Бакалов В.П. Основы теории цепей (3-е издание, 2007).pdf (1095419), страница 84
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18.29496X1,R0(18.16)BpC0à)wá)0â)wÐèñ. 18.30Bê ( w ) = 2 arctgt ãð ( w ) =dB ( w )dw=X1,R0(18.17)2 R01 + ( X1 R0 )2×dX1.dw(18.18)Ôîðìóëû (18.16), (18.17) è (18.18) ïîêàçûâàþò, ÷òî ôàçî÷àñòîòíàÿ õàðàêòåðèñòèêà, ôàçîâàÿ ïîñòîÿííàÿ è õàðàêòåðèñòèêàãðóïïîâîãî âðåìåíè çàïàçäûâàíèÿ êîððåêòîðà çàâèñÿò òîëüêî îòâèäà äâóõïîëþñíèêà X1.Íà ïðàêòèêå èñïîëüçóþòñÿ òèïîâûå çâåíüÿ ïàññèâíûõ ôàçîâûõêîððåêòîðîâ ïåðâîãî è âòîðîãî ïîðÿäêîâ.Íà ðèñ. 18.30, à èçîáðàæåíà ñõåìà ôàçîâîãî êîððåêòîðà 1-ãîïîðÿäêà, â êîòîðîì äâóõïîëþñíèêîì Z1 ÿâëÿåòñÿ èíäóêòèâíîñòüZ1(p) = pL, à äâóõïîëþñíèêîì Z2 åìêîñòü Z2(p) = 1 / (pC).Îïåðàòîðíàÿ ïåðåäàòî÷íàÿ ôóíêöèÿ ýòîãî êîððåêòîðà â ñîîòâåòñòâèè ñ (18.14) èìååò âèä:Hê ( p ) =R0 - pLp - R0 Lp - a1==,R0 + pLp + R0 Lp + a1(18.19)ãäå a1 = R0 / L.Ðàáî÷àÿ ôàçîâàÿ ïîñòîÿííàÿ B(w) è ÃÂÏ â ñîîòâåòñòâèè ñ ôîðìóëàìè (18.17) è (18.18)Bê ( w ) = 2 arctg ( w a 1 ) ,tãð ( w ) =2a 12w +a 12.(18.20)(18.21)Ãðàôè÷åñêîå èçîáðàæåíèå äàííûõ õàðàêòåðèñòèê ïîêàçàíî íàðèñ.
18.30, á è â.Íà ðèñ. 18.31, à èçîáðàæåíà ñõåìà ôàçîâîãî êîððåêòîðà 2-ãîïîðÿäêà, ñ äâóõïîëþñíèêîì Z1, ñîñòîÿùèì èç ïîñëåäîâàòåëüíîãîñîåäèíåíèÿ ýëåìåíòîâ L1 è C1, ò. å. Z1(p) = pL1 + 1 / (pC1).Îïåðàòîðíàÿ ïåðåäàòî÷íàÿ ôóíêöèÿ òàêîãî êîððåêòîðà â ñîîòâåòñòâèè ñ (18.14) èìååò âèä:497w0 = 12pL1C1pw0à)w0á)â)Ðèñ. 18.31p 2 - ( w 0 Qï ) p + w 02R0 - pL1 - 1 ( pC1 ),=- 2Hê ( p ) =R0 + pL1 + 1 ( pC1 )p + ( w 0 Qï ) p + w 02ãäå w 0 = 1 / (L1C1), Qï = 1 / (w 0R0C1) äîáðîòíîñòü ïîëþñà ïåðåäàòî÷íîé ôóíêöèè.Êîìïëåêñíàÿ ïåðåäàòî÷íàÿ ôóíêöèÿ êîððåêòîðà ïîëó÷àåòñÿ ïðèp = jw:2H ê ( jw ) = -w 20 - w 2 - j ( w 0 Qï ) ww 20 - w 2 + j ( w 0 Qï ) w.(18.22)Ìîäóëü ôóíêöèè ðàâåí 1, à ðàáî÷àÿ ôàçîâàÿ ïîñòîÿííàÿ B(w) èÃÂÏ tãð (w) âû÷èñëÿþòñÿ â ñîîòâåòñòâèè ñ (18.17) è (18.18) ïî ôîðìóëàì:Bê ( w ) = p + 2 arctg éë Qï ( w 2 - w 02 ) w 0w ùû ;t ãð ( w ) =2w 0 Qï ( w 2 + w 02 )Qï2(w2-)2w 02+w 02 w 2.(18.23)(18.24)Ãðàôèêè çàâèñèìîñòåé B(w) è tãð (w) ôàçîâîãî êîððåêòîðà 2-ãîïîðÿäêà ïðèâåäåíû íà ðèñ.
18.31, á è â.Åñëè èçâåñòíû êîýôôèöèåíòû ïåðåäàòî÷íîé ôóíêöèè w 0, Qï è íàãðóçêà R 0, òî ïàðàìåòðû ýëåìåíòîâ êîððåêòîðà ðàññ÷èòûâàþòñÿ ïîôîðìóëàìC1 = 1 Qï R0w 0 ;(18.25)L1 = Qï R0 w 0 .(18.26)Ïðèìåð. Ôàçîâûé êîððåêòîð (ðèñ. 18.30, à) èìååò ýëåìåíòû L 1 = 100 ìÃí,R 0 = 500 Îì. Ðàññ÷èòàòü è ïîñòðîèòü ãðàôèêè ÷àñòîòíûõ çàâèñèìîñòåé ôàçîâîé ïîñòîÿííîé B ê (f) è ãðóïïîâîãî âðåìåíè ïðîõîæäåíèÿ t ãð (f) â äèàïàçîíå÷àñòîò îò 0 äî 10 êÃö.Ôàçîâàÿ õàðàêòåðèñòèêà B(w) ðàññ÷èòûâàåòñÿ ïî ôîðìóëå (18.20), ïîýòîìó:498Òàáëèöà 18.4f, êÃö01246810B ê , ðàä01,82,382,752,882,942,98t ãð , ìêñ4001559655382924Bê ( f ) = 2 arctg2p fL1.R0ÃÂÏ t ãð (w) ðàññ÷èòûâàåòñÿ ïî ôîðìóëå (18.21), ïîýòîìó:t ãð ( f ) =2R0 L1.4p 2f 2 L12 + R02Ïîäñòàâëÿÿ â âûðàæåíèÿ äëÿ Bê (f) è tãð (f) çíà÷åíèÿ L1 = 10 ×103 Ãí èR 0 = 500 Îì, ïîëó÷àåì:Bê ( f ) = 2 arctg2p fL12 × 3,14 × 100 × 10 -3f = 2 arctg1,256 × 10 -3 f ,= 2 arctgR05002R0 L12 × 500 × 100 × 10 -3100==t ãð ( f ) =.-6 22 2 2222224p f L1 + R04 × 3,14 × 100 × 10 f + 5000,394f + 25 × 10 4Ðåçóëüòàòû ðàñ÷åòà B ê (f) è t ãð (f) â äèàïàçîíå ÷àñòîò f = 0 ¸ 10 êÃö ïðèâåäåíû â òàáëèöå 18.4, à ãðàôèêè íà ðèñ.
18.32, à è á.Ïðèìåð. Ñõåìà ôàçîâîãî êîððåêòîðà ïðèâåäåíà íà ðèñ. 18.31, à. Ðàññ÷èòàòü è ïîñòðîèòü ãðàôèêè ÷àñòîòíûõ çàâèñèìîñòåé ôàçîâîé ïîñòîÿííîé Bê (f)è ÃÂÏ tãð (f) â äèàïàçîíå ÷àñòîò îò 0 äî 10 êÃö äëÿ äâóõ ñëó÷àåâ:1) R 0 = 600 Îì; L1 = 36 ìÃí, Ñ1 = 0,025 ìêÔ;2) R 0 = 600 Îì; L1 = 36 ìÃí, Ñ1 = 0,05 ìêÔ.Ôàçîâàÿ õàðàêòåðèñòèêà Bê (w) êîððåêòîðà ðàññ÷èòûâàåòñÿ ïî ôîðìóëå(18.23), à ÃÂÏ tãð (w) ïî ôîðìóëå (18.24), ïîýòîìó:Bê ( f ) = p + 2 arctg éë Qï ( 4p 2f 2 - w 02 ) w 0 2p f ùû ,t ãð ( f ) =2w 0 Qï ( 4p 2 f 2 + w 02 )Qï2 ( 4p 2f 2 - w 02 ) 2 +w 02 4p 2 f 2Âê( f ), ðàä,tãð( f ), ìêÑp400300p/220010002 46 8 10à)f, êÃö0246 8 10á)f, êÃöÐèñ.
18.32499ãäå w 02 = 1 / (L1C1), Qï = 1 / (w 0R0C1).Ðàññ÷èòàåì çíà÷åíèÿ w 02 è Qï äëÿ äâóõ ñëó÷àåâ çàäàíèÿ ïàðàìåòðîâ ýëåìåíòîâ êîððåêòîðà:1)2)11== 0,11 × 1010 (ðàä/ñ)2;-3-6L1C1 36 × 10 × 0,025 × 1011== 2.Qï =5w 0 R0C10,11 × 10 × 600 × 0,025 × 10 -6w 02 =1= 0,056 × 1010 (ðàä/ñ)2;-636 × 10 × 0,05 × 101Qï == 1,41 .50,056 × 10 × 600 × 0,05 × 10 -6w 02 =-3Ïîäñòàâëÿÿ çíà÷åíèÿ w 02 è Qï â âûðàæåíèÿ äëÿ ðàñ÷åòà Bê (f) è tãð (f), ðàññ÷èòûâàåì ýòè õàðàêòåðèñòèêè â äèàïàçîíå ÷àñòîò îò 0 äî 10 êÃö è çàíîñèì ðåçóëüòàòû ðàñ÷åòà â òàáëèöó 18.5 äëÿ ñëó÷àÿ 1) è â òàáëèöó 18.6 äëÿ ñëó÷àÿ 2).Ïîñêîëüêó ãðàôèê tãð (w) èìååò ìàêñèìóì (ðèñ. 18.31, â), òî äëÿ îïðåäåëåíèÿ ÷àñòîòû ýòîãî ìàêñèìóìà áåðåì ïðîèçâîäíóþ dtãð (w) è, ïðèðàâíÿâ åå êíóëþ, íàõîäèì:w max = w 0(18.27)4 - Qï2 - 1w04 - Qï2 - 1 = 0 äëÿ ïåðâîãî ñëó÷àÿ (Qï = 2) è fmax = 2,42 êÃö2päëÿ âòîðîãî ñëó÷àÿ (Qï =1,41). îáùåì ñëó÷àå àíàëèç âûðàæåíèÿ (18.27) ïîêàçûâàåò, ÷òî ïðè Qï 3ÃÂÏ èìååò ìàêñèìóì íà ÷àñòîòå f = 0, à ïðè Qï < 3 = 1,73 ìàêñèìóìÃÂÏ íà ÷àñòîòå fmax.Çíà÷åíèå tãð max ðàññ÷èòûâàåòñÿ ïî ôîðìóëå:èëè fmax =t ãð max =12Qï×w 0 4Qï-2 - 1 2Qï - 4Qï-2 - 1.(18.28)Äëÿ âòîðîãî ñëó÷àÿ, êîãäà Q = 1,41, èìååì tãð max = 144 ìêÑ.
Ñëåäóåò òàêæåîòìåòèòü, ÷òî ïðè Qï . 1 ôîðìóëû (18.27) è (18.28) ñóùåñòâåííî óïðîùàþòñÿ:w max = w 0 , t ãð max =4Qïw0.w0(18.29)Òàáëèöà 18.5f, êÃö0245,3810Bê , ðàä01,442,593,144,114,35tãð , ìêñ12010673,56034,826,2Òàáëèöà 18.6500f, êÃö012,423,766810Bê , ðàä00,772,03,144,354,875,19tãð , ìêñ12011714412057,630,818,9Âê, ðàätãð, ìêÑ2p15021p210015002468 10f, êÃöà)02468 10f, êÃöá)Ðèñ. 18.33Ãðàôèêè çàâèñèìîñòåé Bê (w) è tãð (w) äëÿ äâóõ ñëó÷àåâ ïðèâåäåíû íàðèñ. 18.33 (îáîçíà÷åíû öèôðàìè 1 è 2).Ìîñòîâàÿ ñõåìà íå âñåãäà óäîáíà â ðåàëèçàöèè, òàê êàê ÿâëÿåòñÿ óðàâíîâåøåííîé.
Ñóùåñòâóåò ðÿä ýêâèâàëåíòíûõ ñõåì â âèäåíåóðàâíîâåøåííîé ñõåìû, êàê ïîêàçàíî íà ðèñ. 18.34. Çàìåòèì,÷òî íà ïðàêòèêå äîáðîòíîñòü ïîëþñà áîëüøå åäèíèöû è ïîýòîìó÷àùå èñïîëüçóåòñÿ ñõåìà ðèñ. 18.34, à, ÷òî óäîáíî, òàê êàê îíà íåñîäåðæèò ñâÿçàííûõ èíäóêòèâíîñòåé ñ çàäàííûì êîýôôèöèåíòîìñâÿçè. Íåóðàâíîâåøåííûå ñõåìû ïî ñðàâíåíèþ ñ ìîñòîâûìè ñîäåðæàò âäâîå ìåíüøå ýëåìåíòîâ.Àêòèâíûå êîððåêòîðû.
Ïîìèìî ïàññèâíûõ ôàçîâûõ êîððåêòîðîâ ïðèìåíÿþò àêòèâíûå ôàçîâûå êîððåêòîðû. Êðîìå ïàññèâíûõRC èëè RLC-ýëåìåíòîâ ñõåìû àêòèâíûõ êîððåêòîðîâ ñîäåðæàò îïåðàöèîííûå óñèëèòåëè. Ñóùåñòâóþò àêòèâíûå ôàçîâûå çâåíüÿ 1-ãîè 2-ãî ïîðÿäêîâ. Íà ðèñ. 18.35 ïðèâåäåíà ñõåìà ôèëüòðîâîãî çâåíàíà îïåðàöèîííîì óñèëèòåëå. Ïåðåäàòî÷íàÿ ôóíêöèÿ ýòîãî çâåíàâû÷èñëÿåòñÿ ïî ôîðìóëå:Hê ( p ) = -p - 1 ( R1C )p - a1=,p + 1 ( R1C )p + a1à)(18.30)á)Ðèñ.
18.34501RR1tãðÂê(w)Rpa1¥a1a1¢ > a1a1a1¢ > a1U2U10Ðèñ. 18.35wà)0á)wÐèñ. 18.36ãäå a 1 = 1 R1C .Âûðàæåíèå (18.30) àíàëîãè÷íî ôîðìóëå äëÿ ðàñ÷åòà ïåðåäàòî÷íîé ôóíêöèè ïàññèâíîãî ôàçîâîãî êîððåêòîðà (18.19), ò. å.ñõåìà, ïðèâåäåííàÿ íà ðèñ. 18.35, ýòî àêòèâíûé êîððåêòîð 1-ãîïîðÿäêà.Ôàçîâûå õàðàêòåðèñòèêè B(w) è ÃÂÏ äàííîãî çâåíà, òàêæå êàêó ïàññèâíîãî êîððåêòîðà 1-ãî ïîðÿäêà, âû÷èñëÿþòñÿ ïî ôîðìóëàìBê ( w ) = 2 arctg ( w a 1 ) ,2at ãð ( w ) = 2 1 2 .w + a1Ãðàôèê Bê (w) ìîíîòîííî íàðàñòàåò îò Bê (0) = 0 äî Bê (¥) = p, àãðàôèê tãð (w) ìîíîòîííî óáûâàåò îò tãð (0) = 2 / a1 äî tãð (¥) = 0.Íà ðèñ. 18.36 ïîêàçàíû ãðàôèêè Bê (w) è tãð (w), ïîñòðîåííûå äëÿðàçíûõ çíà÷åíèé a1 àêòèâíîãî êîððåêòîðà 1-ãî ïîðÿäêà.Íà ðèñ. 18.37 ïðèâåäåíà åùå îäíà ñõåìà àêòèâíîãî ôàçîâîãîêîððåêòîðà, òàêæå ïîñòðîåííàÿ íà îñíîâå àêòèâíîãî ôèëüòðîâîãîçâåíà.
Åñëè â ñõåìå ðèñ. 18.37 çàäàòü R3 = nR2, R4 = nR2 / (n 1),n > 1, òî ïåðåäàòî÷íàÿ ôóíêöèÿ, ðàññ÷èòàííàÿ, íàïðèìåð, ñ ïîìîùüþ ìåòîäà óçëîâûõ íàïðÿæåíèé, áóäåò èìåòü âèä:H (p) = -R1 - Z ( p )R1 + Z ( p ).(18.31)Ýòî ïåðåäàòî÷íàÿ ôóíêöèÿ ôàçîâîãî êîððåêòîðà (ñðàâíè ñ ôîðìóëîé (18.14)).Åñëè â êà÷åñòâå äâóõïîëþñíèêà Z âûáðàòü åìêîñòü, òî ïåðåäàòî÷íàÿ ôóíêöèÿ (18.31) ïðèíèìàåò âèä (18.30):Hê ( p ) = -R1 - 1 pCp - 1 R1Cp - a1==,R1 + 1 pCp + 1 R1Cp + a1ò.
å. ñõåìà íà ðèñ. 18.37 ýòî ñõåìà ôàçîâîãî êîððåêòîðà 1-ãî ïîðÿäêà.Êîãäà â êà÷åñòâå äâóõïîëþñíèêà Z èñïîëüçóåòñÿ ïîñëåäîâàòåëüíûé LC-êîíòóð, òî ïîëó÷àåòñÿ ïåðåäàòî÷íàÿ ôóíêöèÿ ôàçîâîãîêîððåêòîðà 2-ãî ïîðÿäêà:502Âê(w)R3R12pQï¥ZU1tãðpU2R1QïQï¢Qï¢ < Q ïQï¢Qï¢ < Q ïR40à)Ðèñ. 18.37w0á)wÐèñ.
18.38R1C1+p 2 - ( w 0 Qï ) p + w 02LCLCHê ( p ) = =- 2,2R1C12p+wQp+w( 0 ï)0p +p+LC LC2ãäå w 0 = 1 / (LC), Qï = 1 / (w 0R1C) äîáðîòíîñòü ïîëþñà ïåðåäàòî÷íîé ôóíêöèè.Ãðàôèêè ÷àñòîòíûõ çàâèñèìîñòåé Bê (w) è tãð (w) äàííîãî êîððåêòîðà, ïîëó÷åííûå äëÿ ðàçíûõ çíà÷åíèé Qï, ïðèâåäåíû íàðèñ. 18.38.Õîòÿ àêòèâíûå ARZ-ôàçîâûå êîððåêòîðû èìåþò èíäóêòèâíîñòü,íî ïðåèìóùåñòâîì èõ ïî ñðàâíåíèþ ñ ïàññèâíûìè êîððåêòîðàìèÿâëÿåòñÿ ìåíüøåå êîëè÷åñòâî ýëåìåíòîâ ïðè òîì æå ïîðÿäêå ïåðåäàòî÷íûõ ôóíêöèé.p2 - pÏðèìåð. Îïðåäåëèòü ïåðåäàòî÷íóþ ôóíêöèþ ôàçîâîãî êîððåêòîðà, ïîñòðîåííîãî ïî ñõåìå ðèñ. 18.35, â êîòîðîé â êà÷åñòâå äâóõïîëþñíèêà Z èñïîëüçóåòñÿ ïàðàëëåëüíûé LC-êîíòóð.
Ðàññ÷èòàòü è ïîñòðîèòü êà÷åñòâåííî ÷àñòîòíóþ õàðàêòåðèñòèêó ÃÂÏ tãð (f) êîððåêòîðà â äèàïàçîíå ÷àñòîò îò 0 äî5 êÃö äëÿ ýëåìåíòîâ öåïè R1 = 37,5 Îì, L = 36 ìÃí, C = 1,6 ìêÔ.Íàéäåì ñîïðîòèâëåíèå Z(p) ïàðàëëåëüíîãî LC-êîíòóðà:pL × 1 pCpL= 2.pL + 1 pC p LC + 1Z( p) =Ïîäñòàâèâ Z(p) â ôîðìóëó (18.31), ïîëó÷èì ïåðåäàòî÷íóþ ôóíêöèþ ôàçîâîãî êîððåêòîðà:p 2 - ( w 0 Qï ) p + w 02R1 - Z ( p )R1 - pL ( p 2LC + 1 )H(p) = ==- 2,R1 + Z ( p )p + ( w 0 Qï ) p + w 02R1 + pL ( p 2LC + 1 )ãäå w 02 = 1 / (LC), Qï = w 0R1C.ÃÂÏ ðàññ÷èòûâàåòñÿ ïî ôîðìóëå (18.24), â êîòîðîé w = 2pf,t ãð ( f ) =2w 0 Qï ( 4p 2 f 2 + w 02 )Q2ï( 4p22f -w 02)+w 02 ×4p 2 f 2.Íàõîäèì çíà÷åíèÿ w 02 è Qï:503tãð, ìÑ11==-3LC 36 × 10 × 1,6 × 10 -62= 17,36 × 10 6 ( ðàä ñ ) ,w 02 =432w 0 = 0,416 × 10 4 ðàä/ñ,1f0 = 0,662 êÃö,01234Qï = w 0 R1C == 0,416 × 10 4 × 37,5 × 1,6 × 10 -6 = 0,25.5 f, êÃöÐèñ. 18.39Ïîñêîëüêó Qï < 3 , òî íàõîäèì çíà÷åíèÿ w max è tãð max ïî ôîðìóëàì (18.27)è (18.28):w max = w 0fmax =4 - Qï2 - 1 = 0,416 × 10 44 - 0,25 2 - 1 = 4,16 × 10 3 (ðàä/ñ) ,w max= 0,662 êÃö, tãð max = 3,7 ìÑ.2pÐàññ÷èòûâàåì çíà÷åíèÿ tãð (f) íà ÷àñòîòàõ f1 = 0 è f2 = 5 êÃö ïî ôîðìóëå(18.24).
Ïîëó÷àåì tãð (f1) = 1,92 ìÑ è tãð (f2) = 0,12 ìÑ.Ãðàôèê çàâèñèìîñòè tãð (f) ïðèâåäåí íà ðèñ. 18.39.Ñèíòåç ôàçîâûõ êîððåêòîðîâ. Ïðè ñèíòåçå ôàçîâûõ êîððåêòîðîâ çàäàþòñÿ õàðàêòåðèñòèêà ÃÂÏ êîððåêòèðóåìîé öåïè, ñîïðîòèâëåíèå íàãðóçêè R0, òî÷íîñòü êîððåêöèè è äèàïàçîí ÷àñòîòw í ...
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