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Asachi\", Iasi, Facultatea de Electronica si Telecomunicatii" }} {PARA 19 "" 0 "" {TEXT -1 43 "Laboratorul de Semnale, Circuite si Sist eme" }}{PARA 19 "" 0 "" {URLLINK 17 "http://scs4.etc.tuiasi.ro" 4 "htt p://scs4.etc.tuiasi.ro/scslabs/Simbolice/simbolice.htm" "" }{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Breviar teoretic" }}{PARA 0 "" 0 "" {TEXT 288 16 "Scopul lucrarii:" }}{PARA 0 "" 0 "" {TEXT -1 92 "Familiarizarea cu criteriul Nyquist de analiza a stabilitatii sist emelor liniare cu reactie." }}{PARA 0 "" 0 "" {TEXT 289 17 "Rezumat te oretic:" }}{PARA 0 "" 0 "" {TEXT -1 92 "In esenta, criteriul de analiz a a stabilitatii are la baza teorema Cauchy, enuntata mai jos." }} {PARA 4 "" 0 "" {TEXT -1 14 "Teorema Cauchy" }}{PARA 0 "" 0 "" {TEXT 258 0 "" }{TEXT -1 4 "Fie " }{XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 1 ":" }{XPPEDIT 18 0 "proc (C) options operator, arrow; C end proc;" "6# R6#%\"CG7\"6$%)operatorG%&arrowG6\"F%F*F*F*" }{TEXT -1 50 " o fractie \+ rationala cu coeficienti complecsi, si " }{TEXT 259 1 "C" }{TEXT 260 0 "" }{TEXT -1 90 " un contur inchis in planul complex, care nu trece \+ prin nici un pol sau zerou al functiei " }{XPPEDIT 18 0 "F(s);" "6#-% \"FG6#%\"sG" }{TEXT -1 16 ". Daca conturul " }{TEXT 261 1 "C" }{TEXT -1 27 " contine in interiorul sau " }{XPPEDIT 18 0 "Z;" "6#%\"ZG" } {TEXT -1 12 " zerouri si " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 18 " poli ai functiei " }{XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 34 ", atu nci cand s parcurge conturul " }{TEXT 263 6 "Cauchy" }{TEXT -1 35 " in sens orar, hodograful functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#% \"sG" }{TEXT -1 24 " inconjoara originea de " }{XPPEDIT 18 0 "Z-P;" "6 #,&%\"ZG\"\"\"%\"PG!\"\"" }{TEXT -1 21 " ori in acelasi sens." }} {PARA 0 "" 0 "" {TEXT -1 66 "Sa consideram acum un sistem cu reactie a vand functia de transfer:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Hr(s) = \+ K*A(s)/(1-epsilon*K*A(s)*B(s));" "6#/-%#HrG6#%\"sG*(%\"KG\"\"\"-%\"AG6 #F'F*,&F*F***%(epsilonGF*F)F*-F,6#F'F*-%\"BG6#F'F*!\"\"F6" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "unde " }{XPPEDIT 18 0 "epsilon = 1 ;" "6#/%(epsilonG\"\"\"" }{TEXT -1 28 " pentru reactie pozitiva si " } {XPPEDIT 18 0 "epsilon = -1;" "6#/%(epsilonG,$\"\"\"!\"\"" }{TEXT -1 25 " pentru reactie negativa." }}{PARA 0 "" 0 "" {TEXT -1 9 "Notam cu \+ " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 39 " functia de \+ transfer in bucla deschisa:" }}{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 forma:" }}{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 -1 105 "Stabilitatea sistemului cu re actie este data de pozitia in planul complex a polilor functiei de tra nsfer " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 79 ". Pent ru studiul stabilitati sistemului cu reactie la modificarea amplificar ii " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 94 " se urmareste determi narea legaturii dintre numarul de poli din semiplanul drept ale functi ei " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 17 " si valoa rea lui " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 25 ", folosind terem a Cauchy." }}{PARA 0 "" 0 "" {TEXT -1 28 "In acest scop se noteaza cu \+ " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 20 " numitorul fu nctiei " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 1 ":" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "F(s) = 1-epsilon*Hd(s);" "6#/-%\"FG6# %\"sG,&\"\"\"F)*&%(epsilonGF)-%#HdG6#F'F)!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 265 5 "Obs. " }{TEXT -1 32 "Avand in vedere lega tura dintre " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 4 " \+ si " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 28 ", respecti v legatura dintre " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 4 " si " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 32 ", \+ putem spune ca polii functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"s G" }{TEXT -1 52 " sunt polii functiei de transfer in bucla deschisa, \+ " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 31 ", respectiv \+ zerourile functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 58 " sunt polii functiei de transfer a sistemului cu reactie, " } {XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "Se considera un contur inchis in planul complex " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 81 " care sa contina toti polii si toate zerourile din semiplanul drept ale functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 40 ", dar care sa nu contina pol ii functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 58 " \+ situati pe axa imaginara. Un astfel de contur se numeste " }{TEXT 262 16 "conturul Nyquist" }{TEXT 264 1 "." }}{PARA 4 "" 0 "" {TEXT -1 16 " Conturul Nyquist" }}{PARA 0 "" 0 "" {TEXT -1 122 "Conturul Nyquist est e un contur inchis ce inconjoara semiplanul drept al planului complex \+ in sens orar si este format din:" }}{PARA 15 "" 0 "" {TEXT -1 21 "axa \+ imaginara pentru " }{XPPEDIT 18 0 "omega = -infinity .. infinity;" "6# /%&omegaG;,$%)infinityG!\"\"F'" }{TEXT -1 103 " mai putin in apropiere a polilor de pe axa imaginara care sunt ocoliti fiecare cu un semicerc de raza " }{XPPEDIT 18 0 "r0;" "6#%#r0G" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "proc (r0) options operator, arrow; 0 end proc;" "6#R6#%#r0G7\"6$ %)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 78 ") cu centrul in polul de pe axa imaginara, semicerc situat in semiplanul drept" }}{PARA 15 "" 0 "" {TEXT -1 63 "conturul se inchide cu un semicerc in semiplanul \+ drept de raza " }{XPPEDIT 18 0 "R0;" "6#%#R0G" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "proc (R0) options operator, arrow; 0 end proc;" "6#R6#% #R0G7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 21 ") centrat in origine." }}{PARA 257 "" 0 "" {TEXT -1 28 "Practic, alegerea valorilo r " }{XPPEDIT 18 0 "R0;" "6#%#R0G" }{TEXT -1 4 " si " }{XPPEDIT 18 0 " r0;" "6#%#r0G" }{TEXT -1 26 " se face in felul urmator:" }}{PARA 15 " " 0 "" {XPPEDIT 18 0 "R0;" "6#%#R0G" }{TEXT -1 78 " trebuie sa fie mul t mai mare decat modulul tuturor singularitatilor functiei " } {XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 3 ", (" }{XPPEDIT 18 0 "proc (R0) options operator, arrow; infinity end proc;" "6#R6#%#R 0G7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ")." }} {PARA 15 "" 0 "" {XPPEDIT 18 0 "r0;" "6#%#r0G" }{TEXT -1 101 " trebuie sa fie mult mai mic decat distanta dintre oricare doua singularitati \+ distincte ale functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" } {TEXT -1 3 ", (" }{XPPEDIT 18 0 "proc (r0) options operator, arrow; 0 \+ end proc;" "6#R6#%#r0G7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 2 ")." }}{PARA 258 "" 0 "" {TEXT -1 6 "Notam:" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 29 " numarul de poli ai functie i " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 22 " din semipl anul drept;" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Z;" "6#%\"ZG" }{TEXT -1 54 " numarul de zerouri din semiplanul drept ale functiei " } {XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "Se alege un sens de referinta de exemplu sensul orar . Se noteaza cu " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 60 " numarul de inconjururi pe care le face hodograful functiei " }{XPPEDIT 18 0 " F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 61 " in jurul originii in sensul de referinta fixat, atunci cand " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 117 " parcurge o singura data conturul Nyquist in acelasi sens. Dar numarul de inconjururi pe care le face hodograful lui " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 84 " in jurul originii este acel asi cu numarul de inconjururi pe le face hodograful lui " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 20 " in jurul punctului " } {XPPEDIT 18 0 "1/epsilon;" "6#*&\"\"\"F$%(epsilonG!\"\"" }{TEXT -1 8 " , adica " }{XPPEDIT 18 0 "-1+I*0;" "6#,&\"\"\"!\"\"*&%\"IGF$\"\"!F$F$ " }{TEXT -1 36 " (-1,0) pentru reactie negativa sau " }{XPPEDIT 18 0 " 1+I*0;" "6#,&\"\"\"F$*&%\"IGF$\"\"!F$F$" }{TEXT -1 31 " (1,0) pentru r eactie pozitiva." }}{PARA 0 "" 0 "" {TEXT -1 32 "Hodograful functiei d e transfer " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 7 ", \+ cand " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 38 " parcurge conturul \+ Nyquist se numeste " }{TEXT 267 16 "diagrama Nyquist" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "Avand in vedere teorema Cauchy, " } {XPPEDIT 18 0 "N = Z-P;" "6#/%\"NG,&%\"ZG\"\"\"%\"PG!\"\"" }{TEXT -1 7 ", deci " }{XPPEDIT 18 0 "Z = N+P;" "6#/%\"ZG,&%\"NG\"\"\"%\"PGF'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 5 "Obs. " }{TEXT -1 32 "Avand in vedere legatura dintre " } {XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 4 " si " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 28 ", respectiv legatura dint re " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 4 " si " } {XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 17 ", putem spune c a " }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 91 " reprezinta numarul de poli din semiplanul drept ai functiei de transfer in bucla deschis, \+ " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 6 ", iar " } {XPPEDIT 18 0 "Z;" "6#%\"ZG" }{TEXT -1 98 " reprezinta numarul de poli din semiplanul drept ai functiei de transfer a sistemului cu reactie, " }{XPPEDIT 18 0 "Hr(s);" "6#-%#HrG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 32 "Criteriul de s tabilitate Nyquist" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 0 "" } {TEXT -1 0 "" }{TEXT 270 0 "" }{TEXT -1 87 "Un sistem cu reactie este \+ stabil daca diagrama Nyquist inconjura punctul de referinta (" }{TEXT 291 6 "(-1,0)" }{TEXT -1 26 " pentru reactie negativa, " }{TEXT 290 6 "(+1,0)" }{TEXT -1 29 " pentru reactie pozitiva) de " }{XPPEDIT 18 0 " P;" "6#%\"PG" }{TEXT -1 35 " ori in sens antiorar, atunci cand " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 55 " parcurge o singura data in sens orar conturul Nyquist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Expresia matematica a criteriului de stabilitat e Nyquist este " }{XPPEDIT 18 0 "N = -P;" "6#/%\"NG,$%\"PG!\"\"" } {TEXT -1 33 " si este echivalenta cu conditia " }{XPPEDIT 18 0 "Z = 0; " "6#/%\"ZG\"\"!" }{TEXT -1 80 " (sistemul cu reactie sa nu aiba poli \+ in semiplanul drept, adica sa fie stabil)." }}{PARA 0 "" 0 "" {TEXT 271 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Trasarea diagramei Nyquist ple aca in principiu de la diagrama polara a functiei " }{XPPEDIT 18 0 "Hd (I*omega);" "6#-%#HdG6#*&%\"IG\"\"\"%&omegaGF(" }{TEXT -1 206 ", deoar ece o parte a conturului Nyquist (cea mai semnificativa) este situata \+ pe axa imaginara. Diagrama Nyquist este simetrica fata de axa reala de oarece conturul Nyquist este simetric fata de axa reala si " } {XPPEDIT 18 0 "Hd(conjugate(s)) = conjugate(Hd(s));" "6#/-%#HdG6#-%*co njugateG6#%\"sG-F(6#-F%6#F*" }{TEXT -1 14 " pentru orice " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 53 " complex (pentru sisteme realizabile practic functia " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 24 " are coeficienti reali)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 272 96 "Studiul stabilitatii sistemului cu react ie la variatia amplificarii utilizand criteriul Nyquist." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "In principiu se pleaca de la diagra ma Nyquist trasata pentru amplificare " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 67 " unitara. Daca se modifica valoarea amplificarii, modulu l functiei " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 27 " \+ se rescaleaza cu valoarea " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 135 ", dar argumentul ramane neschimbat. Din acest motiv alura diagram ei Nyquist ramane neschimbata dar se contracta sau se dilata dupa cum \+ " }{XPPEDIT 18 0 "K < 1;" "6#2%\"KG\"\"\"" }{TEXT -1 5 " sau " } {XPPEDIT 18 0 "1 < K;" "6#2\"\"\"%\"KG" }{TEXT -1 63 ". Acest lucru fa ce posibila deducerea numarului de inconjururi " }{XPPEDIT 18 0 "N;" " 6#%\"NG" }{TEXT -1 184 " (in jurul punctului de referinta), la diferit e valori ale amplificarii, din pozitia relativa a punctului de referin ta in raport cu diagrama Nyquist trasata pentru amplificare unitara." }}{PARA 0 "" 0 "" {TEXT -1 98 "In functie de caracterul sistemului la \+ variatia amplificarii se poate face urmatoarea clasificare:" }}{PARA 15 "" 0 "" {TEXT -1 15 "un sistem este " }{TEXT 274 19 "conditionat st abil " }{TEXT -1 4 "sau " }{TEXT 279 20 "conditionat instabil" }{TEXT -1 25 " daca exista doua valori " }{XPPEDIT 18 0 "K1;" "6#%#K1G" } {TEXT -1 4 " si " }{XPPEDIT 18 0 "K2;" "6#%#K2G" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "0 < K1;" "6#2\"\"!%#K1G" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "0 < K2;" "6#2\"\"!%#K2G" }{TEXT -1 55 "), astfel incat sistemul \+ cu reactie este stabil pentru " }{XPPEDIT 18 0 "K = K1;" "6#/%\"KG%#K1 G" }{TEXT -1 20 " si instabil pentru " }{XPPEDIT 18 0 "K = K2;" "6#/% \"KG%#K2G" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 15 "un sistem e ste " }{TEXT 275 19 "neconditonat stabil" }{TEXT 276 0 "" }{TEXT -1 27 " daca pentru orice valoare " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "0 < K;" "6#2\"\"!%\"KG" }{TEXT -1 48 ") a am plificarii sistemul cu reactie este stabil" }}{PARA 15 "" 0 "" {TEXT -1 15 "un sistem este " }{TEXT 277 22 "neconditionat instabil" }{TEXT 278 0 "" }{TEXT -1 27 " daca pentru orice valoare " }{XPPEDIT 18 0 "K; " "6#%\"KG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "0 < K;" "6#2\"\"!%\"KG" } {TEXT -1 51 ") a amplificarii sistemul cu reactie este instabil." }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Mod de lucru" }}{PARA 0 "" 0 "" {TEXT -1 133 "In afara de functiile incluse in libraria standard Maple , in aceasta lucrare vor fi utilizate cateva functii din libraria adit ionala " }{TEXT 284 7 "SCSlib." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Pentru trasarea diagramei polare a functiei de transfer i n bucla deschisa:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 287 69 "Bode[polara] \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws" "Functia Bode [polara]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 92 "Pentru reprez entarea conturului si diagramei Nyquist corespunzatoare unui sistem cu reactie:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 285 15 "Nyq uist[contur]" }{TEXT -1 54 " \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws" "Functia Nyquist[contur]" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 8 "funct ia " }{TEXT 286 16 "Nyquist[diagram]" }{TEXT -1 51 " \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\H elpfile.mws" "Functia Nyquist[diagrama]" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Pe parcursul lucrari i, vom folosi urmatoarele notatii:" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 " Hd(s);" "6#-%#HdG6#%\"sG" }{TEXT -1 69 " pentru functia de transfer in bucla deschisa a sistemului cu reactie" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 47 " pentru amplificare (intervine in ex presia lui " }{XPPEDIT 18 0 "Hd(s);" "6#-%#HdG6#%\"sG" }{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:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Exemp le" }}{PARA 0 "" 0 "" {TEXT -1 139 "Ne propunem studiul stabilitatii u nor sisteme cu reactie la variatia amplificarii. Se porneste de la fun ctia de transfer in bucla deschisa." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Exemplul I" }}{PARA 0 "" 0 "" {TEXT -1 80 "Fie un sistem cu rea ctie pentru care functia de transfer in bucla deschisa este:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hd:=K*(s+1)/(s*(s^2-s+100)): " }}}{PARA 0 "" 0 "" {TEXT -1 175 "Exista un pol in origine care va fi ocolit de catre conturul Nyquist si alti doi poli complex conjugati s ituati in semiplanul drept, poli care vor fi in interiorul conturului. " }}{PARA 0 "" 0 "" {TEXT -1 102 "Reprezentarea diagramei poli zerouri a functiei de transfer in bucla deschisa si a conturului Nyquist:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Nyquist[contur](eval(Hd,K=1) ,repere=[-1,1],compresie=[4,1],tipgrafic=dinamic);" }}{PARA 13 "" 1 " " {GLPLOT2D 250 238 238 {PLOTDATA 2 "6&-%(ANIMATEG6er7/-%'CURVESG6%7$7 $$!+++++5!\"*$!+9w1-T!#>7$$!+7[GBhF1$!+VEY&)HF.-%&COLORG6&%$RGBG$\"#5! \"\"\"\"!F>-%*THICKNESSG6#\"\"\"-F(6%7$7$$F>F>FGF2F7F?-F(6%7$7$$\"+>4K !o*!#6$\"+*eUO$>F.F2F7F?-F(6%7$7$FL$!+*eUO$>F.F2F7F?-F(6%7\\_qF27$$!+f (>v6'F1$!+Q=l#)HF.7$$!+ln[7hF1$!+ez>!)HF.7$$!+nx!o5'F1$!+6\"Hu(HF.7$$! +!yq55'F1$!+\"*>juHF.7$$!+D1M&4'F1$!+-#Q=(HF.7$$!+x'4+4'F1$!+]!R#pHF.7 $$!+@5Z%3'F1$!+H'Ql'HF.7$$!+*oA(ygF1$!+xftjHF.7$$!+G@(H2'F1$!+TA$4'HF. 7$$!+q_.ngF1$!+rw.eHF.7$$!+*p(yhgF1$!+r\"za&HF.7$$!+-%fe0'F1$!+w()e_HF .7$$!+MW))\\gF1$!+Fcn\\HF.7$$!+%40T/'F1$!+fy&o%HF.7$$!+%fQ)QgF1$!+G,HW HF.7$$!+RLbKgF1$!+'pD7%HF.7$$!+*H4s-'F1$!+h,iQHF.7$$!+%3q4-'F1$!+w\"yb 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}}}}{PARA 0 "" 0 "" {TEXT -1 25 "In primul grafic, pentru " }{XPPEDIT 18 0 "K0 < K;" "6#2%#K0G%\"KG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "N = -1;" "6#/%\"NG,$\"\"\"!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "Z = 1;" "6#/%\"ZG\"\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "Z = N+P;" "6#/%\"ZG,&%\"NG\"\"\"%\"PGF'" }{TEXT -1 58 "), deci sist emul cu reactie pozitiva este instabil, avand " }{XPPEDIT 18 0 "1;" "6 #\"\"\"" }{TEXT -1 25 " pol in semiplanul drept." }}{PARA 0 "" 0 "" {TEXT -1 30 "In graficul al doilea, pentru " }{XPPEDIT 18 0 "K0 < K;" "6#2%#K0G%\"KG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "N = 1;" "6#/%\"NG\"\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Z = 3;" "6#/%\"ZG\"\"$" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "Z = N+P;" "6#/%\"ZG,&%\"NG\"\"\"%\"PGF'" } {TEXT -1 57 "), deci sistemul cu reactie pozitiva este instabil avand \+ " }{XPPEDIT 18 0 "3;" "6#\"\"$" }{TEXT -1 25 " pol in semiplanul drept ." }}{PARA 0 "" 0 "" {TEXT -1 52 "Concluzia este ca sistemul cu reacti e pozitiva este " }{TEXT 273 29 "conditionat instabil (stabil)" } {TEXT -1 1 "." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Exemplul II" } }{PARA 0 "" 0 "" {TEXT -1 80 "Fie un sistem cu reactie pentru care fun ctia de transfer in bucla deschisa este:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hd:=K*(s+1)/(s*(s^2+s+100)):" }}}{PARA 0 "" 0 "" {TEXT -1 175 "Exista un pol in origine care va fi ocolit de catre cont urul Nyquist si alti doi poli complex conjugati situati in semiplanul \+ sting, poli care vor fi in exteriorul conturului." }}{PARA 0 "" 0 "" {TEXT -1 102 "Reprezentarea diagramei poli zerouri a functiei de trans fer in bucla deschisa si a conturului Nyquist:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Nyquist[contur](eval(Hd,K=1),repere=[-1,1],compr esie=[4,1],tipgrafic=dinamic);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%(ANIMATEG6er7/-%'CURVESG6%7$7$$!+++++5!\"*$!+9w1- T!#>7$$!+wc;BhF1$!+cXS&)HF.-%&COLORG6&%$RGBG$\"#5!\"\"\"\"!F>-%*THICKN 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Avand in vedere \+ ca functia de transfer in bucla deschisa are " }{XPPEDIT 18 0 "P = 0; " "6#/%\"PG\"\"!" }{TEXT -1 91 " poli in semiplanul drept, deducem ca \+ numarul de zerouri din semiplanul drept ale functiei " }{XPPEDIT 18 0 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT -1 6 " este " }{XPPEDIT 18 0 "Z = 1; " "6#/%\"ZG\"\"\"" }{TEXT -1 3 ". (" }{XPPEDIT 18 0 "Z = N+P;" "6#/%\" ZG,&%\"NG\"\"\"%\"PGF'" }{TEXT -1 85 "). Asadar sistemul cu reactie ne gativa este instabil avand 1 pol in semiplanul drept." }}{PARA 0 "" 0 "" {TEXT -1 45 "In momentul in care se modifica amplificarea " } {XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 86 " alura diagramei Nyquist ra mane aceeasi, dar se \"dilata\" sau se \"contracta\", dupa cum " } {XPPEDIT 18 0 "1 < K;" "6#2\"\"\"%\"KG" }{TEXT -1 5 " sau " }{XPPEDIT 18 0 "K < 1;" "6#2%\"KG\"\"\"" }{TEXT -1 109 ". Avand in vedere poziti a punctului de referinta in raport cu diagrama deducem ca pentru orice valoare a lui " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 14 " are valoarea " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 41 ", deci sistemul cu reactie pozitiva este " }{TEXT 280 22 "neconditionat instabil" }{TEXT 281 0 "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 81 "Acest lucru este confirmat de \+ cele doua diagrame de mai jos, reprezentate pentru " }{XPPEDIT 18 0 "K = 100;" "6#/%\"KG\"$+\"" }{TEXT -1 4 " si " }{XPPEDIT 18 0 "K = .1e-1 ;" "6#/%\"KG$\"\"\"!\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Nyquist[diagrama](eval(Hd,K=100),repere=[+1],compresi e=[2,1],tipgrafic=dinamic);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%(ANIMATEG6er7'-%'CURVESG6%7$7$$\"\"\"\"\"!$F.F.7$$!+Y ;+E$)!#6$\"+NvA4%)!#:-%&COLORG6&%$RGBG$\"#5!\"\"F.F.-%*THICKNESSG6#F-- 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Intrebari" }}{PARA 0 "" 0 "" {TEXT -1 51 "1. Pentru functiile d e transfer analizate, se cere:" }}{PARA 15 "" 0 "" {TEXT -1 48 "sa se \+ traseze diagramele Bode de cistig si faza;" }}{PARA 15 "" 0 "" {TEXT -1 65 "sa se traseze diagrama polara si sa se gradeze in valori ale lu i " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 15 " in intervalul \+ " }{XPPEDIT 18 0 "0 .. infinity;" "6#;\"\"!%)infinityG" }{TEXT -1 1 "; " }}{PARA 15 "" 0 "" {TEXT -1 61 "sa se traseze diagrama Nyquist indic ind sensul de parcurgere;" }}{PARA 15 "" 0 "" {TEXT -1 47 "sa se numer e inconjurul punctelor de referinta." }}{PARA 0 "" 0 "" {TEXT -1 113 " 2. Sa se analizeze stabilitatea sistemelor cu reactie pornind de la fu nctia de transfer in bucla deschisa pentru:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "Hd:=K*s*(s+10)/((s+1)*(s^2+s+100)*(s+100));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*(%\"KG\"\"\"%\"sGF(,&F)F(\"#5 F(F(F(*(,&F)F(F(F(F(,(*$)F)\"\"#F(F(F)F(\"$+\"F(F(,&F)F(F2F(F(!\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Hd:=K*(s^2+10*s+10000)/((s +1)*(s^2+s+100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*&%\"KG\" \"\",(*$)%\"sG\"\"#F(F(*&\"#5F(F,F(F(\"&++\"F(F(F(*&,&F,F(F(F(F(,(F*F( F,F(\"$+\"F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Hd:= K*(s+10)/(s*(s^2+s+100)*(s+100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#HdG*&*&%\"KG\"\"\",&%\"sGF(\"#5F(F(F(*(F*F(,(*$)F*\"\"#F(F(F*F(\"$+ \"F(F(,&F*F(F1F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " Hd:=K*(s+100)/((s+1)*(s-10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Hd G*&*&%\"KG\"\"\",&%\"sGF(\"$+\"F(F(F(*&,&F*F(F(F(F(,&F*F(\"#5!\"\"F(F0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Hd:=K*(s+1)/s;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*&%\"KG\"\"\",&%\"sGF(F(F(F(F( F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Hd:=K*(s-1)*(s-1 0)/((s+1)*(s+10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*(%\"KG \"\"\",&%\"sGF(F(!\"\"F(,&F*F(\"#5F+F(F(*&,&F*F(F(F(F(,&F*F(F-F(F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Hd:=K*(s+3)*(s+4)/(s*(s+1 )*(s+2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*(%\"KG\"\"\",&% \"sGF(\"\"$F(F(,&F*F(\"\"%F(F(F(*(F*F(,&F*F(F(F(F(,&F*F(\"\"#F(F(!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Hd:=K/(s*(s+10)*(s^2+s+ 10000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&%\"KG\"\"\"*(%\"sG F',&F)F'\"#5F'F',(*$)F)\"\"#F'F'F)F'\"&++\"F'F'!\"\"" }}}}}{MARK "4 1 " 0 }{VIEWOPTS 1 1 1 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }