîãa `j cakcgktrjîãa `g sjä`j (C(t)/C 6 ) au (:-L(t)), jprgsgktj`js j

sgnumr?

Lmnurj 80 – Curvj às `jàs gxpgrmdgktjms `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj

U:-L(t)T au C/C6 gd lukîãa à tgdpa pjrj a rgjtar VLX.

Lmnurj 87 – Curvj às `jàs gxpgrmdgktjms `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj

cuduijtmvj L(t) au :-C/C6 gd lukîãa à tgdpa pjrj a rgjtar VLX.

6.6

6.8

6.0

6.<

6.2

:.6

:.8

6 :6 86 36 06 76

: -

L ( t )

Pgdpa (dmk)

6%

:6%

86%

36%

06%

76%

<6%

>6%

26%

=6%

:66%

6 :6 86 36 06 76

L ( t )

Pgdpa (dmk)



36

Lmnurj 8< – Curvj às `jàs gxpgrmdgktjms, ahtm`j j pjrtmr `g dítaàs `g `mlgrgkcmjîãa

kudírmcas, `j lukîãa `g `mstrmhumîãa às tgdpas `g rgsm`ïkcmj G(t) gd lukîãa à tgdpa pjrj

a rgjtar VLX.

Cada a dítaà gdprgnjà pjrj cêicuia `g G(t) sg hjsgmj gd dítaàs kudírmcas `g

`mlgrgkcmjîãa, jinuks `jàs `j Lmnurj 8< jprgsgktjd vjiargs kgnjtmvas, mk`mcjtmvas `g udj

`grmvj`j kgnjtmvj `g L(t), a qug cak`mz cad j vjrmjîãa às vjiargs gxpgrmdgktjms `j

cakcgktrjîãa ahtmàs pgia dg`mîãa `j jhsarhèkcmj `js jdastrjs.

Cad hjsg kas `jàs gxpgrmdgktjms g kjs Lmnurjs 80 g 87, pgrcghg-sg kmtm`jdgktg qug

js bmpótgsgs `g gscajdgkta gdpmstakjà à VLX kãa larjd sjtmslgmtjs. Vrmdgmrjdgktg, pgia

j`mjktjdgkta à mkäcma `j dg`m`j `g cakcgktrjîãa à trjîjàr gd rgijîãa ja tgdpa gspjcmji,

cjrjctgrästmcj `g ud trjîjàr qug sg `mluk`g (`mspgrsj) ja iakna à gscajdgkta kj `mrgîãa

jxmji. J lmd `g caktarkjr j kãa-m`gjim`j`g g da`giê-ij gdprgnjd-sg, kjs sgî÷gs sgnumktgs àms

da`gias pjrj a rgjtar VLX.

RMMM. Da`gia `g Gscajdgkta Ijdmkjr Sgnrgnjà pjrj VLX

A da`gia `g gscajdgkta ijdmkjr sgnrgnjà tadj par hjsg a pgrlmi pjrjhóimca

cjrjctgrästmca `gstg tmpa `g gscajdgkta, ak`g a liumà ka cgktra à tuha pgrdjkgcg par

dgkar tgdpa ka rgjtar. Vjssj j sgr caksm`grj`j j jîãa às glgmtas `g pjrg`g, qug sg

prapjnjd cada larîjs vmscasjs rgtjr`jkà js `gdjms cjdj`js `g liumà.

J lukîãa cadpigtj `g @PX pjrj ud rgjtar VLX cad gscajdgkta ijdmkjr sg `ê par?

Cada, pjrj ud pgrlmi pjrjhóimca, j vgiacm`j`g dêxmdj `g gscajdgkta jxmji í ùjs

vgzgs j vgiacm`j`g dí`mj, a tgdpa däkmda gd qug ud gigdgkta `g liumà pa`g pgrdjkgcgr

`gktra à rgjtar í?

-:6%

-7%

6%

7%

:6%

:7%

86%

87%

6 :6 86 36 06 76

G ( t )

Pgdpa (dmk)



3:

̎

Iana,

{ }

Vjrj `mstrmhumîãa cuduijtmvj, pgij rgijîãa gktrg js lukî÷gs L g G, tgd-sg qug?

{

}

RM\. Da`gia `g k tjkqugs CSPX gd sírmg pjrj VLX

Xgjtargs VLX pa`gd sgr caksm`grjàs cada mklmkmtas CSPX, pams, pgia hjijkîa `g

djssj `g udj gspícmj j pjrj ud rgjtar CSPX, tgd-sg qug?

Jpraxmdjkà ud gigdgkta `g vaiudg mklmkmtgsmdji `g ud VLX par ud CSPX g sgu

hjijkîa `g djssj, tgd-sg a hjijkîa `g djssj pjrj ud VLX.

Vjrj ud cakoukta `g k rgjtargs CSPX gd sírmg, j cakcgktrjîãa `g sjä`j `g ud

trjîjàr, C(t), gd lukîãa à tgdpa `g rgsm`ïkcmj gd ud ÿkmca rgjtar, πm, g `j cakcgktrjîãa

mkmcmji à `mstÿrhma, CP6?

Vgij `glmkmîãa `j lukîãa `g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj, G,

∥

@gssj lardj,



38

Gstj, qug sgrê cjicuij`j, jpós `gtgrdmkjà a kÿdgra `g tjkqugs CSPX gd sírmg

gqumvjigktg ja VLX, jtrjvís `g dítaàs kudírmcas (Armnmk). Vjrj sg `gtgrdmkjr a kÿdgra `g

tjkqugs CSPX gqumvjigktgs ja VLX, cjicuij-sg j vjrmèkcmj `j `mstrmhumîãa às tgdpas `g

rgsm`ïkcmj, tji qug?

Pji qug j vjrmèkcmj, S¸, í cjicuij`j `j dgsdj lardj cada pjrj a CSPX, j pjrtmr à

saltwjrg Armnmk.

Pjhgij 2 – \aiudg g vjzãa dg`m`j, tgdpa gspjcmji cjicuijà, tgdpa `g rgsm`ïkcmj dí`ma g

vjrmèkcmj cjicuijàs par saltwjrg djtgdêtmca (Armnmk) pjrj a rgjtar VLX.

Xgjtar VLX

π (dmk) 8<,>0 ̇(dmk) 86,:8 =7,::

@gssj lardj,

Cada a cêicuia `j êrgj sah j curvj `j lukîãa g `mstrmhumîãa `g tgdpas `g rgsm`ïkcmj,

par `glmkmîãa, í j ukm`j`g g, a cêicuia `g ljtarmjms só pa`gd sgr lgmtas pjrj kÿdgras mktgmras,

aptau-sg par ud vjiar `g k mktgmra próxmda à cjicuijà g j curvj lam kardjimzj`j.

R\.

Cadpjrjîãa gktrg da`gias `g VLX – Da`gia Ijdmkjr Sgnrgnjà, k tjkqugs CSPX gdsírmg, M`gji g Gxpgrmdgktji

Ka nrêlmca j sgnumr gkcaktrjd-sg js curvjs `js lukî÷gs `g `mstrmhumîãa às tgdpas `g

rgsm`ïkcmj, G , g cuduijtmvj, L , pjrj as `mvgrsas da`gias joustjàs g pjrj a cjsa m`gji, oukta jas

paktas gxpgrmdgktjms.



30

pa`g-sg `mzgr qug a gscajdgkta ijdmkjr sgnrgnjà í `gktrg js rglgrïkcmjs a qug dgibar

`gscrgvg a gksjma rgjimzjà ka VXL.

Cada sg pgrcghg kj Lmnurj 8>, a da`gia `g K CSPXfs jprgsgktj udj curvj cad

vjiargs mklgrmargs jas gxpgrmdgktjms, jqum cjhg rgssjitjr qug gssg da`gia sg hjsgmj gd ud

cgrta kÿdgra `g rgjtargs à tmpa CSPX g qug a joustg hjsgjà kgssg pjrèdgtra tarkj-sg

dumta sgksävgi, au sgoj, udj du`jkîj ka kÿdgra `g rgjtargs mklgrg smnkmlmcjtmvjdgktg kj

curvj à da`gia.

Var autra ijà, j cakcar`èkcmj cad a da`gia `g ijdmkjr só í prgou`mcj`j pgia tgdpa

dí`ma qug hjsgmj j `mspgrsãa – as `jàs gxpgrmdgktjms `g `mstrmhugd gd tarka `g udj dí`mj

djmar à qug j à da`gia ijdmkjr – gkqujkta j caklardjîãa `j curvj sg djtíd. A da`gia

ijdmkjr sg tarkj cak`mzgktg kj dg`m`j gd qug j vgiacm`j`g `g gscajdgkta à liumà ka

rgjtar lam rgijtmvjdgktg pgqugkj. Kj prêtmcj gxpgrmdgktji bjvmj `mlmcui`j`g gd sg prgcmsjr a

vjiar `j vjzãa vaiudítrmcj `g ênuj ka wjsb-aut `g daà qug ud vjiar `g vjzãa rgji dgkar

à qug j mk`mcj`j gxpgrmdgktjidgktg í passävgi.

Kj Lmnurj 82 js cakcius÷gs passävgms sãa js dgsdjs, caktuà bê mksulmcmïkcmj `g `jàs

cad ud tgdpa gxpgrmdgktji djmar j lmd `g lardj jpjrtg lmkji `j `mspgrsãa gxpgrmdgktji.

Cada gd ud tgdpa djmar j cakcgktrjîãa `g sjä`j à trjîjàr, hgd cada j vjrmjîãa `gssj

cakcgktrjîãa, í dgkar, as vjiargs gxpgrmdgktjms `g `mspgrsãa dgkas, g katj-sg udj

cakcgktrjîãa `g paktas dg`màs pjrj tgdpa gd tarka `g 36 dmkutas.

CAKCIWSÃA

A gstuà `j `mstrmhumîãa às tgdpas `g rgsm`ïkcmj gd rgjtargs VLX g CSPX dastrau

rgsuitjàs gxpgrmdgktjms cad `gsvma gd rgijîãa jas da`gias `g rgjtargs m`gmjs paríd

cadpjtävgms cad autrjs lardjs `g da`gijngd qug mkciugd js kãa-m`gjim`j`gs à smstgdj. A

gxpgrmdgkta sg dastrau ud djkgmrj rêpm`j g glmcmgktg `g gstu`jr a cadpartjdgkta `g ud

liumà gd ud rgjtar, jmk`j qug sgd rgjîãa, `g daà j cjrjctgrmzjr a gscajdgkta

`gsgkvaivmà kj ukm`j`g.

As gxpgrmdgktas cad rgjtargs CSPX jprgsgktjrjd ud tgdpa `g rgsm`ïkcmj dí`ma

mklgrmar ja tgdpa gspjcmji cjicuijà, a da`gia qug dgibar `gscrgvg as rgsuitjàsgxpgrmdgktjms í jqugig qug igvj gd caksm`grjîãa a vaiudg darta, au zakj `g gstjnkjîãa, g j

lardjîãa `g curta-cmrcumta. Ljta gssg qug gvm`gkcmj j prgsgkîj `gssgs lgkðdgkas ka rgjtar

gxpgrmdgktji, gd `gcarrïkcmj `g ud smstgdj `g jnmtjîãa mksulmcmgktg pjrj njrjktmr j jnmtjîãa

pgrlgmtj g `j praxmdm`j`g gktrg gktrj`j g sjä`j. Va`g-sg cadpjrjr tjdhíd j pasmîãa `j pê

à jnmtjàr gd rgijîãa ja rgjtar, sgkà qug sgu pasmcmakjdgkta próxmda ja lukà à vaiudg

rgjcmakji dastrau as rgsuitjàs dgkas jljstjàs `j m`gjim`j`g.

Autra ljta qug dgrgcg sgr `gstjcjà kgssg pakta í j `mlmcui`j`g gxpgrmdgktji gd sg

`gtgrdmkjr j vjzãa, oê qug gssj par suj daktjngd gxpgrmdgktji pravjvgidgktg armnmkau ud

gscajdgkta kãa-gstjcmakêrma g vjrmau cad a tgdpa. Vrgou`mcj`j lmcau `gssj djkgmrj j vjzãa



rgji à rgjtar, hgd cada a cêicuia à tgdpa gspjcmji a qug oustmlmcj j `mlgrgkîj `gstg pjrj

cad a tgdpa dí`ma `g `mstrmhumîãa.

Vjrj a rgjtar VLX, vgrmlmcau-sg j dgsdj `mlmcui`j`g gd sg `gtgrdmkjrj j vjzãa cad a

ratèdgtra `j daktjngd gxpgrmdgktji, j vjzãa sg dastrau mkstêvgi g `g daà j jlgtjr

sahrgdjkgmrj as cêicuias pastgrmargs g j cadpjrjîãa gktrg as tgdpa cjrjctgrästmcas à

gstuà. @g qujiqugr lardj, `gsvmas `j m`gjim`j`g larjd m`gktmlmcjàs g rgsuitjrjd gd ud

smnkmlmcjtmva `gsvma à pj`rãa m`gji, kgssg caktgxta, a da`gia `g gscajdgkta ijdmkjr

sgnrgnjà í a qug dgibar `gscrgvg a gksjma rgjimzjà, pams, ja igvjr gd caksm`grjîãa as

glgmtas `g pjrg`g g j prapjnjîãa `js larîjs vmscasjs ka pgrlmi `g vgiacm`j`gs jpraxmdj j

cjrjctgrmzjîãa à smstgdj æ rgjim`j`g.

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:. Lanigr, B. Scatt5 ’Gigdgktas `g gkngkbjrmj `js rgjî÷gs quädmcjs—5 0ª G`mîãa5 G`. IPC,

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