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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 76 "Studiul stabilitatii circuitelor
 liniare folosind metoda locului radacinilor" }}{PARA 18 "" 0 "" 
{TEXT -1 31 "Partea IV: Exemple (continuare)" }}{PARA 19 "" 0 "" 
{TEXT -1 86 "Universitatea Tehnica \"Gh. Asachi\", Iasi, Facultatea de
 Electronica si Telecomunicatii" }}{PARA 19 "" 0 "" {TEXT -1 43 "Labor
atorul de Semnale, Circuite si Sisteme" }}{PARA 19 "" 0 "" {URLLINK 
17 "http://scs4.etc.tuiasi.ro" 4 "http://scs4.etc.tuiasi.ro/scslabs/Si
mbolice/simbolice.htm" "" }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 16 "Breviar teoretic" }}{PARA 0 "" 0 "" {TEXT -1 62 "Chestiun
i teoretice sunt descrise in Partea I a laboratorului." }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 46 "Reguli de trasare rapida a locului radaci
nilor" }}{PARA 0 "" 0 "" {TEXT -1 49 "Consideram functia de transfer i
n bucla deschisa " }{XPPEDIT 18 0 "Hd(s) = K*A(s)*B(s);" "6#/-%#HdG6#%
\"sG*(%\"KG\"\"\"-%\"AG6#F'F*-%\"BG6#F'F*" }{TEXT -1 5 "  cu " }
{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 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "Sunt reamintite pe scur
t formulele importante de calcul." }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 
16 "Reactie negativa" }}{PARA 0 "" 0 "" {TEXT -1 50 "Functia de transf
er 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 263 "" 0 "" {TEXT -1 5 "Cele " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG
\"\"\"%\"ZG!\"\"" }{TEXT -1 81 " asimptote are ramurilor care tind spr
e infinit se intalnesc intr-un punct numit " }{TEXT 262 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#/%$yCgG\"\"!" }{TEXT -1 1 "." }}{PARA 264 
"" 0 "" {TEXT -1 5 "Cele " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG
!\"\"" }{TEXT -1 81 " asimptote are ramurilor care tind spre infinit f
ormeaza cu axa reala unghiurile:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "t
heta[i] = (2*i+1)*Pi/(P-Z),i = 1 .. P-Z;" "6$/&%&thetaG6#%\"iG*(,&*&\"
\"#\"\"\"F'F,F,F,F,F,%#PiGF,,&%\"PGF,%\"ZG!\"\"F1/F';F,,&F/F,F0F1" }
{TEXT -1 1 " " }}{PARA 265 "" 0 "" {TEXT -1 29 "Ramurile de pe axa rea
la ale " }{TEXT 258 4 "L.R." }{TEXT 259 0 "" }{TEXT -1 35 " sunt situa
te la stanga unui numar " }{TEXT 264 5 "impar" }{TEXT -1 36 " de singu
laritati (poli si zerouri)." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "R
eactie pozitiva" }}{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 266 
"" 0 "" {TEXT -1 5 "Cele " }{XPPEDIT 18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG
!\"\"" }{TEXT -1 81 " asimptote are ramurilor care tind spre infinit s
e intalnesc intr-un punct numit " }{TEXT 263 18 "centru de greutate" }
{TEXT -1 4 " , (" }{XPPEDIT 18 0 "Cg;" "6#%#CgG" }{TEXT -1 23 "),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#/%$yCgG\"\"!" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 18 "i
n planul complex." }}{PARA 267 "" 0 "" {TEXT -1 5 "Cele " }{XPPEDIT 
18 0 "P-Z;" "6#,&%\"PG\"\"\"%\"ZG!\"\"" }{TEXT -1 78 " asimptote are r
amurilor care tind la infinit formeaza cu axa reala unghiurile" }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "theta[i] = 2*i*Pi/(P-Z),i = 1 .. P-Z;
" "6$/&%&thetaG6#%\"iG**\"\"#\"\"\"F'F*%#PiGF*,&%\"PGF*%\"ZG!\"\"F//F'
;F*,&F-F*F.F/" }{TEXT -1 1 " " }}{PARA 268 "" 0 "" {TEXT -1 29 "Ramuri
le de pe axa reala ale " }{TEXT 260 4 "L.R." }{TEXT 261 0 "" }{TEXT 
-1 35 " sunt situate la stanga unui numar " }{TEXT 265 3 "par" }{TEXT 
-1 36 " de singularitati.(poli si zerouri)." }}}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 12 "Mod de lucru" }}{PARA 0 "" 0 "" {TEXT -1 133 "In afa
ra de functiile incluse in libraria standard Maple, in aceasta lucrare
 vor fi utilizate cateva functii din libraria aditionala " }{TEXT 273 
7 "SCSlib." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Pentru calcu
larea functiei de transfer in bucla deschisa:" }}{PARA 15 "" 0 "" 
{TEXT -1 8 "functia " }{TEXT 276 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 274 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 275 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 11 "Exemplul IV" }}{PARA 0 "" 0 "" {TEXT -1 137 "Ne p
ropunem sa studiem stabilitatea unui sistem cu reactie pentru care fun
ctia de transfer in bucla deschisa la amplificare unitara este:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Hd:=s*(s+3)/((s+5)*(s^2-0.1*
s+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HdG*&*&%\"sG\"\"\",&F'F(
\"\"$F(F(F(*&,&F'F(\"\"&F(F(,(*$)F'\"\"#F(F(*&$F(!\"\"F(F'F(F4F(F(F(F4
" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 62 "Poli si zerourile functiei d
e transfer in bucla deschisa Hd(s)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "p:=PZ[numeric](Hd,s,singularitati=poli):P:= RowDimens
ion(p);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG\"\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "z:=PZ[numeric](Hd,s,sin
gularitati=zerouri):Z:= RowDimension(z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"ZG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "PZ[num
eric](Hd,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")G6q\")-
%'MATRIXG6#7'7$%#z1G$\"\"!F.7$%#z2G$!\"$F.7$%#p1G^$$\"%+]!\"&$\"%()**!
\"%7$%#p2G$F8F.7$%#p3G^$F6$!%()**F;" }}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 16 "Reactie negativa" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 22 
"Trasarea rapida a L.R." }}{PARA 0 "" 0 "" {TEXT -1 94 "Asimptota ramu
rii care tind spre infinitse intersecteaza in centrul de greutate de c
oordonate:" }}{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 258 "" 0 "" {TEXT -1 73 "Asimpto
ta 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 0 "" 0 "" {TEXT -1 25 "Locul radacinilor pentru " }{TEXT 266 
17 "reactiei negativa" }{TEXT -1 6 " este:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "LocRad[plan](Hd,rn,interval=0..10);" }}{PARA 13 "" 
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%Fe`sFfasFaas75F'F<FDFIFUFXFenFbalF[hlFhhlF\\ilFgilF`jlFieoFd[qFa^r-F(
6%FhjpFfasFaas-F(6%F[^rFfasFaas-F(6%Fh`sFfasFaas75F'F<FDFIFUFXFenFbalF
[hlFhhlF\\ilFgilF`jlFieoFd[qFa^r-F(6%F[[qFfasFaas-F(6%F^^rFfasFaas-F(6
%F[asFfasFaas-%*AXESSTYLEG6#%'NORMALG-%&TITLEG6#QJLocul~radacinilor~pe
ntru~reactie~negativaFM" 1 2 0 1 10 0 2 1 1 4 2 1.000000 22.000000 
4.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve \+
6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cur
ve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 1
9" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 23 "Discutia de stabi
litate" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 161 "Pe baza grafi
celor anterioare, observam ca o pereche de poli complex conjugati sunt
 in semiplanul complex drept. La valori mici ale lui K (K<K1) poua ram
uri ale " }{TEXT 272 4 "L.R." }{TEXT -1 86 " sunt situate in semiplanu
l sting. Pentru valori mari ale lui K (K>K2) toate ramurile " }{TEXT 
267 5 "L.R. " }{TEXT -1 76 "sunt situate in semiplanul complex sting. \+
Putem spune asadar ca sistemul cu " }{TEXT 269 16 "reactie negativa" }
{TEXT -1 6 " este " }{TEXT 268 29 "conditionat stabil (instabil)" }
{TEXT -1 1 "." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "Reactie pozitiv
a" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 22 "Trasarea rapida a L.R." }}
{PARA 260 "" 0 "" {TEXT -1 93 "Asimptota ramurii care tinde spre infin
it, are originea in centrul de greutate de coordonate:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "xCg:=(sum('Re(p[k,2])','k'=1..P)-su
m('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 261 "" 0 "" {TEXT -1 73 "Asimptota ramurii care tinde la i
nfinit 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 0 "" 0 "" {TEXT -1 
25 "Locul radacinilor pentru " }{TEXT 270 17 "reactiei pozitiva" }
{TEXT -1 6 " este:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "LocRad
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" "Curve 16" "Curve 17" "Curve 18" "Curve 19" }}}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 23 "Discutia de stabilitate" }{TEXT -1 0 "" }
}{PARA 262 "" 0 "" {TEXT -1 143 "Pe baza graficelor anterior, observam
 ca 2 din cele 3 ramuri sunt in semiplanul drept. Putem spune asadar c
a sistemul cu reactie pozitiva este " }{TEXT 271 22 "neconditionat ins
tabil" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 200 "Specificul exe
mplului este perechea de poli complex conjugati din semiplanul drept. \+
La reactie pozitiva, sistemul este instabil. La reactie negativa, sist
emul poate fi stabil la valori mari ale lui K." }}}}}{MARK "6 10 0 0" 
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