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ÐÀÑÑÅßÍÈÅ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÛÕ ÂÎËÍ ÑÈÑÒÅÌÎÉ ÏËÎÑÊÈÕ ÐÅØÅÒÎÊ ÐÅÇÎÍÀÍÑÍÛÕ ÌÀÃÍÈÒÎÄÈÝËÅÊÒÐÈ×ÅÑÊÈÕ ÑÔÅÐ Ñ ÒÎÏÎËÎÃÈ×ÅÑÊÎÉ ÑÒÐÓÊÒÓÐÎÉ, ÇÀÄÀÂÀÅÌÎÉ ÃÅÎÌÅÒÐÈ×ÅÑÊÎÉ ÏÐÎÃÐÅÑÑÈÅÉ.........................................................................................7 ÔÅÄÎÒΠÔ.Â., ÍÅÐÓÕ À.Ã. ×ÈÑËÅÍÍÎÅ ÈÑÑËÅÄÎÂÀÍÈÅ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Ñ ÍÅÑÒÀÖÈÎÍÀÐÍÛÌ ÄÈÝËÅÊÒÐÈ×ÅÑÊÈÌ ÑËÎÅÌ ÏÐÈ ÏÎÌÎÙÈ ÌÅÒÎÄÀ ÈÍÒÅÃÐÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ ÂÎËÜÒÅÐÐÀ.................................................12 ÁÎÍÄÀÐÅÍÊÎ È.Í. Î ÍÅÊÎÒÎÐÛÕ ÂÎÇÌÎÆÍÎÑÒßÕ ÔÎÐÌÈÐÎÂÀÍÈß ÀÌÏËÈÒÓÄÍÎ- ×ÀÑÒÎÒÍÎÉ ÕÀÐÀÊÒÅÐÈÑÒÈÊÈ Â ÎÁÚÅÌÍÛÕ ÐÅÇÎÍÀÒÎÐÀÕ ÑÒÎß×ÈÕ ÂÎËÍ........................15 ÄÎÐÎØÅÍÊÎ Â.À., ÑÅÌÅÍÎÂÀ Å.Ê. ÐÀÑÑÅßÍÈÅ ÏËÎÑÊÎÉ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÉ ÂÎËÍÛ ÍÅÇÀÌÊÍÓÒÛÌ ÊÎÍÈ×ÅÑÊÈÌ ÎÒÐÀÆÀÒÅËÅÌ..........................................................................18 ÈËÜߨÅÍÊÎ Ë.Í. ×ÈÑËÅÍÍÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ ÏÐÈÇÌÅÍÍÛÕ ÝËÅÌÅÍÒΠÎÏÒÎÝËÅÊÒÐÎÍÍÛÕ ÏÐÈÁÎÐÎÂ...........................................................................................................................23 ÑÈÍÎÒÈÍ À.Ì. ÂËÈßÍÈÅ ÔÎÐÌÛ ÍÀÃÐÅÒÎÉ ÇÎÍÛ ÍÀ ÌÀÊÑÈÌÀËÜÍÛÉ ÏÅÐÅÃÐÅ ÀÏÏÀÐÀÒÀ............................................................................................................................................................................27 ÏÎÑÏÅËΠÁ.Á., ÃÐÓØÅÍÊÎ Ì.Â. ÀËÃÎÐÈÒÌ ÀÄÀÏÒÈÂÍÎÉ ÎÁÐÀÁÎÒÊÈ ÑÈÃÍÀËÎÂ È ÏÎÌÅÕ Â ÀÍÒÅÍÍÛÕ ÑÈÑÒÅÌÀÕ Ñ ÂÂÅÄÅÍÈÅÌ ÈÑÊÓÑÑÒÂÅÍÍÛÕ ØÓÌÎÂ..............................30 ÝËÅÊÒÐÎÍÈÊÀ OLEINICK A.I., AMATORE C.A., SVIR I.B. SIMULATION OF THE DOUBLE HEMICYLINDER ELECTRODE SYSTEM THROUGH CONFORMAL MAPPING. APPLICATION TO STEADY-STATE ELECTROGENERATED CHEMILUMINESCENCE..................................................................................................35 ÑÈÑÒÅÌÛ È ÏÐÎÖÅÑÑÛ ÓÏÐÀÂËÅÍÈß ÒÈÕÎÍΠÂ.À. ÎÁÎÁÙÅÍÍÀß ÌÎÄÅËÜ ÀÂÒÎÐÅÃÐÅÑÑÈÈ ÂÛÑØÈÕ ÐÀÍÃΠÍÅÃÀÓÑÑÎÂÛÕ ÏÐÎÖÅÑÑÎÂ......................................................................................................................................39 ÊÎÐÎËÜÊÎÂÀ Å.Å., ËÀÌÎÍÎÂÀ Í.Ñ., ÏËÈÑÑ È.Ï. ÌÎÄÈÔÈÖÈÐÎÂÀÍÍÛÅ ÀËÃÎÐÈÒÌÛ ÎÁÓ×ÅÍÈß ÍÅÉÐÎÍÍÛÕ ÑÅÒÅÉ ÂÅÊÒÎÐÍÎÃÎ ÊÂÀÍÒÎÂÀÍÈß..........................................................42 ÁÎÄßÍÑÊÈÉ Å.Â., ÏÎËÓØÊÈÍÀ Í.À. ÀËÃÎÐÈÒÌ ÎÁÓ×ÅÍÈß ÔÎÐÌÀËÜÍÎÃÎ ÍÅÉÐÎÍÀ Ñ ÏÎËÈÍÎÌÈÀËÜÍÎÉ ÀÊÒÈÂÀÖÈÎÍÍÎÉ ÔÓÍÊÖÈÅÉ...........................................................................45 ÔÈËÀÒΠÂ.À. ÎÁ ÎÄÍÎÌ ÏÎÄÕÎÄÅ Ê ÏÎÑÒÐÎÅÍÈÞ ÌÀÒÅÌÀÒÈ×ÅÑÊÎÉ ÌÎÄÅËÈ ÏÐÎÃÐÀÌÌÍÎÃÎ ÀÃÅÍÒÀ.............................................................................................................................................48 ÏÎËÎÍÑÊÈÉ À.Ä. ÑÈÍÒÅÇ ÈÍÂÀÐÈÀÍÒÍÛÕ ÊËÀÑÑÈÔÈÊÀÒÎÐÎÂ..................................................51 ÑÈÄÎÐΠÌ.Â. ÎÁ ÎÄÍÎÌ ÏÎÄÕÎÄÅ Ê ×ÈÑËÅÍÍÎÌÓ ÌÎÄÅËÈÐÎÂÀÍÈÞ ÊÎÍÂÅÊÒÈÂÍÛÕ ÂßÇÊÈÕ ÒÅ×ÅÍÈÉ Â ÎÄÍÎÑÂßÇÍÛÕ È ÌÍÎÃÎÑÂßÇÍÛÕ ÎÁËÀÑÒßÕ.............................................................................................................................................................................55 ÐÀÄÈÅÂÑÊÈÉ À.Å. ÇÀÄÀ×À ÄÈÍÀÌÈ×ÅÑÊÎÃÎ ÑÈÍÒÅÇÀ ÏÐÈ ÂÅÊÒÎÐÍÎÌ ÓÏÐÀÂËÅÍÈÈ.....................................................................................................................................................................57 ÃÐÅÁÅÍÍÈÊ È.Â., ËÀÏÊÎ Ä.À. 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Ðàññìàòðèâàåòñÿ ïðåîáðàçîâàíèå ïëîñêîãî ýëåêòðî- ìàãíèòíîãî ãàóññîâîãî èìïóëüñà, âûçâàííîå èçìåíå- íèåì âî âðåìåíè äèýëåêòðè÷åñêîé è ìàãíèòíîé ïðî- íèöàåìîñòåé ïîëóîãðàíè÷åííîé îáëàñòè â èçîòðîï- íîì ìàãíèòîäèýëåêòðè÷åñêîì ïðîñòðàíñòâå. Ìåòî- äîì èíòåãðàëüíûõ óðàâíåíèé ïîëó÷åíû òî÷íûå âûðà- æåíèÿ äëÿ êîìïîíåíòû ýëåêòðè÷åñêîãî ïîëÿ íà âñåé ïîëîæèòåëüíîé ïîëóîñè âðåìåíè â ëþáîé òî÷êå ïðî- ñòðàíñòâà. Ïðîâîäèòñÿ äåòàëüíûé àíàëèç ýâîëþöèè ýëåêòðè÷åñêîãî ïîëÿ âî âðåìåíè. 1. Ââåäåíèå Äàííàÿ ñòàòüÿ ÿâëÿåòñÿ ëîãè÷åñêèì ïðîäîëæåíèåì ðàáîòû [1], ãäå áûëè ïðèâåäåíû è ïðîàíàëèçèðîâà- íû ðåçóëüòàòû àíàëèòè÷åñêèõ èññëåäîâàíèé ïðåîá- ðàçîâàíèÿ êîìïîíåíòû ïëîñêîé ìîíîõðîìàòè÷åñ- êîé ýëåêòðîìàãíèòíîé âîëíû ÒÌ âèäà, èíèöèèðî- âàííîãî èçìåíåíèåì âî âðåìåíè ïàðàìåòðîâ ïîëó- ïðîñòðàíñòâà â áåçãðàíè÷íîé ìàãíèòîäèýëåêòðè- ÷åñêîé èçîòðîïíîé ñðåäå ïî çàêîíó ïðÿìîóãîëüíî- ãî èìïóëüñà.  íàñòîÿùåé ðàáîòå àíàëèòè÷åñêè èññëåäóåòñÿ âëè- ÿíèå èìïóëüñíîãî ñèíõðîííîãî èçìåíåíèÿ äèýëåê- òðè÷åñêîé ) t (  è ìàãíèòíîé ) t (  ïðîíèöàåìîñòåé â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  íà ïðîõîæäåíèå ïëîñêîãî ýëåêòðîìàãíèòíîãî ãàóññîâà èìïóëüñà â ñâîáîäíîì ìàãíèòîäèýëåêòðèêå. Êàê è â äðóãèõ ñòàòüÿõ ïîäîáíîãî ïëàíà [1, 2-4], ïðàêòè÷åñêàÿ íåîáõîäèìîñòü ðàññìîòðåíèÿ ñêà÷êîîáðàçíûõ ôðîí- òîâ èìïóëüñîâ âîçìóùåíèÿ ñðåäû îáóñëîâëåíà òåì, ÷òî â îïòè÷åñêîì äèàïàçîíå èçìåíåíèå ïàðàìåòðîâ ñðåäû ìîæíî ñ÷èòàòü ñêà÷êîîáðàçíûì [5]. 2. Ïîñòàíîâêà çàäà÷è Ïîëîæèì, ÷òî äî íóëåâîãî ìîìåíòà âðåìåíè èìå- åòñÿ îäíîðîäíàÿ è èçîòðîïíàÿ ìàãíèòîäèýëåêòðè- ÷åñêàÿ ñðåäà ñ äèýëåêòðè÷åñêîé è ìàãíèòíîé ïðî- íèöàåìîñòÿìè, ñîîòâåòñòâåííî ðàâíûìè 0  è 0  . Ïóñòü â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  ïðîèñõî- äèò âîçìóùåíèå ïàðàìåòðîâ ñðåäû âî âðåìåíè ïî çàêîíàì         , ) t ( ) t ( ) t ( ) t ( ) t ( , ) t ( ) t ( ) t ( ) t ( ) t ( 1 0 1 0                                 (1) ãäå 1  è 1  – ñîîòâåòñòâåííî äèýëåêòðè÷åñêàÿ è ìàãíèòíàÿ ïðîíèöàåìîñòè îáëàñòè 0 x  â âîçìó- ùåííîì ñîñòîÿíèè íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Ïðè ýòîì â îáëàñòè 0 x  ñðåäà èìååò äèýëåêòðè- ÷åñêóþ è ìàãíèòíóþ ïðîíèöàåìîñòè, ñîîòâåòñòâåííî ðàâíûå 0  è 0  , íà âñåé îñè âðåìåíè ) ; ( t    . Òîãäà ïðåîáðàçîâàíèå ýëåêòðè÷åñêîé êîìïîíåíòû ïåðâè÷íîé âîëíû ) x , t ( E 0 â îáëàñòè 0 x  , âûç- âàííîå âîçìóùåíèåì ïàðàìåòðîâ ýòîé îáëàñòè ïî çàêîíó (1), ñîãëàñíî [6], áóäåò îïèñûâàòüñÿ èíòåã- ðàëüíûì óðàâíåíèåì Âîëüòåððà âòîðîãî ðîäà: , 0 t ), ' x , ' t ( E ) ' x , x , ' t , t ( K ' dx ' dt ) x , t ( E ) x , t ( E 0 t 0 0       (2) ãäå ) ' x , x , ' t , t ( K – ÿäðî èíòåãðàëüíîãî óðàâíåíèÿ (2), êîòîðîå èìååò ñëåäóþùèé âèä [1]:          ) ' x x ( ) ' t t ( ) ' t t ( ] m 1 [ ) ' x , x , ' t , t ( K 2 ) ' x x ) ' t t ( v ( t a 2 m a 1 0 2 2 2         . (3) Çäåñü , a 1 0    1 0 m    , 1  è 1  – âîçìóùåííûå çíà÷åíèÿ, ñîîòâåòñòâåííî, ìàãíèò- íîé è äèýëåêòðè÷åñêîé ïðîíèöàåìîñòåé; 1 1 1 c v    – ñêîðîñòü ðàñïðîñòðàíåíèÿ ñâåòà â ñâîáîäíîì ìàãíèòîäèýëåòðè÷åñêîì ïðîñòðàíñòâå ñ ìàãíèòíîé è äèýëåêòðè÷åñêîé ïðîíèöàåìîñòÿìè, ñîîòâåòñòâåííî ðàâíûìè 1  è 1  ; c – ñêîðîñòü ðàñïðîñòðàíåíèÿ ñâåòà â âàêóóìå. Èíòåãðàëüíîå óðàâíåíèå (1) ðåøàåòñÿ ìåòîäîì ðåçîëüâåíòû [7], ñîãëàñíî êîòîðîìó ðåøåíèå (2) íà èíòåðâàëå âðåìåíè ) ; 0 ( t   ìîæåò áûòü íàéäåíî ïîñðåäñòâîì ðàâåíñòâà  ) x , t ( E (4)         t 0 ), x , t ( E ) ' x , x , ' t , t ( R ' dx ' dt ) x , t ( E 0 0 t 0 0 , ãäå ) ' x , x , ' t , t ( R – ðåçîëüâåíòà èíòåãðàëüíîãî óðàâ- íåíèÿ (2), êîòîðàÿ, ñîãëàñíî [1], èìååò âèä                ) ' x x ( ) ' t t ( ) ' t t ( m 1 1 ) ' x , x , ' t , t ( R 2 ) ' x x ) ' t t ( v ( t m 2 m a 1 1 2 2 2         . 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Ýâîëþöèÿ ïîëÿ íà âñåé ïîëîæèòåëüíîé ïîëóîñè âðåìåíè Ïóñòü ïåðâè÷íîå ýëåêòðè÷åñêîå ïîëå ïðåäñòàâëÿåò ñîáîé ÒÌ âîëíó ñ êîìïîíåíòîé . v k , e ) x , t ( E 0 ) kx t ( 0 2       (6) Ïîäñòàâëÿÿ âûðàæåíèÿ (5) è (6) â êâàäðàòóðíóþ ôîðìóëó (4), ïîëó÷àåì, ÷òî âûðàæåíèå äëÿ ýëåêò- ðè÷åñêîé êîìïîíåíòû ïîëÿ ) x , t ( E â îáëàñòè ïî- ëóïðîñòðàíñòâà 0 x  íà èíòåðâàëå âðåìåíè ) ; 0 ( t   áóäåò èìåòü âèä  ) x , t ( E 1             2 2 ) kx t am ( 1 ) am kx t ( 1 1 e B e A ) v x t ( 2 ) kx t am ( 2 1 1 1 e } B ) v x t ( B ) v x t ( {            ,(7) ãäå am 1 a 2 A 1   , am 2 am 1 B 1    , am 1 ) am 1 ( 2 am B 2 2     . Ñðàâíèâàÿ âûðàæåíèÿ äëÿ ïðåîáðàçîâàííîãî (7) ñ ñîîòâåòñòâóþùèì âûðàæåíèåì äëÿ ïîëÿ ) x , t ( E 1 â ñòàòüå [1], ïðèõîäèì ê èäåíòè÷íîñòè êà÷åñòâåííîé êàðòèíû ïðåîáðàçîâàíèÿ ïîëÿ ) x , t ( E 0 íà ñêà÷êå ïàðàìåòðîâ â îáëàñòè ïîëóïðîñòðàíñòâà 0 x  (çäåñü è äàëåå, êàê è â ðàáîòå [1], óñëîâèìñÿ âîëíû èìåíîâàòü íàçâàíèåì èõ àìïëèòóäíûõ êîýôôèöè- åíòîâ): ñêà÷îê ñâîéñòâ îáëàñòè 0 x  â ìîìåíò âðåìåíè 0 t  ðàñùåïëÿåò ïåðâè÷íûé èìïóëüñ ) x , t ( E 0 íà ïðÿìîé  1 B (êîòîðûé äâèæåòñÿ â íàïðàâëåíèè ïåðâè÷íîãî) è ïðåäñòàâëåí òðåòüèì ñëàãàåìûì â âûðàæåíèè (7), à òàêæå îáðàòíûé  1 B (ðàñïðîñòðàíÿþùèéñÿ ê îáðàçîâàâøåéñÿ ãðàíèöå ðàçäåëà äâóõ ñðåä 0 x  ), êîòîðûé ïðåäñòàâëåí âòîðûì ñëàãàåìûì â âûðàæåíèè (7). Ôàçîâûå õàðàêòåðèñòèêè ýòèõ èìïóëüñîâ îòëè÷àþòñÿ îò ôàçîâûõ õàðàêòåðèñòèê ïåðâè÷íîãî èìïóëüñà ) x , t ( E 0 ; òàê, ñêîðîñòü ðàñïðîñòðàíåíèÿ äàííîé ïàðû èìïóëüñîâ ðàâíà 0 amv .  ìîìåíò âðåìåíè 1 v x t  íà÷èíàåò ñêàçûâàòüñÿ âëèÿíèå ãðàíèöû ðàçäåëà äâóõ ñðåä – åå äîñòèãàåò îáðàòíûé èìïóëüñ  1 B . Ðåçóëüòàòîì îòðàæåíèÿ ïîñëåäíåãî îò ãðàíèöû 0 x  ÿâëÿåòñÿ ïðÿìîé èìïóëüñ 2 B , êîòîðîìó â âûðàæåíèè (7) ñîîòâåò- ñòâóåò ÷åòâåðòîå ñëàãàåìîå. Êðîìå òîãî, â îáëàñòè 0 x  â ìîìåíò âðåìåíè 0 t  ïîÿâëÿåòñÿ ïðÿìîé èìïóëüñ 1 A , êîòîðûé ïî ñóòè ÿâëÿåòñÿ ïðîøåäøèì ïåðâè÷íûì èìïóëüñîì ñ íîâûìè â ñðàâíåíèè ñ ïîñëåäíèì ôàçîâûìè õàðàêòåðèñòèêàìè. Ñðàâíèâàÿ âûðàæåíèÿ äëÿ àìïëèòóä âîëí, îïèñû- âàåìûõ ôîðìóëîé (7), ñ ñîîòâåòñòâóþùèìè âûðà- æåíèÿìè ñòàòüè [1], ïðèõîäèì òàêæå ê èäåíòè÷- íîñòè êîëè÷åñòâåííîé ñòîðîíû ïðîöåññà ïðåîáðà- çîâàíèÿ ìîíîõðîìàòè÷åñêîé âîëíû è ãàóññîâîãî èìïóëüñà íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Ó÷èòûâàÿ êà÷åñòâåííóþ è êîëè÷åñòâåííóþ èäåí- òè÷íîñòü ïðåîáðàçîâàíèÿ ìîíîõðîìàòè÷åñêîé âîë- íû è ãàóññîâîãî èìïóëüñà íà ñêà÷êå ñâîéñòâ ïîëó- îãðàíè÷åííîé îáëàñòè ñâîáîäíîãî ìàãíèòîäèýëåê- òðè÷åñêîãî ïðîñòðàíñòâà, íåòðóäíî ïðåäïîëîæèòü î ñóùåñòâîâàíèè òàêîé æå ïîëíîé èäåíòè÷íîñòè ïîëÿ â íåâîçìóùåííîé îáëàñòè 0 x  íà èíòåðâàëå âðåìåíè ) ; 0 ( t   . Äåéñòâèòåëüíî, íåäîëãèå ïðåîá- ðàçîâàíèÿ ïðèâîäÿò ê ðàâåíñòâó     2 2 2 2 0 2 m a 2 m a 1 ) x , t ( E ) x , t ( E           )) ' t t ( v x , ' t ( E )) ' t t ( v x ( ' dt t 0 1 0 t 0   ) x , t ( E 0 } e C e C ){ v x t ( 2 2 2 2 ) kx t ( m a 2 ) kx t ( 1 0           ,(8) ãäå am 1 am 1 C 1     , 1 2 amC C  . Ðàññóæäàÿ ïî àíàëîãèè ñ àíàëèçîì ñîîòâåòñòâóþ- ùåãî âûðàæåíèÿ â ðàáîòå [1], ïðèõîäèì ê ñëåäóþ- ùåìó àíàëèçó (8).  íóëåâîé ìîìåíò âðåìåíè íàðÿäó ñ ôðîíòîì ïîëÿ t v x 1  îò ãðàíèöû 0 x  îòäåëÿåòñÿ è äâèæåòñÿ ñî ñêîðîñòüþ 0 v â íàïðàâ- ëåíèè, ïðîòèâîïîëîæíîì íàïðàâëåíèþ ïåðâè÷íî- ãî èìïóëüñà, ôðîíò ïîëÿ t v x 0   . Ýòîò ôðîíò îãðàíè÷èâàåò ñëåâà îáëàñòü âíåøíåãî ïîëÿ, âîç- áóæäàåìîãî îòðàæåííûì ïåðâè÷íûì èìïóëüñîì 1 C (âòîðîå ñëàãàåìîå â âûðàæåíèè (8)), ïðîøåä- øèì ÷åðåç ãðàíèöó ðàçäåëà äâóõ ñðåä 0 x  , à òàêæå èìïóëüñîì  1 B , êîòîðûé â îáëàñòè 0 x  èìååò àìïëèòóäó 2 C (òðåòüå ñëàãàåìîå â âûðàæåíèè (8)).  îáëàñòè 0 x  òàêæå ñóùåñòâóåò ïåðâè÷íûé èì- ïóëüñ. Ñêîðîñòè ðàñïðîñòðàíåíèÿ èìïóëüñîâ 1 C è 2 C ñîâïàäàþò ñî ñêîðîñòüþ ðàñïðîñòðàíåíèÿ ïåð- âè÷íîãî èìïóëüñà, ôàçîâûå õàðàêòåðèñòèêè ïåð- âè÷íîãî èìïóëüñà è èìïóëüñà 1 C òàêæå ñîâïàäàþò. 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Êàê âèäíî èç âûðàæåíèÿ (10), ýëåêòðè÷åñêàÿ êîìïîíåíòà ïîëÿ, îáðàçîâàâøàÿñÿ ïîñëå âîçâðàò- íîãî ñêà÷êà ñâîéñòâ îáëàñòè 0 x  (êàê ìîæíî áûëî ïðåäïîëîæèòü), ïðåäñòàâèìà â âèäå ñóïåðïîçèöèè ÷åòûðåõ ïàð ïðÿìûõ è îáðàòíûõ èìïóëüñîâ, à òàêæå ïåðâè÷íîãî èìïóëüñà, êîòîðûé «ïðîäîëæàåò ïîñòó- ïàòü» èç áåñêîíå÷íîñòè. Êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè÷åñêîé âîëíû, àìïëèòóäû îáðàçîâàâ- øèõñÿ èìïóëüñîâ ïðåäñòàâëÿþò ñîáîé ðàçðûâíûå ôóíêöèè ïðîñòðàíñòâåííî-âðåìåííîé îáëàñòè. Äàííàÿ îáëàñòü, ãäå îòëè÷íû îò íóëÿ ýòè àìïëèòó- äû, à çíà÷èò è îáëàñòü ñóùåñòâîâàíèÿ ñîîòâåòñòâó- þùèõ èì èìïóëüñîâ, îãðàíè÷åíà ïëîñêîïàðàëëåëü- íûìè ïëîñêîñòÿìè, äâèæóùèìèñÿ âäîëü îñè x è íàîáîðîò ñî ñêîðîñòÿìè, ðàâíûìè 0 v . Êðîìå òîãî, îáðàçîâàâøèåñÿ ïàðû èìïóëüñîâ õàðàêòåðíû íàëè- ÷èåì íàáåãîâ ôàç, ÷òî îáóñëîâëåíî íåïðåðûâíîñ- òüþ ñïåêòðà ãàóññîâîãî èìïóëüñà. Óêàçàííûå âûøå äâèæóùèåñÿ ïëîñêîïàðàëëåëüíûå îáëàñòè ñóùå- ñòâîâàíèÿ îáðàçîâàâøèõñÿ èìïóëüñîâ â ìîìåíò âðåìåíè   t áûñòðî óõîäÿò íà áåñêîíå÷íîñòü, ïîýòîìó, êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè- ÷åñêîé âîëíû [1], ïðåäñòàâëÿåò èíòåðåñ ïîëó÷åíèå ïîçäíåâðåìåííîãî ïðèáëèæåíèÿ âûðàæåíèÿ (10), ò.å. îïðåäåëåíèå ñîñòàâà èìïóëüñîâ, îñòàþùèõñÿ â êîíå÷íîé îáëàñòè ïðîñòðàíñòâà, ñ ïîñëåäóþùèì îïðåäåëåíèåì âûðàæåíèÿ äëÿ èõ àìïëèòóä. Àíàëèç àðãóìåíòîâ ôóíêöèé Õåâèñàéäà, îïðåäåëÿ- þùèõ îáëàñòè ñóùåñòâîâàíèÿ èìïóëüñîâ â âûðàæå- íèè (10), ïîçâîëÿåò çàïèñàòü ïîçäíåâðåìåííîå ïðèáëèæåíèå äëÿ ýòîãî âûðàæåíèÿ â âèäå 2 ) ) 1 am ( kx t ( 2 2 0 e am 4 m a 1 ) x , t ( E ) x , t ( E           . (11) Êàê âèäíî èç (11), â ñëó÷àå ïåðâè÷íîãî ãàóññîâîãî èìïóëüñà, êàê è â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè- ÷åñêîé âîëíû [1], ïîçäíåâðåìåííîå ïðèáëèæåíèå âûðàæåíèÿ äëÿ ýëåêòðè÷åñêîé êîìïîíåíòû ïîëÿ ïðåäñòàâèìî â âèäå ñóììû ïåðâè÷íîé è îáðàòíîé âîëí, ñ ÷èñëîâûìè õàðàêòåðèñòèêàìè ïåðâè÷íîé âîëíû. Òåì íå ìåíåå î÷åâèäíû êîëè÷åñòâåííûå ðàçëè÷èÿ: îòëè÷àþòñÿ âûðàæåíèÿ äëÿ àìïëèòóä îáðàòíûõ âîëí, â ÷àñòíîñòè, àìïëèòóäà îáðàòíîé âîëíû â ñëó÷àå ïåðâè÷íîãî ãàóññîâîãî èìïóëüñà ÿâíî íå ñîäåðæèò äëèòåëüíîñòè èìïóëüñà âîçìóùå- íèÿ ñâîéñòâ ñðåäû, òàê êàê ïîñëåäíÿÿ ïî ïðè÷èíå íåïðåðûâíîñòè ñïåêòðà ãàóññîâîãî èìïóëüñà âõî- äèò ìíîæèòåëåì â íàáåã ôàç îáðàòíîãî èìïóëüñà. Îäíàêî â îáîèõ óêàçàííûõ ñëó÷àÿõ àìïëèòóäà ñòðåìèòñÿ ê íóëþ ïðè 1 a  è 1 m  , ÷òî ñîîòâåò- ñòâóåò ñëó÷àþ îòñóòñòâèÿ âîçìóùåíèÿ ñâîéñòâ ñðå- äû. Ïîñëåäíèé ôàêò ìîæåò ñëóæèòü íåîáõîäèìûì êðèòåðèåì ïðàâèëüíîñòè ïîëó÷åííûõ âûðàæåíèé äëÿ ïðåîáðàçîâàííûõ ýëåêòðè÷åñêèõ êîìïîíåíò ïîëÿ. Èç âûðàæåíèé (10), (11) âèäíî, ÷òî ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðîìàãíèòíîãî èìïóëüñà â ïîëóîã- ðàíè÷åííîé ìàãíèòîäèýëåêòðè÷åñêîé ñðåäå ïðè èçìåíåíèè âî âðåìåíè å¸ ñâîéñòâ ïî çàêîíó ïðÿìî- óãîëüíûõ èìïóëüñîâ èíâàðèàíòíî îòíîñèòåëüíî èçìåíåíèé ïðîíèöàåìîñòåé, â òî âðåìÿ êàê àíàëî- ãè÷íîå âîçìóùåíèå ïàðàìåòðîâ â ñëó÷àå ïåðâè÷íîé ìîíîõðîìàòè÷åñêîé ýëåêòðîìàãíèòíîé âîëíû íå õàðàêòåðèçóåòñÿ îïèñàííîé âûøå èíâàðèàíòíîñ- òüþ. 4. Çàêëþ÷åíèå Ðàññìîòðåíî ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðî- ìàãíèòíîãî èìïóëüñà ïðè èçìåíåíèè âî âðåìåíè ñâîéñòâ ïîëóîãðàíè÷åííîé ìàãíèòîäèýëåêòðè÷åñ- êîé îáëàñòè ïî çàêîíó ïðÿìîóãîëüíîãî èìïóëüñà. Íàó÷íàÿ íîâèçíà ïîëó÷åííûõ ðåçóëüòàòîâ ñîñòîèò â ñëåäóþùåì: 1.  ïðîñòðàíñòâå îáðàçóåòñÿ ïîëå ñëîæíîé ñòðóê- òóðû, êîòîðîå ïðåäñòàâèìî â âèäå ñóììû ïðÿìûõ è îáðàòíûõ ïëîñêèõ èìïóëüñîâ ñ ÷èñëîâûìè õàðàê- òåðèñòèêàìè ïåðâè÷íîé âîëíû. 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}<{3> |V'^Tx/(/0_z iGicXۢyOcZRI} u?VZ~/=ϙ1+O]:GԸ?a~5[gw)qϘ.qdϗJ3={r)SGȮ2:x'5^׭!nэ͝щ~`쉰5#]:iiK8薮9s`U==Դe?@߬'gtޘaSԳ4Qϙ*9zޭ,U}6{f(gY'~Hld\NhS{hh#_i[7)?8w8~@޿O+rWΖܒp͚8 ę,esا0yHgOM.V^oft>LBp0<;Hid&3L~>F^uy"JS&GNܣqqEǨ}|G3 +W$=m!nMޠv7 mkhNF4҇BW#oTĢ%h{[?Vhi/͛7]=})og^|meGT[9|%.+*U@0NSu7f 9ٿӝ&b'uZsb0xlw{R.T\5,^!L-ṽ>h[As?xr.hw {b{ś_L׵YXMxc=;Ji7VpKZJj8mVWRk^Iz%Gan[̡y; ^X[$t\ХE_ ^-Flmѐzs!7khXB6Ļ6jiÆۜyUXrRp˼Rpk{0x[o0x ;4zy/Z9鲚iso;Zik%aj76Mȭc ,o.PpCZe7[8+7$\Vp )͞Jʕ4f!+ʕ4[Vp1**^\Jn܃^5|r8f͊OweCy;{X_wO: 쨬x]*epCo]ܟ$o5 GC&-:oJnj~_'CSyW;B@Gy; ORx'&x{Ry(tAŠzXJ:-bXY[m^QnJ&]ǹe^|-dxy?Q6[rF}ŵ5FfVdF~:^J)_ G>CQB^m^k^B=7*+M}*Q Vxc[6Jm6D.sxkS|E_oqA-f_4{T?|Dl{} wua΍-vt8Nu눐{Pwx{I鄼nv;>#בVy ,g׭3G%}~>xmUź=ċ˔.&Fy+y2xs 8 ^Qf$_+7+TJ7WQ\ky}EFa/ jWJ g]5ܗu؞uLILɒ]pn2J>G HJ\+ar Ơuw>U+y xՉaDLռ"M^o8nmր-r*=g;ET%y ~֫w;Q@Z{elz  Qw0%x{< ޡ ޡ 4jy{؞훗{]|;0ERz-~{5Ǵ-oG*oBfKx]> Yr` U\Xo/sIWV!_@ ޞnvJgkC^IF`z݋o!J藣XNE'{ }QGo[ՖʻXFy9+z 庼y֎nO_,g՛MlW%%/ ]6 Y2՛ݞM>mUV vgsq\~E掠re$oq='WzQ]R.+!gE.1'y5*yKƳ \ͫ`CeZ毿jVԅA 9QM+^5y[nБ.W6 9і<ԡd[Pȉ2 FtԭQS^ﳖ潮Sf{'ϡz7Xz{!J.ZV6h6Qk%,ށ ^OR}q0%^ˍAm,];4:'Ҟ@mvcCΑU[Ctdqy5xKQM9ѻ= xƈBNyiT{o^si³o6K|si?zQJIiH xFyUWãw=mՀW -eQj"Uz2y\l/3*€Wg ]d1D[9Im`b&/z(mk3K3`q>vQ|>F?3`Ȣ| 猋i..rT.b$aM^aN"^w#AHN'xU?[%b?G]}%Hm^&6-ky6 Û N$j*,dm%HޚJ[SIux"S#՞:_5iwh&4Sjp )mVHM%a,MBT:8 y/g!oO//cH{+R7Qo-dTtzv;X&st-B)އ[|uފyQ1ʹrz*{Nzmūx*]mռݡ śqA}-l7[]o.fŪz_ռ'$ =reռWة)NU8(woyPޏ'y+Du$y{h »;(]YڼIފS:zՄ %-tߞPR)IPIy+ޚռ) j0c~?oIcREjj0EGGXwBűN<|^YʏԬwΎ &3Zp8Tqq;vAѹqs*Ot|HeG]P6]- > tB% d:UE!UsȻ_8+F E_ -=͉щΖyFKڸD=I,,7 ÐÈ, 2003, ¹ 4 2. Ïîñëå ñíÿòèÿ âîçìóùåíèÿ ñâîéñòâ ñðåäû àñèì- ïòîòè÷åñêè â êîíå÷íîé îáëàñòè ïðîñòðàíñòâà îñòà- þòñÿ ïåðâè÷íûé è îáðàòíûé åìó èìïóëüñû. 3. Ïðåîáðàçîâàíèå ïëîñêèõ ìîíîõðîìàòè÷åñêèõ ýëåêòðîìàãíèòíûõ âîëí â ïîëóîãðàíè÷åííîé ìàã- íèòîäèýëåêòðè÷åñêîé ñðåäå ïðè èçìåíåíèè âî âðå- ìåíè å¸ ñâîéñòâ ïî çàêîíó ïðÿìîóãîëüíûõ èìïóëü- ñîâ íå èíâàðèàíòíî, à ïðåîáðàçîâàíèå ãàóññîâîãî ýëåêòðîìàãíèòíîãî èìïóëüñà èíâàðèàíòíî îòíîñè- òåëüíî èçìåíåíèé ïðîíèöàåìîñòåé. Ëèòåðàòóðà: 1. Ñëèï÷åíêî Í.È., Øóëüãà Ë.Í., Ðûáèí O.H. Ïðåîáðàçîâàíèå ýëåêòðîìàãíèòíîãî èìïóëüñà âðåìåííûì âîçáóæäåíèåì ñðåäû â ïîëóïðîñòðàíñòâå / / Ðàäèîýëåêòðîíèêà è èíôîðìàòèêà. 1998. ¹1. C. 31-34. 2. Áîëîòîâñêèé Á.M., Ïëèñ À.È., Ñòîëÿðîâ C.H. Ðàñïðî- ñòðàíåíèå èìïóëüñíîãî èçëó÷åíèÿ â íåñòàöèîíàðíûõ ñðåäàõ // Èçâ. âóçîâ. Ðàäèîôèçèêà. 1976. T. 19, ¹4. C. 567-573. 3. Áîðèñîâ Â.Â. Òðàíñôîðìàöèÿ ýëåêòðîìàã- íèòíîãî ïîëÿ ïðè èçìåíåíèè ïðîâîäèìîñòè ñðåäû âî âðåìåíè // Ãåîìàãíåòèçì è àýðîíîìèÿ. 1989. T.29, ¹5. C. 730-737. 4. Àôàíàñüåâ C.Â. Èçëó÷åíèå ìîäåëè ñâåðõ- ñâåòîâîãî èñòî÷íèêà â íåñòàöèîíàðíîé ñðåäå // Èçâ. âóçîâ. Ðàäèîôèçèêà. 1991. T. 34,¹5. C. 605-607. 5. Áîëî- òîâñêèé Á.Ì., Äàâûäîâ Â.À., Ðîê Â.Å. Îá èçëó÷åíèè ýëåêòðîìàãíèòíûõ âîëí ïðè ìãíîâåííîì èçìåíåíèè ñîñòîÿíèÿ èçëó÷àþùåé ñèñòåìû // ÓÔÍ. 1978. Ò.126. Ñ. 311-321. 6. Íåðóõ À.Ã., Õèæíÿê Í.À. Ñîâðåìåííûå ïðî- áëåìû íåñòàöèîíàðíîé ìàêðîñêîïè÷åñêîé ýëåêòðî- äèíàìèêè. Õ.: ÍÏÎ Òåñò-Ðàäèî, 1991. 280 c. 7. Âëàäèìè- ðîâ Â.Ñ. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ì.: Íà- óêà, 1976. 527 c. Ïîñòóïèëà â ðåäêîëëåãèþ 12.07.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Àñååâ Ã.Ã. Ñëèï÷åíêî Íèêîëàé Èâàíîâè÷, êàíä. òåõí. íàóê, äî- öåíò, ïðîðåêòîð ïî íàó÷íîé ðàáîòå ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèîôèçèêà è ýëåêòðîíèêà. Àäðåñ: Óêðàè- íà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. (0572) 40-90-20. Ðûáèí Îëåã Íèêîëàåâè÷, êàíä. ôèç.-ìàò. íàóê, ñòàð- øèé íàó÷íûé ñîòðóäíèê êàôåäðû ìèêðîýëåêòðîíèêè, ïîëóïðîâîäíèêîâûõ ïðèáîðîâ è óñòðîéñòâ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå â ðàäèîôèçèêå. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëå- íèíà, 14, òåë. (0572) 40-93-62. Øóëüãà Ëàðèñà Íèêàíäðîâíà, àñïèðàíòêà êàôåäðû ìèêðîýëåêòðîíèêè, ïîëóïðîâîäíèêîâûõ ïðèáîðîâ è óñòðîéñòâ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñ- êîå ìîäåëèðîâàíèå â ðàäèîôèçèêå. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. (0572) 40-93-62. Íîâèêîâ Íèêîëàé Èâàíîâè÷, êàíä. âîåííûõ íàóê, 1-é çàìåñòèòåëü íà÷àëüíèêà ôàêóëüòåòà ïî ó÷åáíîé è íàó÷íîé ðàáîòå ÕÈÂÂÑ. Íàó÷íûå èíòåðåñû: ðàäèî- ôèçèêà è ýëåêòðîíèêà. Àäðåñ: Óêðàèíà, 61000, Õàðü- êîâ, óë. Êëî÷êîâñêàÿ, 228, òåë. 30-82-84. ÓÄÊ 621.371.3 ÐÀÑÑÅßÍÈÅ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÛÕ ÂÎËÍ ÑÈÑÒÅÌÎÉ ÏËÎÑÊÈÕ ÐÅØÅÒÎÊ ÐÅÇÎÍÀÍÑÍÛÕ ÌÀÃÍÈÒÎÄÈÝËÅÊÒÐÈ×ÅÑÊÈÕ ÑÔÅÐ Ñ ÒÎÏÎËÎÃÈ×ÅÑÊÎÉ ÑÒÐÓÊÒÓÐÎÉ, ÇÀÄÀÂÀÅÌÎÉ ÃÅÎÌÅÒÐÈ×ÅÑÊÎÉ ÏÐÎÃÐÅÑÑÈÅÉ ÊÎÇÀÐÜ À.È. Îïèñûâàåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðî- ìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ ðåøåòîê ðåçîíàí- ñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð, ïðîñòðàíñòâåí- íîå ðàñïðåäåëåíèå êîòîðûõ ïîä÷èíåíî ãåîìåòðè÷åñ- êîé ïðîãðåññèè. Çàäà÷à ðåøàåòñÿ ñ ïîìîùüþ èíòåã- ðàëüíûõ óðàâíåíèé ýëåêòðîäèíàìèêè. Ïðèâîäÿòñÿ ôîðìóëû äëÿ âíóòðåííèõ ïîëåé ñôåð è ðàññåÿííûõ ïîëåé. Ââåäåíèå Öåëüþ ðàáîòû ÿâëÿåòñÿ ðåøåíèå çàäà÷è î ðàññåÿíèè ýëåêòðîìàãíèòíûõ âîëí ñèñòåìîé ïëîñêèõ îäíî- ñëîéíûõ ðåøåòîê, ïîñòðîåííûõ èç ìàëûõ îäíîðîä- íûõ ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð [1]. Èõ ïðîñòðàíñòâåííîå ðàñïðåäåëåíèå ïîä÷èíåíî ãåîìåòðè÷åñêîé ïðîãðåññèè, îáùèé âèä ÷ëåíà êî- òîðîé äëÿ ñëó÷àÿ, êîãäà ïåðâûé ÷ëåí ïðîãðåññèè 1 a1  , ïðåäñòàâèì â âèäå n1|t| n10 aaq(q1)   , (1) ãäå çíàìåíàòåëü ïðîãðåññèè – 0 qq1;  q0,1,2,3,;n|t|1;|t|0,1,2,3,   .  çàäà÷å äëèíà ðàññåèâàåìîé âîëíû ìîæåò áûòü ñîèçìåðèìà ñ ïîñòîÿííûìè ðåøåòêè, ÷òî ïîçâîëÿåò èçó÷èòü âëèÿíèå ðåøåòî÷íûõ ñòðóêòóðíûõ ðåçî- íàíñîâ ýëåêòðîìàãíèòíîãî âçàèìîäåéñòâèÿ ñôåð íà âíóòðåííèå ðåçîíàíñû ñôåð ðåøåòêè è èõ òîíêóþ ñòðóêòóðó. Ýòî ðåøåíèå îïèñûâàåò îáëàñòè àíî- ìàëüíîé äèñïåðñèè ðåøåòîê. Ïîñòàíîâêà è ðåøåíèå çàäà÷è Ðàññìîòðèì â äåêàðòîâîé ñèñòåìå êîîðäèíàò ïî- ðîæäàþùåå ïðîñòðàíñòâåííóþ ñèñòåìó ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê êîîðäèíàòíîå ïðåä- ñòàâëåíèå (ðèñ. 1) ss1 p,sp,s0 |t| tt1 p,tp,t0 p0p0 x[s{(1)1}0.5]d(1)x (s0,1,2,...,[(q1)1]), y[t{(1)1}0.5]h(1)y (t0,1,2,), zzlzpl (p0,1,2,),             (2) ãäå âåëè÷èíû d, h îïðåäåëÿþòñÿ óñëîâèÿìè x0,xd;y0,yh,  à p,s0p,t0p x,y,z  – êîîð- äèíàòû óçëîâ, íàõîäÿùèõ- ñÿ âíóòðè îáëàñòè: p,s0 p,t0 p 0xd, 0yh, zzz.      (3) Ðèñ. 1 ;x{pU{n Bx&\PhxD@bBGmL@iH V>iJy(/ d(DˣbE*XA-"2"&wÝB̗ۙﵿs`~":A/||&4iL"k_M;ɗ)F.2Ȫi契hNRHVrB+7'I7AZɢHN$׾7̏mM1i&&L_rDdsKc9Ǫx^Cb7]83V1;޲*Nc_Gɲ%4)O[dEՇO;Fwd:YKt##(mq wz9b/L7ݡ턈mߎىkjS&)|ж.sso檏wrJ VH[c7ByɶXNk1uܻRՅqk֣3Ȫpz)S˜&wU^s6|e[8Nݣ݉j `7 sKU~>^pGc+XMd}Bss⛥^u B BSp!y؛-_|u='S$=.me&yˑÕ9rbh)ps߅8 k ٥& 6rgX?&iٛ:l+YGA^=Z֬{@s|qR󯇩U=^||?kفX'q^*ϿYs۰tbNIc,)Y 1?|5'#]A&Hq7ѿLϽ1ށsw/=W22O>CS]_WI as͔D.3/_X˭g84>̔DysE%DI#~w#KU_1^$ha># V7ʱ~,/C>rdu38n3EUu}C; U MHŮ/骞lFfh1p%9{G t_Ʀ1FSԳbǢ+; 'TMe9FYU^jZ"ϔC=灁` xA:|X}w9Eoς5Y_roRHa0f cSΫFx o U}\n9mc`?pY"_ʚ rx \D(1X rSaB& X?w/J"7F&9m&ۛT;$RUWeۏ2K-]p9V+ v1H= ׼x]p[ִ?6>Ǧp --p%vr:)`|ՓkS!RwV$l-/ޫ]ygΤ9{}6~t{5C Gw#Cy_PXpJxW؊Q ' ͷq;_x%kG7sj+Չ<;YSAdg3_S2ova0ֺ(%0TҐ 2rXG< ٣aם94|y ;-ez^J 6[-GDc-uvuZK}Z+DSbvZjÇ+;:ÿq6lwz-W׺ dWO} Gy|.^>\ŘQ D>yYUSTmUJfgTW#¿\{ewk*s$dxe;9|/Xx7|oڵjMcզAf6io^;:ޯ.c_]P̺|.UcԖ?(~]>w &uū+x~ČgRL~]|\k}/OPAvA23nP q}؆34fm:2q܅U߉i6msGf}Ov}Wm̴ c̻ɴ.^+͵j;uAKA{ ?BM/v7mܗ~WV @Xa +V @Xa +\? 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Êàæäîìó óçëó ðåøåòîê (2) ñîïîñòàâëÿåòñÿ óïîðÿäî÷åííàÿ òðîéêà ÷èñåë c(p,s,t)  , âûäåëåííûé óçåë ðåøåòîê áóäåì îáî- çíà÷àòü c'(p',s',t')  . Çíà÷åíèå èíäåêñà ð îïðåäåëÿåò íîìåð ïðîñòîé ïëîñ- êîé ðåøåòêè (ðèñ.2), à çíà÷åíèÿ èíäåêñîâ s, t çàäàþò ïîëîæåíèå óçëà ð-é ïðîñòîé ïëîñêîé ðåøåòêè. Íóæíûé òèï ýëåìåíòàðíîé ÿ÷åéêè ñëîæíîé ïëîñ- êîé îäíîñëîéíîé ðåøåòêè ôîðìèðóþò èç ð ïîðîæ- äàþùèõ ïðîñòûå ïëîñêèå ðåøåòêè óçëîâ âíóòðè îáëàñòè (3), êîòîðóþ ïîâòîðèò çà åå ïðåäåëàìè êîîðäèíàòíîå ïðåäñòàâëåíèå (2) â âèäå îïðåäåëåí- íîé ñëîæíîé ïëîñêîé îäíîñëîéíîé ðåøåòêè. Ðèñ. 2 Çàäàâàÿ ìàêñèìàëüíûå çíà÷åíèÿ ÷èñåë p,s,t , ìîæíî ðàññìàòðèâàòü êîíå÷íûå è áåñêîíå÷íûå ðåøåòêè. Íà ðèñ. 2 èçîáðàæåíà ñèñòåìà ïðîñòûõ ïëîñêèõ îäíîñëîéíûõ ðåøåòîê, êîãäà ïîðîæäàþùèé óçåë ðåøåòêè íàõîäèòñÿ â öåíòðå îáëàñòè (3), äëÿ ñëó÷àÿ – q=1; ð=0, 1, 2; t=0,±1,...,±5. Íà ðèñ. 3 ïðåäñòàâëåíà ñèñòåìà ðåøåòîê, êîãäà ïîðîæäàþùèé óçåë ðåøåòêè íàõîäèòñÿ â öåíòðå îáëàñòè (3), äëÿ ñëó÷àåâ – q=0, p=0, t =0,±1,...,±5 è q=1, p=1; q =2, p=2; q=3, p=3; q=4, p=4 ïðè t=0, ±1, ±2. Ñëó÷àé q=0, p=0, t=0,±1,...,±5 ñîîòâåòñòâó- åò ëèíåéíîé öåïî÷êå óçëîâ. Ðàñïðåäåëåíèå ñôåð â ïëîñêîñòè ðåøåòêè âäîëü îñè õ ïîä÷èíåíî ÷ëåíàì ïðîãðåññèè (1). Êàæäîé òðîéêå ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z ñòàâèòñÿ â ñîîòâåòñòâèå ÷ëåí ïðîãðåññèè, íàïðèìåð, ïðè 1  q òî÷êå   p0,s0,t5  – ÷èñëî 32, à òî÷êå   p2,s0,t3  – ÷èñëî 8 (ðèñ. 2). Ðèñ. 3 Åñëè q =const, òî ìîæíî èçó÷àòü ñèñòåìó îäèíàêî- âûõ ïëîñêèõ ðåøåòîê (ñì. ðèñ. 2). Ìåæäó òðîéêîé ÷èñåë   p,s0,t  ïëîñêîñòè p,tp y,z è ÷ëåíàìè ïðîãðåññèè ñóùåñòâóåò ñâÿçü, îïðåäåëÿ- åìàÿ ôóíêöèåé (1): |t| (q1)  . Òîãäà ÷èñëà s, îïðåäåëÿþùèå êîîðäèíàòû óçëîâ âäîëü îñè õ (2), çàäàþòñÿ ïîñëåäîâàòåëüíîñòüþ |t| 0,1,2,3,,[(q1)1]   . Åñëè èçìåíÿòü êîîðäèíàòû óçëîâ, íàõîäÿùèõñÿ â ïðåäåëàõ îáëàñòè (3), òî, â ñîîòâåòñòâèè ñ êîîðäè- íàòíûì ïðåäñòàâëåíèåì (2), ïîëîæåíèÿ óçëîâ ðå- øåòîê âíå îáëàñòè (3) áóäóò òàêæå ñîîòâåòñòâóþ- ùèì îáðàçîì ñìåùàòüñÿ, ÷òî ïîçâîëÿåò ïåðåñòðàè- âàòü ïðîñòðàíñòâåííóþ êîíôèãóðàöèþ ðåøåòîê. Ïîëàãàåì, ÷òî â óçëû ðåøåòîê (2) ïîìåùàþòñÿ öåíòðû N= p p N  ìàëûõ îäíîðîäíûõ ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð ñ ïðîíèöàåìîñòÿìè cc ,  è ðàäèóñàìè c a(cN)  , ãäå p N – ÷èñëî ñôåð ð-é ïëîñêîé ðåøåòêè. Ïðîíèöàåìîñòè çàïîëíåíèÿ ïðîñòðàíñòâà âíå ñôåð – 00 ,  . Ïîëÿ áóäåì çàïèñûâàòü â âèäå it E(r,t)E(r)e     , it H(r,t)H(r)e     . Ðàññòîÿíèå ìåæäó öåíòðàìè ñôåð c è c  ïðåäñòà- âèì (2) 222 ccp,sp,sp,tp,tpp r(xx)(yy)(zz).   (4) Ñ÷èòàåì, ÷òî âíå ñôåð a/1  , íî âíóòðè ñôåð âîçìîæåí ðåçîíàíñíûé ñëó÷àé g a/~1  , ãäå  – äëèíà âîëíû âíå ñôåð, à g  – äëèíà âîëíû â ñôåðå. Äëÿ ðåøåíèÿ çàäà÷è èñïîëüçóåì èíòåãðàëüíûå óðàâíåíèÿ [2] è ðåøàòü åå áóäåì â äâà ýòàïà. Íà ïåðâîì ýòàïå íàéäåì âíóòðåííåå ïîëå ðàññåèâàþ- ùèõ ñôåð, à íà âòîðîì – ïîëå, ðàññåÿííîå ïðîñòðàí- ñòâåííîé ðåøåòêîé ñôåð. 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Åñëè ó÷åñòü, ÷òî äëÿ âíåøíèõ òî÷åê ñôåðû (rr)   èíòåãðàë ïî îáúåìó ñôåðû îò ôóíêöèè Ãðèíà äëÿ ñâîáîäíîãî ïðîñòðàíñòâà (7) èìååò âèä 00 1 ikrr V ikr 111 3 1 e W(r)dV rr 4e (sinkakacoska), r k             (8) ãäå 100 kk,k2/  , r – ðàññòîÿíèå îò öåíòðà äî âíåøíèõ òî÷åê ñôåðû, òî ìîæíî ïîñòðîèòü óðàâíåíèÿ äëÿ âíóòðåííèõ ïîëåé c  ñôåðû, èñ- ïîëüçóÿ ðåøåíèÿ, ïðåäëîæåííûå â [3,4], â âèäå     1с' 0(p,s,t) 2 c' эф01c'c'эф1c'c'эф0 0 (p,s,t) i 0 cэф 2 э0 00cc(p,s,t) pst 0 (p,s,t)(p,s,t) cэф 0 0 E(r,t) 2i2 E(r,t) 3e 1 (k)1W(r)E(r,t) 4 1 ik,1 4                                       м0 cc(p,s,t) W(r)H(r,t),                  1с' 0(p,s,t) 2 c' эф01c'c'эф1c'c'эф0 0 (p,s,t) i 0 cэф 2 м0 00cc(p,s,t) pst 0 (p,s,t)(p,s,t) cэф 0 0 H(r,t) 2i2 H(r,t) 3e 1 (k)1W(r)H(r,t) 4 1 ik,1 4                                       э0 cc(p,s,t) W(r)E(r,t),              ãäå 0(p,s,t)0(p,s,t) E(r,t),H(r,t)     è 00 (p,s,t)(p,s,t) E(r,t),H(r,t)    – ïîëå ïàäàþùåé âîëíû â ñôåðå è âíóòðåííåå ïîëå ñôåð ñîîòâåòñòâåííî, 222 1c'c'00 ka.  Âåëè÷èíû эм cccc W(r),W(r)   èìåþò âèä (4), (8), (9): 1cc 1cc ikr э cc1c1c1c 3 cc 1 ikr м cc1c1c1c 3 cc 1 4e W(r)(sinkakacoska), r k 4e W(r)(sinkakacoska), r k               à âåëè÷èíû сэфсэф ,  çàïèøåì [4,5,6] сэфcccc сэфcccc F(ka), F(ka),   (11) ãäå ccc ccccccccc 22 cccccccccccc F(ka) 2(sinkakacoska) . (ka1)sinkakacoska     Íà ðèñ. 4 ïðåäñòàâëåíû îñîáåííîñòè ïîâåäåíèÿ ReF()  (ñïëîøíàÿ êðèâàÿ) è ImF()  (ïóíêòèðíàÿ êðèâàÿ) â çàâèñèìîñòè îò  Re ïðè ðàçíûõ çíà÷å- íèÿõ òàíãåíñà óãëà äèýëåêòðè÷åñêèõ ïîòåðü tg   (êðèâàÿ 1 –   tg =0; êðèâàÿ 2 –   tg =0,05; êðèâàÿ 3 –   tg =0,1) è c 1  , çäåñü ccc ka  . Ïåðâûå ñëàãàåìûå ñïðàâà â óðàâíåíèÿõ (9) ñâÿçàíû ñ âíóòðåííèì ïîëåì âûäåëåííîé ñôåðû, à âòîðûå ñëàãàåìûå ó÷èòûâàþò âëèÿíèå íà âûäåëåííóþ ñôåðó âñåõ îñòàëüíûõ ñôåð. Äëÿ îïðåäåëåíèÿ âíóòðåííåãî ïîëÿ, èìååì àëãåáðà- è÷åñêóþ ñèñòåìó 2N âåêòîðíûõ íåîäíîðîäíûõ óðàâíåíèé (9) èëè æå äëÿ x,y,z  ñîñòàâëÿþùèõ 6N óðàâíåíèé ñ 6N íåèçâåñòíûìè. Îñíîâíàÿ ìàòðèöà àëãåáðàè÷åñêîé ñèñòåìû óðàâíå- íèé (9) ñîäåðæèò èíôîðìàöèþ î õàðàêòåðå ýëåêò- ðîìàãíèòíîãî âçàèìîäåéñòâèÿ ìåæäó ñôåðàìè äàí- íîãî òèïà ðåøåòîê. Äëÿ ñôåðû c' ðåøåíèå ñèñòåìû óðàâíåíèé (9) èìååò âèä: 0 (p,s,t) эсэс c0(p,s,t)c0(p,s,t) эм c E(r,t) 1 (gE(r,t)H(r,t)),               (12) 0 (p,s,t) мсмс c0(p,s,t)c0(p,s,t) эм c H(r,t) 1 (H(r,t)gE(r,t)),               ãäå эм  – äåòåðìèíàíò îñíîâíîé ìàòðèöû ñèñòåìû óðàâíåíèé (9). Åñëè ïðåäïîëîæèòü, ÷òî ó âñåõ ñôåð ñ îäèíàêîâûì èíäåêñîì p âíóòðåííåå ïîëå ðàâíî, íàïðèìåð, âíóòðåííåìó ïîëþ ñôåðû (p,s,0,t0)  , òî ñèñòåìó óðàâíåíèé (9) ìîæíî ñâåñòè ê ñèñòåìå   22p1  óðàâíåíèé.  ñëó÷àå, êîãäà âñå ñôåðû ðåøåòêè (9) (10) BIx tEo}%!$kX$ ;AM@YGETpJB#[l*jb}7fΙ=%[u֭UݪsD"R*HKx6'Z$>I*DUBk iE8K X^0X-pETqIȫ*^ g~ttx*`\ 8GDgi}xf[`:arxN]&Ui#W*[0U[xV' %=t3*ndn8I2D^I^&(˽JN)c y g押ST"n[))dɣ暞4,sk+G(yR >sqky7Q v˺:Lx飲 DZaD6Β%KjKK▓*9\:*|'v?`>r:eK~6[/=%usd.&xM]`TY ~NzvԢ|P_TtUg3rl&mPCn܏ݣMe*|& E]iK2=SꐟaNKs5LuYOO/)nRD :]Z7LrI7/MiIk. 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Åñëè ýëåêòðîìàãíèòíûì âçàèìîäåéñòâèåì ìåæäó ñôåðàìè ðåøåòêè ìîæíî ïðåíåáðå÷ü, òî âûðàæåíèÿ äëÿ âíóòðåííåãî ïîëÿ ïðîèçâîëüíîé ñôåðû ðåøåò- êè (12) áóäóò èìåòü âèä 1 i 0 0 coc 2 эф01эф1эф0 3e Е(r,t)Е(r,t), (2)i(2)       1 i 0 0 coc 2 эф01эф1эф0 3e Н(r,t)Н(r,t), (2)i(2)       . Ïîòåíöèàëû Ãåðöà э П  è м П  ðàññåÿííîãî ñèñòåìîé ïëîñêèõ ðåøåòîê ïîëÿ ïî èçâåñòíîìó âíóòðåííåìó ïîëþ (12) îòäåëüíûõ ðàññåèâàòåëåé ïðåäñòàâèì â âèäå ñóïåðïîçèöèè ïîòåíöèàëîâ Ãåðöà îòäåëüíûõ ñôåð ðåøåòîê (6): 1(p,s,t) 1(p,s,t) э 1c1c1c 3 pst 1 ikr cэф 0 c 0(p,s,t) м 1c1c1c 3 pst 1 ikr cэф 0 c 0(p,s,t) 1 П(r,t)(sinkakacoska) k e 1E(r,t), r 1 П(r,t)(sinkakacoska) k e 1H(r,t). r                               (13)  âûðàæåíèÿõ (13) 222 (p,s,t)p,sp,tp r(xx)(yy)(zz),  (14) ãäå êîîðäèíàòû (x,y,z) îïðåäåëÿþò âíå ñôåð òî÷êó íàáëþäåíèÿ ïîëÿ, ðàññåÿííîãî ñèñòåìîé ïëîñêèõ ðåøåòîê, à êîîðäèíàòû p,sp,tp (x,y,z) – òî÷êó íàõîæäåíèÿ öåíòðà ðàññåèâàþùåé ñôåðû ðåøåòêè. Òîãäà, ó÷èòûâàÿ (13), èç (5) íàéäåì ðàññåÿííîå ñèñòåìîé ïëîñêèõ ðåøåòîê ïîëå 1c расс1c1c1c 3 c 1 c эфсэф 00 cc0cc 00 i(tkr) расс1c1c1c 3 c 1 сэф 0 cc 0 1 E(r,t)(sinkakacoska) k 1LE(r)ik1(1)PH(r) e, 1 H(r,t)(sinkakacoska) k 1(1)LH                                   1c сэф 0 0cc 0 i(tkr) (r)ik1PE(r) e,                  ãäå c L  è c P  – ôóíêöèîíàëüíûå ìàòðèöû, êîòîðûå èìåþò âèä 0 zcyc xxcxycxzc 0 cyxcyycyzcczcxc 0 zxczyczzc ycxc 0 L ; P0 . 0                    (16) Âåëè÷èíû, âõîäÿùèå â ôóíêöèîíàëüíûå ìàòðèöû (16), çàïèøåì â âèäå (2), (14): Ðèñ. 4 (15) 8&xX pVv/y IT`BXP H  ha@F-:`""h((e A^rw#8:|9g=sD/1 LJq<hN40͚~?)dx<0/mt +zaix,H9z8Hs7-:"kLQY%\;i: @9Yv`\Jn- T½˙; U/p-09r2w\NnXg;8-qn2ܻE:nS!>7m)߈fȠoZnJcQ}P,ƋR||k)AYul'rݕj DID|C*~~_\F9诧eԫur@|w|6gxD7&n* s^ΧF!6A)8DL|;:_N;+}jS9D[T-nc鄫v66PGP4` 2^_qa"dEqϳ 5QuCjJvZMFy9yxA':T MC(gu;\9.4YskdWyB,;ж\bgh Bo5Nj$uh[~I3<5ȕ4Q.21Ĭ9&<1>x- d\䨭5~kw[ggpXH=acy{j7S,ZܢB{s tE7{6nh8vg'}0Zr=p9$%D q(:kJiqs%oH rU|*J~u` Z'I_8AT"Q lK=*ЇsDO/$Rz}~NBtd)M:Ǥ3a po6 [(0[9'!ėFw9ɶ"uϦ/9@w"DrD}6 T{Q{|OA{ $≺-4=iOvNb; vWо aY[[[~ꊫDWI$On{)!gO8[?+sg&hϭuo!\͸3geh!FOdrl:){LchuSLR/WlDw yJOd'F]#AƫdBǧ.gt ޿ob=Vsȣ䖛hL+eq;LSkb3 wk& ,gZ`rhˬd8ǧ]kS$1^ӔyIB5k=k-z>x(7X]ufHlKxĽ(֞Pt_S +ݕr#20ИіsbTo$9{-cw2Mqz(cXXbnQy{YifZěS3Dzz:mcECp^%gy( :9Du\8gg!^aY&u4a̓dP:t|>wTj& ؽ!ڗ%#4+ Ko6gݕDn{u&}S&[~K{om\mqu%%= ʔlDT|ᨨcss8{朹3]X~Xߵ^{}- `Y mtQ=<AnAI0w%>A_МBvG -. 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Èç äåòåðìèíàíòà óðàâíåíèé (9) íàõîäèì ðåçîíàí- ñíûå óñëîâèÿ äëÿ ñëó÷àÿ, êîãäà cg a/~1  âíóòðè ñôåð. Åñëè cc ,  ñôåð ðåøåòîê äåéñòâèòåëüíû, òî ðåçîíàíñíûå óñëîâèÿ íàõîäèì èç âûðàæåíèÿ [6]: i detRe0 j  , ðàçðåøàÿ åãî îòíîñèòåëüíî ôóíêöèè ccc F(ka)  (11) (ñì. ðèñ. 4), ãäå i j  – îñíîâíàÿ ìàòðèöà ñèñòåìû óðàâíåíèé (9). Çàêëþ÷åíèå Âïåðâûå ðàññìîòðåíî ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí íà ñèñòåìå ïëîñêèõ ðåøåòîê ðåçîíàíñíûõ ñôåð, ïðîñòðàíñòâåííàÿ ñòðóêòóðà êîòîðûõ îïðåäå- ëÿåòñÿ ãåîìåòðè÷åñêîé ïðîãðåññèåé. Ðåçóëüòàòû ðåøåíèÿ äàííîé çàäà÷è ìîãóò áûòü ïîëåçíû äëÿ ñîçäàíèÿ ðàçëè÷íîãî ðîäà óñòðîéñòâ ïî óïðàâëåíèþ ïîëåì èçëó÷åíèÿ ýëåêòðîìàãíèòíûõ èçëó÷àòåëåé [7], â êîòîðûõ ìîãóò èñïîëüçîâàòüñÿ òîíêèå ýô- ôåêòû âçàèìîäåéñòâèÿ ðåøåòî÷íûõ ñòðóêòóðíûõ ðåçîíàíñîâ è âíóòðåííèõ ðåçîíàíñîâ ñôåð ðåøåòêè. Ëèòåðàòóðà: 1. Êîçàðü À.È., Õèæíÿê Í.À. Îòðàæåíèå ýëåêòðîìàãíèòíûõ âîëí îò ðåçîíàíñíîé äèýëåêòðè- ÷åñêîé ñôåðû â âîëíîâîäå // Óêð. ôèç. æóðí. 1970. Ò. 15. Ñ. 847-849. 2. Õèæíÿê Í.À. Èíòåãðàëüíûå óðàâíå- íèÿ ìàêðîñêîïè÷åñêîé ýëåêòðîäèíàìèêè. Êèåâ: Íàóê. äóìêà. 1986. 279 ñ. 3. Êîçàðü À.È. Ðàññåÿíèå ýëåêòðîìàã- íèòíûõ âîëí ñèñòåìîé ïëîñêèõ îäíîñëîéíûõ ðåøåòîê ðåçîíàíñíûõ ìàãíèòîäèýëåêòðè÷åñêèõ ñôåð // Ðàäèî- òåõíèêà. 2002. Âûï. 130. Ñ. 42–53. 4. Ëåâèí Ë. Ñîâðåìåí- íàÿ òåîðèÿ âîëíîâîäîâ. Ì.: Èçä-âî èíîñòð. ëèò. 1954. 216 ñ. 5. Êîçàðü À.È. Ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí â âîëíîâîäå ñ îäíîðîäíûìè ìàãíèòîäèýëåêòðè- ÷åñêèìè ñôåðàìè // Ðàäèîôèçèêà è ýëåêòðîíèêà. Õàðü- êîâ: Èí-ò ðàäèîôèçèêè è ýëåêòðîíèêè ÍÀÍ Óêðàèíû. 2002. 7. Ñïåö. âûïóñê. Ñ. 183-189. Ïîñòóïèëà â ðåäêîëëåãèþ 31.03.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Êðèâóëÿ Ã.Ô. Êîçàðü Àíàòîëèé Èâàíîâè÷, êàíä. ôèç.-ìàò. íàóê, ïðî- ôåññîð êàôåäðû ôèçèêè ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàäèîôèçèêà. 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+V*SYEqdz[BY%8.n冲HGSH"AG=bl,,12 ÐÈ, 2003, ¹ 4 ÓÄÊ 537.87:621.371 ×ÈÑËÅÍÍÎÅ ÈÑÑËÅÄÎÂÀÍÈÅ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Ñ ÍÅÑÒÀÖÈÎÍÀÐÍÛÌ ÄÈÝËÅÊÒÐÈ×ÅÑÊÈÌ ÑËÎÅÌ ÏÐÈ ÏÎÌÎÙÈ ÌÅÒÎÄÀ ÈÍÒÅÃÐÀËÜÍÛÕ ÓÐÀÂÍÅÍÈÉ ÂÎËÜÒÅÐÐÀ ÔÅÄÎÒΠÔ.Â., ÍÅÐÓÕ À.Ã. Ðàññìàòðèâàåòñÿ ÷èñëåííîå èññëåäîâàíèå âçàèìîäåé- ñòâèÿ ýëåêòðîìàãíèòíîãî ïîëÿ ñ íåñòàöèîíàðíûì äèýëåêòðè÷åñêèì ñëîåì ïðè ïîìîùè ìåòîäà èíòåã- ðàëüíûõ óðàâíåíèé Âîëüòåððà (ÌÈÓÂ). Ðàçðàáàòûâà- åòñÿ ñïåöèàëüíîå ïðîãðàììíîå îáåñïå÷åíèå äëÿ àâòî- ìàòèçàöèè ìîäåëèðîâàíèÿ òàêèõ âçàèìîäåéñòâèé íà îñíîâå ðàññìîòðåííîãî ìåòîäà. 1. Ââåäåíèå Ïàðàìåòðè÷åñêèå ÿâëåíèÿ â àêòèâíûõ ñðåäàõ âû- çûâàþò èíòåðåñ â ñâÿçè ñ âîçìîæíîñòüþ ãåíåðàöèè, óñèëåíèÿ è ïðåîáðàçîâàíèÿ ýëåêòðîìàãíèòíûõ âîëí è èìåþò âàæíîå çíà÷åíèå â ñîâðåìåííîé ðàäèîòåõ- íèêå è îïòîýëåêòðîíèêå [1,2]. Èñïîëüçîâàíèå ìå- òîäîâ, îñíîâàííûõ íà ïðåîáðàçîâàíèÿõ Ôóðüå, ìàëîýôôåêòèâíî è çàòðóäíèòåëüíî äëÿ äàííîãî êëàññà çàäà÷ âñëåäñòâèå øèðîêîãî ñïåêòðà èññëå- äóåìûõ ñèãíàëîâ è çàâèñèìîñòè ïàðàìåòðîâ ñðåäû îò âðåìåíè [3].  íàñòîÿùåå âðåìÿ íàèáîëåå ïîïóëÿðíûì ÷èñëåí- íûì ìåòîäîì äëÿ èññëåäîâàíèÿ ýëåêòðîäèíàìè÷åñ- êèõ çàäà÷ âî âðåìåííîé îáëàñòè ÿâëÿåòñÿ ìåòîä êîíå÷íûõ ðàçíîñòåé (FDTD)[4]. Îäíàêî ïðè ðåøå- íèè çàäà÷ ñ ïåðåìåííûìè âî âðåìåíè ïàðàìåòðàìè ñðåäû äàííûì ìåòîäîì âîçíèêàþò ïðîáëåìû ñî ñòàáèëüíîñòüþ ðåøåíèÿ [5,6].  äàííîé ñòàòüå ïðåäëàãàåòñÿ ìåòîä, îñíîâàííûé íà èíòåãðàëüíûõ óðàâíåíèÿõ Âîëüòåððà âî âðåìåííîé îáëàñòè (ÌÈÓÂ) [3,5], êîòîðûå ýêâèâàëåíòíû ýëåê- òðîäèíàìè÷åñêèì óðàâíåíèÿì Ìàêñâåëëà. Îòëè- ÷èòåëüíîé îñîáåííîñòüþ ÌÈÓ ÿâëÿåòñÿ òî, ÷òî íà÷àëüíûå è ãðàíè÷íûå óñëîâèÿ íåÿâíî ó÷èòûâà- þòñÿ â èíòåãðàëüíîì óðàâíåíèè çàäà÷è [5]. Äàííûé ïîäõîä äëÿ ðÿäà çàäà÷ ïðåâîñõîäèò ïî òî÷íîñòè ìåòîä FDTD [6] è îáëàäàåò âûñîêîé ñòàáèëüíîñòüþ ïðè ìîäåëèðîâàíèè íåñòàöèîíàðíûõ ñðåä. 2. Ïîñòàíîâêà çàäà÷è Ðàññìîòðèì îäíîðîäíóþ íåîãðàíè÷åííóþ ñðåäó ñ îòíîñèòåëüíîé äèýëåêòðè÷åñêîé ïðîíèöàåìîñòüþ  , â êîòîðîé â îïðåäåëåííûé íà÷àëüíûé ìîìåíò âðåìåíè 0 t0  îáðàçóåòñÿ íåñòàöèîíàðíûé ñëîé â îáëàñòè 0xL  .  ìîìåíò ñîçäàíèÿ ñëîÿ â ôîíî- âîé ñðåäå, ïåðïåíäèêóëÿðíî ê âîçíèêàþùåìó ñëîþ, ðàñïðîñòðàíÿåòñÿ ýëåêòðîìàãíèòíûé ñèãíàë, çà- äàííûé ôóíêöèåé   0 Et,x .  îáùåì ñëó÷àå ñðåäà â ñëîå îïèñûâàåòñÿ ôóíêöèåé ïîëÿðèçàöèè   Pt,x , ïàðàìåòðû êîòîðîé, ïîä âîçäåéñòâèåì ñòîðîííèõ ôàêòîðîâ, èçìåíÿþòñÿ âî âðåìåíè ñî ñêîðîñòüþ, ñîèçìåðèìîé ñî ñêîðîñòüþ èçìåíåíèÿ ýëåêòðè÷åñ- êîãî ïîëÿ. Ýëåêòðè÷åñêîå ïîëå äëÿ äàííîé çàäà÷è âíóòðè è âíå ñëîÿ îïèñûâàåòñÿ ñëåäóþùèì óðàâíå- íèåì Âîëüòåððà II ðîäà [5]:         tL 0 00 0 1 Et,xEt,xdt'dx' 2vt xx' 11 tt'Pt',x'Et',x' v              ãäå vc/  , c – ñêîðîñòü ñâåòà â âàêóóìå;   t  – äåëüòà – ôóíêöèÿ Äèðàêà; L – ãëóáèíà ñëîÿ; 0  – äèýëåêòðè÷åñêàÿ ïîñòîÿííàÿ. Äëÿ ðåàëèçàöèè ÷èñëåííîãî ñ÷åòà ïåðåéäåì ê áåçðàçìåðíûì êîîð- äèíàòàì, íîðìèðîâàííûì íà ãëóáèíó ñëîÿ: vt/L  , x/L  . Òîãäà ïîñëå âû÷èñëåíèÿ âíóòðåííåãî èíòåãðàëà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå  óðàâíåíèå ñâîäèòñÿ ê âèäó:       0 1 E,E,HL 2    , çäåñü   0' 0'1 HP',(')d'      ,   0' 0'1 LP',(')d'      . Äâà èíòåãðàëà L è H â óðàâíåíèè íà ïðîñòðàíñòâåí- íî-âðåìåííîé ïëîñêîñòè    ñîîòâåòñòâóþò äâóì, ñõîäÿùèìñÿ ê âû÷èñëÿåìîé òî÷êå äèàãîíàëüíûì ëèíèÿì. Äëÿ äèýëåêòðè÷åñêîé ñðåäû ôóíêöèÿ P  â èíòåã- ðàëàõ çàäà¸òñÿ ñëåäóþùèì ñîîòíîøåíèåì:       1 , P,1E,        . 3. Ðåàëèçàöèÿ àëãîðèòìà ÷èñëåííîãî ñ÷åòà Äëÿ ÷èñëåííîãî ïîñòðîåíèÿ ðåøåíèÿ óðàâíåíèÿ öåëåñîîáðàçíî èñïîëüçîâàòü ïðÿìîóãîëüíóþ ïðî- ñòðàíñòâåííî-âðåìåííóþ ñåòêó, â êîòîðîé øàã ïî âðåìåíè â áåçðàçìåðíûõ âåëè÷èíàõ âäâîå ìåíüøå øàãà ïî ïðîñòðàíñòâó [6]. Ðåøåíèå, ïîñòðîåííîå íà òàêîé ñåòêå, îáëàäàåò ëó÷øåé ñòàáèëüíîñòüþ è áîëåå âûñîêîé òî÷íîñòüþ, ïî ñðàâíåíèþ ñ àíàëî- ãè÷íîé êâàäðàòíîé ñåòêîé [6]. Äëÿ ñîâìåùåíèÿ ëèíèé èíòåãðèðîâàíèÿ ñ óçëàìè ñåòêè óçëû íà íå÷åòíûõ âðåìåííûõ ñëîÿõ ñìåùåíû íà ïîëøàãà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå. Ñòðóê- òóðà òàêîé ñåòêè èçîáðàæåíà íà ðèñ.1. Äëÿ ðåøåíèÿ óðàâíåíèÿ âûïîëíåíà äèñêðåòèçàöèÿ ïðîèçâîäíîé êîíå÷íîé ðàçíîñòüþ, à èíòåãðàë ðàññ÷èòàí ïî ôîðìóëå òðàïåöèé. 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Ó÷èòûâàÿ, ÷òî ïóòè èíòåãðèðîâà- íèÿ íà ðàçíûõ âðåìåííûõ èòåðàöèÿõ íàêëàäûâàþò- ñÿ äðóã íà äðóãà, ìîæíî èçáåæàòü ïîâòîðíûõ îïåðàöèé ñóììèðîâàíèÿ äëÿ âû÷èñëåíèÿ èíòåãðà- ëîâ è îïòèìèçèðîâàòü ÷èñëåííûé ðàñ÷åò. Äëÿ ýòîãî äîñòàòî÷íî çàïîìèíàòü âû÷èñëåííóþ èíòåãðàëü- íóþ ñóììó íà òåêóùåé èòåðàöèè è èñïîëüçîâàòü å¸ ïîçæå äëÿ âû÷èñëåíèÿ ñîîòâåòñòâóþùåãî èíòåãðà- ëà íà ñëåäóþùåì âðåìåííîì øàãå.   1  i+2  i+1  i  i-1  i-2  i-3  n-1  n-2  n-4  n-3  n Ðèñ.1. Ïðÿìîóãîëüíàÿ ïðîñòðàíñòâåííî-âðåìåííàÿ ñåòêà. Øàã ïî âðåìåíè âäâîå ìåíüøå ïðîñòðàí- ñòâåííîãî øàãà. Óçëû ñåòêè íà íå÷åòíûõ âðåìåííûõ ñëîÿõ ñìåùåíû íà ïîëøàãà ïî ïðîñòðàíñòâåííîé êîîðäèíàòå äëÿ ñîâìåùåíèÿ ñ ëèíèÿìè èíòåãðèðî- âàíèÿ 4. Òåñòèðîâàíèå àëãîðèòìà Äëÿ ïðîâåðêè òî÷íîñòè è ñòàáèëüíîñòè ïðåäëîæåí- íîãî ìåòîäà ïðîèçâåäåì ÷èñëåííûé ðàñ÷åò ñïåöè- àëüíîé òåñòîâîé çàäà÷è è ñðàâíèì ðåçóëüòàò ñ èçâåñòíûì òî÷íûì àíàëèòè÷åñêèì ðåøåíèåì. Ðàñ- ñìîòðèì ñëó÷àé, êîãäà âîçíèêàþùèé ñëîé, ïîñëå ñîçäàíèÿ, èìååò ñòàöèîíàðíûå ïàðàìåòðû ñðåäû, çíà÷èòåëüíî îòëè÷àþùèåñÿ îò ôîíîâîé ñðåäû, â ÷àñòíîñòè, äèýëåêòðè÷åñêàÿ ïðîíèöàåìîñòü ñðåäû â îáëàñòè ñëîÿ ðåçêî èçìåíÿåòñÿ â ìîìåíò âðåìåíè t0  îò  äî 1  . Ñóùåñòâóþùåå â ôîíîâîé ñðåäå ïîëå ÿâëÿåòñÿ ïëîñêîé âîëíîé è îïèñûâàåòñÿ ñîîò- íîøåíèåì     0 Ecos2L/  , ãäå  – äëèíà âîëíû â ôîíîâîé ñðåäå. Òî÷íîå ðåøåíèå äëÿ äàííîé çàäà÷è èçâåñòíî [5]. Ñëåäóþùèå çíà÷åíèÿ ïàðàìåò- ðîâ èñïîëüçîâàíû äëÿ òåñòèðîâàíèÿ àëãîðèòìà: 9  , 1 11  , íîðìèðîâàííàÿ ÷àñòîòà èñõîäíîé âîëíû 1  . Ñðàâíåíèå òî÷íîãî ðåøåíèÿ äàííîé çàäà÷è ñ ÷èñ- ëåííûìè ðåøåíèÿìè, ïîëó÷åííûìè ìåòîäàìè ÌÈÓ è FDTD, ïðåäñòàâëåíî íà ðèñ.2. Ðåçóëüòèðóþùåå ýëåêòðè÷åñêîå ïîëå â ñëîå ñîäåðæèò ñêà÷îê àìïëè- òóä, ïîÿâèâøèéñÿ âñëåäñòâèå îáðàçîâàíèÿ ñëîÿ. Íà ðèñ.3 ïðèâåäåíû ðåçóëüòàòû àïïðîêñèìàöèè ðåøå- íèÿ ìåòîäîì ÌÈÓ ïðè ðàçëè÷íûõ âåëè÷èíàõ øàãà äèñêðåòèçàöèè. Exact FDTD VIETD  1,5 1 0,5 1 0 -1  0,6 0,58 0,56 0,54 0,52 1,05 1 0,95 0,9 0,85 0,8 FDTD МИУВ Точное решение Ðèñ.2. Ñðàâíåíèå ÷èñëåííûõ ðåçóëüòàòîâ, ðàññ÷è- òàííûõ ìåòîäàìè FDTD è ÌÈÓÂ, ñ òî÷íûì àíàëè- òè÷åñêèì ðåøåíèåì òåñòîâîé çàäà÷è. Ðàçðûâ ïîëÿ âûçâàí ðåçêèì èçìåíåíèåì ïðîíèöàåìîñòè ñðåäû â íóëåâîé ìîìåíò âðåìåíè Exact 0,0005 0,001 0,01  0,65 0,6 0,55 0,5 1 0,9 0,8 Точное решение Ðèñ. 3. Ñðàâíåíèå ÷èñëåííûõ ðåçóëüòàòîâ ìåòîäà ÌÈÓÂ, ðàññ÷èòàííûõ ñ ðàçíûì øàãîì äèñêðåòèçà- öèè h, ñ òî÷íûì ðåøåíèåì çàäà÷è âáëèçè òî÷êè ðàçðûâà. Êà÷åñòâî àïïðîêñèìàöèè çàâèñèò îò âåëè- ÷èíû øàãà h Êàê âèäíî èç ïðèâåäåííûõ ãðàôèêîâ, ÌÈÓ ïðåâîñõîäèò ïî òî÷íîñòè ìåòîä FDTD, à òàêæå îáëàäàåò áîëåå âûñîêîé ñòàáèëüíîñòüþ. Äàæå ïðè âûáîðå äîâîëüíî ãðóáîãî øàãà ìîäåëèðîâàíèÿ, êàê ïîêàçàíî íà ðèñ. 3, ÷èñëåííîå ðåøåíèå, ïîëó÷åííîå ìåòîäîì ÌÈÓÂ, âñ¸ ðàâíî ñõîäèòñÿ ê òî÷íîìó ðåøåíèþ, íî ñîäåðæèò ìåíüøå äåòàëåé â îáëàñòè ñêà÷êà ïîëÿ. Ìåòîä æå FDTD ïðîèçâîäèò ðåçóëü- òàò, ñîäåðæàùèé óãàñàþùèé íåôèçè÷åñêèé ÷èñ- ëåííûé øóì (ñì. ðèñ. 2), âîçíèêàþùèé âñëåäñòâèå ÷èñëåííîãî äèôôåðåíöèðîâàíèÿ ïîëÿ â ðåçêî ìå- íÿþùåéñÿ ôóíêöèè ñðåäû. z+x{pUǿ'I. 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Ðàñïðîñòðàíåíèå ïëîñêîé ýëåêòðîìàãíèòíîé âîëíû â ñëîå ñ ìîäóëèðóåìîé âî âðåìåíè äèýëåêòðè÷åñêîé ïðîíèöàåìîñòüþ Ðàññìîòðèì ïðåîáðàçîâàíèå ïëîñêîé ýëåêòðîìàã- íèòíîé âîëíû ñ íîðìèðîâàííîé àìïëèòóäîé â äèýëåêòðè÷åñêîì ñëîå, äèýëåêòðè÷åñêàÿ ïðîíèöà- åìîñòü êîòîðîãî ìåíÿåòñÿ âî âðåìåíè ïîä âîçäåé- ñòâèåì ñòîðîííèõ ñèë ïî ãàðìîíè÷åñêîìó çàêîíó:   1 /11/9msin2  , ãäå L/  – íîðìèðî- âàííàÿ ÷àñòîòà ìîäóëÿöèè,  – äëèíà âîëíû óïðàâëÿþùåãî ïîëÿ è m – ãëóáèíà ìîäóëÿöèè. Èç- ìåíåíèå ñïåêòðà ñèãíàëà ïîñëå ïðîõîæäåíèÿ ñëîÿ ïðåäñòàâëåíî íà ðèñ. 4 è 5. Частота (THz) глубина - m=1/3 Нормированная частота модуляции  =0,5 0 100 200 300 400 А м п л и т у д а 0,0 0,2 0,4 0,6 0,8 1,0 Ðèñ. 4. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè èçìåíÿþùèìèñÿ âî âðåìåíè ïàðàìåòðàìè: Q=0,5 Äîïîëíèòåëüíûå êðàòíûå ãàðìîíèêè ïîÿâëÿþòñÿ â ñèãíàëå è ðàñòóò ñ óâåëè÷åíèåì ãëóáèíû ìîäóëÿ- öèè. ×àñòîòà äîïîëíèòåëüíûõ ãàðìîíèê çàâèñèò îò ÷àñòîòû ìîäóëÿöèè ñðåäû. Äëèòåëüíîñòü èíòåðâàëà ìîäåëèðîâàíèÿ max  ïðî- öåññà ðàâíà 100,0 (â áåçðàçìåðíûõ âåëè÷èíàõ, ÷òî ñîîòâåòñòâóåò 1 ps), øàã èíòåãðèðîâàíèÿ h0.01  . Äëÿ NX=100 è NT=10 000 ìèíèìàëüíî íåîáõîäèìûé îáú¸ì îïåðàòèâíîé ïàìÿòè ðàâåí 6 Ká, è îáùèé îáú¸ì ïàìÿòè äëÿ õðàíåíèÿ ðåçóëüòàòîâ – 8 Má. Âðåìÿ ìîäåëèðîâàíèÿ ðàâíî ~ 2,8 ñ íà ÏÊ ñ ïðîöåññîðîì Intel Celeron ñ òàêòîâîé ÷àñòîòîé 700 MÃö. 0,0 0,2 0,4 0,6 0,8 1,0 0 100 200 300 400 500 Частота (THz) А м п л и т у д а глубина - m=1/3 Нормированная частота модуляции  =1,25 Ðèñ. 5. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè èçìåíÿþùèìèñÿ âî âðåìåíè ïàðàìåòðàìè: Q=1,25 Ðàññìîòðèì ïîäîáíóþ çàäà÷ó, íî ñî ñðåäîé, ïðîíè- öàåìîñòü êîòîðîé ãàðìîíè÷åñêè ìåíÿåòñÿ â ïðî- ñòðàíñòâå è âî âðåìåíè â ñîîòâåòñòâèè ñ     1 /11/9msin2  . Ðåçóëüòàòû âëèÿ- íèÿ ñðåäû â äàííîì ñëó÷àå ïðåäñòàâëåíû íà ðèñ.6 (à, á): 0 100 200 300 400 600 Частота (THz) 0,0 0,2 0,4 0,6 0,8 700 500 глубина - m=1/3 Нормированная частота модуляции  =0,5 à 0,0 0,2 0,4 0,6 0,8 0 200 400 600 800 Частота (THz) 1200 1000 глубина - m=1/3 Нормированная частота модуляции  1,25 á Ðèñ. 7. Èçìåíåíèå ñïåêòðà ñèãíàëà ïîñëå âçàèìîäåé- ñòâèÿ ñî ñëîåì ñ ãàðìîíè÷åñêè ìîäóëèðóåìûìè âî âðåìåíè è â ïðîñòðàíñòâå ïàðàìåòðàìè  ñðàâíåíèè ñ ïðåäñòàâëåííûì ðàíåå ñëó÷àåì äëÿ ìîäóëèðóåìîé òîëüêî âî âðåìåíè ñðåäû â äàííîì ñëó÷àå êîëè÷åñòâî äîïîëíèòåëüíûõ ãàðìîíèê è èõ àìïëèòóäû çíà÷èòåëüíî âûøå. Ýòî ìîæåò áûòü îáúÿñíåíî äîïîëíèòåëüíîé ôàçîâîé ìîäóëÿöèåé ïîëÿ â êàæäîé òî÷êå ñëîÿ. 6. Çàêëþ÷åíèå Ïðåäñòàâëåí ÷èñëåííî-àíàëèòè÷åñêèé ïîäõîä äëÿ ìîäåëèðîâàíèÿ âçàèìîäåéñòâèÿ ýëåêòðîìàãíèòíî- ãî ñèãíàëà ïðîèçâîëüíîé ôîðìû ñ ïëîñêèì äèýëåê- òðè÷åñêèì ñëîåì, ïàðàìåòðû êîòîðîãî ìåíÿþòñÿ âî âðåìåíè è/èëè â ïðîñòðàíñòâå. 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Ïðèâåäåíû ðåçóëüòàòû ÷èñëåí- íîãî ìîäåëèðîâàíèÿ äëÿ ñëó÷àÿ ñ ãàðìîíè÷åñêè ìîäåëèðóåìîé âî âðåìåíè ñðåäîé, à òàêæå ñî ñðåäîé, ìîäåëèðóåìîé âî âðåìåíè è ïðîñòðàíñòâå. Ëèòåðàòóðà: 1 Hagness S.C., Joseph R.M. and Taflove A., Subpicosecond electrodynamics of distributed Bragg reflector microlasera: Results from finite difference time domain simulations, Radio Science. 1996. Vol. 31, No 4, P. 931-941. 2 F.R.Morgenthaler, Velocity Modulation of Electromagnetic Waves // IRE Trans. on Microwave Theory and Techniques. 1958. MTT-6. P. 167-172. 3 Íåðóõ À.Ã., Õèæíÿê Í.À. Ñîâðåìåííûå ïðîáëåìû íåñòàöèî- íàðíîé ìàêðîñêîïè÷åñêîé ýëåêòðîäèíàìèêè. Òåñò- Ðàäèî: Õàðüêîâ, 1991. 4 Taflove A.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, Boston, Artech House, 1995. 5 Nerukh A.G., Scherbatko I.V., Marciniak M., Electromagnetics of Modulated Media with Applications to Photonics, Nat. Institute of Telecommunic. Publishing House, Warsaw, 2001. 6 Fedotov F.V., Nerukh A.G., Benson T.M., Sewell Ph. Solution of non-stationary electrodynamics boundary value problem by FDTD and Volterra Integral Equation methods. Proc. Of 4 th International Conference on Transparent Optical Networks. April 24-25, 2002, Warsaw, Poland. Vol. 1. P. 180-183. Ïîñòóïèëà â ðåäêîëëåãèþ 24.02.2003 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Ñóõîèâàíîâ È.À. Ôåäîòîâ Ô¸äîð Âëàäèìèðîâè÷, àñïèðàíò êàôåäðû âûñ- øåé ìàòåìàòèêè ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ðàçðà- áîòêà ÷èñëåííûõ àëãîðèòìîâ è ñïåöèàëèçèðîâàííûõ ïðîãðàìì äëÿ ìîäåëèðîâàíèÿ çàäà÷ íåñòàöèîíàðíîé ýëåêòðîäèíàìèêè. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. +38 0572 409372. E-mail: ffedor@ukr.net Íåðóõ Àëåêñàíäð Ãåîðãèåâè÷, ä-ð ôèç.-ìàò. íàóê, ïðî- ôåññîð, çàâåäóþùèé êàôåäðîé âûñøåé ìàòåìàòèêè ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ìàòåìàòè÷åñêàÿ ôèçèêà, íåñòàöèîíàðíàÿ ýëåêòðîäèíàìèêà ñðåä ñ ìåíÿþùè- ìèñÿ âî âðåìåíè ñâîéñòâàìè è äâèæóùèìèñÿ ãðàíèöà- ìè, íîâûå àíàëèòè÷åñêèå è ÷èñëåííûå ìåòîäû ðåøå- íèÿ ýëåêòðîäèíàìè÷åñêèõ çàäà÷. Àäðåñ: Óêðàèíà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë. +38 0572 409372. E- mail: Nerukh@ddan.kharkov.ua ÓÄÊ 621.372 Î ÍÅÊÎÒÎÐÛÕ ÂÎÇÌÎÆÍÎÑÒßÕ ÔÎÐÌÈÐÎÂÀÍÈß ÀÌÏËÈÒÓÄÍÎ- ×ÀÑÒÎÒÍÎÉ ÕÀÐÀÊÒÅÐÈÑÒÈÊÈ Â ÎÁÚÅÌÍÛÕ ÐÅÇÎÍÀÒÎÐÀÕ ÑÒÎß×ÈÕ ÂÎËÍ ÁÎÍÄÀÐÅÍÊÎ È.Í. Íà îñíîâàíèè ïðåäñòàâëåíèÿ ïðîöåññà ôîðìèðîâàíèÿ àìïëèòóäíî-÷àñòîòíûõ õàðàêòåðèñòèê îáúåìíûõ ðå- çîíàòîðîâ ñòîÿ÷èõ âîëí êàê ñóïåðïîçèöèè áîëüøîãî ÷èñëà êîëåáàíèé âîçáóæäàþùåé ýëåêòðîìàãíèòíîé âîëíû ïðåäëàãàåòñÿ âîçäåéñòâîâàòü íà íåãî ñ ïîìî- ùüþ ïðîôèëèðîâàíèÿ ðàáî÷èõ ïîâåðõíîñòåé ðåçîíà- òîðà. Îñíîâíûìè ôàêòîðàìè, îïðåäåëÿþùèìè ÷àñòîòíóþ èçáèðàòåëüíîñòü ðàçëè÷íûõ óñòðîéñòâ, ÿâëÿþòñÿ àìïëèòóäíî-÷àñòîòíûå õàðàêòåðèñòèêè (À×Õ) èõ ýëåìåíòîâ. Ïðåäïîëàãàåòñÿ, ÷òî À×Õ îäèíî÷íîãî êîëåáàòåëüíîãî êîíòóðà ïîëíîñòüþ îïðåäåëÿåòñÿ åãî äîáðîòíîñòüþ, à À×Õ ñâÿçàííûõ êîíòóðîâ – äîáðîòíîñòÿìè, ÷àñòîòàìè íàñòðîéêè è êîýôôèöè- åíòàìè ñâÿçè ìåæäó êîíòóðàìè è âíåøíèìè öåïÿìè. Äëÿ ôîðìèðîâàíèÿ À×Õ çàäàííîé ôîðìû íåîáõî- äèìî, êàê ïðàâèëî, èñïîëüçîâàíèå íåñêîëüêèõ ñâÿçàííûõ êîíòóðîâ, ÷òî âåäåò ê óñëîæíåíèþ êîíñòðóêöèè óñòðîéñòâà â öåëîì, íåîáõîäèìîñòè ââåäåíèÿ äîïîëíèòåëüíûõ ðåãóëèðîâîê è íàñòðîåê, à òàêæå ê ñíèæåíèþ äîáðîòíîñòè ðåçîíàíñíûõ ýëåìåíòîâ è, ñîîòâåòñòâåííî, ê óìåíüøåíèþ êðó- òèçíû ñêàòîâ À×Õ.  ýòîé ñâÿçè ïðåäñòàâëÿåòñÿ àêòóàëüíîé ðàçðàáîòêà ìåòîäîâ ôîðìèðîâàíèÿ À×Õ çàäàííîé ôîðìû ïó- òåì âîçäåéñòâèÿ íà ðåçîíàíñíóþ êðèâóþ îäèíî÷- íîãî êîëåáàòåëüíîãî êîíòóðà. Öåëüþ ðàáîòû ÿâëÿåòñÿ îáîñíîâàíèå ìåòîäà âîç- äåéñòâèÿ íà À×Õ îäèíî÷íîãî îáúåìíîãî êîëåáà- òåëüíîãî êîíòóðà ñ ïîìîùüþ ïðîôèëèðîâàíèÿ åãî ðàáî÷èõ ïîâåðõíîñòåé. Ñóùíîñòü ïðåäëàãàåìîãî ìåòîäà çàêëþ÷àåòñÿ â òîì, ÷òî ïðîöåññ ôîðìèðîâàíèÿ À×Õ â îáúåìíîì ðåçîíà- òîðå ðàññìàòðèâàåòñÿ êàê ñóïåðïîçèöèÿ áîëüøîãî ÷èñëà êîëåáàíèé âîçáóæäàþùåé ýëåêòðîìàãíèòíîé âîëíû, óñëîâèÿ ñëîæåíèÿ êîòîðûõ ñóùåñòâåííî (â ïðåäåëàõ ïîëîñû ÷àñòîò, çàíèìàåìîé À×Õ) çàâèñÿò îò õàðàêòåðèñòèê îòðàæàþùèõ ïîâåðõíîñòåé. Èçâåñòíî, ÷òî ñîáñòâåííóþ äîáðîòíîñòü îáúåìíîãî ðåçîíàòîðà ñ âîçäóøíûì èëè âàêóóìíûì çàïîëíå- íèåì è ñòåíêàìè èç õîðîøî ïðîâîäÿùåãî ìàòåðèàëà ìîæíî íàéòè ñ ïîìîùüþ ñëåäóþùåãî ñîîòíîøåíèÿ [1,2]: п S 2 V 2 п 0 р S 2 0 V 2 0 р 0 R G ds | H | dv | H | R ds | H | 2 dv | H | Q                    ,(1) ãäå p  – ðåçîíàíñíàÿ ÷àñòîòà; 0  – ìàãíèòíàÿ ïðîíèöàåìîñòü âàêóóìà;  – ïðîâîäèìîñòü ìàòå- ðèàëà ñòåíîê ðåçîíàòîðà;     2 R 0 р п – ïîâåð- õíîñòíîå ñîïðîòèâëåíèå ìàòåðèàëà ñòåíîê ðåçîíà- òîðà;         S 2 V 2 0 р ds | H | / dv | H | G – ãåîìåòðè- ÷åñêèé ôàêòîð, çàâèñÿùèé îò ãåîìåòðèè è ðàçìåðîâ ðåçîíàòîðà, à òàêæå îò ñòðóêòóðû ýëåêòðîìàãíèò- íûõ ïîëåé (âèäà êîëåáàíèé). Àíàëèçèðóÿ (1) ïðè       р , 0 <   < p   è   << p  ( p   – ïîëóøèðèíà ïîëîñû ïðîïóñêà- íèÿ ðåçîíàòîðà), ìîæíî ïðèéòè ê âûâîäó, ÷òî çíà÷åíèÿ äîáðîòíîñòè îáúåìíîãî ðåçîíàòîðà íà ÷àñòîòàõ, ëåæàùèõ â ïîëîñå ïðîïóñêàíèÿ, íå îòëè- ÷àþòñÿ îò åãî äîáðîòíîñòè íà ðåçîíàíñíîé ÷àñòîòå. 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M 3b~7|\ӧ\_(LtKP/AWA3g}9iem[2iʞFmoҶj@wgH$:쩌`?"'mF + ůQ<6.sޥ|2go#ѓ)5`wd~l֯?tU(263ZފCgod}kFWL= IJ?IzIn5c@,H {\+4`}্gܠKӈ!|}Dctr&`1M/.zP=+Z+TVAhQq% =+wK|XG>ՅdxsU>|(mG=T6A̿|Yy&䨌k+DB$zܹ`1}_f ~ৎRBAlV^{_T}o'%w85mR7:K[i?~ U9Nz2-Q>&mhq.u%}(#*wpx6~uK]O7{9 {%}enQ۔%k*~;NyIq i3 GLiZvɇ|v 5Uw*!)WR=OJ!Z1qQ%TgsbהeEF{^!?Fme#gpCt;<ĴrqSxvUxVӛj}yz6cnb,_dYq!q27?? ԝv6d_$- oMOY_C}gCwKxzȞi7SErz6I':ýz#%)Y,rJAR׈p0wn|o L;Ք0~|# t~8"?ukXkxV筊Mow| eAA qEZ~0aCVg iW q7ծ6eKLtqЄ[Y!AUMT6AئI,?{yN̎WN^ٯTn mp6 vOpc~_# ?p! Ӟ3 ycݸ\'ҡŜf>R#7 {<?Fi6,,m16 ÐÈ, 2003, ¹ 4 Äåéñòâèòåëüíî, âåëè÷èíà G ïðàêòè÷åñêè íå ìåíÿ- åòñÿ, à çíà÷åíèå ïîâåðõíîñòíîãî ñîïðîòèâëåíèÿ ñëàáî çàâèñèò îò ÷àñòîòû ( 2 / 1 п R   – äëÿ íîð- ìàëüíûõ ìåòàëëîâ è 2 п R   – äëÿ ñâåðõïðîâîä- íèêîâ [2]). Ñëåäîâàòåëüíî, âûðàæåíèå 2 / 1 2 ) x 1 ( ) x ( f    ( р р 2 Q x          ), ïîëó÷åííîå èç ðàññìîò- ðåíèÿ óñëîâèé ðåçîíàíñà â êîëåáàòåëüíîì êîíòóðå ñ ñîñðåäîòî÷åííûìè ïàðàìåòðàìè (R, C, L), ïðè êîòîðûõ ôîðìà À×Õ îïðåäåëÿåòñÿ âåëè÷èíîé è õàðàêòåðîì èçìåíåíèÿ ðåàêòèâíûõ ñîïðîòèâëåíèé (  L è 1/  Ñ), â äàííîì ñëó÷àå íå ñîâñåì ïðèìå- íèìî. Ïîñêîëüêó áîëüøèíñòâî îáúåìíûõ ðåçîíàòîðîâ ÿâëÿþòñÿ ðåçîíàòîðàìè ñòîÿ÷èõ âîëí, òî, ïî-âèäè- ìîìó, èõ ðåçîíàíñíàÿ õàðàêòåðèñòèêà áóäåò çàâèñåòü îò óñëîâèé èíòåðôåðåíöèè êîëåáàíèé íà ðåçîíàíñ- íîé ÷àñòîòå è ÷àñòîòàõ, ëåæàùèõ âáëèçè íå¸. Ðàññìîòðèì óñëîâèÿ ðåçîíàíñà â îäíîìåðíîì ðåçî- íàòîðå ñòîÿ÷èõ âîëí (èç îáúåìíûõ ðåçîíàòîðîâ ê ýòîìó òèïó ìîæíî îòíåñòè, íàïðèìåð, ðåçîíàòîðû íà îòðåçêàõ êîàêñèàëüíûõ ëèíèé, âîçáóæäàåìûõ íà âîëíàõ ÒÅÌ). Ïðåäïîëîæèì, ÷òî â ðåçîíàòîðå îòñóòñòâóþò ïîòå- ðè, à ñàì ðåçîíàòîð ïðåäñòàâëÿåò ñîáîé ëèíèþ, çàêðûòóþ íà êîíöàõ òîðöåâûìè êðûøêàìè. Óñëîâèå ðåçîíàíñà äëÿ òàêîé ñèñòåìû: n 2 l 2 2 1        , (2) ãäå 1  , 2  – èçìåíåíèå ôàçû ñèãíàëà ïðè îòðàæå- íèè îò òîðöåâûõ ñòåíîê ðåçîíàòîðà; фл л v 2       – ïîñòîÿííàÿ ðàñïðîñòðàíåíèÿ ëèíèè;  ë , v ôë – äëèíà è ôàçîâàÿ ñêîðîñòü âîëíû â ëèíèè; l – äëèíà ëèíèè; n = 1,2,3,… .... Ïðè çåðêàëüíîì îòðàæåíèè îò òîðöåâûõ ñòåíîê óñëîâèå ðåçîíàíñà – l = n  ë /2.  äàëüíåéøåì äëÿ ïðîñòîòû áóäåì ðàññìàòðèâàòü ïîëóâîëíîâûé ðåçîíàòîð (l=  ë /2). Äîïóñòèì, ÷òî íà âõîäå ðåçîíàòîðà äåéñòâóåò ñèãíàë ñ ÷àñòîòîé  . Ïðè p    çà âðåìÿ t = T (T – ïåðèîä êîëåáàíèÿ) ýëåêòðîìàãíèòíàÿ âîëíà äâàæäû ïðîé- äåò ïî ðåçîíàòîðó ñ îòðàæåíèåì îò îáåèõ ñòåíîê è ñëîæèòñÿ â ôàçå ñî ñëåäóþùèì êîëåáàíèåì, ïîñòó- ïèâøèì â ðåçîíàòîð.  ïîñëåäóþùèé ïåðèîä ïðî- èçîéäåò ñëîæåíèå óæå óäâîåííîãî çíà÷åíèÿ ñ âíîâü ïîñòóïèâøèì êîëåáàíèåì, è ò.ä. Ïðè p    êàæ- äîå ïîñëåäóþùåå êîëåáàíèå áóäåò àëãåáðàè÷åñêè ñêëàäûâàòüñÿ ñ ïðåäûäóùèìè, ïîñêîëüêó íàáåã ôàçû çà ïåðèîä â ýòîì ñëó÷àå áóäåò îïðåäåëÿòüñÿ îòíîøåíèåì 2l/  , à äëÿ êàæäîé ñîñòàâëÿþùåé – 2nl/  , ãäå n – ñîîòâåòñòâóþùåå êîëè÷åñòâî êîëå- áàíèé. Äëÿ ñèñòåìû áåç ïîòåðü ôóíêöèÿ, îïèñûâàþùàÿ õàðàêòåð èçìåíåíèÿ n-ãî êîëåáàíèÿ â ðåçîíàòîðå, l n 2 j e ) f , t ( S    ; ðåàëüíàÿ ÷àñòü – l n 2 cos ) f , t ( S Re    . Ôóíêöèÿ, îïèñûâàþùàÿ ðåçîíàíñíóþ êðèâóþ, êîòîðàÿ ôîðìèðóåòñÿ çà ñ÷åò ñëîæåíèÿ êîëåáàíèé, ïîñòóïàþùèõ è ñóùåñòâóþùèõ â ïîëóâîëíîâîì ðåçîíàòîðå, ìîæåò áûòü çàïèñàíà ñëåäóþùèì îáðà- çîì:                 n 1 k n 1 k р n 1 k n 1 k р f kf 2 cos k 2 cos f , t S . (3) Òèïè÷íûé ãðàôèê àáñîëþòíûõ çíà÷åíèé ôóíêöèè S(t, f) (3) ïðèâåäåí íà ðèñóíêå. Ïðè óâåëè÷åíèè ÷èñëà n àìïëèòóäà öåíòðàëüíîãî ìàêñèìóìà áóäåò ðàñòè, åãî øèðèíà - óìåíüøàòüñÿ, àìïëèòóäû áîêîâûõ ìàêñèìóìîâ òàêæå áóäóò óìåíüøàòüñÿ, à èõ êîëè÷åñòâî - ðàñòè.  êà÷åñòâå ðåçîíàíñíîé êðèâîé â äàííîì ñëó÷àå ñëåäóåò, ïî-âèäèìîìó, ïðèíÿòü îãèáàþùóþ ãðàôèêà ôóíêöèè S(t,f). Âëèÿíèå àêòèâíûõ ïîòåðü â ðåçîíàòîðå íà ôîðìó åãî ðåçîíàíñíîé êðèâîé ìîæíî ó÷åñòü ïóòåì ââå- äåíèÿ ïîñòîÿííîé çàòóõàíèÿ a, êîòîðàÿ õàðàêòåðè- çóåò ïîòåðè â ñèãíàëå çà âðåìÿ îäíîãî êîëåáàíèÿ. Òîãäà (3) òðàíñôîðìèðóåòñÿ ê òàêîìó âèäó:       . f kf 2 cos e k 2 cos e f , t S n 1 k р n 1 k k 1 n n 1 k р n 1 k k 1 n                        (4) Õàðàêòåð èçìåíåíèÿ ôóíêöèè S(t, f) (4) ïî ñðàâíå- íèþ ñ S(t,f) (3) îñòàåòñÿ òåì æå, ñ òîé ëèøü ðàçíèöåé, ÷òî àìïëèòóäû ìàêñèìóìîâ áóäóò ñóùå- ñòâåííî çàâèñåòü îò âåëè÷èíû a, à ïðè áîëüøèõ åå çíà÷åíèÿõ ôîðìà îãèáàþùåé áóäåò ñëàáî ìåíÿòüñÿ ïðè âîçðàñòàíèè n.  S(t,f)  f р f Òèïè÷íûé ãðàôèê ôóíêöèè |S(t, f)| Ïî-âèäèìîìó, îêîí÷àòåëüíî ñôîðìèðîâàâøåéñÿ ðåçîíàíñíóþ õàðàêòåðèñòèêó ìîæíî ñ÷èòàòü òîãäà, êîãäà ìîùíîñòü ïîòåðü â ðåçîíàòîðå çà ïåðèîä ñòàíåò ðàâíîé ìîùíîñòè ïîñòóïàþùåãî âíåøíåãî ñèãíàëà, ò. å. ïîñëå îêîí÷àíèÿ ïåðåõîäíûõ ïðîöåñ- ñîâ óñòàíîâëåíèÿ êîëåáàíèé â ðåçîíàòîðå. 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Çàêîí ðàñïðåäåëåíèÿ ïëîòíîñòè âåðîÿòíîñòè äëÿ ðàçìåðîâ ðåçîíàòîðà â ïðèíöèïå ìîæåò èìåòü íå îäèí, à íåñêîëüêî ìàêñèìóìîâ. Ýòî ïðèâåäåò ê òîìó, ÷òî ãëàâíûé ìàêñèìóì À×Õ ðåçîíàòîðà, êîòîðûé, ñîáñòâåííî, ãëàâíûì îáðàçîì è õàðàêòå- ðèçóåò ðåçîíàíñíóþ êðèâóþ, áóäåò ðàñøèðÿòüñÿ èëè äàæå ïðèîáðåòàòü äîïîëíèòåëüíûå ìàêñèìó- ìû, ñòåïåíü âûðàæåííîñòè êîòîðûõ çàâèñèò îò ñîîòâåòñòâóþùåé ôóíêöèè ðàñïðåäåëåíèÿ ïëîòíî- ñòè âåðîÿòíîñòåé (ò.å. îò êîëè÷åñòâà è ñòåïåíè âûðàæåííîñòè åå ìîä). Ñëåäîâàòåëüíî, åñëè èñêóññòâåííûì îáðàçîì( íà- ïðèìåð, îáðàáîòêîé) ñîçäàòü óñëîâèÿ äëÿ ôîðìèðî- âàíèÿ ôóíêöèè ðàñïðåäåëåíèÿ ñ íåñêîëüêèìè ìàê- ñèìóìàìè, òî ìîæíî, ñîîòâåòñòâåííî, èçìåíèòü è ôîðìó À×Õ ðåçîíàòîðà. Äëÿ îäíîìåðíîãî ðåçîíàòîðà ñòîÿ÷èõ âîëí ýòîãî ìîæíî äîñòè÷ü ïðîôèëèðîâàíèåì òîðöåâîé ïîâåð- õíîñòè, ïðè÷åì ðàçíîñòü âûñîò ïðîôèëÿ äîëæíà ïðåâûøàòü ñðåäíèé ðàçìåð øåðîõîâàòîñòè, îïðåäå- ëÿåìûé êëàññîì ÷èñòîòû îáðàáîòêè ïîâåðõíîñòè, è íå ïðåâûøàòü âåëè÷èíû, êîòîðàÿ ìîæåò ïðèâåñòè ê âîçíèêíîâåíèþ â ðåçîíàòîðå íåæåëàòåëüíûõ âèäîâ êîëåáàíèé. Ïëîùàäü ïîâåðõíîñòè, çàíèìàå- ìàÿ òåì èëè èíûì ðàçìåðîì ïðîôèëÿ, áóäåò îïðå- äåëÿòüñÿ ñòåïåíüþ òðåáóåìîãî âûäåëåíèÿ òîãî èëè èíîãî ìàêñèìóìà â À×Õ ðåçîíàòîðà. Íàèáîëåå ÷àñòî èñïîëüçóåìûå ðåçîíàòîðû ñòîÿ÷èõ âîëí ïðåäñòàâëÿþò ñîáîé, êàê ïðàâèëî, ïîëóâîëíî- âûå çàìêíóòûå îòðåçêè ïðÿìîóãîëüíûõ èëè êðóã- ëûõ âîëíîâîäîâ.  òàêèõ ðåçîíàíñíûõ ñèñòåìàõ çíà÷åíèå ðåçîíàíñíîé ÷àñòîòû áóäåò îïðåäåëÿòüñÿ â îáùåì ñëó÷àå âñåìè ðàçìåðàìè ðåçîíàòîðà (äëÿ ïðèçìàòè÷åñêîãî ðåçîíàòîðà: ðàçìåðîì øèðîêîé ñòåíêè – a, óçêîé – b è äëèíîé – l; äëÿ öèëèíä- ðè÷åñêîãî: ðàäèóñîì – R è äëèíîé – l). 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Ïðåäëîæåííûå â ðàáîòå ïîäõîäû ïîçâîëÿþò: áîëåå äåòàëüíî ðàññìîòðåòü ïåðåõîäíûå ïðîöåññû â ðåçî- íàòîðàõ ñòîÿ÷èõ âîëí è ñâÿçàòü èõ ñ ôîðìèðîâàíèåì À×Õ, èçó÷èòü õàðàêòåð èçìåíåíèÿ çà âðåìÿ ïåðåõîä- íûõ ïðîöåññîâ ñèãíàëà, ïðîõîäÿùåãî ÷åðåç âûñîêî- äîáðîòíûé ðåçîíàòîð, ðàçðàáîòàòü ìåòîäû ñèíòåçà ôèëüòðóþùèõ óñòðîéñòâ íà îñíîâå îäèíî÷íûõ âû- ñîêîäîáðîòíûõ ðåçîíàòîðîâ ñ çàäàííûìè À×Õ. Ñóùåñòâóþùèå â íàñòîÿùåå âðåìÿ ôèëüòðû ÑÂ× ñîçäàþòñÿ íà îñíîâå ïðèíöèïîâ, èçëîæåííûõ â ðàáî- òàõ [7-9], è ÿâëÿþòñÿ, êàê ïðàâèëî, ìíîãîýëåìåíòíûå è, ñîîòâåòñòâåííî, èìåþò âñå íåäîñòàòêè, óïîìÿíóòûå â íà÷àëå ñòàòüè. Ïðåäëîæåííûé ïóòü ïîñòðîåíèÿ óñòðîéñòâ ñ çàäàííûìè À×Õ âåäåò ê çíà÷èòåëüíîìó óïðîùåíèþ èõ êîíñòðóêöèè è óëó÷øåíèþ îñíîâíûõ õàðàêòåðèñòèê ïî èçáèðàòåëüíîñòè çà ñ÷åò ñîõðàíåíèÿ âûñîêîé äîáðîòíîñòè íà ðàáî÷èõ ÷àñòîòàõ. Òàêèì îáðàçîì, ïðîâåäåííîå â ðàáîòå ðàññìîòðåíèå ïðîöåññà ôîðìèðîâàíèÿ À×Õ â ðåçîíàòîðàõ ñòîÿ÷èõ âîëí ïîçâîëÿåò âûÿâèòü äîïîëíèòåëüíûå âîçìîæíî- ñòè äëÿ àíàëèçà ïðîöåññà óñòàíîâëåíèÿ êîëåáàíèé â âûñîêîäîáðîòíûõ ðåçîíàòîðàõ è ñèíòåçà èçáèðàòåëü- íûõ óñòðîéñòâ ñ óëó÷øåííûìè õàðàêòåðèñòèêàìè. Ëèòåðàòóðà: 1. Ôåäîðîâ È.Í. Îñíîâû ýëåêòðîäèíàìèêè. Ì.: Âûñø. øê., 1980. 399ñ. 2. Ìåíäå Ô.Ô., Ñïèöûí À.È. Ïîâåðõíîñòíûé èìïåäàíñ ñâåðõïðîâîäíèêîâ. Ê.: Íàóê. äóìêà, 1985. 240ñ. 3. Àëüòìàí Äæ. Óñòðîéñòâà ñâåðõâû- ñîêèõ ÷àñòîò. Ì.: Ìèð, 1968. 487ñ. 4. Ëåáåäåâ È.Â. Òåõíèêà è ïðèáîðû ÑÂ×. Ò.1. Ì.: Âûñø. øê., 1970. 440ñ. 5. Áàðàíîâñêèé Ì.À., Ìîëî÷êîâ À.Â. Ñïðàâî÷íèê òîêàðÿ. Ìèíñê: Ãîñèçäàò ÁÑÑÐ,1962. 492ñ. 6. Ùèãîëåâ Á.Ì. Ìàòåìàòè÷åñêàÿ îáðàáîòêà íàáëþäåíèé. Ì.: ÃÈÔÌË, 1962. 344ñ. 7. Ìîäåëü À.Ì. Ôèëüòðû ÑÂ× â ðàäèîðåëåé- íûõ ñèñòåìàõ. Ì.: Ñâÿçü, 1967. 352ñ. 8. Ìàòòåé Ä.Ë., ßíã Ë., Äæîíñ Å.Ì.Ò. Ôèëüòðû ÑÂ×, ñîãëàñóþùèå öåïè è öåïè ñâÿçè. Ò.1. Ì.: Ñâÿçü, 1971. 440ñ. 9. Ìàòòåé Ä.Ë., ßíã Ë., Äæîíñ Å.Ì.Ò. Ôèëüòðû ÑÂ×, ñîãëàñóþùèå öåïè è öåïè ñâÿçè. Ò.2. Ì.: Ñâÿçü, 1971. 496ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 24.02.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê Ìåíäå Ô.Ô. Áîíäàðåíêî Èãîðü Íèêîëàåâè÷, êàíä. òåõí. íàóê, äî- öåíò êàôåäðû àâèàöèîííûõ ñðåäñòâ ñâÿçè Õàðüêîâñêî- ãî èíñòèòóòà ÂÂÑ. Íàó÷íûå èíòåðåñû: êðèîãåííàÿ ðàäèîôèçèêà, òåõíèêà ÑÂ×, ñèñòåìû ðàäèîñâÿçè. Àä- ðåñ: Óêðàèíà, 61165, Õàðüêîâ, óë. Êëî÷êîâñêàÿ, 228, êàô. ¹306, òåë. 30-82-16. ÓÄÊ 517.958:537.8 ÐÀÑÑÅßÍÈÅ ÏËÎÑÊÎÉ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÉ ÂÎËÍÛ ÍÅÇÀÌÊÍÓÒÛÌ ÊÎÍÈ×ÅÑÊÈÌ ÎÒÐÀÆÀÒÅËÅÌ ÄÎÐÎØÅÍÊÎ Â.À., ÑÅÌÅÍÎÂÀ Å.Ê. Ðàññìàòðèâàåòñÿ çàäà÷à äèôðàêöèè ïëîñêîé ýëåêòðî- ìàãíèòíîé âîëíû íà êîíóñå ñ ïåðèîäè÷åñêèìè ïðî- äîëüíûìè ùåëÿìè. Îïèñûâàåòñÿ ÷èñëåííîå ðåøåíèå ýòîé çàäà÷è è ñòðîÿòñÿ äèàãðàììû ðàññåÿíèÿ â ãîðè- çîíòàëüíîé ïëîñêîñòè. Èçó÷àåòñÿ âëèÿíèå ùåëåé íà îñíîâíûå õàðàêòåðèñòèêè. 1. Ââåäåíèå Íåîäíîðîäíûå êîíè÷åñêèå ñòðóêòóðû ïðèìåíÿþò- ñÿ íå òîëüêî â àíòåííîé òåõíèêå, íî è â ðàäèîëî- êàöèè. Íà èõ îñíîâå ìîãóò ñîçäàâàòüñÿ îòðàæàòåëè ñ îïðåäåëåííûìè ñâîéñòâàìè [1, 2]. Ðàññìàòðèâà- åìàÿ êîíè÷åñêàÿ ïîâåðõíîñòü èìååò íåîäíîðîäíî- ñòè â âèäå ïðîäîëüíûõ ùåëåé è ÿâëÿåòñÿ ìîäåëüþ êîíè÷åñêîãî îòðàæàòåëÿ ñ óïðàâëÿåìûìè õàðàêòå- ðèñòèêàìè â çàâèñèìîñòè îò óãëà ðàñêðûâà êîíóñà, ðàçìåðîâ ùåëåé è èõ êîëè÷åñòâà. Ðåçóëüòàòû òåî- ðåòè÷åñêèõ èññëåäîâàíèé ãðàíè÷íûõ ýëåêòðîäèíà- ìè÷åñêèõ çàäà÷ äëÿ ñîîòâåòñòâóþùèõ ñòðóêòóð íåîáõîäèìû íà ñòàäèè ïðîåêòèðîâàíèÿ è ðàçðàáîò- êè ñèñòåì, êîòîðûå ñîäåðæàò ýòè ñòðóêòóðû â êà÷åñòâå ýëåìåíòîâ.  ðàáîòàõ [3, 4] ïðåäëîæåí è ðàçâèò ïîäõîä äëÿ ðåøåíèÿ çàäà÷ ðàññåÿíèÿ ïîëÿ ñîñðåäîòî÷åííîãî èñòî÷íèêà íà êîíóñå ñ ïðîäîëüíûìè ùåëÿìè, à â [5] íàéäåíî àíàëèòè÷åñêîå ðåøåíèå çàäà÷è äèôðàêöèè ïëîñêîé ýëåêòðîìàãíèòíîé âîëíû íà êîíóñå ñ ïðîäîëüíûìè ùåëÿìè â ÷àñòíûõ ñëó÷àÿõ áîëüøîãî ÷èñëà ùåëåé; êîíóñà ñ óçêèìè ùåëÿìè è óçêèõ êîíè÷åñêèõ ñåêòîðîâ (ëåíò). Íà îñíîâå ïîëó÷åííî- ãî àñèìïòîòè÷åñêîãî ðåøåíèÿ ïðîâåäåí êà÷åñòâåí- íûé àíàëèç ñïåêòðà ãðàíè÷íîé çàäà÷è, ñòðóêòóðû ðàññåÿííîãî ïîëÿ è åãî ïîâåäåíèÿ âáëèçè âåðøèíû êîíóñà â ýòèõ ïðåäåëüíûõ ñëó÷àÿõ. Öåëüþ äàííîé ðàáîòû ÿâëÿåòñÿ ÷èñëåííîå ðåøåíèå çàäà÷è ðàññåÿíèÿ ïîëÿ ïëîñêîé âîëíû êîíóñîì ñ ïðîäîëüíûìè ùåëÿìè ïðîèçâîëüíûõ ðàçìåðîâ è èçó÷åíèå èõ âëèÿíèÿ íà îñíîâíûå õàðàêòåðèñòèêè ðàññåÿíèÿ. 2. Ïîñòàíîâêà çàäà÷è è ìåòîä ðåøåíèÿ Ïëîñêàÿ ýëåêòðîìàãíèòíàÿ âîëíà, ïîëå êîòîðîé èçìåíÿåòñÿ ïî ãàðìîíè÷åñêîìó çàêîíó it e  , ïàäàåò íà íåçàìêíóòóþ êîíè÷åñêóþ ñòðóêòóðó (ðèñ. 1) è ðàñïðîñòðàíÿåòñÿ âäîëü åå îñè. Ðàññìàòðèâàåìàÿ êîíè÷åñêàÿ ñòðóêòóðà ïðåäñòàâëÿåò ñîáîé ïîëóáåñ- êîíå÷íûé èäåàëüíî ïðîâîäÿùèé êðóãîâîé êîíóñ ñ ïåðèîäè÷åñêè ïðîðåçàííûìè âäîëü îáðàçóþùèõ N ùåëÿìè. Îáîçíà÷èì ÷åðåç 2  óãîë ðàñêðûâà êîíóñà, d –óãëîâóþ øèðèíó ùåëåé, l2/N  – ïåðèîä êî- íè÷åñêîé ñòðóêòóðû. 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Tfϸq\Eãw17+ӿUOlyB_g@&幎?_o_^Kg;RxyծuCO{QkS>lU|Nht0>5mz_?kwjw3]y|+N?T3>_Q9Y9]ب5lj\ie͏^YJSޮM!h_7.=G;vBk( j֣5݇u%+7|GW"|Ds[wg޴U*۾z[?yLmRykB֋iwZNc7MtEI `dK(}bmpO;eh }nt-{D .P&y+^4I?M1sKa^A8~= nIG/W<~^ie[ no{!t)=/ 5+Npo?j9G~~gP_!Nqx|Qqbq`ԧ=M?O9Bp:=S>~C^F"*bleE;Ny'(/ VaÙ '<<ٝ_Sxԯ2 L&+q0 W O_)~"~}BrÂɿm\1Qoڿzg Ưh:G 㼳:3Cx}6o?igʂ)Vy|@HP>6ȟ-S>4(ƒ0|I.}p! }T}k;}m3ex &|sbK*>`-i] ߝ[χE'?zӾsO͌1ۙq8 eoλןT|ϻ7<~M~Jg{Mge>)D$m~'=5E'{蟧|k;BѺsNǿ$zCos޾U}SK {INodxEO(H _3c/gt?E *tc1}ǂ|eC C[9no|g#qY~ꊔwdw/7Kݻ:^0nLSy?|[u lSac!>>9!߇Yk0s@~xe܇f+>l܇}R s  >LrV踂챬هRx(I#!C\߷}ŸҶf1cio>RDapÞR[ Jׄ,F}XxGp w}XP؇湖3 wyw o|(Sd`*0}Ewe=R߇a?=ݺw]\]GSʛ3ϔ}zgN76zC{,%؇VPaȹ/_3e}q:(z_0zya[lM@>yhz^$zo74þpaNw*+d<dz#̇+VwS~plb{xi)qu)؇0>7߇QIBJl,9xطMwa,,520 ÐÈ, 2003, ¹ 4 |mnN| (j) (j)i(mnN) 0 m mnN j1 n |mnN| 0 j1 P(cos) Ue d P(cos) d              ,(12) ãäå m 1/2i P(cos)   – ïðèñîåäèíåííàÿ ôóíêöèÿ Ëå- æàíäðà ïåðâîãî ðîäà, à íåèçâåñòíûå êîýôôèöèåí- òû (j) p x è (j) p  ñâÿçàíû ñîîòíîøåíèåì (j)(j) pp i1/2 limx   . Âåðõíèå çíàêè â (10), (12) ñîîòâåòñòâóþò îáëàñòè 0  , à íèæíèå –  . Òàêèì îáðàçîì, äëÿ îïðåäåëåíèÿ ïîòåíöèàëîâ Äå- áàÿ (4), (8)-(12) òðåáóåòñÿ íàéòè èõ êîýôôèöèåíòû Ôóðüå (j) p x . Ïàðíûå ñóììàòîðíûå óðàâíåíèÿ äëÿ íèõ ïîëó÷àþòñÿ â ðåçóëüòàòå ïðèìåíåíèÿ êðàåâîãî óñëîâèÿ (3) íà êîíè÷åñêèõ ñåêòîðàõ è óñëîâèÿ íåïðåðûâíîñòè ïîëÿ â ùåëÿõ, è èìåþò âèä:   0 j imN inN m,n n xee       , dlN  , (13)   (j) (j)(j)inN m,nm,n n n N(n)(1)xe0 n       , (14) Ndl  , (j)(j) m,n n [N(n)](1) n   (n)Nj1 1(j) (1)ch(1/2i(n)N) (1/2i(n)N) (sin)       j1j1 (n)N(n)N 1/2i1/2i j1j1 1 , dd P(cos)P(cos) dd        (15) 0 m m N  , 1/21/2  , j1 (j)(1)   , 0 m – áëè- æàéøåå ê m/N öåëîå ÷èñëî; (z)  ãàììà-ôóíê- öèÿ. Êîýôôèöèåíòû ñèñòåìû (j) m,n  îïðåäåëåíû â (15), íå çàâèñÿò îò âîëíîâîãî ÷èñëà k è ïðè (n)N1  èìåþò îöåíêó (j) m,n 22 1 O (n)N       . Íåçàâèñèìîñòü (j) m,n  îò k âëå÷åò çà ñîáîé íåçàâèñè- ìîñòü îò ÷àñòîòíîãî ïàðàìåòðà è íåèçâåñòíûõ êî- ýôôèöèåíòîâ (j) m,n x , ÷òî óäîáíî äëÿ ïîñòðîåíèÿ äèàãðàìì ðàññåÿíèÿ. Ïîñêîëüêó ñèñòåìà (13), (14) ïåðâîãî ðîäà, òî îíà ìàëî ïðèãîäíà äëÿ íàõîæäåíèÿ (j) m,n x . Îäíàêî, èñïîëüçóÿ àëãîðèòì ðåãóëÿðèçàöèè, áàçèðóþùèéñÿ íà ïðèìåíåíèè ìåòîäà çàäà÷è Ðèìà- íà-Ãèëüáåðòà, ñèñòåìó (13), (14) óäàåòñÿ ñâåñòè ê ñèñòåìå ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé (ÑËÀÓ) âòîðîãî ðîäà ôðåäãîëüìîâñêîãî òèïà. Ïðè- âåäåì ÑËÀÓ äëÿ (2) m,n x : (2) m 0 m m,0 0 B(u)xDV(u)   m (2)(2)p 0 m,ppm,p p |p| (x)V(u) p     , (16) mm1 (2) 00 m m,00n1 0 xDV(u)    p1 m (2)(2) 0 m,ppm,p n1 p |p| (x)V(u) p       (2)(2) 1 nn1 0m,0 x[P(u)V(u)]    , n0  , (17) 1 1 2P(u) 1 B(u) P(u)P(u)       , (2) 0 m m,m 0 0 0 |m| D(1) m  , d ucos l   , m 0 p 0,mp 1,mp       , p1 n1 V(u)   , p V(u) , 1 n1 V(u)   – èçâåñòíûå ôóíêöèè [10]. Òàê êàê ìàò- ðè÷íûé îïåðàòîð ñèñòåìû (16), (17) âïîëíå íåïðå- ðûâíûé, åå ðåøåíèå ìîæíî ïîëó÷èòü ÷èñëåííî ìåòîäîì ðåäóêöèè. Ýòîò ìåòîä è áûë èñïîëüçîâàí äëÿ ðåøåíèÿ (16), (17). 3. ×èñëåííûå ðåçóëüòàòû Ïðîâåäåì àíàëèç çàâèñèìîñòè êîýôôèöèåíòîâ (j) m,n x îò øèðèíû ùåëè. Ïîñêîëüêó êîíè÷åñêàÿ ñòðóêòóðà ñèììåòðè÷íà ïî  è ïåðåõîäèò ñàìà â ñåáÿ ïðè ïîâîðîòå âîêðóã îñè íà óãîë, ðàâíûé ïåðèîäó l , ðàññìîòðèì ñëó÷àé êîíóñà ñ îäíîé ùåëüþ ( N1  ). Íà ðèñ.2-9 ïðèâåäåíû êðèâûå çàâèñèìîñòè êîýô- ôèöèåíòîâ (2) m,n |x| , ÿâëÿþùèõñÿ ðåøåíèåì ÑËÀÓ (16), (17), îò óãëîâîé øèðèíû ùåëè d äëÿ ðàçëè÷- íûõ èíäåêñîâ m è n , è 1  . 0 60 120 180 240 300 360 0,0 0,2 0,4 0,6 0,8 1,0 (2) d 1 2 3 1,1 |X (d)| Ðèñ. 2. Êðèâûå çàâèñèìîñòè (2) 1,1 |x(d)| äëÿ ðàçëè÷íûõ çíà÷åíèé  : 1 – /16  , 2 – /8  , 3 – /2  0 60 120 180 240 300 360 0,0 0,2 0,4 0,6 0,8 1,0 (2) d 1,0 |X (d)| Ðèñ. 3. Êðèâàÿ çàâèñèìîñòè (2) 1,0 |x(d)| äëÿ /8  Ïðèíèìàÿ âî âíèìàíèå ñèììåòðèþ ñòðóêòóðû, íàïðàâëåíèå ðàñïðîñòðàíåíèÿ ïàäàþùåé âîëíû è êðèâûå íà ðèñ. 2-9, ïðèõîäèì ê âûâîäó, ÷òî (2)(2) m,nm,n xx   , (2)(2) m,nm,n xx   . 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Ïðè øèðèíå ùåëè, áîëüøåé o 240 , âïàäèíà èñ÷åçàåò è äèàãðàììà ïðåîáðàçóåòñÿ â äèàãðàììó îäèíî÷íîãî ðàññåèâàòå- ëÿ â âèäå èãëû (óçêèé ñïëîøíîé êîíóñ), îñü êîòîðîãî ñìåùåíà ïî îòíîøåíèþ ê îñè OZ íà óãîë /8  . 1,0 0,8 0,6 0,4 0,2 0,0 0,2 0,4 0,6 0,8 1,0 0 30 60 90 120 150 180 210 240 270 300 330 =0 1 2 3 4 Ðèñ. 10. Äèàãðàììû ðàññåÿíèÿ â ãîðèçîíòàëüíîé ïëîñêîñòè äëÿ êîíóñà ñ îäíîé ùåëüþ: 1– o d5  , 2– o d30  , 3– o d60  , 4– o d90  1,0 0,8 0,6 0,4 0,2 0,0 0,2 0,4 0,6 0,8 1,0 0 30 60 90 120 150 180 210 240 270 300 330 1 2 3 4 =0 5 6 7 8 Ðèñ. 11. Äèàãðàììû ðàññåÿíèÿ â ãîðèçîíòàëüíîé ïëîñêîñòè äëÿ êîíóñà ñ îäíîé ùåëüþ: 1– o d120  , 2– o d150  , 3– o d180  , 4– o d210  , 5– o d240  , 6– o d270  , 7– o d300  , 8– o d330  1,0 0,5 0,0 0,5 1,0 0 30 60 90 120 150 180 210 240 270 300 330 =0 Ðèñ. 12. Äèàãðàììà ðàññåÿíèÿ â ãîðèçîíòàëüíîé ïëîñêî- ñòè äëÿ ñïëîøíîãî êîíóñà 4. Çàêëþ÷åíèå Ïðèâåäåíû ðåçóëüòàòû èññëåäîâàíèÿ çàäà÷è ðàññå- ÿíèÿ ïîëÿ ïëîñêîé âîëíû íà êîíóñå ñ ïåðèîäè÷åñ- êèìè ùåëÿìè. Íà îñíîâå ïîëó÷åííîãî ÷èñëåííîãî ðåøåíèÿ èçó÷åíî âëèÿíèå ùåëåé íà êîýôôèöèåíòû Ôóðüå. Èçìåíåíèåì øèðèíû ùåëåé ìîæíî óïðàâ- ëÿòü êàê ôîðìîé äèàãðàììû ðàññåÿíèÿ, òàê è ñòåïåíüþ âîçäåéñòâèÿ ùåëè íà ðàññåÿíèå âäîëü åå îñè. Ïðèâåäåííûå ðåçóëüòàòû õîðîøî ñîãëàñóþòñÿ ñ ðåçóëüòàòàìè, ïîëó÷åííûìè äðóãèìè àâòîðàìè [2, 11, 12] â ÷àñòíûõ ñëó÷àÿõ äàííîé êîíè÷åñêîé ñòðóêòóðû. Ëèòåðàòóðà:1.Êîáàê Â.Î. Ðàäèîëîêàöèîííûå îòðàæàòå- ëè. Ì.: Ñîâ.ðàäèî, 1975. 248ñ. 2. Ãîøèí Ã.Ã. Ãðàíè÷íûå çàäà÷è ýëåêòðîäèíàìèêè â êîíè÷åñêèõ îáëàñòÿõ. Òîìñê. Èçä-âî Òîìñê. óí-òà, 1987. 127ñ. 3. Ñîëîãóá Â.Ã., Õàð÷åâ- íèêîâà Ò.È. Äèôðàêöèÿ ñôåðè÷åñêèõ âîëí íà êîíè÷åñ- êîé ïîâåðõíîñòè ñïåöèàëüíîãî âèäà // Ðàäèîòåõíèêà. Âûï.20. 1972. Ñ.52-58. 4. Äîðîøåíêî Â.À., Ñîëîãóá Â.Ã. Âîçáóæäåíèå íåçàìêíóòîé êîíè÷åñêîé ïîâåðõíîñòè ýëåêòðè÷åñêèì ðàäèàëüíûì äèïîëåì // Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1990. Ò.35, ¹12. Ñ. 2624-2626. 5 . Doroshenko V.A., Kravchenko V.F. The scattering of plane electromagnetic waves from a cone with longitudinal slots // Journal of communications Technology and Electronics, 2001. Vol. 46, N.3. P.271-278. 6. Õåíë Õ., Ìàóý Ë., Âåñòïôàëü Ê. Òåîðèÿ äèôðàêöèè. Ì: Ìèð, 1964. 428ñ. 7. Èëüèíñêèé À.Ñ., Êðàâöîâ Â.Â., Ñâåøíèêîâ À.Ã. Ìàòåìàòè÷åñêèå ìîäåëè ýëåêòðîäèíàìèêè. Ì.: Âûñø.øê., 1991. 224ñ. 8 . Ñîáîëåâ Ë. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ì.: Íàóêà, 1992. 432ñ. 9. Êîíòîðîâè÷ Ì.È., Ëåáåäåâ Í.Í. ÆÝÒÔ, 1939. Ò.9. Âûï. 6. Ñ.729-741. 10. Øåñòîïàëîâ Â.Ï. Ñóììàòîðíûå óðàâíåíèÿ â ñîâðåìåííîé òåîðèè äèôðàêöèè. Êèåâ: Íàóêîâà äóìêà, 1983. 252ñ. 11. Ãîðÿèíîâ À.Ñ. Äèôïàêöèÿ ïëîñêîé ýëåêòðîìàãíèòíîé âîëíû, ðàñïðîñòðàíÿþ- ùåéñÿ âäîëü îñè êîíóñà // Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1961. Ò.6, ¹4. Ñ.47-57. 12. Áåëè÷åíêî Â.Ï., Ãîøèí Ã.Ã., Äìèòðåíêî À.Ã. è äð. Ìàòåìàòè÷åñêèå ìåòîäû â ãðàíè÷- íûõ çàäà÷àõ ýëåêòðîäèíàìèêè. Òîìñê: Èçä-âî Òîìñê. óí-òà, 1990. 171ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 20.04.2003 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Íåðóõ À.Ã. Äîðîøåíêî Âëàäèìèð Àëåêñååâè÷, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ÂÌ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ýëåê- òðîäèíàìèêà, ìàòåìàòè÷åñêàÿ ôèçèêà. Àäðåñ: Óêðàè- íà, 61166, Õàðüêîâ, ïð.Ëåíèíà,14, òåë.40-93-72. Email: vlad_doroshenko@yahoo.com Ñåìåíîâà Åëåíà Êîíñòàíòèíîâíà, àñïèðàíò êàôåäðû ÂÌ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: ýëåêòðîäèíàìèêà, ìàòåìàòè÷åñêàÿ ôèçèêà. Àäðåñ: Óêðàèíà, 61166, Õàðü- êîâ, ïð.Ëåíèíà, 14, òåë. 711-65-09. 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Ïîäðîáíî ðàññìàòðèâàþòñÿ ïðèçìåííûå ýëåìåíòû îï- òîýëåêòðîííûõ ïðèáîðîâ è ñâÿçàííàÿ ñ íèìè çàäà÷à äèôðàêöèè ýëåêòðîìàãíèòíûõ âîëí íà äèýëåêòðè÷åñ- êîé òðåóãîëüíîé ïðèçìå. Êàê ÷àñòíûå ñëó÷àè, èçó÷àþò- ñÿ îáîðà÷èâàþùàÿ, ïîâîðîòíî-îáîðà÷èâàþùàÿ è îáî- ðîòíàÿ ïðèçìû. Ðàññìàòðèâàþòñÿ òàêæå ðàâíîñòîðîí- íèå, âûïóêëûå è âîãíóòûå êðèâîëèíåéíûå ïðèçìû, ïðèâîäÿòñÿ ðåçóëüòàòû ðàñ÷åòîâ äèàãðàìì íàïðàâëåí- íîñòè, ïîïåðå÷íèêîâ ïîëíîãî è îáðàòíîãî ðàññåÿíèÿ, èíòåíñèâíîñòè ïîëÿ â áëèæíåé çîíå. Óñòàíàâëèâàåòñÿ ðåçîíàíñíûé õàðàêòåð ýòèõ ÷àñòîòíûõ õàðàêòåðèñòèê. Ââåäåíèå Ïðèçìåííûå ýëåìåíòû èñïîëüçóþòñÿ ïðè êîíñòðó- èðîâàíèè îïòîýëåêòðîííûõ ïðèáîðîâ íà ïðîòÿæå- íèè ìíîãèõ ëåò.Èç-çà âîçíèêàþùèõ ïðè ïåðåäà÷å èíôîðìàöèè ïîìåõ äëÿ ñâåòîâûõ âîëí ïðèìåíÿåòñÿ òåõíîëîãèÿ, óæå èçâåñòíàÿ äëÿ ìèêðîâîëí, à èìåí- íî, òåõíèêà âîëíîâîäîâ, ïðè ýòîì íîñèòåëåì èí- ôîðìàöèè ÿâëÿåòñÿ ñâåò, à ïðèçìåííûé ýëåìåíò ñâÿçè ñëóæèò äëÿ ïðåîáðàçîâàíèÿ ïàäàþùåé âîëíû â ïîâåðõíîñòíóþ âîëíó âîëíîâîäà. Ïðèçìó îáû÷íî èñïîëüçóþò â ýêñïåðèìåíòàõ äëÿ ïîëó÷åíèÿ ñïåê- òðîâ. Åùå Íüþòîí âïåðâûå ðàçëîæèë ïðèçìîé ñîëíå÷íûé ëó÷.  åãî îïûòå äâå òðåóãîëüíûå ïðèçìû ðàñïîëàãàëèñü òàê, ÷òîáû îíè ðàçëàãàëè ïðîõîäÿùèé ïó÷îê ñâåòà â ñïåêòð â äâóõ âçàèìíî ïåðïåíäèêóëÿðíûõ íàïðàâëåíèÿõ (ìåòîä “ ñêðå- ùåííûõ ïðèçì ” ). È ïî ñåé äåíü ïðèçìû èãðàþò îñíîâíóþ ðîëü ïðè ñîçäàíèè ñïåêòðàëüíûõ ïðèáî- ðîâ, ñåíñîðîâ, äåëèòåëåé è ò.ä. Òàê êàê äëÿ âñåõ ïðîçðà÷íûõ âåùåñòâ ïîêàçàòåëü ïðåëîìëåíèÿ óâå- ëè÷èâàåòñÿ ñ óìåíüøåíèåì äëèíû âîëíû, òî ïðèçìà èñïîëüçóåòñÿ â êà÷åñòâå äèñïåðãèðóþùåãî ýëåìåí- òà â êîðîòêîâîëíîâîé îáëàñòè ñïåêòðà. Ïðè èññëå- äîâàíèè ñëîæíûõ ÿâëåíèé ïðèðîäû íàèáîëåå èí- òåðåñíûì ÿâëÿåòñÿ èçó÷åíèå äèôðàêöèè, êîãäà äëèíû âîëí ñðàâíèìû ñ ãåîìåòðè÷åñêèìè ðàçìåðà- ìè ïðèçìû.  äàííîì ñëó÷àå ïðèìåíåíèå çàêîíîâ ãåîìåòðè÷åñêîé îïòèêè ìàëîýôôåêòèâíî, è çàäà÷à ñâîäèòñÿ ê ðàññìîòðåíèþ çàäà÷è äèôðàêöèè ýëåê- òðîìàãíèòíûõ âîëí íà äèýëåêòðè÷åñêîì ïðåïÿò- ñòâèè. Ïîäðîáíîå ðàññìîòðåíèå ïðèçìåííûõ ýëå- ìåíòîâ îïòîýëåêòðîííûõ ïðèáîðîâ è ñâÿçàííîé ñ íèìè çàäà÷è äèôðàêöèè âîëí íà äèýëåêòðè÷åñêîé ïðèçìå ÿâëÿåòñÿ öåëüþ äàííîé ðàáîòû. Ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí íà öèëèíäðè÷åñêèõ òåëàõ âûçûâàåò çíà÷èòåëüíûé èíòåðåñ ó èññëåäîâàòåëåé, êàê ñ òåîðåòè÷åñêîé, òàê è ñ ÷èñòî ïðàêòè÷åñêîé òî÷åê çðåíèÿ.  íàñòîÿùåå âðåìÿ ðàçðàáîòàíû äîñòàòî÷íî ýôôåêòèâíûå ìåòîäû ðåøåíèÿ çàäà÷ äèôðàêöèè âîëí, ãëàâíûì îáðàçîì íà èäåàëüíî ïðîâîäÿùèõ òåëàõ ðàçëè÷íîé êîíôèãóðàöèè [1]. ×èñëåííî ýôôåêòèâíûé è ýêîíîìè÷íûé àëãîðèòì ðåøåíèÿ çàäà÷ íà äèýëåêòðè÷åñêèõ òåëàõ áûë ïðåä- ëîæåí â ðàáîòå [2] è øèðîêî ïðèìåíÿëñÿ ïðè ðåøåíèè çàäà÷ äèôðàêöèè íà öèëèíäðàõ ýëëèïòè- ÷åñêîãî è “ ñóïåð-ýëëèïòè÷åñêîãî ” ïîïåðå÷íîãî ñå÷åíèÿ. Èñïîëüçîâàíèå ìåòîäà êîíôîðìíûõ îòî- áðàæåíèé äëÿ ïîëó÷åíèÿ ãëîáàëüíîé ïàðàìåòðèçà- öèè êîíòóðà ìíîãîóãîëüíîãî ïîïåðå÷íîãî ñå÷åíèÿ, ïðåäëîæåííîãî â ðàáîòå [3], ïîçâîëèëî ðàñøèðèòü îáëàñòü ïðèìåíåíèÿ ýòîãî àëãîðèòìà, ñäåëàâ åãî ïðèãîäíûì äëÿ ðåøåíèÿ çàäà÷ íà ïðîèçâîëüíîì ìíîãîãðàííîì öèëèíäðå. Ïîñòàíîâêà çàäà÷è è ìåòîä ðåøåíèÿ Çàäà÷ó äèôðàêöèè ïëîñêîé ýëåêòðîìàãíèòíîé âîë- íû íà äèýëåêòðè÷åñêîé ïðèçìå, íå îãðàíè÷èâàÿ îáùíîñòè, ìîæíî ñâåñòè ê ðàññìîòðåíèþ çàäà÷è ðàññåÿíèÿ âîëíû íà îäèíî÷íîì áåñêîíå÷íîì ìíî- ãîóãîëüíîì öèëèíäðå, íàõîäÿùåìñÿ â îäíîðîäíîé ñðåäå. Ýëåêòðîìàãíèòíàÿ âîëíà ïàäàåò ïîä óãëîì  ê îñè àáñöèññ:     sin iky cos ikx i e U (ðèñ. 1). Ðèñ.1. Ãåîìåòðèÿ çàäà÷è Çäåñü U – êîìïîíåíòà ïîëÿ E z èëè H z â ñëó÷àå E- è H-ïîëÿðèçàöèè ñîîòâåòñòâåííî. Ïðè ýòîì êîì- ïîíåíòû E x, E y, H x è H y ïîëíîãî ïîëÿ âûðàæàþòñÿ ÷åðåç êîìïîíåíòû E z è H z ïðè ïîìîùè óðàâíåíèé Ìàêñâåëëà [4]: y E i H z x      , x E i H z y     , y H i E z x     , x H i E z y      . Ïîëíîå ïîëå â ïðèñóòñòâèè îäíîðîäíîãî öèëèíäðà ñ îòíîñèòåëüíîé äèýëåêòðè÷åñêîé ïðîíèöàåìîñ- òüþ e ïðåäñòàâèì â ñëåäóþùåì âèäå:             , r , U , r , U U U sc i tot   ãäå  – îáëàñòü, îãðàíè÷åííàÿ êîíòóðîì S; U i – ïàäàþùåå ïîëå; U sc – ïîëå, ðàññåÿííîå íà öèëèíäðå; U  – ïîëå âíóòðè ïðåïÿòñòâèÿ. Ôóíêöèè ïîëÿ óäîâëåòâîðÿþò óðàâíåíèþ Ãåëüìãîëüöà ñ êóñî÷íî- ïîñòîÿííûì êîýôôèöèåíòîì. 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Ïîñëå ïðåäñòàâëåíèÿ èñêîìûõ êîì- ïîíåíò ïîëÿ â âèäå ïîòåíöèàëîâ ïðîñòîãî ñëîÿ íà ïîâåðõíîñòè öèëèíäðà, óäîâëåòâîðåíèÿ ãðàíè÷íûõ óñëîâèé, óñëîâèÿ èçëó÷åíèÿ Çîììåðôåëüäà [5] çàäà÷à ñâîäèòñÿ ê ðåøåíèþ äâóõ èíòåãðî-äèôôå- ðåíöèàëüíûõ óðàâíåíèé ñ íåèçâåñòíûìè ôóíêöè- ÿìè ïëîòíîñòè ðàñïðåäåëåíèÿ ïîëÿ ïî ïîâåðõíîñòè öèëèíäðà [6]. Ïðè ïîìîùè êîíôîðìíîãî îòîáðà- æåíèÿ âíåøíîñòè ìíîãîóãîëüíèêà íà âíåøíîñòü êðóãà ïîëó÷åíû ïàðàìåòðè÷åñêèå óðàâíåíèÿ êîí- òóðà ïîïåðå÷íîãî ñå÷åíèÿ öèëèíäðà, ïîçâîëèâøèå ñâåñòè êðèâîëèíåéíûå èíòåãðàëû ê îïðåäåëåííûì [3]. Ïîñëå âûäåëåíèÿ è ïîñëåäóþùåãî àíàëèòè÷åñ- êîãî îáðàùåíèÿ ñèíãóëÿðíîé ÷àñòè ÿäðà èíòåãðàëü- íîãî îïåðàòîðà ýòè óðàâíåíèÿ ñâåäåíû ê áåñêîíå÷- íîé ñèñòåìå ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé Ôðåäãîëüìà âòîðîãî ðîäà, êîòîðàÿ ðåøàåòñÿ ÷èñ- ëåííî ñ ëþáîé íàïåðåä çàäàííîé òî÷íîñòüþ. Ïðè ýòîì ïðèìåíÿåòñÿ ìåòîä Ãàëåðêèíà ñ áàçèñîì â âèäå ñîáñòâåííûõ îðòîãîíàëüíûõ ôóíêöèé ñèíãóëÿð- íîé ÷àñòè èíòåãðàëüíîãî îïåðàòîðà. Ìàòðè÷íûå ýëåìåíòû âû÷èñëÿþòñÿ ïðè ïîìîùè äâîéíîãî äèñ- êðåòíîãî ïðåîáðàçîâàíèÿ Ôóðüå. Îñíîâíûå óðàâ- íåíèÿ è ìåòîä ðåøåíèÿ ïîäîáíîé çàäà÷è ïîäðîáíî èçó÷åíû â [7].  äàííîé ðàáîòå èññëåäîâàíû òàêèå õàðàêòåðèñòèêè, êàê äèàãðàììà íàïðàâëåííîñòè ðàññåÿííîãî ïîëÿ â äàëüíåé çîíå, èíòåíñèâíîñòü ïîëÿ, à òàêæå ïîïåðå÷íèêè ïîëíîãî è îáðàòíîãî ðàññåÿíèÿ êàê äëÿ Å-, òàê è äëÿ Í-ïîëÿðèçàöèè. ×èñëåííûå ðåçóëüòàòû è îáñóæäåíèå Ñ òåõ ïîð êàê âîëíîâîäû ñòàëè èñïîëüçîâàòü äëÿ ïåðåäà÷è èíôîðìàöèè, îäíà èç âàæíûõ ïðîáëåì çàêëþ÷àëàñü â òîì, ÷òîáû ââåñòè âîëíó â âîëíîâîä, ò.å. ñâÿçàòü âíåøíåå ïîëå âîëíîâîäà ñ âíóòðåííèì. Íà ïðîòÿæåíèè ìíîãèõ ëåò íàèáîëåå àêòèâíî èñ- ïîëüçóþòñÿ òðè îñíîâíûå âîçìîæíîñòè îñóùåñòâ- ëåíèÿ ñâÿçè ñ îïòè÷åñêèì âîëíîâîäîì: ïðÿìàÿ ñâÿçü ïðè ïîìîùè ëèíçû, ñâÿçü ñ ïîìîùüþ ðåøåò- êè è ñâÿçü ïðè ïîìîùè ïðèçìû. Íàèáîëåå øèðîêî ïðèìåíÿåòñÿ â îïòèêå òðåóãîëüíàÿ ïðèçìà, îñîáåí- íî ïðèçìû, â êîòîðûõ íàáëþäàåòñÿ ïîëíîå âíóò- ðåííåå îòðàæåíèå (ðèñ 2, à, á, â). ßâëåíèå ïîëíîãî âíóòðåííåãî îòðàæåíèÿ âîçíèêàåò ïðè ïåðåõîäå âîëíû èç îïòè÷åñêè ìåíåå ïëîòíîé ñðåäû â îïòè- ÷åñêè áîëåå ïëîòíóþ ïðè óãëàõ ïàäåíèÿ, ïðåâîñõî- äÿùèõ ïðåäåëüíîå çíà÷åíèå, ò.å.  arcsin( ï 1 /n 2 ) . Çíàê ðàâåíñòâà ñîîòâåòñòâóåò ñëó÷àþ, êîãäà âîëíà âûõîäèò ïî êàñàòåëüíîé ê ãðàíèöå ðàçäåëà äâóõ ñðåä, ïðè áîëüøèõ óãëàõ ïàäåíèÿ âîëíà âîîáùå íå âûõîäèò, âîçâðàùàÿñü â ïåðâóþ ñðåäó. Äàííîå çíà÷åíèå ïðåäåëüíîãî óãëà ïîëó÷åíî ïî çàêîíó Ñíåëëèóñà. Ðåçóëüòàòû ðàñ÷åòîâ ïðèâåäåíû äëÿ ñëó÷àÿ, êîãäà ïðèçìû èçãîòîâëåíû èç ñòåêëà ñ äèýëåêòðè÷åñêîé ïðîíèöàåìîñòüþ  2 =2,25 è íàõî- äÿòñÿ â ñâîáîäíîì ïðîñòðàíñòâå  1 =1.  äàííîì ñëó÷àå çíà÷åíèå ïðåäåëüíîãî óãëà ðàâíî 41,81°. Òàêèå ïðèçìû òàêæå èçãîòàâëèâàþò èç ãåðìàíèÿ (n=4,0), ôîñôèäà ãàëèÿ (n=3,4), ðóòèëà (n=2,5) è ò.ä. Ãðàíè îòðàæàþùèõ ïðÿìîóãîëüíûõ òðåõãðàí- íûõ ïðèçì èíîãäà áûâàþò ïîñåðåáðåííûìè è ñëóæàò çåðêàëàìè. Ðèñ. 2. Ïîâîðîòíàÿ (à), ïîâîðîòíî-îáîðà÷èâàþùàÿ (á) è îáîðà÷èâàþùàÿ (â) ïðèçìû Äëÿ èçó÷åíèÿ ïîâåäåíèÿ ïîëÿ â äàëüíåé çîíå è ïîñòðîåíèÿ äèàãðàìì íàïðàâëåííîñòè ðàññåÿííîãî ïîëÿ áóäåì èñïîëüçîâàòü ôîðìóëó:   2 ()cos()sin int 0 1 () 2 ikxtyt n n peedt           . Çäåñü x(t) è y(t) – ïàðàìåòðè÷åñêèå óðàâíåíèÿ òðåóãîëüíîãî êîíòóðà ïîïåðå÷íîãî ñå÷åíèÿ [3]. Ïðè ïîëó÷åíèè ýòîé ôîðìóëû çíà÷åíèå ðàññåÿííî- ãî ïîëÿ, îïðåäåëÿåìîå êàê ïîòåíöèàë ïðîñòîãî ñëîÿ, çàìåíåíî íà åãî àñèìïòîòè÷åñêîå ïðåäñòàâëå- íèå ñ èñïîëüçîâàíèåì ñâîéñòâ ôóíêöèè Õàíêåëÿ ïðè áîëüøèõ çíà÷åíèÿõ àðãóìåíòà [8]. Íà ðèñ. 3 ïðèâåäåíû íîðìèðîâàííûå äèàãðàììû íàïðàâëåííîñòè ðàññåÿííîãî ïîëÿ ïëîñêîé Í-ïî- ëÿðèçîâàííîé âîëíû â äàëüíåé çîíå ïðè íåêîòîðûõ çíà÷åíèÿõ âîëíîâîãî ïàðàìåòðà a=kd=2  d/  Ðèñ. 3,à ñîîòâåòñòâóåò ðàññåÿíèþ Í -ïîëÿðèçîâàí- íîé âîëíû, ïàäàþùåé ïåðïåíäèêóëÿðíî íà íèæ- íþþ ãðàíü ïðèçìû.  äàííîì ñëó÷àå ïðåëîìëåííàÿ âîëíà, îòðàæàÿñü îò áîêîâûõ ãðàíåé ïðèçìû, âû- õîäèò èç íåå â íàïðàâëåíèè, îáðàòíîì íàïðàâëåíèþ ïàäàþùåé âîëíû (ïîâîðîòíàÿ ïðèçìà). Íîðìàëüíîå ïàäåíèå íà áîêîâóþ ãðàíü ñîîòâåò- ñòâóåò ïîâîðîòíî-îáîðà÷èâàþùåé ïðèçìå (ðèñ. 3, á). Òàêèå ïðèçìû èñïîëüçîâàëèñü íà ïðîòÿæåíèè ìíîãèõ ëåò ïðè êîíñòðóèðîâàíèè ïðèçìàòè÷åñêèõ çðèòåëüíûõ òðóá, îäíèì èç ïðèìåðîâ êîòîðûõ ÿâëÿåòñÿ ñèñòåìà îáðàùàþùèõ ïðèçì Ïîððî. Ñëó- ÷àé îáîðà÷èâàþùåé äèýëåêòðè÷åñêîé ïðèçìû ðàñ- ñìîòðåí íà ðèñ. 3, â. Àíàëîãè÷íûå ðåçóëüòàòû ìîãóò áûòü ïîëó÷åíû äëÿ äèýëåêòðè÷åñêèõ ïðèçì, îáëó- ÷àåìûõ ïëîñêîé Å -ïîëÿðèçîâàííîé âîëíîé. Îäíàêî óòâåðæäåíèå, ÷òî ïîòîê ýëåêòðîìàãíèòíîé ýíåðãèè íå ïîïàäàåò âî âòîðóþ ñðåäó, ïîëíîñòüþ îòðàæàÿñü îò ãðàíèöû ðàçäåëà, íåëüçÿ ñ÷èòàòü òî÷- íûì. Íàëè÷èå áîêîâûõ ëåïåñòêîâ â äèàãðàììàõ íàïðàâëåííîñòè ïîäòâåðæäàåò èçâåñòíîå óòâåðæäå- íèå î òîì, ÷òî ïðè ïîëíîì âíóòðåííåì îòðàæåíèè, ò. å. êîãäà óãîë ïàäåíèÿ âîëíû ïðåâûøàåò ïðåäåëü- íîå çíà÷åíèå, âî âòîðîé ñðåäå ïîÿâëÿåòñÿ ýëåêòðî- ìàãíèòíàÿ âîëíà, ðàñïðîñòðàíÿþùàÿñÿ âäîëü ãðàíè- öû ðàçäåëà (íàïðèìåð, ïðè  =45, 135, 225 è 315° äëÿ ïîâîðîòíîé ïðèçìû, ðèñ. 3, à). Ïðàêòè÷åñêè ýòà âîëíà ñóùåñòâóåò ëèøü â ïîâåðõíîñòíîì ñëîå âòîðîé ñðåäû, òîëùèíà êîòîðîãî ïðèìåðíî ðàâíà äëèíå âîëíû èññëåäóåìîãî èçëó÷åíèÿ. 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Ïðîíèêíîâåíèå ýëåêòðîìàãíèòíîé ýíåðãèè âî âòîðóþ ñðåäó ìîæíî ñâÿçàòü ñ ôèçè÷åñ- êèìè ÿâëåíèÿìè, ïðîòåêàþùèìè ïðè óñòàíîâëåíèè ïðîöåññà, è èñïîëüçîâàòü â ïðàêòè÷åñêèõ öåëÿõ. à á â Ðèñ. 3. Äèàãðàììû íàïðàâëåííîñòè ïîëÿ â äàëüíåé çîíå ïðè íåêîòîðûõ ÷àñòîòàõ, Í-ïîëÿðèçàöèÿ: à – ïîâîðîòíàÿ ïðèçìà; á – ïîâîðîòíî-îáîðà÷èâàþùàÿ ïðèçìà; â – îáîðà÷èâàþùàÿ ïðèçìà Îäíîé èç èçó÷àåìûõ õàðàêòåðèñòèê ïîëÿ ÿâëÿåòñÿ èíòåíñèâíîñòü â áëèæíåé çîíå (ðèñ. 4, 5). Èíòåíñèâíîñòü ïîëÿ Í-ïîëÿðèçîâàííîé âîëíû (ñì. ðèñ. 4) çíà÷èòåëüíî ïðåâûøàåò èíòåíñèâíîñòü ïîëÿ Å-ïîëÿðèçîâàííîé âîëíû (ñì. ðèñ. 5). Ïðèçìà ìîæåò èñïîëüçîâàòüñÿ êàê âõîäíîé è âû- õîäíîé ýëåìåíòû ñâÿçè â îòêðûòûõ âîëíîâîäàõ, à èíîãäà äàæå êàê äåëèòåëü äëÿ ââåäåíèÿ âîëíû â äâà âîëíîâîäà. Òàêîå ìîæåò äîñòèãàòüñÿ ïðè âåðòè- êàëüíîì ïàäåíèè âîëíû íà ðåáðî ðàâíîáåäðåííîé òðåóãîëüíîé ïðèçìû. Íà ðèñ. 6 ïðèâåäåíû äèàãðàì- ìû íàïðàâëåííîñòè ïîëÿ â äàëüíåé çîíå ïðè íåêîòîðûõ çíà÷åíèÿõ íîðìèðîâàííîé ÷àñòîòû a=kd. Ðèñ. 4. Äâóìåðíûé ñëàéä èíòåíñèâíîñòè ïîëÿ â áëèæíåé çîíå, ïîâîðîòíàÿ ïðèçìà, Í-ïîëÿðèçàöèÿ, à=10 Ðèñ. 5. Äâóìåðíûé ñëàéä èíòåíñèâíîñòè ïîëÿ â áëèæíåé çîíå, ïîâîðîòíàÿ ïðèìà, Å-ïîëÿðèçàöèÿ, à=10 Äèàãðàììû íà ðèñ. 6, à ïîêàçûâàþò, ÷òî ðàññåÿííîå ïîëå ðàñïðîñòðàíÿåòñÿ ïðåèìóùåñòâåííî â äâóõ ïðîòèâîïîëîæíûõ íàïðàâëåíèÿõ, ñèììåòðè÷íûõ îòíîñèòåëüíî âûñîòû ïîïåðå÷íîãî ñå÷åíèÿ. Äëÿ äàííîé öåëè ìîæåò îêàçàòüñÿ áîëåå ýôôåêòèâíîé êðèâîëèíåéíàÿ âîãíóòàÿ òðåóãîëüíàÿ ïðèçìà.  ýòîì ñëó÷àå ðàññåÿíèå â ïðÿìîì è îáðàòíîì íàïðàâ- ëåíèè çíà÷èòåëüíî ìåíüøå, ÷åì äëÿ ïðèçìû ñ ïðÿìîëèíåéíûìè ãðàíÿìè (ðèñ. 6, á). Äëÿ òðåó- ãîëüíîé âûïóêëîé ïðèçìû (ðèñ. 6, â) ðàññåÿíèå â îáðàòíîì íàïðàâëåíèè çíà÷èòåëüíî ñèëüíåå, ÷åì äëÿ âîãíóòîé ïðèçìû, îäíàêî èñïîëüçîâàíèå òàêîé ïðèçìû òàêæå ìîæåò îêàçàòüñÿ ýôôåêòèâíûì, è â çàâèñèìîñòè îò öåëè òåõíè÷åñêîå âûïîëíåíèå ìî- æåò áûòü î÷åíü ðàçíîîáðàçíî. 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Èçìåíåíèå ïàðàìåòðà m ïîçâîëÿåò âàðü- èðîâàòü âûñîòó ðàâíîáåäðåííîãî òðåóãîëüíèêà. à á â Ðèñ. 6. Íîðìèðîâàííûå äèàãðàììû íàïðàâëåííîñòè ïîëÿ â äàëüíåé çîíå äëÿ òðåóãîëüíîé ïðèçìû (à), êðèâîëèíåéíîé âîãíóòîé òðåóãîëüíîé ïðèçìû (á), êðèâîëèíåéíîé âûïóêëîé òðåóãîëüíîé ïðèçìû (â), Í-ïîëÿðèçàöèÿ Ðàññìîòðèì ïîëíóþ ýíåðãèþ ïîëÿ, ðàññåÿííóþ âî âñåõ íàïðàâëåíèÿõ âî âíåøíåå ïðîñòðàíñòâî âîêðóã ïðèçìû. Òàêóþ âåëè÷èíó íàçûâàþò ïîëíûì ïîïå- ðå÷íèêîì ðàññåÿíèÿ. Åùå îäíîé ïðàêòè÷åñêè âàæ- íîé õàðàêòåðèñòèêîé ÿâëÿåòñÿ ñå÷åíèå îáðàòíîãî ðàññåÿíèÿ, êîòîðîå õàðàêòåðèçóåò ýíåðãèþ ïîëÿ, ðàññåÿííîãî â íàïðàâëåíèè, ïðîòèâîïîëîæíîì íà- ïðàâëåíèþ ïàäàþùåé âîëíû. Äàííóþ âåëè÷èíó èíîãäà íàçûâàþò ðàäèîëîêàöèîííûì ïîïåðå÷íè- êîì ðàññåÿíèÿ. Ïîëíûé è ðàäèîëîêàöèîííûé ïî- ïåðå÷íèêè ðàññåÿíèÿ âû÷èñëÿþò ïî ôîðìóëàì:         2 0 2 t d ) ( k 2 , 2 b ) ( k 4       . Îñîáûé èíòåðåñ ïðåäñòàâëÿþò ÷àñòîòíûå çàâèñèìî- ñòè ïîïåðå÷íèêîâ ïîëíîãî è îáðàòíîãî ðàññåÿíèÿ, êîòîðûå ïîñòðîåíû íà ðèñ. 7 äëÿ ðàâíîñòîðîííåé ïðèçìû. ×àñòîòû, ïðè êîòîðûõ ïîëíàÿ ðàññåÿííàÿ ýíåðãèÿ äîñòèãàåò ìàêñèìàëüíûõ çíà÷åíèé, ñîîò- âåòñòâóþò ðåçîíàíñàì, òîãäà êàê îáðàòíîå ðàññåÿíèå èìååò ýêñòðåìóìû, íîñÿùèå õàðàêòåð êàê ìèíèìó- ìîâ, òàê è ìàêñèìóìîâ íà ðåçîíàíñíûõ ÷àñòîòàõ. Ïîñêîëüêó îáùàÿ ýíåðãèÿ ðàññåÿíèÿ â ðåçîíàíñàõ âñåãäà ïðèíèìàåò ìàêñèìàëüíîå çíà÷åíèå, òî óìåíü- øåíèå ïîïåðå÷íèêà îáðàòíîãî ðàññåÿíèÿ ñâÿçàíî ñ òåì, ÷òî îñíîâíàÿ äîëÿ ýíåðãèè èçëó÷àåòñÿ â íàïðàâ- ëåíèÿõ, îòëè÷íûõ îò îáðàòíîãî. Àíàëèçèðóÿ äàííûå ãðàôèêè, íåòðóäíî çàìåòèòü, ÷òî ïðè èñïîëüçîâà- íèè ðàâíîñòîðîííåé ïðèçìû â îáðàòíîì íàïðàâëå- íèè ýíåðãèÿ ðàññåèâàåòñÿ çíà÷èòåëüíåé â ñëó÷àå Í- ïîëÿðèçàöèè, ÷åì â ñëó÷àå Å-ïîëÿðèçàöèè. Ðèñ. 7. 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Ðàññìîòðåíû îñ- íîâíûå âèäû òðåóãîëüíûõ ïðèçì, êîòîðûå ÿâëÿþò- ñÿ êëþ÷åâûìè ýëåìåíòàìè ìíîãèõ îïòè÷åñêèõ ïðèáîðîâ. Ïðîâåäåí ôèçè÷åñêèé àíàëèç äèôðàê- öèîííîãî ïîëÿ â ñëó÷àå Å- è Í-ïîëÿðèçàöèè êàê â äàëüíåé, òàê è â áëèæíåé çîíàõ â ðåçîíàíñíîé îáëàñòè ÷àñòîò. Íà ïðàêòèêå òàêèå òðåóãîëüíûå ïðèçìû èñïîëüçóþòñÿ êàê ïðèçìåííûå ýëåìåíòû ñâÿçè íå òîëüêî äëÿ âîçáóæäåíèÿ âîëíîâîäíûõ ìîä, íî è äëÿ âûâîäà èçëó÷åíèÿ èç âîëíîâîäà. Äëÿ ïðîñòîòû ðàññìîòðåíèå îãðàíè÷èëîñü äèôðàêöèåé ïëîñêèõ ýëåêòðîìàãíèòíûõ âîëí íà òðåóãîëüíîì öèëèíäðè÷åñêîì äèýëåêòðè÷åñêîì ïðåïÿòñòâèè. Îäíàêî äàííûé ìåòîä ìîæåò áûòü ëåãêî îáîáùåí íà ñëó÷àé, êîãäà ïàäàþùåå ïîëå ÿâëÿåòñÿ âîëíîé ñ ãàóññîâûì ïðîôèëåì àìïëèòóäû, à äèýëåêòðè÷åñ- êàÿ ïðèçìà ñ ïðîèçâîëüíûì ÷èñëîì ñòîðîí ïîãðó- æåíà â ïðîèçâîëüíóþ ñëîèñòóþ èçîòðîïíóþ ñðåäó. Ðàñ÷åòû ïðîèçâåäåíû íà êîìïüþòåðå AMD Duron 650 ïðè èñïîëüçîâàíèè ñèñòåìû MATLAB 6.1, âðåìÿ âû÷èñëåíèÿ íåçíà÷èòåëüíî çàâèñèò îò âîë- íîâûõ ðàçìåðîâ ïðèçìû, äèýëåêòðè÷åñêîé ïðîíè- öàåìîñòè ñðåäû, òî÷íîñòè âû÷èñëåíèé, ïîëÿðèçà- öèè è äðóãèõ ïàðàìåòðîâ çàäà÷è è â ñðåäíåì ñîñòàâëÿåò 2 ñåêóíäû. ×èñëåííûå ðåçóëüòàòû áûëè ñðàâíåíû ñ ðåçóëüòàòàìè ðàñ÷åòîâ ìåòîäîì äîïîë- íèòåëüíûõ èñòî÷íèêîâ, îïèñàííûì â ðàáîòàõ [9, 10], ñîâïàäåíèå ñîñòàâëÿåò 99,68%. Ðàáîòà ÷àñòè÷íî ôèíàíñèðóåòñÿ ìåæäóíàðîäíîé îðãà- íèçàöèåé INTAS, ãðàíò ¹ YSF 2001/2-11. Àâòîð áëàãîäàðèò Òáèëèññêèé Ãîñóäàðñòâåííûé Óíè- âåðñèòåò çà ïðåäîñòàâëåíèå ðåçóëüòàòîâ òåñòîâûõ ðàñ- ÷åòîâ äëÿ ñðàâíåíèÿ, ïðîô. À.Ã. Íåðóõà çà èäåþ ðàñ- ñìîòðåíèÿ êðèâîëèíåéíûõ ïðèçì, ñâîåãî íàó÷íîãî ðóêîâîäèòåëÿ ä-ðà ôèç .- ìàò. íàóê À.È. Íîñè÷à, à òàêæå ïðîô. Á.Ì. Áóëãàêîâà, ä-ðà ôèç.-ìàò. íàóê. Ç.Ò. Íàçàð- ÷óêà çà ïîëåçíûå ñîâåòû è îáñóæäåíèå. Ëèòåðàòóðà: 1. Veliev E.I., Veremey V.V. Numerical- analytical approach for the solution to the wave scattering by polygonal cylinders and flat strip structures, in Hashimoto M. et. al. (Eds.) // Analytical and Numerical Methods in Electromagnetic Wave Theory. Tokyo: Science House Publ. 1993. Ð. 471-514. 2. Áîðèñêèíà Ñ.Â., Íîñè÷ À.È. Ìåòîä àíàëèòè÷åñêîé ðåãóëÿðèçàöèè â çàäà÷àõ äèôðàêöèè âîëí íà äèýëåêòðè÷åñêèõ öèëèíäðàõ ïðî- èçâîëüíîãî ïîïåðå÷íîãî ñå÷åíèÿ // Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ. 1998. Ò. 3, ¹3. Ñ. 310-318. 3. Illyashenko L.N., Nosich A.I. and Benson T.M., Integral equation analysis of a polygonal semiconductor microcavity, Proc. Int. Conf. on Numerical Simulation of Semiconductor Optoelectronic Devices (NUSOD 02), Zurich, 2002. P. 72- 73. 4. Ãàëèøíèêîâà Ò.Í., Èëüèíñêèé À.Ñ. ×èñëåííûå ìåòîäû â çàäà÷àõ äèôðàêöèè. Ì.: Èçä-âî ÌÃÓ, 1987. 208ñ. 5. Õåíë Õ., Ìàóý À., Âåñòïôàëü Ê. Òåîðèÿ äèôðàê- öèè. Ì.: Ìèð, 1964. 428 ñ. 6. Colton D., Kress R. Integral Equation Methods in Scattering Theory. New York: Wiley. 1983. 7. Èëüÿøåíêî Ë.Í. Ðåøåíèå çàäà÷ äèôðàêöèè âîëí íà äèýëåêòðè÷åñêîì öèëèíäðå ñ ïðèìåíåíèåì òåîðèè êîíôîðìíîãî îòîáðàæåíèÿ // Ðàäèîôèçèêà è ýëåêòðî- íèêà, ÈÐÝ ÍÀÍÓ, 2002. Ò. 7, ¹ 3. 8. Ñïðàâî÷íèê ïî ñïåöèàëüíûì ôóíêöèÿì ñ ôîðìóëàìè, ãðàôèêàìè è ìàòåìàòè÷åñêèìè òàáëèöàìè / Ïîä ðåä. Àáðàìîâèöà Â.À., Ñòèãàí È. Ì.: Íàóêà, 1979. 830 ñ. 9. Öàðèäçå Ð.Ñ., Êàðêàøèäçå Ä.Ä. è äð. Èññëåäîâàíèå âîçìîæíîñòåé ìåòîäà äîïîëíèòåëüíûõ èñòî÷íèêîâ äëÿ ðåøåíèÿ äâó- ìåðíûõ çàäà÷ ýëåêòðîäèíàìèêè // Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1991. Ò.26, ¹ 2. 10. Zaridze R.S., Karkashidze D.D. Method of auxiliary sourses and wave field singularities // 9 th school-seminar on diffractyion and wave propagation, Kazan, 1988. Ïîñòóïèëà â ðåäêîëëåãèþ 17.04.2003 Ðåöåíçåíò: ä-ð ôèç.- ìàò. íàóê, ïðîô. ßöóê Ë.Ï. Èëüÿøåíêî Ëþäìèëà Íèêîëàåâíà, àñïèðàíòêà èíñòè- òóòà ðàäèîôèçèêè è ýëåêòðîíèêè èì. À.ß. Óñèêîâà Íàöèîíàëüíîé Àêàäåìèè Íàóê Óêðàèíû, îòäåë âû÷èñ- ëèòåëüíîé ýëåêòðîäèíàìèêè. Àäðåñ: Óêðàèíà, 61000, Õàðüêîâ, òåë. 428-278, e-mail: lyusi@vil.com.ua ÓÄÊ 658.51.011.56 ÂËÈßÍÈÅ ÔÎÐÌÛ ÍÀÃÐÅÒÎÉ ÇÎÍÛ ÍÀ ÌÀÊÑÈÌÀËÜÍÛÉ ÏÅÐÅÃÐÅ ÀÏÏÀÐÀÒÀ ÑÈÍÎÒÈÍ À.Ì. Ïðèâîäÿòñÿ àíàëèòè÷åñêèå çàâèñèìîñòè, ïîçâîëÿþ- ùèå ó÷èòûâàòü âëèÿíèå ôîðìû íàãðåòîé çîíû íà ìàêñèìàëüíûé ïåðåãðåâ àïïàðàòà. Îïèñûâàþòñÿ ðå- çóëüòàòû ýêñïåðèìåíòàëüíûõ äàííûõ ïðè ðàçëè÷íûõ çíà÷åíèÿõ êðèòåðèÿ   і . 1. Àêòóàëüíîñòü, ïîñòàíîâêà öåëè è çàäà÷è èññëåäîâàíèÿ Ñîâðåìåííûå òåõíè÷åñêèå óñòðîéñòâà âñå áîëåå íàñûùàþòñÿ ðàäèîýëåêòðîííîé àïïàðàòóðîé ðàç- ëè÷íîãî íàçíà÷åíèÿ. Çíà÷èòåëüíî âîçðàñòàåò êîëè- ÷åñòâî âõîäÿùèõ â ðàäèîýëåêòðîííûé àïïàðàò ýëå- ìåíòîâ, ñîâåðøåíñòâóåòñÿ ýëåìåíòíàÿ áàçà ïðè ïîñòîÿííîì ñòðåìëåíèè ìàêñèìàëüíî ñíèçèòü ìàñ- ñó è ãàáàðèòíûå ðàçìåðû àïïàðàòóðû. Ýíåðãåòè÷åñêèé êîýôôèöèåíò ïîëåçíîãî äåéñòâèÿ ðàäèîýëåìåíòîâ àïïàðàòà, êàê ïðàâèëî, íåâåëèê, è çíà÷èòåëüíàÿ äîëÿ ýíåðãèè ïèòàíèÿ ïðåâðàùàåòñÿ â òåïëîâóþ ýíåðãèþ, ÷òî ïðèâîäèò ê ïåðåãðåâó àïïà- ðàòóðû â öåëîì. 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Òàê, äëÿ êâàäðàò- íîãî áðóñà ïðè ïåðåõîäå îò 1    (êóá) ê 2 , 0    ïàðàìåòð ф F ìèíèìèçèðóåòñÿ äî 0,4 ïðè     і è äî 0,7 ïðè 1 і    . Äëÿ ôîðìû êâàäðàòíîé ïëàñòèíû ïðàêòè÷åñêè óæå ïðè 1    і ïàðàìåòð ôîðìû ñîõðàíÿåò ïîñòîÿííîå çíà÷åíèå, áëèçêîå ê ïðåäåëüíîìó, ò.å. 1 F ф  . Òàêèì îáðàçîì, äëÿ ìèíèìèçàöèè ïàðàìåòðà ôîð- ìû íåîáõîäèìî íàãðåòîé çîíå àïïàðàòà ïðèäàâàòü ôîðìó êâàäðàòíîãî áðóñà [1 –3]. Ïðè ýòîì ïëàòû ñ ýëåìåíòàìè ìîãóò èìåòü êâàäðàòíóþ ôîðìó (ðèñ. 2,à) ñ íàèìåíüøèìè ðàçìåðàìè min l 2 min l 2  èëè ôîðìó ïðÿìîóãîëüíèêà y l 2 min l 2  (ðèñ. 2,á), ðàñ- ïîëàãàÿñü ñîîòâåòñòâåííî ïåðïåíäèêóëÿðíî ê áîëü- øîé èëè ìàëîé îñè êâàäðàòíîãî áðóñà. n n-1 2 3 n 1 n- 1 2 3 а) б) 1 2l z 2l x 2 l y 2 l z 2 l x à á Ðèñ. 2. Ðàçìåùåíèå ïëàò ñ ýëåìåíòàìè â íàãðåòîé çîíå îïòèìàëüíîé ôîðìû êâàäðàòíîãî “ áðóñà ” Ïðàêòè÷åñêè äåôîðìèðîâàíèå íàãðåòîé çîíû ê ôîðìå êâàäðàòíîãî áðóñà îãðàíè÷åíî íàèáîëüøèì ëèíåéíûì ðàçìåðîì ýëåìåíòîâ э max l : э max min l l  , (5) ò.å. ïðåäåëüíîå çíà÷åíèå      äîëæíî áûòü çàìåíåíî ñ ó÷¸òîì îãðàíè÷åíèÿ (5) 1         ; . min       (6) Âûðàæåíèå (6) ÿâëÿåòñÿ ïðåäåëüíûì óñëîâèåì ìèíèìèçàöèè ïàðàìåòðà ф F , à ïðàêòè÷åñêè îçíà- ÷àåò, ÷òî âñå ýëåìåíòû ðàñïîëàãàþòñÿ â ðÿä íà îäíîé óñëîâíîé ïëàòå, âûòÿíóòîé âäîëü îñè  . Òàêèì îáðàçîì, øèðèíà x l è âûñîòà z l îïðåäåëÿ- þòñÿ ëèíåéíûìè ðàçìåðàìè ýëåìåíòà э x l è э z l , à ïðîòÿæ¸ííîñòü (äëèíà ïëàòû) – êîëè÷åñòâîì ýëå- ìåíòîâ (n), âõîäÿùèõ â ñîñòàâ íàãðåòîé çîíû ñîãëàñíî ýëåêòðè÷åñêîé ñõåìå àïïàðàòà, è ëèíåé- íûì ðàçìåðîì ýëåìåíòà э y l . 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Ww ߁*)!XMͲ>s5:O#E\NŇqK{s J;quJ0G:N7+.9hc;JFkn*xkE \Mc-N|HĚ^Ԟ T6v"mvsثi;e'Czm]=nMM3&;NNd{Jn\,ɢudقߛرO᡻'㷘-Ky Y~"6Y?bo.d񹘒֣7rD\k̵p,`.BowkT fS]_)tnm* ReѬnt|fͳEǯ^t|m,W~_³6YSJ]SWݴyusfz6YEu'>M>^DpX$?MfzK{}iLz_Dڿ4. ɀ+&?YDrq?`1~IR\06>&8Ǽu?{1ll-17ɯgo:>/>S{ǜP1+>&2IOl&xC|~Lq64T vYqT;x큾FԖspIs>uNT&_C}U0U݂]6K/ {8u_Ir||z3NbZ?m`hih>Ӱ 61q@ְy>:@p O8F:uݨKA5|ߟ)bV6Cm?񕰦Z܌ߓVe3Rgߒr2<Ƈ_>|alk| .ȋLL3W^C\Уnӵʰo4v~;NubP~Lsߩud=0hKy! i885#5KعbE#~?](b;,7\}"9ke Nen!΃DEm,,?29 ÐÈ, 2003, ¹ 4 ýëåìåíòîâ ñîñòàâëÿåò ñîòíè, òûñÿ÷è è áîëåå, ïðàê- òè÷åñêè íå ïðèåìëåìî. Óñëîâíóþ ïëàòó (ðèñ. 3,à) íåîáõîäèìî ðàçáèòü íà ó÷àñòêè, èç êîòîðûõ ìîãóò áûòü ñîñòàâëåíû îäíà èëè, êàê ïðàâèëî, íåñêîëüêî ïëàò, ðàñïîëîæåííûõ äðóã ïîä äðóãîì èëè ðÿäîì (ñì. ðèñ. 2). Äëÿ ïåðåõîäà îò óñëîâíîé ïëàòû ê íàáîðó ðåàëüíûõ ïëàò íåîáõîäèìî ïðåäóñìîòðåòü çàçîð x  ïî øè- ðèíå ïëàòû è z  ïî âûñîòå ýëåìåíòà â ñëó÷àå ïåðåõîäà ê ìíîãîïëàòíîé êîíñòðóêöèè íàãðåòîé çîíû (ðèñ. 3,á). Ýòî îçíà÷àåò, ÷òî êàæäûé ýëåìåíò çàíèìàåò íåêîòîðóþ ÿ÷åéêó îáú¸ìîì эі V  : ) l ( ) l ( ) l ( V і і і і і і z zэ y yэ x xэ эі           . (7) n n-1 n n- 1 n- 2 n- 3 6 7 8 5 1 2 3 4 1 2 3 l хэ а) б) х l у э à á Ðèñ. 3. Ñòðóêòóðíàÿ ñõåìà ðÿäíîãî ðàñïîëîæåíèÿ íà óñëîâíîé ïëàòå ñ ïåðåõîäîì ê ðåàëüíûì ïëàòàì 1,2,3, n – ýëåìåíòàðíàÿ ÿ÷åéêà ñ îäíèì ýëåìåíòîì: à – óñëîâíàÿ ïëàòà; á – ðàçëè÷íûå ïëàòû Èíòåðåñóþùèé íàñ îáú¸ì íàãðåòîé çîíû ìîæåò áûòü ïîëó÷åí ñóììèðîâàíèåì ýëåìåíòàðíûõ îáú¸- ìîâ âñåõ ýëåìåíòîâ: . V V эі n 1 і     (8) Äëÿ ñëó÷àÿ, êîãäà íàãðåòàÿ çîíà ñîáèðàåòñÿ èç íåñêîëüêèõ ïîäïîñëåäîâàòåëüíîñòåé îäíîòèïíèõ ïî ðàçìåðàì ýëåìåíòîâ: эі n 1 і і V m V      , (9) ãäå і m – êîëè÷åñòâî îäíîòèïíèõ ýëåìåíòîâ â ïîäïîñëåäîâàòåëüíîñòè;  n – êîëè÷åñòâî ïîäïîñëå- äîâàòåëüíîñòåé. Ïðåîáðàçóåì âûðàæåíèå (7), ââåäÿ ïîíÿòèå î êîýô- ôèöèåíòå ïëîòíîñòè ìîíòàæà: і max l м    , (10) çäåñü эі max l max l  , . z , y , x і  (11) C ó÷¸òîì (10) âûðàæåíèå äëÿ îáú¸ìà ýëåìåíòàðíîé ÿ÷åéêè ïðèìåò âèä: * l V z y x м м м 3 max э        ) 1 )( 1 )( 1 ( * nz э z ny yэ nx xэ          . (12) Åñëè ëèíåéíûå ðàçìåðû ýëåìåíòà è çàçîðû i  áëèçêè ìåæäó ñîáîé, òî 1 э э э z y x       ; n z м м у х        è âûðàæåíèå (12) óïðîùàåòñÿ: 3 n n 3 max э ) 1 ( l V      . (13) Ñîîòâåòñòâåííî, îáú¸ìû íàãðåòîé çîíû áóäóò îï- ðåäåëÿòñÿ ïî ôîðìóëàì: – ìîíòàæ âûñîêîé ïëîòíîñòè ( 1 м   ) ; l m V 3 max n 1 і і і     (14) – ìîíòàæ ñðåäíåé ïëîòíîñòè ( 1 м   ) ) ( f l m V n 3 max n 1 і і і       , (15) ãäå 3 n n n ) 1 ( ) ( f      ; (16) – ìîíòàæ ìàëîé ïëîòíîñòè ) 1 ( м      8 і m 8 v 3 max і l . (17) Ïðè çàäàííîì äîïóñòèìîì íàèìåíüøåì ëèíåéíîì ðàçìåðå íàãðåòîé çîíû min l 2 â ôîðìå ïàðàëëåëèïè- ïåäà å¸ ïîâåðõíîñòü âûðàæàåòñÿ ñëåäóþùåé çàâè- ñèìîñòüþ ÷åðåç îáú¸ì:   . l v s 0 0 0 z y x min        (18) Äëÿ îïòèìàëüíîé ôîðìû êâàäðàòíîãî áðóñà 1 0 0 y x     è âûðàæåíèå (18) ïðèìåò âèä   ; 2 l v s 0 z min    (19) v l 8 3 mіі z 0   . (20) Åñëè 2 l m³n íå çàäàíî, òî â ïåðâîì ïðèáëèæåíèè ïîâåðõíîñòü íàãðåòîé çîíû îïðåäåëÿåì òàê: 3 2 v 6 s  (21) èëè ïîâåðõíîñòü êîæóõà èç ñóììàðíîé ðàññåèâàå- ìîé ìîùíîñòè è äîïóñòèìîé ñðåäíåé òåìïåðàòóðû íàãðåòîé çîíû [1]:   4 p S k . 3. Âûâîäû 1. 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Ðàññìîòðèì ÀÀÑ, â êîòîðîé ðåàëèçîâàí àëãîðèòì àäàïòàöèè ÌÑÊÎ. Äëÿ ââåäåíèÿ èñêóññòâåííûõ øóìîâ âîñïîëüçóåìñÿ öåïüþ îòðèöàòåëüíîé îáðàò- íîé ñâÿçè ñ íåêîòîðûì êîýôôèöèåíòîì óñèëåíèÿ  , äîïîëíèòåëüíî ââîäèìîé â ðàññìàòðèâàåìûé àëãîðèòì. Òàêîé ñïîñîá ââåäåíèÿ èñêóññòâåííîãî øóìà ïðåäñòàâëÿåòñÿ íàèáîëåå ïðîñòûì è êîíñò- ðóêòèâíûì. Äëÿ íåïðåðûâíûõ ñèãíàëîâ àëãîðèòì àäàïòàöèè i -ãî âåñîâîãî êîýôôèöèåíòà i w ñ ó÷åòîì ââåäåíèÿ èñêóññòâåííîãî øóìà è ñìåùåíèÿ íàïðÿæåíèÿ i  ïðè ïåðåìíîæåíèè ñèãíàëîâ ìîæåò áûòü ïðåäñòàâëåí â âèäå: i i i i w ] ) t ( x ) t ( [ k 2 dt dw       . (1) Çäåñü     i i w ) t ( x ) t ( v ) t ( – ñèãíàë îøèáêè; ) t (  – ýòàëîííûé ñèãíàë;  ) t ( x i – ñèãíàë íà âûõîäå i-é àíòåííû. Ñ ó÷åòîì (1) äëÿ ñðåäíåãî çíà÷åíèÿ âåêòîðà âåñîâûõ êîýôôèöèåíòîâ W ñïðàâåäëèâî ìàòðè÷íîå óðàâíå- íèå ) S ( k 2 W W k 2 dt dW        , ãäå  – êîððåëÿöèîííàÿ ìàòðèöà ñèãíàëîâ íà âõîäå ñèñòå- ìû àäàïòèâíîé îáðàáîòêè ñ ýëåìåíòàìè )} t ( x ) t ( x { M f j i j i  ; S – âåêòîð îïîðíîãî ñèãíà- ëà ñ ýëåìåíòàìè )} t ( x ) t ( v { M ) t ( s i i  ;  – âåêòîð ñìåùåíèé íàïðÿæåíèé ñ ýëåìåíòàìè ) t ( i  . Ïðåäñòàâèì êîððåëÿöèîííóþ ìàòðèöó âõîäíûõ ñèãíàëîâ  â âèäå äâóõ ñîñòàâëÿþùèõ, îïðåäåëÿ- åìûõ âíåøíèìè ìåøàþùèìè ñèãíàëàìè, âíåøíè- ìè øóìàìè ) ( c  è âíóòðåííèìè øóìàìè ÀÀÑ n  , ò.å. n c      .  ñëó÷àå íåçàâèñèìûõ è îäèíàêîâûõ ïî óðîâíþ âíóòðåííèõ øóìîâ â êàíà- ëàõ ìàòðèöà I 2 n n    , ãäå 2 n  – äèñïåðñèÿ âíóòðåííåãî øóìà â êàæäîì êàíàëå àäàïòèâíîé ñèñòåìå. Ó÷èòûâàÿ ñêàçàííîå, ) S ( k 2 W ) k 2 ( W k 2 dt dW 2 n c          . (2) Èç óðàâíåíèÿ (2) ñëåäóåò, ÷òî â ðàññìàòðèâàåìîé êîíôèãóðàöèè óâåëè÷åíèå êîýôôèöèåíòà îáðàò- íîé ñâÿçè  ýêâèâàëåíòíî äîáàâëåíèþ âíóòðåííèõ øóìîâ â ÀÀÑ. Ïðè ýòîì ýêâèâàëåíòíàÿ äèñïåðñèÿ äîáàâî÷íûõ øóìîâ k 2 2 d    . Ââåäåì ìàòðèöó Q , äèàãîíàëèçèðóþùóþ êîððåëÿ- öèîííóþ ìàòðèöó  . Ïóòåì ñîîòâåòñòâóþùåãî ïðåîáðàçîâàíèÿ     Q Q 1 ïîëó÷èì âìåñòî  åå äèàãîíàëüíîå îòîáðàæåíèå  – äèàãîíàëüíóþ ìàòðèöó, ýëåìåíòàìè êîòîðîé ÿâëÿþòñÿ ñîáñòâåí- íûå çíà÷åíèÿ ìàòðèöû  .  ñîîòâåòñòâèè ñ ââåäåí- íîé ìàòðèöåé Q îïðåäåëèì ñëåäóþùèå âåêòîðû: W Q H 1   ; S Q P 1   ;    1 Q D . Ñ ó÷åòîì ââåäåííûõ îáîçíà÷åíèé âûðàæåíèå (2) ïðèìåò âèä: ) D P ( k 2 H ] I ) ( [ k 2 dt dH 2 a 2 n          . Äëÿ i h -é êîìïîíåíòû âåêòîðà H áóäåì èìåòü ) d p ( k 2 h ) ( k 2 dt dh i i i 2 a 2 n i i         , (3) ãäå i  – i-å ñîáñòâåííîå çíà÷åíèå ìàòðèöû  ; i i d , p – i-å êîìïîíåíòû âåêòîðîâ P è D ñîîòâåò- ñòâåííî.  óñòàíîâèâøåìñÿ ðåæèìå, ñîãëàñíî (3), ïîëó÷èì, ÷òî 2 a 2 n i i i i d p ) ( h         . Ìîùíîñòü ñèãíàëà íà âûõîäå àäàïòèâíîé ñèñòåìû â ýòîì ñëó÷àå áóäåò îïðåäåëÿòüñÿ âåëè÷èíîé W ) I ( W P 2 n c T      . (4) Ñ ó÷åòîì ìàòðèöû ïðåîáðàçîâàíèÿ Q ñîîòíîøåíèå (4) ïðèìåò âèä:           ) ( h H ) I ( H P 2 n i 2 i 2 n T . Âåëè÷èíà ìîùíîñòè  P îïðåäåëÿåòñÿ â îáùåì ñëó÷àå ñîñòàâëÿþùèìè ïîëåçíîãî ñèãíàëà, âíå- øíèõ ìåøàþùèõ ñèãíàëîâ è ïîìåõ, à òàêæå òåïëî- âûìè øóìàìè, âêëþ÷àÿ øóìû ïðèåìà. Ïðè ýòîì áóäåì ïîëàãàòü, ÷òî ïîëåçíûé ñèãíàë îïðåäåëÿåòñÿ òîé ÷àñòüþ âõîäíîãî ñèãíàëà ) t ( x i , êîòîðàÿ êîð- ðåëèðóåò ñ ýòàëîííûì ñèãíàëîì ) t ( v . Ñ ó÷åòîì ýòîãî ìîùíîñòü ïîëåçíîãî ñèãíàëà íà âûõîäå ñèñ- òåìû ìîæåò áûòü îïðåäåëåíà âåëè÷èíîé 2 max 2 T c V ) S W ( P  . Òîãäà îòíîøåíèå ìîùíîñ- òè ïîëåçíîãî ñèãíàëà ê ìîùíîñòè ïîìåõ è øóìîâ íà âûõîäå áóäåò îïðåäåëÿòüñÿ âåëè÷èíîé c c P P P     . (5) Ó÷èòûâàÿ ïðåîáðàçîâàíèå ñîîòâåòñòâóþùèõ êîì- ïîíåíò ñ ó÷åòîì ìàòðèöû Q , ñîîòíîøåíèå (5) ìîæåò áûòü ïðåäñòàâëåíî â âèäå:                            2 2 a 2 n i i i i 2 2 a 2 n i 2 n i 2 i i 2 2 a 2 n i i i i ) ( p ) d p ( ) ( ) ( ) d p ( ) ( p ) d p ( . (6) Ïîêàæåì, ÷òî â ñëó÷àå êîëèíåàðíîñòè âåêòîðîâ D è P ñìåùåíèÿ íàïðÿæåíèé íå âëèÿþò íà âûõîäíîå ÎÑÏØ. Ïóñòü i i Cp d  , òîãäà ïðîèçâåäÿ óêàçàí- íóþ çàìåíó â (6), ïîñëå ïðåîáðàçîâàíèÿ ïîëó÷èì, ÷òî Tx tUE뾬I@Q@ 8+*(:# 2"A0 "qёG|uIx swNꥺݪADj'AKEV#A*'r]شN*+bHLhlJ&M^vMo" ,pV6r Z]T- h9J^ܧ JBRYwH4ظRMђ) K7,C?e@d0٭*[_B}3|)έ-+d{lvB7GEv%rF3m4ԯF2թ,CN{n|FП! 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(7) Ïðè ýòîì ñëåäóåò çàìåòèòü, ÷òî âûðàæåíèå (7) íå çàâèñèò îò Ñ. Èç (6) ñëåäóåò, ÷òî ïðè áîëüøîì ðàçáðîñå â âåëè÷èíå ñîáñòâåííûõ çíà÷åíèé i  ñìåùåíèÿ, ñîîòâåòñòâóþùèå áîëüøèì ñîáñòâåí- íûì çíà÷åíèÿì, èìåþò îòíîñèòåëüíî ìàëîå âëèÿ- íèå.  ñëó÷àå ìàëûõ ñîáñòâåííûõ çíà÷åíèé è ìàëîé âåëè÷èíû ) ( 2 a 2 n    äàæå ìàëîå ñìåùåíèå i d áóäåò îêàçûâàòü ñóùåñòâåííîå âëèÿíèå íà âåëè÷è- íó âûõîäíîãî ÎÑÏØ. Äëÿ àíàëèçà (6) ìîäåëèðîâàëàñü ÷åòûðåõýëåìåíò- íàÿ ýêâèäèñòàíòíàÿ àíòåííà ñ ïîëóâîëíîâûì ðàñ- ñòîÿíèåì ìåæäó ýëåìåíòàìè ïðè âîçäåéñòâèè èñ- òî÷íèêà ïîëåçíîãî ñèãíàëà è äâóõ èñòî÷íèêîâ âíåøíèõ ïîìåõ. Íàïðàâëåíèå ïðèõîäà ïîëåçíîãî ñèãíàëà ñîâïàäàëî ñ îïòè÷åñêîé îñüþ ðàññìàòðèâà- åìîé àíòåííîé ñèñòåìû, à íàïðàâëåíèå ïðèõîäà ïåðâîé è âòîðîé ïîìåõè õàðàêòåðèçîâàëîñü çíà÷å- íèÿìè óãëîâ  , ðàâíûõ 20 è 40 ãðàäóñîâ ñîîòâåò- ñòâåííî, ãäå  – óãîë, îòñ÷èòûâàåìûé îò îïòè÷åñ- êîé îñè àíòåííîé ñèñòåìû. Ìîùíîñòü èñòî÷íèêà ïîëåçíîãî ñèãíàëà âûáèðàëàñü òàêîâîé, ÷òî íà âûõîäàõ àíòåííûõ ýëåìåíòîâ äåéñòâîâàëè ñèãíàëû, ìîùíîñòü êîòîðûõ ñîñòàâëÿëà 2,5 åäèíèöû. Èñòî÷- íèêè ïåðâîé è âòîðîé ïîìåõè ñîçäàâàëè íà âûõîäå àíòåííûõ ýëåìåíòîâ ïîìåõè, ìîùíîñòü êîòîðûõ íà 10 è 20äÁ ñîîòâåòñòâåííî ïðåâûøàëà ìîùíîñòü ïîëåçíîãî ñèãíàëà.  òàáëèöå ïðèâåäåíû â ïîðÿäêå óáûâàíèÿ ñîá- ñòâåííûå çíà÷åíèÿ ìàòðèöû c  äëÿ ýòîãî ñëó÷àÿ è ñîîòâåòñòâóþùèå èì çíà÷åíèÿ i p , áåç ó÷åòà âíóò- ðåííèõ øóìîâ ïðèåìà. Íà ðèñ.1-4 ïðèâåäåíû çàâèñèìîñòè  îò óðîâíÿ øóìîâ ïðèåìà 2 n  ïðè îòñóòñòâèè èñêóññòâåííî ââîäèìûõ â ÀÀÑ øóìîâ ) 0 ( 2 a   . Çàâèñèìîñòü íà ðèñ.1 ñîîòâåòñòâóåò îòñóòñòâèþ ñìåùåíèÿ, à çàâèñèìîñòè íà ðèñ.2-4 èëëþñòðèðóþò âëèÿíèå ñìåùåíèé, ðàâíûõ åäèíèöå, äëÿ ñîîòâåò- ñòâóþùèõ ñîáñòâåííûõ çíà÷åíèé. Èç àíàëèçà ïðè- âåäåííûõ çàâèñèìîñòåé ñëåäóåò, ÷òî äëÿ íèçêèõ óðîâíåé øóìîâ ïðèåìà 2 n  íàëè÷èå ñìåùåíèÿ âûçûâàåò ñóùåñòâåííîå ñíèæåíèå âåëè÷èíû  . 0 2 4 10 0 10   n2 0  ( )  n2  0 0 0 0              Ðèñ.1 0 2 4 10 0 10   n2 0  ( )  n2  1 0 0 0              Ðèñ.2 0 2 4 10 0 10   n2 0  ( )  n2  0 1 0 0              Ðèñ. 3 0 2 4 10 0 10   n2 0  ( )  n2  0 0 1 0              Ðèñ. 4 Ïðè ýòîì åñëè ñîîòâåòñòâóþùèå ñîáñòâåííûå çíà÷å- íèÿ äîñòàòî÷íî âåëèêè, òî íàëè÷èå ñìåùåíèé îêàçû- âàåò íåçíà÷èòåëüíîå âëèÿíèå íà óìåíüøåíèå  . Àíàëîãè÷íûå çàâèñèìîñòè ïðèâåäåíû íà ðèñ. 5 äëÿ ñîáñòâåííîãî çíà÷åíèÿ 4  . 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<~Kqn։9'=ټb3Uy\9z늯7)gV&AbX)uI#3 mW@uGaKZ~(odg(#_eu7*ŵ 4_)m^5,:] Nl4ycH>2 K9QtF0oԵ^tPFOµ p9[^qzp_g(Ü*wlxY >(gc9Mlq OPXmci˓]_HkmD}',,B33 ÐÈ, 2003, ¹ 4 ñîîòâåòñòâåííî. Îäíàêî ïðè äàëüíåéøåì óâåëè÷å- íèè ìîùíîñòè èñêóññòâåííîãî øóìà íàáëþäàåòñÿ îáùåå ñíèæåíèå âåëè÷èíû  . Íà ðèñ. 6 ïðèâåäåíû çàâèñèìîñòè  îò ìîùíîñòè èñêóññòâåííîãî øóìà ïðè äàëüíåéøåì óâåëè÷åíèè 2 a  . Çàâèñèìîñòè ñîîòâåòñòâóþò çíà÷åíèÿì 2 a  , ðàâíûì 20, 50 è 100 ñîîòâåòñòâåííî. Ñëåäóåò îòìåòèòü, ÷òî ñíèæåíèå âåëè÷èíû  íàñòóïàåò, êîãäà ìîùíîñòü èñêóññò- âåííîãî øóìà ïðåâûøàåò ìîùíîñòü ïîëåçíîãî ñèãíàëà íà âõîäå ÀÀÑ. Ïðè îòñóòñòâèè ñìåùåíèé óâåëè÷åíèå ìîùíîñòè èñêóññòâåííîãî øóìà, ââî- äèìîãî äîïîëíèòåëüíî, ïðîÿâëÿåòñÿ àíàëîãè÷íî íàëè÷èþ ñìåùåíèé. 0 2 4 5 0 5 10   n2 0.0  ( )   n2 1.0  ( )   n2 2.5  ( )   n2 5.0  ( )   n2 10.0  ( )  n2  0 0 0 3              Ðèñ. 5 0 2 4 10 0 10   n2 20.0  ( )   n2 50.0  ( )   n2 100  ( )  n2  0 0 0 3              Ðèñ. 6 Íà ðèñ. 7 è 8 ïðèâåäåíû àíàëîãè÷íûå çàâèñèìîñòè ïðè òîé æå âåëè÷èíå ñìåùåíèÿ, íî ñîîòâåòñòâóþ- ùåãî ðàçëè÷íûì ñîáñòâåííûì çíà÷åíèÿì êîððåëÿ- öèîííîé ìàòðèöû. Èç àíàëèçà ïðèâåäåííûõ çàâè- ñèìîñòåé ñëåäóåò, ÷òî íàëè÷èå ñìåùåíèÿ íàïðÿæå- íèé â óìíîæèòåëÿõ ÀÀÑ ÿâëÿåòñÿ ñåðüåçíîé ïðî- áëåìîé äëÿ âûðîæäåííîé èëè ïëîõî îáóñëîâëåííîé êîððåëÿöèîííîé ìàòðèöû ñèãíàëîâ è ïîìåõ, ïðè- íèìàåìûõ íà áîðòó ËÀ. Îäíîé èç ïðè÷èí ïëîõîé îáóñëîâëåííîñòè ìàòðèöû ìîæåò áûòü íåäîñòàòî÷- íûé óðîâåíü øóìîâ ïðèåìà.  ýòîì ñëó÷àå ãðàäè- åíòíûå àëãîðèòìû àäàïòàöèè îêàçûâàþòñÿ ìàëî- ýôôåêòèâíûìè.  ðàññìàòðèâàåìîé ñèòóàöèè ââå- äåíèå äîïîëíèòåëüíî â ÀÀÑ èñêóññòâåííûõ øóìîâ ÷àñòè÷íî ïîçâîëÿåò íàèáîëåå ïðîñòûì ïóòåì ðå- øèòü óêàçàííóþ ïðîáëåìó. Ïðàâäà, ñóùåñòâóåò îãðàíè÷åíèå íà ìàêñèìàëüíûé óðîâåíü òàêèõ øó- ìîâ â óñòàíîâèâøåìñÿ ðåæèìå ôóíêöèîíèðîâàíèÿ àäàïòèâíîé ñèñòåìû. 0 2 4 10 0 10   n2 0.0  ( )   n2 1.0  ( )   n2 2.5  ( )   n2 10.  ( )   n2 15.  ( )  n2  0 0 3 0              Ðèñ. 7 0 2 4 10 0 10   n2 0.0  ( )   n2 1.0  ( )   n2 2.5  ( )   n2 10.  ( )   n2 15.  ( )  n2  3 0 0 0              Ðèñ. 8 Òàêèì îáðàçîì, â ðàáîòå ðàññìîòðåí àëãîðèòì àäàïòèâíîé îáðàáîòêè ñèãíàëîâ è ïîìåõ â àíòåí- íûõ ñèñòåìàõ ñ èñïîëüçîâàíèåì èñêóññòâåííûõ øóìîâ, äîïîëíèòåëüíî ââîäèìûõ â ñèñòåìó. Ïîêà- çàíî, êàê êîíñòðóêòèâíî ýòî âîçìîæíî îñóùå- ñòâèòü. Ñëåäóåò îòìåòèòü, ÷òî äàëüíåéøèì ðàçâè- òèåì ðàññìàòðèâàåìîãî ïîäõîäà íà îñíîâå ââåäåíèÿ èñêóññòâåííûõ øóìîâ â ÀÀÑ ÿâëÿåòñÿ ïðèìåíåíèå èñêóññòâåííûõ øóìîâ ñ èçìåíÿþùèìñÿ âî âðåìåíè óðîâíåì â ïðîöåññå àäàïòàöèè, íà÷èíàÿ ñ ìîìåíòà åãî íà÷àëà è äî ìîìåíòà äîñòèæåíèÿ óñòàíîâèâøå- ãîñÿ çíà÷åíèÿ. Òåõíè÷åñêè ýòó çàäà÷ó ìîæíî ðå- øèòü ïóòåì ñîîòâåòñòâóþùåãî èçìåíåíèÿ â ïðîöåñ- ñå àäàïòàöèè êîýôôèöèåíòà óñèëåíèÿ  , ò.å.   A T , 0 t ), t (     , ãäå A T – ìîìåíò íàñòóïëå- íèÿ óñòàíîâèâøåãîñÿ ðåæèìà àäàïòàöèè. Ïðè ýòîì äèñïåðñèÿ k 2 ) t ( ) t ( 2 a    â íà÷àëüíûé ìîìåíò ìî- æåò äîñòèãàòü çíà÷åíèé, ñóùåñòâåííî ïðåâûøàþ- ùèõ óðîâåíü ïîëåçíîãî ñèãíàëà, ÷òî îáåñïå÷èò áîëåå áëàãîïðèÿòíûå óñëîâèÿ äëÿ ðàáîòû ãðàäèåí- òíûõ àëãîðèòìîâ. Çàòåì âåëè÷èíà 2 a  äîëæíà óìåíüøàòüñÿ äî óðîâíÿ, ìåíüøåãî óðîâíÿ ïîëåçíî- ãî ñèãíàëà, íî áîëüøåãî óðîâíÿ øóìîâ ïðèåìà.  ýòîì ñëó÷àå ìîæíî äîáèòüñÿ ïîâûøåíèÿ ýôôåê- òèâíîñòè ïðîñòûõ è êîíñòðóêòèâíûõ ãðàäèåíòíûõ àëãîðèòìîâ àäàïòàöèè áîðòîâûõ ÀÀÑ â óñëîâèÿõ, õàðàêòåðèçóåìûõ ïëîõîé îáóñëîâëåííîñòüþ êîððå- ëÿöèîííîé ìàòðèöû ïðèíèìàåìûõ ñèãíàëîâ è íà- ëè÷èåì ñìåùåíèé. ^<xypUU&BXQ5gF1`!e*ARDqǩB1HZ.+ .qAE\, KDT4(ücn=I@)9}ӧO{"*" @[d_'F"7Wgf+R߾'S3\ٚKd>ТtةL-*eKGi%hϦG'Rc*s('"-./W;UֲvejQ m2%"@"/.cJH9io0-uZ2 7ʘocANv TmrkoݺHz_<~M$9].wu\`de)CYg?dعǏܟ/v]p@|EM I? ~)ڜ[R.~LVXfs4md$L,܎o!C7y# kM}L7GE7&D>Fd7vv.'=NxIPْQ$LT[$L" f?E쳾݊S(Bo!~ǫj[mGJwlb.*pl|_@vlO#u+7^< bhp>{I5Nt4#:1O߸>t}k}ug[wΖ0/&E6Ol>s}$kqNccDsR椽TAC @|;1y9jvke>;_oy9o6>U3tė!|v[pOB6 %wlrtJUݟ4wߑп<{\V\?x7K/6nm"@xc\p:qem E؞o|"~yN Y/\cdbc4u;C5 I ֒.%2x*:u'[ 'JC*D~5z jd"G»/oo#vb% Q+>j'Ko~d{.jcN6'2tGLp8~6 N1{)cpGV/_Dޒ|xK3WcT sIˌ[Cvmu¼cYU:=Ռ/Z l.΋ؘ cܗJ=M!f䫡_ b|\x{b_EL/r֠{x*䂍Jqh'盫 ]^}85F5)5ϭMuߏE5:#̅*9y` dtz}Z4j;<~HӪ1Syh18|~/0;[q6$z55M K$fnCr/~~.Wik\zMIZ:m[t۶5۶P۷诏Z~~|T:N1vjYZm[V-OQfudPY"uDc^v 5eL";7Ehkrݻqjۉʃ+H'2\m}0)x W zn+`~s_ۍ=_o#7 ܶsqA<OVoBz @B=z @聣' =z @B=ɎMicrosoft Office Document Image Writer 11.03.1897.00 2005:12:01 14:57:59  rH bj(162&/Hx1/*{tPx͛}luǯ 45kӎ DKn]LrVyR:J;6SiLI7їT[?PiLcҪi,˖L1Ru lyiB}?>_N>>s<%W0 `n P~~(ٻB( %¿A`GsRKkBacT/|2 cgYz˳|QucYK@lTZKTn&/ Cc?"} Zb (*a2ۮetVN5FJ{~ϬE1=_k7rgKmޒۺw,}oó>2xn}0h{̫Mi%ۣ~dXF< 6_ /޷T&An-vaCöe?4a&[?DoNJ|ujy~M <9:%9ϹkgohE柘i:CW W,f9r]KՎCF0v.dLר5\Z\q)7 Osgϻ9`I'<|[}G0GXx5j>~CWrx\#kq]>2SaE(R?]-t7ƦY>^QN<[%y"4y >?[mނy~RSw)l9y~?!7.wR_j/ɫۥ%푯*nOͻ8u l9y~vIxq۩> Luqvtښ:oC)n{ۏq{Q em70q 7L~r@ӂA@@}VyS'Op.gC EX:~Hy\-Szm3Y .zI ՕX!`|!+y|xu#3a47^Rnq~Ug&7nїᗷ +З ^mpec=/[o6#oP?<^Io7GCcs}+߆|ahu_#&}"\ aG(Ba#xDc [ea6̙DX'Z7˻;W@fiqyV1`U{I\27ul^is'.-|q]!7Ш`dwE{:": t#) F oްz?0{1{nfl[i[;6r:㨓ǢL`ձ>Q5ܨ̌B/Vs3n7j커1oK%.wJ3aX ~9vg"D2)4ςp{]km܃`#Sv{&6b3|mY@trvGW6Kea6qDˣ=+C: \>%xZ1k@:6Nhm^li-1Qg[`"fRw,h52g ˢt0\dnlzu2Ͽ8xCZus<,`Hrj/0b*[Oľ!bw(O2!db@x>FOf 8Ge)1N;;7;e?0gz}~~8#~9!ߟ)߱:\ ~<¶;yE^lh &@G';5Xmhu֪3I+< VOCUOO>=~`nNٖO C~?grmY%CG>_!k&>VT>ǟ;NS<Wvܨpj`6G.پӎ6n/_7wȵhGձs ЭK>-_P#ӝq6 fx<$Cɛ'@)0\4tzIkrDȰ:b f>>ZWH QUox}UN@czRTݼ<] ?:3:l=y`nour]|+~yP!S:<*k8{ 8 O/79JOwpK;3؅9Mц^ڑ?=?y]]G>}c]:|0>1X(ZX/Zc]СN}`붛GS֑2mݷGe@!|YMb^VM] 7ΊS:rW.WIRArw]]tobۘcX 4_@_'nˀ9ky=. ^UnhYX3_A#\=ݰcQnRXi2<"7Y<]J_s} {`X>_#aM4P;6rA|'8VbB)`Y-~Yբx|W]'M_ qyg]sl8<Ǭ#bZoJ)vavQ\&d_Ж:/9ٯȄF|hdX.?ׅ:~vXX9d_jSCqZ OE7d9@vPo*_OsM}ӭ`zke\}K'mlb~b+ll1/ܖ:k0~r\g_!/C(BK({Zb>6&Z_oo9mL֤j!SV&r}s#-r$g+uˢw1D|]uMCP|r 9Y&2!.Ǚc u>ޕ0݂Onz,D[1;ձ1|}x=9}NYLܻ22~9_S+[2FīQ(Guܦ聎4Ə{4ƴ Âs`P8ŢMw~A;ΈJ8c-[|__Ĵf:5{ I$D_Fo6SLN?dxfm3j49so87 v~yє!'9.5)(vgtH\ݠaN6|i[=tG99sݘwN9UunͻA=o Eh1OOeTwq[I^kf;cNҡt'vWghWm<2~8`(,hG2,>?.χ~VVC?:9SY<;ل;4}sgradogmLN_iX@/oںv_ܕ뷊vhS:H{<閙wE|&9&7y5 O\x"~`[ǶAs^;||G:v '^eXim|lgYsgi ]Ѧ1~[nJhn[eqӶ*DzqWC|;m,cd_oc篗潟?*Zdt 4fcZYiɹ%nvD2:_whW?WSN:àuCOtbq.ٶ C*ev1Gv.G 8o8GU3yg8y+M ߥ.^Wl vplqhK_Աx0gy5 O&ھYSC)fގD9r!cclb-?vǝ]oͯ_יvѬLh-ZtW+-Q:=E|Gn9D_ ۀ-<h2`SZh?g]7p~<yA6rN,ϓ+J_oj덢Mw\X=֟}(6-Qe+q.A-9Ѿ?)y@xGne W|2@:VG9 %E'_.UNI?^ԧ.g}Ir\SgW/?6W؀]D>ŗ~!1yo^ѿG[p?Y#?jYUań׹n~}1?,^XD[*Z],3qo*ڲ78k'rzv5~. 3828SM8`m<=~Gª~_xv|.X#+9; !7ׇz^F~uk4r7,,K 34 ÐÈ, 2003, ¹ 4 Ëèòåðàòóðà: 1. Ìîíçèíãî Ð.À., Ìèëëåð Ò.Ó. Àäàïòèâíûå àíòåííûå ðåøåòêè. Ââåäåíèå â òåîðèþ. Ì.: Ðàäèî è ñâÿçü, 1986. 448 ñ. 2. Óèäðîó Á., Ñòèðíç Ñ. Àäàïòèâíàÿ îáðàáîòêà ñèãíàëîâ: Ïåð. ñ àíãë. Ì.: Ðàäèî è ñâÿçü, 1989. 440ñ. 3. Àäàïòèâíàÿ êîìïåíñàöèÿ ïîìåõ â êàíàëàõ ñâÿçè/ Ïîä ðåä. Þ.È.Ëîñåâà. Ì.: Ðàäèî è ñâÿçü, 1988. 208ñ. 4. Ðîäèìîâ À. Ï., Ïîïîâñêèé Â.Â. Ñòàòèñòè÷åñêàÿ òåîðèÿ ïîëÿðèçàöèîííî-âðåìåííîé îáðàáîòêè ñèãíàëîâ è ïî- ìåõ â ëèíèÿõ ñâÿçè. Ì.: Ðàäèî è ñâÿçü, 1984. 272ñ. 5. Âåíñêàóñêàñ Ê.Ê. Êîìïåíñàöèÿ ïîìåõ â ñóäîâûõ ðàäèî- òåõíè÷åñêèõ ñèñòåìàõ. Ë.: Ñóäîñòðîåíèå, 1989. 264ñ. 6. Àäàïòèâíûå ðàäèîòåõíè÷åñêèå ñèñòåìû ñ àíòåííûìè ðåøåòêàìè / Æóðàâëåâ À. Ê., Õëåáíèêîâ Â. À., Ðîäèìîâ À. Ï. è äð. Ë.: Èç-âî Ëåíèíãð. óí-òà, 1991. 544ñ. 7. Ïîñïåëîâ Á.Á. Èññëåäîâàíèå ýôôåêòèâíîñòè àäàïòèâíûõ àíòåí- íûõ ñèñòåì ïðè íåñòàöèîíàðíîñòè êîíôèãóðàöèè «ïðè- åìíèê–ïåðåäàò÷èê»// Ðàäèîýëåêòðîíèêà. 2000. Ò.43, ¹3- 4. Ñ. 73–76. 8. Ïîñïåëîâ Á.Á. Èññëåäîâàíèå öåëåâîé ôóíêöèè â çàäà÷àõ àäàïòèâíîé îáðàáîòêè ñèãíàëîâ â àíòåííûõ ðåøåòêàõ // ñá. íàó÷íûõ ìàòåðèàëîâ. Âûï. 1. Õ.: Õàðüêîâñêîå ÂÂÀÓÑ, 1981. Ïîñòóïèëà â ðåäêîëëåãèþ 09.09.2003 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Õàõàíîâ Â.È. Ïîñïåëîâ Áîðèñ Áîðèñîâè÷, êàíä. òåõí. íàóê, äîöåíò, ïðîôåññîð êàôåäðû áîåâîãî ïðèìåíåíèÿ óçëîâ ñâÿçè, ÐÒÇ è áîðòîâûõ ðàäèîýëåêòðîííûõ êîìïëåêñîâ Õàðü- êîâñêîãî èíñòèòóòà ÂÂÑ Óêðàèíû. Íàó÷íûå èíòåðå- ñû: ïðîáëåìû àäàïòèâíîé îáðàáîòêè ñèãíàëîâ â ðà- äèîòåõíè÷åñêèõ ñèñòåìàõ ðàçëè÷íîãî íàçíà÷åíèÿ. Àäðåñ: Óêðàèíà, 61000, Õàðüêîâ, óë. Áàëàêèðåâà, 46, êâ.90, òåë. 30-82-20. Ãðóøåíêî Ìèõàèë Âèêòîðîâè÷, íà÷àëüíèê ó÷åáíîãî îòäåëà Õàðüêîâñêîãî èíñòèòóòà ÂÂÑ Óêðàèíû. Íàó÷- íûå èíòåðåñû: ïðîáëåìû ïîìåõîçàùèùåííîñòè àâèà- öèîííûõ ðàäèîëèíèé. Àäðåñ: Óêðàèíà, 61000, Õàðüêîâ, óë. 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APPLICATION TO STEADY-STATE ELECTROGENERATED CHEMILUMINESCENCE OLEINICK A.I., AMATORE C.A., SVIR I.B. The analytical expressions for steady state currents and electrogenerated chemiluminescence (ECL) intensities are derived for two-hemicylinder assemblies operating in the ECL generation mode based on a specific conformal mapping transform. This transform is also used for fast and accurate simulations of transient currents and ECL intensities leading to an excellent agreement at long times with the analytical steady state solutions. These results are compared to those relative to two-band assemblies to investigate the effect of hemicylindrical electrode protuberance on ECL efficiencies. This provides evidence that electrode protuberance results in a drastic increase of the ECL generation and shortening of commutation times when the gap is not too large as compared to the electrode radii. 1. Introduction Using conformal mapping techniques for the simulation of two dimensional electrochemical problems at microelectrodes represents the most efficient and easiest way of obtaining an accurate numerical solution for diffusion at microelectrodes or microelectrode assemblies. Conformal transforms allow mapping of a two- dimensional real space, where the flux lines may be extremely curved or present singularities, onto a space in which the flux lines become (almost) parallel [1,2]. It is probably the best numerical approach for the symmetrical electrode problems like diffusion at a disk [3], spherical/hemispherical [4,5] and cylindrical/ hemicylindrical [6-8] electrodes. The conformal mapping approach has been successfully used for solution of band electrode and double band microelectrode problems [4,9-11]. These all transformations then were applied by many authors for the simulation of different electrochemical problems. Furthermore, conformal mappings have the important own properties and often allow obtaining the exact solution of the problem. There are a few analytical solutions for the steady state current in double band and channel double band microelectrode generator-collector systems using the conformal mapping [4,11-13] and electrogenerated chemiluminescence (ECL) steady state intensity for the double band microelectrode system [11]. Recently we used the conformal transformation for the simulation of the double hemicylinder generator- collector microelectrode assembly [8], which to apply here for investigation of ECL in cell with two parallel- hemicylindrical microelectrodes. The analytical expressions for the steady state current and ECL intensity in cell with two microelectrode hemicylinders are presented here. Using of the paired microelectrodes or arrays of such electrodes is popular in modern electrochemistry because it offers the possibil