Диссертация (1145368), страница 18
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äëÿ g(t) ≡ 0), L êîýôôèöèåíò óñèëåíèÿÏÃ.Ñëåäóþùèé ïðèìåð ïîêàçûâàåò âàæíîñòü àíàëèòè÷åñêîãî èññëåäîâàíèÿóñòîé÷èâîñòè. íåì ïîêàçàíî, ÷òî èñïîëüçîâàíèå çíà÷åíèé ïàðàìåòðîâìîäåëèðîâàíèÿ ïî óìîë÷àíèþ ìîæåò ïðèâîäèòü ê êà÷åñòâåííî íåâåðíûìâûâîäàì î ãëîáàëüíîé óñòîé÷èâîñòè ÔÀÏ, è, íàïðèìåð, î ïîëîñå çàõâàòà.100001scarrier frequency1Integrator3sinTr igonometricFunction2Pr oduct0.5x' = Ax+B uy = Cx+D uG ain1L oop filter1thetacosTr igonometricFunction3Add1Pr oduct1cosvco phaseTr igonometricFunction4sinTr igonometricFunction51sIntegrator2feedback2250* 2VC O input gain110000-(2*89.27478)vco free-running frequency1Ðèñóíîê 3.16: Model of two-phase PLL in MatLab Simulink136Ðàññìîòðèì ôèëüòð ñ ïåðåäàòî÷íîé ôóíêöèåé H(s) =τ2 = 0.0185.
è ñîîòâåòñòâóþùèìè ïàðàìåòðàìè A =1τ1 +τ2 ,h=τ2τ1 +τ2 .1+sτ21+s(τ1 +τ2 ) , τ1 = 0.0448,12− τ1 +τ, b = 1 − τ1τ+τ,c=22Ìîäåëü äâóõôàçíîé ÔÀÏ â MatLab ïîêàçàíà íà Ðèñ. 3.16 (áîëååïîäðîáíîå îáñóæäåíèå ìîäåëèðîâàíèÿ ÔÀÏ â MatLab ïðèâåäåíî, íàïðèìåð,â [64, 277, 278]).Íà Ðèñ. 3.16 èñïîëüçîâàí áëîê Looplter, ÷òîáû ó÷åñòü íà÷àëüíîå ñîñòîÿíèåôèëüòðà x(0) (çäåñü A → A, b → B , c → C , h → D); íà÷àëüíàÿ ðàçíîñòü ôàçθΔ (0) óñòàíàâëèâàåòñÿ â ñâîéñòâå initialdataáëîêàIntergator.Íà Ðèñ. 3.17 ïðåäñòàâëåíû ðåçóëüòàòû ìîäåëèðîâàíèÿ äâóõôàçíîé ÔÀÏ:ïðè çíà÷åíèè "relative tolerance" ðàâíîì 1e-3 èëè ìåíüøå ìîäåëü íåâòÿãèâàåòñÿ â ñèíõðîíèçì (÷åðíûé öâåò), â òî âðåìÿ êàê ïðè ìîäåëèðîâàíèèäëÿ ïðåäóñòàíîâëåííûõ ñòàíäàðòíûõ ïàðàìåòðîâ (relative tolerance èìååòçíà÷åíèå auto) ìîäåëü âòÿãèâàåòñÿ â ñèíõðîíèçì (êðàñíûé öâåò).Çäåñü÷àñòîòà âõîäíîãî ñèãíàëà 10000, ñâîáîäíàÿ ÷àñòîòà Ïà ω2free = 10000 − 178.9,êîýôôèöèåíò óñèëåíèÿ Ïà L = 500, íà÷àëüíîå ñîñòîÿíèå ôèëüòðà x0 = 0.131810è íà÷àëüíàÿ ðàñôàçèðîâêà θΔ (0) = 0.Ïðîâåäåì àíàëîãè÷íîå ìîäåëèðîâàíèå â èíæåíåðíîì ïàêåòå SIMetrix,êîòîðûé ÿâëÿåòñÿ îäíîé èç êîììåð÷åñêèõ âåðñèé SPICE.Ðàññìîòðèì SIMetrix ðåàëèçàöèþ äâóõôàçíîé ÔÀÏ íà Ðèñ.
3.18. Âõîäíîéñèãíàë è âûõîä ôàçîâîãî âðàùàòåëÿ íà Ðèñ. 3.14 ìîäåëèðóþòñÿ ïðèïîìîùè ñèíóñîèäàëüíîãî èñòî÷íèêà íàïðÿæåíèÿ V1 (ïàðàìåòð ÷àñòîòû èìååòçíà÷åíèå 1.5915494k ) è V2 (ïàðàìåòð ÷àñòîòû èìååò çíà÷åíèå 1.5915494kè ôàçû 90) (sin_input and cos_input).Ïåðåìíîæèòåëü êîìïëåêñíûõ÷èñåë íà Ðèñ. 3.15 ìîäåëèðóåòñÿ ïðè ïîìîùè äâóõ èñòî÷íèêîâ ARB1 èARB2 ñ õàðàêòåðèñòèêàìè V (N 1)∗ V (N 2).×òîáû ïðîèçâåñòè âû÷èòàíèåèñïîëüçóåòñÿ Voltage Controlled Voltage Source (E3). Êîýôôèöèåíò óñèëåíèÿÔÄ (E5) ðàâåí12.Ôèëüòð (Loop Filter) íà Ðèñ.
3.14 ðåàëèçîâàí êàêïàññèâíûé ïðîïîðöèîíàëüíî-èíòåãðèðóþùèé ôèëüòð ñ ñîïðîòèâëåíèåì R2,êîíäåíñàòîðîì C2, è ñîïðîòèâëåíèåì R1.Óñèëèòåëü Ïà (E6) ðàâåí −500.Ñîáñòâåííàÿ ÷àñòîòà (DC èñòî÷íèê íàïðÿæåíèÿ V3) 9.8211k . Óïðàâëÿåìûéíàïðÿæåíèåì èñòî÷íèê íàïðÿæåíèÿ E2 ñóììèðóåò ñîáñòâåííóþ ÷àñòîòó ÏÃè óïðàâëÿþùèé ñèãíàë îò E6. Ñîïðîòèâëåíèå R1b1 (10u), êîíäåíñàòîð C110 Ïî÷òèäëÿ âñåõ íà÷àëüíûõ ñîñòîÿíèé èç èíòåðâàëà[1, 2]ìîäåëèðîâàíèå äàåò îäèíàêîâûé ðåçóëüòàò.137relative tolerance `auto`relative tolerance `1e-3`g(t)tÐèñóíîê 3.17: Ìîäåëèðîâàíèå äâóõôàçíîé ÔÀÏ. Âûõîä ôèëüòðà g(t) äëÿíà÷àëüíûõ äàííûõ x0 = 0.1318, θΔ (0) = 0 ïîëó÷åííûé äëÿ auto relativetolerance (êðàñíûé öâåò) âòÿãèâàåòñÿ â ñèíõðîíèçì, äëÿ çíà÷åíèÿ relativetolerance 1e-3(çåëåíûé öâåò) íå âòÿãèâàåòñÿ â ñèíõðîíèçì.(5G), è óñèëèòåëü E1(50k ) èç èíòåãðàòîðà11.
Ôîðìà ñèãíàëà Ïà îïðåäåëÿåòñÿáëîêîì ARB3 ñ õàðàêòåðèñòèêîé sin(V (N 1)) è áëîêîì ARB4 ñ õàðàêòåðèñòèêîécos(V (N 1))).Âíèçó ïðèâåäåí Netlist, ñãåíåðèðîâàííûé äëÿ ìîäåëè SIMetrix:∗#SIMETRIXV1 sin_Input 0 0 S i n e ( 0 1 1 . 5 9 1 5 4 9 4 k 0 0 )V2 cos_input 0 0 S i n e ( 0 1 1 . 5 9 1 5 4 9 4 k − 157.03518u 0 )V3 vco_frequency 0 9 . 8 2 1 1 kR1 C2_N 0 1 . 8 5 kR2 f i l t e r _ o u t PD_output 4 . 4 8 kX$ARB1 sin_Input vco_cos_output ARB1_OUT11 Àíàëîãè÷íûåðåçóëüòàòû ïîëó÷åíû äëÿ ñîïðîòèâëåíèÿ 10K è åìêîñòè êîíäåíñàòîðà 5138sin_InputV(N1)*V(N2)ARB1PD_output14.48kN1 OUTN20 Sine(0 1 1.5915494k 0 0)V1filter_out500mR2E3E5+filter_outC210u IC=185mR11.85kvco_frequencycos_inputV(N1)*V(N2)ARB20 Sine(0 1 1.5915494k -157.03518u 0)V2N1 OUTN2vco_sin_output9.8211kV3sin(V(N1))ARB3integrator_outintegrator_inOUT N1150k10uvco_cos_outputcos(V(N1))ARB4-500R1b1OUT N1E1C15GE2E6Ðèñóíîê 3.18: SPICE ðåàëèçàöèÿ äâóõôàçíîé ÔÀÏ â SIMetrix$$arbsourceARB1 pinnames : N1 N2 OUT.
s u b c k t $$arbsourceARB1 N1 N2 OUTB1 OUT 0 V=V(N1) ∗V(N2). endsX$ARB2 cos_input vco_sin_output E3_CN$$arbsourceARB2 pinnames : N1 N2 OUT. s u b c k t $$arbsourceARB2 N1 N2 OUTB1 OUT 0 V=V(N1) ∗V(N2). endsX$ARB3 i n t e g r a t o r _ o u t vco_sin_output$$arbsourceARB3 pinnames : N1 OUT. s u b c k t $$arbsourceARB3 N1 OUTB1 OUT 0 V=s i n (V(N1 ) ). endsX$ARB4 i n t e g r a t o r _ o u t vco_cos_output$$arbsourceARB4 pinnames : N1 OUT. s u b c k t $$arbsourceARB4 N1 OUTB1 OUT 0 V=c o s (V(N1 ) ). endsE1 i n t e g r a t o r _ o u t 0 E1_CP 0 50 kE2 i n t e g r a t o r _ i n 0 vco_frequency E2_CN 1C1 E1_CP 0 5G139C2 f i l t e r _ o u t C2_N 10u IC=185m BRANCH={IF (ANALYSIS=2 ,1 ,0)}E3 E3_P 0 ARB1_OUT E3_CN 1E5 PD_output 0 E3_P 0 500mE6 E2_CN 0 f i l t e r _ o u t 0 −500R1b1 i n t e g r a t o r _ i n E1_CP 10u.GRAPH f i l t e r _ o u t c u r v e L a b e l= f i l t e r _ o u tnowarn=t r u e y l o g=auto x l o g=auto d i s a b l e d=f a l s e.TRAN 0 5 0 1m UIC.
OPTIONS minTimeStep=1m+tnom=27Íà Ðèñ. 3.19 ïîêàçàíû ðåçóëüòàòû ìîäåëèðîâàíèÿ â SPICE, êîòîðûåñîîòâåòñòâóþò ðåçóëüòàòàì ìîäåëèðîâàíèÿ â MatLab Simulink (íà Ðèñ. 3.17).Äëÿ ïàðàìåòðîâ ìîäåëèðîâàíèÿ ïî óìîë÷àíèþ â SIMetrix ìîäåëü âòÿãèâàåòñÿâ ñèíõðîíèçì (êðàñíàÿ êðèâàÿ).Îäíàêî äëÿ øàãà äèñêðåòèçàöèè 1mìîäåëèðîâàíèå ïîêàçûâàåò êîëåáàíèÿ (çåëåíàÿ êðèâàÿ).filter_out / mV3002001000-10001234Time/Secs51Secs/divÐèñóíîê 3.19: Ñêðûòûå êîëåáàíèÿ â SPICEÒåïåðü ðàññìîòðèì ñîîòâåòñòâóþùóþ ìàòåìàòè÷åñêóþ ìîäåëü.
Äâóõôàçíàÿìîäåëü ÔÀÏ îïèñûâàåòñÿ óðàâíåíèÿìè (3.48), (3.50) è (3.49) èç êîòîðûõ140ïîëó÷àåòñÿ ñëåäóþùàÿ ñèñòåìà óðàâíåíèébsin(θΔ ),2Lhsin(θΔ ),θ̇Δ = ωΔ − Lc∗ x −2θΔ (t) = θ1 (t) − θ2 (t), ωΔ = ω1 − ω2f ree .ẋ = Ax +Äëÿ H(s) =1+sτ21+s(τ1 +τ2 )(3.51)ñèñòåìà (3.51) ïðèíèìàåò âèä−1τ21x + (1 −) sin(θΔ ),τ1 + τ2τ1 + τ2 21τ2 Lsin(θΔ ).x −θ̇Δ = ωΔ − Lτ1 + τ2τ1 + τ 2 2ẋ =(3.52)Ñîñòîÿíèÿ ðàâíîâåñèÿ (3.52) îïðåäåëÿþòñÿ èç ðàâåíñòâτ1sin(θΔ ),2ωΔsin(θeq ) = 2 .Lxeq =(3.53)Äëÿ τ1 = 0.0448, L = 500 è ωΔ = 178.9 ïîëó÷èìxeq = 0.016,kθeq = (−1) 0.7975 + πk,(3.54)k ∈ N.Ðàññìîòðèì ôàçîâûé ïîðòðåò ñîîòâåòñòâóþùèé ìîäåëè â ïðîñòðàíñòâåôàç ñèãíàëîâ (ñì.Ðèñ.
3.20).Ñïëîøíàÿ ñèíÿÿ ëèíèÿ íà Ðèñ. 3.20ñîîòâåòñòâóåò òðàåêòîðèè ñ íà÷àëüíûì ñîñòîÿíèåì ôèëüòðà x(0) = 0.005è íóëåâûì ñäâèãîì ôàçû ÏÃ. Ýòà êðèâàÿ ïðèòÿãèâàåòñÿ ê ïåðèîäè÷åñêîéòðàåêòîðèè è ñëåäîâàòåëüíî íå âòÿãèâàåòñÿ â ñèíõðîíèçì. Âñå òðàåêòîðèè ïîäñèíåé ëèíèåé (ñì., íàïðèìåð, çåëåíóþ òðàåêòîðèþ ñ íà÷àëüíûì ñîñòîÿíèåìôèëüòðà x(0) = 0) òàêæå ñòðåìÿòñÿ ê òîé æå ïåðèîäè÷åñêîé òðàåêòîðèè.Ñïëîøíàÿêðàñíàÿëèíèÿñîîòâåòñòâóåòòðàåêòîðèèñíà÷àëüíûìñîñòîÿíèåì ôèëüòðà 0.00555 è íóëåâûì íà÷àëüíûì ñäâèãîì ôàçû ÏÃ.Ýòà òðàåêòîðèÿ ëåæèò âûøå íåóñòîé÷èâîé ïåðèîäè÷åñêîé òðàåêòîðèè èïðèòÿãèâàåòñÿ ê îäíîìó èç ñîñòîÿíèé ðàâíîâåñèÿ.âòÿãèâàåòñÿ â ñèíõðîíèçì. ýòîì ñëó÷àå ñèñòåìà141Xequilibrium point0.0150.010.00500204060Ðèñóíîê 3.20: Ôàçîâûé ïîðòðåò ñ óñòîé÷èâûì è íåóñòîé÷èâûì ïðåäåëüíûìèöèêëàìèreal trajectoriessimulationXÐèñóíîê 3.21: Phase portrait of the classical PLL with stable and unstableperiodic trajectories142Âñå òðàåêòîðèè ìåæäó óñòîé÷èâûì è íåóñòîé÷èâûì ïðåäåëüíûìè öèêëàìèñòðåìÿòñÿ ê óñòîé÷èâîìó öèêëó.
Åñëè çàçîð ìåæäó öèêëàìè ìåíüøå øàãàèíòåãðèðîâàíèÿ, òî ÷èñëåííàÿ ïðîöåäóðà ìîæåò ïåðåñêî÷èòü ÷åðåç óñòîé÷èâûéöèêë (ñì. òàêæå îáñóæäåíèå ïàðàìåòðîâ èíòåãðèðîâàíèÿ ýëåêòðîííûõ ñõåìâ [43, 258]). Ýòîò ñëó÷àé ñîîòâåòñòâóåò ñóùåñòâîâàíèþ ñêðûòîãî êîëåáàíèÿ(óñòîé÷èâûé öèêë) è ðîæäåíèþ ïîëóóñòîé÷èâîé òðàåêòîðèè [20, 119, 180, 235].3.4 Âûâîä ìàòåìàòè÷åñêèõ ìîäåëåé êëàññè÷åñêîéñõåìû ÊîñòàñàÑõåìà Êîñòàñà áûëà èçîáðåòåíà â 50õ ãîäàõ ïðîøëîãî âåêà [83, 84] èïðåäíàçíà÷åíà äëÿ äåìîäóëÿöèè è âîññòàíîâëåíèÿ íåñóùåé ñèãíàëîâ.
Ñõåìàøèðîêî ïðèìåíÿåòñÿ â ñèñòåìàõ ñâÿçè è ñïóòíèêîâîé íàâèãàöèè (íàïðèìåð,GPS è ÃËÎÍÀÑ) [123, 147, 239].Ðàáîòà êëàññè÷åñêîé ñõåìû Êîñòàñà îïèñûâàåòñÿ íåëèíåéíîé íåàâòîíîìíîéñèñòåìîé äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ðàçðûâíîé ïðàâîé ÷àñòüþ, ÷èñëåííîåèññëåäîâàíèå êîòîðîé ÿâëÿåòñÿ òðóäíîé çàäà÷åé, ïîòîìó ÷òî íåîáõîäèìîðàññìàòðèâàòü îäíîâðåìåííî áûñòðî ìåíÿþùèåñÿ ñèãíàëû ãåíåðàòîðîâ èìåäëåííî ìåíÿþùèéñÿ ñèãíàë îøèáêè.Ïîýòîìó èíæåíåðàìè øèðîêîïðèìåíÿåòñÿ êëàññè÷åñêàÿ óïðîùåííàÿ ìîäåëü ðàáîòû ñõåìû, îïèñûâàåìàÿàâòîíîìíîé ñèñòåìîé äèôôåðåíöèàëüíûõ óðàâíåíèé ñ ãëàäêîé ïðàâîé ÷àñòüþ[57, 83, 110, 213, 275]. Äàëåå áóäåò ïîêàçàíî, ÷òî ïðåäïîëîæåíèÿ, èñïîëüçóåìûåèíæåíåðàìè äëÿ âûâîäà óïðîùåííîé ìîäåëè, âîîáùå ãîâîðÿ ìîãóò íåâûïîëíÿòüñÿ è òðåáóþò äîïîëíèòåëüíûõ îáîñíîâàíèé.Äàëåå, ñëåäóÿ [227],ïðèâåäåí âûâîä è ñòðîãîå îáîñíîâàíèå íåëèíåéíîé ìîäåëè ñõåìû Êîñòàñà.Ðàññìîòðèì êëàññè÷åñêóþ ñõåìó Êîñòàñà [83, 84] íà óðîâíå ôèçè÷åñêîéðåàëèçàöèè â ïðîñòðàíñòâå ñèãíàëîâ (ñì.
Ðèñ. 3.22).Çäåñü âõîäîì ñõåìû ÿâëÿåòñÿ ïðîèçâåäåíèå ñèãíàëà äàííûõ m(t) ∈ {+1, −1}è íåcóùåé sin(θ1 (t)); sin(θ2 (t)) ñèãíàë ïîäñòðàèâàåìîãî ãåíåðàòîðà (ÏÃ);θ1,2 (t) ôàçû; θΔ (t) = θ1 (t) − θ2 (t) ðàçíîñòü ôàç ñèãíàëîâ; ⊗ ïåðåìíîæèòåëü; ÔÍ×1, ÔÍ×2 ôèëüòðû íèçêèõ ÷àñòîò; Ô ëèíåéíûéôèëüòð; áëîê 90o ñäâèãàåò ôàçó ñèãíàëà íà π2 ; g1,2 (t) âûõîäû ôèëüòðîâ íèçêèõ÷àñòîò; g(t) ñèãíàë íà óïðàâëÿþùåì âõîäå ÏÃ.
Íà âõîäû ôèëüòðîâ íèçêèõ143ɜɯɨɞPWVLQș1(t))=ɞɚɧɧɵɟij1(t)= 1 PWFRVșǻ(t))-cos(ș1Wș2(t)))2sin(ș2(t))g(t)ɉȽɎɎɇɑg1(t)ijWoɧɟɫɭɳɚɹ90ij2(t)= 1 PWVLQșǻ(t))+sin(ș1Wș2(t)))2Ɏɇɑg2(t)Ðèñóíîê 3.22: Ñõåìà Êîñòàñà â ïðîñòðàíñòâå ñèãíàëîâ íà óðîâíå ôèçè÷åñêîéðåàëèçàöèè.÷àñòîò ïîñòóïàþò ïðîèçâåäåíèÿ1m(t) cos(θΔ (t)) − m(t) cos(θ1 (t) + θ2 (t)) ,21ϕ2 (t) = m(t) sin(θ1 (t)) cos(θ2 (t)) = m(t) sin(θΔ (t)) + m(t) sin(θ1 (t) + θ2 (t)) .2ϕ1 (t) = m(t) sin(θ1 (t)) sin(θ2 (t)) =(3.55)Ñîîòíîøåíèÿ ìåæäó âõîäàìè ϕ(t), ϕ1,2(t) è âûõîäàìè g(t), g1,2(t)ñîîòâåòñòâåííî äëÿ ôèëüòðà (Ô) è íèçêî÷àñòîòíûõ ôèëüòðîâ (ÔÍ×1, ÔÍ×2)èìåþò âèä tdx= Ax + bϕ(t), g(t) = c∗ x + hϕ(t) = hϕ(t) + c∗ eAt x(0) + ñ∗ eA(t−τ ) bϕ(τ )dτ,0dtdx1= A1 x1 + b1 ϕ1 (t), g1 (t) = c∗1 x1 ,dtdx2= A2 x2 + b2 ϕ2 (t), g2 (t) = c∗2 x2 ,dt(3.56)ãäå A, A1,2 ïîñòîÿííûå óñòîé÷èâûå ìàòðèöû; x(t), x1,2(t) âåêòîðûñîñòîÿíèÿ ôèëüòðîâ; b, b1,2, c, c1,2, ïîñòîÿííûå âåêòîðû; h êîíñòàíòà.
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