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}{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 3 0 3 0 2 2 15 2 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Helv etica" 1 16 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "List Item" -1 257 1 {CSTYLE "" -1 -1 "Helvetic a" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 3 0 3 0 2 2 14 5 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 76 "Studiul stabilitatii circuitelor liniare folosind metoda locului radacinilor" }}{PARA 18 "" 0 "" {TEXT -1 47 "Partea I: Reguli de trasare rapida & Exemplul I" }}{PARA 19 "" 0 "" {TEXT -1 86 "Universitatea Tehnica \"Gh. Asachi\", Iasi, Fa cultatea de Electronica si Telecomunicatii" }}{PARA 19 "" 0 "" {TEXT -1 43 "Laboratorul de Semnale, Circuite si Sisteme" }}{PARA 19 "" 0 " " {URLLINK 17 "http://scs4.etc.tuiasi.ro" 4 "http://scs4.etc.tuiasi.ro /scslabs/Simbolice/simbolice.htm" "" }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Breviar teoretic" }}{PARA 0 "" 0 "" {TEXT -1 77 " Schema bloc generala a unui sistem cu reactie este cea din figura de m ai jos." }}{PARA 0 "" 0 "" {TEXT -1 49 "Functia de transfer a sistemul ui cu reactie este:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Hr(s) = K*A(s) /(1-epsilon*K*A(s)*B(s));" "6#/-%#HrG6#%\"sG*(%\"KG\"\"\"-%\"AG6#F'F*, &F*F***%(epsilonGF*F)F*-F,6#F'F*-%\"BG6#F'F*!\"\"F6" }{TEXT -1 1 "," } }{PARA 0 "" 0 "" {TEXT -1 6 " unde " }{XPPEDIT 18 0 "epsilon = 1;" "6# /%(epsilonG\"\"\"" }{TEXT -1 28 " pentru reactie pozitiva si " } {XPPEDIT 18 0 "epsilon = -1;" "6#/%(epsilonG,$\"\"\"!\"\"" }{TEXT -1 25 " pentru reactie negativa." }}{PARA 0 "" 0 "" {TEXT -1 62 "Avand in vedere ca functia de transfer in bucla deschisa este:" }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "Hd(s) = K*A(s)*B(s);" "6#/-%#HdG6#%\"sG*(%\"KG\" \"\"-%\"AG6#F'F*-%\"BG6#F'F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 68 "putem exprima functia de transfer a sistemului cu reactie sub f orma:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Hr(s) = K*A(s)/(1-epsilon*Hd (s));" "6#/-%#HrG6#%\"sG*(%\"KG\"\"\"-%\"AG6#F'F*,&F*F**&%(epsilonGF*- %#HdG6#F'F*!\"\"F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 112 "St abilitatea sistemului cu reactie este data asadar de pozitia in planul complex a polilor functiei de transfer " }{XPPEDIT 18 0 "Hr(s);" "6#- %#HrG6#%\"sG" }{TEXT -1 37 ", care reprezinta zerourile functiei " } {XPPEDIT 18 0 "F(s) = 1-epsilon*Hd(s);" "6#/-%\"FG6#%\"sG,&\"\"\"F)*&% (epsilonGF)-%#HdG6#F'F)!\"\"" }{TEXT -1 157 ". Pentru studiul stabilit atii sistemului cu reactie la variatia amplificarii este suficient asa dar studiul pozitiei in planul complex a radacinilor ecuatiei " } {XPPEDIT 18 0 "1-epsilon*Hd(s) = 0;" "6#/,&\"\"\"F%*&%(epsilonGF%-%#Hd G6#%\"sGF%!\"\"\"\"!" }{TEXT -1 27 ", la variatia amplificarii." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 282 17 "Locul Radacinilor" } {TEXT -1 2 " (" }{TEXT 288 4 "L.R." }{TEXT -1 53 ") reprezinta locul g eometric al radacinilor ecuatiei " }{XPPEDIT 18 0 "1-K*A(s)*B(s) = 0; " "6#/,&\"\"\"F%*(%\"KGF%-%\"AG6#%\"sGF%-%\"BG6#F+F%!\"\"\"\"!" } {TEXT -1 34 ", la variatia amplificarii pentru " }{XPPEDIT 18 0 "K = 0 .. infinity;" "6#/%\"KG;\"\"!%)infinityG" }{TEXT -1 23 ". Radacinile \+ ecuatiei " }{XPPEDIT 18 0 "1-K*A(s)*B(s) = 0;" "6#/,&\"\"\"F%*(%\"KGF %-%\"AG6#%\"sGF%-%\"BG6#F+F%!\"\"\"\"!" }{TEXT -1 4 " ( " }{XPPEDIT 18 0 "1-epsilon*Hd(s) = 0;" "6#/,&\"\"\"F%*&%(epsilonGF%-%#HdG6#%\"sGF %!\"\"\"\"!" }{TEXT -1 44 " ) sunt tocmai polilor functiei de transfer " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 25 " a sistemul ui cu reactie." }}{PARA 0 "" 0 "" {TEXT 295 10 "Observatii" }{TEXT -1 2 ": " }}{PARA 14 "" 0 "" {TEXT 289 3 "1. " }{TEXT 290 4 "L.R." } {TEXT 292 1 " " }{TEXT -1 30 "se traseaza in planul complex " }{TEXT 291 1 "s" }{TEXT -1 37 " si se gradeaza in valori ale lui K. " }{TEXT 293 1 " " }{TEXT 294 4 "L.R." }{TEXT -1 115 " poate fi format din mai \+ multe ramuri, aceeasi valoare a lui K va fi scrisa in cite un punct de pe fiecare ramura. " }}{PARA 14 "" 0 "" {TEXT 337 2 "2." }{TEXT -1 61 " Analiza de stabilitate se determina prin simpla inspectie a " } {TEXT 296 4 "L.R." }{TEXT -1 217 " Sistemul cu reactie va fi stabil pe ntru acele valori ale lui K pentru care toate ramurile se afla in semi planul sting si instabil pentru toate valorile lui K pentru care macar una din ramuri este in semiplanul drept." }}{PARA 14 "" 0 "" {TEXT 338 2 "3." }{TEXT -1 181 " Discutia de stabilitate pentru situatia lim ita cu poli ai functiei de transfer in bucla inchisa pe axa imaginara \+ duce la concluzia ca circuitul este stabil daca polii sunt simpli." } }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "Reguli de trasare rapida a locul ui radacinilor" }}{PARA 0 "" 0 "" {TEXT -1 74 "Vom presupune in cele c e urmeaza ca functia de transfer in bucla deschisa " }{XPPEDIT 18 0 "H d(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "P;" "6#% \"PG" }{TEXT -1 14 " poli, notati " }{XPPEDIT 18 0 "p[k],k = 1 .. P;" "6$&%\"pG6#%\"kG/F&;\"\"\"%\"PG" }{TEXT -1 4 " si " }{XPPEDIT 18 0 "Z; " "6#%\"ZG" }{TEXT -1 17 " zerouri, notate " }{XPPEDIT 18 0 "z[l],l = \+ 1 .. Z;" "6$&%\"zG6#%\"lG/F&;\"\"\"%\"ZG" }{TEXT -1 160 ". Proprietati le matematice ale locului radacinilor sunt concretizate in regulile de trasare rapida atit pentru reactie pozitiva cit si pentru reactie neg ativa. " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 17 "Reactie negativa " } {TEXT 298 1 "(" }{XPPEDIT 299 0 "epsilon = -1;" "6#/%(epsilonG,$\"\"\" !\"\"" }{TEXT 300 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 50 "Functia de tran sfer a circuitului cu reactie este:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Hr(s) = K*A(s)/(1+K*A(s)*B(s));" "6#/-%#HrG6#%\"sG*(%\"KG\"\"\"-%\" AG6#F'F*,&F*F**(F)F*-F,6#F'F*-%\"BG6#F'F*F*!\"\"" }{TEXT -1 1 "." }} {PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 266 3 "1. " }{TEXT 271 4 "L.R." }{TEXT -1 5 " are " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 30 " ramur i care pleaca din polii " }{XPPEDIT 18 0 "p[k]" "6#&%\"pG6#%\"kG" } {TEXT -1 25 " ai functiei de transfer " }{XPPEDIT 18 0 "Hd(s);" "6#-%# HdG6#%\"sG" }{TEXT -1 8 " pentru " }{XPPEDIT 18 0 "K = 0;" "6#/%\"KG\" \"!" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT 263 3 "2. " }{XPPEDIT 18 0 "Z;" "6#%\"ZG" }{TEXT -1 12 " ramuri ale " }{TEXT 339 4 "L.R." } {TEXT -1 20 " ajung in zerourile " }{XPPEDIT 18 0 "z[l];" "6#&%\"zG6#% \"lG" }{TEXT -1 26 " ale functiei de transfer " }{XPPEDIT 18 0 "Hd(s); " "6#-%#HdG6#%\"sG" }{TEXT -1 5 " iar " }{XPPEDIT 18 0 "P-Z;" "6#,&%\" PG\"\"\"%\"ZG!\"\"" }{TEXT -1 44 " ramuri tind asimptotic spre infinit pentru " }{XPPEDIT 18 0 "proc (K) options operator, arrow; infinity e nd proc;" "6#R6#%\"KG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT 272 2 "3." }{TEXT -1 10 " Ramu rile " }{TEXT 264 4 "L.R." }{TEXT -1 34 " sunt simetrice fata de axa r eala." }}{PARA 14 "" 0 "" {TEXT 273 2 "4." }{TEXT -1 6 " Cele " } {XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 81 " asimpt ote are ramurilor care tind spre infinit se intalnesc intr-un punct nu mit " }{TEXT 280 19 "centru de greutate," }{TEXT -1 2 " (" }{XPPEDIT 18 0 "Cg;" "6#%#CgG" }{TEXT -1 25 "), care are coordonatele:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "xCg = (Sum(Real(p[k]),k = 1 .. P)-Sum(Real( z[l]),l = 1 .. Z))/(P-Z);" "6#/%$xCgG*&,&-%$SumG6$-%%RealG6#&%\"pG6#% \"kG/F0;\"\"\"%\"PGF3-F(6$-F+6#&%\"zG6#%\"lG/F<;F3%\"ZG!\"\"F3,&F4F3F? F@F@" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "yCg = 0;" "6#/%$yCgG\"\"!" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT 274 2 "5." }{TEXT -1 6 " Cele \+ " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 81 " asi mptote are ramurilor care tind spre infinit formeaza cu axa reala ungh iurile:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "theta[i] = (2*i+1)*Pi/(P-Z ),i = 1 .. P-Z;" "6$/&%&thetaG6#%\"iG*(,&*&\"\"#\"\"\"F'F,F,F,F,F,%#Pi GF,,&%\"PGF,%\"ZG!\"\"F1/F';F,,&F/F,F0F1" }{TEXT -1 1 " " }}{PARA 14 " " 0 "" {TEXT 275 2 "6." }{TEXT -1 30 " Ramurile de pe axa reala ale " }{TEXT 265 4 "L.R." }{TEXT 267 0 "" }{TEXT -1 35 " sunt situate la sta nga unui numar " }{TEXT 304 5 "impar" }{TEXT -1 36 " de singularitati \+ (poli si zerouri)." }}{PARA 14 "" 0 "" {TEXT 297 3 "7. " }{TEXT -1 61 "Ramurile care parasesc axa reala o fac perpendicular pe axa. " }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 17 "Reactie pozitiva " }{TEXT 301 1 " (" }{XPPEDIT 257 0 "epsilon = 1;" "6#/%(epsilonG\"\"\"" }{TEXT 302 1 " )" }}{PARA 0 "" 0 "" {TEXT -1 50 "Functia de transfer a circuitului cu reactie este:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Hr(s) = K*A(s)/(1-K *A(s)*B(s));" "6#/-%#HrG6#%\"sG*(%\"KG\"\"\"-%\"AG6#F'F*,&F*F**(F)F*-F ,6#F'F*-%\"BG6#F'F*!\"\"F5" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 269 3 "1. " }{TEXT 276 4 "L.R." }{TEXT -1 5 " are " } {XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 51 " ramuri care pleaca din pol ii functiei de transfer " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" } {TEXT -1 8 " pentru " }{XPPEDIT 18 0 "K = 0;" "6#/%\"KG\"\"!" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT 283 3 "2. " }{XPPEDIT 18 0 "Z;" "6#% \"ZG" }{TEXT -1 12 " ramuri ale " }{TEXT 340 4 "L.R." }{TEXT -1 15 " a jung in cele " }{XPPEDIT 18 0 "Z;" "6#%\"ZG" }{TEXT -1 12 " zerouri si " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 44 " ra muri tind asimptotic spre infinit pentru " }{XPPEDIT 18 0 "proc (K) op tions operator, arrow; infinity end proc;" "6#R6#%\"KG7\"6$%)operatorG %&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT 285 3 "3. " }{TEXT -1 9 "Ramurile " }{TEXT 284 4 "L.R." }{TEXT -1 34 " sunt simetrice fata de axa reala." }}{PARA 14 "" 0 "" {TEXT 277 3 "4. " }{TEXT -1 5 "Cele " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\" %\"ZG!\"\"" }{TEXT -1 81 " asimptote are ramurilor care tind spre infi nit se intalnesc intr-un punct numit " }{TEXT 281 18 "centru de greuta te" }{TEXT -1 4 " , (" }{XPPEDIT 18 0 "Cg;" "6#%#CgG" }{TEXT -1 23 "), care are coordonatele" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "xCg = (Sum(R eal(p[k]),k = 1 .. P)-Sum(Real(z[l]),l = 1 .. Z))/(P-Z);" "6#/%$xCgG*& ,&-%$SumG6$-%%RealG6#&%\"pG6#%\"kG/F0;\"\"\"%\"PGF3-F(6$-F+6#&%\"zG6#% \"lG/F<;F3%\"ZG!\"\"F3,&F4F3F?F@F@" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y Cg = 0;" "6#/%$yCgG\"\"!" }{TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 18 "in planul complex." }}{PARA 14 "" 0 "" {TEXT 278 3 "5. " }{TEXT -1 5 "Cele " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 78 " asimptote are ramurilor care tind la infinit formeaza cu axa r eala unghiurile" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "theta[i] = 2*i*Pi/ (P-Z),i = 1 .. P-Z;" "6$/&%&thetaG6#%\"iG**\"\"#\"\"\"F'F*%#PiGF*,&%\" PGF*%\"ZG!\"\"F//F';F*,&F-F*F.F/" }{TEXT -1 1 " " }}{PARA 14 "" 0 "" {TEXT 279 3 "6. " }{TEXT -1 29 "Ramurile de pe axa reala ale " }{TEXT 268 4 "L.R." }{TEXT 270 0 "" }{TEXT -1 35 " sunt situate la stanga unu i numar " }{TEXT 305 3 "par" }{TEXT -1 36 " de singularitati.(poli si \+ zerouri)." }}{PARA 14 "" 0 "" {TEXT 303 3 "7. " }{TEXT -1 67 "Ramurile care parasesc/vin pe axa reala o fac perpendicular pe axa." }}{PARA 14 "" 0 "" {TEXT -1 58 "Obs: Diferit fata de reactie pozitiva sunt num ai regulile " }{TEXT 307 1 "5" }{TEXT -1 14 " (formula) si " }{TEXT 306 1 "6" }{TEXT -1 2 ". " }}}{PARA 258 "" 0 "" {TEXT 256 11 "Observat ii." }}{PARA 15 "" 0 "" {TEXT -1 17 "In cazul in care " }{XPPEDIT 18 0 "Z = P;" "6#/%\"ZG%\"PG" }{TEXT -1 17 ", toate ramurile " }{TEXT 286 0 "" }{TEXT 287 4 "L.R." }{TEXT -1 24 " pleaca din poli pentru " } {XPPEDIT 18 0 "K = 0;" "6#/%\"KG\"\"!" }{TEXT -1 28 " si ajung in zero uri pentru " }{XPPEDIT 18 0 "proc (K) options operator, arrow; infinit y end proc;" "6#R6#%\"KG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 83 ". In acest caz nu exista ramuri divergente la infinit si nu se calculeaza asimtote." }}{PARA 15 "" 0 "" {TEXT -1 119 "Atit pen tru reactie negativa cit si pentru reactie pozitiva asimptotele se int ersecteaza in acelasi centru de greutate " }{XPPEDIT 18 0 "Cg" "6#%#Cg G" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "Reprezentar e spatiala a locului radacinilor" }}{PARA 258 "" 0 "" {TEXT -1 0 "" } {TEXT 341 4 "L.R." }{TEXT -1 96 " se poate reprezenta si spatial in co ordonate \{K, Re(p), Im(p)\} la variatia amplificarii pentru " } {XPPEDIT 18 0 "K = 0 .. infinity;" "6#/%\"KG;\"\"!%)infinityG" }{TEXT -1 2 ". " }{TEXT 342 4 "L.R." }{TEXT -1 5 " are " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 60 " ramuri spatiale care pleaca din polii functie i de transfer " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 8 " pentru " }{XPPEDIT 18 0 "K = 0;" "6#/%\"KG\"\"!" }{TEXT -1 2 ". " } {XPPEDIT 18 0 "Z;" "6#%\"ZG" }{TEXT -1 12 " ramuri ale " }{TEXT 343 4 "L.R." }{TEXT -1 33 " 'ajung asimptotic' in zerourile " }{XPPEDIT 18 0 "z[l];" "6#&%\"zG6#%\"lG" }{TEXT -1 26 " ale functiei de transfer " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 5 " iar " } {XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 44 " ramuri tind asimptotic spre infinit pentru " }{XPPEDIT 18 0 "proc (K) option s operator, arrow; infinity end proc;" "6#R6#%\"KG7\"6$%)operatorG%&ar rowG6\"%)infinityGF*F*F*" }{TEXT -1 155 ". Centrul de greutate este de coordonate \{0, xCg, yCg\} iar asimptotele sunt curbe spatiale care s e intersecteaza in centrul de greutate si au coordonatele:" }}{PARA 256 "" 0 "" {TEXT -1 6 "\{ K , " }{XPPEDIT 18 0 "K^(1/(P-Z))*cos(theta [j]);" "6#*&)%\"KG*&\"\"\"F',&%\"PGF'%\"ZG!\"\"F+F'-%$cosG6#&%&thetaG6 #%\"jGF'" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "K^(1/(P-Z))*sin(theta[j]); " "6#*&)%\"KG*&\"\"\"F',&%\"PGF'%\"ZG!\"\"F+F'-%$sinG6#&%&thetaG6#%\"j GF'" }{TEXT -1 4 " \}, " }{XPPEDIT 18 0 "i = 1 .. P-Z" "6#/%\"iG;\"\" \",&%\"PGF&%\"ZG!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "K = 0 .. infi nity" "6#/%\"KG;\"\"!%)infinityG" }{TEXT -1 0 "" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 12 "Mod de lucru" }}{PARA 0 "" 0 "" {TEXT -1 133 "In a fara de functiile incluse in libraria standard Maple, in aceasta lucra re vor fi utilizate cateva functii din libraria aditionala " }{TEXT 344 7 "SCSlib." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Pentru c alcularea functiei de transfer in bucla deschisa:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 347 67 "PZ[numeric] \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\ Helpfile.mws" "Functia PZ[numeric]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 100 "Pentru reprezentarea locului radacinilor, in plan si in \+ spatiu, corespunzator sistemului cu reactie:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 345 12 "LocRad[plan]" }{TEXT -1 54 " \+ " }{HYPERLNK 17 "help " 1 "..\\Help\\Helpfile.mws" "Functia LocRad[plan]" }{TEXT -1 1 " " }} {PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 346 14 "LocRad[spatiu]" }{TEXT -1 51 " " } {HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws" "Functia LocRad[spatiu] " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "libname:=\"../SCSlib\",libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "rp := 'reactie = pozitiva':rn := 'reactie = negativa' :" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Exemple" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Exemplul I" }}{PARA 0 "" 0 "" {TEXT -1 137 "Ne pr opunem sa studiem stabilitatea unui sistem cu reactie pentru care func tia de transfer in bucla deschisa la amplificare unitara este:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Hd:=(s^2+0.1*s+4)/(s*(s^2+s+ 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&,(*$)%\"sG\"\"#\"\"\"F +*&$F+!\"\"F+F)F+F+\"\"%F+F+*&F)F+,(F'F+F)F+F+F+F+F." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 62 "Poli si zerourile functiei de transfer in bucla deschisa Hd(s)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "p:= PZ[numeric](Hd,s,singularitati=poli):P:= RowDimension(p);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "z:=PZ[numeric](Hd,s,singularitati=zerouri): Z:= RowDimension(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZG\"\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "PZ[numeric](Hd,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")wzp\")-%'MATRIXG6#7'7$% #z1G^$$!%+]!\"&$\"%**>!\"$7$%#z2G^$F.$!%**>F37$%#p1G$\"\"!F<7$%#p2G^$$ F/!\"%$\"%g')FA7$%#p3G^$F@$!%g')FA" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Reactie negativa" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 22 "Trasa rea rapida a L.R." }}{PARA 0 "" 0 "" {TEXT -1 7 "Pentru " }{TEXT 310 16 "reactie negativa" }{TEXT -1 16 ", proprietatile " }{TEXT 308 4 "L. R." }{TEXT -1 0 "" }{TEXT 309 0 "" }{TEXT -1 18 " sunt urmatoarele:" } }{PARA 14 "" 0 "" {TEXT -1 3 "1. " }{TEXT 314 4 "L.R." }{TEXT 313 0 " " }{TEXT -1 89 " are P ramuri care pleaca din cei P poli ai functiei d e transfer in bucla deschisa Hd(s)." }}{PARA 14 "" 0 "" {TEXT -1 25 "2 . Din cele P ramuri ale " }{TEXT 312 4 "L.R." }{TEXT 317 0 "" }{TEXT -1 3 ", Z" }{TEXT 316 1 " " }{TEXT -1 126 "ramuri tind spre cele Z zer ouri ale functiei de transfer in bucla deschisa Hd(s), iar P-Z ramuri \+ tind asimptotic spre infinit." }}{PARA 14 "" 0 "" {TEXT -1 12 "3. Ramu rile " }{TEXT 315 4 "L.R." }{TEXT 318 0 "" }{TEXT -1 33 " sunt simtric e fata de axa reala." }}{PARA 14 "" 0 "" {TEXT -1 97 "4. Asimptota ram urii care tind spre infinitse intersecteaza in centrul de greutate de \+ coordonate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "xCg:=(sum('Re (p[k,2])','k'=1..P)-sum('Re(z[l,2])','l'=1..Z))/(P-Z);yCg:=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xCgG$!&++*!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yCgG\"\"!" }}}{PARA 14 "" 0 "" {TEXT -1 76 "5. Asimp tota ramurii care tinde la infinit formeaza cu axa reala unghiurile:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "theta:=[seq((2*i+1)*Pi/(P- Z),i=0..P-Z-1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG7#%#PiG" }}}{PARA 14 "" 0 "" {TEXT -1 12 "6. Ramurile " }{TEXT 311 4 "L.R." } {TEXT -1 75 " situate pe axa reala sunt la stanga unui unui numar impa r de singularitati" }}{PARA 14 "" 0 "" {TEXT -1 66 "7. 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Pentru K = (K1, K2)doua ramuri ramurile sunt in semip lanul drept. Putem spune asadar ca sistemul cu " }{TEXT 322 16 "reacti e negativa" }{TEXT -1 6 " este " }{TEXT 321 29 "conditionat stabil (in stabil)" }{TEXT -1 1 "." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "React ie pozitiva" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 22 "Trasarea rapida a L.R." }}{PARA 0 "" 0 "" {TEXT -1 7 "Pentru " }{TEXT 325 16 "reactie p ozitiva" }{TEXT -1 16 ", proprietatile " }{TEXT 323 4 "L.R." }{TEXT -1 0 "" }{TEXT 324 0 "" }{TEXT -1 18 " sunt urmatoarele:" }}{PARA 14 " " 0 "" {TEXT -1 3 "1. " }{TEXT 328 4 "L.R." }{TEXT 327 0 "" }{TEXT -1 89 " are P ramuri care pleaca din cei P poli ai functiei de transfer i n bucla deschisa Hd(s)." }}{PARA 14 "" 0 "" {TEXT -1 25 "2. Din cele P ramuri ale " }{TEXT 326 4 "L.R." }{TEXT 331 0 "" }{TEXT -1 3 ", Z" } {TEXT 330 1 " " }{TEXT -1 127 "ramuri tind spre cele Z zerouri ale fun ctiei de transfer in bucla deschisa Hd(s), iar P-Z ramura tinde asimpt otic spre infinit." }}{PARA 14 "" 0 "" {TEXT -1 12 "3. Ramurile " } {TEXT 329 4 "L.R." }{TEXT 332 0 "" }{TEXT -1 33 " sunt simtrice fata d e axa reala." }}{PARA 14 "" 0 "" {TEXT -1 96 "4. Asimptota ramurii car e tinde spre infinit, are originea in centrul de greutate de coordonat e:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "xCg:=(sum('Re(p[k,2])' ,'k'=1..P)-sum('Re(z[l,2])','l'=1..Z))/(P-Z);yCg:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xCgG$!&++*!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$yCgG\"\"!" }}}{PARA 14 "" 0 "" {TEXT -1 76 "5. Asimptota ramurii c are tinde la infinit formeaza cu axa reala unghiurile:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "theta:=[seq(2*i*Pi/(P-Z),i=0..P-Z-1 )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG7#\"\"!" }}}{PARA 14 " " 0 "" {TEXT -1 12 "6. Ramurile " }{TEXT 333 4 "L.R." }{TEXT -1 74 " s ituate pe axa reala sunt la stanga unui unui numar par de singularitat i." }}{PARA 14 "" 0 "" {TEXT -1 47 "7. 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