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Îïèñûâàåòñÿ óðàâíåíèå èçìåðåíèÿ îñåâîãî äèïîëüíî- ãî ìàãíèòíîãî ìîìåíòà ìîäóëüíîé ìàãíèòîìåòðè÷åñ- êîé ñèñòåìîé èç ÷åòûðåõ îñåâûõ äàò÷èêîâ. Ïðèâîäèòñÿ ìàòåìàòè÷åñêàÿ ìîäåëü ìóëüòèïîëüíîé ïîãðåøíîñòè. 1. Àêòóàëüíîñòü èññëåäîâàíèÿ Ýëåêòðî- è ðàäèîòåõíè÷åñêèå óñòðîéñòâà ÿâëÿþòñÿ îñíîâíûìè èñòî÷íèêàìè ïîìåõîíåñóùèõ âíåøíèõ ìàãíèòíûõ ïîëåé (ÂÌÏ), êîòîðûå íàðóøàþò íîð- ìàëüíîå ôóíêöèîíèðîâàíèå ðàçëè÷íûõ ìàãíèòî- ÷óâñòâèòåëüíûõ óñòðîéñòâ è îêàçûâàþò íåæåëà- òåëüíîå âîçäåéñòâèå íà áèîëîãè÷åñêèå îáúåêòû, íàõîäÿùèåñÿ â çîíå âëèÿíèÿ ýòèõ ïîëåé [1, 2]. Ýòî òðåáóåò ðåøåíèÿ çàäà÷è êîíòðîëÿ ýëåêòðîìàãíèò- íîé îáñòàíîâêè â çîíàõ ýíåðãåòè÷åñêèõ è ïðîìûø- ëåííûõ îáúåêòîâ. Íåîáõîäèìîñòü ðåãëàìåíòàöèè óðîâíÿ íàïðÿæåí- íîñòè ìàãíèòíîãî ïîëÿ â îêðóæàþùåì ïðîñòðàí- ñòâå [3, 4] ñóùåñòâåííî ïîâûñèëà òðåáîâàíèÿ ê òî÷íîñòè êîíòðîëÿ ÂÌÏ èñòî÷íèêîâ, ê îöåíêå ïîãðåøíîñòè ìàãíèòîìåòðè÷åñêèõ ìåòîäîâ è ñðåäñòâ èçìåðåíèÿ ïàðàìåòðîâ ÂÌÏ òåõíè÷åñêèõ îáúåêòîâ. Íà çíà÷èòåëüíûõ ðàññòîÿíèÿõ îò èñòî÷íèêà åãî ÂÌÏ â îêðóæàþùåì ïðîñòðàíñòâå íîñèò äèïîëü- íûé õàðàêòåð. Òàêîå ïîëå ïðîïîðöèîíàëüíî ýêâè- âàëåíòíîìó äèïîëüíîìó ìîìåíòó, êîòîðûé ÿâëÿåò- ñÿ ïîñòîÿííîé âåëè÷èíîé è â îòëè÷èå îò íàïðÿæåí- íîñòè ìàãíèòíîãî ïîëÿ íå çàâèñèò îò êîîðäèíàò òî÷åê íàáëþäåíèÿ. Ýòî ïîçâîëÿåò ïî èçìåðåííûì çíà÷åíèÿì êîìïîíåíò äèïîëüíîãî ìàãíèòíîãî ìî- ìåíòà èñòî÷íèêà ëåãêî îïðåäåëèòü íàïðÿæåííîñòü äèïîëüíîé ñîñòàâëÿþùåé ìàãíèòíîãî ïîëÿ â çàäàí- íûõ òî÷êàõ âíåøíåãî ïðîñòðàíñòâà. Èçìåðåíèå êîìïîíåíò äèïîëüíîãî ìàãíèòíîãî ìîìåíòà òåõíè÷åñêîãî îáúåêòà áàçèðóåòñÿ íà ìóëü- òèïîëüíîé ìîäåëè ÂÌÏ èñòî÷íèêà. Ñîãëàñíî òà- êîé ìîäåëè ñêàëÿðíûé ìàãíèòíûé ïîòåíöèàë èñ- òî÷íèêà ïîëÿ äëÿ îáëàñòè âíåøíåãî ïðîñòðàíñòâà, ãäå ïëîòíîñòü òîêà ðàâíà íóëþ è 0 rotH  , ìîæíî àíàëèòè÷åñêè îïèñàòü ñôåðè÷åñêèì ãàðìîíè÷åñ- êèì ðÿäîì Ãàóññà â âèäå ñóììû ìóëüòèïîëåé äèïîëüíîé, êâàäðóïîëüíîé, îêòóïîëüíîé è ò.ä. ñîñòàâëÿþùèõ ìàãíèòíîãî ïîëÿ [5]: ) cos ( P ) m sin h m cos g ( R 1 4 1 U 1 n n 0 m m n nm nm 1 n             , (1) ãäå   , , R – ñôåðè÷åñêèå êîîðäèíàòû òî÷êè íàáëþäåíèÿ; nm nm h , g – ïîñòîÿííûå êîýôôèöèåí- òû ãàðìîíè÷åñêîãî ðÿäà, ðàâíûå ìóëüòèïîëüíûì ìàãíèòíûì ìîìåíòàì èñòî÷íèêà ïîëÿ; n-ïîðÿäêî- âûé íîìåð ÷ëåíà ñôåðè÷åñêîãî ðÿäà Ãàóññà; m- ïîðÿäêîâûé íîìåð ýëåìåíòàðíîãî ìóëüòèïîëÿ n-ãî ïîðÿäêà;    cos P m n – ïðèñîåäèíåííûå ïîëèíîìû Ëåæàíäðà ïåðâîãî ðîäà. Àíàëèç ðÿäà (1) ïîêàçûâàåò, ÷òî ìàãíèòíîå ïîëå èñòî÷íèêà èìååò ñëîæíûé ïðîñòðàíñòâåííî-ãàðìî- íè÷åñêèé õàðàêòåð. Ïîýòîìó çàäà÷à èçìåðåíèÿ êîýôôèöèåíòà 10 g ïåðâîé ãàðìîíèêè, ðàâíîãî îñåâîìó äèïîëüíîìó ìîìåíòó z M èñòî÷íèêà ïîëÿ, ñâîäèòñÿ ê èçáèðàòåëüíîìó âûäåëåíèþ èç ïîëíîãî ñïåêòðà ìóëüòèïîëåé ÂÌÏ äèïîëüíîé ïîñòîÿííîé ñîñòàâëÿþùåé, íå çàâèñÿùåé îò óãëîâîé êîîðäèíà- òû ,  è èçìåðåíèþ ïàðàìåòðîâ ïîëÿ çîíàëüíîé ãàðìîíèêè äèïîëüíîé ñîñòàâëÿþùåé ) 0 m , 1 n (   . Öåëü èññëåäîâàíèÿ - îïðåäåëåíèå îñíîâíûõ ìåòðî- ëîãè÷åñêèõ õàðàêòåðèñòèê ìîäóëüíîé èçìåðèòåëü- íîé ñèñòåìû, ðåàëèçóþùåé ÷åòûðåõòî÷å÷íûé ìàã- íèòîìåòðè÷åñêèé ìåòîä êîíòðîëÿ îñåâîãî äèïîëü- íîãî ìàãíèòíîãî ìîìåíòà èñòî÷íèêîâ ìàãíèòíîãî ïîëÿ, ê êîòîðûì ïðåäúÿâëÿþòñÿ òðåáîâàíèÿ ïî îãðàíè÷åíèþ óðîâíÿ íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ âî âíåøíåì ïðîñòðàíñòâå. Äëÿ äîñòèæåíèÿ ïîñòàâëåííîé öåëè íåîáõîäèìî ðåøèòü ñëåäóþùóþ çàäà÷ó: ðàçðàáîòàòü ìàòåìàòè- ÷åñêèå ìîäåëè ïîëåçíîãî ñèãíàëà, ðåçóëüòèðóþùåé ìóëüòèïîëüíîé ïîãðåøíîñòè è åå íàèáîëåå âåñî- ìûõ ñîñòàâëÿþùèõ, îáóñëîâëåííûõ ïîìåõîíåñó- ùèìè ìóëüòèïîëÿìè ïðîñòðàíñòâåííûõ ãàðìîíèê ÂÌÏ. 2. Ðåøåíèå çàäà÷è Ðàññìîòðèì ÷åòûðåõòî÷å÷íûé ìåòîä êîíòðîëÿ îñå- âîãî äèïîëüíîãî ìàãíèòíîãî ìîìåíòà èñòî÷íèêîâ ÂÌÏ, êîòîðûé ðåàëèçóåòñÿ èçìåðèòåëüíîé ñèñòå- ìîé â âèäå ìîäóëÿ èç ÷åòûðåõ îñåâûõ äàò÷èêîâ. Äàò÷èêè ýòîé ñèñòåìû ðàñïîëîæåíû âîêðóã èñïû- òóåìîãî èñòî÷íèêà ïîëÿ â ýêâàòîðèàëüíîé ïëîñêî- ñòè ) 90 (    íà îêðóæíîñòè ðàäèóñà R â òî÷êàõ ñ óãëîâîé êîîðäèíàòîé  90 ) 1 k ( k    , ãäå k=1...4 - íîìåð îñåâîãî äàò÷èêà, îðèåíòèðîâàííîãî ñâîèìè ìàãíèòíûìè îñÿìè ïàðàëëåëüíî ïîëÿðíîé îñè Z ïðèíÿòîé ñèñòåìû êîîðäèíàò, íà÷àëî êîòîðîé ñâÿ- çàíî ñ èçìåðÿåìûì èñòî÷íèêîì ïîëÿ. 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(2) Äëÿ óïðîùåíèÿ àíàëèçà çàïèøåì ðÿä  H (2) â ðàçâåðíóòîì âèäå, îãðàíè÷èâ åãî ãàðìîíèêîé n=5:                                  50 7 43 43 41 41 6 32 32 30 5 21 21 4 10 3 g 8 1 [ R 4 15 )] 3 sin h 3 cos g ( 7 ) sin h cos g ( 2 1 [ R 4 1 )] 2 sin h 2 cos g ( 15 g 2 3 [ R 4 1 ) sin h cos g ( 3 R 4 1 g R 4 1 ) 90 ( H       ) 2 sin h 2 cos g ( 2 7 52 52 ... )] 4 sin h 4 cos g ( 63 54 54      (3) Êàòóøêè ÷åòûðåõ äàò÷èêîâ ìîäóëüíîé èçìåðè- òåëüíîé ñèñòåìû ýëåêòðè÷åñêè ñîåäèíåíû ïî ïî- ëåçíîìó ñèãíàëó ìàãíèòíîãî ïîëÿ çîíàëüíîé ãàð- ìîíèêè äèïîëüíîé ïîñòîÿííîé ñîñòàâëÿþùåé. Ïîýòîìó ðåçóëüòèðóþùèé ýëåêòðè÷åñêèé ñèãíàë ðàâåí ñóììå ñèãíàëîâ, íàâåäåííûõ îñåâîé êîìïî- íåíòîé íàïðÿæåííîñòè ïîëÿ  H ) 90 (    â ýòèõ êàòóøêàõ:        4 1 k k k f z ) , 90 ( H k 1 E  , (4) ãäå f k , ìÝ/ì - ïîñòîÿííàÿ êàòóøåê äàò÷èêîâ. Ïîñëå ïîäñòàíîâêè îñåâîé êîìïîíåíòû íàïðÿæåí- íîñòè ìàãíèòíîãî ïîëÿ (3) ñ ó÷åòîì çíà÷åíèé óãëîâîé êîîðäèíàòû k  â óðàâíåíèå (4) ïîëó÷èì ðåçóëüòèðóþùèé ñèãíàë â âèäå ñóììû ñèãíàëîâ îò ìóëüòèïîëåé ÂÌÏ íå÷åòíîãî ïîðÿäêà:    5 f 30 3 f 10 z R k / g 6 R k / g 4 E ..., E E E ... R k 2 / g 15 5 z 3 z 1 z 7 f 50       (5) ãäå 3 f 10 1 z R k / g 4 E  – ïîëåçíûé ñèãíàë, ïðîïîðöè- îíàëüíûé êîýôôèöèåíòó 10 g çîíàëüíîé ãàðìîíè- êè äèïîëüíîé ïîñòîÿííîé ñîñòàâëÿþùåé ìàãíèò- íîãî ïîëÿ; 5 f 30 3 z R k / g 6 E   – ñèãíàë ïîìåõè, ïðîïîðöèîíàëüíûé êîýôôèöèåíòó 30 g çîíàëüíîé ãàðìîíèêè îêòóïîëüíîé ñîñòàâëÿþùåé ìàãíèòíî- ãî ïîëÿ îáúåêòà (n=3); 7 f 50 5 z R k 2 / g 15 E  – ñèãíàë ïîìåõè, ñîçäàâàåìûé ãàðìîíèêîé ïÿòîãî ïîðÿäêà. Àíàëèç óðàâíåíèÿ (5) ïîêàçûâàåò, ÷òî â èçìåðÿå- ìîì ýëåêòðè÷åñêîì ñèãíàëå z E îòñóòñòâóþò ñèãíà- ëû ïîìåõ îò ïðîñòðàíñòâåííûõ ãàðìîíèê ÂÌÏ ÷åòíîãî ïîðÿäêà, ÷òî ïîäòâåðæäàåò èçáèðàòåëü- íîñòü ðàññìàòðèâàåìîé ìàãíèòîìåòðè÷åñêîé ñèñòå- ìû äàò÷èêîâ, êîòîðàÿ îñóùåñòâëÿåòñÿ ñ òî÷íîñòüþ äî ïîìåõîíåñóùåãî ìóëüòèïîëÿ òðåòüåãî ïîðÿäêà. Òàêèì îáðàçîì, ñèãíàë z E (5) èçìåðÿåòñÿ ñ ïîãðåø- íîñòüþ, îáóñëîâëåííîé ïîìåõîíåñóùèìè íå÷åòíû- ìè ãàðìîíèêàìè ÂÌÏ âûñøåãî ïîðÿäêà, íà÷èíàÿ ñ ãàðìîíèêè n=3. Ñîãëàñíî (5) ÷óâñòâèòåëüíîñòü ìàãíèòîìåòðè÷åñ- êîé ñèñòåìû îñåâûõ äàò÷èêîâ ê íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ çîíàëüíîé ãàðìîíèêè äèïîëüíîé ñîñòàâëÿþùåé ðàâíà f z k / 4 S  , à çíà÷åíèå îñåâîãî äèïîëüíîãî ìîìåíòà îïðåäåëÿåòñÿ âûðàæåíèåì . R k E 4 1 M g 3 f z z 10   Î÷åâèäíî, ÷òî ìóëüòèïîëüíàÿ ñîñòàâëÿþùàÿ ìåòî- äè÷åñêîé ïîãðåøíîñòè èçìåðåíèÿ, îáóñëîâëåííàÿ ïðîñòðàíñòâåííûìè ãàðìîíèêàìè ÂÌÏ íå÷åòíîãî ïîðÿäêà, ñîçäàåòñÿ â îñíîâíîì ìóëüòèïîëÿìè ïî- ìåõîíåñóùèõ ãàðìîíèê, êîòîðûå èìåþò íàèìåíü- øóþ ñòåïåíü ñïàäàíèÿ íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ ïî ñðàâíåíèþ ñ íàïðÿæåííîñòüþ ïîëÿ ãàðìî- íèê áîëåå âûñîêèõ ïîðÿäêîâ. Îöåíèì âåñ òðåòüåé è ïÿòîé ãàðìîíèê â ðåçóëüòè- ðóþùåé ïîìåõå, îáðàçîâàííîé â ìîäóëüíîé ìàãíè- òîìåòðè÷åñêîé ñèñòåìå èçìåðèòåëüíûõ äàò÷èêîâ. Äëÿ ýòîãî îïðåäåëèì òî÷íîå çíà÷åíèå îñåâîé êîì- ïîíåíòû íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ èñòî÷- íèêà, çàòåì çíà÷åíèå ðåçóëüòèðóþùåé ïîìåõè è, â êîíå÷íîì ñ÷åòå, ïîãðåøíîñòü èçìåðåíèÿ äèïîëü- íîãî ìîìåíòà z M , ðàñïîëîæåííîãî íà ïîëÿðíîé îñè ñî ñìåùåíèåì íà 0 z îòíîñèòåëüíî íà÷àëà êîîðäèíàò. Çà ñ÷åò ýêñöåíòðè÷íîñòè îñåâîãî äè- ïîëüíîãî ìîìåíòà èñòî÷íèêà ñîçäàåòñÿ ïîëíûé ñïåêòð ïðîñòðàíñòâåííûõ ãàðìîíèê, ìàòåìàòè÷åñ- êè îïèñûâàåìûõ ñôåðè÷åñêèì ãàðìîíè÷åñêèì ðÿ- äîì (1). Òî÷íîå çíà÷åíèå ñêàëÿðíîãî ìàãíèòíîãî ïîòåíöèàëà òàêîãî èñòî÷íèêà ïîëÿ ðàâíî 2 / 3 2 0 2 2 0 z z ] ) z z ( y x [ 4 ) z z ( M U       , ãäå x, y, z – òåêóùèå êîîðäèíàòû òî÷êè íàáëþäå- íèÿ: . cos R z , sin sin R y , sin cos R x         Òîãäà îñåâàÿ ñîñòàâëÿþùàÿ íàïðÿæåííîñòè ìàã- íèòíîãî ïîëÿ â ýòîì ñëó÷àå îïèñûâàåòñÿ â ýêâàòî- ðèàëüíîé ïëîñêîñòè, â êîòîðîé ðàñïîëîæåíà ñèñòå- ìà èçìåðèòåëüíûõ äàò÷èêîâ, óðàâíåíèåì 2 / 5 2 0 2 2 0 2 z ' ) z R ( 4 ) z 2 R ( M ) 90 ( H         . Ýòà ñîñòàâëÿþùàÿ ìàãíèòíîãî ïîëÿ âîçäåéñòâóåò íà ìàãíèòíûå îñè êàòóøåê äàò÷èêîâ è íàâîäèò â íèõ ýëåêòðè÷åñêèé ñèãíàë f ' ' z k / ) 90 ( H 4 E      . (6) Èç ðàâåíñòâà ýëåêòðè÷åñêèõ ñèãíàëîâ z E (5) è ' z E (6) íàõîäèì ðåçóëüòèðóþùóþ ïîìåõó, ðàâíóþ àá- ñîëþòíîé ïîãðåøíîñòè, êîòîðàÿ ñîçäàåòñÿ ïîëíûì ñïåêòðîì ïîìåõîíåñóùèõ ãàðìîíèê ÂÌÏ íå÷åòíî- ãî ïîðÿäêà (n=3,5,7...): Qx tU}! 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Z,,M%7 ÐÈ, 2005, ¹ 2 ÓÄÊ 621.385.6 Î ÊÎËÅÁÀÍÈßÕ ÇÀÌÊÍÓÒÎÃÎ ÝËÅÊÒÐÎÍÍÎÃÎ ÏÎÒÎÊÀ  ÑÊÐÅÙÅÍÍÛÕ ÏÎËßÕ ÃÐÈÖÓÍΠÀ.Â. Ïðè ïîìîùè âû÷èñëèòåëüíîãî ýêñïåðèìåíòà èçó÷à- þòñÿ êîëåáàíèÿ çàìêíóòîãî â àçèìóòàëüíîì íàïðàâ- ëåíèè ýëåêòðîííîãî ïîòîêà âáëèçè ðàñïðåäåëåííîé ýìèòòèðóþùåé ïîâåðõíîñòè â ñêðåùåííûõ ïîëÿõ. Îíè ÿâëÿþòñÿ îñíîâíûìè ïðè÷èíàìè øóìîâ â ñòàòè÷åñ- êîì è äèíàìè÷åñêîì ðåæèìàõ ÑÂ×-ïðèáîðîâ Ì-òèïà.  ñòàòè÷åñêîì ðåæèìå âûäåëÿþòñÿ: ñîáñòâåííûå òàí- ãåíöèàëüíûå êîëåáàíèÿ ýëåêòðîííîãî îáëàêà; âòî- ðè÷íî-ýìèññèîííàÿ ãðóïïèðîâêà âòóëêè; ñîëèòîíû â ýëåêòðîííîì ïîòîêå.  äèíàìè÷åñêîì ðåæèìå ýòî òðàíçèòíûå è óñòîé÷èâûå êîíâåêòèâíûå îáëàêà â ýëåê- òðîííûõ ñïèöàõ. Èññëåäóåòñÿ âëèÿíèå óêàçàííûõ íåóñ- òîé÷èâîñòåé íà ñïåêòð êîëåáàíèé ýëåêòðîííîãî ïîòî- êà â ïðèáîðàõ ñî ñêðåùåííûìè ïîëÿìè. 1. Ââåäåíèå Àêòóàëüíîé ïðîáëåìîé ñîâðåìåííîé ðàäèîôèçèêè è ôèçè÷åñêîé ýëåêòðîíèêè ÿâëÿåòñÿ îáåñïå÷åíèå ýëåêòðîìàãíèòíîé ñîâìåñòèìîñòè (ÝÌÑ) ðàäèî- ýëåêòðîííûõ ñèñòåì. Ýòî ñâÿçàíî ñ ðàçâèòèåì ðàäèîëîêàöèîííîé òåõíèêè, ÑÂ×-ýíåðãåòèêè, ñðåäñòâ òåëåêîììóíèêàöèé (â òîì ÷èñëå ìîáèëü- íûõ), óâåëè÷åíèåì ñêîðîñòè ïåðåäà÷è äàííûõ â íèõ è ïåðåõîäîì íà öèôðîâûå ñòàíäàðòû îáìåíà èí- ôîðìàöèåé. Ââèäó çíà÷èòåëüíîñòè ýíåðãåòè÷åñêî- ãî ïîòåíöèàëà ðàäèîëîêàöèîííûõ è òåõíîëîãè÷åñ- êèõ ÑÂ×-óñòàíîâîê âàæíîé çàäà÷åé ÿâëÿåòñÿ óìåíü- øåíèå óðîâíÿ øóìîâûõ è ïîáî÷íûõ êîëåáàíèé àêòèâíûõ ýëåìåíòîâ, ñðåäè êîòîðûõ çíà÷èòåëüíóþ ÷àñòü ñîñòàâëÿþò ïðèáîðû Ì-òèïà. Èññëåäîâàíèå ïðè÷èí, âûçûâàþùèõ óõóäøåíèå ñïåêòðà âûõîäíîãî ñèãíàëà ìàãíåòðîííûõ ïðèáî- ðîâ, ÿâëÿåòñÿ òåìàòèêîé ìíîãî÷èñëåííûõ ïóáëèêà- öèé (îáçîð ëèòåðàòóðû ïî äàííîé òåìå èìååòñÿ, íàïðèìåð, â ðàáîòå [1]). Íåñìîòðÿ íà ýòî, íåêîòîðûå «ýëåìåíòàðíûå» è â òî æå âðåìÿ ôóíäàìåíòàëüíûå âîïðîñû, íàïðèìåð, îñîáåííîñòè ïîâåäåíèÿ ìàã- íåòðîííîãî äèîäà â çàêðèòè÷åñêîì ðåæèìå (â ÷àñòíîñòè, ìåõàíèçì âîçíèêíîâåíèÿ àíîäíîãî òîêà â äàííûõ óñëîâèÿõ) îñòàâàëèñü íåîáúÿñíåííûìè äî ñåðåäèíû 1960-õ ãîäîâ. Çàòåì îíè áûëè ÷àñòè÷íî èñòîëêîâàíû â õîäå ÷èñëåííûõ ýêñïåðèìåíòîâ, îäíàêî áîëüøèíñòâî äàííûõ ÿâëåíèé ïî-ïðåæíåìó íå íàõîäèëî àäåêâàòíîãî îáúÿñíåíèÿ. Ïðè÷èíà çàêëþ÷àëàñü â íåñîîòâåòñòâèè ïåðâûõ ìîäåëåé ÷àñòèö ôèçè÷åñêèì ïðîöåññàì, ïðîèñõîäÿùèì â ìàãíåòðîííûõ äèîäàõ. Ïðè àíàëèçå òîëüêî ðàáî÷åãî âèäà êîëåáàíèé çà- ìåäëÿþùåé ñèñòåìû (ÇÑ) ïðèáîðà Ì-òèïà ìîæíî ïðåäïîëîæèòü, ÷òî ýëåêòðîííûé ïîòîê èìååò ïðî- äîëüíóþ èëè àçèìóòàëüíóþ ïåðèîäè÷íîñòü (êâàçè- ïåðèîäè÷íîñòü), ñîâïàäàþùóþ ñ ñîîòâåòñòâóþùèì ïåðèîäîì ñèíõðîííîé ïðîñòðàíñòâåííîé ãàðìîíè- êè Â×-ïîëÿ ÇÑ. Íàïðîòèâ, ïðè èññëåäîâàíèè øóìîâ è ïîáî÷íûõ êîëåáàíèé àçèìóòàëüíûé ïåðè- îä çàìêíóòîãî ýëåêòðîííîãî îáëàêà äîëæåí áðàòüñÿ ðàâíûì 2  . Ïîýòîìó ìîäåëè, îñíîâàííûå íà ñàìî- ñîãëàñîâàííîì ó÷åòå ïðîöåññîâ âçàèìîäåéñòâèÿ â ïðåäåëàõ îäíîé äëèíû âîëíû ÇÑ, êàê ñòàöèîíàð- íûå, òàê è íåñòàöèîíàðíûå, íå â ñîñòîÿíèè àäåê- âàòíî ïðîãíîçèðîâàòü ïîâåäåíèå ïîòîêà ïðè íàëè- ÷èè íåñêîëüêèõ âèäîâ êîëåáàíèé. Ñèòóàöèÿ èçìåíèëàñü ëèøü â íà÷àëå 1980-õ ãîäîâ, êîãäà áûëè ðàçðàáîòàíû ïåðâûå ïðîãðàììû äëÿ ïîëíîôîðìàòíîãî íåñòàöèîíàðíîãî ìîäåëèðîâàíèÿ ïðîöåññîâ â ñêðåùåííûõ ïîëÿõ ìåòîäîì ÷àñòèö [2– 4]. Ïîÿâèëàñü âîçìîæíîñòü êîððåêòíî ìîäåëèðî- âàòü äèíàìèêó ðàçâèòèÿ ðàçíîîáðàçíûõ êîëåáàíèé ýëåêòðîííîãî îáëàêà â ìàãíåòðîííûõ ïðèáîðàõ ñ ðàñïðåäåëåííûì êàòîäîì è âûÿñíèòü ìåõàíèçìû èõ âîçíèêíîâåíèÿ.  ðåçóëüòàòå íà ñåãîäíÿøíèé äåíü óñòàíîâëåíî è êëàññèôèöèðîâàíî òðè âèäà íåóñòîé÷èâîñòåé âòóëêè, à òàêæå äâà âèäà íåóñòîé- ÷èâîñòåé ñïèö [5, 6]. Öåëüþ äàííîãî èññëåäîâàíèÿ ÿâëÿåòñÿ êîìïëåêñ- íîå èçó÷åíèå êîëåáàòåëüíûõ ÿâëåíèé â ýëåêòðîí- íîì îáëàêå â ñêðåùåííûõ ïîëÿõ ìåòîäîì âû÷èñëè- òåëüíîãî ýêñïåðèìåíòà ñ èñïîëüçîâàíèåì ñïåêò- ðàëüíîãî ïîäõîäà [7], âêëþ÷àþùåå àíàëèç ôîðìû îáëàêà, ñïåêòðîâ êîëåáàíèé è ôèçè÷åñêóþ èíòåð- ïðåòàöèþ ðåçóëüòàòîâ. Îñíîâíûå çàäà÷è èññëåäîâàíèÿ: – òåîðåòè÷åñêîå èçó÷åíèå äèíàìè÷åñêèõ ÿâëåíèé â çàìêíóòîì ýëåêòðîííîì ïîòîêå â ñêðåùåííûõ ïîëÿõ ïðè îòñóòñòâèè ñîëåíîèäàëüíîãî ýëåêòðî- ìàãíèòíîãî ïîëÿ ÇÑ (ñòàòè÷åñêèé ðåæèì) äëÿ âûÿâëåíèÿ ïðè÷èí, âûçûâàþùèõ ïîâûøåííûé óðîâåíü øóìîâ è ïîáî÷íûõ êîëåáàíèé â ýòîì ðåæèìå; – àíàëèç êîëåáàòåëüíûõ ïðîöåññîâ ïðè âçàèìîäåé- ñòâèè çàìêíóòîãî ýëåêòðîííîãî ïîòîêà â ñêðåùåí- íûõ ïîëÿõ ñ ñîëåíîèäàëüíûì ýëåêòðîìàãíèòíûì ïîëåì ÇÑ (äèíàìè÷åñêèé ðåæèì) äëÿ âûÿñíåíèÿ âëèÿíèÿ òóðáóëåíòíîñòåé îáúåìíîãî çàðÿäà íà ñïåêòð âûõîäíîãî ñèãíàëà. 2. Ñòàòè÷åñêèé ðåæèì ýëåêòðîííîãî îáëàêà Ñòàòè÷åñêèé ðåæèì ïðèáîðîâ Ì-òèïà íà ïðîòÿæå- íèè âîñüìè äåñÿòèëåòèé ïðèâëåêàåò âíèìàíèå êàê ôèçèêîâ, òàê è ñïåöèàëèñòîâ â îáëàñòè ýëåêòðîíè- êè, òåîðåòèêîâ è ýêñïåðèìåíòàòîðîâ. Îäíà èç ïðè÷èí òàêîãî èíòåðåñà çàêëþ÷àåòñÿ â òîì, ÷òî çà âíåøíåé ïðîñòîòîé êîíñòðóêöèè ìàãíåòðîííîãî äèîäà êðîåòñÿ èñêëþ÷èòåëüíàÿ ñëîæíîñòü è ìíîãî- îáðàçèå ÿâëåíèé, êîòîðûå ìîãóò â íåì ïðîòåêàòü. Äðóãàÿ ïðè÷èíà ñîñòîèò â âàæíîñòè ðåçóëüòàòîâ èññëåäîâàíèé äëÿ ïðàêòèêè, ïîñêîëüêó áîëüøèå ïðèíöèïèàëüíî äîñòèæèìûå çíà÷åíèÿ ïåðâåàíñà çàðÿæåííûõ ïó÷êîâ â ñêðåùåííûõ ïîëÿõ îáóñëîâ- ëèâàþò øèðîêîå ïðèìåíåíèå ìàãíåòðîííûõ ñèñòåì íå òîëüêî â ýëåêòðîíèêå ÑÂ×, íî è â óñêîðèòåëü- íîé òåõíèêå (ìàãíåòðîííûå èíæåêòîðû), òåõíîëî- ãè÷åñêèõ ïðîöåññàõ (ìàãíåòðîííûå ðàñïûëèòåëü- íûå ñèñòåìû) è  ò.ï. 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{_ X]%m]>(#hlxsޏbƊ9fS6߂9Om,`Gp.6y Gs2Ex^ڹ;>j~0f˙ ƍpQ]QHYm c>TЁ7 y>D!C46|(h}o;Oó3'!dd= !óL4a*b/-˓r1"ucLYY 88~(ǫk' ~q ϩ ~Oj8UAp3'BHÚ" ,+)dzD:F(r(GvOn:eWC '&^#|y oNgpU#M= hqe:77  &s%NY|my0VK9/7 ,}Dz`,Dp_*5 pxǍs:0 s?p=]B }``ƼiAH2SƝ|Hy(K=Bo=A9ˌ>ːƂ)/vxT.U^8#(6'= s?cZG_':f\+eE7yb~ڥ`m:=/,tH~C+Nտ5\[CAQ9L,'x=Bp˼䃁q \,F[X9 !䶈˜剗ENb/we罂̻Kd]v%J3Y&F\Q|/.qڅg$Nԧc9P8u6{Ԛ̶rJHf1;Exlzb>S;`xs`B搊<ﱊ'"m^$)e刣Ik2|%LEƒ2[0J*-om=yTzgO`mq5*_Cʁ7+{i y +0IE |#hcAubu"s1;A"[+s&yQ8~S节k,\aR.V\/s5`5F.M3ysqOy@e4>]>|u{ x]9EF{ l16pU2K2n=?lBq !0!>zo0w`촱=- <]@Hٲ[YO ޞn.R^8KY# Wf(.;Ӳ9罌u8וKeL m7b5t9x(7|èv=|.)ݛgT>Dq_*:@{>Ȍawuz@mozwy,HKp|<$m͍ʋ=m,Wͭۙټ?81whZ6GcȜ/!uq]M>|=hoxiRc;^/Q^x.[]{zz߳<Q=v"ĸf*l;cg2Wxq@Ȉ<)匟oK=݆a 48au[o/UE8~By^H&v\G΢} c87( l)_i˄8uNJʢRofrx3Te{,agoxlՂmXO!R+ؼl0B_otB>Q. F+@.C[g/4(C_2Ef WGsxzl_OC@tF{779k+y;{*)3{c?ݑw\";Q{7M a{ $PvYH'!dcCEkp|Jw/0pC-z>L.:#5OmDŽ.Z1"e'6>6<μ\ɣy0❨ESy0%t?DgAa?_?ϫ;ڟ6˞2^#{<>eȣi`?zoG=ʃu8367K~*b ƚofzxǩNKaY7؇e'7ox#̛3#=7X91#t' H<s*nQ,UU oVNP" *ayA_r^>SX+5r?n2cj\2[MϴkiezKE#:S%c;$^w+/Iv OJ''؄pF WW<9ߍ q070g,|2zt9Lj@>VR?wy/MY|`>r@w*_2@>$k~N R7:{ À.b#+Oc|4L#~J'gx MHt2]g^82ﶨCeSa[oīOe0/tĺV!1Ŀжq#4>`8N{ml;KaS}W身?\S>*{*kb3N:6`N|5ᙖ;[3Ɩ͙̍ rh<>y{SX_Bmq\OYU||?M❨sDoSL,@y~W43ֺe26)dk:H,l^1'OD|eҲ:a( e lC:1QUcB{,+ě#1EwٿnݭZQN[ =90f9ϳ̕2 OO@'N8Qr<|AƩ,m._S,,c"8 ÐÈ, 2005, ¹ 2 ïðîèçîøëî âîçáóæäåíèå êîëåáàíèé (èìåë ìåñòî ïðîïóñê èìïóëüñà), ñ òî÷êè çðåíèÿ ôèçè÷åñêèõ ïðîöåññîâ â ýëåêòðîííîì ïîòîêå ïðàêòè÷åñêè íå îòëè÷àåòñÿ îò ìàãíåòðîííîãî äèîäà. Õîòÿ îáðàçóþ- ùèåñÿ âñëåäñòâèå ñàìîãðóïïèðîâêè ñãóñòêè ýëåê- òðîííîãî ïîòîêà îáû÷íî íå ñèíõðîííû ñ íàâîäè- ìîé èìè âîëíîé â ÇÑ, íåîäíîðîäíàÿ â àçèìóòàëü- íîì íàïðàâëåíèè âòóëêà ìîæåò ÿâëÿòüñÿ ìîùíûì ãåíåðàòîðîì øóìîâ. Ìíîãî÷èñëåííûå ýêñïåðèìåíòàëüíûå ðàáîòû 1950- 1970-õ ãîäîâ ïî èññëåäîâàíèþ øóìîâ â ìàãíåòðî- íàõ âûÿâèëè îñíîâíûå çàêîíîìåðíîñòè ÿâëåíèé, èìåþùèõ ìåñòî â ñòàòè÷åñêîé âòóëêå îáúåìíîãî çàðÿäà â ñêðåùåííûõ ïîëÿõ: – ìàãíåòðîííûé äèîä â çàêðèòè÷åñêîì ðåæèìå îáû÷íî íàõîäèòñÿ â êîëåáàòåëüíîì ñîñòîÿíèè. Ñïåêòð ýòèõ êîëåáàíèé èìååò áîëåå èëè ìåíåå ÷åòêî âûðàæåííûå ïèêè (ìàêñèìóìû) íà äèñêðåò- íûõ ÷àñòîòàõ h  . Äàííûå ÷àñòîòû äîñòàòî÷íî õîðîøî îïèñûâàþòñÿ ôîðìóëîé Õàðòðè-Áàíåìàíà [8]: 2 ext 1 h 2 2 2 2 2 exta 1 22 22 Bh r 1 2 r Bh2h r 1, 2 rr                  (1) ãäå h – íîìåð ïèêà;  – îòíîøåíèå çàðÿäà ýëåêòðîíà ê åãî ìàññå; 1 r – ðàäèóñ êàòîäà; 2 r – ðàäèóñ àíîäà; ext B – èíäóêöèÿ âíåøíåãî ìàãíèò- íîãî ïîëÿ; a  – àíîäíîå íàïðÿæåíèå; – âî ìíîãèõ ñëó÷àÿõ ÷àñòîòû ïèêîâ êðàòíû íåêî- òîðîé îñíîâíîé ÷àñòîòå 1  , ò.å. h ÿâëÿåòñÿ öåëûì ïîëîæèòåëüíûì ÷èñëîì. Îäíàêî èíîãäà ýòî ïðàâè- ëî íå âûïîëíÿåòñÿ, ò.å. ÷àñòîòû ñïåêòðàëüíûõ ìàêñèìóìîâ âûðàæàþòñÿ íåöåëûìè ÷èñëàìè h . Íàèáîëüøàÿ àìïëèòóäà ïèêîâ îáû÷íî èìååò ìåñòî ïðè h  = 3 … 5; – âîëüòàìïåðíàÿ õàðàêòåðèñòèêà (ÂÀÕ) ìàãíåòðîí- íîãî äèîäà â çàêðèòè÷åñêîì ðåæèìå òàêæå ÷àñòî (íî íå âñåãäà) èìååò íåìîíîòîííûé õàðàêòåð, ò.å. íà- áëþäàåòñÿ ðÿä ïèêîâ àíîäíîãî òîêà ïðè óâåëè÷å- íèè àíîäíîãî íàïðÿæåíèÿ. Ïèêàì àíîäíîãî òîêà îáû÷íî ñîîòâåòñòâóþò ìàêñèìóìû âûñîêî÷àñòîò- íûõ øóìîâ; – â ðÿäå ñëó÷àåâ èçìåíåíèå òåìïåðàòóðû êàòîäà ñóùåñòâåííî âëèÿåò íà ïðîöåññû â îáëàêå îáúåìíî- ãî çàðÿäà. Èíîãäà íàáëþäàþòñÿ äâà ðåçêî îòëè÷à- þùèõñÿ êîëåáàòåëüíûõ ñîñòîÿíèÿ. Ïðè ìàëîé òåì- ïåðàòóðå òèïè÷íî ñîñòîÿíèå ïî÷òè ìîíîõðîìàòè- ÷åñêèõ êîëåáàíèé ñ óçêèìè ñïåêòðàëüíûìè ïèêà- ìè. Ñ óâåëè÷åíèåì òåìïåðàòóðû ìîùíîñòü âûñîêî- ÷àñòîòíîãî ñèãíàëà íà âûõîäå çîíäà ñêà÷êîîáðàçíî óìåíüøàåòñÿ ïðèìåðíî íà ïîðÿäîê, è ïðîèñõîäèò ïåðåõîä ê øóìîâîìó ðåæèìó äèîäà ñ øèðîêèì ñïåêòðîì êîëåáàíèé; – â ýëåêòðîííîì îáëàêå èìååòñÿ çíà÷èòåëüíîå ÷èñëî ýëåêòðîíîâ ñ ýíåðãèåé, ñóùåñòâåííî ïðåâû- øàþùåé èõ íà÷àëüíóþ ýíåðãèþ â ìîìåíò ýìèññèè èç êàòîäà; – â ðåæèìå áîëüøèõ âíåøíèõ ìàãíèòíûõ ïîëåé è àíîäíûõ íàïðÿæåíèé â ìàãíåòðîííîì äèîäå íà- áëþäàåòñÿ ðåçêîå óâåëè÷åíèå ýìèññèè èç êàòîäà, âåðîÿòíî, âñëåäñòâèå âîçðàñòàíèÿ âòîðè÷íîé ýìèñ- ñèè; – ïðè ïåðåõîäå îò ñòàòè÷åñêîãî ê äèíàìè÷åñêîìó ðåæèìó ìîùíîñòü îáðàòíîé áîìáàðäèðîâêè êàòîäà ïàäàåò â íåñêîëüêî ðàç; – â ìàãíåòðîííîì äèîäå çíà÷èòåëüíàÿ ÷àñòü îáúåì- íîãî çàðÿäà âðàùàåòñÿ íà ðàäèóñàõ, ïðåâûøàþùèõ ðàäèóñ ãðàíèöû áðèëëþýíîâñêîé âòóëêè. Ýòà ÷àñòü îáðàçóåò íåîäíîðîäíûé â àçèìóòàëüíîì íàïðàâëå- íèè ýëåêòðîííûé ïîòîê, íàïîäîáèå ñïèö, èìåþ- ùèõ ìåñòî â äèíàìè÷åñêîì ðåæèìå ïðèáîðîâ Ì- òèïà.  áîëåå ïîçäíèõ ýêñïåðèìåíòàëüíûõ ðàáîòàõ (ñì., íàïðèìåð, [9]) èçó÷åíû ñïåêòðàëüíûå õàðàêòåðèñ- òèêè øóìîâ â ìàãíåòðîííîì äèîäå. Íà ñïåêòðî- ãðàììå ïðîñëåæèâàåòñÿ ìíîæåñòâî (íå ìåíåå äåñÿ- òè) ÷åòêî âûðàæåííûõ ãàðìîíèê íåêîòîðîé îñíîâ- íîé ÷àñòîòû, ðàâíîé ïðèìåðíî 90 ÌÃö. Íàèáîëü- øóþ àìïëèòóäó èìååò âòîðàÿ ãàðìîíèêà, îäíàêî îñòàëüíûå ãàðìîíèêè ìåíüøå åå ëèøü íà 15 … 20 äÁ. Îòìå÷åíî, ÷òî ïðè óìåíüøåíèè òåìïåðàòóðû êàòî- äà óðîâåíü øóìîâ ñíèæàåòñÿ, ò.å. øóìû âîçíèêàþò ãëàâíûì îáðàçîì ïðè îãðàíè÷åíèè òîêà êàòîäà îáúåìíûì çàðÿäîì. Îïèñàííûå ðåçóëüòàòû ïðåäîñòàâëÿþò øèðîêóþ ýêñïåðèìåíòàëüíóþ áàçó äëÿ òåîðåòè÷åñêèõ îáîá- ùåíèé è ÷èñëåííîãî ìîäåëèðîâàíèÿ.  ÷àñòíîñòè, èç íèõ ñòàíîâèòñÿ ïîíÿòíûì, ÷òî îáúÿñíèòü âñå ìíîãîîáðàçèå íàáëþäàþùèõñÿ øóìîâûõ è êîëåáà- òåëüíûõ ÿâëåíèé â ðàìêàõ êàêîãî-ëèáî îäíîãî âèäà íåóñòîé÷èâîñòè ýëåêòðîííîãî ïîòîêà âðÿä ëè âîçìîæíî. Îñíîâûâàÿñü íà ðåçóëüòàòàõ âû÷èñëè- òåëüíûõ ýêñïåðèìåíòîâ, ìîæíî âûäåëèòü òðè îñ- íîâíûõ ìåõàíèçìà ñàìîãðóïïèðîâêè çàìêíóòîãî ýëåêòðîííîãî îáëàêà âáëèçè ðàñïðåäåëåííîãî êàòî- äà â ñêðåùåííûõ ïîëÿõ [5]: – ñîáñòâåííûå òàíãåíöèàëüíûå (àçèìóòàëüíûå) êîëåáàíèÿ ýëåêòðîííîãî îáëàêà; – âòîðè÷íî-ýìèññèîííàÿ ãðóïïèðîâêà âòóëêè; – ñîëèòîíû â ýëåêòðîííîì ïîòîêå. Ðàññìîòðèì ïåðå÷èñëåííûå ÿâëåíèÿ áîëåå ïîäðîá- íî. Òàíãåíöèàëüíûå êîëåáàíèÿ. Ñîãëàñíî ðàííèì òåîðå- òè÷åñêèì ðàáîòàì [10], â êîòîðûõ äåëàëèñü ïîïûò- êè àíàëèòè÷åñêè îáúÿñíèòü îïèñàííûå âûøå «àíîìàëüíûå» ÿâëåíèÿ â çàêðèòè÷åñêîì ðåæèìå ìàãíåòðîííîãî äèîäà, â íåì ìîãóò ñóùåñòâîâàòü äâà âèäà êîëåáàíèé: àçèìóòàëüíî-îäíîðîäíûå (ðàäè- àëüíûå) è àçèìóòàëüíî-íåîäíîðîäíûå (òàíãåíöè- àëüíûå). Ïåðâûå ñâÿçàíû ñ äâóõïîòî÷íûì ñîñòîÿ- íèåì âòóëêè, ò.å. ðàäèàëüíîé ñîñòàâëÿþùåé ñêîðî- ñòè ýëåêòðîíîâ. 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Ñêà÷êîîáðàçíîå èçìåíåíèå íîìåðà ìîäû ïðè íå- çíà÷èòåëüíîì èçìåíåíèè àíîäíîãî íàïðÿæåíèÿ, à òåì áîëåå ïðè ïîñòîÿííîì a  , â ñîîòâåòñòâèè ñ ôîðìóëîé (1) ïðèâîäèò ê ñóùåñòâåííîìó èçìåíå- íèþ ÷àñòîòû ñèãíàëà íà âûõîäå çîíäà. Ïðè ýòîì óãëîâàÿ ÷àñòîòà âðàùåíèÿ îáëàêà 1  ìåíÿåòñÿ ìàëî, à ñäâèã ÷àñòîòû ïðîèñõîäèò çà ñ÷åò ïåðåðàñïðåäå- ëåíèÿ îòíîñèòåëüíûõ àìïëèòóä ñïåêòðàëüíûõ ïè- êîâ, ò.å. äîìèíèðóþùåé ñòàíîâèòñÿ äðóãàÿ ãàðìî- íèêà ÷àñòîòû 1  . Âîçíèêàåò âîïðîñ î ñîïîñòàâëå- íèè äàííîãî ÿâëåíèÿ ñ ïðèâåäåííûìè âûøå ýêñ- ïåðèìåíòàëüíûìè ðåçóëüòàòàìè. Òåîðåòè÷åñêè âîç- ìîæíû äâà âàðèàíòà: – èçìåíåíèå íîìåðà ìîäû èìååò îäíîðàçîâûé õàðàêòåð (ïîñëå òîãî, êàê m èçìåíèëîñü, âîçâðàòà ê åãî ïðåäûäóùåìó çíà÷åíèþ íå ïðîèñõîäèò).  ýòîì ñëó÷àå îíî îòâåòñòâåííî çà íåìîíîòîííîñòü ÂÀÕ äèîäà è çàâèñèìîñòü àìïëèòóäû øóìîâ îò a  ; – èçìåíåíèå m ïðîèñõîäèò ìíîãîêðàòíî (ïåðèî- äè÷åñêè èëè ñòîõàñòè÷åñêè íîìåð ìîäû óâåëè÷èâà- åòñÿ èëè óìåíüøàåòñÿ). Òàêîå ÿâëåíèå äîëæíî âûðàæàòüñÿ â ñóùåñòâåííîì ðàñøèðåíèè ñïåêòðà ñèãíàëà íà âûõîäå çîíäà çà ñ÷åò åãî ÷àñòîòíîé ìîäóëÿöèè. Ïîñêîëüêó â ýêñïåðèìåíòàõ íàáëþäàþòñÿ îáà óêà- çàííûõ ñëåäñòâèÿ, îñòàåòñÿ íåÿñíûì, êàêàÿ èç èõ ïðè÷èí èìååò ìåñòî íà ñàìîì äåëå è äåéñòâèòåëüíî ëè ñîáñòâåííûå òàíãåíöèàëüíûå êîëåáàíèÿ ïðèâî- äÿò ê ýòèì ñëåäñòâèÿì. Âîçìîæíî, â ðàçëè÷íûõ ðåæèìàõ ðàáîòû ìàãíåòðîííîãî äèîäà íàáëþäàåòñÿ ëèáî ïåðâîå, ëèáî âòîðîå ÿâëåíèå. Ñ òåîðåòè÷åñêîé òî÷êè çðåíèÿ âàæíà òàêæå ïðîáëåìà óñòîé÷èâîñòè òàíãåíöèàëüíûõ ìîä. Âûøå ãîâîðèëîñü î òîì, ÷òî ïðè ñêîëü-íèáóäü ñóùåñòâåííîì âîçäåéñòâèè íà äàííûå êîëåáàíèÿ âîçìóùàþùèõ ôàêòîðîâ (íà- ïðèìåð, äðóãèõ âèäîâ ñàìîãðóïïèðîâêè ïîòîêà) ïðîèñõîäèò èõ íåìåäëåííîå ðàçðóøåíèå. Îäíàêî îñòàåòñÿ îòêðûòûì âîïðîñ, â òå÷åíèå êàêîãî âðåìå- íè òàíãåíöèàëüíûå ìîäû ìîãóò ñóùåñòâîâàòü ïðè îòñóòñòâèè òàêèõ ôàêòîðîâ.  ðÿäå êîíñòðóêöèé ìàãíåòðîííûõ äèîäîâ è ðåæèìîâ èõ ðàáîòû ýòè ìîäû ñîõðàíÿþòñÿ â âû÷èñëèòåëüíîì ýêñïåðèìåí- òå íåîãðàíè÷åííîå âðåìÿ. Òåì íå ìåíåå, â äðóãèõ ñëó÷àÿõ ÷åðåç íåñêîëüêî ñîòåí öèêëîòðîííûõ ïå- ðèîäîâ ïîñëå ôîðìèðîâàíèÿ ñîáñòâåííûõ òàíãåí- öèàëüíûõ êîëåáàíèé ýëåêòðîííîãî îáëàêà èìååò ìåñòî òåíäåíöèÿ ê ÷àñòè÷íîìó èëè ïîëíîìó èõ ðàçðóøåíèþ èëè ïåðåðîæäåíèþ â ñîëèòîíîïîäîá- íûå ñãóñòêè. Ñ ïðàêòè÷åñêîé òî÷êè çðåíèÿ ñóùåñòâåíåí âîïðîñ, íàñêîëüêî ñîáñòâåííûå ìîäû âòóëêè ñïîñîáíû óõóäøàòü ïàðàìåòðû ïðèáîðîâ ñî ñêðåùåííûìè ïîëÿìè. Îïèñàííûå âûøå èññëåäîâàíèÿ ïîçâîëÿ- þò ñäåëàòü âûâîä, ÷òî, âîïðåêè ðåçóëüòàòàì ïðåäû- äóùèõ òåîðåòè÷åñêèõ ðàáîò, ðîëü òàíãåíöèàëüíûõ ìîä â ãåíåðàöèè øóìîâ è ïîáî÷íûõ êîëåáàíèé, à òàêæå äðóãèõ íåãàòèâíûõ ÿâëåíèé â ïðèáîðàõ Ì- òèïà, ïî-âèäèìîìó, íåçíà÷èòåëüíà. Äëÿ ðàçâèòèÿ äàííûõ ìîä íåîáõîäèìî äëèòåëüíîå âðåìÿ è îñîáûå óñëîâèÿ, êàê ïðàâèëî, îòñóòñòâóþùèå â ðåàëüíûõ ìàãíåòðîííûõ ñèñòåìàõ. Êðîìå òîãî, îíè íå âûäåð- æèâàþò êîíêóðåíöèè ñî ñòîðîíû äðóãèõ âèäîâ ãðóïïèðîâêè ïîòîêà.  ìàãíåòðîííûõ ïóøêàõ, ãäå, â îòëè÷èå îò ìàãíåòðîííîãî äèîäà, ýëåêòðîííîå îáëàêî íåïðåðûâíî äðåéôóåò â àêñèàëüíîì íàïðàâ- ëåíèè, òàíãåíöèàëüíûå êîëåáàíèÿ åùå ìåíåå âåðî- ÿòíû. Îäíàêî îíè ìîãóò ñóùåñòâîâàòü â çàäíåì ïðîñòðàíñòâå ïóøêè, ãäå àêñèàëüíàÿ äðåéôîâàÿ ñêîðîñòü ïîòîêà ìàëà. Âòîðè÷íî-ýìèññèîííàÿ ãðóïïèðîâêà.  îòëè÷èå îò òàíãåíöèàëüíûõ êîëåáàíèé ýëåêòðîííîãî îáëàêà, õàðàêòåðíàÿ íåðåãóëÿðíàÿ ãðóïïèðîâêà ïîòîêà â ñêðåùåííûõ ïîëÿõ âáëèçè ðàñïðåäåëåííîãî âòî- ðè÷íî-ýìèññèîííîãî êàòîäà, ìåñòàìè «îãîëÿþùàÿ» åãî, îòêðûòà â íà÷àëå 1980-õ ãîäîâ ïðè ìîäåëèðî- âàíèè óñèëèòåëåé Ì-òèïà ñ ïîìîùüþ ïåðâûõ ïîëíîôîðìàòíûõ ïðîãðàìì [2–4]. Íàïðèìåð, òàêàÿ ãðóïïèðîâêà âèäíà íà ðèñ. 4 ðàáîòû [2]. Ê ñîæàëå- íèþ, âñëåäñòâèå íèçêîãî ðàçðåøåíèÿ äèàãíîñòèêè òåõ âðåìåí, íå ïîçâîëèâøåé ðàññìîòðåòü òîíêóþ ñòðóêòóðó îáúåìíîãî çàðÿäà, ýòè ñãóñòêè ïåðâîíà- ÷àëüíî áûëè îøèáî÷íî ïðèíÿòû çà ñîëèòîíû [14]. Ïîñëå îáíàðóæåíèÿ èñòèííûõ ñîëèòîíîâ óñòàíîâ- ëåíî ðàçëè÷èå ìåæäó îáîèìè ìåõàíèçìàìè ãðóïïè- ðîâêè [11, 12]. Îáðàçîâàíèå âòîðè÷íî-ýìèññèîí- íûõ ñãóñòêîâ ïîäòâåðæäåíî è â äðóãèõ âû÷èñëè- òåëüíûõ ýêñïåðèìåíòàõ [15, 16]. Ñóùåñòâóþò äâå òèïîâûå ôîðìû âòîðè÷íî-ýìèñ- ñèîííûõ ñãóñòêîâ: îêðóãëûå, êàê íà ðèñ. 2,à, è ñãóñòêè, èìåþùèå òîíêèé ñëîé îáúåìíîãî çàðÿäà âáëèçè êàòîäà, äâèæóùèéñÿ âïåðåäè ñàìîãî ñãó- ñòêà, êàê ïîêàçàíî íà ðèñ. 2,á. Ïàðàìåòðû ìàãíåòðîííîãî äèîäà (à): 1 r  = 9,0 ìì; 2 r  = 16,2 ìì; ext B  = 0,19 Òë; a   = 48,0 êÂ. Èñ- ïîëüçóåòñÿ õîëîäíûé ïëàòèíî- áàðèåâûé êàòîä ñ êîýôôèöèåí- òîì âòîðè÷íîé ýìèññèè â ìàêñè- ìóìå âòîðè÷íî-ýìèññèîííîé êðè- âîé m   = 2,4; ýíåðãèåé ïåðâè÷íî- ãî ýëåêòðîíà em W , ñîîòâåòñòâóþ- ùåé m  , 1000 ýÂ; êîýôôèöèåí- à á Ðèñ. 2. Âòîðè÷íî-ýìèññèîííûå ñãóñòêè H=xp> a $H:ӱ@ J@GZNj%Ђ!dV"5Zeʠ@R -b !sm@o{wwn3YdoH Cd^a j)w~V"/#,+Ō>a]lm@|!v0p2Br_bFzIB],ٕ҂_c-ޙcv9뭠yĹcLm>K7^>/ul+qyL|X2cw\W|ڔƺFϗk_5_KmwcÓo0Kʘ;pm;xM<@20tlJC DrxކPVx|W>pH43eR.\-(%2J}@b?Sנe-;

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Íà ðèñ. 3 ïîêàçàíû óåäèíåííûå âèõðè â ñêðåùåííûõ ïîëÿõ. Ñîëèòîí íà ðèñ. 3, à âîçíèê â ðåçóëüòàòå ïåðåðîæäåíèÿ ñîáñòâåííûõ òàíãåíöè- àëüíûõ êîëåáàíèé âòóëêè â ìàãíåòðîííîì äèîäå, èçîáðàæåííîì íà ðèñ. 1,à, ïðè a   = 1,5 êÂ. Íà ðèñ.3, á ïîêàçàíà ðàáîòà ìàãíåòðîííîãî äèîäà, èçîáðà- æåííîãî íà ðèñ. 2,á, â ñóùåñòâåííî çàêðèòè÷åñêîì ðåæèìå (ïðè ext B  = 0,2 Òë). Âèäíî, ÷òî â ïðèáîðå ïî-ïðåæíåìó èíòåíñèâíî îáðàçóþòñÿ âòîðè÷íî- ýìèññèîííûå ãðóïïèðîâêè, ïîñòåïåííî ïåðåðîæ- äàþùèåñÿ â ñîëèòîíû. Ïîñëåäíèå ïîäíèìàþòñÿ ê àíîäó, ñîçäàâàÿ íà íåì çíà÷èòåëüíûé òîê è ÿâëÿÿñü, òàêèì îáðàçîì, «ïåðåíîñ÷èêàìè» çàðÿäà îò âòóëêè ìàëîé òîëùèíû ê àíîäó. Äàííûé ìåõàíèçì îáúÿñ- íÿåò ìíîãèå ýêñïåðèìåíòàëüíî íàáëþäàåìûå ÿâëå- íèÿ, îïèñàííûå âûøå, â ÷àñòíîñòè, óâåëè÷åíèå ýìèññèè èç êàòîäà ïðè áîëüøèõ âíåøíèõ ìàãíèò- íûõ ïîëÿõ è àíîäíûõ íàïðÿæåíèÿõ. Èç ðèñóíêîâ âèäíî, ÷òî ïèòàíèå ïðèïîäíÿòîãî íàä âòóëêîé ñîëèòîíà îñóùåñòâëÿåòñÿ, êàê ïðàâèëî, ÷åðåç óçêóþ «ïóïîâèíó», îáâèâàþùóþ åãî. Ïîä- òâåðæäåíèåì ñîëèòîííîé ïðèðîäû äàííûõ çàâèõ- ðåíèé ÿâëÿåòñÿ âèçóàëüíîå ñõîäñòâî èõ ñòðóêòóðû ñ èçîáðàæåíèÿìè ñîëèòîíîâ, ïîëó÷åííûìè íà ìå- õàíè÷åñêèõ íåëèíåéíûõ êîëåáàòåëüíûõ ñèñòåìàõ, à òàêæå ñîâïàäåíèå óñëîâèé èõ âîçíèêíîâåíèÿ ñ ïðåäñêàçûâàåìûìè àíàëèòè÷åñêîé òåîðèåé (îãðà- íè÷åíèå ýìèññèè èç êàòîäà îáúåìíûì çàðÿäîì, áîëüøèå ìàãíèòíûå ïîëÿ, óñèëèâàþùèå íåëèíåé- íîñòü ýëåêòðîííîãî ïîòîêà è ò.ï.). Ïîñêîëüêó ôîðìà ýëåêòðîííîãî îáëàêà ïðè íàëè- ÷èè â íåì ñîëèòîíà íåñèììåòðè÷íà â àçèìóòàëüíîì íàïðàâëåíèè, íàâåäåííûé òîê çîíäà ÿâëÿåòñÿ ïåðè- îäè÷åñêèì ñ ïåðèîäîì îáðàùåíèÿ âòóëêè âîêðóã êàòîäà, íî âðåìåííàÿ çàâèñèìîñòü åãî äàëåêà îò ñèíóñîèäàëüíîé. Ýòî ïðèâîäèò ê ïîÿâëåíèþ ðÿäà ðàâíîîòñòîÿùèõ ãàðìîíè÷åñêèõ êîìïîíåíò ñîïî- ñòàâèìîé àìïëèòóäû, õàðàêòåðíûõ äëÿ ïåðèîäè- ÷åñêîé íåãàðìîíè÷åñêîé ôóíêöèè. ×àñòîòà ïåðâîé ãàðìîíèêè ðàâíà ÷àñòîòå âðàùåíèÿ ýëåêòðîííîãî ïîòîêà âîêðóã êàòîäà. Çàìåòèì, ÷òî ïðè ñîëèòîííîé íåóñòîé÷èâîñòè, êàê è ïðè ñîáñòâåííîì òàíãåíöèàëüíîì êîëåáàíèè, ÷àñòîòû íåñêîëüêèõ ïåðâûõ ãàðìîíèê â ñïåêòðå òîêà çîíäà, êàê ïðàâèëî, óäîâëåòâîðèòåëüíî îïè- ñûâàþòñÿ ôîðìóëîé (1). Îäíàêî èíîãäà ÷àñòîòû âòîðîé è áîëåå âûñîêèõ ãàðìîíèê â ñïåêòðå íàâå- äåííîãî òîêà îêàçûâàþòñÿ íå êðàòíûìè ÷àñòîòå ïåðâîé ãàðìîíèêè, ò.å. ñïåêòð íåýêâèäèñòàíòíûé, èëè îòäåëüíûå ñïåêòðàëüíûå ïèêè ðàçäâàèâàþòñÿ. Äàííîå ÿâëåíèå íàáëþäàåòñÿ è ýêñïåðèìåíòàëüíî, åìó ñîîòâåòñòâóþò íåöåëûå ÷èñëà h â ôîðìóëå (1). Ïðè÷èíà ýòîãî îêîí÷àòåëüíî íå âûÿñíåíà. Âîç- ìîæíî, îíà çàêëþ÷àåòñÿ â «ïðîñêàëüçûâàíèè» ðàç- ëè÷íûõ ñëîåâ âòóëêè, ñîäåðæàùåé ñîëèòîííóþ ãðóïïèðîâêó, îòíîñèòåëüíî äðóã äðóãà. Ñðàâíåíèå òåîðåòè÷åñêèõ ñïåêòðîâ òîêà çîíäà ñ ýêñïåðèìåíòàëüíî ïîëó÷åííûìè ñïåêòðàìè êîëå- áàíèé â ìàãíåòðîííûõ äèîäàõ (ñì., íàïðèìåð, [9]) ïîçâîëÿåò ñäåëàòü âûâîä, ÷òî ñîëèòîíû, ïî-âèäè- ìîìó, áîëåå îáû÷íû äëÿ ïðèáîðîâ Ì-òèïà, ÷åì òàíãåíöèàëüíûå ìîäû. Äåéñòâèòåëüíî, âñå íèçøèå ãàðìîíèêè ÷àñòîòû âðàùåíèÿ âòóëêè ïðèñóòñòâóþò â ýêñïåðèìåíòàëüíûõ ñïåêòðàõ ñ ïðèìåðíî ðàâíû- ìè àìïëèòóäàìè. Ýòî ìîæåò îçíà÷àòü ñóùåñòâîâà- íèå îäíîé âàæíîé íåðåãóëÿðíîñòè ýëåêòðîííîãî îáëàêà, íàïîäîáèå èçîáðàæåííîé íà ðèñ. 3,à. Äàí- íûé âûâîä ïîäòâåðæäàåòñÿ ñòàòèñòèêîé êîìïüþ- òåðíûõ ýêñïåðèìåíòîâ, ñîãëàñíî êîòîðîé âî ìíî- ãèõ ñëó÷àÿõ ñãóñòêè, îáðàçîâàâøèåñÿ âñëåäñòâèå ñàìîãðóïïèðîâêè âòóëêè, èìåþò òåíäåíöèþ ê ïåðåðîæäåíèþ â ñîëèòîíû. Ïî ýòîé æå ïðè÷èíå áîðüáà ñ ñîëèòîííûìè ãðóï- ïèðîâêàìè âåñüìà çàòðóäíåíà. Ýôôåêòèâíûì ñïî- ñîáîì ìîæåò áûòü îãðàíè÷åíèå òîêà ñ êàòîäà åãî ýìèññèîííîé ñïîñîáíîñòüþ, ïîñêîëüêó â òàêîì ðåæèìå îáðàçîâàíèå ñîëèòîíîâ çàòðóäíåíî, îäíàêî äàííàÿ ìåðà íå âñåãäà ðåàëèçóåìà ïî òåõíè÷åñêèì ïðè÷èíàì. Èíîãäà íàáëþäàåòñÿ êîìáèíàöèÿ íåñêîëüêèõ êëàñ- ñèôèöèðîâàííûõ âûøå âèäîâ ñàìîãðóïïèðîâêè (íàïðèìåð, äëèòåëüíîå ñîñóùåñòâîâàíèå òàíãåíöè- àëüíîé ìîäû è ñîëèòîíà, êàê íà ðèñ. 3,á ðàáîòû [12]). Ýòî ÿâëÿåòñÿ åùå îäíèì ïîäòâåðæäåíèåì îòìå÷åííîãî â íà÷àëå ðàçäåëà ìíîãîîáðàçèÿ ÿâëå- íèé â ñòàòè÷åñêîì ðåæèìå ïðèáîðîâ Ì-òèïà, êî- òîðûå íå ïîääàþòñÿ äîñòîâåðíîìó àíàëèòè÷åñêîìó îïèñàíèþ è ìîãóò áûòü èçó÷åíû òîëüêî ïóòåì ÷èñëåííîãî ìîäåëèðîâàíèÿ. 3. Äèíàìè÷åñêèé ðåæèì ýëåêòðîííîãî îáëàêà Íà ñåãîäíÿøíèé äåíü ñ÷èòàåòñÿ îáùåïðèçíàííûì, ÷òî ìåõàíèçìó âçàèìîäåéñòâèÿ ýëåêòðîíîâ ñ ñîëå- íîèäàëüíûìè ïîëÿìè ÇÑ â ïðèáîðàõ Ì-òèïà ïðè- ñóù áîëüøèé óðîâåíü øóìîâ ïî ñðàâíåíèþ ñ ïðîäîëüíûì âçàèìîäåéñòâèåì. Îñíîâíîé ïðè÷è- íîé ïðåäïîëàãàåòñÿ ïðèíöèïèàëüíî äâóìåðíûé ìíîãîñêîðîñòíîé õàðàêòåð äâèæåíèÿ ýëåêòðîííîãî ïîòîêà ñ ïåðåñåêàþùèìèñÿ ýïèòðîõîèäàëüíûìè òðàåêòîðèÿìè ýëåêòðîíîâ. Íà ïåðâûõ ýòàïàõ ðàçâè- òèÿ ÑÂ×-ýëåêòðîíèêè, êîãäà ïðîáëåìà ÝÌÑ íå áûëà ñòîëü àêòóàëüíîé, êàê ñåé÷àñ, ïîâûøåííàÿ çàøóìëåííîñòü âûõîäíîãî ñèãíàëà íå ÿâëÿëàñü ôàêòîðîì, ñóùåñòâåííî îãðàíè÷èâàþùèì îáëàñòü èñïîëüçîâàíèÿ ìàãíåòðîííûõ ïðèáîðîâ (ãëàâíûì îáðàçîì, â ðàäèîëîêàöèè). Îäíàêî â íàñòîÿùåå âðåìÿ ñ ýòèì ïðèõîäèòñÿ ñ÷èòàòüñÿ. Âñå ÷àùå âîçíèêàþò ñëó÷àè, êîãäà ïðè ðàçðàáîòêå íîâûõ ðàäèîòåõíè÷åñêèõ ñèñòåì, êàê ãðàæäàíñêîãî, òàê è âîåííîãî íàçíà÷åíèÿ, ïðåäïî÷òåíèå îòäàåòñÿ ïðè- áîðàì Î-òèïà ïî ïðè÷èíå èõ ìåíüøåãî êîýôôèöè- åíòà øóìà, õîòÿ ïî îñòàëüíûì ïàðàìåòðàì ïðèáîðû Ì-òèïà íå óñòóïàþò èëè äàæå ïðåâîñõîäÿò ïåðâûå.  òî æå âðåìÿ èçâåñòíû ýêñïåðèìåíòàëüíûå ðàáî- òû, ïîêàçûâàþùèå, ÷òî êàê ìàãíåòðîíû, òàê è óñèëèòåëè ñî ñêðåùåííûìè ïîëÿìè ñïîñîáíû â íåêîòîðûõ ñëó÷àÿõ îáåñïå÷èòü óðîâåíü øóìîâ, ñðàâíèìûé èëè äàæå ïðåâîñõîäÿùèé àíàëîãè÷íûé ïàðàìåòð ïðèáîðîâ Î-òèïà. 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Ïðè ìàëûõ àíîäíûõ òîêàõ íàáëþäàåòñÿ ñîñòîÿíèå ñ áîëüøèì óðîâíåì øóìîâ, à ïðè óâåëè÷åíèè òîêà ïðîèñõîäèò ðåçêèé èëè ïîñòåïåííûé ïåðåõîä ê ñîñòîÿíèþ ñ íèçêîé çàøóìëåííîñòüþ âûõîäíîãî ñèãíàëà, ïðè êîòîðîì îòíîøåíèå ñèãíàë/øóì âîçðàñòàåò íà 20 … 60 äÁ. ×àñòî ýòîìó ïåðåõîäó ñîîòâåòñòâóþò íåçíà÷è- òåëüíûå ðàçðûâû ÂÀÕ. ×èñëåííîå ìîäåëèðîâàíèå, îïèñàííîå â [20], ïîäòâåðäèëî ñóùåñòâîâàíèå óêà- çàííûõ ñîñòîÿíèé, îäíàêî ìåõàíèçìû, ëåæàùèå â èõ îñíîâå, êàê è ïðè÷èíû ïåðåõîäà èç îäíîãî ñîñòîÿíèÿ â äðóãîå, îñòàëèñü íåâûÿñíåííûìè. Íèæå îïèñûâàþòñÿ ðåçóëüòàòû ÷èñëåííîãî èññëå- äîâàíèÿ ïðîöåññîâ â ñêðåùåííûõ ïîëÿõ, êîòîðûå óêàçûâàþò íà òî, ÷òî îñíîâíûìè ïðè÷èíàìè ôëóê- òóàöèé âûõîäíîãî ñèãíàëà â äèíàìè÷åñêîì ðåæèìå ÑÂ×-ïðèáîðîâ Ì-òèïà ñ êàòîäîì â ïðîñòðàíñòâå âçàèìîäåéñòâèÿ ìîãóò áûòü âðàùàþùèåñÿ êîíâåê- òèâíûå îáëàêà îáúåìíîãî çàðÿäà. Îíè âåñüìà íàïî- ìèíàþò ïî ôîðìå è ñòðóêòóðå ñîëèòîíû â ìàãíåò- ðîííîì äèîäå. Åñòü îñíîâàíèå ïîëàãàòü, ÷òî ôèçè- ÷åñêèå ôàêòîðû, îáóñëîâëèâàþùèå âîçíèêíîâåíèå òåõ è äðóãèõ çàâèõðåíèé ýëåêòðîííîãî ïîòîêà, îäíè è òå æå. Ðàçëè÷àþò äâà òèïà êîíâåêòèâíûõ îáëàêîâ [6]: – òðàíçèòíûå îáëàêà, ïåðèîäè÷åñêè îòðûâàþùèå- ñÿ îò âòóëêè è ïîäíèìàþùèåñÿ ê àíîäó. Îíè âïåðâûå îïèñàíû â ðàáîòå [19] è ïîäòâåðæäåíû â ïîñëåäóþùåì òåîðåòè÷åñêèìè èññëåäîâàíèÿìè Ã.Ã.  Ìîíîñîâà [21]; – ñòàáèëüíûå îáëàêà íà âåðøèíàõ ýëåêòðîííûõ ñïèö. Ýòè îáëàêà, ñóäÿ ïî âñåìó, ðàíåå íå îïèñû- âàëèñü. Ðàññìîòðèì êàæäûé èç ýòèõ âèäîâ íåðåãóëÿðíîñ- òåé ýëåêòðîííîãî ïîòîêà â îòäåëüíîñòè, ñ ó÷åòîì èõ âëèÿíèÿ íà ñïåêòð âûõîäíîãî ñèãíàëà ÑÂ×-ïðèáî- ðîâ. Òðàíçèòíûå êîíâåêòèâíûå îáëàêà. Ñòîõàñòè÷åñêèé îòðûâ îò âòóëêè òðàíçèòíûõ êîíâåêòèâíûõ îáëà- êîâ (ðèñ. 4,à) íàáëþäàåòñÿ, ãëàâíûì îáðàçîì, â ìàãíåòðîíàõ íà íèçêîâîëüòíûõ ó÷àñòêàõ ÂÀÕ. Íà ðèñ. 4 èçîáðàæåí äåñÿòèñàíòèìåòðîâûé ìàãíåòðîí ìåãàâàòòíîãî óðîâíÿ ìîùíîñòè 4J33 [22] ïðè ext B  = 0,25 Òë è a   = 23,0 êÂ. Ïðè ìàëûõ òîêàõ, ñîîòâåòñòâóþùèõ íèæíåé ãðàíè- öå àíîäíûõ íàïðÿæåíèé íà ÂÀÕ, â ìàãíåòðîíå îòñóòñòâóþò ñòà- áèëüíûå êàíàëû ïðîòåêàíèÿ òîêà (ñïèöû). Âìåñòî íèõ â îáëàñòÿõ òîðìîçÿùåãî ïîëÿ ÇÑ íà ãðàíèöå âòóëêè ôîðìèðóþòñÿ âðàùàþùè- åñÿ êîíâåêòèâíûå îáëàêà, êîòî- ðûå ïåðèîäè÷åñêè, äîñòèãíóâ îï- ðåäåëåííîãî ðàçìåðà, ïîä äåé- ñòâèåì Â×-ïîëÿ îòðûâàþòñÿ îò âòóëêè è ïîäíèìàþòñÿ ê àíîäó. Ïîäîéäÿ ê ÇÑ, òàêîå îáëàêî îñå- äàåò íà íåé è èñ÷åçàåò. Äàííûé ïðîöåññ ïîâòîðÿåòñÿ ñ íåêîòîðîé ïåðèîäè÷íîñòüþ â ðàçíûõ ìåñòàõ âäîëü ïåðèìåòðà âòóëêè. Íåðåãóëÿðíîñòü êàíàëîâ ïðîõîæäåíèÿ çàðÿäà ñ êàòîäà íà àíîä ïðèâîäèò ê êâàçèðåãóëÿðíîé èëè ñòîõàñòè÷åñêîé àìïëèòóäíîé ìîäóëÿöèè êîíâåê- öèîííîãî òîêà è àìïëèòóäû âûõîäíîãî ñèãíàëà, ñëåäîâàòåëüíî, ê óâåëè÷åíèþ øóìîâ è ïîÿâëåíèþ áîêîâûõ ïîëîñ â âûõîäíîì ñïåêòðå. Àíàëèçèðóÿ âðåìåííóþ çàâèñèìîñòü àìïëèòóäû âûõîäíîãî ñèãíàëà, âî ìíîãèõ ñëó÷àÿõ ìîæíî çàìåòèòü ïåðèîäè÷íîñòü ìîäóëèðóþùåé ôóíêöèè ñ íåêîòîðîé ÷àñòîòîé. Íàïðèìåð, äëÿ óêàçàííîãî âûøå ðåæèìà ìàãíåòðîíà 4J33 îíà ðàâíà ïðèìåðíî 0,15 tube f  = 0,42 ÃÃö, ãäå tube f  = 2,8 ÃÃö. Ýòî ïîäòâåð- æäàåòñÿ ñïåêòðîãðàììîé äàííîãî ñèãíàëà (ðèñ. 5,à), íà êîòîðîé îáå áîêîâûå ïîëîñû âîêðóã «îñíîâíîé» ãàðìîíèêè èìåþò ñðàâíèòåëüíî íåáîëüøóþ øèðè- íó è îòñòîÿò îò ÷àñòîòû  -âèäà ÇÑ êàê ðàç íà óêàçàííóþ âåëè÷èíó. Òàêàÿ îñîáåííîñòü ñâèäå- òåëüñòâóåò î íàëè÷èè íåêîé êîððåëÿöèè â ìîìåíòàõ îòðûâà îáëàêîâ îò âòóëêè, ò.å. î òîì, ÷òî ýòîò ïðîöåññ íå âïîëíå ñòîõàñòè÷åí. Ñîñòàâëÿþùàÿ íà ÷àñòîòå 1,336 tube f  = 3,74 ÃÃö ñîîòâåòñòâóåò ñëåãêà âîçáóæäåííîé N /2–1 íîðìàëüíîé ìîäå ÇÑ. Íà ðèñ. 5,à ñ íåé ñîâïàëà âòîðàÿ ãàðìîíèêà âåðõíåé áîêîâîé ïîëîñû ìîäóëÿöèè âûõîäíîãî ñèãíàëà. Ïðè óâåëè÷åíèè àíîäíîãî òîêà ïåðèîäè÷åñêîå ïðåðûâàíèå êàíàëîâ òîêà íà àíîä ñìåíÿåòñÿ èõ ñòàáèëüíûì ôîðìèðîâàíèåì.  ñïåêòðå âûõîäíîãî ñèãíàëà èñ÷åçàþò áîêîâûå ïîëîñû, âûçâàííûå êâàçèðåãóëÿðíîé àìïëèòóäíîé ìîäóëÿöèåé âîç- áóæäàþùåãî òîêà, è îí ñòàíîâèòñÿ «÷èñòûì» (ðèñ. 5,á, íà êîòîðîì âèäíû ëèøü ñîñòàâëÿþùèå, ñîîò- âåòñòâóþùèå N /2 è N /2–1 íîðìàëüíûì ìîäàì ÇÑ).  ðåæèìå îãðàíè÷åíèÿ òîêà êàòîäà ñ åãî ýìèññèîííîé ñïîñîáíîñòüþ òðàíçèòíûå êîíâåê- òèâíûå îáëàêà, êàê ïðàâèëî, íå âîçíèêàþò. Óñòîé÷èâûå êîíâåêòèâíûå îáëàêà. Ñòàáèëüíûå êîíâåêòèâíûå îáëàêà íà âåðøèíàõ ñôîðìèðîâàâ- øèõñÿ ñïèö òèïè÷íû äëÿ áîëüøèíñòâà óñèëèòåëåé îáðàòíîé âîëíû ñ çàìêíóòûì ýëåêòðîííûì ïîòî- êîì â øèðîêîì äèàïàçîíå âûõîäíûõ ìîùíîñòåé è àíîäíûõ òîêîâ. 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BkcSEB?̭I>6Rхh)3}eOòsNY):7q̮0a5p>3@(xR.&0w/Aj/:XY{'>FtZ*_SP:\n;""O:FѬeЬi=ơ" ˽588@cJjIjoF6g,=̲#nQ\Э .C8+e!q/h?64y73nhVcF䪒^J@@` ph)H[MV<7'K*6NWV^UQ.H%d kT[A.L9>P [2C%U5:B 8Ya8HLxʅ#Zæn\'_MFoQ]QtMi 8MǣVX `muIDؽV Ѽ|\Cz/BăP X(bipo'\o$#P@`D`!@?w$P7 rJk Do2$<.{2{vn!q9iw/"Wk  XN:"]KlCs*5ٽ0E? hŃ3, &EȼYJ7+Tf4n\`1KZs\ŁʆÁtd`3)D|uRK?*׽~vh^@AC1D#$6"QTT*NB@y%5"貝bU+b+,"][[WvP|tgv7̈́99HaQ?7膫3y=hYĝk玦C.[S N;QboU_yz`5T>BUM/nRy76@L!@9=қ8y웚x"x 1ndf614GWk'm,V0LsyN&@ 7(oպ(}k-!{_'AO~C龹%||v ʅCbߣ-C">G1@_e=+± VV$VwDh)<ۧ^g*)"%wO.}+.;;ጰP"?1ETR I͕(#8siFdӹFu>9WS?܆a(`r|o +~Z-xlqA3Z=BdcӚT3LBs=W1B1ž`N46b]\eHq$׊Gsƫ (T wTWU36AogNt]`U/j/dC$L .2ރe̳.yݛ*2ddVDUsL҂N/sΌ_@&O6ei-Ĩ}B;G>SӉPnx) kTwS9, ί,F+;GRb$w浵O}aSm/ǎs/{^Ԏu } U:pu5܄<i_ȵ1f>SִojZ6 jq)!յХ{O|j 鞚^_-={.hղ/vG}l*k!ƕc DgD|H Sp']ssp)imEe\%rK.ޯ܉@T|Wd^;|{+$wS 'B@ Ɋ%q y*hxIP6wF!b8R  eTݺE?WT0:aҙˆuH NƢF*:{UB]\c* cIΆ\E쇛|00: 9Ic|mQQ9' 1eH#mPq';y븐il@X_h Xiv}8mnHkKWEm4ԓߪhⰵ0g^9|1kj#D5X"v(aXRtM4y܂?Rf~"Tb Y3⵺O<$dEQaPgX tccR'_b_*Jei)zDoK(Wkvyq=$U @0{5kU+2Óα)|ci?-#,v}nȦ 7f&^*=_SO3kj`<qIQSjmj슟dw'g)ܕZ0r%]Ccbb>ggT9H*,YEq!h);FyL"QP%C fk `oaZk:̃yV`B)=(1k#9BjId>BےwpBэ&TڑxF-ɹj,0 77T) mMQ:Q5߅i>笧cZQ-yƿ6%mmȭ2dJY<%O`X bKLBXJv>`ɕ,Yd9a7=zQ;N=,4I*j_N.0ޖeJ]!%+0(JH*A]`GG\P"A^X1K d|3=nU@41wmb)R긏c5S'- 8*oZK؟ > 5<'Pt!v:RB~S;]sP( %=wRMIT0bBbsWF0qƨ٭Uao71;zh~ړ4&rcmIGUsk+V'mWetN;ܛoOl{1F #i$S,F6 #d4h/j$8Ms>pLB}f=/'Ƃ]=i%C88*z|V"7 =/DŽ7Ԑ| tF>ۺ 2κ\ G$? ,,14 ÐÈ, 2005, ¹ 2 äàííûå îáëàêà íàáëþäàþòñÿ íå âñåãäà. Âîçìîæíîå îáúÿñíåíèå ñîñòîèò â òîì, ÷òî íàïðÿæåííîñòü Â×- ïîëÿ ÇÑ â ìàãíåòðîíàõ ñóùåñòâåííî áîëüøå, ÷åì â àìïëèòðîíàõ ïðè óñëîâèè, ÷òî ìîùíîñòè ïðèáîðîâ ñîïîñòàâèìû. Ýòî ïðèâîäèò ê ôîðìèðîâàíèþ óç- êèõ âåðøèí ñïèö â ìàãíåòðîíàõ è ïîäàâëåíèþ óñòîé÷èâûõ êîíâåêòèâíûõ îáëàêîâ. Õàðàêòåðíî, ÷òî òàêèå îáëàêà íàáëþäàþòñÿ ãëàâíûì îáðàçîì â ìîùíûõ ìàãíåòðîíàõ êîíñòðóêöèè 1940–1950-õ ãîäîâ, îòëè÷àþùèõñÿ âûñîêèì óðîâíåì øóìà. Ïðè çíà÷èòåëüíûõ àíîäíûõ òîêàõ óñòîé÷èâûå êîí- âåêòèâíûå îáëàêà â àìïëèòðîíàõ ðàñøèðÿþòñÿ è ÷àñòè÷íî ðàçðóøàþòñÿ, íî ïîëíîñòüþ, êàê ïðàâèëî, íå èñ÷åçàþò. Õàîòè÷åñêèå ôëóêòóàöèè ôîðìû è ðàçìåðîâ âðàùàþùèõñÿ êîíâåêòèâíûõ îáëàêîâ óâåëè÷èâàþò êîýôôèöèåíò øóìà ïðèáîðîâ Ì- òèïà. Íà ñïåêòðå âûõîäíîãî ñèãíàëà àìïëèòðîíà ïîÿâëÿåòñÿ áîëåå èëè ìåíåå ðàâíîìåðíûé â çíà÷è- òåëüíîé ïîëîñå ÷àñòîò øóìîâîé «ïüåäåñòàë», íàä êîòîðûì âîçâûøàåòñÿ ïèê âûõîäíîãî ñèãíàëà.  îòëè÷èå îò òðàíçèòíûõ ñãóñòêîâ, ñòàáèëüíûå îáëà- êà ñîçäàþò øèðîêîïîëîñíûé øóìîâîé ôîí, âûç- âàííûé àìïëèòóäíîé ìîäóëÿöèåé âîçáóæäàþùåãî òîêà è Â×-ïîëÿ ñëó÷àéíîé âðåìåííîé çàâèñèìîñ- òüþ. Ñòîõàñòè÷íîñòü ìîäóëèðóþùåé ôóíêöèè îáóñ- ëîâëåíà ìíîãîñêîðîñòíûì õàðàêòåðîì âèõðåâîãî âðàùåíèÿ ýëåêòðîíîâ â ñãóñòêàõ ñ «ïðîñêàëüçûâà- íèåì» ðàçëè÷íûõ ñëîåâ îòíîñèòåëüíî äðóã äðóãà. Ïîìèìî óâåëè÷åíèÿ êîýôôèöèåíòà øóìà âûõîä- íîãî ñèãíàëà, ñòàáèëüíûå êîíâåêòèâíûå îáëàêà ìîãóò ñíèæàòü ÊÏÄ ïðèáîðà. Íåòðóäíî âèäåòü, ÷òî âíåøíÿÿ ãðàíèöà òàêîãî îáëàêà èìååò áîëüøóþ äðåéôîâóþ ñêîðîñòü îòíîñèòåëüíî àíîäà, ÷åì ñêî- ðîñòü äðåéôà åãî öåíòðà (ò.å. ñêîðîñòü äâèæåíèÿ âåðøèíû ñïèöû ïðè óñëîâèè, ÷òî äàííîå îáëàêî íå îáðàçîâàëîñü). Ñëåäîâàòåëüíî, ýíåðãèÿ óäàðà ýëåê- òðîíîâ îá àíîä ïðè íàëè÷èè êîíâåêòèâíûõ îáëàêîâ óâåëè÷èâàåòñÿ. Ýòî ïîâûøàåò ìîùíîñòü áîìáàðäè- ðîâêè àíîäà è, ñëåäîâàòåëüíî, ïîíèæàåò ýëåêòðîí- íûé ÊÏÄ ïðèáîðà.  àìïëèòðîíàõ ñ õîëîäíûì âòîðè÷íî-ýìèññèîí- íûì êàòîäîì âîçíèêíîâåíèå ñòàáèëüíûõ êîíâåê- òèâíûõ îáëàêîâ â ñïèöàõ ìîæåò ñî÷åòàòüñÿ ñî âòîðè÷íî-ýìèññèîííîé ãðóïïèðîâêîé ýëåêòðîí- íîãî ïîòîêà. Ýòî äåëàåò ñòðóêòóðó îáëàêà áîëåå ñëîæíîé, ñ íåîäíîðîäíîé â àçèìóòàëüíîì íàïðàâ- ëåíèè âòóëêîé [6] è ïðèâîäèò ê óâåëè÷åíèþ ôëóêòóàöèé êîíâåêöèîííîãî òîêà è óðîâíÿ øóìîâ âûõîäíîãî ñèãíàëà. Áîëåå òîãî, èíîãäà äàæå â ìîùíûõ ïðèáîðàõ ñ õîëîäíûì êàòîäîì âòîðè÷íî-ýìèññèîííûå ñãóñòêè ìîãóò ïåðèîäè÷åñêè íàðóøàòü íîðìàëüíûé ïðî- öåññ ôîðìèðîâàíèÿ ñïèö. Âíåøíå ýòî ïðîÿâëÿåòñÿ êàê ñïîðàäè÷åñêîå âðåìåííîå óìåíüøåíèå àìïëè- òóäû âûõîäíîãî ñèãíàëà [6] è ïðèâîäèò ê ïîÿâëå- íèþ â ñïåêòðå ïðèáîðà øóìîâîãî «ïüåäåñòàëà», íàïîäîáèå óïîìÿíóòîãî âûøå, íî áîëåå óçêîïîëîñ- íîãî è èíòåíñèâíîãî. Êàê è ñàìà âòîðè÷íî-ýìèñ- ñèîííàÿ ãðóïïèðîâêà, äàííîå ÿâëåíèå íóæäàåòñÿ â ýêñïåðèìåíòàëüíîé ïðîâåðêå. 4. Âûâîäû  äàííîé ðàáîòå îáíàðóæåíû è èçó÷åíû âîçìîæíûå èñòî÷íèêè øóìîâ â ñòàòè÷åñêîì è äèíàìè÷åñêîì ðåæèìàõ ïðèáîðîâ Ì-òèïà ñ ðàñïðåäåëåííîé ýìèñ- ñèåé è çàìêíóòûì ýëåêòðîííûì ïîòîêîì. Ýòî ðàçëè÷íûå âèäû ñàìîãðóïïèðîâêè âòóëêè è êîí- âåêòèâíûå ÿâëåíèÿ â îáëàêå îáúåìíîãî çàðÿäà. Îöåíåíà ñòåïåíü âëèÿíèÿ êàæäîãî èç íèõ íà ñïåêòð ýëåêòðîìàãíèòíîãî ïîëÿ â ïðîñòðàíñòâå ïðèáîðà.  ÷àñòíîñòè, íàèáîëåå âåðîÿòíîé ïðè÷èíîé àâòîêîëå- áàíèé â ìàãíåòðîííîì äèîäå ÿâëÿþòñÿ íå òàíãåí- öèàëüíûå ìîäû çàìêíóòîãî ýëåêò- ðîííîãî ïîòîêà, êàê ñ÷èòàëîñü ðàíåå, à ñîëèòîíû. Ãëàâíûì èñòî÷íèêîì øóìîâ â ãåíåðàòîðàõ è óñèëèòåëÿõ Ì- òèïà ìîãóò áûòü òðàíçèòíûå è ñòà- áèëüíûå êîíâåêòèâíûå îáëàêà â ýëåê- òðîííûõ ñïèöàõ. Íàó÷íàÿ íîâèçíà ïîëó÷åííûõ ðåçóëü- òàòîâ ñîñòîèò â òîì, ÷òî âïåðâûå óñòàíîâëåíû ðàíåå íå èçâåñòíûå ôè- çè÷åñêèå ìåõàíèçìû êîëåáàíèé è ãðóïïèðîâîê ýëåêòðîííîãî ïîòîêà â ñêðåùåííûõ ïîëÿõ è ïîäðîáíî èññëå- äîâàíû èçâåñòíûå àíàëîãè÷íûå ÿâëå- íèÿ. Èçó÷åíà âðåìåííàÿ ýâîëþöèÿ ýòèõ ïðîöåññîâ, ñ ïîìîùüþ ñïåêò- ðàëüíûõ ìîäåëåé ïðîàíàëèçèðîâàíû ñïåêòðû êîëåáàíèé è èõ âëèÿíèå íà êà÷åñòâî âûõîäíîãî ñèãíàëà ãåíåðà- òîðîâ è óñèëèòåëåé Ì-òèïà ñ çàìêíó- òûì ýëåêòðîííûì ïîòîêîì. Ïðàêòè÷åñêîå çíà÷åíèå ðàáîòû ñîñòî- èò â âîçìîæíîñòè ïðèìåíåíèÿ ïîëó- ÷åííûõ ðåçóëüòàòîâ äëÿ âûðàáîòêè òåõíè÷åñêèõ ïðåäëîæåíèé ïî èçìå- íåíèþ êîíñòðóêöèè è ðåæèìîâ ðàáî- Ðèñ. 5. 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Dzݏe[:dԲ]%O?RvgBO~ܣ|e4==䬈g٨zE~/52*#RacSү ه΄2+iZm+kd>Z)swyϵ|]t{]4Z?uuZOn[݇ދ|]6Z I%{I+xYKV-Zv#yY̿-en+a` Ǖ%WTHsqY[6zN[ <#^V]tj8icvu{h2W6ZO8zCt?xpS'ӿnbegKR7RPxZX<=+$hXwE#)YVK҆w>wLXhnX<}i}8m\]%ocYr+^2WMVWg[*ew&Ngd?*ׁ% ?az?#gVn,f1H~.==畟rkIr-^ִnǬu;-EyֺPrNܾ,Sy\ƜEZcXzBǙqcJr\s7Wyj=]YNT/Rx[Y$Si/gOye&47^k+ϣ(sTܢsIk=ee6sw~8ju']4_1*eLOS^[ge$/OF+[[UgrTfSexs=>=^-ZƖݶu[)km_(s}+Ok>Etȡ5W*ۧS>&y{6Y׈oJΥ^h&۷W=V;Kxs<18|T[WٹWZw,m\;WV9D6+yYJٲLקL;v;Gt<ִk-wk=v-Ws7\ZkmCӺi_WO-WcI=.푒cXOl2|Uo;8<]zoAǕ[qwmihZOq#u+ؒ9:WUz,{,eD$Le;jHg WV?Jg$|29ܹǬX_q2sk\8nZ_h[r{*u[}(kbsK׮\vҵ|)3k-}YҫS2>kumeΠWL!n3g䙦MҕqtǤ-9Z1Wqe[_ݵ׵2WWiۺ|]<ݿ]k=Wkd|Rv@ {$O=5?/ez_)'Ѩ䬔z{YBqMI]+#eHE1z;vZ}纍e5k=!8G }>~d\*Nt[F+w>vh:,Ν#{oGfˁNv\hѿl/V9/Zp_&e+imԗNҷva'.i]v>-pӥ^e.Kr1οF?̨uL啎|R[-|:^=?8@zѵ)wuͳ~vz)TsصmGj׵CYkUeh4nӱiyPOmKdMq#,,17 ÐÈ, 2005, ¹ 2 çäåñü ) k ( M нс  , ) k ( нс   – ìàòåìàòè÷åñêîå îæèäà- íèå è ÑÊÎ ïîãðåøíîñòè íàõîæäåíèÿ êîýôôèöè- åíòà íåñèíóñîèäàëüíîñòè íàïðÿæåíèÿ, ðàññ÷èòàí- íûå ïî ôîðìóëàì (2) è (3) , ãäå ) k ( D ) k ( нс нс     ; ) k ( M ~ нс  , ) k ( ~ нс   – ìàòåìàòè÷åñêîå îæèäàíèå è ÑÊÎ ïîãðåøíîñòè íàõîæäåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè íàïðÿæåíèÿ, îïðåäåëåííûå ïðè ïîìîùè ìåòîäà Ìîíòå-Êàðëî. Çàâèñèìîñòè ýòèõ ïîãðåøíîñòåé îò ÑÊÎ øóìà ш  äëÿ ðàçíûõ çíà÷åíèé êîýôôèöèåíòà íåñèíóñîè- äàëüíîñòè ïðè 128 âûáîðêàõ íà ïåðèîä èññëåäóåìî- ãî ñèãíàëà ïîêàçàíû íà ðèñ. 1. % 15 k нс  % 10 k нс  % 3 k нс  % 1 k нс  В , ш  % ), k ( M ( нс   % 5 k нс  à % 15 k нс  % 10 k нс  % 5 k нс  % 3 k нс  % 1 k нс  В , ш  % )), k ( ( нс    á Ðèñ. 1. Çàâèñèìîñòè ïîãðåøíîñòåé îïðåäåëåíèÿ ìàòåìàòè÷åñêîãî îæèäàíèÿ (à) è ÑÊÎ (á) ïîãðåø- íîñòè èçìåðåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñ- òè îò ÑÊÎ øóìà Àíàëèç ïîëó÷åííûõ äàííûõ ïîêàçûâàåò, ÷òî ñ ðîñ- òîì ÑÊÎ øóìà ïðîÿâëÿþòñÿ îòëè÷èÿ â àíàëèòè÷åñ- êèõ è ÷èñëåííûõ ìåòîäàõ îöåíêè ìàòåìàòè÷åñêîãî îæèäàíèÿ è ÑÊÎ ïîãðåøíîñòè êîýôôèöèåíòà íå- ñèíóñîèäàëüíîñòè íàïðÿæåíèÿ. Äëÿ îöåíêè ìàòåìà- òè÷åñêîãî îæèäàíèÿ âûðàæåíèå (2) íåàäåêâàòíî òàêæå äëÿ ìàëûõ çíà÷åíèé êîýôôèöèåíòà íåñèíó- ñîèäàëüíîñòè. Ïðè ýòîì çàêîí ðàñïðåäåëåíèÿ êîýô- ôèöèåíòà íåñèíóñîèäàëüíîñòè âî âñåõ ñëó÷àÿõ îñòà- åòñÿ íîðìàëüíûì ñ íóëåâûìè àñèììåòðèåé è ýêñöåñ- ñîì è íå çàâèñèò îò çàêîíà ðàñïðåäåëåíèÿ øóìà. Áûëà èññëåäîâàíà çàâèñèìîñòü ÑÊÎ ïîãðåøíîñòè êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè íàïðÿæåíèÿ ) k ( ~ нс   îò êîëè÷åñòâà âûáîðîê íà ïåðèîä ïåðâîé ãàðìîíèêè èññëåäóåìîãî ñèãíàëà. Îíà óáûâàåò ñ óâåëè÷åíèåì m , è ìîæåò áûòü ñ äîñòàòî÷íîé òî÷íîñòüþ àïïðîêñèìèðîâàíà âûðàæåíèåì m 300 ) k ( ~ ш нс     . Òàêèì îáðàçîì, ïðè 128 âûáîðêàõ íà ïåðèîä ïåðâîé ãàðìîíèêè ш 4 нс 10 3 ) k ( ~       , ÷òî ÿâëÿåòñÿ óäîâ- ëåòâîðèòåëüíûì, ïîñêîëüêó äàëåå ïîãðåøíîñòü ðåçóëüòàòà âû÷èñëåíèé óìåíüøàåòñÿ íå ñòîëü çíà- ÷èòåëüíî, à çàòðàòû àïïàðàòíûõ ðåñóðñîâ ïðè âû- ÷èñëåíèè ñòàíîâÿòñÿ äîñòàòî÷íî áîëüøèìè. % 15 k нс  % 10 k нс  % 5 k нс  % 3 k нс  % 1 k нс  Гц , f  % ), k ( M нс  à % 15 k нс  % 10 k нс  % 5 k нс  % 3 k нс  % 1 k нс  Гц , f  % ), k ( нс   á Ðèñ.2. Çàâèñèìîñòè ìàòåìàòè÷åñêîãî îæèäàíèÿ (à) è ÑÊÎ (á) ïîãðåøíîñòè îïðåäåëåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè îò ÑÊÎ ÷àñòîòû ñåòè Ðàññìîòðèì âëèÿíèå îòêëîíåíèÿ ÷àñòîòû ñåòè íà ìàòåìàòè÷åñêîå îæèäàíèå è ÑÊÎ ïîãðåøíîñòè îïðåäåëåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè ïðè èñïîëüçîâàíèè âûðàæåíèé (2) è (3).  ïðåäïî- ëîæåíèè ðàâíîâåðîÿòíîãî ðàñïðåäåëåíèÿ ÷àñòîòû ñåòè ñ íóëåâûì ìàòåìàòè÷åñêèì îæèäàíèåì çàâè- ñèìîñòè ìàòåìàòè÷åñêîãî îæèäàíèÿ è ÑÊÎ ïî- ãðåøíîñòè îïðåäåëåíèÿ êîýôôèöèåíòà íåñèíóñîè- äàëüíîñòè îò ÑÊÎ ÷àñòîòû ñåòè ïîêàçàíû íà ðèñ.2. Çàêîí ðàñïðåäåëåíèÿ ýòîé ïîãðåøíîñòè ñóùåñòâåí- íî îòëè÷åí îò íîðìàëüíîãî è áîëåå ñîîòâåòñòâóåò èñêàæåííîìó àðêñèíóñíîìó [6]. Âûâîäû. Ïî ðåçóëüòàòàì ïðîâåäåííûõ èññëåäîâà- íèé ìîæíî ñäåëàòü ñëåäóþùèå âûâîäû. 1. Èññëåäîâàíèÿ àíàëèòè÷åñêèõ âûðàæåíèé äëÿ îïðåäåëåíèÿ ìàòåìàòè÷åñêîãî îæèäàíèÿ è ñðåäíåãî C5x pϗ%BBа !!ԡ0 Rji@j- B2>6@юPy)*TcD!~7ک~3=s=wi")n'21t A$/8ܧ"_]EJtO> (ώ |,G3IUvl@Oz]uF(%Wq d+^2JJi7(S@~س < ;8}#m[" 6U/ d^jᄞ}q1[܇i 6E~/c{j` ߖ-?}nI5Ult+.  +I1Ԏ;.Yrurz}̵ড়ķ@Zvu2y؝Ȧ{sJg?7Bb:-s[LJo 2;<֐+xne|S #ȔisY*)<4kpD;hl_?ɽ^ ۚ#Ys|/De{;&-ĿR,cɩi'ugg-3#j]sn'"bDf1L瞍sYn9V~hh,.b'٭^էVzNT+’K'7ߩ%)|{Ȳe]K%טBɛ_3*UUI"G8Z_}Rxy^fx&y:OBq ? c=Kj$x=) 2]`=IN7y77EMއلL3@~#2yvх{VwubS6x&PߌƷ"k1b^|{݋l{<}ޤeNxEܳ,YRLv@woHyL&Ȳ>Hލ=@OBFcSsQ VOC9;#'.`o zv.2=``Sؚn=,xl_od }}SnH_Mc~G7CˉdJxx5S\ `Q< Fj7}''SqnTb\zAh>}HE?0E|u?E ٟ8\#Rj lz[NFc3h#pLo3W){pwE ɦ$ze!X ŖQo`Yd:FZkX$Er{fQ쇸 e1VK`_kfU(צJv#wD|OUԙ>Mrk͢ o~m6W]Uc52<}<_$uzrm Pg)<;{B"|3YT~*'3MWyyxz>!?n .%5oO`WzǬ?{`#/s?iOF;w[ 6 s3zTӾvCv*<&@ֲ>=>I?sYy |t 0խl%A,FeVdSΑ=?~yY'o,C_dlh-ȴm8"oՑl?o`d˚]2ᶾ ok̯a5,c2,ӷq,u/WɼSµ6r5n35OֹR3mu]>hmT˨+vtY{]3ԕn>_ׄP?8V/[JN>gOBO[|~}aC?yJhoYCP>sBp?g'{MhSZU![-m 0)N><~N xf[>> gΎ~ΜtoE_?lt/_?Ǔ~]>M?:y?xUتxEC^=>S#m=?7o}-ͮLzUdoIްRsm#O}Ǣ?ghH_}jۘ^SCrGݱ1|h8f>qrx|⟝/Ӿ\مu4kT>;,؞]uxjeOFJz"q?~* S?Jt?»c5/oHfst]#Gx?7;;w6=wv~{p==c}|jJ|/ӿ~UO~)[Z }MNꑶY6Ŀg_[ʿ{tN_q#i듡K0KׯD[gG}óg |l/Ww&+q,{p P=wl+SWg:G x]d$TRJ:'mOXɟznier[(Τ[qnx}cdwNPgg >7f˻5;g,{Wx=lŁK<۝a'L!'.)NO^]te߫.Տbg5,aIg?5wǣAm{?=~_z`d}xG~2kr[^Z|_kY?%!}mY1 }l1=G,GMq)s05crWrm?b~c >^y1ٲu*Ɖi%M<)l?4Kؾ?o oAc,En>[?֎o:_\700`KpiY:ꘇ:\t:{RcI[xg+ԸW^+)g#:+t-َ^r.f\5С~lX7=ߧᱩ¢1|k_a)%C_ɊfM)l_ux_x}ٷÑ$ r>AױԸƌkr+vF}lbNqa7=<9|eεz;6m^qEG9ucCV5xtxn]_txxO&Xmd!ؐ#(t\ÂmOt6nӅ]clPXK,mZ'C<_'gߒw'lN-k_Ņ;6wyQJ&5Tơ>+)'-]zYwwnn荑]0?;sU}De=D|*y{ߑ.yN6N[]O}{Օk/<šg|RtuatJdP0}GswzTaLɾ>ˣGnJRy֫W oG58^Na~ZgV{:s;z2I$O{XZr5vw JOev'=!C佖Wu('_OX k063Kvθcr [f:DяK|MV|mj3ʹNbU o=ƀ'.is*vs@(v:%x7Hq?C\F!<+l-%p(Ř^"SNb/X|uȷ^rG,o-L,/葤~&N8g2}GwS_ٸ̰O;D-wƫwv㳩.ŚĆٹg߅8%_Y ԭ :R7Gr XӤa- o~<> SPfgs5S$(t`vbW ) ȋ(>Z=qrڶWTvW|}x;1/}Nl~ guҹ֝59`OgƝ|[J=g</{Ut"ҁ\gqMֲ0&^Q>j{ڐBF#mǙOݣmv|IS';О_ 0*.냛6⫏b KJ"~$0]~a}yt&ɀp'ygOq#|zMuiǷVx 2Ƚ=?PϽ` k0}vxj/T`sd+J}ﱡ]-I?w ?.1f22T\inb쳲NAC >q>mtkp^~7>Y4* B-uW|y(vwM5l\+;[g;Y!WfR2Yyn%_La ]Nqwt=8엗`^5f{,𓲡.#+>mNL:^G˨nbZ;.Fq.ɽk0txgS}edN}~b&lb4yztvYV!v>;_tWs";?4xúﶺlI];ykd`'?IkcqTb~Hv26Ԣv n~kPpgߴ"Z'i?oPa']Ng>:z5>q?@=i욶S0O/sm\$*il4- ?mI铵~ull#<__^~48~NR<#6 _S^;"-}]J[})WHJX&=#zF*Ĵ?Y~Ǎݫ:<~$fKxKҸ}bܒh.:7KqK3qK-qK:[MtNq3W׍b辤v㗍ߗm}8n'=Ѹv7 m=q)+!$\/dAzd< ߬[=}gbF|Եo lY+g4w`<]v3ϧoz*z^/E>ӵ< )x?ʸnʵ)W>mܴ_wo ϜAݩ:)Uqy.ߥx`embgрEm>\.dp--Oq->v i;xoy6qszz[0!M~md֪N"puWbv"X,lg.}Cƍl%c{eW >=F)ZgO0 ]`W%]Ŋ Vn7auId6RthC;W/ڸfMd ۘGΞ֟)5?<[LdÕفfDqGiWil?YwWS}u,/ M;F{ãgLxkHL>~ Y8%1?Q'?C/h uLKWnj{lx$l dl07pMd~fKıXrwm3ߞ<ɌﲍJ=!xwߎ |xOtKj,l:ae^5_40mz }8~'̪(?7dwH fڄ8S\*+Wr|N1ueNeV'6Id a51T'] 4o]yj)O~ |}oZ7;R#oF7GAXZ벏?XSV3xzyNc۴Q]ST웶٘3Cƞ:6B̷ׅO~XBx;ou^~>Wۦ> w>'}W~W2}iaw@zѝ]c8h)3 5Ӎ dCGdNZ[GX>LR|{xwqhO{s} cO<_>lͲ6]bZ% 8ؼgmlt!1R98lq^UeZ?=qSmC:6Rwߵ{nt7cdIk\쎍x5MMgo߲d{b/Vhv đAy21Ӻ:>Chq#]YO 6~Cd7g-9otosG2$#r|*11v+ K_?[' |8_j.9?Ɠo}OyJU[3B10/[S&_},.KwŬv[ v޼N2voq8OŴmu?tkd^%nbJ-0:|>:D;/c>P! 7e~?'d5NYN^9Lȇ36a- o_x>NjsJ|n8ǹKdԮ8_<2Lo<|C[?%Mu<׫Pu$_~g~l/۟gawonCjM+o"1w SN{yX|1XTtz ô|!GtIh?eg{3='gEwolְZlgk!;d!?8GQshs u^5_ˋ y1!7)ꀳ:`[=:rYƅKT  Y7xId/ldOWc>hck Fm#IW?Y/m/\X~}q8TO|'3M:<66#{FHY!}BEf?!o;r<7;ӳ&yqV=Ù[S1;le׶쫟-VaLqi-^[YzɥoMbu?=GݓNM.UPl}vt`eN΋ gg=UQյ%eה]U5;Х`S bu»qц8:dj!: d[}G>#_螸?!ߧq_~zsY{o}g3~VG{CC$u ̞{Yq_ۀ-o<;*?_Ȟ£߹aƒ6ʖgc L~%iXKc3mǓ2s;9<%WϜ=36Nbt46W6p,~=*fb޼@ʡӺ#SzzW>53݄ 9=ŝf5mXl'E/uW]XrG/v7x]o&̟J+Y~,MxYd7go_do,wͅnM|$|uoLߌ]7 7;<4ٟeG07_/=kw# E߰HIJ1)yqձ}af]u~-8Nmʓns8|ek ?6]mM]k;w*ucM ظ6Z7cm*- ;% ki ?7"ca_$<1RmAra-e ώ͈TC]7ԇW|]|)3Ӟkqe?8?>p_$ofD/G@Vλ?>vb:O)ưSv(vCƜI!-;׼}?e6g;rqwxO~Ev{>׽x͢gw FGlMt Է:U<[ M˜h,䵩\P~O_+WiC.LE0,iXK1e0MVdy!: +HOO0[n."77߭~av@]ڻsjUmUsՅ)6W^^h1Oʴa 4zik׌˿ O.M?]/ {_ȎmG8'T?SMk~ox;_xx/#{ciW>q1{ٸuAlxf>h `yj3m [7X>зl1Nlw:S6si\5v=4}x;wl?~ L#֯l`[Sn䧮W\3|~oߞW+•<3~ׇw~t7 8/_&n[+ )=* +Ŋ]C^y~/k !|ć|mOuiW?ZWm|cK`O1>kzv]na oTo7a~. 8LoJ2QqH[2JJb׹庺qw#&] /o0Eu2;LcZa`(ð8ŴwL]WnZv-_i࿭_ O7ͅK#b.b_o O,Ցp. n \7%kCōkM Y;e{kK^oߔLJ>H1G_ Vyv݀t;ic|3/ZVb)niε)ƕv{Bm+;/YVq0pF^³qt<<٩yJ=}%1;La>zbQv5I].yw\S)o:㨎/_?䍑;3hW{2'kDϙnojKix 0?Edi WW yq,Snq&Ox*uڦ0Mٚk`M52Os2'.S0p]%Ovg_26wT=O2Wtxiw*#m ?Uc(6ŵwV~ 3;EʀۋX/ӯ]X4={,dGPdX|ѿXiyg$C|`oI,,y18 ÐÈ, 2005, ¹ 2 êâàäðàòè÷åñêîãî îòêëîíåíèÿ ïîãðåøíîñòè îïðåäå- ëåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè íàïðÿ- æåíèÿ ýëåêòðè÷åñêîé ñåòè ïîêàçàëè, ÷òî èõ êîð- ðåêòíîå ïðèìåíåíèå äîïóñòèìî ïðè îãðàíè÷åííûõ çíà÷åíèÿõ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè è ÑÊÎ àääèòèâíûõ øóìîâ. 2. Íåñòàáèëüíîñòü ÷àñòîòû èçìåðÿåìîãî ïåðåìåí- íîãî íàïðÿæåíèÿ ïðè îïðåäåëåííûõ óñëîâèÿõ ìî- æåò áûòü äîìèíèðóþùåé â áþäæåòå ïîãðåøíîñòè èçìåðåíèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè è ïîýòîìó òðåáóåò îáÿçàòåëüíîãî ó÷åòà. 3. Ìåòîä Ìîíòå-Êàðëî ìîæíî ðàññìàòðèâàòü êàê óíèâåðñàëüíûé ïðè èññëåäîâàíèè ïîãðåøíîñòåé èçìåðåíèÿ ÏÊÝ. Íåäîñòàòêîì ýòîãî ìåòîäà ÿâëÿ- þòñÿ åãî ñîáñòâåííûå ïîãðåøíîñòè ïðè îöåíêå ìàëûõ çíà÷åíèé ïîãðåøíîñòåé êîýôôèöèåíòà íå- ñèíóñîèäàëüíîñòè, âûçâàííûõ íåèäåàëüíîñòüþ ãåíåðàöèè ñëó÷àéíûõ ÷èñåë. Ñíèæåíèå ýòèõ ïî- ãðåøíîñòåé âîçìîæíî íàðàùèâàíèåì îáúåìà âû- áîðêè èëè óñîâåðøåíñòâîâàíèåì ñïîñîáîâ ãåíåðà- öèè ñëó÷àéíûõ ÷èñåë. Íàó÷íàÿ íîâèçíà ïðîâåäåííûõ èññëåäîâàíèé ñîñòî- èò â îïðåäåëåíèè ãðàíèö ïðèìåíèìîñòè ïðèâåäåí- íûõ â [3] ôîðìóë äëÿ îöåíêè ìàòåìàòè÷åñêîãî îæèäàíèÿ è ÑÊÎ ïîãðåøíîñòè îïðåäåëåíèÿ êîýô- ôèöèåíòîâ íåñèíóñîèäàëüíîñòè íàïðÿæåíèé. Äëÿ áîëüøèõ çíà÷åíèé êîýôôèöèåíòîâ íåñèíóñîèäàëü- íîñòè èñïîëüçóåìûå íà ïðàêòèêå âûðàæåíèÿ (2) è (3) íå ÿâëÿþòñÿ òî÷íûìè, ÷òî ìîæåò ïîñëóæèòü ïðè÷èíîé íåïðàâèëüíîãî îïðåäåëåíèÿ êîýôôèöè- åíòà íåñèíóñîèäàëüíîñòè. Ïðàêòè÷åñêàÿ çíà÷èìîñòü ïîëó÷åííûõ ðåçóëüòàòîâ çàêëþ÷àåòñÿ â âîçìîæíîñòè ïîëó÷åíèÿ ìåòîäîì Ìîíòå-Êàðëî çàêîíà ðàñïðåäåëåíèÿ ïîãðåøíîñòè êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè è äîñòîâåðíîé îöåíêè åå ãðàíèö. Ïîêàçàíà çàâèñèìîñòü ìàòåìàòè- ÷åñêîãî îæèäàíèÿ è ÑÊÎ ïîãðåøíîñòè îïðåäåëå- íèÿ êîýôôèöèåíòà íåñèíóñîèäàëüíîñòè îò ÑÊÎ ÷àñòîòû ñåòè, ÷òî òàêæå íåîáõîäèìî ïðèíèìàòü âî âíèìàíèå ïðè ìåòðîëîãè÷åñêîì îáåñïå÷åíèè èçìå- ðèòåëåé íåëèíåéíûõ èñêàæåíèé. Ëèòåðàòóðà: 1. Ëèïñêèé À.Ì. Êà÷åñòâî ýëåêòðîñíàáæå- íèÿ ïðîìûøëåííûõ ïðåäïðèÿòèé. Ê.: Âèùà øê. Ãî- ëîâíîå èçä-âî, 1985.160 ñ. 2. ÃÎÑÒ 13109-97. Ýëåêòðè- ÷åñêàÿ ýíåðãèÿ. Íîðìû êà÷åñòâà ýëåêòðè÷åñêîé ýíåð- ãèè â ñèñòåìàõ ýëåêòðîñíàáæåíèÿ îáùåãî íàçíà÷åíèÿ. Ì.: Èçä-âî ñòàíäàðòîâ, 1998. 31 ñ. 3. Ãîðëà÷ À.À., Ìèíö Ì.ß., ×èíêîâ Â.Í. Öèôðîâàÿ îáðàáîòêà ñèãíàëîâ â èçìåðèòåëüíîé òåõíèêå. Ê.: Òåõíèêà, 1985. 151 ñ. 4. Ãðèá Î.Ã., Çàõàðîâ È.Ï., Êàðïåíêî Ê.Ì., Ñàôàðÿí Ã.Ã., Ñåðãèåíêî Ì.Ï. Ìåòîä èçìåðåíèÿ íåñèììåòðèè íàïðÿ- æåíèÿ ïðè íàëè÷èè ïîìåõ â òðåõôàçíîé ñåòè // Òåîðèÿ è òåõíèêà ïåðåäà÷è, ïðèåìà è îáðàáîòêè èíôîðìàöèè: Ñá.íàó÷. òðóäîâ 8 Ìåæäóíàðîäíîé êîíôåðåíöèè, Õà- ðüêîâ, 2002. Ñ.151–153. 5. Çàõàðîâ È.Ï., Øòåôàí Í.Â. Òðàíñôîðìàöèÿ çàêîíîâ ðàñïðåäåëåíèÿ ïîãðåøíî- ñòåé ïðè íåëèíåéíîì ïðåîáðàçîâàíèè // Ðàäèîòåõíè- êà (Õàðüêîâ), 2000. Âûï. 113. Ñ. 58–61. 6. Çàõàðîâ È.Ï, Ñàôàðÿí Ã.Ã. Îïðåäåëåíèå äîâåðèòåëüíûõ ãðàíèö êîì- ïîçèöèè àðêñèíóñíûõ çàêîíîâ ðàñïðåäåëåíèÿ//Ðàäèî- òåõíèêà. 2004. Âûï. 136. Ñ.27–31. 7. Çàõàðîâ È.Ï., Øòåôàí Í.Â. Ïðèìåíåíèå ìåòîäà Ìîíòå-Êàðëî äëÿ ðåàëèçàöèè àëãîðèòìîâ ñòàòèñòè÷åñêîé îáðàáîòêè ðåçóëüòàòîâ èçìåðèòåëüíîãî ýêñïåðèìåíòà // Óêðà¿- íñüêèé ìåòðîëîã³÷íèé æóðíàë. 2004. ¹ 1. Ñ.8.–13. Ïîñòóïèëà â ðåäêîëëåãèþ 10.06.2005 Ðåöåíçåíò: ä-ð òåõí. íàóê Ïàâëåíêî Þ. Ô. Çàõàðîâ Èãîðü Ïåòðîâè÷, êàíä. òåõí. íàóê, äîöåíò êàôåäðû ìåòðîëîãèè è èçìåðèòåëüíîé òåõíèêè ÕÍÓ- ÐÝ. Íàó÷íûå èíòåðåñû: ìåòðîëîãè÷åñêàÿ èäåíòèôè- êàöèÿ ìíîãîïàðàìåòðè÷åñêèõ ñðåäñòâ èçìåðåíèé. Àä- ðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. 702- 13-31. Ñàôàðÿí Ãðèãîðèé Ãàãèêîâè÷, àñïèðàíò êàôåäðû ìåò- ðîëîãèè è èçìåðèòåëüíîé òåõíèêè ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: èññëåäîâàíèå ïîãðåøíîñòåé âû÷èñëèòåëü- íûõ îïåðàöèé ïðè öèôðîâîé îáðàáîòêå ñèãíàëîâ. Àä- ðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà,14, òåë. 702- 13-31. Ñåðãèåíêî Ìàðèíà Ïåòðîâíà, àñïèðàíòêà êàôåäðû ìåòðîëîãèè è èçìåðèòåëüíîé òåõíèêè ÕÍÓÐÝ. Íàó÷- íûå èíòåðåñû: èññëåäîâàíèå ìåòîäîâ è àëãîðèòìîâ èçìåðåíèÿ äèíàìè÷åñêèõ õàðàêòåðèñòèê àïåðèîäè÷åñ- êèõ ñðåäñòâ èçìåðåíèé.. Àäðåñ: Óêðàèíà, 61166, Õàðü- êîâ, ïð. 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Ðåøàåòñÿ çàäà÷à ñèíòåçà ðåøåò÷àòîãî ôèëüòðà ñ èñ- ïîëüçîâàíèåì ñòàòèñòèê âûñøèõ ïîðÿäêîâ. Ïðèâîäÿò- ñÿ âûðàæåíèÿ äëÿ ðàñ÷åòà êîýôôèöèåíòîâ îòðàæåíèÿ ðåøåò÷àòîãî ôèëüòðà íà îñíîâå ìîìåíòíûõ ôóíêöèé òðåòüåãî ïîðÿäêà. Ïðåäëàãàåòñÿ ñòðóêòóðà ñèíòåçèðî- âàííîãî ðåøåò÷àòîãî ôèëüòðà. 1. Ââåäåíèå Öèôðîâûå ðåøåò÷àòûå ôèëüòðû (ÐÔ) øèðîêî ïðè- ìåíÿþòñÿ â àíàëèçå êîððåëèðîâàííûõ ñëó÷àéíûõ ïðîöåññîâ [1], íàïðèìåð, â ñèñòåìàõ êîäèðîâàíèÿ- äåêîäèðîâàíèÿ ðå÷åâûõ ñèãíàëîâ â öèôðîâûõ ñè- ñòåìàõ ñâÿçè [2], äëÿ çàùèòû îò ïîìåõ [3], â óñòðîéñòâàõ êîððåêöèè ìåæñèìâîëüíûõ èñêàæå- íèé [4]. Ïàðàìåòðàìè ÐÔ ÿâëÿþòñÿ êîýôôèöèåíòû îòðàæåíèÿ è ÷èñëî çâåíüåâ ôèëüòðà. Êîýôôèöè- åíòû îòðàæåíèÿ îäíîçíà÷íî ñâÿçàíû ñ ïàðàìåòðàìè àâòîðåãðåññèè (ÀÐ) è îïðåäåëÿþòñÿ, â êîíå÷íîì ñ÷åòå, êîððåëÿöèîííîé ôóíêöèåé ñëó÷àéíîãî ïðî- öåññà. Ñòàòèñòè÷åñêèå ñâÿçè îòñ÷åòîâ ñëó÷àéíîãî ïðî- öåññà, îïèñûâàåìûå êîððåëÿöèîííûìè ôóíêöèÿ- ìè, ÿâëÿþòñÿ íàèáîëåå ñóùåñòâåííûìè õàðàêòåðè- ñòèêàìè ñëó÷àéíûõ ïðîöåññîâ. Âìåñòå ñ òåì, äëÿ ðåøåíèÿ ðÿäà çàäà÷ ñòàòèñòè÷åñêîãî àíàëèçà íåãà- óññîâûõ ïðîöåññîâ íåîáõîäèìî àíàëèçèðîâàòü ñòà- òèñòè÷åñêèå ñâÿçè áîëåå âûñîêèõ ïîðÿäêîâ [5-9]. Èñïîëüçîâàíèå íåãàóññîâûõ ñâîéñòâ ñëó÷àéíûõ ïðîöåññîâ ïîçâîëÿåò ïîâûñèòü ýôôåêòèâíîñòü ìåòîäîâ îáðàáîòêè ïðè ðåøåíèè íåêîòîðûõ ïðè- êëàäíûõ çàäà÷ [10-12]. Ïðîáëåìà ñèíòåçà ëèíåéíûõ ôèëüòðîâ ñ ó÷åòîì íåãàóññîâûõ ñâîéñòâ ñëó÷àéíûõ ïðîöåññîâ ñëàáî îòðàæåíà â íàó÷íîé ëèòåðàòóðå.  èññëåäîâàíèÿõ ïî íåãàóññîâûì ïðîöåññàì, êàê ïðàâèëî, îãðàíè÷èâà- þòñÿ àíàëèçîì ïðåîáðàçîâàíèÿ íåãàóññîâûõ ïðî- öåññîâ â ëèíåéíûõ è íåëèíåéíûõ ñèñòåìàõ [8-9, 13]. Ïîýòîìó îñòàåòñÿ àêòóàëüíîé ïðîáëåìà ñèíòåçà ëèíåéíûõ ôèëüòðîâ ïî ñòàòèñòè÷åñêèì õàðàêòåðè- ñòèêàì, îïèñûâàþùèì íåãàóññîâû ñâîéñòâà ñëó- ÷àéíûõ ïðîöåññîâ. Öåëü èññëåäîâàíèÿ – ðàçðàáîòêà òåîðèè ñèíòåçà ðåøåò÷àòûõ ôèëüòðîâ, ïàðàìåòðû êîòîðûõ ðàññ÷èòû- âàþòñÿ ïî ìîìåíòíûì ôóíêöèÿì òðåòüåãî ïîðÿäêà. Çàäà÷àìè èññëåäîâàíèÿ ÿâëÿþòñÿ: ïîëó÷åíèå ñ ïîìîùüþ ïðåäëîæåííûõ óðàâíåíèé ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ ïîëíîé ìîìåíòíîé ìàòðè- öû; âûâîä ñîîòíîøåíèé, ñâÿçûâàþùèõ îáîáùåí- íûå êîýôôèöèåíòû îòðàæåíèÿ è îáîáùåííûå êî- ýôôèöèåíòû àâòîðåãðåññèè; ïîëó÷åíèå ðåêóððåí- òíûõ âûðàæåíèé äëÿ îöåíêè îáîáùåííûõ êîýôôè- öèåíòîâ îòðàæåíèÿ; ñèíòåç ñòðóêòóðû ôèëüòðà; ïîëó÷åíèå ôîðìóë äëÿ îöåíêè îáîáùåííûõ êîýô- ôèöèåíòîâ îòðàæåíèÿ ÷åðåç îøèáêè ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ. 2. Ñèíòåç îáîáùåííîãî ðåøåò÷àòîãî ôèëüòðà òðåòüåãî ðàíãà Ðàññìîòðèì çàäà÷ó ñèíòåçà ôèëüòðà ðåøåò÷àòîé ñòðóêòóðû íà îñíîâå îáîáùåííîé ìîäåëè àâòîðåãðåñ- ñèè (ÎÀÐ) òðåòüåãî ðàíãà, êîýôôèöèåíòû êîòîðîé íàõîäÿòñÿ ïî ìîìåíòíûì ôóíêöèÿì òðåòüåãî ïîðÿä- êà [14]. Çàäà÷à ñèíòåçà îáîáùåííîãî ðåøåò÷àòîãî ôèëüòðà (ÎÐÔ) (ðèñóíîê) âêëþ÷àåò â ñåáÿ íàõîæ- äåíèå âûðàæåíèé äëÿ ïðåîáðàçîâàíèÿ âõîäíîãî ñèãíàëà ëèíåéíûì ôèëüòðîì, îïðåäåëåíèå ñòðóêòó- ðû ÎÐÔ è ñïîñîáà âû÷èñëåíèÿ êîýôôèöèåíòîâ îòðàæåíèÿ. Áóäåì íàçûâàòü «ÎÐÔ òðåòüåãî ðàíãà» ÐÔ, ðàññ÷èòàííûì ïî ìîìåíòíûì ôóíêöèÿì òðå- òüåãî ïîðÿäêà. Ïóñòü çàäàíà âûáîðêà íåãàóññîâà ïðîöåññà ] t [ x ñ íåíóëåâîé ìîìåíòíîé ôóíêöèåé òðåòüåãî ïîðÿäêà. Íàéäåì ñèñòåìó ðåêóððåíòíûõ óðàâíåíèé äëÿ âû÷èñ- ëåíèÿ êîýôôèöèåíòîâ îòðàæåíèÿ ôèëüòðà ñ ðåøåò÷àòîé ñòðóêòóðîé ïî çíà÷åíèÿì ìîìåíòíîé ôóíêöèè òðåòüåãî ïîðÿäêà íåãàóññîâà ïðîöåññà. Ïðè âûâîäå ñèñòåìû óðàâíåíèé ïîëàãàåòñÿ, ÷òî âûõîä ôèëüòðà èìååò íóëåâóþ ìîìåíòíóþ ôóíê- öèþ òðåòüåãî ðàíãà. Ýòîò êðèòåðèé îïòèìàëüíîñòè ïðèìåíÿåòñÿ òàêæå äëÿ âûáîðà ÷èñëà çâåíüåâ ÎÐÔ. Äëÿ ñèíòåçà ÎÐÔ òðåòüåãî ðàíãà èñïîëüçóþòñÿ âûðàæåíèÿ ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ íåãàóññîâà ñëó÷àéíîãî ïðîöåññà ìîäåëüþ ÎÀÐ. Âûðàæåíèå äëÿ ïðÿìîãî ïðåäñêàçàíèÿ íåãàóññîâà ïðîöåññà ñ íåíóëåâîé ìîìåíòíîé ôóíêöèåé òðåòüå- ãî ïîðÿäêà îïèñûâàåòñÿ óðàâíåíèåì ÎÀÐ , ] i t [ x ] i [ ] t [ x ] t [ a p 1 i p p       (1) ãäå p  – êîýôôèöèåíòû ÎÀÐ; p – ïîðÿäîê ìîäåëè ÎÀÐ; ] t [ a p – îøèáêè ïðÿìîãî ïðåäñêàçà- íèÿ ìîäåëè ÎÀÐ, óäîâëåòâîðÿþùèå ñîîòíîøåíèþ 0 ]} l t [ a ] j t [ a ] t [ a { E p p p    , ïðè 0 l , 0 j   . Òàêèì îáðàçîì, â ìîäåëè ÎÀÐ òðåòüåãî ðàíãà îøèáêè ïðåäñêàçàíèÿ íå èìåþò ñòàòèñòè÷åñêèõ ñâÿçåé âòîðîãî ïîðÿäêà, îïèñûâàåìûõ ìîìåíòíû- ìè ôóíêöèÿìè òðåòüåãî ïîðÿäêà. 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F: {~(7`c~k6&+a/|FpQ%FڄZg^Ab"k 6nC<ŭnG!Fu7.Jbvdp)TyٹsZ^mI7DY{'YpއG U'YGda`1.DC] 0-2At.n1=|8htc>rO)t |KƁq?`3Ģ}9 ڟH7n 0aԅ:^sw7qH~/ݳobCPo'e*3,0׭p;jBeN¾EL!GI{[al|~߯c1tC=9|G1۽Ta@q[ 60 VY>_ftY wR `pS uCYLgg?sOc2 fr>`T@s[DFGZ``FoP/t:%G?WCKKL/nRyF/t#(΅[zTqš:*_n[zT8BG5ңꞿH![z{c uK=3H'݈As:t/dSu^0/v/tX_q' 7}/OwUG̀iZFgzߨ$CtYw;UzÿuP/`S@{lna1%C_x PЅ'|OATK~ln GSMb\MLĜ[I,JA;1+ҡC_L ȣbRs{ނTk=CؗXc<і^u:ˠ! `\8'1;M`u>dA22& .PKڲo膩F ݐEg;n5񡦉'iЍH˭.̂PǭS0O`lhtlwlӡBG4!nQ1(ȇ2ogeNji^A:xGO9/ Ƥb/'b %`k`9|D 1HϹpnV얜/$+"7Oa s8`o1̗'pu%q>C|GZ&Hq tcHTݘN0`(2 lzJ ?ÍG =ݗ]~'X, YO$G ߱Xd\' O ԭښ)dF#&&t=n2?=LaJM#)rW1zW{:{@ӛUr.'mOby&y G q!$ 잘;fq0֨SaT́e MgXv74w ˌC~$we={΃~t1?w= mC~c?rzd"7ntu:p,|X*2_ [zT%F/.Y "gO]a:]q 㟨r̺߳ϦLo^:5?fwCp.vWrΆ@x@#V0 I,PRrgwT:jथ/T2ᜎz,R0,#mЍ4;ix$-ӫ~uC8Ǵgp}r?g̽ 3`Q|ԵvE[ጟ)k )bcow\3n=8A!t7 r~$͞P^ g.*`\:6~`3go{]z+~jBsP6GA CL~:I"C̍g40.bRBqBd6 ~C r1`ć_-|y:}+gk~!˅mF^WՃqk$0t$")?nx3IT[ ;~R?FCh_  ;bD \iĵH11@7cX1$x ~ӱC}HٚxN>47]]x;퉿D.=[jG MݤrbG}ˎn4`H  `p\LϑXMZZ!ҁvt+kQg|n.dBGrP)n!6&; z6knQ M0^pKZ^PZK-7:7 @P-IBgB ՟L\q rōʿjntTYBң^rz]nQZ /UꅎZ\+nQǍZ ([ ΍ FG1СrBϠoxA9 ]_B2/)'8WsF?H70`Qzr)lk~krU2upny: FG={s"qGzT\(7:jڨYtnS狯G1w` ;H,3[dG1 t\ehc<33Lu\8^ "_ڞ]a+t3Y a4(\0`u0l%a:`pOGF6`I,mzð 0s-lQQo@PPg;U]nıdCM$iE,,21 ÐÈ, 2005, ¹ 2 Ìàòðè÷íûå óðàâíåíèÿ (12) è (11) ïîçâîëÿþò ïîëó- ÷èòü ïîëíûé íàáîð óðàâíåíèé äëÿ íàõîæäåíèÿ îöåíîê êîýôôèöèåíòîâ îòðàæåíèÿ, à òàêæå ôîðìó- ëû îáíîâëåíèÿ ïî ïîðÿäêó êîýôôèöèåíòîâ ] i [ p  è ] i [ D p . Èç (11) ñëåäóþò ôîðìóëû, ñâÿçûâàþùèå êîýôôèöèåíòû îòðàæåíèÿ ÎÐÔ è êîýôôèöèåíòû ÎÀÐ: . ] 1 p [ ] 1 p [ K , ] 1 p [ D ] 1 p [ K 1 p d 1 p a            (13) Òàêèì îáðàçîì, êîýôôèöèåíòû îòðàæåíèÿ p - çâåííîãî ÎÐÔ ðàâíû ïîñëåäíèì êîýôôèöèåíòàì ÎÀÐ ìîäåëåé ïîðÿäêîâ p , , 2 , 1  . Èñïîëüçóÿ (11), ïðèõîäèì òàêæå ê ðåêóððåíòíûì âûðàæåíèÿì îáíîâëåíèÿ êîýôôèöèåíòîâ ÎÀÐ: . ] 1 p [ D ] i 1 p [ ] i [ D ] i 1 p [ ] 1 p [ K ] i [ D ] i [ D , ] i 1 p [ D ] 1 p [ ] i [ ] 1 i p [ D ] 1 p [ K ] i [ ] i [ 1 p p p p a p 1 p p 1 p p p d p 1 p                                 (14) C ïîìîùüþ ìàòðè÷íîãî óðàâíåíèÿ (12) ìîæíî íàéòè ñëåäóþùèå ôîðìóëû îáíîâëåíèÿ ìîìåíòîâ îøèáîê ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ: . m m ] 1 p [ D , m m ] 1 p [ K , m ] 1 p [ m , m ] 1 p [ K m 1 p , d 3 p , d 3 1 p , a 3 1 p 1 p , d 3 p , d 3 1 p , a 3 a 1 p , a 3 1 p , a 3 1 p p , a 3 1 p , a 3 1 p , d 3 d p , a 3                             (15) Âèä íàéäåííûõ âûðàæåíèé (14) è (15) óêàçûâàåò íà ïîëíóþ àíàëîãèþ ñ ïðîöåäóðîé Ëåâèíñîíà-Äóðáè- íà [1]. Èç (12) òàêæå ñëåäóþò óðàâíåíèÿ, ñâÿçûâàþùèå âåëè÷èíû 1 p , a 3   è 1 p , d 3   ñ êîýôôèöèåíòàìè îòðàæåíèÿ: . 0 m ] 1 p [ D , 0 m ] 1 p [ K , 0 m ] 1 p [ , 0 m ] 1 p [ K 1 p , d 3 p , a 3 1 p 1 p , d 3 p , a 3 a p , d 3 1 p 1 p , a 3 p , d 3 d 1 p , a 3                         (16) Äëÿ ïîëó÷åíèÿ (16) ó÷èòûâàëèñü ñîîòíîøåíèÿ (13). Èç (16) ïîëó÷àåì âûðàæåíèÿ äëÿ êîýôôèöèåíòîâ îòðàæåíèÿ: , m ] 1 p [ K , m ] 1 p [ K p , a 3 1 p , d 3 a p , d 3 1 p , a 3 d         (17) ãäå âåëè÷èíû 1 p , a 3   è 1 p , d 3   îïðåäåëÿþòñÿ èç (10). ×òîáû íàéòè ñòðóêòóðó ÎÐÔ òðåòüåãî ðàíãà, óìíî- æèì (11) íà T ]] 1 p t [ x ],..., t [ x [   .  ðåçóëüòàòå ïîëó- ÷èì â ìàòðè÷íîì âèäå âûðàæåíèå, îïèñûâàþùåå ïðîõîæäåíèå ñëó÷àéíîãî ïðîöåññà â ÎÐÔ: . ] t [ d ] t [ a ] 1 t [ d ] t [ a 1 ] 1 p [ K ] 1 p [ K 1 1 p 1 p p p a d                                 (18) Èç (18) ïîëó÷àåì, ÷òî ñîîòâåòñòâóþùàÿ ñèñòåìà óðàâíåíèé èìååò âèä , ] t [ a ] 1 t [ d ] 1 p [ K ] t [ a 1 p p d p      (19à) . ] t [ d ] t [ a ] 1 p [ K ] 1 t [ d 1 p p a p      (19á) Íà âûõîäå íóëåâîãî çâåíà ÎÐÔ, ò.å. ïðè 0 p  , èìååì , ] t [ x ] t [ a 0  ] 1 t [ x ] t [ d 0   . Êàê ñëåäóåò èç (18), ñòðóêòóðà ïîëó÷åííîãî ñ ïîìîùüþ ìîäåëè ÎÀÐ òðåòüåãî ðàíãà ÎÐÔ òðåòüåãî ðàíãà íå îòëè- ÷àåòñÿ îò ñòðóêòóðû îáû÷íîãî ÐÔ âòîðîãî ðàíãà, ðàáîòà êîòîðîãî îïèñûâàåòñÿ ñèñòåìîé óðàâíåíèé, ïîäîáíîé (19à), (19á). Îòëè÷èå ñîñòîèò â ñïîñîáå âû÷èñëåíèÿ êîýôôèöèåíòîâ îòðàæåíèÿ. Îáîáùåííûé ðåøåò÷àòûé ôèëüòð 3. Âûâîä ôîðìóë äëÿ îöåíêè êîýôôèöèåíòîâ îòðàæåíèÿ Ïîëó÷èì ôîðìóëû, ïîçâîëÿþùèå èíòåðïðåòèðî- âàòü âåëè÷èíû 1 p , a 3   è 1 p , d 3   , à òàêæå äàþùèå óäîáíûé ñïîñîá ðàñ÷åòà êîýôôèöèåíòîâ îòðàæå- íèÿ. Äëÿ ýòîãî ïðèâåäåì íåñêîëüêî ïîëåçíûõ ñâîéñòâ îøèáîê ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ [1]. Ìîäåëü ÎÀÐ ) 1 p (  -ãî ïîðÿäêà îïèñûâàåòñÿ óðàâíåíèåì . ] i t [ x ] i [ ] t [ x ] t [ a 1 p 1 i 1 p 1 p          (20) Ñðàâíèâàÿ (1) è (20), âèäèì, ÷òî ïðîãíîçû îòñ÷åòà ] t [ x â íèõ ñîñòàâëåíû èç îáùèõ çíà÷åíèé îòñ÷åòîâ ] p t [ x , ,... ] 2 t [ x , ] 1 t [ x    , ñòàòèñòè÷åñêè íå ñâÿçàí- íûõ ñ ] t [ a p .  òî æå âðåìÿ èç (20) ñëåäóåò, ÷òî îøèáêà ïðåäñêàçàíèÿ ] t [ a p áóäåò ñòàòèñòè÷åñêè ñâÿçàíà ñ ] 1 p t [ x   .  ñâîþ î÷åðåäü, ñ ] 1 p t [ x   ñòàòèñòè÷åñêè ñâÿçàíà îøèáêà îáðàòíîãî ïðåäñêà- çàíèÿ ] 1 t [ d p  , ÷òî âèäíî èç (3). Òàêèì îáðàçîì, ñïðàâåäëèâî ñîîòíîøåíèå . ]} 1 t [ d ] 1 t [ d ] t [ a { E ]} 1 p t [ x ] 1 p t [ x ] t [ a { E p p p p         (21) pex eA`A`@Pv O: OEf}4>ɺzk6sSS׹z۝F接}me~Bטh\\F4M N:դ#1@1#My +_]g9iyC^ ґϠ˴ ڿ#V Ӹi-o Kϥ),λF\KԌu>T7jAkY;ڇӭB=Uۦ%'^Tcd|SM۩w3VFG׎{WO\~jݾʳ3 Ff穕V]%UؗLtԎʆrSrW'^GZo)Gv[lGJ|oYtp29~`um|>_ /Qj-/86{ Paڻ_ߢgdkNӽ$/Ne^Y?W}#W7WÜM=v^]dg5'Bp N'-bks(SFⳇ~}ΜysY[6ʶ=SŬb U8ŤXvY7@oX&ab0,ɲo;wAi8<{b}|9\W`z3_w(9~)iKy7<t *}IfϠ}u]<#[I=);[7qǪi)zl]ݠ렷 VtA伻t*0s>3wϿ-oy/aޭ=Vɘϩ:/6^ňŊLXI'g⮘Ѷ.^;yGv6nH 쨼EDsW>ywW;3h$W{sueqz\o45;OUZֵo52ϠT~e]domY,O}^Oq9#jmP,Ť5ܞ4q+X~^7͜ߦ.NTg s6c.=Gq\KYKyg+ٕ[MCoDOǟuo ゲڶ tTP1g|ͯKC|B߄˥^li,c}@ucor=e]@? y5jܺ|۵>[5LʾϬaa56Zfd:ew;`]3M^Jvc)z+ŀH9Tڮ77ߖ|]CϝCcֵ%DMbX(iL\۠V0&1q7ՉyMzR[bޗy3bOLye Cfie'6Fݳwَ0 }~h|^X;|;=ܫ,M i[s\/wyu]s9ꁽMpלS{Zʗ|%6_V"̽+dm}W[V;}1ƫ$䞿sBHogx'Q¿P-^֋}̀2i`uͪW`ǭ\ M}q9Rn8f}ŜC9k,黥Tĺךw*lҷQ%dSTnވ㺃軇 W2Gs}ziDmR=S:w.9xgy\U۷WqƬv]V9cO } ici{%z{ckkgzCh-mWz܉z(&R޶sgOzGvgv4[kYqM~)Pysh,2vy @"D<y @G? 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Èç ýòèõ ðàññóæäåíèé ïðèõîäèì ê âûðàæå- íèþ äëÿ ñòàòèñòè÷åñêèõ ñðåäíèõ: . ]} t [ a ] t [ a ] 1 t [ d { E ]} t [ x ] t [ x ] 1 t [ d { E p p p p    (23) Îøèáêà îáðàòíîãî ïðåäñêàçàíèÿ ìîäåëè ÎÀÐ ) 1 p (  -ãî ïîðÿäêà îïèñûâàåòñÿ óðàâíåíèåì , ] 1 i t [ x ] i 1 p [ D ] 1 p t [ x ] t [ d 1 p 1 i p 1 p             èç êîòîðîãî ñëåäóåò, ÷òî ] t [ d 1 p  çàâèñèò îò ] 1 p t [ x   , ÷òî ïðèâîäèò ê ñòàòèñòè÷åñêîé íåçàâèñèìîñòè ] t [ d 1 p  è ] t [ x : 0 ]} t [ x ] t [ x ] t [ d { E 1 p   . (24) Óìíîæèâ âûðàæåíèå (19à) íà ] 1 p t [ x ] 1 p t [ x     , à âûðàæåíèå (19á) íà ] t [ x ] t [ x è óñðåäíèâ, ïîëó÷èì ñ ó÷åòîì ñòàòèñòè÷åñêèõ ñðåäíèõ (21)-(24) , 0 } ]) 1 t [ d {( E ] 1 p [ K ]} 1 t [ d ] 1 t [ d ] t [ a { E 3 p d p p p       (25à) . 0 } ]) t [ a {( E ] 1 p [ K ]} t [ a ] t [ a ] 1 t [ d { E 3 p a p p p     (25á) Ïîäñòàâëÿÿ (25à), (25á) â (16), â ðåçóëüòàòå ïðåîá- ðàçîâàíèé íàéäåì . ]} 1 t [ d ] 1 t [ d ] t [ a { E ]} 1 t [ d ] t [ a ] t [ a { E p p p p p p 1 p , d 3 1 p , a 3                         Ïîëó÷åííîå âûðàæåíèå ïîêàçûâàåò, ÷òî âåëè÷èíû 1 p , a 3   è 1 p , d 3   èíòåðïðåòèðóþòñÿ êàê âçàèìíûå ìîìåíòíûå ôóíêöèè îøèáîê ïðÿìîãî è îáðàòíîãî ïðåäñêàçàíèÿ Âûðàæåíèÿ (25à), (25á) äàþò óäîáíûé ñïîñîá îöåí- êè êîýôôèöèåíòîâ ] 1 p [ K a  è ] 1 p [ K d  . Êàê ñëåäóåò èç ýòèõ âûðàæåíèé, äëÿ îöåíêè êîýôôè- öèåíòîâ îòðàæåíèÿ ìîæíî èñïîëüçîâàòü ñîîòíîøå- íèÿ , ]) t [ a ( ] 1 t [ d ]) t [ a ( ] 1 p [ K N 1 t 3 p N 2 t p 2 p a        (26à) . ]) 1 t [ d ( ]) 1 t [ d ]( t [ a ] 1 p [ K N 1 t 3 p N 2 t 2 p p d         (26á) Êîýôôèöèåíòû îòðàæåíèÿ îöåíèâàþòñÿ òàêèì ñïîñîáîì ïîýòàïíî. Ñíà÷àëà îöåíèâàþòñÿ ] 1 [ K d è ] 1 [ K a ïî ôîðìóëàì (26à),(26á), â êîòîðûõ ïîëàãà- åòñÿ ] t [ x ] t [ a 0  , ] 1 t [ x ] 1 t [ d 0    . Çàòåì âûáîðêà äàííûõ ïðîïóñêàåòñÿ ÷åðåç îäíîçâåííûé ÐÔ ñ íàéäåííûìè êîýôôèöèåíòàìè îòðàæåíèÿ, è ïî îøèáêàì ïðåäñêàçàíèÿ ] t [ a 1 è ] 1 t [ d 1  îïðåäåëÿ- þòñÿ ] 2 [ K a è ] 2 [ K d . Àíàëîãè÷íàÿ ïðîöåäóðà âû÷èñëåíèÿ êîýôôèöèåíòîâ îòðàæåíèÿ ïîâòîðÿ- åòñÿ âïëîòü äî p -ãî ïîðÿäêà. 4. Çàêëþ÷åíèå Ìîìåíòíàÿ ìàòðèöà ñ êîìïîíåíòàìè â âèäå çíà÷å- íèé äâóìîìåíòíûõ ôóíêöèé íåãàóññîâûõ ñëó÷àé- íûõ ïðîöåññîâ èìååò òåïëèöåâó ñòðóêòóðó. Ýòî ïîçâîëÿåò ñèíòåçèðîâàòü ÎÐÔ, ïîäîáíûé ïî ñòðóê- òóðå îáû÷íîìó ÐÔ. Òàêèì îáðàçîì, áûë ïîëó÷åí ïîëíûé íàáîð ðåêóððåíòíûõ óðàâíåíèé äëÿ îáíîâ- ëåíèÿ ïî ïîðÿäêó êîýôôèöèåíòîâ ÎÀÐ. Êîýôôèöèåíòû îòðàæåíèÿ ÎÐÔ òðåòüåãî ðàíãà íàõîäÿòñÿ ïî ìîìåíòíûì ôóíêöèÿì òðåòüåãî ïî- ðÿäêà. Èõ òàêæå ìîæíî îöåíèâàòü ïî íîðìèðîâàí- íîé âçàèìíîé ôóíêöèè îøèáîê ïðÿìîãî è îáðàò- íîãî ïðåäñêàçàíèÿ. Íîðìèðîâêà ïðîâîäèòñÿ äåëå- íèåì íà òðåòüè ìîìåíòû îøèáîê ïðÿìîãî è îáðàò- íîãî ïðåäñêàçàíèÿ. Êîýôôèöèåíòû îòðàæåíèÿ ìåíåå ÷óâñòâèòåëüíû ê ïîìåõàì, ó êîòîðûõ ìîìåíòíûå ôóíêöèè òðåòüåãî ïîðÿäêà ðàâíû íóëþ, íàïðèìåð ê ãàóññîâûì ïîìåõàì.  äàííîé ðàáîòå âïåðâûå ñèíòåçèðîâàíà ñòðóêòóðà îáîáùåííîãî ðåøåò÷àòîãî ôèëüòðà è ïîëó÷åíû âûðàæåíèÿ äëÿ ðàñ÷åòà êîýôôèöèåíòîâ ôèëüòðà ïî ìîìåíòíûì ôóíêöèÿì òðåòüåãî ïîðÿäêà.  îòëè÷èå îò àíàëîãè÷íûõ ðåøåò÷àòûõ ôèëüòðîâ, ïàðàìåòðû êîòîðûõ ðàññ÷èòûâàþòñÿ ïî êîððåëÿöè- îííûì ôóíêöèÿì, îáîáùåííûå ðåøåò÷àòûå ôèëü- òðû ïîçâîëÿþò èçâëåêàòü è èñïîëüçîâàòü íåãàóññî- âû ñâîéñòâà îáðàáàòûâàåìûõ ñëó÷àéíûõ ïðîöåññîâ. Ñèíòåçèðîâàííûå ôèëüòðû áîëåå óñòîé÷èâû ê ãàóññîâûì ïîìåõàì. Îíè ìîãóò èñïîëüçîâàòüñÿ ïðè àäàïòèâíîé îáðàáîòêå íåãàóññîâûõ ïðîöåññîâ, â çàäà÷àõ îáåëåíèÿ è ïîäàâëåíèÿ ãàóññîâûõ è íåãà- óññîâûõ ïîìåõ, äëÿ ïîëó÷åíèÿ äîïîëíèòåëüíûõ ïðèçíàêîâ â çàäà÷å ðàñïîçíàâàíèÿ íåãàóññîâûõ ïðîöåññîâ. Äàëüíåéøèå èññëåäîâàíèÿ ìîãóò áûòü íàïðàâëåíû íà ðàçðàáîòêó òåîðèè ÎÐÔ ðàíãîâ áîëüøå òðåõ, ñèíòåç ÎÐÔ ïî ðåàëüíûì íåãàóññîâûì ïðîöåññàì. Ëèòåðàòóðà: 1. Àäàïòèâíûå ôèëüòðû /Ïîä ðåä. Ê.Ô.Í. Êîóýíà è Ï.Ì. Ãðàíòà. Ì.: Ìèð, 1988. 392ñ. 2. Êîðîòàåâ Ã.À. Ýôôåêòèâíûé àëãîðèòì êîäèðîâàíèÿ ðå÷åâîãî ñèãíàëà íà ñêîðîñòè 4,8 êáèò/ñ è íèæå // Çàðóáåæíàÿ ðàäèîýëåêòðîíèêà.1996. ¹3. Ñ.57-68. 3. Haykin S. Radar signal processing. IEEE ASSP Magazine, 1985. Vol. 2. P.2- 18. 4. Êóðåøè Ø.Ó.Õ. Àäàïòèâíàÿ êîððåêöèÿ // ÒÈÈÝÐ. 1985. Ò. 73, ¹9. Ñ.5-497. 5. Áðèëëèíäæåð Ä.Ð. Âðåìåí- íûå ðÿäû. Îáðàáîòêà äàííûõ è òåîðèÿ. Ì.: Ìèð, 1980. 536 ñ. 6. Øèðÿåâ À.Í. Íåêîòîðûå âîïðîñû ñïåêòðàëü- íîé òåîðèè ñòàðøèõ ìîìåíòîâ. Òåîðèÿ âåðîÿòíîñòè è åå ïðèìåíåíèå. 1960, 5, 3, Ñ. 293-313. 7. Ëåîíîâ Â.Ï. Íåêîòîðûå ïðèìåíåíèÿ ñòàðøèõ ñåìèèíâàðèàíòîâ â òåîðèè ñòàöèîíàðíûõ ñëó÷àéíûõ ïðîöåññîâ.Ì.: Íà- óêà, 1964.124 ñ. 8. Øåëóõèí Î.È. Áåëÿåâ È.Â. Íåãàóññîâ- ñêèå ïðîöåññû. ÑÏá.: Ïîëèòåõíèêà, 1992. 312 ñ. 9. Ìàëàõîâ À.Í. Êóìóëÿíòíûé àíàëèç ñëó÷àéíûõ íåãàóñ- ñîâûõ ïðîöåññîâ è èõ ïðåîáðàçîâàíèé.Ì.: Ñîâ. ðàäèî, cx xU HI $D#Q! 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Êóí÷åíêî Þ.Ï. Íåëèíåéíàÿ îöåíêà ïàðàìåòðîâ íåãàóññîâñêèõ ðàäèîòåõíè÷åñêèõ ñèãíà- ëîâ. Ê.: Âûùà øê.,1987. 191 ñ. 11. Âàëååâ Â.Ã., Äàíèëîâ Â.À. Îïòèìàëüíîå îáíàðóæåíèå ñèãíàëîâ íà ôîíå íåãàóññîâñêèõ êîððåëèðîâàííûõ ðàäèîïîìåõ// Èçâ. âóçîâ. Ðàäèîýëåêòðîíèêà. 1991. ¹7. Ñ. 30-34. 12. Êèñå- ëåâ Í.Â. Ìåðîäû ïîñòðîåíèÿ ñèñòåì ðàñïîçíàâàíèÿ è êëàññèôèêàöèè íåãàóññîâûõ ñèãíàëîâ. Ë.: Èçä-âî Ëå- íèíãð. óí-òà, 1986. 188 ñ. 13. Øåëóõèí Î.È. Íåãàóññîâ- ñêèå ïðîöåññû â ðàäèîòåõíèêå. Ì.: Ðàäèî è ñâÿçü, 1999. 361 ñ. 14. Òèõîíîâ Â.À. Îáîáùåííàÿ ìîäåëü àâòîðåãðåñ- ñèè íåãàóññîâûõ ïðîöåññîâ // Ðàä³îòåõí³êà. 2003. Ïîñòóïèëà â ðåäêîëëåãèþ 26.01.2005 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Êðàâ÷åíêî Í.È. Òèõîíîâ Âÿ÷åñëàâ Àíàòîëüåâè÷, êàíä. òåõí. íàóê, äî- öåíò êàôåäðû ÐÝÑ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèî- ëîêàöèÿ, ðàñïîçíàâàíèå îáðàçîâ, ñòàòèñòè÷åñêèå ìî- äåëè. Àäðåñ: Óêðàèíà, 61726, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. 70-215-87. ÓÄÊ 621.317.7 ÍÅËÈÍÅÉÍÀß ÄÈÍÀÌÈÊÀ ÏÎËÓÊÂÀÍÒÎÂÛÕ ÁÀËËÈÑÒÈ×ÅÑÊÈÕ ÝËÅÊÒÐÎÍΠ ÑÂÅÐÕÐÅØÅÒÊÀÕ ÏÐÈ ÑÈËÜÍÎÌ ÑÂ× ÂÎÇÁÓÆÄÅÍÈÈ ÞÐ×ÅÍÊÎ Ë.Â., ÞÐ×ÅÍÊÎ Â.Á. Îïèñûâàåòñÿ íåëèíåéíàÿ äèíàìèêà áàëëèñòè÷åñêèõ ýëåêòðîíîâ â êëàññè÷åñêèõ è ïîëóêâàíòîâûõ ñâåðõðå- øåòêàõ ñ ïîìîùüþ íîâîãî ýôôåêòèâíîãî ïðèáëèæå- íèÿ ïîëóêâàíòîâîé äèíàìèêè. ×èñëåííîå ìîäåëèðî- âàíèå îáíàðóæèëî âîçìîæíîñòü ïðîÿâëåíèÿ õàîòè- ÷åñêîé äèíàìèêè ïîâåäåíèÿ êëàññè÷åñêèõ è ÷àñòè÷íî êâàíòîâûõ ýëåêòðîíîâ â áàëëèñòè÷åñêèõ ñâåðõðåøåò- êàõ ïðè ñèëüíîì ÑÂ× âîçáóæäåíèè. Òàêàÿ ñèñòåìà ìîæåò ñëóæèòü èñòî÷íèêîì øóìîâûõ êîëåáàíèé. Ïðèâîäÿòñÿ ïàðàìåòðû ñèñòåìû, êîòîðûå ïîçâîëÿþò ïðåîáðàçîâàòü êîãåðåíòíîå âûñîêî÷àñòîòíîå ïîëå â õàîòè÷åñêèé òîê àíñàìáëÿ áàëëèñòè÷åñêèõ ýëåêòðî- íîâ. 1.Ââåäåíèå Äàííîå èññëåäîâàíèå ïîñâÿùåíî èçó÷åíèþ õàîòè- ÷åñêîé äèíàìèêè ýëåêòðîíîâ â áàëëèñòè÷åñêèõ ãåòåðîñòðóêòóðàõ â óñëîâèÿõ, ïîãðàíè÷íûõ ìåæäó êëàññè÷åñêèì è êâàíòîâûì ðåæèìàìè äâèæåíèÿ ÷àñòèö, ÷òî òèïè÷íî äëÿ ïðèáîðîâ ìåçîñêîïè÷åñ- êèõ ðàçìåðîâ è ïðåäñòàâëÿåò â íàñòîÿùåå âðåìÿ çíà÷èòåëüíûé íàó÷íûé èíòåðåñ [1]. Îíî òàêæå ñâÿçàíî ñ ïîèñêîì íîâûõ ìåòîäîâ ñîçäàíèÿ èñòî÷- íèêîâ øóìîâûõ êîëåáàíèé, êîòîðûå íåîáõîäèìû â ñîâðåìåííîé øóìîâîé ðàäèîëîêàöèè è ñèñòåìàõ ñâÿçè ñ ðàñøèðåííûì ñïåêòðîì [2]. Êâàíòîâûå ãåòåðîñòðóêòóðû ïðåäñòàâëÿþò îñîáûé èíòåðåñ äëÿ ýòèõ öåëåé. Îíè îáíàðóæèâàþò áîëüøîå ðàçíîîá- ðàçèå íåñòàáèëüíûõ ýôôåêòîâ è ìîãóò áûòü ïîòåí- öèàëüíî èñïîëüçîâàíû êàê èñòî÷íèêè øóìà. Àíàëèç ýëåêòðîííîãî ïåðåíîñà â êâàíòîâûõ è ìåçîñêîïè÷åñêèõ ñòðóêòóðàõ ÿâëÿåòñÿ äîñòàòî÷íî ñëîæíîé ïðîáëåìîé, ïîñêîëüêó äàæå äëÿ îäíîãî ýëåêòðîíà íàäî ðåøàòü íåñòàöèîíàðíîå óðàâíåíèå Øðåäèíãåðà â ÷àñòíûõ ïðîèçâîäíûõ. Ïîýòîìó äëÿ èçó÷åíèÿ áàëëèñòè÷åñêîãî ïåðåíîñà ìû èñïîëüçóåì â ýòîé ðàáîòå ïðèáëèæåíèå ïîëóêâàíòîâîé äèíà- ìèêè [3].  îòëè÷èå îò îáû÷íîãî êâàçèêëàññè÷åñ- êîãî ïðèáëèæåíèÿ, ýòî ïðèáëèæåíèå ÿâëÿåòñÿ áîëåå óñîâåðøåíñòâîâàííûì. Îíî îïèñûâàåò ýâî- ëþöèþ ãàóññîâûõ âîëíîâûõ ïàêåòîâ ýëåêòðîíîâ ñ ó÷åòîì êâàíòîâûõ ïîïðàâîê áîëåå âûñîêîãî ïîðÿä- êà, à òàêæå ñ ó÷åòîì ðàñïëûâàíèÿ ïàêåòîâ íå ïðîñòî â câîáîäíîì ïðîñòðàíñòâå, à â ïðîèçâîëüíîì ïîòåí- öèàëå V(x) . Öåëüþ ýòîé ðàáîòû ÿâëÿåòñÿ èçó÷åíèå âîçìîæíîñòè âîçáóæäåíèÿ äèíàìè÷åñêîãî õàîñà â êëàññè÷åñêèõ è êâàíòîâûõ ñâåðõðåøåòêàõ ñ ïîìîùüþ ÑÂ× ïîëÿ. Õîòÿ ê ïðîáëåìå âûñîêî÷àñòîòíîãî âîçáóæäåíèÿ ðàçëè÷íîãî âèäà ñâåðõðåøåòîê îáðàùàëèñü ìíîãèå àâòîðû [4], ìû ðàññìîòðèì íåêîòîðûå íîâûå ýô- ôåêòû, âîçìîæíûå â ñïåöèàëüíîãî âèäà áîêîâûõ ñâåðõðåøåòêàõ, ñôîðìèðîâàííûõ âäîëü ãåòåðîãðà- íèöû â äâóìåðíîì ýëåêòðîííîì ãàçå î÷åíü âûñî- êîé ïîäâèæíîñòè. Ìû ïîêàæåì, ÷òî äèíàìèêà êëàññè÷åñêèõ è ÷àñòè÷- íî-êâàíòîâûõ áàëëèñòè÷åñêèõ ýëåêòðîíîâ â ïåðè- îäè÷åñêîì ïîòåíöèàëå ñâåðõðåøåòêè ÿâëÿåòñÿ õà- îòè÷åñêîé, êîãäà ýëåêòðîíû âîçáóæäàþòñÿ ñèëü- íûì ÑÂ× ïîëåì îïðåäåëåííîé ÷àñòîòû f . Êâàíòî- âûå ýôôåêòû èìåþò òåíäåíöèþ ðàçðóøàòü õàîñ, êîãäà ýëåêòðîíû äåëîêàëèçîâàíû èëè â ðåàëüíîì ïðîñòðàíñòâå (çàíèìàþò äâà èëè áîëåå ïåðèîäîâ ñâåðõðåøåòêè L ), èëè æå ïî ýíåðãèè (â ýòîì ñëó÷àå îíè îãðàíè÷åíû â ðåàëüíîì ïðîñòðàíñòâå âíóòðè îáëàñòè, ìàëîé ïî ñðàâíåíèþ ñ ïåðèîäîì ñâåðõðå- øåòêè). Îäíàêî åñëè ýëåêòðîíû ÷àñòè÷íî ëîêàëè- çîâàíû è ïî ýíåðãèè, è â ðåàëüíîì ïðîñòðàíñòâå, ò.å. èõ òåïëîâàÿ ýíåðãèÿ èëè ýíåðãèÿ Ôåðìè  íàõîäèòñÿ â èíòåðâàëå 2 0 2 (2) V 2mL    , ãäå  – ïîñòîÿííàÿ Ïëàíêà, m – ýôôåêòèâíàÿ ìàññà ýëåêòðîíà, 0 V – àìïëèòóäà ïîòåíöèàëà V(x) ñâåðõ- ðåøåòêè, òî íåîáõîäèìî íåêîòîðîå âðåìÿ  äëÿ ïåðåõîäà îò íåðåãóëÿðíîãî êâàçèêëàññè÷åñêîãî äâè- æåíèÿ ê ðåãóëÿðíîìó êâàçèêâàíòîâîìó. Òàêèì îáðàçîì, åñëè 1/f  è âðåìÿ ðåëàêñàöèè ýíåðãèè è èìïóëüñà e  è i  ñîîòâåòñòâåííî ñðàâíèìî èëè áîëüøå âðåìåíè ïåðåõîäà  , òî äèíàìèêà ýëåêòðî- íà â îñíîâíîì ÿâëÿåòñÿ êëàññè÷åñêîé, áàëëèñòè÷åñ- êîé è õàîòè÷åñêîé. Òðåáîâàíèÿ ê ñèñòåìå, ïðè êîòîðûõ ïîÿâëÿåòñÿ óêàçàííûé õàîñ, âåñüìà æåñòêèå. Ñóùåñòâåííûì óñëîâèåì ÿâëÿåòñÿ äîñòàòî÷íî áîëüøàÿ àìïëèòóäà âîçáóæäàþùåãî ÑÂ× ïîëÿ. Íàðÿäó ñ ýòèì íåîáõî- äèìî òàêæå ñîçäàíèå ñëîÿ äâóìåðíîãî ãàçà âûñîêîé ïîäâèæíîñòè ñ áîêîâîé ñâåðõðåøåòêîé ïðè íèçêèõ òåìïåðàòóðàõ, êîãäà âîçìîæíî áàëëèñòè÷åñêîå äâè- æåíèå ýëåêòðîíîâ ñ äëèíîé ñâîáîäíîãî ïðîáåãà áîëüøåé, ÷åì 100 ìêì. 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"9wt;) )3۬effƇy.R84k`G2sh 3ͶyeO,,!25 ÐÈ, 2005, ¹ 2 Òàêèì îáðàçîì, ðåøåòêà âñå æå êëàññè÷åñêàÿ, íî ïîñêîëüêó åå ïåðèîä íåáîëüøîé – ðåøåòêà áàëëè- ñòè÷åñêàÿ (íåò ðàññåÿíèÿ ýëåêòðîíîâ, i e l , l L  ). Ïîñêîëüêó ýëåêòðîí îáëàäàåò êâàíòîâûìè ñâîé- ñòâàìè, òî ïðè äîñòàòî÷íî äëèòåëüíîì åãî ñâîáîä- íîì äâèæåíèè îí íà÷èíàåò «ðàñïëûâàòüñÿ» è ïðî- ÿâëÿòü êâàíòîâîå ïîâåäåíèå ïðè íå ñëèøêîì áîëü- øèõ L , èëè æå, åñëè íàáåðåò äîñòàòî÷íî ýíåðãèè, òî ñìîæåò òóííåëèðîâàòü ÷åðåç áàðüåð, ÷òî óæå íàõîäèòñÿ íà ãðàíè ôîðìèðîâàíèÿ êâàíòîâûõ ñâåðõ- ðåøåòîê ñ ìèíèçîííîé ñòðóêòóðîé ýíåðãèé è çîí Áðèëëþýíà. Êðîìå òîãî, â ýòîé ðàáîòå èçó÷àåòñÿ ëèøü äèíàìèêà îòäåëüíûõ ýëåêòðîíîâ è íå ðàñ- ñìàòðèâàåòñÿ èõ àíñàìáëü è óñðåäíåíèå âñåõ äâè- æåíèé ïî àíñàìáëþ (òåì ñàìûì çäåñü íå ó÷èòûâà- þòñÿ âîçìîæíûå ýôôåêòû êîððåëÿöèè â äâèæåíèè ýëåêòðîíîâ â àíñàìáëå). Äëÿ ðåàëèçàöèè óñëîâèé, íàçâàíûõ âûøå, íåîáõî- äèìî îáåñïå÷èòü äëèíó ñâîáîäíîãî ïðîáåãà ýëåêò- ðîíà â ñâåðõðåøåòêå áîëüøå òåõ äëèí, íà êîòîðûõ ïðîèñõîäÿò âñå ýòè ýôôåêòû. Ñîîòâåòñòâåííî âðå- ìÿ ðàññåÿíèÿ êàê ýíåðãèè, òàê è èìïóëüñà ýëåêò- ðîíà äîëæíî áûòü áîëüøå âñåãî âðåìåíè ýòîãî ïðîöåññà è, ðàçóìååòñÿ, åùå áîëüøå, ÷åì ïåðèîä âîëíû. Ýòî ñðàçó æå òðåáóåò âîëíû äîñòàòî÷íî âûñîêî÷àñòîòíîé, íî ñèëüíî óâåëè÷èâàòü ÷àñòîòó íåëüçÿ ïî äðóãèì ïðè÷èíàì. Äàëåå, â îáû÷íûõ ìàòåðèàëàõ  A 1000 700 ~ l i  (ò.å. ìåíüøå 1 ìêì), â ÷èñòûõ – áîëüøå, à â äâóìåðíîì ãàçå ïðè 78 Ê, êîãäà ïîäâèæíîñòü ýëåêòðîíîâ e  äîñòèãàåò 62 10cм/Вс (îáû÷íî ïðè 300 Ê â êðåìíèè 3 e 10 ~  , à â GaAs 4 e 10 ~  ), äîñòèæèìû äëèíû ñâîáîäíîãî ïðîáåãà i l1 мкм  (èíîãäà мкм 10 l i  ).  [5] ïðèâåäåíû äàííûå, ÷òî ïîäâèæíîñòü ýëåêò- ðîíîâ â 2D-ãàçå óæå äîñòèãàåò 7 e 10   , à мкм 100 l i  (ýòî äîâîëüíî òèïè÷íî, îñîáåííî ïðè T<78 Ê, êîãäà i l или e l   , õîòÿ ïðè комн T èëè íàîáîðîò 4 Ê ñíîâà ie ll  ). Áàëëèñòè÷åñêèé ðåæèì äâèæåíèÿ ýëåêòðîíîâ ýêñïåðèìåíòàëüíî íàáëþäà- åòñÿ ïðè óìåíüøåíèè òåìïåðàòóðû äî óðîâíÿ àçîò- íîé è íèæå â òîíêèõ ãåòåðîñëîÿõ ïîëóïðîâîäíèêî- âûõ ñîåäèíåíèé IIIV AB òîëùèíîé t0,11  ìêì, ãäå îí îáû÷íî èçó÷àåòñÿ ïðè òîêå ïîïåðåê ñëîÿ. 3. ×èñëåííûå ðåçóëüòàòû Äëÿ ðåøåíèÿ óðàâíåíèé (7), (8), à òàêæå (9) â êëàññè÷åñêîì ïðåäåëå ìû èñïîëüçóåì ìåòîä Äîð- ìàíà-Ïðèíöà [6] ñ àâòîìàòè÷åñêèì èçìåíåíèåì øàãà èíòåãðèðîâàíèÿ, îáåñïå÷èâàþùèì ëîêàëü- íóþ òî÷íîñòü ðåøåíèÿ äî 12 10   .  îòñóòñòâèå âîçáóæäåíèÿ åñòü òî÷íîå àíàëèòè÷åñ- êîå ðåøåíèå óðàâíåíèÿ (9), à ïðè íàëè÷èè âîçáóæ- äåíèÿ â òàêîé ñèñòåìå ðàíî èëè ïîçäíî ïîÿâëÿåòñÿ õàîñ. Òî÷íåå, õàîñ ïîÿâëÿåòñÿ ñðàçó æå, ïðè ìàëåé- øåì âîçáóæäåíèè, íî òîëüêî â î÷åíü óçêîé îáëàñòè ïàðàìåòðîâ: â îêðåñòíîñòè ñåïàðàòðèññû íà ôàçî- âîì ïîðòðåòå ìàÿòíèêà, êîãäà îí ïðè ñëàáîì âîçäåéñòâèè çàïóñêàåòñÿ èç ñîñòîÿíèÿ ïîêîÿ âáëè- çè âåðõíåé òî÷êè. Äëÿ ñèëüíîãî âîçáóæäåíèÿ íåò òî÷íîãî àíàëèòè÷åñêîãî ðåøåíèÿ, à âîçìîæíî òîëü- êî ÷èñëåííîå ìîäåëèðîâàíèå. Ïðè ýòîì â èññëåäî- âàíèÿõ îñíîâíîå âíèìàíèå îáû÷íî óäåëÿåòñÿ îêðå- ñòíîñòè ñåïàðàòðèñ íà ôàçîâîì ïîðòðåòå, òàê êàê èìåííî òàì áîëåå âñåãî ðàçâèâàåòñÿ õàîñ [7, 8]. Ïîñêîëüêó ýëåêòðîíû îáû÷íî íàõîäÿòñÿ íå íà âåðõóøêå ïîòåíöèàëüíûõ áàðüåðîâ, à íà äíå, òî íå âñåãäà ÿñíî â äåòàëÿõ, êàê îíè ñåáÿ áóäóò âåñòè â óñëîâèÿõ, èçó÷àåìûõ â äàííîé ðàáîòå. Ðÿä ñèòóà- öèé, âîçíèêàþùèõ â ýòèõ ñëó÷àÿõ, áûë íàìè ïðîìîäåëèðîâàí ïóòåì ÷èñëåííîãî ðåøåíèÿ óðàâ- íåíèé (7), (8), è íåêîòîðûå ðåçóëüòàòû ïðèâåäåíû íà ðèñ. 1-4. Íà ðèñ. 1 ïîêàçàíû òðàåêòîðèè ïîëóêâàíòîâîãî è êëàññè÷åñêîãî ýëåêòðîíîâ. Íà÷àëüíûå óñëîâèÿ ïðåäïîëàãàþò, ÷òî ýëåêòðîíû íàõîäÿòñÿ â ïîêîå ïðè ìèíèìóìå ïîòåíöèàëà ( x0   ), ïðè÷åì ïîëó- êâàíòîâûå ýëåêòðîíû ñëåãêà äåëîêàëèçàâàíû ( 0,1   0,1   ). Ðèñ.1 ïîêàçûâàåò ñëîæíóþ õàîòè- ÷åñêóþ òðàåêòîðèþ êëàññè÷åñêîãî ýëåêòðîíà x(t)  . Ôîðìà åãî òðàåêòîðèè î÷åíü ÷óâñòâèòåëüíà ê òî÷- íîñòè ðåøåíèÿ. Ïîëóêâàíòîâûé ýëåêòðîí âíà÷àëå ñëåäóåò êëàññè÷åñêîé õàîòè÷åñêîé òðàåêòîðèè íå- êîòîðîå âðåìÿ  , íî ïîòîì îòêëîíÿåòñÿ îò íåå è â äàëüíåéøåì ðàñïðîñòðàíÿåòñÿ ïî ñâîåìó ñîáñòâåí- íîìó ïóòè. Ïðè ýòîì åãî äâèæåíèå íå çàâèñèò îò òî÷íîñòè ðåøåíèÿ. Ýòè ðåçóëüòàòû ïîëó÷åíû äëÿ ïîòåíöèàëà ñâåðõðåøåòêè ñ ïåðèîäîì 1 ìêì è àìïëèòóäîé 20 ìý ïðè íàïðÿæåííîñòè ÑÂ× ïîëÿ 1,2 êÂ/ñì è ÷àñòîòå 60 ÃÃö [9]. Ðèñ. 1. Òðàåêòîðèè ïîëóêâàíòîâîãî (ñïëîøíàÿ êðèâàÿ) è êëàññè÷åñêîãî ýëåêòðîíîâ ïðè íà÷àëüíûõ óñëîâèÿõ: x0   , p0   , 0,1   , 0   è òî÷íîñòè ðåøåíèÿ 10 10   (ïóíêòèðíàÿ êðèâàÿ) è 9 10   (òî÷å÷íàÿ êðèâàÿ). Ïàðàìåòðû ìîäåëè L1 мкм  , 0 V20мэВ  , 0 E1,2кВ/см  è f60ГГц  , ÷òî ñîîòâåòñòâóåò 0 V1   , e1   è 3 10     Ðèñ. 2 èëëþñòðèðóåò âàæíîñòü èñïîëüçîâàíèÿ ÑÂ× ïîëÿ âûñîêîé ýíåðãèè. Çäåñü ïðåäñòàâëåíû ðåçóëü- òàòû äëÿ òîãî æå ñàìîãî ñëó÷àÿ, ÷òî è íà ðèñ. 1, íî ïðè àìïëèòóäå ÑÂ× ïîëÿ, â äâà ðàçà ìåíüøåé.  ýòîì ñëó÷àå ïðèçíàêîâ õàîñà äëÿ ýëåêòðîíîâ, ñòàð- òóþùèõ ñ äîëèíû ïîòåíöèàëüíîãî ïðîôèëÿ, åñòå- ñòâåííî, íå íàáëþäàåòñÿ. Fx{tUIAyZBT(MhZE4Y/D^]cuBh!<ĠRYfUEl]0{!bυof={f9ĉ8}E"}:p#Qy3 0D$;VV.Ж2GB:ix@:'\% 8GAwq+5)JLS6I2ϴ&QJv^jIaRe.p`uDhulsq/k$˗61q'߉|>SۈuH7`S$d\|ANO1]2-O|y4rhYtŒPEY՝L`j4NYZNB6P]us>Jʑ#簯t-`&\`.Z;7o7ٲ1mH?^~.v{Ճ=y)RnrU9>0*!tJPazQMƾHӶ3 pox+(oUdΙ _n(W "6YҜdi."|:9CJ9g݄$O !3s4<.G7Tm 2lP.Iyܾg1>B7z{px&| :ʟ2Ө|F:<<,)SB>Bm[0yu-Jas9po5p/dr3̹?2%Ȥ~k"Ə 0kuu`1{.@ڨ}w0rxrl~?S6Oai)ɉ9Oc 5~#t?A=5"ۍ8fp ivd )#j6t2hYwd>cd)c[Lg,8:3\@f~YJ̧G}mM^O1|1uϐi`pWMCR|}A] ^ ;Z!Vg8+@e3ev_ -R[,KP2XVч=1 ?v,ԇRӝ4!MrG]O#ӿUV]][։{] {""_3Y?O΃uk\6X%G}բE}bhR?gig3p Pz7ةp LsD D T)saEwQt%Ȼ'@#bBN_ u܀+u{ƨcxC7 ~;#e,[s x=u{-R07 pe YJ8;@Ml3 wwu@1OA`(t/i`x2tǖNtjy=A /[O/Ч!d;>B"PȹOOtCk\]➉];'"2ȓum+Ccn"#ԁޭyJf#}.\zԍkT.Uc\1`>4:D|'[N,pN}o[qMfPFE>![FۙA A]Xw׌er` (\gË,lt_ӯ>'ޫL'6ׂ{4.47Q DדOf=2 `8O뽮 ~c*dRp. 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Ïåðåéäåì îò çàäà÷è îïòèìèçàöèè (3) ê ýêâèâàëåí- òíîé çàäà÷å ñ âûïóêëîé (ñèëüíî âûïóêëîé ñ ïàðà- ìåòðîì 0  , äèôôåðåíöèðóåìîé) öåëåâîé ôóíê- öèåé. Äëÿ ýòîãî íåîáõîäèìî âûïîëíèòü ñëåäóþùèå äåéñòâèÿ. 1. Ïîñòðîèòü ìíîæåñòâî 1 1 *vv wkw wk E{E,E,S}  , ïî- ðîæäåííîå ìíîæåñòâîì D , ñîîòâåòñòâóþùåå ìíî- æåñòâó E , ïðè÷åì * EE  . 2. Äëÿ âñåõ * xE\E  äîîïðåäåëèòü ôóíêöèþ (x)  . Ïðè èçâåñòíîì àíàëèòè÷åñêîì âèäå (x)  ýòî ìîæíî ñäåëàòü ñ ñîõðàíåíèåì âûðàæåíèÿ äëÿ (x)  . 3. Èñïîëüçóÿ èçâåñòíûå ìåòîäû [6-10], ïîñòðîèòü, åñëè ýòî âîçìîæíî, âûïóêëîå (ñèëüíî âûïóêëîå ñ ïàðàìåòðîì 0  , äèôôåðåíöèðóåìîå) ïðîäîëæå- íèå conv  ôóíêöèè (x)  íà âûïóêëîå çàìêíó- òîå ìíîæåñòâî N VR  . Ïðè ýòîì * convEV  , à çíà÷èò EV  . Ïîñòðîåííàÿ òàêèì îáðàçîì ôóíêöèÿ (x)  áóäåò âûïóêëûì (ñèëüíî âûïóêëûì ñ ïàðàìåòðîì 0  , äèôôåðåíöèðóåìûì) ïðîäîëæåíèåì ôóíêöèè öåëè (x)  çàäà÷è (3) íà âûïóêëîå çàìêíóòîå ìíîæåñòâî N VR  , EV  . Çàäà÷à îïòèìèçàöèè, ýêâèâàëåí- òíàÿ (3), ïðèìåò âèä: (x)min  , xE  , (4) ãäå (x)  âûïóêëà (ñèëüíî âûïóêëà) íà N VR  , EV  . 4. Îöåíêè ìèíèìóìà âûïóêëûõ ôóíêöèé íà ìíîæåñòâàõ nk EI , m nk EI , m n SI  ðàáîòàõ [4, 6, 7, 11-15] èçëàãàåòñÿ îáùèé ïîäõîä ê ïîñòðîåíèþ îöåíîê ìèíèìóìà âûïóêëûõ è ñèëüíî âûïóêëûõ ôóíêöèé íà åâêëèäîâûõ êîìáè- íàòîðíûõ ìíîæåñòâàõ.  ðàìêàõ ýòîãî ïîäõîäà ïîëó÷åíû îöåíêè ìèíèìóìà ôóíêöèé, çàäàííûõ íà ìíîæåñòâàõ ïåðåñòàíîâîê, ðàçìåùåíèé, ñî÷åòà- íèé è äðóãèõ. Ýòè ðåçóëüòàòû èñïîëüçóþòñÿ ïðè ðåàëèçàöèè äåêîìïîçèöèîííûõ ìåòîäîâ ðåøåíèÿ çàäà÷ îïòèìèçàöèè íà óêàçàííûõ êëàññàõ êîìáèíà- òîðíûõ ìíîæåñòâ. Ïðèìåíÿÿ äàííûé ïîäõîä, ïîñò- ðîèì îöåíêè è äîñòàòî÷íûå óñëîâèÿ ìèíèìóìà âûïóêëûõ è ñèëüíî âûïóêëûõ ôóíêöèé íà ìíîæå- ñòâàõ nk EI , m nk EI , m n SI . Ïðè ýòîì áóäåì îïèðàòüñÿ íà óòâåðæäåíèÿ, äîêàçàííûå â [15]. Ëåììà 1 [15]. Ïóñòü ôóíêöèÿ (x)  âûïóêëà è äèôôåðåíöèðóåìà íà âûïóêëîì çàìêíóòîì ìíîæå- ñòâå V , ãäå N EVR  . Òîãäà xV  N i yEyE i i1 (x) min(y)(x)((x),x)miny x       . (5) Òåîðåìà 1. Ïóñòü ôóíêöèÿ (x)  âûïóêëà è äèôôå- ðåíöèðóåìà íà âûïóêëîì çàìêíóòîì ìíîæåñòâå nk VEI  , ãäå ìíîæåñòâî w nk EIR  , w2n  , ïî- ðîæäåíî ìíîæåñòâàìè iii Z{a,b}  , n iJ  . Òîãäà äëÿ ëþáîãî xV  nk w * j yEI j j1 (x) min(y)(x)((x),x)y x       , (6) ãäå j * 2j1i ya   , j * 2ji yb  , jt ii  , ij  , jn iJ  , n jJ  ,(7) à   12n i,i,...,i óäîâëåòâîðÿåò ñîîòíîøåíèþ 1122nn iiiiii ccc (a,b)(a,b)...(a,b)  , (8) ãäå jjkkjk iiii2j12k1ii c ((a,b)(a,b))((cc)(aa)    jk 2j2kii (cc)(bb)0)  ïðè c(x)  . Äîêàçàòåëüñòâî. Îöåíêó (6) ïîëó÷èì èç ñîîòíî- øåíèÿ (5) ïðè nk EEI  , Nw  , w2n  . Òîãäà çàäà÷ó îïòèìèçàöèè â ïðàâîé ÷àñòè íåðàâåíñòâà (5) ìîæíî ðåøèòü ñ ïîìîùüþ óòâåðæäåíèÿ, äîêàçàí- íîãî â [16], ïîñêîëüêó ìèíèìèçèðóåìàÿ ôóíêöèÿ ÿâëÿåòñÿ ëèíåéíîé ñ êîýôôèöèåíòàìè iw i (x) c,iJ x    . Óïîðÿäî÷èâàÿ ïàðû jj ii (a,b) â ñîîòâåòñòâèè ñ ñîîòíîøåíèåì (8) íà îñíîâàíèè [16] ïðè c(x)  , ïðèõîäèì ê ñïðàâåäëèâîñòè óòâåð- æäåíèÿ òåîðåìû 1. Òåîðåìà 2. Ïóñòü ôóíêöèÿ (x)  âûïóêëà è äèôôå- ðåíöèðóåìà íà âûïóêëîì çàìêíóòîì ìíîæåñòâå m nk VEI  , ãäå ìíîæåñòâî mv nk EIR  , v2m  , ïî- ðîæäåíî ìíîæåñòâàìè iii Z{a,b}  , n iJ  . Òîãäà äëÿ ëþáîãî xV  : 1) m nk v * j yEI j j1 (x) min(y)(x)((x),x)y x       , (9) ãäå M v *i j iJ j j1 (x) yargminy x       , (10) (i) (i) mk i v i j j j1 yEI(G) (x) yargminy x       , à (i) G èìååò âèä [16]:   1122mm (i) jjjjjj G(a,b),(a,b),...,(a,b)    1122nn (a,b),(a,b),...,(a,b)  , t r i 2r1j ya   , t r i 2rj yb  , rs tt jj  ïðè sr  , r tn jJ  , rm tJ  , m rJ  à   12m tttn j,j,...,jJ  òàêîâî, ÷òî tttttt 1122mm jjjjjj ccc (a,b)(a,b)...(a,b)  (11) ïðè c(x)  ; 2) m nk mm ** iiii xEI i1i1 min(y)(x)((x),x)cycz      ,(12) ãäå i 2i1 (x) c x     , i 2i (x) c x     , m iJ  , Mx pU@?, ѠȚEA("RTtdQDARC, " #:*hw_}SLի7ù˹{vDL7WI?;LZ5hAn%Ii7gH#< aHhddՃ !drzizMid:ɔRyvL/:R쾗V䮗S͕2GĊ/4cR^Niǥ' *R92іJe:}$W16~&,ͩr-r¾Re6"zWJCSv4cٳ/ا;1r o馘xyt;6;A-6O(-EdSތ_x MQ3#bR~/r3k>r-W]d5}9'F7_k̛"\+Glc3+CS phب_ ͕n<'kˇ9LFoMZrKwFy9}okUB5c2T֘6"OKzӮSVGMPbW0_o*3WRw OqE2bw:11]fRj4Μ ! 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Ll'r+1%[`{ƐclyǕR<,g:osX7w&uΤ2 F!tׂ$?ɋe^~Ƴli#qӈ+Q8v@9"_O,.qsKθ=_c {脩ܿ\;"ci/ʸroaؗԟ#xaix.yRxG=eޒVH'[<-daXx¹;Y~cfL=wCV-Mg禳-ij1_Ob,Ixl]qn<u'̻.Geta^̃anL+/q~ ~s|Yp ƞ7qn-fz>u2I2Ut~#G~ʈeQ&(P8H;0N _qVwYX_zZ3% ;ɲΛ{Q(# kO-}~^SƥșzKR)c:33R.epvY)ţ2 0̃={->ĀNǾD,<SW]|r6gZôb)~ė8pΏ:g_-V:\$sx8v89,K<#uXo%ނ#uEN 3pLGC} G=7ς/Xo0h_Ol;8O{{oEg ė7Gԓ""xJ78R_ԈeL+/|A?,yəXvrDGxN|fx/=\ y>(l=z܁ nB6L~C.b();d9K_~A$u>q>)tGѕ2DX1O3u"{&ˤӐ7X6qK\_f qDuؔ2|, khĖXs Gm '0xXWzSgQFG}3Ors,1c>rlY{)aFXx8<[ |0?_rE8؞ qq3h 3qOxƇv>|_kN ۀuH:+KZ' %o2p,2%=n:@c?Ϙùi!3ߛcR^wu֢5j [#Hօ.(3o<#rQ'k~M'kۘ2-Wxc2n_3hvo~C?9K\?0Bf,,26 ÐÈ, 2005, ¹ 2 Ðèñ. 2. Òðàåêòîðèè ïîëóêâàíòîâîãî (ñïëîøíàÿ êðèâàÿ) è êëàññè÷åñêîãî (ïóíêòèðíàÿ êðèâàÿ) ýëåêòðîíîâ ïðè 0 E0,6кВ/см  ( e0,5   ; îáå êðèâûå ïðàêòè÷åñêè ñîâïàäàþò). Îñòàëüíûå ïàðàìåòðû òàêèå æå, êàê è íà ðèñ.1 Íà ðèñ. 3 ìû ñðàâíèâàåì òðàåêòîðèè ïîëóêâàíòî- âûõ è êëàññè÷åñêèõ ýëåêòðîíîâ äëÿ äâóõ ñëó÷àåâ, êîãäà íà÷àëüíîå ïîëîæåíèå ýëåêòðîíîâ x0,2   (ïåðèîä ñâåðõðåøåòêè L6,28   â îòíîñèòåëüíûõ åäèíèöàõ). Îñòàëüíûå ïàðàìåòðû âûáðàíû òàêèìè æå, êàê è â ïðåäûäóùèõ ñëó÷àÿõ. ×èñëåííîå ìîäåëèðîâàíèå îáíàðóæèëî ýôôåêò ÷óâñòâèòåëü- íîé çàâèñèìîñòè òðàåêòîðèé êëàññè÷åñêîãî ýëåêò- ðîíà îò òî÷íîñòè ðåøåíèÿ è íà÷àëüíûõ óñëîâèé. Ïîëóêâàíòîâûé ýëåêòðîí òàêæå îáíàðóæèâàåò íå- êîòîðóþ ÷óâñòâèòåëüíîñòü ê íà÷àëüíûì óñëîâèÿì, ïîñêîëüêó îí âíà÷àëå ñëåäóåò ïðåæíåé êëàññè÷åñ- êîé òðàåêòîðèè. Âìåñòå ñî ñëîæíîé ôîðìîé ýëåê- òðîííûõ òðàåêòîðèé, äåìîíñòðèðóþùåé ïðîèç- âîëüíûå ïðûæêè ýëåêòðîíà ñ îäíîãî ïîòåíöèàëü- íîãî ìèíèìóìà â äðóãîé, çà÷àñòóþ âåñüìà óäàëåí- íûé, ýôôåêò ÷óâñòâèòåëüíîé çàâèñèìîñòè îáåñïå- ÷èâàåò äîïîëíèòåëüíîå äîêàçàòåëüñòâî íàëè÷èÿ äèíàìè÷åñêîãî õàîñà â íåëèíåéíîé ýëåêòðîííîé ñèñòåìå. Ðèñ. 3. Ñðàâíåíèå òðàåêòîðèé ïîëóêâàíòîâîãî è êëàññè÷åñêîãî ýëåêòðîíîâ, êîãäà íà÷àëüíàÿ ïîçè- öèÿ x0,2   (ñïëîøíàÿ è äëèííàÿ ïóíêòèðíàÿ êðèâàÿ ñîîòâåòñòâåííî) è x0,2   (êîðîòêàÿ ïóíê- òèðíàÿ è òî÷å÷íàÿ êðèâàÿ) ïðè 10 10   . Äðóãèå ïàðàìåòðû òàêèå æå, êàê íà ðèñ.1 ( x0,2   – ïîëî- âèíà øèðèíû àíñàìáëÿ íåâûðîæäåííûõ ýëåêòðîíîâ â äîëèíå ñâåðõðåøåòêè ïðè òåìïåðàòóðå T =4 K) Íàêîíåö, íà ðèñ.4 ìîæíî óâèäåòü èçìåíåíèå ìî- ìåíòà ïîëóêâàíòîâîãî è êëàññè÷åñêîãî ýëåêòðîíà äëÿ ñëó÷àÿ õàîòè÷åñêîãî äâèæåíèÿ, ïðåäñòàâëåííî- ãî íà ðèñ. 1. Âîçðàñòàíèå èçìåíåíèé ìîìåíòà, òèïè÷íîå äëÿ õàîòè÷åñêîãî äâèæåíèÿ, âûçûâàåò óñèëåíèå ôëóêòóàöèé òîêà è, ñëåäîâàòåëüíî, ïîÿâ- ëåíèå äîïîëíèòåëüíîãî øóìà â áàëëèñòè÷åñêîé ñòðóêòóðå. Ðèñ. 4. Èçìåíåíèå èìïóëüñà ïîëóêâàíòîâîãî (ñïëîøíàÿ êðèâàÿ) è êëàññè÷åñêîãî (ïóíêòèðíàÿ ëèíèÿ) ýëåêòðîíîâ äëÿ ñëó÷àÿ, ïðåäñòàâëåííîãî íà ðèñ.1 ïðè 10 10   Ïîëó÷åííûå ðåçóëüòàòû òðåáóþò íåêîòîðîãî ïîÿñ- íåíèÿ. Åñëè ìîùíîñòü âûñîêî÷àñòîòíîãî ïîëÿ íå äîñòèãàåò ïîðîãà (åãî âåëè÷èíà îïðåäåëÿåòñÿ óñëî- âèåì 0 eV    ), òî ýëåêòðîíû îñòàþòñÿ â äîëèíàõ ïîòåíöèàëà V(x) è ñëåãêà êîëåáëþòñÿ. Ýòîò ñëó÷àé ðåàëèçóåòñÿ, åñëè ïðè ÷àñòîòå 60 ÃÃö íàïðÿæåí- íîñòü Å ïîëÿ â âîëíå ñîñòàâëÿåò ìåíåå 1,2 êÂ/ñì. Åñëè äîñòè÷ü ïîðîãà ïî ìîùíîñòè ( 0 eV    ), òî äàæå ýòè ýëåêòðîíû ìîæíî çàñòàâèòü äâèãàòüñÿ â ðåæèìå õàîòè÷åñêîé äèíàìèêè. ×òî êàñàåòñÿ òèïè÷íûõ êâàíòîâûõ ñâåðõðåøåòîê ñ êâàíòîâûìè ýëåêòðîíà- ìè, òî èõ ïîâåäåíèå â òàêèõ óñëîâèÿõ ìàëî èçó÷åíî.  ñëó÷àå æå ïîëóêâàíòîâûõ ðåøåòîê, ðàññìîòðåí- íûõ âûøå, ïîëó÷åííûå ðåøåíèÿ îáóñëîâëåíû ñëå- äóþùèì ñöåíàðèåì ïîâåäåíèÿ ýëåêòðîíîâ. Åñëè ïîëóêâàíòîâûå ýëåêòðîíû âíà÷àëå íàõîäÿòñÿ íà äíå «ÿìîê» (íå ñèëüíî ëîêàëèçîâàíû â òî÷êó, íî è íå ñèëüíî ðàçìûòû, òàê ÷òî ýëåêòðîííûå âîëíî- âûå ôóíêöèè â ñîñåäíèõ «ÿìêàõ» íå ïåðåêðûâàþò- ñÿ, à ïåðèîä ñâåðõðåøåòêè Бройля де L    , íî íå ñèëüíî), òî íåêîòîðîå âðåìÿ ïîä äåéñòâèåì ïîëÿ îíè áóäóò äâèãàòüñÿ ïî òîìó æå äèíàìè÷åñêè- õàîòè÷åñêîìó çàêîíó, ÷òî è ýëåêòðîíû êëàññè÷åñ- êèå, à èìåþùàÿñÿ íåêàÿ êâàíòîâàÿ «ðàçìàçêà» íå ìåøàåò èõ äâèæåíèþ, íåñìîòðÿ íà òî, ÷òî ýòî óæå íå òî÷êà ñ ôèêñèðîâàííûì ïîëîæåíèåì è ñêîðîñ- òüþ. Ïðè ïîñòåïåííîì âîçðàñòàíèè «ðàçìàçêè», êîãäà âîëíîâàÿ ôóíêöèÿ ðàñøèðÿåòñÿ çà ïðåäåëû îäíîé «ÿìêè», ýëåêòðîí ïåðåõîäèò â ðåæèì ðåãó- ëÿðíîãî âîëíîîáðàçíîãî ïðîäâèæåíèÿ âäîëü ñâåðõ- ðåøåòêè ñ íåêîòîðîé ñðåäíåé ñêîðîñòüþ äðåéôà â îäíîì èëè â äðóãîì íàïðàâëåíèè. Ýòî åñòü êâàçè- ñòàöèîíàðíûé ïîëóêâàíòîâûé ðåæèì äâèæåíèÿ, êîòîðûé áóäåò ïðîäîëæàòüñÿ äî ìîìåíòà ðàññåÿíèÿ ýëåêòðîíà ôîíîíîì.  ýòîì ñëó÷àå óæå íåò õàîñà, îäíàêî ìîìåíò ïåðåõîäà â òàêîé ðåæèì, âîçíèêøàÿ ñðåäíÿÿ ñêîðîñòü äðåéôà è ñàìî íàïðàâëåíèå äâè- æåíèÿ âäîëü ðåøåòêè âåñüìà ñëó÷àéíû è î÷åíü ÷óâñòâèòåëüíû ê íà÷àëüíûì óñëîâèÿì äâèæåíèÿ, à òàêæå ê òî÷íîñòè èíòåãðèðîâàíèÿ óðàâíåíèé äâè- æåíèÿ. 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Èíòåðåñíûì è âàæíûì ýôôåêòîì ñìåøàííîé õà- îòè÷åñêîé è êâàçè-ðåãóëÿðíîé äèíàìèêè ýëåêòðî- íîâ â ïîëóêâàíòîâûõ áàëëèñòè÷åñêèõ ñâåðõðåøåò- êàõ ÿâëÿåòñÿ âîçíèêíîâåíèå èçáûòî÷íîãî øóìà ÷èñòî äèíàìè÷åñêîãî ïðîèñõîæäåíèÿ. Ïî ñðàâíå- íèþ ñ îáû÷íûì òåïëîâûì øóìîì ýòîò äèíàìè÷åñ- êèé èçáûòî÷íûé øóì â áàëëèñòè÷åñêîé ñòðóêòóðå äîëæåí áûòü âåñüìà áîëüøèì. Äåëî â òîì, ÷òî õàîòè÷åñêàÿ äèíàìèêà áàëëèñòè÷åñêîãî ýëåêòðîíà îçíà÷àåò íàëè÷èå ñîîòâåòñòâóþùåé ôëóêòóàöèè (ñì. ðèñ.1,3, ãäå ðåçêî ìåíÿåòñÿ êîîðäèíàòà, è ðèñ.4, ãäå âàðüèðóåòñÿ èìïóëüñ), ÷òî ïðèâîäèò ê èçáûòî÷- íîìó øóìó, îáóñëîâëåííîìó â ýòèõ ñòðóêòóðàõ âîçäåéñòâèåì ïîëÿ. ×èñòî äèíàìè÷åñêîå äâèæåíèå ýëåêòðîíîâ áóäåò êîãäà-òî ïðåðûâàòüñÿ ðàññåÿíè- åì, è ïðè ñèëüíîì ðàññåÿíèè õàîñ ìîæåò íå óñïåòü ïðîÿâèòüñÿ. Îäíàêî â áàëëèñòè÷åñêèõ ñòðóêòóðàõ äèíàìè÷åñêèå ôëóêòóàöèè ìîãóò ðàçâèâàòüñÿ äî- ñòàòî÷íî äîëãî, íå ðàçðóøàÿñü ðàññåÿíèåì. Íàëè÷èå íåêîòîðîãî èçáûòî÷íîãî øóìà â áàëëèñòè÷åñêèõ ñòðóêòóðàõ íàáëþäàëîñü ýêñïåðèìåíòàëüíî [5], õîòÿ áîëåå òèïè÷íûì ÿâëÿåòñÿ óìåíüøåíèå øóìà çà ñ÷åò êóëîíîâñêèõ è êâàíòîâûõ êîððåëÿöèé [10, 11]. Íåîáõîäèìî îòìåòèòü, ÷òî íàëè÷èå èçáûòî÷íîãî øóìà íå òàê î÷åâèäíî, ïîñêîëüêó íàäî âûïîëíèòü óñðåäíåíèå ïî àíñàìáëþ âñåõ ýëåêòðîíîâ, à îíè óæå èçíà÷àëüíî èç-çà òåïëîâîãî äâèæåíèÿ ðàñïðåäåëå- íû õàîòè÷åñêè è âñå ôëóêòóàöèè áóäóò óñðåäíÿòü- ñÿ, ò.å. ñðåäíÿÿ àìïëèòóäà îáû÷íî óìåíüøàåòñÿ â N ðàç, ãäå N – ÷èñëî ýëåêòðîíîâ. Åñëè òåìïåðà- òóðà ìàëà, à ïîëå ñèëüíîå, òî íà÷àëüíûé òåïëîâîé øóì ìàë è â áàëëèñòè÷åñêèõ ñâåðõðåøåòêàõ áóäóò óñðåäíÿòüñÿ ôëóêòóàöèè ãîðàçäî áîëüøåé àìïëè- òóäû, ÷åì ïðîñòî òåïëîâûå. Êðîìå ýòîãî, åñòü åùå íåìàëàÿ êîððåëÿöèÿ (êîëëåêòèâíîñòü) ó ñàìèõ õàîòè÷åñêèõ êîëåáàíèé, òàê êàê âäîëü ïîòåíöèàëü- íîé êàíàâêè â ñâåðõðåøåòêå (â íàïðàâëåíèè ïîïå- ðåê ïîëÿ) íàõîäèòñÿ ìíîãî ýëåêòðîíîâ è îíè ìåæäó ñîáîé îòêëèêàþòñÿ íà âîëíó áîëåå-ìåíåå ñîãëàñî- âàííî. Ïîýòîìó ýôôåêò ñëîæåíèÿ äèíàìè÷åñêèõ ôëóêòóàöèé, ñîâåðøàåìûõ ñèíõðîííî ýëåêòðîíà- ìè, ëåæàùèìè âî âñåé ïëîñêîñòè, ïîïåðå÷íîé ê îñè ñâåðõðåøåòêè è ê ïîëþ âîëíû, âñå æå åñòü è, ïî âñåé âèäèìîñòè, äîâîëüíî âåëèê. Óñëîâèÿ, íåîáõîäèìûå äëÿ íàáëþäåíèÿ îïèñàííûõ ýôåêòîâ, âåñüìà òðóäíî ðåàëèçîâàòü ýêñïåðèìåí- òàëüíî. Íåñìîòðÿ íà î÷åíü óçêèé äèàïàçîí, íàì òåì íå ìåíåå óäàëîñü ïîäîáðàòü ïàðàìåòðû ñèñòåìû, ïðè êîòîðûõ ýêñïåðèìåíòàëüíàÿ ðåàëèçàöèÿ ïðåä- ñòàâëÿåòñÿ âîçìîæíîé. Ðàññìîòðèì âûáîð ýòèõ óñëîâèé äåòàëüíåå. Àìïëèòóäó ñâåðõðåøåòî÷íîãî ïîòåíöèàëà ( 0 V20мэВ  ) íåëüçÿ óâåëè÷èòü, òàê êàê åñëè ýëåê- òðîí íàáèðàåò â ñèëüíîì ïîëå ýíåðãèþ, ïîçâîëÿþ- ùóþ åìó äîñòè÷ü âåðõóøêè áàðüåðà è âûøå, òî óæå ïðè 30 ìý áóäåò ïðîèñõîäèòü ñèëüíîå ðàññåÿíèå íà îïòè÷åñêèõ ôîíîíàõ, ïîòåðÿ ñâîáîäíîãî áàëëè- ñòè÷åñêîãî äâèæåíèÿ è êîãåðåíòíîñòè ñîñåäíèõ ýëåêòðîíîâ, ÷òî â ñâîþ î÷åðåäü ïðèâåäåò ê îòñóò- ñòâèþ íåîáõîäèìîãî äèíàìè÷åñêîãî øóìà. Ïðè ìåíüøèõ 0 V äàæå ïðè 4 Ê ìíîãî ýëåêòðîíîâ áóäåò íà ñàìîì äíå (4 Ê = 0,33 ìýÂ), è ïî êîîðäèíàòå x  «ðàçìàçêà» ñîñòàâëÿåò â îòíîñèòåëüíûõ åäèíèöàõ x0,2   ( x2x/L   ), ÷òî íåæåëàòåëüíî. Åñòü åùå âîçìîæíîñòü óìåíüøàòü L (äî 0,1ìêì è ìåíüøå) è îäíîâðåìåííî óâåëè÷èâàòü ÷àñòîòó f (äî 12 10 Гц ), òîãäà è ïðè ñðàâíèòåëüíî ìàëûõ i l , e l = 0,5ìêì è ñîîòâåòñòâåííî ìàëûõ âðåìåíàõ ñâîáîäíîãî ïðîáå- ãà i  , c 10 ~ 11 e   ìîæíî ïîëó÷èòü òðåáóåìûé ðåçóëüòàò. Îäíàêî çäåñü íà÷èíàþòñÿ ïðîáëåìû ñ êâàíòîâûì äâèæåíèåì: òèïè÷íàÿ äëèíà âîëíû äå- Áðîéëÿ îêàçûâàåòñÿ ñëèøêîì áëèçêà ê L. Ïðè ýòîì ðàñòåò áåçðàçìåðíàÿ âåëè÷èíà 2 2/(mLf)    , ÷òî òóò æå ñêàçûâàåòñÿ íà äèíàìèêå – òàêîé ýëåêòðîí ñëèøêîì áûñòðî «ðàñïëûâàåòñÿ» è íå óñïåâàåò äâèãàòüñÿ õàîòè÷åñêè. Òåì ñàìûì ìû ïîëó÷èëè îöåíêó òîãî, íàñêîëüêî áûñòðî êâàíòîâûå ïîïðàâ- êè ðàçðóøàþò õàîñ. Ñ ýëåêòðè÷åñêèì ïîëåì ïðè ìåíüøèõ L òîæå âîçíèêàþò ïðîáëåìû: äëÿ õàîñà, êîòîðûé ïîëó÷à- åòñÿ äàæå äëÿ ýëåêòðîíîâ, ñòàðòóþùèõ ñî äíà çîíû, íåîáõîäèìî îáåñïå÷èòü 00 VEL/2  , à èìåÿ îãðà- íè÷åíèå íà 0 V «ñíèçó» , ÷òîáû èçáåæàòü òåìïåðà- òóðíîé «ðàçìàçêè», êîòîðàÿ äàæå ïðè 78 Ê ñëèøêîì âåëèêà (ïðè íàëè÷èè «ïîòîëêà» 0 V2030мэВ  ), óìåíüøåíèå L ïîòðåáóåò óñèëåíèÿ ïîëÿ äî íåðå- àëüíûõ çíà÷åíèé (çäåñü è òàê óæå íåîáõîäèìî îáåñïå÷èòü 1ê äà åùå â óñëîâèÿõ ãèãàíòñêîé áàëëèñòè÷åñêîé ïîäâèæíîñòè äî 72 10cм/Вс ). Ñëå- äóåò, âïðî÷åì, îòìåòèòü, ÷òî òîê áóäåò ñèëüíî îãðàíè÷åí íàëè÷èåì áàðüåðîâ, òàê ÷òî îáùàÿ ïðî- âîäèìîñòü ñòðóêòóðû, íåñìîòðÿ íà ãèãàíòñêóþ ïîäâèæíîñòü ýëåêòðîíîâ, íå äîëæíà áûòü áîëü- øîé. Èñõîäÿ èç èçëîæåííîãî âûøå, íåîáõîäèìî îáåñïå- ÷èòü L =1ìêì, ÷àñòîòó f =60ÃÃö, îáùóþ äëèíó ñòðóêòóðû ïîðÿäêà 100 L =100ìêì è, ñëåäîâàòåëü- íî, äëèíó ñâîáîäíîãî ïðîáåãà íå ìåíåå 100ìêì. Ýòî äîñòèãàåòñÿ â ñïåöèàëüíûõ óñëîâèÿõ ïðè òåìïåðà- òóðå ïîðÿäêà 4 Ê íà äâóìåðíîì ãàçå, ò.å. â ñëîå ýëåêòðîíîâ, çàæàòîì â ïîòåíöèàëüíîé ÿìå øèðè- íîé îêîëî o 2050A  âäîëü ãåòåðîãðàíèöû GaAs- AlGaAs, òàê ÷òî ïîïåðåê ãðàíèöû äâèæåíèå êâàí- òîâàíî (íåîáõîäèìî òàêæå, ÷òîáû 0 V áûëî ìåíüøå ðàññòîÿíèÿ ìåæäó óðîâíÿìè ýòîãî «ãåîìåòðè÷åñêî- ãî êâàíòîâàíèÿ»), à âäîëü ãðàíèöû, â 2D ïëîñêîñòè, îñòàåòñÿ ñâîáîäíûì êëàññè÷åñêèì – èìåííî â òàêîì ãàçå òîëüêî è âîçìîæíû òå ãèãàíòñêèå i l100мкм  . Òîãäà ïîëó÷àåòñÿ, ÷òî ñâåðõðåøåòêà äîëæíà èäòè âäîëü ïîâåðõíîñòè, ò.å. îñü ñâåðõðåøåòêè õ, âäîëü êîòîðîé ñîçäàíà ïåðèîäè÷íîñòü 0 V(x)Vcos(x)  , èäåò âäîëü ãåòåðîãðàíèöû. Ýòî ìîæåò áûòü îáåñïå- ÷åíî íàíåñåíèåì íà ïîâåðõíîñòü AlGaAs (ïîä åãî òîíêèì ñëîåì è íàõîäèòñÿ ãðàíèöà ñ 2D ãàçîì) ïåðèîäè÷íîé êîíòàêòíîé ñåòêè ñ ïåðèîäîì L =1ìêì, íà êîòîðóþ ïîäàí ñîîòâåòñòâóþùèé ïîòåíöèàë, âûòåñíÿþùèé ýëåêòðîíû â 2D ãàçå èç ó÷àñòêîâ ïîä êîíòàêòàìè, òàê ÷òîáû ïîòåíöèàëüíûé áàðüåð è áûë 0 V =20ìýÂ. 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ѬCsZExߜ'F- J#4/e]~ y?lm?wa8l™OGt5n(Os\rS5G.~  mf+Sa1ܞž \i[CZeVKC}:OG~7O|vYZ$Bt݁%1qc UY TIEdϹ6C^ImC1}Hi~H(jh=NwIx>[+Ek)=Ţd:q-5:u=Z34oTrQɩ bV<_{, H=Kౚ_~|NI8*tD A饈FF AG (9T$ӶOMB$҆n#% 3&tz@dZ{J8l~~qJao`=F398wc 7qo m|IIX/%>OR4ydZ5ҨxΌBٸR^e:Pq ԑ+ tX鷪X΁/?\Bbp.VT&qBCo$9K"vtQ{Afy~ m"C7 6nyj(҉=f3`|ꗱi21UBXJ{ Āp>w;{G1>S-(E^eJ-[1\/pډnS7hGUR~o4gQI'!FgY7KU`ul^P ?a+ 7ib#JyDkn1q$z0Ĭ@-P>h+*Ce)֠G, vJ77e,FKo6Kݩ<'+KtJ)//OJY;ǯ_!v7z\j8laKh@=%!"Q'G%T(BJDb;9LN{K7<Ɲ"瓋[U4r$ u!S7 uA!rIʁS||i~6lOG#f13AQtI>l!xc{-CA1x(Ymy甅b;9F{߆n/TixF4y1<|Ҿ!A'.N-4"B G@0cUIf%,?>R诌ExWz_zN~#hX7N_!mf0t8R|[ hPTfںgfIA::xlg\2(u`$&Tip۰x b}uiIvuI7Qy_l:͓i,v|]2/82pŝzȢ;J@G\J̃0#]ލ1Ο^NV?>ljcu)N Ot]xݞu6M(Cr+pJأVdxu܇. 0Lĝ{ 4Xnq.R̐㤙V7P 5]l5$IFdhRS 9 "mK ߂GV=vX^ub>B yi+^LR8F-\kXܜ"-:ե]AW'EUs{}R(" 0!s="O9ׅ$3Z`'tfs89TdV8D @S.0XUQK7hg5&SÊCA32*,'X+ΣW;+"{%VH.WQ~jv:'PO~jҎ lӤmܓd8K aU,8Z'WO'dp.,\22,Z# #9Ϝ'B%W8U|lG 3c$rno%˵2"J׸QE 8g;/e/| Prh56~zlo|P"YD sAQ%{#at .$~8j6%QeY}sG1jm] -Ԋ=ۈU3n.i֝BD::' e?D"D?0="ZO?mQk&UeHt,,"28 ÐÈ, 2005, ¹ 2 [8]. Òîãäà âîëíà ïàäàåò ïî íîðìàëè íà ãðàíèöó è ïîëÿðèçîâàíà òàê, ÷òî åå ïîëå E ïàðàëëåëüíî îñè x . Äðóãîé ñèñòåìîé, ãäå â ïðèíöèïå âîçìîæíî íàáëþ- äåíèå òàêîãî ðîäà ýôôåêòîâ, ÿâëÿåòñÿ ñèñòåìà õîëîäíûõ èîíîâ, çàõâà÷åííûõ ïåðèîäè÷åñêèì ïî- òåíöèàëîì îïòè÷åñêîé ðåøåòêè ñòîÿ÷åé âîëíû ëàçåðíîãî èçëó÷åíèÿ [12]. 4. Çàêëþ÷åíèå ×èñëåííîå ìîäåëèðîâàíèå îáíàðóæèëî âîçìîæ- íîñòü ïðîÿâëåíèÿ õàîòè÷åñêîé äèíàìèêè ïîâåäå- íèÿ êëàññè÷åñêèõ è, ÷àñòè÷íî, ïîëóêâàíòîâûõ ýëåêòðîíîâ â áàëëèñòè÷åñêèõ ñâåðõðåøåòêàõ ïðè ñèëüíîì ÑÂ× âîçáóæäåíèè. Äëÿ ïîÿâëåíèÿ õàîñà ÷àñòîòà ÑÂ× ïîëÿ äîëæíà óäîâëåòâîðÿòü ñîîòíîøå- íèþ: 1 1 i 1 e , , f        è òàêèå óñëîâèÿ ìîãóò áûòü âûïîëíåíû â áîêîâûõ ñâåðõðåøåòêàõ äâóìåðíîãî ýëåêòðîííîãî ãàçà ïðè î÷åíü íèçêèõ òåìïåðàòóðàõ, êîãäà ýëåêòðîííàÿ ïîäâèæíîñòü äîñòèãàåò 72 10cм/Вс , à áàëëèñòè÷åñêîå äâèæåíèå âîçìîæíî âïëîòü äî 100 ìêì [5].  ýêñïåðèìåíòå äèíàìè÷åñêèé õàîñ ýòîãî òèïà ìîæåò íàáëþäàòüñÿ êàê ôëóêòóàöèè òîêà àíîìàëü- íî âûñîêîãî óðîâíÿ è ìîæåò áûòü èñïîëüçîâàí â êà÷åñòâå èñòî÷íèêà øóìà, êîòîðûé äåéñòâóåò ïî ïðèíöèïó ïðåîáðàçîâàíèÿ êîãåðåíòíûõ ãàðìîíè- ÷åñêèõ (ñèíóñîèäàëüíûõ) êîëåáàíèé ìîùíîé âû- ñîêî÷àñòîòíîé âîëíû, ïàäàþùåé íà ñòðóêòóðó, â íåêîãåðåíòíûå (õàîòè÷åñêèå, øóìîâûå) êîëåáàíèÿ ýëåêòðè÷åñêîãî òîêà áàëëèñòè÷åñêèõ ýëåêòðîíîâ íà âûõîäå ïðèáîðà (íà âíåøíèõ êîíòàêòàõ). Âîçìîæíûì ïðîÿâëåíèåì ýôôåêòà â òàêèõ ñòðóê- òóðàõ ìîæåò áûòü íå òîëüêî íåîáû÷íûé øóì (êàê ïî âåëè÷èíå, òàê è ïî ñâîéñòâàì), íî òàêæå è àíîìàëèè ñîïðîòèâëåíèÿ.  òî æå âðåìÿ ðåàëèçî- âàòü ýòîò ýôôåêò âåñüìà ñëîæíî èç-çà î÷åíü ñïåöèàëüíûõ óñëîâèé åãî âîçíèêíîâåíèÿ. Áîëåå âàæíîå çíà÷åíèå ïîëóêâàíòîâîãî õàîñà ñîñòîèò, âèäèìî, â òîì, ÷òî îí ìîæåò ñîçäàâàòü àíîìàëüíûé èçáûòî÷íûé øóì â òåõ óñëîâèÿõ, êîãäà â îòñóòñòâèå ðàññåÿíèÿ ýëåêòðîíîâ, íî ïðè ðàáîòå â âûñîêî÷à- ñòîòíûõ ðåæèìàõ, â áàëëèñòè÷åñêèõ ñòðóêòóðàõ íàäåþòñÿ ïîëó÷èòü ïîäàâëåíèå øóìà. Óðàâíåíèÿ ïîëóêâàíòîâîé äèíàìèêè, ïðåîáðàçî- âàííûå ê âèäó (7),(8), óñòàíàâëèâàþò ïðÿìóþ ñâÿçü ìåæäó êëàññè÷åñêèì äèíàìè÷åñêèì õàîñîì â ñèñ- òåìå, îïèñûâàåìîé ñòàíäàðòíûì óðàâíåíèåì íåëè- íåéíîãî ìàÿòíèêà (9), è ñëàáî-êâàíòîâûì ïîâåäå- íèåì ýòîé ñèñòåìû, êîãäà êâàíòîâûå ïîïðàâêè ïîñòåïåííî ðàçðóøàþò êëàññè÷åñêèå íåóñòîé÷èâûå (õàîòè÷åñêèå) òðàåêòîðèè ýëåêòðîíîâ è ïðèâîäÿò ê ðåãóëÿðíîé äèíàìèêå ðàçìûòûõ âîëíîâûõ ïàêåòîâ. Òàêèì îáðàçîì, ïðîâåäåííîå èññëåäîâàíèå îáåñïå- ÷èâàåò áîëåå ãëóáîêîå ïîíèìàíèå ïåðåõîäíûõ ñè- òóàöèé ìåæäó ÷èñòî êëàññè÷åñêèì (â ÷àñòíîñòè, äèíàìèêî-õàîòè÷åñêèì) è êâàíòîâûì ðåæèìàìè äâèæåíèÿ ÷àñòèö, à òàêæå ïðåäëàãàåò íîâûé ïîäõîä ê ìîäåëèðîâàíèþ òàêèõ ñèòóàöèé. Ïîëó÷åííûå ðåçóëüòàòû áóäóò ïîëåçíû ïðè ðàçðàáîòêå è ïðîåê- òèðîâàíèè ðåàëüíûõ èñòî÷íèêîâ õàîòè÷åñêèõ êî- ëåáàíèé ðàçëè÷íûõ äèàïàçîíîâ, ïðèìåíÿåìûõ â ñîâðåìåííîé øóìîâîé ðàäèîëîêàöèè [2]. Ëèòåðàòóðà: 1. Nakamura K. Introduction to Chaos and Quantum Transport // Chaos, Solitons and Fractals. 1997. V.8, ¹.7-8. Ð.971-993. 2. Lukin K.A. Millimeter Wave Noise Radar Applications: Theory and Experiment. Proc. 4 th Int. Kharkov Symposium “ Physics and Engineering of Millimeter and Sub-Millimeter Waves, ” June 4-9, 2001. V.1. Ð.68-73. 3. Pattanayak A.K. and Schieve W.C. Gaussian Wave-Packet Dynamics. Semiquantal and Semiclassical Phase-Space Formalism // Phys. Rev. E. 1994. V.50, ¹5. Ð.3601-3614. 4. Wacker A. Semiconductor Superlattices: A Model System for Nonlinear Transport // Phys. Rep. 2002. V.357, ¹1. Ð.1-111. 5. Weisbuch C. and Vinter B. Quantum Semiconductor Structures. Fundamentals and Applications. New York: Academic Press, 1991. 252 p. 6. Hairer E.,. Norsett S.P. and Wanner G. Solving Ordinary Differential Equations. I. Nonstiff Problems. Berlin: Springer-Verlag, 1987.480 ð. 7. Hilborn R.C. Chaos and Nonlinear Dynamics. An Introduction for Scientists and Engineers. Oxford: 2000. 650 p. 8. Wagenhuber J., Geisel T., Niebauer P. and Obermair G. Chaos and Anomalous Diffusion of Ballistic Electrons in Latteral Surface Superlattices // Phys. Rev. B. 1992. V. 45, ¹ 8. Ð.4372-4383. 9. Yurchenko L.V. and Yurchenko V.B. Semiquantal Dynamics of Electrons in Quantum Heterostructures // NATO Science Series. 3. High Technology. The Netherlands. Kluwer Acad.Publ. 1998. V. 48. Ð.83-86. 10. Blanter Ya.M. and Buttiker M. Short Noise in Mesoscopic Conductors // Phys. Rep. 2000. V. 336. Ð.1-99. 11. Bulashenko O.M. and Rubi J.M. Self- Consistent Theory of Current and Voltage Noise in Multimode Ballistic Conductors // Phys. Rev. B. 2002. V. 66, ¹ 4. Ð.045310-16. 12. Fromhold T.M., Tench C.R., Bujkiewicz S., Wilkinson P.B. and Sheard F.W. Quantum Chaos for Cold Atoms in an Optical Lattice with a Tilted Harmonic Trap // J. Opt. B. Quantum Semiclass. Opt. 2000. V. 2, ¹ 5. Ð.628-632. Ïîñòóïèëà â ðåäêîëëåãèþ 23.04.2005 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. ×óðþìîâ Ã.È. Þð÷åíêî Ëèäèÿ Âàëåðüåâíà, êàíä. ôèç.-ìàò. íàóê, ñòàðøèé íàó÷íûé ñîòðóäíèê îòäåëà íåëèíåéíîé äè- íàìèêè ýëåêòðîííûõ ñèñòåì ÈÐÝ èì. À.ß.Óñèêîâà ÍÀÍ Óêðàèíû. Íà- ó÷íûå èíòåðåñû: ìîäåëèðîâàíèå äè- íàìè÷åñêîãî õàîñà â ýëåêòðîííûõ ñèñòåìàõ, àâòîãåíåðàöèè øèðîêî- ïîëîñíûõ øóìîâûõ ñèãíàëîâ è óëü- òðà-êîðîòêèõ èìïóëüñîâ. Àäðåñ: Óêðàèíà, 61085, Õàðüêîâ, óë. Àê. Ïðîñêóðû, 12, òåë.:+380-577-448349, email: yurchenk@ire.kharkov.ua Þð÷åíêî Âëàäèìèð Áîðèñîâè÷, ä-ð ôèç.-ìàò. íàóê, ñòàðøèé íàó÷íûé ñîòðóäíèê, ôàêóëüòåò ýêñïåðè- ìåíòàëüíîé ôèçèêè Íàöèîíàëüíîãî Óíèâåðñèòåòà Èðëàíäèè, ã. Ìåéíóò. Íàó÷íûå èíòåðåñû: ðàíåå - òåîðèÿ ïåðåíîñà ãîðÿ÷èõ ýëåêòðîíîâ â ïîëóïðîâîä- íèêîâûõ ïðèáîðàõ, ôîòî- è òåð- ìî-ýëåêòðè÷åñêèå ýôôåêòû, ÿâ- ëåíèÿ íåñòàáèëüíîñòè è õàîñ; â ïîñëåäíåå âðåìÿ – íåëèíåéíàÿ äèíàìèêà ýëåêòðîííûõ ñèñòåì, ðàñïðîñòðàíåíèå âîëí è ìîäåëè- ðîâàíèå àíòåíí. Àäðåñ: Experimental Physics Department, NUI Maynooth, Co.Kildare, Ireland. 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Ïðåäëàãàåòñÿ ìåòîä àíàëèçà äîïóñêîâ äëÿ íåìîíîòîí- íûõ âûõîäíûõ ôóíêöèé, îñíîâàííûé íà ìåòîäå Áðîé- äåíà–Ôëåò÷åðà–Ãîëüäôàðáà–Øåííî, ìîäèôèöèðî- âàííîì ñ ó÷åòîì îñîáåííîñòåé äàííîé çàäà÷è. Ìåòîä ýôôåêòèâíî ïðèìåíÿåòñÿ êàê ê ìîíîòîííûì, òàê è íåìîíîòîííûì âûõîäíûì ôóíêöèÿì. Âûñîêàÿ ñêî- ðîñòü ñõîäèìîñòè è íèçêèå âû÷èñëèòåëüíûå çàòðàòû ïîäòâåðæäåíû ÷èñëåííûìè ýêñïåðèìåíòàìè. 1. Ââåäåíèå Àíàëèç äîïóñêîâ ÿâëÿåòñÿ îäíèì èç ýòàïîâ ïðîåê- òèðîâàíèÿ ýëåêòðîííûõ àïïàðàòîâ. Äîñòîâåðíàÿ èíôîðìàöèÿ îá îòêëîíåíèÿõ âûõîäíûõ ïàðàìåòðîâ ïîçâîëÿåò îöåíèòü òî÷íîñòü è íàäåæíîñòü ðàäèî- ýëåêòðîííîé àïïàðàòóðû, ïðèíèìàòü ïðàâèëüíûå ðåøåíèÿ â ïðîöåññå ïðîåêòèðîâàíèÿ. Èçâåñòåí öåëûé ðÿä ìåòîäîâ ðàñ÷åòà äîïóñêîâ, òàêèõ êàê ïåðåáîð âîçìîæíûõ êîìáèíàöèé ïàðà- ìåòðîâ, ïåðåäàòî÷íûõ êîýôôèöèåíòîâ, ôóíêöèé ÷óâñòâèòåëüíîñòè [1,2]. Ïåðâûé èç íèõ òðåáóåò áîëüøîãî îáúåìà âû÷èñëåíèé, òî÷íîñòü æå îñòàëü- íûõ íåâåëèêà. Ïîãðåøíîñòü ðàñ÷åòà ðàñòåò ñ óâåëè- ÷åíèåì íåëèíåéíîñòè âûõîäíîé ôóíêöèè. Íàèáîëåå ýôôåêòèâíîé è òî÷íîé ÿâëÿåòñÿ ìåòîäè- êà, èñïîëüçóþùàÿ ëèíåéíûå èíòåðâàëüíûå ìîäåëè  [3,4]. Ïðîâåäåíèå äâîéíîé èíòåðïîëÿöèè ïðè ïî- ñòðîåíèè ìîäåëè ïîçâîëÿåò ó÷åñòü íåëèíåéíîñòü âûõîäíîé ôóíêöèè è îáåñïå÷èâàåò ïîëó÷åíèå òî÷- íîãî ðåçóëüòàòà. Ãëàâíûì íåäîñòàòêîì, ñóùåñòâåí- íî îãðàíè÷èâàþùèì îáëàñòü ïðèìåíåíèÿ äàííîãî ìåòîäà, ÿâëÿåòñÿ îáÿçàòåëüíîå óñëîâèå ìîíîòîííî- ñòè âûõîäíîé ôóíêöèè.  òî æå âðåìÿ, ñóùåñòâóåò áîëüøîå êîëè÷åñòâî óñòðîéñòâ (íàïðèìåð, óñòðîé- ñòâà ÷àñòîòíîé ñåëåêöèè), âûõîäíûå õàðàêòåðèñòè- êè êîòîðûõ ñîäåðæàò ýêñòðåìóìû. Ïîâûøåíèå òî÷íîñòè è ýôôåêòèâíîñòè ïðîöåññà àíàëèçà äî- ïóñêîâ òàêèõ óñòðîéñòâ ÿâëÿåòñÿ àêòóàëüíîé çàäà- ÷åé. Öåëü äàííîãî èññëåäîâàíèÿ – ðàçðàáîòêà ìåòîäà, ïîçâîëÿþùåãî ðåøèòü çàäà÷ó àíàëèçà äîïóñêîâ äëÿ íåìîíîòîííûõ âûõîäíûõ ôóíêöèé. 2. Àíàëèç äîïóñêîâ äëÿ íåìîíîòîííûõ âûõîäíûõ ôóíêöèé Àíàëèç äîïóñêîâ ïðåäñòàâëÿåò ñîáîé çàäà÷ó íàõîæ- äåíèÿ ìàêñèìàëüíîãî è ìèíèìàëüíîãî çíà÷åíèÿ âûõîäíîé ôóíêöèè â îáëàñòè, îãðàíè÷åííîé èí- òåðâàëîì äîïóñêà âõîäíûõ ïàðàìåòðîâ:       n , 1 i , x x x ) X ( f овать максимизир и вать минимизиро i i i , (1) ãäå ) X ( f – âûõîäíàÿ ôóíêöèÿ;   n 1 x ,... x X  – âåêòîð âõîäíûõ ïàðàìåòðîâ; i i x , x – íèæíÿÿ è âåðõíÿÿ ãðàíèöû i -ãî âõîäíîãî ïàðàìåòðà; n – êîëè÷åñòâî âõîäíûõ ïàðàìåòðîâ.  ñëó÷àå ìîíîòîííîé âûõîäíîé ôóíêöèè åå ìàê- ñèìàëüíîå è ìèíèìàëüíîå çíà÷åíèÿ ñîäåðæàòñÿ â òî÷êå ) x ,... x ( X max n max 1 max i  è ) x ,... x ( X min n min 1 min i  , ãäå max i x è min i x ïðèíè- ìàþò çíà÷åíèå i x ëèáî i x .  ñëó÷àå íåìîíîòîííîé âûõîäíîé ôóíêöèè çàäà÷à íàõîæäåíèÿ åå ìàêñè- ìàëüíîãî è ìèíèìàëüíîãî çíà÷åíèÿ óñëîæíÿåòñÿ è òðåáóåò íàõîæäåíèÿ ýêñòðåìóìîâ âûõîäíîé ôóíê- öèè íà èíòåðâàëå äîïóñêà âõîäíûõ ïàðàìåòðîâ.  òåîðèè îïòèìèçàöèè çàäà÷à (1) ïîëó÷èëà íàçâà- íèå çàäà÷è óñëîâíîé îïòèìèçàöèè ñ îãðàíè÷åíèÿ- ìè â âèäå íåðàâåíñòâ. Ñóùåñòâóåò áîëüøîå êîëè- ÷åñòâî ìåòîäîâ îïòèìèçàöèè äèôôåðåíöèðóåìûõ ôóíêöèé ïðè ëèíåéíûõ îãðàíè÷åíèÿõ [5,6]. Íàè- áîëåå ïðîñòûì ñïîñîáîì ðåøåíèÿ çàäà÷è (1) ÿâëÿ- åòñÿ ïðèìåíåíèå ìîäèôèöèðîâàííîãî ìåòîäà íàè- ñêîðåéøåãî ñïóñêà [7]. Íåäîñòàòêîì äàííîãî ìåòî- äà ÿâëÿåòñÿ âîçìîæíàÿ ñëàáàÿ ýôôåêòèâíîñòü âáëèçè ñòàöèîíàðíîé òî÷êè âñëåäñòâèå ýôôåêòà çèãçàãà [5,6]. Êðîìå òîãî, ñêîðîñòü ñõîäèìîñòè íåâûñîêà, äëÿ ôóíêöèé îáùåãî âèäà ïðîöåññ ñõîäèòñÿ â ëó÷øåì ñëó÷àå ëèíåéíî. Áîëåå âûñîêóþ ñêîðîñòü ñõîäèìîñòè èìåþò íüþòîíîâñêèå, êâàçèíüþòî- íîâñêèå ìåòîäû è ìåòîäû ñîïðÿæåííûõ íàïðàâëå- íèé [7]. Íàèëó÷øèì êâàçèíüþòîíîâñêèì ìåòîäîì îáùåãî íàçíà÷åíèÿ ñ÷èòàåòñÿ ìåòîä Áðîéäåíà– Ôëåò÷åðà–Ãîëüäôàðáà–Øåííî (ÁÔÃØ) [7], êîòî- ðûé èñïîëüçóåò èòåðàöèè: ) k ( ) k ( ) k ( ) 1 k ( h X X      , (2) ãäå ) 1 k ( X  è ) k ( X – èñêîìàÿ òî÷êà íà 1 k  è k -é èòåðàöèè; ) k (  – ïîëîæèòåëüíûé øàãîâûé ìíîæè- òåëü; ) X ( f D h ) k ( ) k ( ) k (    – íàïðàâëåíèå ñïóñêà; ) k ( D – ïîëîæèòåëüíî îïðåäåëåííàÿ ìàòðèöà, ÿâëÿ- þùàÿñÿ àïïðîêñèìàöèåé ìàòðèöû Ãåññå; ) X ( f ) k (  – ãðàäèåíò âûõîäíîé ôóíêöèè íà k -é èòåðàöèè. Çíàê «–» â âûðàæåíèè (2) îïðåäåëÿåò ïðîöåäóðó ìèíèìèçàöèè, çíàê «+» – ìàêñèìèçàöèè âûõîäíîé ôóíêöèè. Ìàòðèöà ) k ( D ïåðåñ÷èòûâàåòñÿ ïî õîäó âû÷èñëèòåëüíîãî ïðîöåññà ïî ñëåäóþùèì ôîðìó- ëàì: T k T k T k T T ) k ( ) 1 k ( v v q D q D q q D q p p p D D                ; ) k ( ) 1 k ( X X p    ; ) X ( f ) X ( f q ) k ( ) 1 k (      ; q p q D q T ) k ( T      ;     q D p v ) k ( . (3) Äëÿ âûáîðà øàãîâîãî ìíîæèòåëÿ èñïîëüçóþòñÿ îäíîìåðíàÿ îïòèìèçàöèÿ, îäíîìåðíàÿ îïòèìèçà- öèÿ ñ îãðàíè÷åíèåì, âûáîð ïî ïðàâèëó Àðìèõî, ïî ïðàâèëó Ãîëäñòåéíà è ò.ä. [5 – 7]. Ìåòîä ÁÔÃØ ïðåäíàçíà÷åí äëÿ ïðîâåäåíèÿ áåçóñ- ëîâíîé îïòèìèçàöèè. 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?cv+i z e]~ozee|ﲩ`T1y;CѮNquW}}* Tdߗ~>Yr69&!OtdIɋ4G>%A?re)&Ņfjj7Ҕ Plep?P X|Mq?7%T_57ٗ%})JӛԸfq$pp_#×|nZNzڌEy;"5a%yx&g N) hڮ0x/Fen0جs7A-ڌ{VdhQѝҀzt m'Z›az9t0\ \Gm>?Ođ.}Kcڞ0ҶKqD'[K%kE;)~^qgw ϡ*K{ʸoqclqldz4 {5ҟ&>t,#iz$FǗ~uǏTh{sxph׌g}hgz ear;_߯NWP8Zxb2G<}G/22>w`ot^~Cv Y@,ި 5mRjMgFoa0tu#aН YCO?9Jwɭ?E8֟GF/!%±Y_ڦv@jܱީ|υ??VHv1/=:^/},jg=skuuYKӮ>.̆OdeV.T;?C2W9;P5C\hA}tZtsTry"]fIt˾X>Oy4Aa:.W;49dwD0נosrqtPƙO׳bʸVBZY8х֡zm>%l3z[[Qoō-:Mqvv:-ӋH"ǵuWC|l{ٖ+ߩ$hs kax`5VUD9f~g{~] \@}"VuxR+A ˱>UpK|Myo-šmݥ'vg= jŦU@ l(=/i/y5-= =i=x7I~ y8we2(ὐ&/G]W;_S 76!?agQa0kIS_9<{%Z h=e?k{7Q {uh?k%W_,@4|}fD2`A*K}?{^9s+šՕ*jgmG:ǽ_~y(te(32eSPZ @64Vg+Gk$GЏ{J.c֏0W |2y2Jp3=p>;% x*Ր +Վ%勩EM5-üV*rL[^;lY:p}~<,ڑ]~%$pzyߩul!]E?hm{n7(5tO/E[yvQ `Yrÿ6CnAhK[BcZ4q#q}Dߒ~u~ mӁ=3><-3vAfɲYizWI!k/yYKY藁=g*w6dj|Ef:iqBY&/fI{|ĝ-PX`WRFPI}b DaBЌgI@ᥲ`Ǒx&e.#Mݠti~0An7*n ݒ;]+~VT/Zf|]h'<'h8; _@'F1S\v~VK>O;h3m-nM[E* }¦*S>W|M? ɾ~LU!^gaWY )U鼃:jD_Qw?-t&?޷n\w\Ni屄ܰ dI@ZumR[  zQ9*nVN߬#TNw{vM{B[cCWӘt?ִ$z0.8ai!=8g~@|ZjAaԣS>O(3Sx &t綀ٌ.l y.gn?6NzIn'a6O93B"wV7%2n9^.c8?[e9޲<:EVx]=_ )F'tyii:)E,U2"ޠ4>/*jg9޶ꣁw1 =]6r-ȣ0`U3yo?$oㅼs=,,30 ÐÈ, 2005, ¹ 2 ìîäèôèêàöèþ, îáåñïå÷èâàþùóþ íàõîæäåíèå òî÷- êè ) 1 k ( X  â îáëàñòè çàäàííûõ èíòåðâàëîâ. Ýòî ìîæíî îáåñïå÷èòü çàìåíîé (2) íà ñîîòíîøåíèå: ) z ( in X ) 1 k (   ; (4) çäåñü ) k ( ) k ( ) k ( h X z     ; ) z ( in – îïåðàòîð, îáåñïå- ÷èâàþùèé íàõîæäåíèå z âíóòðè çàäàííûõ èíòåð- âàëîâ:           i i i i i i i i i i x z x x z x z x z x ) z ( in ; n , 1 i  . (5) Öåëåñîîáðàçíûì òàêæå ÿâëÿåòñÿ èñêëþ÷åíèå èç ïðîöåññà ïåðåñ÷åòà ìàòðèöû òåõ âõîäíûõ ïàðàìåò- ðîâ, çíà÷åíèÿ êîòîðûõ íå èçìåíèëèñü ïî ñðàâíå- íèþ ñ ïðåäûäóùåé èòåðàöèåé. Ýòî äàåò âîçìîæ- íîñòü óìåíüøèòü ðàçìåðíîñòè âåêòîðîâ è ìàòðèö â âûðàæåíèÿõ (3) è, ñîîòâåòñòâåííî, ñýêîíîìèòü ðåñóðñû íà èõ òðàíñïîíèðîâàíèå è ïåðåìíîæåíèå. Òîãäà äëÿ ôîðìèðîâàíèÿ íàïðàâëåíèÿ ñïóñêà ) k ( h èñïîëüçóåòñÿ âûðàæåíèå:          n , 1 i otherwise h I i dx ) X ( df h ) k ( fj ) k ( i ) k ( ) k ( i , (6) ãäå ) k ( I – ìíîæåñòâî èíäåêñîâ âõîäíûõ ïàðàìåò- ðîâ, ôîðìèðóþùååñÿ èç óñëîâèÿ: } n i , x x | N i , i { I ) 1 k ( i ) k ( i ) k (      ; (7) ) X ( f D h ) k ( f ) k ( f ) k ( f    ; (8) ) k ( f D – ìàòðèöà, âêëþ÷àþùàÿ ñòðîêè è ñòîëáöû ìàòðèöû ) k ( D , èíäåêñû êîòîðûõ íå âõîäÿò âî ìíîæåñòâî ) k ( I ; ) X ( f ) k ( f  – âåêòîð, âêëþ÷àþùèé ñòðîêè âåêòîðà ) X ( f ) k (  , èíäåêñû êîòîðûõ íå âõîäÿò âî ìíîæåñòâî ) k ( I ; m , 1 j  , m – êîëè÷åñòâî íàòóðàëüíûõ ÷èñåë, äëÿ êîòîðûõ âûïîëíÿåòñÿ óñ- ëîâèå: ) i m ( and ) 0 m ( and ) I m ( ) k (    . (9) Èñïîëüçîâàíèå âûðàæåíèÿ (6) èñêëþ÷àåò âëèÿíèå âõîäíûõ ïàðàìåòðîâ, âûøåäøèõ çà ãðàíèöó îãðà- íè÷åíèé, íà ôîðìèðîâàíèå íàïðàâëåíèÿ ñïóñêà ïî îñòàëüíûì ïàðàìåòðàì, à òàêæå óìåíüøàåò îáúåì âû÷èñëåíèé ïðè ïåðåñ÷åòå ìàòðèöû k D íà êàæäîé èòåðàöèè. Ìàòðèöà ) k ( f D èñïîëüçóåòñÿ äëÿ ïåðå- ñ÷åòà ìàòðèöû ) 1 k ( D  â õîäå èòåðàöèîííîãî ïðîöåñ- ñà ïðè ïîìîùè âûðàæåíèÿ:   ) 1 k ( j , i D                 otherwise , 1 1 j 1 j , 1 1 i 1 i D ) I j or I i ( and j i , 0 1 j , 1 i ) 1 k ( f ) 1 k ( ) 1 k ( , , (10) n , 1 j , i  , m , 1 1 j , 1 i  . Âûõîäíûå ôóíêöèè ïðîåêòèðóåìûõ óñòðîéñòâ ðåä- êî ñîäåðæàò ãëîáàëüíûé ìàêñèìóì èëè ìèíèìóì. Ãîðàçäî ÷àùå âñòðå÷àþòñÿ ÷àñòíûå ýêñòðåìóìû ïî îòäåëüíûì âõîäíûì ïåðåìåííûì. Íåëèíåéíûå ñîñòàâëÿþùèå âûõîäíîé ôóíêöèè ïî îñòàëüíûì ïàðàìåòðàì ìîãóò áûòü íåçíà÷èòåëüíûìè, è âûõîä ýòèõ ïàðàìåòðîâ çà îáëàñòü îãðàíè÷åíèé ÿâëÿåòñÿ ïðîãíîçèðóåìûì. Ïðàâèëüíûé âûáîð íà÷àëüíûõ çíà÷åíèé òàêèõ ïàðàìåòðîâ íà ãðàíèöå îáëàñòè îãðàíè÷åíèé ìîæåò ñïîñîáñòâîâàòü áîëåå áûñòðîìó èõ âûõîäó çà îãðàíè÷åíèÿ è óìåíüøåíèþ êîëè÷å- ñòâà èòåðàöèé è îáúåìà âû÷èñëåíèé íà êàæäîé èç íèõ. Âûáîð íà÷àëüíîé òî÷êè ìîæåò ïðîèçâîäèòüñÿ ñ ïîìîùüþ ëèíåéíîé èíòåðâàëüíîé ìîäåëè [4]:              n 0 1 i i i n 0 1 i i i 0 i i | | dual | | a a x a x a a y , (11) ãäå , ] y ; y [ y  , ] a ; a [ a i i i  , ] x , x [ x i i i  – èíòåðâàëû çíà÷åíèé âûõîäíîé ôóíêöèè, êîýôôèöèåíòîâ ìîäåëè è ïàðàìåòðîâ ýëåìåíòîâ; y , a , x , y , a , x i i i i – íèæíèå è âåðõíèå ãðàíèöû èíòåðâàëîâ ïàðàìåòðîâ ýëåìåíòîâ, êîýôôèöèåíòîâ ìîäåëè è âûõîäíîé ôóíêöèè;  – îïåðàòîð êîììóòàöèîííîãî óìíî- æåíèÿ èíòåðâàëîâ; ] x ; x [ ) x dual(  – äóàëüíûé èí- òåðâàë. Îòîáðàæåíèå (11) â ïðîñòðàíñòâî äåéñòâèòåëüíûõ ÷èñåë ïîçâîëÿåò çàïèñàòü äâà óðàâíåíèÿ, îïèñûâà- þùèõ ïîâåäåíèå âûõîäíîé ôóíêöèè â çîíàõ, ðàçäåëåííûõ íîìèíàëüíûì çíà÷åíèåì:          n 0 i a 1 i n 0 i a 1 i i i i i 0 x a x a a y , ) y y ( н  , (12)          n 0 i a 1 i n 0 i a 1 i i i i i 0 x a x a a y , ) y y ( н  , (13) ãäå н y – íîìèíàëüíîå çíà÷åíèå âûõîäíîé ôóíê- öèè. Êîýôôèöèåíòû ìîäåëè, ïîëó÷åííûå â ðåçóëüòàòå ïðîâåäåíèÿ âíóòðåííåé èíòåðïîëÿöèè, îáîçíà÷à- þòñÿ êàê ri a , âíåøíåé èíòåðïîëÿöèè – bi a . Åñëè 0 a , a , a , a bi bi ri ri  , òî âåðîÿòíîñòü òîãî, ÷òî âûõîä- íàÿ ôóíêöèÿ íå ñîäåðæèò ÷àñòíûõ ýêñòðåìóìîâ ïî i -ìó âõîäíîìó ïàðàìåòðó, äîâîëüíî âûñîêà. Óâå- ëè÷åíèå ýòîãî ïàðàìåòðà ïðèâîäèò ê ðîñòó âûõîä- íîé ôóíêöèè, åå ìèíèìóì áóäåò îïðåäåëÿòüñÿ åãî íèæíåé ãðàíèöåé, à ìàêñèìóì – âåðõíåé. Åñëè 0 a , a , a , a bi bi ri ri  , òî âåðîÿòíîñòü íàëè÷èÿ ÷àñò- íîãî ýêñòðåìóìà ïî i -ìó âõîäíîìó ïàðàìåòðó òîæå íåâûñîêà. Óâåëè÷åíèå ýòîãî ïàðàìåòðà ïðèâîäèò ê óìåíüøåíèþ âûõîäíîé ôóíêöèè, åå ìèíèìóì áóäåò îïðåäåëÿòüñÿ åãî âåðõíåé ãðàíèöåé, à ìàêñè- ìóì – íèæíåé. Ðàçëè÷èå çíàêîâ âíåøíèõ è âíóòðåííèõ êîýôôè- öèåíòîâ ìîäåëè îäíîçíà÷íî ñâèäåòåëüñòâóåò î íà- ëè÷èè íåìîíîòîííîñòè ïî i -ìó âõîäíîìó ïàðàìåò- \x tl/( !P "[PUE*(*qj"s,*ڢ+*R! 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Ðåêîìåíäóåìûé âûáîð íà÷àëüíîãî çíà÷åíèÿ âõîäíûõ ïàðàìåòðîâ ïðèâåäåí â òàáë. 1. Âî âñåõ îñòàëüíûõ ñëó÷àÿõ ðåêîìåíäóåòñÿ â êà÷åñòâå íà- ÷àëüíîãî çíà÷åíèÿ âõîäíîãî ïàðàìåòðà âûáèðàòü åãî íîìèíàëüíîå çíà÷åíèå.  ñëó÷àå ñòðîãî ìîíîòîííîé âûõîäíîé ôóíêöèè ïðèìåíåíèå òàêîãî ïîäõîäà ê âûáîðó íà÷àëüíîé òî÷êè ïîèñêà ïðèâåäåò ê òîìó, ÷òî ìåòîä ÁÔÃØ, ìîäèôèöèðîâàííûé ïî ïðàâèëàì (4)–(10), îñòàíî- âèò ñâîþ ðàáîòó ïîñëå ïåðâîé èòåðàöèè. Ýòî ïîçâîëÿåò ýôôåêòèâíî ïðèìåíÿòü åãî äëÿ àíàëèçà äîïóñêîâ êàê íåìîíîòîííûõ, òàê è ìîíîòîííûõ âûõîäíûõ ôóíêöèé. Ïðåäëîæåííûé ìåòîä â ðå- çóëüòàòå ïðåäñòàâëÿåòñÿ àëãîðèòìîì: Øàã 1. Ïîñòðîåíèå ëèíåéíîé èíòåðâàëüíîé ìîäåëè (10). Øàã 2. Âûáîð íà÷àëüíîé òî÷êè ïîèñêà (òàáë. 1). Øàã 3. 0 k  . Âûáîð ìàòðèöû ) 0 ( D êàê åäèíè÷íîé. Âûáîð òî÷íîñòè îïòèìèçàöèè  . Øàã 4. Îïðåäåëåíèå ) X ( f ) k (  , ) X ( f ) k ( f  . Åñëè    || ) X ( f || ) k ( f – êîíåö ðàáîòû. Øàã 5. Îïðåäåëåíèå íàïðàâëåíèÿ îïòèìèçàöèè ) X ( h ) k ( ïî ïðàâèëàì  (6)–(8) . Øàã 6. Îïðåäåëåíèå øàãîâîãî ìíîæèòåëÿ ) k (  . Øàã 7. Ïåðåõîä ê ñëåäóþùåé òî÷êå ) 1 k ( X  ïî ïðàâèëàì (4), (5). Øàã 8. Åñëè ) 1 k ( ) k ( X X   – êîíåö ðàáîòû. Øàã 9. Îïðåäåëåíèå ìíîæåñòâà ) 1 k ( I  ñîãëàñíî óñëîâèþ (7), îïðåäåëåíèå ìàòðèöû ) k ( f D . Øàã 10. Îïðåäåëåíèå ) X ( f ) k (  , ) X ( f ) k ( f  . Åñëè    || ) X ( f || ) k ( f – êîíåö ðàáîòû. Øàã 11. Ïåðåñ÷åò ìàòðèöû ) 1 k ( f D  ñîãëàñíî âûðà- æåíèÿì (3). Øàã 12. Âíåñåíèå èçìåíåíèé â ìàòðèöó ) 1 k ( D  ñîãëàñíî âûðàæåíèþ (10). Øàã 13. 1 k k   . Âîçâðàò ê øàãó 5. 3. Ïðèìåíåíèå ìîäèôèöèðîâàííîãî ìåòîäà ÁÔÃØ Ðàçðàáîòàííàÿ ìåòîäèêà áûëà èñïîëüçîâàíà äëÿ ðåøåíèÿ çàäà÷è àíàëèçà äîïóñêîâ íà êîýôôèöèåíò ïåðåäà÷è êîëåáàòåëüíîãî êîíòóðà è êîýôôèöèåíò çàòóõàíèÿ ôèëüòðà íèæíèõ ÷àñòîò. Ñõåìà êîëåáà- òåëüíîãî êîíòóðà ïðèâåäåíà íà ðèñ. 1. Êîýôôèöè- åíò åãî ïåðåäà÷è îïðåäåëÿåòñÿ âûðàæåíèåì: 2 2 1 2 2 1 ) L 1 C ( R ) R / R 1 ( 1 H          ,(14) ãäå  – êðóãîâàÿ ÷àñòîòà; C , R , R 2 1 è L – ïàðàìåòðû ýëåìåíòîâ. Ïðè àíàëèçå íîðìèðîâàííûå íîìèíàëüíûå çíà÷å- íèÿ âõîäíûõ ïàðàìåòðîâ ñîñòàâëÿëè 05 . 1 R 1  ; 05 . 1 R 2  ; 05 . 1 L  ; 05 . 1 C  ñ äîïóñêîì % 10  . Ðàñ- ÷åò ïðîâîäèëñÿ íà ÷àñòîòå 05 . 1   . Íîìèíàëüíûì çíà÷åíèÿì âõîäíûõ ïàðàìåòðîâ ñîîòâåòñòâîâàë êîýôôèöèåíò ïåðåäà÷è êîëåáàòåëüíîãî êîíòóðà 4974 . 0 y н  . Ðèñ. 1 Çíà÷åíèÿ êîýôôèöèåíòîâ ëèíåéíîé èíòåðâàëüíîé ìîäåëè ïðèâåäåíû â òàáë. 2. Òàáëèöà 2 –0,228 –0,252 –0,228 –0,249 0,247 0,223 0,234 0,225 –0,035 –0,02 –0,04 –0,062 –0,025 –0,019 –0,032 –0,049 r a r a b a b a 1 R 2 R C L Ìàêñèìàëüíîå è ìèíèìàëüíîå çíà÷åíèÿ âûõîäíîé ôóíêöèè íàõîäèëèñü ïðè ïîìîùè ìîäèôèöèðî- âàííîãî ìåòîäà íàèñêîðåéøåãî ñïóñêà [7] è ÁÔÃØ. Çíà÷åíèå øàãîâîãî ìíîæèòåëÿ ) k (  âûáèðàëîñü ïî ìîäèôèöèðîâàííîìó ïðàâèëó Àðìèõî [5,7] ïðè 1 s  , 4 . 0   , 0001 . 0   , 4 10 1     .Ìèíèìàëüíîå çíà÷åíèå êîýôôèöèåíòà ïåðåäà÷è êîëåáàòåëüíîãî êîíòóðà ñîñòàâëÿåò 4411 . 0 y  è íàõîäèòñÿ â òî÷êå .155) 45;1.155;1 (1.155;0.9 ) x , x , x , x ( 4 2 2 1  . Äëÿ åå íàõîæäåíèÿ ìåòîäîì íàèñêîðåéøåãî ñïóñêà ïîíàäîáèëîñü 10 èòåðàöèé. Ìîäèôèöèðîâàííûé ìåòîä ÁÔÃØ ðåøèë çàäà÷ó íàõîæäåíèÿ ìèíèìóìà çà 2 èòåðàöèè. Ìàêñèìàëüíîå çíà÷åíèå êîýôôèöè- åíòà ïåðåäà÷è êîëåáàòåëüíîãî êîíòóðà ñîñòàâëÿåò 55 . 0 y  è íàõîäèòñÿ â òî÷êå 5) 5;0.95;0.9 0.945;1.15 ( ) x , x , x , x ( 5 4 2 1  . 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S쩇rweH\ 'il;ȣtqCr`}*"ەFSXU^W^8]FnVQѩJXEhSY'|'YWAC8ixVd1)>'P˝c~ɛ4)~WkYlՎw<#)EHa0;SBd7~I7-A)v9hŘay^-^åj#Ұ)z֏vJ֖ >; O-&!}H}OwHn>GMi8I#lۙf,ų'mDsO~dD&[)+$x7LӲ's-i8/,_HJPZ_V'aaD.c[Umxz@Tß҆ok5+P#zMcÌ˃3.Q!<K:t ߎ8z#Qg}jϘxV"`AآxW0ߠs;ˌٌ1$ށguƅ'|čim t|cF4&X^)O:4ġsy >,vQ s@)l7*o6vd4g `@fek4bja_E*oUS] y6i\ۓС0wN:)[tkKv æ,tq ޣFk;twIra `XHש"'7cG㰏!ըaa<-7c4x)޿X( rw}J]7XD!q^-΅ró$<P#9:/uvm%>V5c/i+6 ̋<\s8?#O}m@x<Ж~YG,n0nC-g].7*qlxl+{z*|XnҮtU\Zh2i%Ur+o|x"2y@]96poyA.7ƇtH9 yQ4iwyPҗ}(_AOQЮY}:3G›W JYTO,,32 ÐÈ, 2005, ¹ 2 Äëÿ åå íàõîæäåíèÿ ïîíàäîáèëîñü 25 èòåðàöèé ìåòîäîì íàèñêîðåéøåãî ñïóñêà è 5 èòåðàöèé ìåòî- äîì ÁÔÃØ ïðè èñïîëüçîâàíèè íîìèíàëüíîé òî÷- êè â êà÷åñòâå íà÷àëüíîé. Èñïîëüçîâàíèå ëèíåéíîé èíòåðâàëüíîé ìîäåëè äëÿ âûáîðà íà÷àëüíîé òî÷êè ïîèñêà ñîêðàòèëî êîëè÷åñòâî èòåðàöèé äî 3. Äî- ïóñê íà âûõîäíóþ ôóíêöèþ ñîñòàâèë %) 57 . 10 %, 32 . 11 ( y   . Ñõåìà ôèëüòðà íèæíèõ ÷àñòîò ïðèâåäåíà íà ðèñ.  2. Íîðìèðîâàííûå íîìèíàëüíûå çíà÷åíèÿ âõîäíûõ ïàðàìåòðîâ ïðèíèìàëèñü 1 R 1  , 1 R 2  , 368557 . 0 C 1  , 368557 . 0 C 3  , 737714 . 0 L 2  ñ îò- êëîíåíèÿìè % 10  . Ðàñ÷åò êîýôôèöèåíòà ïåðåäà- ÷è ïðîâîäèëñÿ íà ÷àñòîòå 1   . Ðèñ. 2 Çíà÷åíèÿ êîýôôèöèåíòîâ ëèíåéíîé èíòåðâàëüíîé ìîäåëè ïðèâåäåíû â òàáë. 3. Ðàçëè÷èå çíàêîâ êîýôôèöèåíòîâ ïî ïàðàìåòðó 3 C ñâèäåòåëüñòâóåò î íåìîíîòîííîñòè âûõîäíîé ôóíêöèè è íåàäåêâàò- íîñòè ëèíåéíîé èíòåðâàëüíîé ìîäåëè. Òàáëèöà 3 0,4833 0,4834 0,4822 0,492 –0,47 –0,575 –0,44 –0,634 –0,013 –0,023 0,021 –0,078 –0,013 –0,023 –0,024 –0,014 0,017 0,04 0,0006 0,059 r a r a b a b a 1 R 2 R 3 C 4 C 2 L Ìèíèìàëüíîå çíà÷åíèå êîýôôèöèåíòà çàòóõàíèÿ ôèëüòðà íèæíèõ ÷àñòîò ðàâíÿåòñÿ 9058 . 0 y  è íàõîäèòñÿ â òî÷êå 54;0.6639) .3317;0.40 (0.9;1.1;0 ) x , x , x , x , x ( 5 4 3 2 1  . Äëÿ åå íàõîæäåíèÿ ïîíàäîáèëîñü 18 èòåðàöèé ìåòîäîì íàèñêîðåéøåãî ñïóñêà. Ìîäèôèöèðîâàí- íûé ìåòîä ÁÔÃØ ðåøèë çàäà÷ó íàõîæäåíèÿ ìèíè- ìóìà çà 5 èòåðàöèé. Ìàêñèìàëüíîå çíà÷åíèå êîýô- ôèöèåíòà çàòóõàíèÿ ôèëüòðà íèæíèõ ÷àñòîò ðàâíÿ- åòñÿ 1167 . 1 y  è íàõîäèòñÿ â òî÷êå 17;0.8115) .3317;0.33 (1.1;0.9;0 ) x , x , x , x , x ( 5 4 3 2 1  . Äëÿ åå íàõîæäåíèÿ ïîíàäîáèëîñü 11 èòåðàöèé ìåòîäîì íàèñêîðåéøåãî ñïóñêà è 8 èòåðàöèé ìåòî- äîì ÁÔÃØ ïðè èñïîëüçîâàíèè íîìèíàëüíîé òî÷- êè â êà÷åñòâå íà÷àëüíîé. Èñïîëüçîâàíèå ëèíåéíîé èíòåðâàëüíîé ìîäåëè äëÿ âûáîðà íà÷àëüíîé òî÷êè ïîèñêà ñîêðàòèëî êîëè÷åñòâî èòåðàöèé äî 4. Äî- ïóñê íà âûõîäíóþ ôóíêöèþ ñîñòàâèë %) 52 . 11 %, 54 . 9 ( y   . 4. Âûâîäû Ðàçðàáîòàí ìåòîä ðàñ÷åòà äîïóñêîâ äëÿ íåìîíîòîí- íûõ âûõîäíûõ ôóíêöèé, ñâîéñòâåííûõ ìíîãèì ðàäèîýëåêòðîííûì óñòðîéñòâàì. Îñîáåííîñòè çà- äà÷è ïðèâåëè ê íåîáõîäèìîñòè ìîäåðíèçàöèè ñó- ùåñòâóþùèõ ìåòîäîâ îïòèìèçàöèè. Äëÿ óëó÷øå- íèÿ ñõîäèìîñòè èñïîëüçóþòñÿ èíòåðâàëüíûå ìîäå- ëè. Íàèáîëåå ýôôåêòèâíûì îêàçàëñÿ ìîäèôèöè- ðîâàííûé ìåòîä ÁÔÃØ. Ñõîäèìîñòü åãî âûøå ïî ñðàâíåíèþ ñ ìåòîäîì íàèñêîðåéøåãî ñïóñêà, ÷òî ïîäòâåðæäàåòñÿ ïðîâåäåííûìè ðàñ÷åòàìè. Èñïîëü- çîâàíèå ëèíåéíîé èíòåðâàëüíîé ìîäåëè äëÿ âûáî- ðà íà÷àëüíîé òî÷êè ïîèñêà ïîçâîëèëî óìåíüøèòü îáúåì âû÷èñëåíèé íà êàæäîé èòåðàöèè è çíà÷è- òåëüíî ïîâûñèòü ñêîðîñòü ñõîäèìîñòè äëÿ îòíîñè- òåëüíî ãëàäêèõ ôóíêöèé. Ðàçðàáîòàííûé àëãîðèòì ìîæåò áûòü ýôôåêòèâíî ïðèìåíåí êàê äëÿ ìîíî- òîííûõ ôóíêöèé, òàê è äëÿ ôóíêöèé ñ ýêñòðåìó- ìàìè. Ëèòåðàòóðà: 1. Öâåòêîâ  À.Ô. Ìåòîäû ðàñ÷åòà äîïóñêîâ â ðàäèîýëåêòðîííîé àïïàðàòóðå.  Ðÿçàíü: ÐÐÒÈ, 1970. 131ñ. 2. Ôðèäëåíäåð È.Ã. Ðàñ÷åòû òî÷íîñòè ìàøèí ïðè ïðîåêòèðîâàíèè. Êèåâ–Äîíåöê: Âèùà øê., 1980. 184 ñ. 3. Âàñèëåãà  Í.Ì., Ãàïîíåíêî  Ã.Í., Ãàïîíåíêî Í.Ï. Ìåòîä èíòåðâàëüíîãî àíàëèçà äëÿ äîïóñêîâûõ çàäà÷ ïðîåêòè- ðîâàíèÿ è ïðîèçâîäñòâà // Èíôîðìàòèêà–ìàøèíî- ñòðîåíèå. 1999, ¹2(24). Ñ.3–7. 4. Øèëî Ã.Ì. Ôîðìóâàí- íÿ ³íòåðâàëüíèõ ìîäåëåé äëÿ îá÷èñëåííÿ äîïóñê³â // Ðàä³îåëåêòðîí³êà. ²íôîðìàòèêà. Óïðàâë³ííÿ. 2002. ¹1. Ñ.90–95. 5. Ïîëëàê Ý. ×èñëåííûå ìåòîäû îïòèìèçà- öèè. Åäèíûé ïîäõîä. Ì: Ìèð, 1974. 374 ñ. 6. Áàçàðà Ì., Øåòòè Ê. Íåëèíåéíîå ïðîãðàììèðîâàíèå. Òåîðèÿ è àëãîðèòìû. Ì: Ìèð, 1982. 583 ñ. 7. Áåðòñåêàñ Ä. Óñëîâ- íàÿ îïòèìèçàöèÿ è ìåòîäû ìíîæèòåëåé Ëàãðàíæà. Ì: Ðàäèî è ñâÿçü, 1987. 398 ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 03.02.2005 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Ïèçà Ä.Ì. Ãàïîíåíêî Íèêîëàé Ïðîêîôüåâè÷, êàíä. òåõí. íàóê, äîöåíò êàôåäðû ïðîèçâîäñòâà ðàäèîàïïàðàòóðû ÇÍÒÓ. Íàó÷íûå èíòåðåñû: êîíñòðóèðîâàíèå ýëåêòðîííûõ àïïàðàòîâ. Àäðåñ: Óêðàèíà, 69063, Çàïîðîæüå, óë. Æó- êîâñêîãî, 64, òåë. 698-252. Âîðîïàé Àëåêñåé Þðüåâè÷, àñïèðàíò êàôåäðû ïðîèç- âîäñòâà ðàäèîàïïàðàòóðû ÇÍÒÓ. Íàó÷íûå èíòåðåñû: ñèñòåìû àâòîìàòèçàöèè ïðîåêòíûõ ïðîöåäóð. Àäðåñ: Óêðàèíà, 69063, Çàïîðîæüå, óë. 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Å. Èññëåäóåòñÿ ïðîöåäóðà ðàçðàáîòêè ìàòåìàòè÷åñêîãî îáåñïå÷åíèÿ ïðîöåäóðû äèíàìè÷åñêîãî ñèíòåçà â êëàññå çàäà÷ àíàëèòè÷åñêîãî êîíñòðóèðîâàíèÿ îïòèìàëüíûõ ðåãóëÿòîðîâ äëÿ êâàçèëèíåéíîãî îáüåêòà óïðàâëåíèÿ. 1. Ââåäåíèå Íà ñîâðåìåííîì ýòàïå àâòîìàòèçàöèè òåõíîëîãè- ÷åñêèõ ïðîöåññîâ â òàêèõ îáëàñòÿõ êàê ýíåðãåòèêà, ìàøèíîñòðîåíèå, ìåòàëëóðãèÿ, õèìèÿ, íåôòåõè- ìèÿ, óïðàâëåíèå ïîäâèæíûìè îáúåêòàìè, ñîâðå- ìåííûå ñðåäñòâà âåäåíèÿ âîîðóæåííîé áîðüáû îïðåäåëÿþùåé â íàó÷íî-òåõíè÷åñêîì àñïåêòå ÿâ- ëÿåòñÿ ïðîáëåìà äèíàìè÷åñêîãî ñèíòåçà [1]. Ñïå- öèôèêà ìîäåëåé, èñïîëüçóþùèõñÿ â êà÷åñòâå îáúåê- òîâ óïðàâëåíèÿ (ÎÓ), â çàäà÷àõ äèíàìè÷åñêîãî ñèíòåçà ïðîÿâëÿåòñÿ, â ÷àñòíîñòè, â ñâÿçè ñ èõ íåëèíåéíîñòüþ. Ïîñëåäíåå îáóñëàâëèâàåò íåëèíå- éíûé õàðàêòåð ôóíêöèîíèðîâàíèÿ ñèíòåçèðîâàí- íûõ ñèñòåì óïðàâëåíèÿ(ÑÓ). Ïðè èññëåäîâàíèè ïîñëåäíèõ øèðîêîå ðàñïðîñòðàíåíèå ïîëó÷èëè ìåòîäû ëèíåàðèçàöèè [2]: çàìåíà èñõîäíîé íåëè- íåéíîé ìîäåëè åå êâàçèëèíåéíûì àíàëîãîì [3]. Îäíèì èç ìåòîäîâ, èñïîëüçóþùèõñÿ ïðè ðåàëèçà- öèè ïðîöåäóðû äèíàìè÷åñêîãî ñèíòåçà, ÿâëÿåòñÿ àíàëèòè÷åñêîå êîíñòðóèðîâàíèå îïòèìàëüíûõ ðå- ãóëÿòîðîâ (ÀÊÎÐ) [4]. Çàäà÷è ÀÊÎÐ äëÿ êâàçèëè- íåéíîãî ÎÓ ðàññìàòðèâàëîñü â [5-9], ãäå èññëåäó- þòñÿ âîïðîñû ñóùåñòâîâàíèÿ íåïðåðûâíîãî ðåøå- íèÿ óðàâíåíèÿ Ëÿïóíîâà-Áåëëìàíà [5], íàõîæäå- íèÿ ñóáîïòèìàëüíîãî óïðàâëåíèÿ íà îñíîâå ïðè- áëèæåííîãî ðåøåíèÿ óðàâíåíèÿ Ãàìèëüòîíà-ßêî- áè-Áåëëìàíà [6-8], ñõåìà ïîñëåäîâàòåëüíûõ ïðè- áëèæåíèé ìåòîäà âîçìóùåíèé [9]. Öåëüþ èññëåäîâàíèÿ ÿâëÿåòñÿ ðàçðàáîòêà ìàòåìàòè- ÷åñêîãî îáåñïå÷åíèÿ ïðîöåäóðû äèíàìè÷åñêîãî ñèíòåçà â êëàññå çàäà÷ ÀÊÎÐ äëÿ êâàçèëèíåéíîãî ÎÓ. 2. Ïîñòàíîâêà è îñîáåííîñòè çàäà÷è Ïðîöåäóðà äèíàìè÷åñêîãî ñèíòåçà, ðåàëèçóåìàÿ â êëàññå çàäà÷ ÀÊÎÐ, ìîæåò áûòü ñôîðìóëèðîâàíà ñëåäóþùèì îáðàçîì. Íà äâèæåíèÿõ ÎÓ ) t , , u , x ( F dt dx   (1) îïòèìèçèðóåòñÿ êðèòåðèé êà÷åñòâà   1 t 0 t dt ) t , q , u , x ( W ) u ( J (2) ïðè íàëè÷èè îãðàíè÷åíèÿ E ) t , q , , u , x (   (3) è ãðàíè÷íûõ óñëîâèé 1 1 0 0 P ) t , x ( , P ) t , x (   , (4) ãäå n E ) t ( x x   -ñîñòîÿíèå; r E ) t ( u u   – óïðàâ- ëåíèå; E – îáëàñòü îïðåäåëåíèÿ çàäà÷è; r n E , E – íåêîòîðûå ïðîñòðàíñòâà; ) : ( E max         - ïàðàìåòð ÎÓ (1); ) q q : q ( E q max q    – ïàðàìåòð êðèòåðèÿ êà÷åñòâà (2); 0 const max    , 0 const q max   – çàäàííûå ÷èñëà; i P – ìíîãîîáðà- çèÿ, ] 1 , 0 [ i  ; 1 1 0 R ] t , t [ t   – âðåìÿ, ] t , t [ 1 0 – èíòåð- âàë óïðàâëåíèÿ, 1 R – ÷èñëîâàÿ ïðÿìàÿ. 3. Îñîáåííîñòè ïðîöåäóðû äèíàìè÷åñêîãî ñèíòåçà â êëàññå çàäà÷ ÀÊÎÐ Îäíîé èç îñîáåííîñòåé ïðîöåäóðû äèíàìè÷åñêîãî ñèíòåçà â êëàññå çàäà÷ ÀÊÎÐ ÿâëÿåòñÿ íåîáõîäè- ìîñòü âûáîðà ïàðàìåòðà q êðèòåðèÿ êà÷åñòâà (2 ) [4]. Ïîñëåäíåå îáóñëàâëèâàåòñÿ òåì, ÷òî ôóíêöè- îíèðîâàíèå ÑÓ, ñèíòåçèðîâàííûõ â êëàññå çàäà÷ ÀÊÎÐ, îöåíèâàåòñÿ ïîñðåäñòâîì “ âòîðè÷íûõ ” ïî- êàçàòåëåé êà÷åñòâà [10], êîòîðûå àïðèîðè íå ìîãóò áûòü ó÷òåíû â èñõîäíîé ñòðóêòóðå êðèòåðèÿ êà- ÷åñòâà (2 ). Ïîýòîìó öåëåñîîáðàçíîé ÿâëÿåòñÿ ñëå- äóþùàÿ ïðîöåäóðà äèíàìè÷åñêîãî ñèíòåçà â êëàññå çàäà÷ ÀÊÎÐ [11]: – îïðåäåëåíèå ñòðóêòóðû óïðàâëÿþùåãî óñòðîé- ñòâà (ÓÓ) ïðè çàäàííîé ñòðóêòóðå ÎÓ (1), êðèòåðèÿ êà÷åñòâà (2), îãðàíè÷åíèÿ (3) è ãðàíè÷íûõ óñëîâèé (4) (ñòðóêòóðíûé ñèíòåç); – â ðàìêàõ ñèíòåçèðîâàííîãî ÓÓ âûáîð ïàðàìåòðà q êðèòåðèÿ êà÷åñòâà (2), ïðè çíà÷åíèè êîòîðîãî äâèæåíèå ñèíòåçèðîâàííîé ÑÓ ïðè îòðàáîòêå òðå- áóåìîãî çàäàíèÿ óäîâëåòâîðÿåò íàïåðåä çàäàííûì ïîêàçàòåëÿì êà÷åñòâà (ïàðàìåòðè÷åñêèé ñèíòåç).  íàñòîÿùåå âðåìÿ ïðè ðåàëèçàöèè ïðîöåäóðû ñòðóêòóðíîãî ñèíòåçà â çàäà÷å (1)-(4) èñïîëüçóþòñÿ ïðèíöèï ìàêñèìóìà, ìåòîä äèíàìè÷åñêîãî ïðî- ãðàììèðîâàíèÿ, ìåòîä ââåäåíèÿ íîâûõ ïåðåìåí- íûõ, èòåðàöèîííûå ìåòîäû, ïðèìåíåíèå êîòîðûõ ïðåäïîëàãàåò íåîáõîäèìîñòü ïðèâëå÷åíèÿ ÷èñëåí- íûõ ìåòîäîâ.  ñëó÷àå ëèíåéíîãî ÎÓ ðåøåíèå çàäà÷è ñòðóêòóðíîãî ñèíòåçà ñâîäèòñÿ ê ïðîöåäóðå ïîñòðîåíèÿ ñòàáèëèçèðóþùåãî ðåøåíèÿ ìàòðè÷íî- ãî óðàâíåíèÿ Ðèêêàòè. Íåâîçìîæíîñòü ðåàëèçàöèè àíàëèòè÷åñêîãî ðåøåíèÿ çàäà÷è ñòðóêòóðíîãî ñèí- òåçà íå ïîçâîëÿåò ðåàëèçîâàòü ðåøåíèå çàäà÷è ïàðàìåòðè÷åñêîãî ñèíòåçà â ïëàíå ñâÿçè ïàðàìåòðà q êðèòåðèÿ êà÷åñòâà (2) è “ âòîðè÷íûõ ” ïîêàçàòåëåé êà÷åñòâà.  íàñòîÿùåé ðàáîòå èññëåäîâàíèå çàäà÷è (1)-(4) áàçèðóåòñÿ íà ïîëîæåíèÿõ ôóíêöèîíàëüíî- àíàëèòè÷åñêîãî ïîäõîäà ðåøåíèÿ ýêñòðåìàëüíûõ çàäà÷ [12]. 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Ñòðóêòóðà àëãîðèòìà óïðàâëåíèÿ Ïóñòü 1 q 1 0 r 1 0 n R ) t , t ( L ) t , t ( C E          , ãäå ) t , t ( C 1 0 n – ïðîñòðàíñòâî n -ìåðíûõ íåïðåðûâíûõ íà ] t , t [ 1 0 ôóíêöèé ) t ( x c íîðìîé , ) t ( x max x  ] t , t [ t 1 0   , ) x x : x ( Q x max    , 0 const x max   – çàäàííîå ÷èñëî,   Q int ; ) t , t ( L 1 0 r  – ïðîñòðàí- ñòâî r -ìåðíûõ ñóùåñòâåííî îãðàíè÷åííûõ íà ] t , t [ 1 0 èçìåðèìûõ ôóíêöèé ) t ( u ñ íîðìîé ) t ( u sup vrai u  , ] t , t [ t 1 0   , ) u u : u ( U u max    , 0 const u max   – çàäàííîå ÷èñëî,   U int ; 0 0 x ) t ( x  , 0 ) t ( x 1  ; 1 t – êîíå÷íûé, íå ôèêñèðî- âàííûé ìîìåíòîì âðåìåíè; ) u , x ( 0 0 – ðåøåíèå çàäà÷è (1)-(4). Åñëè ìíîæåñòâî ïàð âàðèàöèé ) u , x ( , óäîâëåòâîðÿ- þùèõ ïîäûíòåãðàëüíîìó âûðàæåíèþ êðèòåðèÿ êà÷åñòâà (2), ïðèíàäëåæèò êîíóñó óáûâàíèÿ 0  , òî ôóíêöèîíàë   0 0 K ) u , x ( f ïîëó÷èì â âèäå     1 t 0 t 0 0 ' u 0 0 ' x 0 dt ] u ), t , q , u , x ( W x ), t , q , u , x ( W [ ) u , x ( f , ãäå ) t , q , u , x ( W ), t , q , u , x ( W 0 0 ' u 0 0 ' x – ïðîèçâîäíûå â òî÷êå ) u , x ( 0 0 ïî x è u îò ) t , q , u , x ( W ñîîòâåò- ñòâåííî;  – ñèìâîë îïåðàöèè ñîïðÿæåíèÿ. Åñëè 1 K êîíóñ âîçìîæíûõ íàïðàâëåíèé â òî÷êå 0 u è ôóíêöèîíàë   1 1 K ) u ( f , òî )) u ( f , 0 ( ) u ( f ' 1 1  , ãäå ) t , t ( L ) u ( f 1 0 r ' 1    è ÿâëÿåòñÿ îïîðíûì ê ìíîæåñòâó U â òî÷êå 0 u . Ïóñòü E H 1  – ìíîæåñòâî ïàð âàðèàöèé ) u , x ( , óäîâëåòâîðÿþùèõ âûðàæåíèþ , u ), t , , u , x ( F x ), t , , u , x ( F dt x d 0 0 ' u 0 0 ' x     0 ) t ( x 0  ,(5) à E H 2  – 0 ) t ( x 1  , (6) ãäå ) t , , u , x ( F 0 0 ' x  , ) t , , u , x ( F 0 0 ' u  – ïðîèçâîäíûå â òî÷êå ) u , x ( 0 0 ïî x è u îò ) t , , u , x ( F  ñîîòâåòñòâåí- íî. Òîãäà 1 H è 2 H – ïîäïðîñòðàíñòâà, à êàñàòåëüíîå ïðîñòðàíñòâî 2 1 2 H H K   ,      2 1 2 H H K . Ïîýòî- ìó, åñëè ôóíêöèîíàë   1 2 H ) u , x ( f , òî 0 ) u , x ( f 2  , è åñëè ôóíêöèîíàë   2 3 H ) u , x ( f , òî ) , 0 ( ) u , x ( f 3   , n R   , n R – n -ìåðíîå åâêëèäîâî ïðîñòðàíñòâî. Åñëè 4 K – êîíóñ âîçìîæíûõ íàïðàâëåíèé â òî÷êå 0 x è   4 4 K ) x ( f , òî )) x ( f , 0 ( ) x ( f ' 4 4  , ãäå ] t , t [ C ) x ( f 1 0 n ' 4   è ÿâëÿåòñÿ îïîðíûì ê ìíîæåñòâó Q â òî÷êå 0 x . Òîãäà óðàâíåíèå Ýéëåðà çàïèøåì â âèäå 0 ) u , x ( f 4 0 i i    èëè, ó÷èòûâàÿ âèä íàéäåííûõ ôóíêöèîíàëîâ   4 , 0 i ), u , x ( f i  ,    1 t 0 t 0 0 ' x ' 1 x ), t , q , u , x ( W [ ) u ( f ) x ( f dt ] u ), t , q , u , x ( W ' 4 0 0 ' u   . Ïóñòü [12] ) t , q , u , x ( W ) t ( )) t , , u , x ( F ( dt ) t ( d 0 0 ' x T 0 0 ' x      ïðè 0 ) t ( 1   . (7) Ó÷èòûâàÿ âûðàæåíèÿ (5)-(7), ìîæíî çàïèñàòü   1 t 0 t 0 0 ' x dt x ), t , q , u , x ( W        1 t 0 t T 0 0 ' x dt x )), t ( )) t , , u , x ( F ( dt ) t ( d ( dt x , dt ) t ( d 1 t 0 t         1 t 0 t T 0 0 ' x dt x ), t ( )) t , , u , x ( F ( =      dt ) dt x d ), t ( ( t t x ), t ( 1 t 0 t 0 1      1 t 0 t T 0 0 ' x dt x ), t ( )) t , , u , x ( F (         1 t 0 t T 0 0 ' x 1 t 0 t dt x ), t ( )) t , p , w , u , x ( F ( dt ) dt x d ) t ( (       1 t 0 t 0 0 ' x x ), t , , u , x ( F )[ t (   dt ] u ), t , , u , x ( F 0 0 ' u      1 t 0 t 0 0 ' x dt x ), t ( ) t , , u , x ( F (      1 t 0 t T 0 0 ' u dt u ), t ( )) t , , u , x ( F ( . Òîãäà       1 t 0 t T 0 0 ' u ' 1 ) t ( )) t , , u , x ( F ( [ ) u ( f ) x ( f dt ] u ), t , q , u , x ( W ' 4 0 0 ' u   . 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KۖM9po6W5PuB^;żҿSeUfb^鵬q̋r>Sd-k$_hf,# 8>x?뻠>N~wEޙX;\܂/cł|2UF.J豹:5ln95$q )skL.Nh} uг^k0 bn-?A׉@[rkFu}ikGL"_}:k?5,kDP7xtj^GEboRM):b<#Ƚ h[d_҈ rrַGfs J9_,,D35 ÐÈ, 2005, ¹ 2 äåéñòâèé, r n  ; Mu u Rx x ) t , q , u , x ( W T T   , n 1 i r diag R  , r 1 i m diag M  ; u F x F ) t , , u , x ( F u x    . Òîãäà àëãîðèòì óïðàâëåíèÿ (ÀÓ) ïîëó÷èì â âèäå [12]              max max max max max max u ) t ( L при u u ) t ( L u при ) t ( L u ) t ( L при u ) t ( u , (8) ãäå ) t ( )) t , , u , x ( F ( M 2 1 ) t ( L T 0 0 ' u 1     , . T 0 0 ' x Rx 2 ) t ( )) t , , u , x ( F ( dt ) t ( d      ïðè 0 ) t ( 1   . (9) Óìíîæàÿ âûðàæåíèå (9) ñëåâà íà ìàòðèöó 1 R 2 1  , ïîëó÷àåì . T 0 0 ' x 1 1 x ) t ( )) t , , u , x ( F ( R 2 1 dt ) t ( d R 2 1        . (10) Ïðèíèìàÿ âî âíèìàíèå âûðàæåíèå (10), ìîæíî çàïèñàòü        T 0 0 ' x 1 1 x )) t , , u , x ( F ( R 2 1 [ dt ) t ( d R F 2 1 dt dx ) t ( ] )) t , , u , x ( F ( M F 2 1 T 0 0 ' u 1 u     , ïîäñòàâëÿÿ êîòîðîå â âûðàæåíèå dt dx R 2 dt ) t ( d )) t , , u , x ( F ( dt ) t ( d T 0 0 ' x 2 2      , ïîëó÷àåì âåêòîðíîå îäíîðîäíîå äèôôåðåíöèàëü- íîå óðàâíåíèå âòîðîãî ïîðÿäêà: 0 ) t ( C dt ) t ( d C dt ) t ( d 2 1 2 2       ïðè 0 ) t ( 1   , (11) ãäå 1 x T 0 0 ' x 1 R RF )) t , , u , x ( F ( C     ;      T ' x 1 x 2 )) t , , u , x ( F ( R RF C T 0 0 ' u 1 u )) t , , u , x ( F ( M RF    . Îäíîðîäíîå ðåøåíèå óðàâíåíèÿ (11) çàïèøåì â âèäå ) t ( ) t ( ) t ( ) t ( ) t ( ) t ( ) t ( ) t ( 0 2 0 1 22 21 12 11 2 1          , ãäå ) t ( ) t ( 1    , ) t ( dt ) t ( d 2 1    ; ) t ( ij  – n n  - ìåð- íûå ìàòðèöû, ] 2 , 1 [ j , i  . Òàê êàê I ) t ( ) t ( 1 22 1 11     , O ) t ( ) t ( 1 21 1 12     ,  O и I – n n  -ìåðíûå åäèíè÷íàÿ è íóëåâàÿ ìàòðè- öû), à 0 ) t ( 1   , òî 0 ) t ( ) t ( 0 0 1     .Ïîýòîìó ñîïðÿæåííóþ ñèñòåìó äèôôåðåíöèàëüíûõ óðàâíå- íèé ) СДУ ( (9) ìîæíî çàïèñàòü [14] â “ ïðÿìîì ” âðåìåíè: ) t ( Rx 2 ) t ( )) t , , u , x ( F ( dt ) t ( d 0 . T 0 0 ' x       ïðè 0 ) t ( 0   . (12) Ïóñòü ) x ( f u x dt dx       , ãäå n 1 ij a A  , r 1 ij b B  – ïîñòîÿííûå ìàòðèöû, ïðè÷åì ìàòðèöà n 1 ij a A  – óñòîé÷èâàÿ,     n 1 i x f x f  – àíàëèòè÷åñêàÿ âåêòîð- ôóíêöèÿ,  - ìàëûé ïàðàìåòð. Òîãäà ÑÄÓ (12) çàïèøåì â âèäå     x f ) t ( Rx 2 t dt ) t ( d 0 . T         , ïðè 0 ) t ( 0   , à åå ðåøåíèå – â âèäå   x f ) t ( Z ) t ( Rx ) t ( Z 2 ) t ( c 0 0 c 0 c     , ãäå          d e ) t ( Z t 0 t t T c 0 . Äëÿ îòêðûòîé îáëàñòè ïîëó÷èì )) x ( f 5 . 0 ) t ( Rx )( t ( Z B M ) t ( L 0 c 0 T 1 c     , ))) x ( f 5 . 0 ) t ( Rx )( t ( Z ( B ) x ( f Ax dt dx 0 c 0 T 1          ,    ) t ( x ) R ) t ( Z e ( ) t ( x 0 c 02 At + ) x ( f )) t ( Z 5 . 0 ) t ( Z ( c 02 c 01   , ãäå         d e ) t ( Z t 0 t t c 01 ,              d t Z e ) t ( Z t 0 t c 0 T 1 t c 02 . Ïóñòü     ) t , x ( f u t x t dt dx       , ãäå ýëåìåíòû ìàò- ðèö n 1 ij || ) t ( a || ) t ( A  è r 1 ij || ) t ( b || ) t ( B  – íåïðåðûâ- íî äèôôåðåíöèðóåìûå ôóíêöèè íà ] t , t [ 1 0 ,  – ìàëûé ïàðàìåòð, n 1 i || ) t , x ( f || ) t , x ( f  – àíàëèòè÷åñ- êàÿ âåêòîð-ôóíêöèÿ. Òîãäà ÑÄÓ (12) çàïèøåì â âèäå ) t , x ( f ) t ( Rx 2 ) t ( ) t ( dt ) t ( d 0 . 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Äëÿ îòêðûòîé îáëàñòè ïîëó÷èì )) t ( 5 . 0 ) t ( Rx ) t ( Z ( ) t ( B M ) t ( L нс 2 0 нc 1 T 1 нc      ,     ) t , x ( f x ) t ( A dt dx ))) t ( 5 . 0 ) t ( Rx ) t ( )( t ( )( t ( B нс 2 0 нс 1 T 1         ,       ) t ( x ) R ) t ( Z ) t ( ( t x 0 нc 11 )) t ( Z 5 . 0 ) t ( Z ( нc 21 нc 22    , ãäå                        d t Z t ) t ( Z t 0 t нc 1 T 1 1 нc 11 ,                        d t Z t ) t ( Z t 0 t нc 2 T 1 1 нc 21 ,               d , x f t ) t ( Z t 0 t 1 нc 22 . 6. Çàêëþ÷åíèå  êëàññå çàäà÷ ÀÊÎÐ ñôîðìóëèðîâàíà çàäà÷à äèíà- ìè÷åñêîãî ñèíòåçà äëÿ êâàçèëèíåéíîãî îáúåêòà óïðàâëåíèÿ. Ïðîâåäåííîå èññëåäîâàíèå ïîçâîëèëî ïîëó÷èòü ñëåäóþùèå íîâûå ðåçóëüòàòû, èìåþùèå íàó÷íîå è ïðèêëàäíîå çíà÷åíèå: – íà îñíîâàíèè èññëåäîâàíèÿ ìíîæåñòâà äîïóñòè- ìûõ óïðàâëåíèé ïîêàçàíî, ÷òî ñèíòåçèðîâàííûé ÀÓ îòíîñèòñÿ ê êëàññó íåëèíåéíûõ, ïðåäåëüíî- ëèíåéíîãî òèïà; – ïîëó÷åíî àíàëèòè÷åñêîå ðåøåíèå çàäà÷è ñòðóê- òóðíîãî ñèíòåçà, ÷òî ïîçâîëèò íà ýòàïå ïàðàìåòðè- ÷åñêîãî ñèíòåçà ñâÿçàòü àíàëèòè÷åñêîé çàâèñèìî- ñòüþ ïàðàìåòðû îïòèìèçèðóåìîãî êðèòåðèÿ êà÷å- ñòâà (2) è “ âòîðè÷íûå ” ïîêàçàòåëè êà÷åñòâà. Ïðàêòè÷åñêàÿ çíà÷èìîñòü ðåçóëüòàòîâ èññëåäîâà- íèÿ îïðåäåëÿåòñÿ âîçìîæíîñòüþ èõ èñïîëüçîâàíèÿ â êà÷åñòâå ìàòåìàòè÷åñêîãî îáåñïå÷åíèÿ ïðè ïðî- åêòèðîâàíèè ÑÓ â ðàññìàòðèâàåìîì êëàññå çàäà÷. Ëèòåðàòóðà: 1. Ðàäèåâñêèé À.Å. Ïðîáëåìàòèêà ñîâðå- ìåííîãî ýòàïà àâòîìàòèçàöèè òåõíîëîãè÷åñêèõ ïðî- öåññîâ // Àâòîìàòèçàö³ÿ âèðîáíè÷èõ ïðîöåñ³â.2004. ¹1(18). Ñ.126-132. 2. Ìåòîäû èññëåäîâàíèÿ íåëèíåé- íûõ ñèñòåì àâòîìàòè÷åñêîãî óïðàâëåíèÿ / Ïîä ðåä. Ð. À. Íåëåïèíà. Ì.: Íàóêà, 1975. 448ñ. 3. Ìîèñååâ Í.Í. Àñèìïòîòè÷åñêèå ìåòîäû íåëèíåéíîé ìåõàíèêè. Ì.: Íàóêà, 1969. 380ñ. 4. Ñïðàâî÷íèê ïî òåîðèè àâòîìàòè- ÷åñêîãî óïðàâëåíèÿ / Ïîä ðåä. À. À. Êðàñîâñêîãî. Ì.: Íàóêà, 1987.712ñ. 5. Àëüáðåõò Ý.Ã. Î ñóùåñòâîâàíèè îï- òèìàëüíîé ôóíêöèè Ëÿïóíîâà è íåïðåðûâíîãî îïòè- ìàëüíîãî óïðàâëåíèÿ äëÿ îäíîé çàäà÷è îá àíàëèòè÷åñ- êîì êîíñòðóèðîâàíèè ðåãóëÿòîðîâ // ÄÓ. 1965.Ò.1, ¹10. Ñ. 1301-1313. 6. Garrard W.L., McClamroch N.H., Clark L.G. An approach to suboptimal feedback control for nonlinear system // Int.J.Control. 1967.V.5, No5.P.425- 435. 7. Garrard W.L. Additional result on suboptimal feedback control of nonlinear system //Int.J.Con-trol. 1969. V.10, No6. P.657-663. 8. Garrard W.L. Suboptimal fe-edback control for nonlinear system // Automatica. 1972. V.8, No2. P.219-221. 9. Êîëìàíîâñêèé Â.Á. Ïðèìå- íåíèå ìåòîäà âîçìóùåíèé ê íåêîòîðûì çàäà÷àì îïòè- ìàëüíîãî óïðàâëåíèÿ // ÏÌÌ.1975.Ò.39, ¹15.Ñ.788- 796. 10. Ëåòîâ À.Ì. Äèíàìèêà ïîëåòà è óïðàâëåíèå. Ì.: Íàóêà, 1969. 360ñ. 11. Ëè Ý.Á., Ìàðêóñ Ë. Îñíîâû òåîðèè îïòèìàëüíîãî óïðàâëåíèÿ: Ïåð. ñ àíãë. Ì.: Íàóêà,1972. 576ñ. 12. Ãèðñàíîâ È.Â. Ëåêöèè ïî ìàòåìà- òè÷åñêîé òåîðèè ýêñòðåìàëüíûõ çàäà÷. Ì.: Èçä-âî ÌÃÓ,1970.117ñ. 13. Êîëìîãîðîâ À.Í., Ôîìèí Ñ.Â. Ýëå- ìåíòû òåîðèè ôóíêöèé è ôóíêöèîíàëüíîãî àíàëèçà. Ì.: Íàóêà,1968.496ñ.14. Àðíîëüä Â.È. Îáûêíîâåííûå äèôôåðåíöèàëüíûå óðàâíåíèÿ. Ì.:Íàóêà,1971. 240ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 01.11.2004 Ðåöåíçåíò: ä-ð òåõí.íàóê, ïðîô. Êóçíåöîâ Á.È. Ðàäèåâñêèé Àíàòîëèé Åâãåíüåâè÷, çàâåäóþùèé ëàáîðà- òîðèåé Õàðüêîâñêîãî ÍÈÈ êîìïëåêñíîé àâòîìàòèçà- öèè (ÕÈÊÀ) Ìèíòîïýíåðãî Óêðàèíû. Íàó÷íûå èíòå- ðåñû: ìàòåìàòè÷åñêàÿ òåîðèÿ ýêñòðåìàëüíûõ çàäà÷, äèíàìè÷åñêèå çàäà÷è ìíîãîêðèòåðèàëüíîé îïòèìèçà- öèè. Àäðåñ: Óêðàèíà, 61003, Õàðüêîâ, ïåð. Êóçíå÷íûé, 2, òåë. 731-35-67, 731-41-80. ÓÄÊ 519.6:514.1 ÝÊÑÒÐÅÌÀËÜÍÛÅ ÑÂÎÉÑÒÂÀ ÔÓÍÊÖÈÉ ÍÀ ÊÎÌÏÎÇÈÖÈÎÍÍÛÕ ÎÁÐÀÇÀÕ ÊÎÌÁÈÍÀÒÎÐÍÛÕ ÌÍÎÆÅÑÒ ÃÐÅÁÅÍÍÈÊ È.Â. Èññëåäóþòñÿ ýêñòðåìàëüíûå ñâîéñòâà âûïóêëûõ è ñèëüíî âûïóêëûõ ïðîäîëæåíèé ôóíêöèé, çàäàííûõ íà êîìïîçèöèîííûõ îáðàçàõ êîìáèíàòîðíûõ ìíîæåñòâ – ìíîæåñòâàõ ïàðíûõ ïåðåñòàíîâîê, ïàðíûõ ðàçìå- ùåíèé è ïàðíûõ ñî÷åòàíèé ñ ïîâòîðåíèÿìè. Ââåäåíèå. Ìíîãèå êëàññû çàäà÷, âîçíèêàþùèõ â ïðîåêòèðîâàíèè, óïðàâëåíèè, êîíòðîëå, îïèñûâà- þòñÿ ìîäåëÿìè êîìáèíàòîðíîé îïòèìèçàöèè [1]. Îáëàñòè äîïóñòèìûõ ðåøåíèé ýòèõ çàäà÷ ÷àñòî ïðåäñòàâëÿþòñÿ êëàññè÷åñêèìè êîìáèíàòîðíûìè ìíîæåñòâàìè [2]. Ðàçðàáîòêà àäåêâàòíûõ ìîäåëåé ðÿäà çàäà÷ òðåáóåò ïîñòðîåíèÿ êîìáèíàòîðíûõ ìíîæåñòâ ñ áîëåå ñëîæíîé ñòðóêòóðîé, îòðàæàþ- ùåé êîìáèíàòîðíóþ ïðèðîäó ðåøàåìîé çàäà÷è. Ýòî ñïðàâåäëèâî, â ÷àñòíîñòè, ïðè ðåøåíèè ìíîãèõ ýêñòðåìàëüíûõ êîìáèíàòîðíûõ çàäà÷ ãåîìåòðè÷åñ- êîãî ïðîåêòèðîâàíèÿ [3,4]. Äëÿ ïîñòðîåíèÿ ýôôåêòèâíûõ ìîäåëåé çàäà÷ óêà- çàííîãî êëàññà â [5] ââîäèòñÿ íîâûé êëàññ êîìáè- íàòîðíûõ ìíîæåñòâ ñî ñëîæíîé ñòðóêòóðîé – êîìïîçèöèîííûå îáðàçû ( k -îáðàçû) êîìáèíàòîð- íûõ ìíîæåñòâ.  ñâÿçè ñ ýòèì àêòóàëüíîé çàäà÷åé ÿâëÿåòñÿ àíàëèç ðàçëè÷íûõ îïòèìèçàöèîííûõ ìî- äåëåé íà k -îáðàçàõ êîìáèíàòîðíûõ ìíîæåñòâ. Öåëüþ íàñòîÿùåé ðàáîòû ÿâëÿåòñÿ èññëåäîâàíèå ñâîéñòâ íåêîòîðûõ êëàññîâ çàäà÷ îïòèìèçàöèè íà k -îáðàçàõ êîìáèíàòîðíûõ ìíîæåñòâ. 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Îïðåäåëåíèå ìíîæåñòâ ïàðíûõ ïåðåñòàíîâîê, ïàðíûõ ðàçìåùåíèé è ïàðíûõ ñî÷åòàíèé ñ ïîâòîðåíèÿìè Ðàññìîòðèì ìíîæåñòâà   12n Aa,a,...,a    k a 12 a 12 k g,g,...,g    , ãäå in J{1,2,...,n}  , a k iJ  – êðàòíîñòè ýëåìåíòîâ ìíîæåñòâà A , a 12k ...n  ; a 12k gg...g  è   12n Bb,b,...,b  , 1 i aR  , 1 i bR  . Ïîñòðîèì ìíî- æåñòâà 12n Z,Z,...,Z âèäà   iiin Z(a,b),iJ  . Ïðè ýòîì k ìíîæåñòâ i Z èç n ÿâëÿþòñÿ ðàçëè÷íûìè, a kkn  . Èñïîëüçóÿ ïîäõîä, ïðèâåäåííûé â [5], ñôîðìèðóåì ñëåäóþùèå êëàññû k -îáðàçîâ êîìáè- íàòîðíûõ ìíîæåñòâ. 1. Êîìïîçèöèîííûé îáðàç êîìáèíàòîðíûõ ìíî- æåñòâ nk P , 12n Z,Z,...,Z , ïîðîæäåííûé ìíîæåñòâà- ìè       1122nn a,b,a,b,...,a,b , ãäå nk P – ìíîæåñòâî ïåðåñòàíîâîê èç n ýëåìåíòîâ, k èç êîòîðûõ ðàç- ëè÷íû [3,4]. Îáîçíà÷èì òàêîé k -îáðàç ìíîæåñòâ ÷åðåç nk12n PI(Z,Z,...,Z) èëè nk PI è íàçîâåì ìíîæå- ñòâîì ïàðíûõ ïåðåñòàíîâîê. 2. Êîìïîçèöèîííûé îáðàç êîìáèíàòîðíûõ ìíî- æåñòâ m nk A , 12n Z,Z,...,Z , ïîðîæäåííûé ìíîæåñòâà- ìè       1122nn a,b,a,b,...,a,b . Çäåñü m nk A – îáùåå ìíîæåñòâî ðàçìåùåíèé èç n ýëåìåíòîâ, k èç êîòîðûõ ðàçëè÷íû, ïî m [4]. Òàêîé k -îáðàç ìíî- æåñòâ îáîçíà÷èì ÷åðåç m nk12n AI(Z,Z,...,Z) èëè m nk AI è íàçîâåì ìíîæåñòâîì ïàðíûõ ðàçìåùåíèé. 3. Êîìïîçèöèîííûé îáðàç êîìáèíàòîðíûõ ìíî- æåñòâ m n C , 12n Z,Z,...,Z , ïîðîæäåííûé ìíîæåñòâà- ìè       1122nn a,b,a,b,...,a,b , ãäå   iii Z(a,b)  ðàç- ëè÷íû, n iJ  . Çäåñü m n C – ìíîæåñòâî ñî÷åòàíèé ñ ïîâòîðåíèÿìè èç n ðàçëè÷íûõ ýëåìåíòîâ ïî m [4], ïðè÷åì äëÿ ëþáîãî 12m m iiin h(g,g,...,g)C  ñïðà- âåäëèâî 12m iii gg...g  . Ïðåäïîëîæèì ïðè ýòîì, ÷òî êðàòíîñòè i  ýëåìåíòîâ a 12k g,g,...,g ìíîæåñòâà A ðàâíû a 12k ...m  , à   12n Bb,b,...,b    k a 12 a 12 k e,e,...,e    . Ïðè ýòîì ìíîæåñòâà i Z ìîæíî ïðåäñòàâèòü êàê   12m11 ZZ...Z(g,e)  , m1m2 ZZ...     2m22 Z(g,e)  , ... , nm1nm2n ZZ...Z     aa kk (g,e)  . Ýòîò k -îáðàç ìíîæåñòâ îáîçíà÷èì m n12n CI(Z,Z,...,Z) èëè m n CI è íàçîâåì ìíîæåñòâîì ïàðíûõ ñî÷åòàíèé ñ ïîâòîðåíèÿìè. 2. Ïîñòàíîâêà çàäà÷è Ïóñòü X – k -îáðàç êîìáèíàòîðíûõ ìíîæåñòâ [5],   mm nknkn XPI,AI,CI  . Ðàññìîòðèì çàäà÷ó îïòèìèçà- öèè (h)min  , hX  , (1) ãäå 1 :XR  – íåêîòîðûé ôóíêöèîíàë.  ðåçóëüòàòå ïîãðóæåíèÿ f ìíîæåñòâà X â N R [3, 4] ñôîðìóëèðóåì çàäà÷ó îïòèìèçàöèè ôóíêöèè (x)  , ýêâèâàëåíòíóþ (1): (x)min  , xE  , (2) ãäå 1 :ER  – ôóíêöèÿ N ïåðåìåííûõ, îïðåäå- ëåííàÿ íà ìíîæåñòâå N ER  , Ef(X)  – îáðàç ìíîæåñòâà X â ïðîñòðàíñòâå N R , (x)(h)  ïðè xf(h)  , hH  . Èññëåäóåì ñëó÷àé, êîãäà â çàäà÷å (2) (x)  – âûïóêëàÿ (ñèëüíî âûïóêëàÿ ñ ïàðàìåòðîì 0  ) íà âûïóêëîì çàìêíóòîì ìíîæåñòâå VconvE  ôóí- êöèÿ, à â êà÷åñòâå ìíîæåñòâà E âûñòóïàþò îáðàçû ìíîæåñòâ ïàðíûõ ïåðåñòàíîâîê nk EI , ïàðíûõ ðàç- ìåùåíèé m nk EI , ïàðíûõ ñî÷åòàíèé ñ ïîâòîðåíèÿìè m n SI â N R . Âîïðîñû èññëåäîâàíèÿ è ðåøåíèÿ çàäà÷ îïòèìèçà- öèè âûïóêëûõ è ñèëüíî âûïóêëûõ ôóíêöèé íà åâêëèäîâûõ êîìáèíàòîðíûõ ìíîæåñòâàõ ðàññìàò- ðèâàëèñü âî ìíîãèõ ïóáëèêàöèÿõ. Òàê, ðàáîòû [6- 10] ïîñâÿùåíû îïèñàíèþ òåîðèè è êîíñòðóêòèâ- íûõ ìåòîäîâ ïîñòðîåíèÿ âûïóêëûõ è ñèëüíî âû- ïóêëûõ äèôôåðåíöèðóåìûõ ïðîäîëæåíèé äëÿ êëàñ- ñîâ ôóíêöèé, çàäàííûõ íà ðàçëè÷íûõ êîìáèíàòîð- íûõ ìíîæåñòâàõ.  ðàáîòàõ [4, 6, 11-15] èññëåäóþò- ñÿ äåêîìïîçèöèîííûå ìåòîäû îïòèìèçàöèè, èñ- ïîëüçóþùèå îöåíêè ìèíèìóìà âûïóêëûõ ôóíê- öèé íà åâêëèäîâûõ êîìáèíàòîðíûõ ìíîæåñòâàõ è èõ ïîäìíîæåñòâàõ. Ðàñïðîñòðàíèì íåêîòîðûå ðå- çóëüòàòû, ïðèâåäåííûå â ýòèõ ðàáîòàõ, íà ìíîæå- ñòâà nk EI , m nk EI , m n SI . 3. Ïîñòðîåíèå âûïóêëûõ ïðîäîëæåíèé ôóíêöèé, çàäàííûõ íà ìíîæåñòâàõ nk EI , m nk EI , m n SI Èç êîìáèíàòîðíûõ ñâîéñòâ ìíîæåñòâ nk EI , m nk EI , m n SI , ïîðîæäåííûõ ìíîæåñòâàìè   iii Za,b  , n iJ  , ñëåäóåò, ÷òî ìíîæåñòâà nk EI , m nk EI , m n SI ÿâëÿþòñÿ ïîäìíîæåñòâàìè ñîîòâåòñòâåííî ìíî- æåñòâ ïåðåñòàíîâîê nk E , ðàçìåùåíèé m nk E è ñî÷å- òàíèé ñ ïîâòîðåíèÿìè m n S , ïîðîæäåííûõ ìíîæå- ñòâîì   12n12n Da,a,...,a,b,b,...,b  . Äðóãèìè ñëî- âàìè, 1 nkwk EIE  , 1 mv nk wk EIE  , mv nw SIS  , ãäå w2n  , v2m  , 1 k - êîëè÷åñòâî ðàçëè÷íûõ ýëå- ìåíòîâ ìíîæåñòâà D . Ýòî çíà÷èò, ÷òî ìåòîäû ïîñòðîåíèÿ âûïóêëûõ è ñèëüíî âûïóêëûõ ïðîäîë- æåíèé ôóíêöèé, çàäàííûõ íà ìíîæåñòâàõ 1 wk E , 1 v wk E è v w S , ïîðîæäåííûõ ìíîæåñòâîì D , ñïðàâåä- ëèâû è äëÿ ôóíêöèé, çàäàííûõ íà èõ ïîäìíîæå- ñòâàõ nk EI , m nk EI , m n SI , ïîðîæäåííûõ ìíîæåñòâàìè   iii Za,b  , n iJ  . 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