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WANzlc_6O.h:Ex^{0Ei6lcKOa??ٖ13[jﱥv2[n~y^p(cxO'bru劶ʲ4şf{^񺓽4s?سvZrYtvM[ )W%7>qq9^<<ꤍeu>P>}Ц?Z/d#*Aޖ+Xix CܗXp~aRkYfɳc=Ǹ}!Wޏ%p7,,4 ÐÈ, 2003, ¹ 1 ÐÀÄÈÎÒÅÕÍÈÊÀ ÓÄÊ 621.396 ÑÈÍÒÅÇ ÑÂÅÐÕÐÎÁÀÑÒÍÛÕ ÎÖÅÍÎÊ ÍÀ ÎÑÍÎÂÅ ÁÓÒÑÒÐÅÏÀ ÊÎËßÄÈÍ Â.Ë. Ðàññìàòðèâàåòñÿ îáùèé ïîäõîä ê ñèíòåçó îöåíîê, îáëàäàþùèõ ñóùåñòâåííî áîëåå âûñîêîé ðîáàñòíîñ- òüþ, ÷åì ìåäèàíà âûáîðêè. Ïðåäëàãàåòñÿ ñåìåéñòâî òàêèõ îöåíîê è ìåòîä àäàïòèâíîãî âûáîðà îöåíêè íà îñíîâå áóòñòðåïà. Ïóòåì ñòàòèñòè÷åñêîãî ìîäåëèðî- âàíèÿ ïîêàçûâàåòñÿ, ÷òî ñèíòåçèðîâàííûå òàêèì îá- ðàçîì àäàïòèâíûå îöåíêè ñîõðàíÿþò ðàáîòîñïîñîá- íîñòü ïðè íàëè÷èè â âûáîðêå 90% è áîëåå àíîìàëüíûõ èçìåðåíèé. Ïðè ðåøåíèè øèðîêîãî êðóãà ïðàêòè÷åñêèõ çàäà÷ ñòàòèñòè÷åñêîé îáðàáîòêè èçìåðèòåëüíîé èíôîð- ìàöèè òðåáóåòñÿ ðîáàñòíîñòü, ò.å. óñòîé÷èâîñòü ê èçìåíåíèþ ñòàòèñòè÷åñêèõ ñâîéñòâ èñõîäíûõ äàí- íûõ â øèðîêèõ ïðåäåëàõ.  áîëüøèíñòâå ñëó÷àåâ íóæíà óñòîé÷èâîñòü ê íàëè÷èþ ó ðàñïðåäåëåíèÿ øóìà òÿæåëûõ õâîñòîâ, ò.å. ê íàëè÷èþ ñèëüíûõ âûáðîñîâ (àíîìàëüíûõ èçìåðåíèé). Èçâåñòíî, ÷òî êëàññè÷åñêèå ïðîöåäóðû îöåíèâàíèÿ ïàðàìåòðîâ, îïòèìàëüíûå äëÿ íîðìàëüíîãî ðàñïðåäåëåíèÿ è ñâîäÿùèåñÿ îáû÷íî ê ìåòîäó íàèìåíüøèõ êâàäðà- òîâ (ÌÍÊ), ðîáàñòíûìè íå ÿâëÿþòñÿ. Íàëè÷èå äàæå íåçíà÷èòåëüíîãî êîëè÷åñòâà àíîìàëüíûõ èç- ìåðåíèé ñ ñóùåñòâåííî áîëüøåé äèñïåðñèåé, ÷åì ó “ íîðìàëüíûõ ” èçìåðåíèé, ðåçêî ñíèæàåò ïîêàçàòå- ëè òî÷íîñòè [1]. Èìåííî ïîýòîìó íà ïðàêòèêå ïðåäïî÷èòàþò èñïîëüçîâàòü ðîáàñòíûå ïðîöåäóðû, íàïðèìåð ìåòîä íàèìåíüøèõ ìîäóëåé (ÌÍÌ). Ïðîñòåéøèì ïðèìåðîì îöåíèâàíèÿ ïî ÌÍÌ, ñî- îòâåòñòâóþùèì îöåíèâàíèþ îäíîãî ñêàëÿðíîãî ïàðàìåòðà ïî ïðÿìûì èçìåðåíèÿì, ÿâëÿåòñÿ ìåäè- àíà âûáîðêè. Ìåäèàíó ñ÷èòàþò íàèáîëåå ðîáàñòíîé ïðîöåäóðîé îöåíèâàíèÿ äëÿ ýòîé çàäà÷è [1].  äàííîé ðàáîòå ââîäèòñÿ è èññëåäóåòñÿ íîâûé êëàññ ïðîöåäóð îöåíèâàíèÿ (äàëåå - ïðîñòî îöå- íîê), îáëàäàþùèõ ñóùåñòâåííî áîëåå âûñîêîé ðîáàñòíîñòüþ, ÷åì ìåäèàíà, è äîïóñêàþùèõ îöåíè- âàíèå â óñëîâèÿõ, êîãäà ìåäèàíà óæå ïðàêòè÷åñêè íåðàáîòîñïîñîáíà. Òàêèå îöåíêè, èìåíóåìûå íèæå “ ñâåðõðîáàñòíûìè ” , ñîõðàíÿþò ðàáîòîñïîñîáíîñòü ïðè íàëè÷èè 90% è áîëåå àíîìàëüíûõ èçìåðåíèé. Ïîä àíîìàëüíûìè çäåñü ïîíèìàþòñÿ èçìåðåíèÿ, èìåþùèå ñóùåñòâåííî áîëüøóþ äèñïåðñèþ îøèá- êè è ïðàêòè÷åñêè íå íåñóùèå èíôîðìàöèè îá îöåíèâàåìûõ ïàðàìåòðàõ. Ðàññìîòðèì çàäà÷ó îöåíèâàíèÿ îäíîãî ñêàëÿðíîãî ïàðàìåòðà  ïî íàáîðó X x i N i   { , ,..., } 1 åãî ïðÿ- ìûõ íåçàâèñèìûõ èçáûòî÷íûõ èçìåðåíèé: x n i i        : ~ ( ): "нормальные" измерения; n аномальные измерения. i (1) Çäåñü n i – ñëó÷àéíûé àääèòèâíûé øóì “ íîðìàëü- íûõ ” èçìåðåíèé, ~ n i – ñëó÷àéíûå ðåçóëüòàòû àíî- ìàëüíûõ èçìåðåíèé. Àíîìàëüíûå èçìåðåíèÿ îáëà- äàþò ñóùåñòâåííî áîëüøåé äèñïåðñèåé, ÷åì øóìû “ íîðìàëüíûõ ” èçìåðåíèé, è ìîãóò âîîáùå íå çàâèñåòü îò èñòèííîãî çíà÷åíèÿ îöåíèâàåìîãî ïà- ðàìåòðà  . Îáùèì ïðèíöèïîì ôîðìèðîâàíèÿ îöåíîê ïî ðå- çóëüòàòàì ïðÿìûõ èçáûòî÷íûõ èçìåðåíèé ÿâëÿåòñÿ ïðèíöèï ìèíèìóìà íåêîòîðîé ôóíêöèè F( )  îò íåâÿçîê èçìåðåíèé: d x i N i i     ; ,...,  1 . ×àùå âñåãî â êà÷åñòâå F( )  èñïîëüçóþò L p -íîðìó íåâÿçîê: F d d d p N i p i N ( ,..., ) | | ; 1 1 1     . (2) Ïðè ýòîì ñëó÷àé p  2 (ÌÍÊ) ñîîòâåòñòâóåò îöåíè- âàíèþ ïî ñðåäíåìó çíà÷åíèþ. Òàêàÿ îöåíêà îïòè- ìàëüíà ïî êðèòåðèþ ìèíèìóìà ñðåäíåãî êâàäðàòà îøèáêè äëÿ íîðìàëüíîãî ðàñïðåäåëåíèÿ àääèòèâ- íûõ îøèáîê, íî îíà íå îáëàäàåò óñòîé÷èâîñòüþ ê àíîìàëüíûì èçìåðåíèÿì. Ïîýòîìó äëÿ ðîáàñòíîãî îöåíèâàíèÿ îáû÷íî èñïîëüçóþò p  1 (ÌÍÌ), ÷òî â äàííîé çàäà÷å ñîîòâåòñòâóåò îöåíèâàíèþ ïî ìåäèàíå âûáîðêè. Ìåäèàíà âûáîðêè äîñòàòî÷íî óñòîé÷èâà ê íàëè÷èþ àíîìàëüíûõ èçìåðåíèé. Îäíàêî ïðè íàëè÷èè èõ â âûáîðêå áîëåå 50% ìåäèàíà â îáùåì ñëó÷àå òåðÿåò óñòîé÷èâîñòü. Ïî- ýòîìó ïðàêòè÷åñêèé è òåîðåòè÷åñêèé èíòåðåñ ïðåä- ñòàâëÿåò ñèíòåç îöåíîê, ñîõðàíÿþùèõ ðàáîòîñïî- ñîáíîñòü ïðè íàëè÷èè â âûáîðêå äîëè àíîìàëüíûõ èçìåðåíèé, çàìåòíî ïðåâîñõîäÿùåé 50%. Ðàññìîòðèì êëàññ îöåíîê, îñíîâàííûõ íà ìèíèìè- çàöèè ôóíêöèè îò íåâÿçîê:  ( ) ,     k q X  k,q 1 N arg min F (x - ,..., x - ) , (3) ãäå k è q – ïàðàìåòðû, à ôóíêöèÿ F k q , ( )  îïðå- äåëÿåòñÿ âûðàæåíèåì F d d d d q N k q N k m N m N , ( ,..., ) (| | ,...,| |; , ) min 1 1     . (4) Çäåñü ôóíêöèÿ  m ( )  èìååò âèä  m N m D D q N M D D ( ,..., ; , ) [ ( ,..., ) ( ) ( ) 1 1 2 2     qS D D m N m D m i ( ,..., ) / ] / ; ( ) ( ) / 1 2 2 1 2 0 , (5) ãäå D i ( ) – i-é â ïîðÿäêå âîçðàñòàíèÿ ýëåìåíò ìíîæåñòâà àðãóìåíòîâ { ,..., } D D N 1 ; M ( )  – âûáî- ðî÷íîå ñðåäíåå; S( )  – êîðåíü êâàäðàòíûé èç âûáîðî÷íîé äèñïåðñèè. Ôóíêöèÿ  m ( )  ÿâëÿåòñÿ ñâîåîáðàçíîé îöåíêîé ñðåäíåêâàäðàòè÷íîãî çíà÷å- íèÿ àðãóìåíòîâ { , ,..., } D i N i  1 , ïîëó÷åííîé ïî m N  íàèìåíüøèì çíà÷åíèÿì. 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Ïîýòîìó âåëè÷èíà   / i äëÿ èñïîëüçóåìîé ìîäåëè èçìåðåíèé ïðèíöèïèàëüíî íå ìîæåò áûòü íèæå åäèíèöû. Íà ðèñóíêå ïîêàçàíà çàâèñèìîñòü îòíîñèòåëüíîé îøèáêè   / i îò êîëè÷åñòâà N a àíîìàëüíûõ èçìå- ðåíèé â âûáîðêå (ïðè íàëè÷èè “ íîðìàëüíûõ ” èçìåðåíèé N n  15 ) äëÿ îöåíêè “ ìåäèàíà âûáîð- êè ” è ïðåäëîæåííîé âûøå ñâåðõðîáàñòíîé îöåíêè. Îò÷åòëèâî íàáëþäàåòñÿ áîëåå âûñîêàÿ ðîáàñòíîñòü ïðåäëàãàåìîé îöåíêè, ÷åì ó ìåäèàíû. Ïðè íàëè÷èè â âûáîðêå 14 è áîëåå àíîìàëüíûõ èçìåðåíèé îøèáêà îöåíèâàíèÿ ïî ìåäèàíå âûáîðêè ðåçêî âîçðàñòàåò.  òî æå âðåìÿ äëÿ ïðåäëàãàåìîé ñâåðõ- ðîáàñòíîé îöåíêè òàêîãî ïîðîãîâîãî ýôôåêòà íå íàáëþäàåòñÿ è ïðè çàìåòíî áîëüøåì ÷èñëå àíî- ìàëüíûõ èçìåðåíèé. 0 2 4 6 8 10 0 5 10 15 20 Na относит. ошибка медиана сверхробастная Áîëåå ïîäðîáíûå ÷èñëåííûå ðåçóëüòàòû ìîäåëèðî- âàíèÿ ïðèâåäåíû â òàáëèöå, âêëþ÷àÿ îòíîñèòåëü- íóþ òî÷íîñòü îöåíîê ïî ñðåäíåìó çíà÷åíèþ. Èç òàáëèöû îò÷åòëèâî âèäíî, ÷òî ïðåäëîæåííàÿ àäàïòèâíàÿ îöåíêà çíà÷èòåëüíî ïðåâîñõîäèò ìåäè- àíó ïî óñòîé÷èâîñòè ê àíîìàëüíûì èçìåðåíèÿì. Êîãäà äîëÿ àíîìàëüíûõ îöåíîê äîñòèãàåò ïðèáëè- çèòåëüíî 50% îò îáùåãî ÷èñëà èçìåðåíèé, îøèáêà ìåäèàíû ðåçêî âîçðàñòàåò (â îñíîâíîì çà ñ÷åò ñìåùåíèÿ). Ôàêòè÷åñêè ïðè ýòîì ìåäèàíà ñòàíîâèò- ñÿ íåðàáîòîñïîñîáíîé.  òî æå âðåìÿ ïðåäëîæåííàÿ îöåíêà ïîçâîëÿåò ïðîâîäèòü îöåíèâàíèå äàæå ïðè íàëè÷èè 90% àíîìàëüíûõ èçìåðåíèé, ëèøü íåçíà- ÷èòåëüíî (ïîðÿäêà 1,5 ðàçà) ïðîèãðûâàÿ èäåàëüíîé îöåíêå. Ñâåðõðîáàñòíûå îöåíêè äàæå ïðè îòñóò- ñòâèè àíîìàëüíûõ èçìåðåíèé ( N a  0 ) óñòóïàþò îïòèìàëüíîé äëÿ ýòîãî ñëó÷àÿ îöåíêå ïî ñðåäíåìó çíà÷åíèþ íå áîëåå, ÷åì â 1,5 ðàçà. Îòìåòèì òàêæå, ÷òî óñòîé÷èâîñòü ñâåðõðîáàñòíûõ îöåíîê ïîâûøàåòñÿ ñ óâåëè÷åíèåì äèñïåðñèè àíî- ìàëüíûõ èçìåðåíèé. Õîòÿ òàêîå ñâîéñòâî ïðåäñòàâ- ëÿåòñÿ ïàðàäîêñàëüíûì ñ òî÷êè çðåíèÿ êëàññè÷åñ- êîé òåîðèè îöåíèâàíèÿ, îíî èìååò ýëåìåíòàðíîå êà÷åñòâåííîå îáúÿñíåíèå. ×åì âûøå äèñïåðñèÿ àíîìàëüíûõ èçìåðåíèé, òåì ëåã÷å íà èõ ôîíå âûäåëèòü “ íîðìàëüíûå ” èçìåðåíèÿ. Âûâîäû Ïðåäëîæåíî ñåìåéñòâî îöåíîê, îáëàäàþùèõ ñóùå- ñòâåííî áîëüøåé óñòîé÷èâîñòüþ ê àíîìàëüíûì èçìåðåíèÿì, ÷åì ìåäèàíà âûáîðêè, òðàäèöèîííî ñ÷èòàþùàÿñÿ íàèáîëåå ðîáàñòíîé îöåíêîé. Îöåíêè èç ïðåäëîæåííîãî ñåìåéñòâà èìåþò òåíäåíöèþ ïðèíèìàòü çíà÷åíèÿ â òî÷êàõ ìàêñèìàëüíîé ïëîò- íîñòè èçìåðåíèé, ÷òî ñîçäàåò ïðåäïîñûëêè äëÿ ñâåðõðîáàñòíîãî îöåíèâàíèÿ. Ïîñêîëüêó òî÷íîñòü ýòîãî êëàññà îöåíîê çàâèñèò îò íåèçâåñòíûõ âåðî- ÿòíîñòíûõ õàðàêòåðèñòèê èçìåðåíèé, òðåáóåòñÿ àäàïòèâíàÿ ïðîöåäóðà âûáîðà îöåíêè. Äëÿ ðåàëè- çàöèè òàêîé ïðîöåäóðû öåëåñîîáðàçíî èñïîëüçî- âàòü ñðàâíèòåëüíî íîâóþ ñòàòèñòè÷åñêóþ òåõíèêó - áóòñòðåï. Ïðèìåíåíèå áóòñòðåïà ïîçâîëÿåò îöå- íèòü òî÷íîñòü êàæäîé îöåíêè èç èñïîëüçóåìîãî íàáîðà ïåðâè÷íûõ îöåíîê ïî åäèíñòâåííîé ðåàëè- çàöèè âåêòîðà èçìåðåíèé. Çàòåì â êà÷åñòâå ðåçóëü- òèðóþùåé âûáèðàåòñÿ òà ïåðâè÷íàÿ îöåíêà, äëÿ êîòîðîé áóòñòðåïîì ïîëó÷åíà íàèëó÷øàÿ îöåíêà òî÷íîñòè. Ñòàòèñòè÷åñêîå ìîäåëèðîâàíèå ïîêàçû- âàåò, ÷òî ñèíòåçèðîâàííûå àäàïòèâíûå îöåíêè îáëàäàþò ãîðàçäî áîëüøåé óñòîé÷èâîñòüþ ê àíî- ìàëüíûì èçìåðåíèÿì, ÷åì ìåäèàíà âûáîðêè, è ñîõðàíÿþò ðàáîòîñïîñîáíîñòü ïðè íàëè÷èè 90% è áîëåå àíîìàëüíûõ èçìåðåíèé. Ëèòåðàòóðà: 1. Õüþáåð Ï. Ðîáàñòíîñòü â ñòàòèñòèêå. Ì.: Ìèð, 1984. 304 ñ. 2. Ýôðîí Á. Íåòðàäèöèîííûå ìåòîäû ìíîãîìåðíîãî ñòàòèñòè÷åñêîãî àíàëèçà: Ñáîðíèê ñòà- òåé. Ì.: Ôèíàíñû è ñòàòèñòèêà, 1988. 262 ñ. 3. Êîëÿäèí Â.Ë. Ñèíòåç àäàïòèâíûõ ðîáàñòíûõ îöåíîê íà îñíîâå òåõíèêè áóòñòðåï // Ðàäèîýëåêòðîíèêà è èíôîðìàòè- êà. 2001. ¹ 4. Ñ.18-22. 4. Davison A.C., Hinkley D.V. Bootstrap methods and their application. Cambridge: Cambridge University Press, 1997. 582 p. Ïîñòóïèëà â ðåäêîëëåãèþ 21.12.2002 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Êîñòåíêî Ï.Þ. Êîëÿäèí Âëàäèìèð Ëåîíèäîâè÷, êàíä. òåõí. íàóê, äîê- òîðàíò êàôåäðû “ Àâèàöèîííî-êîñìè÷åñêèå ðàäèîòåõ- íè÷åñêèå ñèñòåìû ” Íàöèîíàëüíîãî àýðîêîñìè÷åñêî- ãî óíèâåðñèòåòà “ ÕÀÈ ” . Íàó÷íûå èíòåðåñû: íåêëàñ- ñè÷åñêèå ìåòîäû ñòàòèñòè÷åñêîãî àíàëèçà äàííûõ, âêëþ÷àÿ îáðàáîòêó ñèãíàëîâ è èçîáðàæåíèé. Óâëå÷å- íèÿ è õîááè: èñòîðèÿ íàóêè, òåííèñ. Àäðåñ: Óêðàèíà, 61129, Õàðüêîâ, ïð. Òðàêòîðîñòðîèòåëåé ,162-Ã, êâ.128, òåë.14-81-44. 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Ïðèâîäèòñÿ ìàòåìàòè÷åñêàÿ ìîäåëü ìóëüòèïîëüíîé ñîñòàâëÿþùåé ìåòîäè÷åñêîé ïîãðåøíîñòè äâóõòî÷å÷- íîãî ìàãíèòîìåòðè÷åñêîãî ìåòîäà èçìåðåíèÿ îñåâîãî äèïîëüíîãî ìîìåíòà è îïðåäåëÿåòñÿ îñíîâíîé âêëàä ïîìåõîíåñóùèõ ìóëüòèïîëåé âíåøíåãî ìàãíèòíîãî ïîëÿ èñòî÷íèêà â ñîçäàíèå ýòîé ïîãðåøíîñòè. 1. Ïîñòàíîâêà çàäà÷è è àíàëèç ïîñëåäíèõ èññëåäîâàíèé Ïðè èññëåäîâàíèè ýëåêòðîìàãíèòíîé ñîâìåñòèìîñòè ðàçëè÷íîãî ðîäà èñòî÷íèêîâ âíåøíåãî ìàãíèòíîãî ïîëÿ (ÂÌÏ) è ýëåêòðîííîãî îáîðóäîâàíèÿ âàæíî çíàòü óðîâåíü íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ â çàäàííûõ òî÷êàõ îêðóæàþùåãî ïðîñòðàíñòâà íà ïëîñêîñòè èëè â îïðåäåëåííîì îáúåìå âáëèçè èñòî÷íèêà ïîëÿ. Ïîäîáíûå çàäà÷è âîçíèêàþò, êîãäà íåîáõîäèìî îöåíèòü âëèÿíèå ìàãíèòíîãî ïîëÿ èñòî÷íèêà íà ìàãíèòî÷óâñòâèòåëüíûå óñòðîéñòâà, ðàñïîëîæåííûå â çîíå âîçäåéñòâèÿ ÂÌÏ [1,2]. Âàæíûì ïàðàìåòðîì ìàãíèòíîãî ïîëÿ ðàññåÿíèÿ ÿâëÿåòñÿ ìàãíèòíûé ìîìåíò, îáóñëîâëåííûé òîêàìè ðàçëè÷íûõ óñòðîéñòâ, êîòîðûìè îñíàùåí èñòî÷íèê ïîëÿ. Ïîýòîìó îäíèì èç îñíîâíûõ êîíòðîëèðóåìûõ ïàðàìåòðîâ ôèçè÷åñêèõ òåë – èñòî÷íèêîâ ÂÌÏ åñòü äèïîëüíûé ìàãíèòíûé ìîìåíò [3], êîòîðûé ÿâëÿåòñÿ ïîñòîÿííîé âåëè÷èíîé â îòëè÷èå îò íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ è íå çàâèñèò îò êîîðäèíàò òî÷åê íàáëþäåíèÿ. Òàêèì îáðàçîì, çàäà÷à êîíòðîëÿ íàïðÿæåííîñòè ÂÌÏ èñòî÷íèêà ñîñòîèò â èçìåðåíèè ïðîñòðàí- ñòâåííîé ãàðìîíèêè ïåðâîãî ïîðÿäêà, ò.å. äèïîëü- íîé ñîñòàâëÿþùåé ïîëÿ, è îïðåäåëåíèè ïî ðåçóëü- òàòàì èçìåðåíèé êîìïîíåíò äèïîëüíîãî ìàãíèòíî- ãî ìîìåíòà. Ýòî äàåò âîçìîæíîñòü ëåãêî âû÷èñëèòü ïî íàéäåííûì çíà÷åíèÿì äèïîëüíîãî ìîìåíòà íàïðÿæåííîñòü ÂÌÏ èñòî÷íèêà â ïðîèçâîëüíûõ òî÷êàõ âíåøíåãî ïðîñòðàíñòâà. Ìàãíèòíîå ïîëå èñòî÷íèêà èìååò ñëîæíûé ïðî- ñòðàíñòâåííî-ãàðìîíè÷åñêèé õàðàêòåð. Íà çíà÷è- òåëüíûõ ðàññòîÿíèÿõ ïî ñðàâíåíèþ ñ ãàáàðèòíûìè ðàçìåðàìè èñòî÷íèêà åãî ÂÌÏ ìîæíî ïðåäñòàâèòü ñ äîñòàòî÷íîé äëÿ ïðàêòèêè òî÷íîñòüþ äèïîëüíîé ìîäåëüþ, ïîëå êîòîðîé ïðîïîðöèîíàëüíî äèïîëü- íîìó ìàãíèòíîìó ìîìåíòó èñòî÷íèêà. Îäíàêî îñóùåñòâèòü èçìåðåíèå äèïîëüíîãî ìàãíèòíîãî ìîìåíòà èñòî÷íèêà ïîëÿ íà òàêèõ ðàññòîÿíèÿõ âåñüìà ïðîáëåìàòè÷íî èç-çà âíåøíèõ ýëåêòðîìàã- íèòíûõ ïîìåõ, óðîâåíü êîòîðûõ ìîæåò ïðåâûøàòü ïîëåçíûé ñèãíàë íà ïîðÿäîê è áîëåå.  ñâÿçè ñ ýòèì âîçíèêëà íåîáõîäèìîñòü â ðàçðàáîòêå ìàãíèòîìåò- ðè÷åñêèõ ìåòîäîâ èçìåðåíèÿ äèïîëüíîãî ìîìåíòà ñ ïîìîùüþ òî÷å÷íûõ äàò÷èêîâ – èíäóêöèîííûõ ïðåîáðàçîâàòåëåé, ðàñïîëîæåííûõ îïðåäåëåííûì îáðàçîì âîêðóã èñòî÷íèêà ïîëÿ âáëèçè åãî ðàáî÷åé ïîâåðõíîñòè.  ýòîì ñëó÷àå, ñ îäíîé ñòîðîíû, óìåíüøàåòñÿ ïîìåõîâàÿ ïîãðåøíîñòü, ïîñêîëüêó çíà÷èòåëüíî óâåëè÷èâàåòñÿ ñîîòíîøåíèå ïîëåçíî- ãî ñèãíàëà ê ïîìåõîíåñóùåìó, ñîçäàííîìó âíåøíè- ìè èñòî÷íèêàìè, ðàñïîëîæåííûìè âáëèçè çîíû ìàãíèòîìåòðè÷åñêîãî èñïûòàòåëüíîãî ñòåíäà. Ñ äðóãîé ñòîðîíû, ïðè ðàñïîëîæåíèè äàò÷èêîâ âáëè- çè èçìåðÿåìîãî èñòî÷íèêà óñëîæíÿåòñÿ âûäåëåíèå äèïîëüíîé ñîñòàâëÿþùåé ïîëÿ èç ÂÌÏ èñòî÷íèêà, ñêàëÿðíûé ìàãíèòíûé ïîòåíöèàë êîòîðîãî äëÿ îáëàñòè âíåøíåãî ïðîñòðàíñòâà, ãäå ïëîòíîñòü òîêà ðàâíà íóëþ è 0  H rot  , ìîæíî ïðåäñòàâèòü ñôåðè- ÷åñêèì ãàðìîíè÷åñêèì ðÿäîì â âèäå ñóììû ìóëü- òèïîëåé – äèïîëÿ êâàäðóïîëÿ, îêòóïîëÿ è ò.ä. [4]:         1 0 1 ) (cos ) sin cos ( 1 4 1 n n m m n nm nm n P m h m g R U     , (1) ãäå   , , R – ñôåðè÷åñêèå êîîðäèíàòû òî÷åê íàáëþäåíèÿ; nm g , nm h – êîýôôèöèåíòû ðÿäà, ðàâíûå ìàãíèòíûì ìîìåíòàì ýëåìåíòàðíûõ m-õ ìóëüòèïîëåé ïðîñòðàíñòâåííîé ãàðìîíèêè n-ãî ïîðÿäêà; ) (cos  m n P – ïðèñîåäèíåííûå ïîëèíîìû Ëåæàíäðà ïåðâîãî ðîäà. Ýëåìåíòàðíûå ìóëüòèïîëÿ âûñøèõ ïðîñòðàíñòâåí- íûõ ãàðìîíèê ) 2 (  n ðÿäà (1) ÿâëÿþòñÿ ïîìåõîíå- ñóùèìè ïî îòíîøåíèþ ê èçìåðÿåìîé äèïîëüíîé ãàðìîíèêå (n=1), âñëåäñòâèå ÷åãî ïîÿâëÿåòñÿ ìóëü- òèïîëüíàÿ ñîñòàâëÿþùàÿ ìåòîäè÷åñêîé ïîãðåøíî- ñòè èçìåðåíèÿ îñåâîãî äèïîëüíîãî ìîìåíòà, êîòî- ðàÿ ìîæåò áûòü çíà÷èòåëüíîé ïî âåëè÷èíå. Âûäåëåíèå íåðåøåííûõ ðàíåå ÷àñòåé çàäà÷è. Àíà- ëèç òî÷íîñòè âûäåëåíèÿ äèïîëüíîé ñîñòàâëÿþùåé ìàãíèòíîãî ïîëÿ èç ïîëíîãî ñïåêòðà ïîìåõîíåñó- ùèõ ãàðìîíèê ÂÌÏ íåâîçìîæåí áåç ïîëó÷åíèÿ àäåêâàòíîé ìàòåìàòè÷åñêîé ìîäåëè èçìåðåíèÿ äè- ïîëüíîãî ìàãíèòíîãî ìîìåíòà, îáåñïå÷èâàþùåé àíàëèòè÷åñêîå îïèñàíèå ôóíêöèîíèðîâàíèÿ ìàã- íèòîèçìåðèòåëüíîé ñèñòåìû äàò÷èêîâ è îïðåäåëå- íèå òàêèõ åå ìåòðîëîãè÷åñêèõ õàðàêòåðèñòèê, êàê óðàâíåíèå èçìåðåíèÿ äèïîëüíîãî ìîìåíòà, ÷óâ- ñòâèòåëüíîñòè è ïîãðåøíîñòè èçìåðåíèÿ. Îäíàêî òàêàÿ èíôîðìàöèÿ ïî èçìåðèòåëüíîé ñèñòåìå ñ ìèíèìàëüíûì êîëè÷åñòâîì ïàð îñåâûõ äàò÷èêîâ â ïóáëèêàöèÿõ îòñóòñòâóåò. Öåëü. Íàñòîÿùàÿ ñòàòüÿ ïîñâÿùåíà îöåíêå ïåðå- ÷èñëåííûõ ìåòðîëîãè÷åñêèõ õàðàêòåðèñòèê ìàãíè- òîìåòðè÷åñêîé ñèñòåìû èç äâóõ îñåâûõ äàò÷èêîâ, ïðåäíàçíà÷åííîé äëÿ èçìåðåíèÿ îñåâîãî äèïîëüíî- ãî ìàãíèòíîãî ìîìåíòà èñòî÷íèêà ïîëÿ. 2. 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Y  R 180  90  0  270  Ðèñ.1. Ñõåìà ðàñïîëîæåíèÿ îñåâûõ äàò÷èêîâ âîêðóã èñòî÷íèêà ÂÌÏ Àíàëèç ïîêàçûâàåò, ÷òî â ðåçóëüòèðóþùåì ñèãíàëå z E (2) îòñóòñòâóþò ýëåêòðè÷åñêèå ñèãíàëû ïîìåõ 4 z 2 z E , E è ò.ä., ñîçäàâàåìûå ïîìåõîíåñóùèìè ìóëü- òèïîëÿìè ïðîñòðàíñòâåííûõ ãàðìîíèê ÷¸òíîãî ïîðÿäêà (n=2,4,...). Îáúÿñíÿåòñÿ ýòî òåì, ÷òî ìàã- íèòíûå ïîëÿ ïîìåõîíåñóùèõ ìóëüòèïîëåé ÷¸òíûõ ãàðìîíèê íàâîäÿò â êàòóøêàõ îñåâûõ äàò÷èêîâ 1,2 ðàâíûå ïî âåëè÷èíå è ïðîòèâîôàçíûå ñèãíàëû ïîìåõ, êîòîðûå ïîëíîñòüþ êîìïåíñèðóþò äðóã äðóãà, ïîñêîëüêó êàòóøêè äàò÷èêîâ ñîåäèíåíû ïîñëåäîâàòåëüíî – ñîãëàñíî. Òàêèì îáðàçîì, èçìåðèòåëüíàÿ ñèñòåìà èç äâóõ îñåâûõ äàò÷èêîâ îáëàäàåò ñâîéñòâîì ìóëüòèïîëü- íîãî ãàðìîíè÷åñêîãî àíàëèçà ÂÌÏ è îñóùåñòâëÿåò âûäåëåíèå äèïîëüíîé ïîñòîÿííîé ñîñòàâëÿþùåé ìàãíèòíîãî ïîëÿ ñ òî÷íîñòüþ äî ïîìåõîíåñóùåé ãàðìîíèêè òðåòüåãî ïîðÿäêà. Ïîëåçíûé ýëåêòðè÷åñêèé ñèãíàë 1 z E (2), íàâåäåí- íûé â êàòóøêàõ äàò÷èêîâ 1,2 äèïîëüíîé îñåâîé ñîñòàâëÿþùåé íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ ` 3 10 1 R 4 / g H    , ðàâåí , E g R k 2 E z 10 3 f 1 z   (2) îòêóäà ïîëó÷àåì óðàâíåíèå èçìåðåíèÿ îñåâîãî äèïîëüíîãî ìîìåíòà 3 f z z 10 R k E 2 1 M g   (3) è çíà÷åíèå ÷óâñòâèòåëüíîñòè èçìåðèòåëüíîé ñèñòå- ìû ê íàïðÿæåííîñòè äèïîëüíîé ñîñòàâëÿþùåé ìàãíèòíîãî ïîëÿ, íå çàâèñÿùåé îò óãëîâîé êîîðäè- íàòû :  . мЭ / мВ , K / 2 S f 10  Èç óðàâíåíèé (2), (3) ñëåäóåò, ÷òî äèïîëüíûé ìîìåíò Ì Z èçìåðÿåòñÿ ñ ïîãðåøíîñòüþ, îáóñëîâ- ëåííîé ïîìåõîíåñóùèìè ãàðìîíèêàìè ÂÌÏ íå- ÷åòíîãî ïîðÿäêà, îñíîâíîé âêëàä â êîòîðóþ âíîñÿò ñèãíàëû 3 z E è 5 z E , ñîçäàâàåìûå òðåòüåé è ïÿòîé ãàðìîíèêàìè ÂÌÏ, ïîñêîëüêó íàïðÿæåííîñòü ìàãíèòíîãî ïîëÿ ýòèõ ãàðìîíèê óáûâàåò áîëåå ìåäëåííî ñ ðàññòîÿíèåì ïî ñðàâíåíèþ ñ ìàãíèòíûì ïîëåì ãàðìîíèê âûñøèõ ïîðÿäêîâ (n=7,9,...): % 100 E E E 1 z 5 z 3 z z    . (4) Ñèãíàë ïîìåõè 3 z E ,ñîçäàâàåìûé îêòóïîëüíîé îñåâîé ñîñòàâëÿþùåé íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ (n=3)                  ) 2 sin h 2 cos g ( 15 g 2 3 R 4 1 ) 90 ( H 32 32 30 5 0 3 , îïðåäåëÿåòñÿ ïî ôîðìóëå:         ) 90 , 0 ( H k 1 E 0 0 3 1 f 3 z  . g 5 g 2 1 R k 6 ) 90 , 180 ( H 32 30 5 f 0 0 5 2                 (5) Ïîìåõà 5 z E , â ñâîþ î÷åðåäü, ñîçäàåòñÿ ýëåìåíòàð- íûìè ìóëüòèïîëÿìè ïðîñòðàíñòâåííîé ãàðìîíèêè ÂÌÏ ïÿòîãî ïîðÿäêà:                ) 90 , 180 ( H ) 90 , 0 ( H k 1 E 0 0 5 2 0 0 5 1 f 5 z ), g 63 g 8 7 g 8 1 ( R k 30 54 52 50 7 f    (6) ãäå           ) 180 ( 2 ), 0 ( 1 k ), 90 , ( H 0 0 0 5 k îñåâàÿ ñîñòàâëÿþùàÿ íàïðÿæåííîñòè ìàãíèòíîãî ïîëÿ ïÿòîé ãàðìîíèêè ÂÌÏ èñòî÷íèêà, àíàëèòè- ÷åñêè îïèñûâàåìàÿ óðàâíåíèåì :       7 0 5 R 4 15 ) 90 ( H          ) 2 sin h 2 cos g ( 2 7 g 8 1 52 52 50 +  . ) 4 sin h 4 cos g ( 63 54 54    Çíà÷åíèÿ êîýôôèöèåíòîâ ), 2 , 0 m ( g m 3  ) 4 , 2 , 0 m ( g m 5  â âûðàæåíèÿõ (5), (6) çàâèñÿò îò ïàðàìåòðîâ ýêñöåíòðè÷íîñòè èçìåðÿåìîãî îñåâîãî äèïîëüíîãî ìàãíèòíîãî ìîìåíòà . M z Òàê, äëÿ íàèáîëåå íåáëàãîïðèÿòíîãî ñëó÷àÿ, êîãäà äèïîëü- íûé ìîìåíò z M ñìåùåí âíóòðè ðàáî÷åãî îáúåìà èñòî÷íèêà ÂÌÏ íà êîîðäèíàòó x 0 îòíîñèòåëüíî íà÷àëà êîîðäèíàò, êîýôôèöèåíòû m 3 g è m 5 g íàéäåì ðàñ÷åòíûì ïóòåì äëÿ ãàðìîíèêè n=3: 2 0 z 32 2 0 z 30 x M 4 1 g , x M 2 3 g    ; äëÿ ãàðìîíèêè n=5: 4 0 z 54 4 0 z 52 4 0 z 50 x M 192 1 g , x M 8 1 g , x M 8 15 g     . Jx xUG,)x,(V; ʢK ||"PEԲHI@->n˻{kB@د77ߜ3g̜s˝. 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(12) Ïîïåðå÷íàÿ êîìïîíåíòà îáùåãî ýëåêòðè÷åñêîãî ïîëÿ z ) z , x ( H i 1 ) z , x ( E y 0 x     , (13) çäåñü 0  – ýëåêòðè÷åñêàÿ ïîñòîÿííàÿ ñâîáîäíîãî ïðîñòðàíñòâà, äîëæíà áûòü íåïðåðûâíà íà âîëíî- âîäíîé àïåðòóðå è îáðàùàòüñÿ â íîëü íà ðåáðàõ. Ýòî óñëîâèå è îðòîãîíàëüíîñòü ôóíêöèé (4) ïðèâîäèò ê ñëåäóþùåìó ñîîòíîøåíèþ [1]:        L 1 z i q q q 0 q q e T T l l l       1 m * mq m m m Q ) R A ( , ,... 2 , 1 , 0 q    , (14) ãäå * – ñèìâîë êîìïëåêñíîãî ñîïðÿæåíèÿ. Ðàññìîòðèì ãðàíè÷íûå óñëîâèÿ äëÿ ýëåêòðîìàã- íèòíîãî ïîëÿ â îáëàñòè ëåíò. Ìàãíèòíîå ïîëå (3) äîëæíî áûòü íåïðåðûâíî ìåæäó ëåíòàìè è äîëæíî óäîâëåòâîðÿòü óñëîâèþ ñêà÷êà ïðè ïåðåõîäå ÷åðåç ëåíòû:           q q q y y ) x ( T 2 ) 0 z , x ( H ) 0 z , x ( H l l l         , 2 / b | x | 2 / s 0 ; 2 / s | x | ), x ( J x l l l , L ,..., 2 , 1  l . (15) Äëÿ òîãî ÷òîáû ïðèìåíèòü ìåòîä ñóììèðîâàíèÿ ê ðÿäàì â (11), ðàçëîæèì íà ìíîæèòåëè çíàìåíàòåëü â (12):      2 2 q ] ) 1 m [( ) a ( ] ) 1 m ( q 2 u [ ] q x [ 2 m           , (16) ãäå       2 u ) 1 m ( x m , b / a   . Ïåðåïèøåì (12) ñ ó÷åòîì (16):       ) a i ( ) 2 ( Q q m 1 mq ) q x )( 2 ( 1 W m mq     , (17) çäåñü              ) 1 m ( q 2 u e ) 1 ( e W 2 / a i 1 m 2 / a i mq q q . Âûäåëèì âîñïðîèçâîäÿùåå ÿäðî â (17):       mq q m 1 mq W ) a i ( ) 2 ( Q ) q x ( ) q x ( sin x sin q cos ) 2 ( 1 m m m          . (18) Âîñïîëüçóåìñÿ (8), (10).  ïåðâûé ðÿä èç (11) ïîäñòàâèì mq Q â ôîðìå (18):                m m 1 q q m q 0 x sin ) 2 ( 1 ) ia ( ) 2 ( Q T               q m m q q m q q 0 ) q x ( ) q x ( sin ) 1 ( W T . (19) Âû÷èñëèì ðÿä â ïðàâîé ÷àñòè (19) ñ ó÷åòîì (8), (10):               m 0 m 1 q q m q 0 T 4 ) 2 ( i Q T               2 / ] u ) 1 m cos[( 2 / ] u ) 1 m sin[( ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i            . (20) Äëÿ âòîðîãî áèëàòåðàëüíîãî ðÿäà èç (11) àíàëîãè÷- íî ìîæíî ïîëó÷èòü åãî ñóììó:                4 ) 2 ( i T e Q T m 1 m q z i q m q q l l l               2 / ] u ) 1 m cos[( 2 / ] u ) 1 m sin[(             ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i 2 2 2 ) 1 m ( ) ka ( a z i e      l . (21) Òåïåðü ñ ó÷åòîì (20) è (21) óðàâíåíèå (11) ïðèíè- ìàåò âèä:                        2 / ] u ) 1 m cos[( 2 / ] u ) 1 m sin[( 4 ) 2 ( i R A m 1 m m             ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i ] e T T [ L 1 ) 1 m ( ) ka ( a z i m m 0 2 2 2           l l l , ,... 2 , 1 m   (22) Ïåðåéäåì ê ïðåîáðàçîâàíèþ óðàâíåíèÿ (14). Ðàñ- ñìîòðèì ïðàâóþ ÷àñòü è îïðåäåëèì * mq Q :         ) a ( ) 2 ( i Q q m 1 * mq 2 2 q 2 / a i 2 / a i ] ) 1 m [( ) a ( e ) m (cos e q q           . (23) Âîñïîëüçóåìñÿ ðàçëîæåíèåì çíàìåíàòåëÿ â (22) íà ìíîæèòåëè â ñëåäóþùåé ôîðìå:      2 2 q ] ) 1 m [( ) a ( ] ) 1 m ( a ][ m / ) a [( q q            . (24) Ñôîðìèðóåì âîñïðîèçâîäÿùåå ÿäðî ñ ó÷åòîì (24):            ] ) 1 m ( a ][ m / ) a [( 1 q q  ix xϼ@Hl1 U_H"UѺ|*RB,BEYZ"E@I @"(Z*mxyN3L VN|ns3/s""QC2^hTX[A= e[C_e-0\8/̃RVkڹWeX|R$e&Jd ʐSV؍eu]w9?@2X#CRLyJ)̱> KVVl˫A. 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(25) Òåïåðü äëÿ ñóììèðîâàíèÿ ïðàâîé ÷àñòè (14) âîñ- ïîëüçóåìñÿ ôîðìóëàìè (7), (9). Ïîäñòàâèì (25) â (23):   ) 2 ( mq ) 1 ( mq * mq V V Q ] m / ) a [( ] m / ) a [( ] m / ) a [( sin m 2 q q q                   , (26) ãäå ïðèíÿòû ñëåäóþùèå îáîçíà÷åíèÿ:        ) a ( ) 2 ( i V q m 1 ) 1 ( mq ] e ) m (cos e [ 2 / a i 2 / a i q q       , ) a ( sin ] ) 1 m ( a [ m 2 ] m / ) a )[( m (cos V q q q ) 2 ( mq                . Ïîäñòàâëÿÿ (26) â (14) è èñïîëüçóÿ ñóììèðîâàíèå ïî (7), (9) ïîëó÷àåì:        L 1 z i q q q 0 q q e T T l l l         / ) a ( m ) 2 ( mq ) 1 ( mq m q q q ] V V [ ) R A ( , (27) ,... 2 , 1 , 0 q    . Îïðåäåëèâ ïîäñòàíîâêè â ïðàâîé ÷àñòè (27), ïåðå- ïèøåì ñèñòåìó (14) â âèäå:           ) 2 ( i ) R A ( e T T q 1 q q L 1 z i q q 0 q l l l        ] e ) a (cos e [ 2 / a i q 2 / a i q q ) a ( sin 2 ) a cos( q q       , ,... 2 , 1 , 0 q    . (28) Äëÿ êîýôôèöèåíòîâ q T l èçâåñòíî [1]:      1 n * q n q q Q ~ C T l l l , (29) ãäå       2 / s 2 / s q n q n dx ) x ( ) x ( ~ Q ~ l l l l 2 2 q 2 / s i 2 / s i n ) n ( ) s ( e e ) 1 ( n q q           l l l . (30) Èç ïðàâîé ÷àñòè (30) ñôîðìèðóåì âîñïðîèçâîäÿùåå ÿäðî òèïà (7): 1 q 2 2 q Ker ) s sin( 2 n cos ) n ( ) s ( n l l        , çäåñü ) n / s )( n / s ( ) n / s sin( n 2 Ker q q q 1           l l l . (31) Ó÷èòûâàÿ (31), èç (30) òåïåðü èìååì 1 q 2 / s i 2 / s i n q n Ker ) s sin( 2 n cos ] e e ) 1 [( Q ~ q q l l l l         . (32) Èç (32) íàéäåì * q n Q ~ l : 1 q 2 / s i 2 / s i * q n Ker ) s sin( 2 n cos ] e e ) n [(cos Q ~ q q l l l l         .(33) Ïîäñòàâèì (33) â (29):             1 n q 2 / s i 2 / s i n q ) s sin( 2 ] e e ) n [(cos C T q q l l l l l ) n / s ( ) n / s sin( ) n / s ( ) n (cos n 2 q q q            l l l . (34) Èç (34) ñëåäóåò: ) 3 ( q V C T lq lq l  , (35) ãäå ââåäåíî îáîçíà÷åíèå ) s sin( 2 e ) s cos( e V q 2 / s i q 2 / s i ) 3 ( q q l l lq l l        . Ïîäñòàâèì (35) â (22) è (28):     m m R A                     ] 2 / ) u ) 1 m cos[(( ] 2 / ) u ) 1 m sin[(( 4 ) 2 ( i m 1             ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i ] e V C T [ L 1 ) 1 m ( ) ka ( a z i ) 3 ( m m m 0 2 2 2            l l l l , (36) ,... 2 , 1 m   ,           ) 2 ( i ) R A ( e V C T q 1 q q L 1 z i ) 3 ( q q q 0 q l l l l        ] e ) a (cos e [ 2 / a i q 2 / a i q q ) a ( sin 2 ) a cos( q q       , ,... 2 , 1 , 0 q    . (37) Òàíãåíöèàëüíàÿ êîìïîíåíòà îáùåãî ýëåêòðè÷åñ- êîãî ïîëÿ äîëæíà îáðàùàòüñÿ â íîëü íà ëåíòàõ, êàê óêàçàíî â [1]. 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Èç ïðàâîé ÷àñòè (30) ñôîð- ìèðóåì âîñïðîèçâîäÿùåå ÿäðî ïî q :     2 2 q ) n ( ) s ( 1 l b / s 2 b / u s n sin b s 2 ] b ) q 2 u ( s n [ Ker ) q (cos 2 l l l l          , (40) ãäå ] q b / s 2 b / u s n [ ] q b / s 2 b / u s n [ sin Ker 2            l l l l . Ñ ó÷åòîì (40) ïðåäñòàâëåíèå (30) ïðèíèìàåò âèä: 2 2 / s i 2 / s i q n Ker b / s 2 b / u s n sin b s 2 ] b ) q 2 u ( s n [ q cos ] e e ) n [(cos n Q ~ q q l l l l l l l                . Ïîñëå ñóììèðîâàíèÿ áèëàòåðàëüíûõ ðÿäîâ (39) ïðèíèìàåò âèä:             2 2 2 ) b s / ) u s b n ( b / u ( k iz n 0 e T l l l l 0 e T L 1 ) b s / ) u s b n ( b / u ( k | z z | i n 2 2 2                 l l l l l l l . (42) Äëÿ íàõîæäåíèÿ êîýôôèöèåíòîâ èìååì óðàâíåíèÿ (28), (36), (35) è (42), èç êîòîðûõ â îáëàñòè ïåðåñå÷åíèÿ èíäåêñîâ q è m ïîëó÷àåì / ) a ( tg ) e V C V C ( A m L 1 z i ) 3 ( m m ) 3 ( m 0 m 0 m m                 l l l l              ] e ) a (cos e [ ) 2 ( i / 2 / a i m 2 / a i m 1 m m                    ] 2 / ) u ) 1 m cos[(( ] 2 / ) u ) 1 m sin[(( 8 ) 2 ( i m 1             ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i ] e V C V C [ L 1 ) 1 m ( ) ka ( z i ) 3 ( m m ) 3 ( m 0 m 0 2 2 2             l a l l l , / ) a ( tg ) e V C V C ( R m L 1 z i ) 3 ( m m ) 3 ( m 0 m 0 m m                  l l l l              ] e ) a (cos e [ ) 2 ( i / 2 / a i m 2 / a i m 1 m m                    ] 2 / ) u ) 1 m cos[(( ] 2 / ) u ) 1 m sin[(( 8 ) 2 ( i m 1             ] e ) m (cos e [ ) 1 m ( 2 i ) 1 m ( 2 i ] e V C V C [ L 1 ) 1 m ( ) ka ( z i ) 3 ( m m ) 3 ( m 0 m 0 2 2 2             l a l l l . Êîýôôèöèåíòû q T îïðåäåëÿþòñÿ èç (35), ãäå q C l ÿâëÿþòñÿ ðåøåíèÿìè óðàâíåíèÿ (42). 5. Ðåçóëüòàòû è âûâîäû Òàêèì îáðàçîì, ïîëó÷åíî òî÷íîå àíàëèòè÷åñêîå ðåøåíèå çàäà÷è î âîçáóæäåíèè ôàçèðîâàííîé àí- òåííîé ðåøåòêè ñëîæíîé ñòðóêòóðû. Çàäà÷à ðåøåíà ìåòîäîì ÷àñòè÷íûõ îáëàñòåé ñ ïðèìåíåíèåì ãàðìî- íèê Ôëîêå, êàê â [1], íî â ñî÷åòàíèè ñ ìåòîäîì ñóììèðîâàíèÿ ðÿäîâ ïî âûáîðî÷íûì çíà÷åíèÿì â ãèëüáåðòîâîì ïðîñòðàíñòâå ñ âîñïðîèçâîäÿùèì ÿäðîì. Ïîñëåäíèé èñïîëüçîâàëñÿ äëÿ âûâîäà àìï- ëèòóäíûõ êîýôôèöèåíòîâ ïîëÿ. Äëÿ íèõ ïîëó÷åíû ïðåäñòàâëåíèÿ â àíàëèòè÷åñêîé ôîðìå, íå ñîäåðæà- ùèå ðÿäîâ, â òî âðåìÿ êàê â [1] àìïëèòóäû îïðåäåëÿþòñÿ èç áåñêîíå÷íûõ ñèñòåì ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé, âêëþ÷àþùèõ êðàòíûå ðÿäû, â òîì ÷èñëå, áèëàòåðàëüíûå. Ðåçóëüòàòû ìîãóò áûòü èñïîëüçîâàíû íà ïðàêòèêå ïðè ìîäåëè- ðîâàíèè óêàçàííîé ñòðóêòóðû. Ïîñêîëüêó ðÿäû âèäà (7), (8) âîçíèêàþò ïðè ðåøåíèè çàäà÷ òåîðèè àíòåíí, ïðåäëîæåííûé ìåòîä ìîæåò áûòü îáîáùåí è ïðèìåíåí ïðè ðàñ÷åòå ÔÀÐ îïðåäåëåííîãî òèïà. Èññëåäîâàíèÿ â óêàçàííîì íàïðàâëåíèè ÿâëÿþòñÿ äîñòàòî÷íî ïåðñïåêòèâíûìè [1, 8]. Ëèòåðàòóðà: 1. Sergey P. Skobelev. Shaping of Flat-Topped Element Patterns in an Array of Slow-Wave Strip Structures Excited by Parallel-Plate Waveguides // IEEE Transactions on antennas and propagation. Vol. 49, N12, December 2001. P.1763-1768. 2. Âàéíøòåéí Ë.À. Ýëåêòðîìàãíèò- íûå âîëíû. Ì.: Ñîâ. ðàäèî, 1988. 580 ñ. 3. Øåñòîïàëîâ Â.Ï., Ëèòâèíåíêî Ë.Í., Ìàñàëîâ Ñ.À., Ñîëîãóá Â.Ã. Äèôðàêöèÿ âîëí íà ðåøåòêàõ. Õàðüêîâ: Èçä-âî Õàðüê. óí-òà, 1973. 288ñ. 4. Øåñòîïàëîâ Â.Ï. Äèôðàêöèîííàÿ ýëåêòðîíèêà. Õàðüêîâ: Âûùà øê., 1976. 232 ñ. 5. Øåñòî- ïàëîâ Â.Ï., Êèðèëåíêî À.À., Ìàñàëîâ Ñ.À. Ìàòðè÷íûå óðàâíåíèÿ òèïà ñâåðòêè â òåîðèè äèôðàêöèè. Ê.: Íàóê. äóìêà, 1984. 296ñ. 6. Ìèòòðà Ð., Ëè Ñ. Àíàëèòè÷åñêèå ìåòîäû òåîðèè âîëíîâîäîâ: Ïåð. ñ àíãë. Ì.: Ìèð, 1974. 328ñ. 7. Àìèòåé Í., Ãàëèíäî Â., Âó ×. Òåîðèÿ è àíàëèç ôàçèðîâàííûõ àíòåííûõ ðåøåòîê: Ïåð. ñ àíãë. Ì.: Ìèð, 1974. 455ñ. 8. Veliev E.I., Oksasoglu A. Bessel functions series in two dimensional diffraction problems / J. of Electromagnetic and Applications. 1996. Vol. 10. N4. P. 493-507. 9. Íîáë Á. Ìåòîä Âèíåðà-Õîïôà: Ïåð. ñ àíãë. Ì.: Èçä-âî èíîñòð. ëèò., 1962. 280ñ. 10. Ôóíêöèè ñ äâîéíîé îðòîãîíàëüíîñòüþ â ðàäèîýëåêòðîíèêå è îï- òèêå: Ïåð. Ì.Ê. Ðàçìàõíèíà è Â.Ï. ßêîâëåâà. Ì.: Ñîâ. ðàäèî, 1971. 256ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 20.01.2003 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. ×óðþìîâ Ã.È. ×óìà÷åíêî Ñâåòëàíà Âèêòîðîâíà, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ÀÏÂÒ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñêàÿ ôèçèêà. Àäðåñ: Óêðàèíà, 61166, Õàðü- êîâ, ïð. Ëåíèíà, 14, òåë. 70-21-326. 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