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ÌÀÒÅÌÀÒÈ×ÅÑÊÀß ÌÎÄÅËÜ ÍÅËÈÍÅÉÍÎÃÎ ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÑÂ× ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß ÐÅÇÎÍÀÒÎÐÍÛÕ ÑÈÑÒÅÌ Ñ ÏÎËÓÏÐÎÂÎÄÍÈÊÎÂÛÌÈ ÑÒÐÓÊÒÓÐÀÌÈ.....................................................................................................................................................................9 ÂËÀÑÅÍÊÎ Ë.À. ×ÈÑËÅÍÍÎ-ÀÍÀËÈÒÈ×ÅÑÊÀß ÌÎÄÅËÜ ÝÂÎËÞÖÈÈ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Â ÂÎËÍÎÂÎÄÅ Ñ ÄÈÑÏÅÐÑÍÎÉ ÑÐÅÄÎÉ..........................................14 ÁÀÐÀÍΠÍ.Ã., ÊËÞ×ÍÈÊ È.È., ËÎÄÛÃÈÍ Ì.À. ÊÐÈÒÅÐÈÈ ÎÖÅÍÊÈ ÏÎÑÒÎßÍÑÒÂÀ ÂÛÁÎÐÎÊ ÄËß ÀËÃÎÐÈÒÌΠÎÁÐÀÁÎÒÊÈ ÑÈÃÍÀËΠ ÖÈÔÐÎÂÛÕ ÎÑÖÈËËÎÃÐÀÔÀÕ.......20 ÂÀÑßÍÎÂÈ× À.Â., ÃÐÈÖÓÍΠÀ.Â., ËÅÁÅÄÅ  Î.Ã., ×ÓÐÞÌΠÃ.È. 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Ðèñ. 1. Ãåîìåòðèÿ çàäà÷è Ïîëå ) t , r ( E   , ) t , r ( H   ïðåäñòàâèì â âèäå ) t , r ( E ) t , r ( E ) t , r ( E 1 0        , ) t , r ( H ) t , r ( H ) t , r ( H 1 0         , (4) ãäå 0 E  , 0 H  – ïîëå èñòî÷íèêà, à 1 E  , 1 H  – ïîëå, îáóñëîâëåííîå ïðèñóòñòâèåì êîíè÷åñêîé ñòðóêòó- ðû. Ïîñêîëüêó êîíóñ ÿâëÿåòñÿ êîîðäèíàòíîé ïîâåðõ- íîñòüþ ñôåðè÷åñêîé ñèñòåìû êîîðäèíàò, à ïîëå ýëåêòðè÷åñêîãî (ìàãíèòíîãî) ðàäèàëüíîãî äèïîëÿ ïðåäñòàâëÿåò ñîáîé ïîëå Å-òèïà (Í-òèïà) îòíîñè- òåëüíî ðàäèàëüíîé êîîðäèíàòû äëÿ ðåøåíèÿ ãðà- íè÷íîé çàäà÷è ñ êîíè÷åñêîé ãåîìåòðèåé, óäîáíî ââåñòè ýëåêòðè÷åñêèé ) t , r ( ) 1 (   è ìàãíèòíûé ) t , r ( ) 2 (   ïîòåíöèàëû Äåáàÿ. Ñîñòàâëÿþùèå ýëåêòðîìàãíèò- íîãî ïîëÿ âûðàæàþòñÿ ÷åðåç ïîòåíöèàëû Äåáàÿ ïî ôîðìóëàì: à) äëÿ ïîëÿ E-òèïà (â ñëó÷àå ýëåêòðè÷åñêîãî ðàäè- àëüíîãî äèïîëÿ, 1   )   ) 1 ( 2 2 2 2 r r t r E                 0 H r  ,   ) 1 ( 2 r r r 1 E        , ) 1 ( 2 t sin H          , (5)   ) 1 ( 2 r r sin r 1 E         , ) 1 ( 2 t H          ; á) äëÿ ïîëÿ H- òèïà (â ñëó÷àå ìàãíèòíîãî ðàäèàëü- íîãî äèïîëÿ, 2   ) 0 E r  ,   2 2 2 2 2 r r t r H                  , ) 2 ( 2 t sin E            ,   ) 2 ( 2 r r r 1 H          , (6) ) 2 ( 2 t E         ,   ) 2 ( 2 r r sin r 1 H           . Ïîòåíöèàëû Äåáàÿ ) t , r ( ) (    , 2 , 1   , äëÿ ïîëíîãî ïîëÿ E  , H  (5), (6) â êàæäûé ìîìåíò âðåìåíè óäîâëåòâîðÿþò: 1) ÷åòûðåõìåðíîìó âîëíîâîìó óðàâíåíèþ ) t , r ( F ˆ ) t , r ( t a 1 ) ( ) ( 2 2 2                    ,   r  , ) t t ( f ) r r ( M r 1 ) t , r ( F ˆ 0 0 ) ( r 1 2 ) (                , (7) 2 a 1   ; 2) íà÷àëüíîìó óñëîâèþ t 0 ) ( ) (         , 0 t t  ; (8) 3) êðàåâîìó óñëîâèþ 0 t n ) ( 1 1                     , (9) 4) óñëîâèþ îãðàíè÷åííîñòè ýíåðãèè          V 2 2 ) | | | t (| . (10) Êðàåâàÿ çàäà÷à ìàòåìàòè÷åñêîé ôèçèêè â òàêîé ïîñòàíîâêå èìååò åäèíñòâåííîå ðåøåíèå.  ñîîòâåòñòâèè ñî ñòðóêòóðîé ïîëÿ (4) èñïîëüçóåì ïðåäñòàâëåíèå ) t , r ( ) t , r ( ) t , r ( ) ( 1 ) ( 0 ) (            , (11) ) R a 1 t t ( ) R a 1 t t ( f R 1 r 4 M 0 0 1 2 0 ) ( ) ( 0                  ïîòåíöèàë Äåáàÿ èñòî÷íèêà, ) (   – ôóíêöèÿ Õåâè- ñàéäà, 0 r r R     ,   1 a . 3. Ìåòîä ðåøåíèÿ Äëÿ ðåøåíèÿ çàäà÷è (7)-(10) ìîæíî âîñïîëüçîâàòü- ñÿ ïîäõîäîì, îñíîâàííûì íà ñâåäåíèè êðàåâîé çàäà÷è äëÿ íåîäíîðîäíîãî âîëíîâîãî óðàâíåíèÿ (7) ê ñîîòâåòñòâóþùåé êðàåâîé çàäà÷å äëÿ óðàâíåíèÿ Ãåëüìãîëüöà (ñòàöèîíàðíîé çàäà÷å) ñ ïîñëåäóþ- ùèì îáðàùåíèåì ðåøåíèÿ ïîñëåäíåé [6]. 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G[J]'gU2cG&| iw',W}лaEl-ubWݣu++zE]<5udOu|ȳ_$=fC}|&ɸ*Җl|[Ƨ/tSJ ٽAqvvM'Ǯx33;ќqvەc7oG`W8\v3k!;;U:3-9ksu}?v/\ĭ]iP{sg3}}G})qO;{>s0;sN'KDw$/ns&e+k r4yu)/n3JisOdulԏn|K٣%UarؙۖϬiS5!k28YgOIS \-$|/Υazû.1Rrt,,ÐÈ, 2005, ¹ 1 7                 n N ) n ( i ) ( n , m ) ( j , n ) ( 0 e z ˆ ) 1 ( n n )] n ( N [ , ùåëè j  , , 2 , 1 j  (25)                1 J 1 2 N ) n ( ), ( i 1 ), ( n , m ) ( n , m )] , ( h ˆ [ z z ˆ j 2 2 1 N ) n ( ), ( i 2 ), ( n , m )] , ( h ˆ [ z          , 1 j ) ( n , m j 2 ) ( n , m 2 j ) ( n , m 1 j ) ( n , m j ), ( n , m ] y [ ] x [ y x z              , ) y (cos P dy d ) x (cos P dx d ) y , x ( h ˆ N ) n ( i 2 / 1 1 1 N ) n ( i 2 / 1 1 1 N ) n ( ), ( i                        , ) , ( U d d b ˆ 1 ) ( g ˆ 0 ) 0 ( m 1 1 p ), ( i p ), ( i               , ) (cos P d d ˆ x 1 N ) n ( i 2 / 1 1 1 1 ) ( mn ) ( 0 m n , m                 , ) (cos P d d ˆ y 2 N ) n ( i 2 / 1 1 2 1 ) ( mn ) ( 0 m n , m                 ,         ) 1 ( n n )] n ( N [ ) ( nj ) (                           ) N ) n ( i 2 / 1 ( ) N ) n ( i 2 / 1 ( ) (sin ch ) 1 ( ) ( 1 j 1 N ) n (                          ) cos ( P d d ) (cos P d d 1 j N ) n ( i 2 / 1 1 j 1 j N ) n ( i 2 / 1 1 j 1 ) , ( C 1 1 2 1 N ) n ( ), ( i         ,                          ) x cos ( P dx d ) x (cos P dx d ) y , x ( C N ) n ( i 2 / 1 1 1 N ) n ( i 2 / 1 1 1 N ) n ( ), ( i ) y (cos P dy d ) y cos ( P dy d N ) n ( i 2 / 1 1 1 N ) n ( i 2 / 1 1 1                     . Ðèñ. 2. Ñïëîøíîé áèêîíóñ 4. ×àñòíûå ñëó÷àè ñòðóêòóðû. Àíàëèòè÷åñêîå ðåøåíèå 1) Ñïëîøíîé áèêîíóñ (ðèñ.2).  ýòîì ñëó÷àå ïîòåí- öèàë Äåáàÿ èìååò ñëåäóþùèé âèä ( 2 0 1      ):                              m 0 0 i m ), ( i 1 2 1 ) ( m ) ( m im ) ( 1 d ) t t ( d d ) ( B ~ ~ e r 1 ) t , r (  , 2 1      , ) i m 2 / 1 ( ) i m 2 / 1 ( th e r 4 aM ) 1 ( ~ 0 im 1 2 2 0 ) ( r 1 m ) ( m                          ,                         ) (cos P ) cos ( P d d ) , ( ) ( B ~ m i 2 / 1 2 m i 2 / 1 1 2 1 0 1 m ), ( i ) ( m ) cos ( P ) (cos P d d ) , ( m i 2 / 1 1 m i 2 / 1 1 1 1 0 2 m ), ( i                     ,                      ) cos ( P ) (cos P d d ) , ( 2 m i 2 / 1 1 m i 2 / 1 1 1 1 2 1 m ), ( i ). (cos P ) cos ( P d d 2 m i 2 / 1 1 m i 2 / 1 1 1 1                 ñëó÷àå îäèíî÷íîãî êîíóñà ïîëó÷åííûå ðåçóëüòà- òû ñîâïàäàþò ñ ðåçóëüòàòàìè ðàáîòû[9]. 2) Îäèíî÷íûé êîíóñ ñ ïðîäîëüíûìè ùåëÿìè, ) t t ( ) t t ( f 0 0     (ðèñ.3). Ðèñ. 3. Êîíóñ ñ ïðîäîëüíûìè ùåëÿìè  ñëó÷àå îäèíî÷íîãî êîíóñà ñ ïåðèîäè÷åñêè ïðî- ðåçàííûìè âäîëü îáðàçóþùèõ ùåëÿìè ñèñòåìó ëèíåéíûõ óðàâíåíèé ïåðâîãî ðîäà (24), (25) ñëåäóåò ðåãóëÿðèçîâàòü è ñâåñòè ê ñèñòåìå ëèíåéíûõ àëãåá- ðàè÷åñêèõ óðàâíåíèé âòîðîãî ðîäà, êîòîðàÿ â îòëè- ÷èå îò ñèñòåìû (24), (25) ÿâëÿåòñÿ óñòîé÷èâîé. 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(cos P ) cos ( P m i 2 / 1 0 m i 2 / 1            ñëó÷àå ðàñïîëîæåíèÿ èñòî÷íèêà íà îñè êîíóñà (    0 ) ïðåäñòàâëåíèå (26) óïðîùàåòñÿ è ïðèîáðå- òàåò âèä               c r r t rr 4 A ~ ) t , r ( 0 0 ) 1 ( 1                     0 i 2 / 1 2 i 2 / 1 i ) cos ( P ) (cos P D th      d ) chb ( P i 2 / 1 .      , (27)               Qch 2 ) cos ( P ) (cos P D i 2 / 1 i 2 / 1 i . Äëÿ ïîëó÷åíèÿ îäíîìîäîâîãî ðåæèìà â ïðåäñòàâ- ëåíèè (27) íåîáõîäèìî ïåðåéòè ê èíòåãðèðîâàíèþ ïî ìíèìîé îñè è âîñïîëüçîâàòüñÿ îñíîâíîé òåîðå- ìîé î âû÷åòàõ äëÿ ðàçëîæåíèÿ èíòåãðàëà â ðÿä ïî âû÷åòàì ïîäûíòåãðàëüíîé ôóíêöèè [11]. Åñëè ïîìåñòèòü èñòî÷íèê ó âåðøèíû êîíóñà, òî äîìèíè- ðóþùèì â ðÿäå áóäåò ïåðâûé ÷ëåí, êîòîðûé è õàðàêòåðèçóåò ðàñïðåäåëåíèå ïîëÿ â ýòîì ñëó÷àå. 5. Çàêëþ÷åíèå Âïåðâûå ïðåäëîæåí ÷èñëåííî-àíàëèòè÷åñêèé ïîä- õîä äëÿ ðåøåíèÿ çàäà÷è ðàññåÿíèÿ ýëåêòðîìàãíèò- íûõ âîëí íà ñëîæíîé íåçàìêíóòîé êîíè÷åñêîé ñòðóêòóðå âî âðåìåííîé îáëàñòè. Ýòîò ïîäõîä îñíîâàí íà èñïîëüçîâàíèè èíòåãðàëüíîãî ïðåîáðà- çîâàíèÿ Ìåëåðà-Ôîêà è ìåòîäà ïîëóîáðàùåíèÿ. Åãî ïðèìåíåíèå çíà÷èòåëüíî óïðîùàåò ðåøåíèå ÷åòûðåõìåðíîé ýëåêòðîäèíàìè÷åñêîé çàäà÷è è äàåò âîçìîæíîñòü, ìèíóÿ ïðîöåäóðó îáðàùåíèÿ ðåøå- íèÿ ñîîòâåòñòâóþùåé çàäà÷è â ÷àñòîòíîé îáëàñòè, ñâåñòè åå ê ñèñòåìå ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé âòîðîãî ðîäà îòíîñèòåëüíî êîýôôèöè- åíòîâ íåèçâåñòíîé òðàíñôîðìàíòû ïðåîáðàçîâàíèÿ Ìåëåðà-Ôîêà â ðÿä Ôóðüå ïî àçèìóòàëüíîé êîîð- äèíàòå ñôåðè÷åñêîé ñèñòåìû. Ðåøåíèå ñèñòåìû ìîæåò áûòü ïîëó÷åíî êàê ÷èñëåííî (ïðè ëþáûõ ñîîòíîøåíèÿõ ìåæäó ïàðàìåòðàìè ñòðóêòóðû), òàê è àíàëèòè÷åñêè (â ñëó÷àå áîëüøîãî ÷èñëà ùåëåé è ñðàâíèìîñòè èëè èõ ìàëîñòè ïî ñðàâíåíèþ ñ ïåðèîäîì ñòðóêòóðû). Àíàëèòè÷åñêîå ðåøåíèå ïî- çâîëÿåò íàãëÿäíî è êà÷åñòâåííî ïðîâåñòè àíàëèç ñïåêòðà ãðàíè÷íîé çàäà÷è è ñòðóêòóðû ïîëÿ.  ÷àñòíîì ñëó÷àå îäèíî÷íîãî ñïëîøíîãî êîíóñà ðå- øåíèå çàäà÷è ñîâïàäàåò ñ èçâåñòíûìè ðåçóëüòàòàìè äðóãèõ àâòîðîâ [9]. Ïðåäëîæåííûé ïîäõîä ìîæåò áûòü èñïîëüçîâàí äëÿ èññëåäîâàíèÿ çàäà÷è ðàññå- ÿíèÿ âîëí íà ñëîæíûõ êîíè÷åñêèõ è êëèíîâèäíûõ ñòðóêòóðàõ, êîòîðûå ÿâëÿþòñÿ ìîäåëÿìè ýëåìåíòîâ ñîâðåìåííûõ àíòåííûõ ñèñòåì, ðàäèîëîêàöèîííûõ êîìïëåêñîâ è àïïàðàòóðû äëÿ äèñòàíöèîííîãî çîíäèðîâàíèÿ. Ëèòåðàòóðà: 1. Ëåðåð À.Ì. Ðåãóëÿðèçàöèÿ â äâóìåðíûõ çàäà÷àõ äèôðàêöèè êîðîòêèõ ýëåêòðîìàãíèòíûõ èì- ïóëüñîâ // Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1998. Ò.43, ¹8. Ñ.915-920. 2. Brit Ch. L. Solution of electromagnetic scattering problems using time domain techniques // IEEE Trans. Antennas Propagat. 1989. Vol.37, No.9. Ð.1181-1192. 3. Lee J.F., Lee R., Cangellaris A. Time-domain finite element methods // IEEE Trans. Antenna Propagat. 1997. Vol.45, No.3. Ð .430-441. 4. Shanker B., Ergin A.A., Aygun K., Michielssen E. Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time- domain algorithm // IEEE Trans. Antennas Propagat. 1997.Vol.45, No.3 Ð.430-441. 5. Ôîê Â.À. Î ðàçëîæåíèè ïðîèçâîëüíîé ôóíêöèè â èíòåãðàë ïî ôóíêöèÿì Ëå- æàíäðà ñ êîìïëåêñíûì çíà÷êîì //Äîêë. ÀÍ ÑÑÑÐ. 1943. Ò.39, ¹7. Ñ.279-283. 6. Äîðîøåíêî Â.À. Âîçáóæäå- íèå áèêîíóñà ñ ïðîäîëüíûìè ùåëÿìè òî÷å÷íûì èñ- òî÷íèêîì, ïîëå êîòîðîãî ïðîèçâîëüíî ìåíÿåòñÿ âî âðåìåíè // Ðàäèîòåõíèêà. 2002. Âûï.129. Ñ.14-21. 7. Ïîëîæèé Ã.Í. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè. Ì.: Âûñø. øê., 1964. 560ñ. 8. Ôåëñåí Ë., Ìàðêóâèö Í. Èçëó÷åíèå è ðàññåÿíèå âîëí. Ì.: Ìèð, 1978. Ò.1. 550ñ. Ò.2. 558ñ. 9. Chan K.-K., Felsen L. Transient and time- harmonic diffraction by a semi-infinite cone// IEEE Trans. Antennas Propagat., 1977. Vol.25, No.6. P.802-806. 10. Øåñòîïàëîâ Â.Ï. Ñóììàòîðíûå óðàâíåíèÿ â ñîâðå- ìåííîé òåîðèè äèôðàêöèè. Êèåâ: Íàóê. äóìêà, 1983. 252ñ. 11. Ìàðêóøåâè÷ À.È. Òåîðèÿ àíàëèòè÷åñêèõ ôóí- êöèé. Ì.: Íàóêà, 1967. Ò.1. 488 ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 25.12.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Íåðóõ À.Ã. Äîðîøåíêî Âëàäèìèð Àëåêñååâè÷, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ÂÌ ÕÍÓÐÝ. 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Ââåäåíèå  ðàáîòàõ [1-3] ïîñòàâëåíà è ðåøåíà çàäà÷à ðàñ÷åòà ðåçîíàíñíûõ ÷àñòîò è äîáðîòíîñòè ðåçîíàòîðà êî- àêñèàëüíîãî òèïà, ñîäåðæàùåãî èçîòðîïíûé èëè àíèçîòðîïíûé äèýëåêòðèê, â öåëÿõ äèàãíîñòèêè åãî ïàðàìåòðîâ. Îäíàêî ïîäîáíàÿ ïîñòàíîâêà íå ïîçâîëÿåò ó÷åñòü ýôôåêò íåîäíîðîäíîãî ðàñïðåäå- ëåíèÿ ðàâíîâåñíîé êîíöåíòðàöèè ñâîáîäíûõ íîñè- òåëåé çàðÿäà åñòåñòâåííîãî èëè èñêóññòâåííîãî ïðîèñõîæäåíèÿ, ÷òî òàêæå ìîæåò áûòü èñïîëüçîâà- íî â äèàãíîñòè÷åñêèõ öåëÿõ. Êðîìå òîãî, ðåøåíèå ïðàêòè÷åñêèõ çàäà÷ ÑÂ× äèàãíîñòèêè ïîëóïðîâîä- íèêîâûõ ìàòåðèàëîâ ñîïðÿæåíî ñ âûñîêîé ñòåïå- íüþ ëîêàëèçàöèè è, ñëåäîâàòåëüíî, êîíöåíòðàöèè ïîëÿ, ÷òî íåèçáåæíî âåäåò ê ïðîÿâëåíèþ íåëèíåé- íûõ ýôôåêòîâ â ïîëóïðîâîäíèêîâîì îáðàçöå, êî- òîðûå è íåîáõîäèìî ó÷èòûâàòü ïðè ðàçðàáîòêå ìàòåìàòè÷åñêîé ìîäåëè. 2. Ïîñòàíîâêà çàäà÷è Ðàññìîòðèì îñåñèììåòðè÷íóþ ïîëóþ ýëåêòðîäè- íàìè÷åñêóþ ñèñòåìó (ïðîäîëüíîå ñå÷åíèå èçîáðà- æåíî íà ðèñóíêå) ñ êîëåáàíèÿìè êâàçè ÒÅÌ òèïà, ÷àñòè÷íî çàïîëíåííóþ ïîëóïðîâîäíèêîì, ïîïå- ðå÷íûé ðàçìåð êîòîðîé çíà÷èòåëüíî ìåíüøå äëèíû âîëíû êîëåáàíèé, à àìïëèòóäà êîëåáàíèé íå îãðà- íè÷åíà ïðèáëèæåíèåì ìàëîñòè ÑÂ× ñèãíàëà. Ïðîäîëüíîå ñå÷åíèå îáðàçöà Èñõîäíûìè äëÿ ðåøåíèÿ çàäà÷è î ñîáñòâåííûõ êîëåáàíèÿõ ÿâëÿþòñÿ óðàâíåíèÿ Ìàêñâåëëà â êâà- çèñòàòè÷åñêîì ïðèáëèæåíèè è óðàâíåíèÿ äâèæå- íèÿ ñâîáîäíûõ çàðÿäîâ â ïîëóïðîâîäíèêå ï -òèïà â ãèäðîäèíàìè÷åñêîì ïðèáëèæåíèè:                             , n n v m e v t v ), N n ( e 4 , 0 ) v n ( div t n 0 2 T g    (1) ãäå v , , n   – íåèçâåñòíûå ïëîòíîñòü ñâîáîäíîãî çàðÿäà, ïîòåíöèàë ýëåêòðîìàãíèòíîãî ïîëÿ è ñêî- ðîñòü çàðÿäà, ñîîòâåòñòâåííî; ôèçè÷åñêèå õàðàêòå- ðèñòèêè ïîëóïðîâîäíèêà: m , e , , n , N , , v 0 g T   – ñðåäíÿÿ òåïëîâàÿ ñêîðîñòü ñâîáîäíîãî çàðÿäà, ñðåä- íÿÿ ÷àñòîòà ñòîëêíîâåíèé çàðÿäà ñ èîíàìè ðåøåòêè, êîíöåíòðàöèÿ äîíîðîâ, êîíöåíòðàöèÿ ñâîáîäíûõ çàðÿäîâ, íåâîçìóùåííûõ ïîëåì, äèýëåêòðè÷åñêàÿ ïðîíèöàåìîñòü ïîëóïðîâîäíèêà, çàðÿä è ìàññà ýëåêòðîíà, ñîîòâåòñòâåííî. Èç-çà ïðåäïîëàãàåìîé íåëèíåéíîñòè âçàèìîäåé- ñòâèÿ ýëåêòðîìàãíèòíîãî ïîëÿ ñ ïîëóïðîâîäíèêîì ôóíêöèè v , , n   ïðåäñòàâëÿþòñÿ ðàçëîæåíèÿìè â ðÿäû Ôóðüå ïî ãàðìîíèêàì îñíîâíîé ÷àñòîòû – ÷àñòîòû âîçáóæäåíèÿ ýëåêòðîäèíàìè÷åñêîé ñèñòå- ìû [4]:                                   . e ) k ( v v , e ) k ( , e ) k ( n n k t ik k k t ik k k t ik k   (2) Ýëåêòðîìàãíèòíîå ïîëå â ïîëîé ñèñòåìå, â îáùåì, äîëæíî áûòü ïðåäñòàâëåíî ñóïåðïîçèöèåé ïîëåé ñîáñòâåííûõ êîëåáàíèé, îäíàêî âáëèçè ðåçîíàíñ- íîé ÷àñòîòû îñíîâíîãî êîëåáàíèÿ àìïëèòóäû âûñ- øèõ ãàðìîíèê ïðåíåáðåæèìî ìàëû. Êðîìå òîãî, èñõîäÿ èç ìàëîñòè ïîïåðå÷íîãî ñå÷åíèÿ êîàêñèàëü- íîãî îòðåçêà ïî ñðàâíåíèþ ñ äëèíîé âîëíû, ñòðóê- òóðó ïåðåìåííîãî ýëåêòðè÷åñêîãî ïîëÿ è òîêà ïðîâîäèìîñòè íà ðåçîíàíñíîé ÷àñòîòå w â ïëîñêî- ñòè z =0 ìîæíî ïðåäñòàâèòü àíàëîãè÷íî ðàñïðåäå- ëåíèþ ïîëÿ ÒÅÌ âîëíû â êîðîòêîçàìêíóòîì îòðåçêå â ñå÷åíèè g z   :               ], R , [ r для ) t sin( r 1 E r ), , 0 [ r для 0 E 0 0 0 0 0 0 z   (3) ), t cos( w 1 E z J 0 0 0 0 0 z        (4) ãäå , g k 0 0   c k 0   – âîëíîâîå ñîïðîòèâëå- íèå ñâîáîäíîãî ïðîñòðàíñòâà; 0 E – àìïëèòóäà ïîëÿ; 0 0 z , r   – êîîðäèíàòíûå îðòû. 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Ñ ó÷åòîì ýòèõ ïðåäñòàâëåíèé ãðàíè÷íûå óñëîâèÿ äëÿ ïîòåíöèàëà îñíîâíîé ãàðìîíèêè èìåþò âèä:               F r r , F r r 0 z 1 0 0 z 0     , (6) à äëÿ ïîëíîãî òîêà, èñõîäÿ èç óñëîâèÿ åãî íåïðå- ðûâíîñòè â ñå÷åíèè 0  z – ñëåäóþùèé:                  R 0 0 z 1 0 R 0 0 z 1 0 I rdr z z i ; I rdr z z i     .(7) Äëÿ ïîëó÷åíèÿ óðàâíåíèé îòíîñèòåëüíî àìïëèòóä ãàðìîíèê âíåñåì ïðåäñòàâëåíèÿ (2) â óðàâíåíèÿ (1), êîòîðûå ïîñëå ââåäåíèÿ çàìåí: , v N m e 4 , v m e , N N n 2 e 2 T g 2 k 2 T k k g k         èìåþò âèä: ,... 1 , 0 k , N N ik v v ); N ( , N N N ) s k ( i 1 N v k i 0 k k 2 T k k 0 k 2 e k s 0 s k s k s k 2 T                                                          (8) Èç ðàâåíñòâ (6)-(8) íåòðóäíî óñòàíîâèòü: ,... 2 , 1 k , N N , k k k k          . (9) Ñ ó÷åòîì (9) ãðàíè÷íûå óñëîâèÿ (6) è (7) ïðèâîäÿò- ñÿ ê âèäó:                 ]. R , [ r для sin F r 1 ), , 0 [ r для 0 r Re 0 0 0 0 0 z 1 (10.1) , cos k 1 F rdr z Re 0 0 0 0 z R 0 1         (10.2) ãäå 0 F ó÷èòûâàåò ïåðåõîä ê áåçðàçìåðíîìó ïîòåí- öèàëó. Êðîìå òîãî, äîëæíû óäîâëåòâîðÿòüñÿ ãðàíè÷íûå óñëîâèÿ íà èäåàëüíî ïðîâîäÿùèõ ìåòàëëè÷åñêèõ ãðàíèöàõ: 0 r , 0 z h z 1 R r 1           (11) è îòñóòñòâèå íîðìàëüíîé ñîñòàâëÿþùåé ñêîðîñòè ñâîáîäíûõ çàðÿäîâ íà ãðàíèöàõ ïîëóïðîâîäíèêà: . r N r N 0 z N z N R r 1 0 1 h , 0 z 1 0 1                             (12) Êàê ñëåäóåò èç (8), ïîëó÷åíà áåñêîíå÷íàÿ ñèñòåìà äèôôåðåíöèàëüíûõ óðàâíåíèé ñ âçàèìîñâÿçàííû- ìè àìïëèòóäàìè ãàðìîíè÷åñêèõ ñîñòàâëÿþùèõ ïëîòíîñòè çàðÿäà è ïîòåíöèàëà. Îãðàíè÷èâàÿñü ó÷åòîì òîãî èëè èíîãî êîëè÷åñòâà ãàðìîíèê â ýòîé ñèñòåìå, ìîæíî ïîëó÷àòü ñîîòâåòñòâóþùèå ïðè- áëèæåíèÿ ðåøåíèé ïîñòàâëåííîé çàäà÷è. Îñòàâëÿÿ ëèøü íàèáîëåå ñóùåñòâåííûå ñâÿçè ìåæäó ãàðìî- íèêàìè è îòáðàñûâàÿ â (8) ìàëûå ÷ëåíû âûñøèõ ãàðìîíèê, èìååì:                               . N , 0 N N v i N N 1 2 e 1 1 0 2 e 2 T 2 0 1 1 (13) Çàìåòèì, ÷òî ñèñòåìà óðàâíåíèé äëÿ  1 N è   1 îòëè÷àåòñÿ îò (13) ïåðåìåíîé çíàêà  íà ïðîòèâî- ïîëîæíûé, ò.å. ïåðåõîäîì ê êîìïëåêñíî ñîïðÿæåí- íîìó óðàâíåíèþ. Âõîäÿùàÿ â (13) ôóíêöèÿ 0 N , â ñîîòâåòñòâèè ñ (8), èìååò ïðîñòîé àíàëèòè÷åñêèé âèä: , e N 0 0   (14) à ôóíêöèÿ 0 Ф óäîâëåòâîðÿåò óðàâíåíèþ: ) 1 N ( 0 2 е 0     (15) è ãðàíè÷íûì óñëîâèÿì: R r 0 h z 0 0 0 z 0 0 , R r , V r 0 , 0                   . (16) Óñëîâèå 0 n v 0 0     â ñîîòâåòñòâèè ñ (8) è (14) óäîâëåòâîðÿåòñÿ òîæäåñòâåííî. Òàêèì îáðàçîì, äëÿ ðåøåíèÿ ñèñòåìû óðàâíåíèé (13) ñ ãðàíè÷íûìè óñëîâèÿìè (9)-(12) òðåáóåòñÿ ïðåäâàðèòåëüíîå ðåøåíèå çàäà÷è (15)-(16). Ïîëó- ÷åííûå, â ðåçóëüòàòå, ïåðâûå ïðèáëèæåíèÿ ) 1 ( 1 ) 1 ( 1 , N  èñïîëüçóþòñÿ çàòåì äëÿ îïðåäåëåíèÿ âòîðîé ãàðìîíèêè: 2 N , 2 Ф â ïåðâîì ïðèáëèæå- íèè ïóòåì ðåøåíèÿ äèôôåðåíöèàëüíûõ óðàâíå- íèé: . N ; N N N i 1 N N v ) 2 ( i ) 2 ( N N 2 2 e 2 0 1 1 1 2 2 2 2 0 2 e 2 T 2 0 2 2                                                              (17) ñ ãðàíè÷íûìè óñëîâèÿìè, àíàëîãè÷íûìè (11) è (12).  ïîñëåäóþùåì ïîëó÷åííûå ïðèáëèæåííûå ðåøå- íèÿ 2 N , 2 Ф ìîãóò áûòü èñïîëüçîâàíû â óðàâíåíèè (8) îòíîñèòåëüíî 1 N , 1 Ф äëÿ óòî÷íåíèÿ îïðåäåëåí- íîãî ðàíåå ïðèáëèæåííîãî çíà÷åíèÿ ðåçîíàíñíîé ÷àñòîòû  . 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Àëãîðèòì ðåøåíèÿ çàäà÷è î ðåçîíàòîðå Ñôîðìóëèðîâàííàÿ â (9)-(13) çàäà÷à ïîçâîëÿåò â ðåçóëüòàòå åå ðåøåíèÿ îïðåäåëèòü â ïåðâîì ïðè- áëèæåíèè ðåçîíàíñíûå ÷àñòîòû êâàçèñòàòè÷åñêèõ òèïîâ êîëåáàíèé è ñîîòâåòñòâóþùèå èì ðàñïðåäå- ëåíèÿ ïëîòíîñòè çàðÿäà è ïîòåíöèàëà. Ñîñðåäîòî- ÷èì âíèìàíèå íà ìåòîäå åå ðåøåíèÿ. Ïðåäñòàâèì ïåðâîå óðàâíåíèå (13) â âèäå: 0 CN z B r A N 1 1 1 1            , (29) ãäå r N А 0    , z N B 0    , 2 T 2 0 2 e v i N C       – èçâåñòíûå ôóíêöèè êîîðäèíàò. Ðåøåíèå çàäà÷è áóäåì èñêàòü â âèäå:       M 1 m m , 1 m 0 1 ) z ( ) r ( J ) z , r ( (30) è ) z ( ) lr ( J ) z ( ) r ( J ) z , r ( N 0 0 M 1 m m , 2 m 0 1        , (31) çäåñü m m m p ), R p (   – êîðíè ôóíêöèè Áåññåëÿ íóëåâîãî ïîðÿäêà; l – ïðîèçâîëüíîå öåëîå ÷èñëî, íå ðàâíîå l p m (íàïðèìåð R 1 l  ). Òàêîé âûáîð èñêîìûõ ðåøåíèé îáåñïå÷èâàåò âû- ïîëíåíèå ãðàíè÷íîãî óñëîâèÿ (11) íà öèëèíäðè÷åñ- êîé ïîâåðõíîñòè R r  è âîçìîæíîñòü óäîâëåòâî- ðèòü ãðàíè÷íûì óñëîâèÿì (12). Ïîäñòàíîâêà ïðåä- ñòàâëåíèé (30) è (31) â (13) ïîçâîëÿåò ïîëó÷èòü óðàâíåíèå äëÿ êîýôôèöèåíòîâ Ôóðüå-ðàçëîæåíèé ïî ñèñòåìå ôóíêöèé   ) r ( J m 0  : , 0 ) l ( b ) q ( c , M ,..., 2 , 1 m , 0 b 0 2 0 m M 1 k k , 2 mk k , 1 mk k , 1 mk 0 m m , 2 2 m m , 2 0 m m , 2 2 e m , 1 2 m m , 1                                         (32) ãäå                                          . rdr ) lr ( J ) r ( J ) z , r ( C ) N 1 ( c ; rdr ) r ( J ) r ( J ) z , r ( C ) N 1 ( q ; rdr ) r ( J ) r ( J ) z , r ( B ) N 1 ( ; rdr ) r ( J ) r ( J ) z , r ( A ) N ( ; ) p ( J ) lR ( J ) lR ( p p 2 b R 0 0 m 0 2 m , 0 mk R 0 k 0 m 0 2 m , 0 mk R 0 k 0 m 0 2 m , 0 mk R 0 k 1 m 0 2 m , 0 m mk m 1 0 2 2 m m 2 e m Ïîäñòàâëÿÿ (30) è (31) âî âòîðîå ðàâåíñòâî (12) è ó÷èòûâàÿ, ÷òî 1 ) z , R ( N 0  , ïîëó÷àåì: , ) ( ~ ) z ( M 1 k k , 2 k , 1 k 0         (33) ãäå ) lR ( lJ ) p ( J ~ 1 k 1 k k    . Ïîäñòàâèâ ) z ( 0  â ïåðâîå ðàâåíñòâî (32), ïîëó÷èì ñâÿçü ìåæäó 1 ) z , R ( N 0  : . 0 ) ( ~ b M 1 k k , 2 k , 1 k m m , 2 2 e m , 1 2 m m , 1                  (34) Äàëåå, ñ ïîìîùüþ (33) è (34), èñêëþ÷èâ èç âòîðîãî ðàâåíñòâà â (32) m , 1 0 0 и ,        , ïîëó÷èì: , M ,..., 2 , 1 m , R b 1 ~ m m , 2 m M 1 k k , 2 k             (35) ãäå ). A ~ ( ~ b ~ ; b q ~ ~ b ~ q ~ ; b ~ ; b c l ~ ; b ~ A ~ ; b ~ ~ ) A ~ ( ~ ; ) q ~ ~ ~ ( b R 2 e k k m mk k m k mk m mk mk m m 2 m M 1 s s s m mk k m 2 m m mk M 1 k k , 2 mk k , 1 mk k , 1 mk m , 2 m 2 m m                                              Àëãåáðàè÷åñêàÿ ñèñòåìà óðàâíåíèé (35) ìîæåò áûòü ðàçðåøåíà îòíîñèòåëüíî k , 2    â âèäå: , R 1 b ~ R b ~ b m M 1 k k k M 1 k k k k m m , 2                           (36) ÷òî äàåò âîçìîæíîñòü âìåñòî âòîðîãî ðàâåíñòâà â (32) ïîëó÷èòü ñèñòåìó äèôôåðåíöèàëüíûõ óðàâíå- íèé âòîðîãî ïîðÿäêà â êàíîíè÷åñêîé ôîðìå: , 0 ) c b a ( M 1 k k , 2 mk k , 1 mk k , 1 mk m , 2 2 m m , 2                 (37) ãäå . ~ q ~ b ~ c ~ , A ~ 1 c ~ q ~ b c ; ~ b ~ b ~ , A ~ 1 b ~ ~ b b ; ~ b ~ a ~ , A ~ 1 a ~ ~ b a k 2 k M 1 n nk n n 0 k 0 k mk m mk M 1 n kn n n 0 k 0 k mk m mk M 1 n kn n n 0 k 0 k mk m mk                                                     Äàëåå ïðåäñòîèò ðåøèòü ñèñòåìó M 2 ëèíåéíûõ äèôôåðåíöèàëüíûõ óðàâíåíèé (35) è (37) îòíîñè- òåëüíî íåèçâåñòíûõ ôóíêöèé m , 1  , m , 2  , M ,..., 2 , 1 m  . 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Ðåàëèçàöèÿ ïåðâîãî óñëîâèÿ (12) äëÿ 0 z  ñ ó÷åòîì òîãî, ÷òî               R r , , r 0 , 0 0 , r N e 0 V 0 , äà¸ò: . 0 ) 0 ( ) lr ( J ) 0 ( ) r ( J ) 0 , r ( N ) 0 ( ) r ( J 0 0 M 1 m m , 1 m 0 0 M 1 m m , 2 m 0                 (40) Íàõîäÿ èç (33) ) 0 ( 0   è ïîäñòàâëÿÿ åãî â (40) äëÿ êîýôôèöèåíòîâ Ôóðüå-ðàçëîæåíèé ïî ñèñòåìå ôóíêöèé )} r ( J { m 0  , ïîëó÷àåì ñèñòåìó óðàâíåíèé: , 0 )] 0 ( ~ A ) 0 ( ) D ~ A [( ) 0 ( M 1 k k , 2 k m k , 1 mk k m m , 2               (41) ãäå , ) ( J ) ( J ) ( J ) ( J ) p ( J R ) e 1 ( 2 D , b A 2 m 2 k m 1 k 0 m k 1 m 0 k k 2 1 2 V mk 2 e m m 0                    ïðè÷åì ïðè k m  äëÿ mk D èìååò ìåñòî ðàâåíñòâî )] ( J ) ( J [ R ) p ( J e 1 e D m 2 0 m 2 1 2 2 m 2 1 V V mm 0 0          . Àíàëîãè÷íî èç òîãî æå óñëîâèÿ (12) äëÿ h z  ñ ó÷åòîì ðàâåíñòâà 1 ) h , r ( N 0  ñëåäóåò: . 0 )] h ( ~ A ) h ( ) ~ A [( ) h ( M 1 k k , 2 k m k , 1 km k m m , 2                (42) Òàêèì îáðàçîì, ñôîðìóëèðîâàíû M 4 êðàåâûå óñ- ëîâèÿ (38), (41), (42), äîñòàòî÷íûå äëÿ îäíîçíà÷íî- ãî íàõîæäåíèÿ èñêîìûõ ôóíêöèé ) z ( m , 1  , ) z ( m , 2  íà îòðåçêå ] h , 0 [ z  . Ââåäÿ îáîçíà÷åíèÿ m m , 1 y   , , у т М т , 1     , у т М 2 т , 2    , у т М 3 т , 2     ( , М ,..., 1 т  ), ïðåäñòàâèì (34), (37) â ñòàíäàðòíîé ôîðìå îäíîðîäíîé ñèñòåìû 4M ëèíåéíûõ äèôôå- ðåíöèàëüíûõ óðàâíåíèé ïåðâîãî ïîðÿäêà:                                                   0 ) y c y b y a ( y y ; 0 y y ; 0 ) y y ( ~ b y y y ; 0 y y M 1 k k M 2 mk k M mk k mk m M 2 2 m m M 3 m M 3 m M 2 M 1 k k M 2 k k m m M 2 2 e m 2 m m M m M m è ñîîòâåòñòâóþùèå êðàåâûå óñëîâèÿ (38), (41), (42):                                          , 0 )] h ( y ~ A ) h ( y ) ~ A [( ) h ( y ; 0 )] 0 ( y ~ A ) 0 ( y ) D ~ A [( ) 0 ( y M ,..., 2 , 1 m , 0 ) h ( y , f ) 0 ( y M 1 k k M 3 k m k M km k m m M 3 M 1 k k M 3 k m k M mk k m m M 3 m m m ( 44) ãäå 0 0 m m sin F d f   , km  – ñèìâîë Êðîíåêåðà. Ïðåäñòàâëÿÿ îáùåå ðåøåíèå ) y ,..., y , y ( ) z ( Y M 4 2 1  ëèíåéíîé êîìáèíàöèåé ôóíäàìåíòàëüíîé ñèñòåìû ðåøåíèé (43): ), z ( g c ... ) z ( g c ) z ( g c ) z ( Y M 4 M 4 2 2 1 1     (45) ãäå ) z ( g k – âåêòîð-ôóíêöèÿ ñ êîìïîíåíòàìè ) z ( g m , k , ïðè÷åì m , k m , k ) 0 ( g   , ïîëó÷àåì ñîîò- íîøåíèÿ äëÿ îïðåäåëåíèÿ êîíñòàíò k c :                                                     . 0 )] h ( g ~ A ) h ( g ) ~ A [( ) h ( g c 0 ] c ~ A c ) D ~ A [( c , 0 c ) h ( g , M ,..., 2 , 1 m , f c M 1 n n M 3 , k n m n M , k mn n m M 4 1 k m M 3 , k k M 1 k k M 3 k m k M mk k m m M 3 M 4 1 k k m , k m m Î÷åâèäíî, ÷òî êîíñòàíòû k c , óäîâëåòâîðÿþùèå ÑËÀÓ (46), èìåþò îáùèé ìíîæèòåëü 0 0 sin F  , ò.å. ñîäåðæàò íàðÿäó ñ ïðîèçâîëüíîé ïîñòîÿííîé 0 F è íåîïðåäåëåííóþ âåëè÷èíó  .  äàëüíåéøåì, äëÿ îïðåäåëåííîñòè, âìåñòî k c áóäåì ïèñàòü k c 0 0 sin F  , íàïðèìåð:           M 1 m m 0 m M 0 0 0 z 1 ). r ( J c sin F z Äëÿ îïðåäåëåíèÿ  ðåàëèçóåì óñëîâèå çàìêíóòîñòè ïîëíîãî òîêà (10), â ðåçóëüòàòå ÷åãî âîçíèêàåò õàðàêòåðèñòè÷åñêîå óðàâíåíèå: , 0 p ) p ( J c R k ) g k ( ctg M 1 m m m 1 m M 2 0 0       (47) â êîòîðîì íåèçâåñòíîå  ïðèñóòñòâóåò êàê ÿâíî, âõîäÿ â 0 k , òàê è íåÿâíî ÷åðåç ïîñðåäñòâî êîýôôè- öèåíòîâ ) ( c m M   , ïîñêîëüêó èñõîäíîå óðàâíåíèå (29) äëÿ 1 Ф ñîäåðæèò ôóíêöèþ ) ( С  . Çàìåòèì òàêæå, ÷òî âõîäÿùèé â ôîðìóëû àëãîðèòìà (45) ïîâåðõíîñòíûé ïîòåíöèàë ïîëóïðîâîäíèêà 0 V ÷åðåç ïîñðåäñòâî âåëè÷èíû D òàêæå, î÷åâèäíî, îêàçûâàåò âëèÿíèå íà ÷àñòîòíûé ñïåêòð è ïîòåðè ðåçîíàòîðà. , (46) (43) , yx xUEܰ !B" ʒ^Xl^ V@}LsN xz.gж??m!EF qds8Ș2[@=:R zX2G+ڋ-eܣo1|u%}6 d0Ư}M6N==&˾B~`@)MŶv3_O";؊q2h 6, h UWg*C{CxFu,p^b>G0eo>xa0v]sy` #5<8+c1Y%X8ژEJ_L2N8fbDZ@nyz|R}یoJdcPjV{/E]h󌽿 O׀ydkllEy.|kbcI+?bl1~cɴAm픒Voc/Nfw$H'$~d}ֆۋR>ɗ~L}eAU]: 4dSz\c(?m?6uOsJ|l|DIed0XikiU}N0ڧ_= b]KD4do0_Okllİ16s45WǬ+Kc?%H/o$n#Iz95m IwL#gP4/DYi$}BO:4Oߦdɝ@w.a?\~ x՗B.i}Y[嵌ki nB*;P[M8称Rۣ nk>蛡Oa%22j[{,s$_{+]]/iuNE;.nCԣvi'ƭb ihtL ?`-xkVXo\'.ey o>c/:Y UMG^=?Zw/9/=R΋iMr䳶bb:_K xOSpږ9㬷"t-#}3RmKI#%ԡ' X<OaO*ߔ/uQ@x9؞R$b%qH-95C~Z*NnYг3u]D?M~>VǏסYot-8|[H1~'x sG9ފN }AhecSع8wjx|?-mySyڏ<AmcRYWq$|aDv0x'JgspGMFdl{S]@hu¦ST9HߓlMY+y'5? 3#ճ9$r.nIf395D|֓(%Wz>-~ %#mi쉉;u}kg1A6LO97>Ԍy{ ;YF]1O܏`iNXrNflrzYv.ݺr3e2cl앙URX}Hs[T}}_+kU}^se.b: =hlf' -}o݀ӌ}!m&?n7Я&F9 뼤:2pwtx9;rn=z;U; ZGFgcBn#cJK=ԣzr˖la pK'Pu~X|99GzqK(3@imӇb`?456ܜ4Yl!w|4iG(V;3=[ʬ/ǧ~/`۩ we@[C/VR"h+T?^[ ~8[H'j-`!:Ogz6u|2v<*6ܟE_ӖfS&kT'vֆi+<і1?XN2QP}1ߋ=DKgQΖQ<HSG&*˳FXTDsg鿮im59FE>@xdU1x/p) 0u^l2نnGGJծfQYMWOz4{?1o2sqŐugKOc)_)oXmf{T~iZF޶UNa#>eTVh~_ՑպuǡO} ^tĿwSw_ N'bj~O״xNwȞ._xQ|ve~ʟwYCv=Ӗ_U}s [gMNFg蚞GiZ%Yb'xǦ~RXg]fz2+M۲mZle֖ulƣFl^|bKlVatoe ~Fߤ{77<;|+-2 [ŧ]%y}_i9ߙwg~ Wl=`)hӞ;׏ޮ(|"v3=V⫫ȳ2vZV+M+]8LvWV$TqVYUú/>@U׆S~9u؆|;辞PkX.ao oyם~HH`_+Ӥ56_&ڿ PbmfL:控Hwg՟? 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Íåîáõîäèìûå äëÿ ýòîãî ñîîòíîøå- íèÿ âîçíèêàþò èç óñëîâèé, àíàëîãè÷íûõ (9), (10): . 2 cos k 2 1 F rdr r Re ]; R , [ r для 2 sin F r r ), , 0 [ r для 0 r Re 0 0 02 0 z R 0 2 0 0 02 0 0 0 z 2                          (48) Ðåàëèçàöèÿ ïðèâåäåííûõ ñîîòíîøåíèé â ïðîöåäó- ðå ðåøåíèÿ çàäà÷è äëÿ 2 Ф è 2 N , àíàëîãè÷íîé èçëîæåííîé, ïðèâåäåò ê óñòàíîâëåíèþ ìàòåìàòè- ÷åñêîé âçàèìîñâÿçè ìåæäó âåëè÷èíîé 02 F è 2 0 F , îïèñûâàþùåé ðåçóëüòàò íåëèíåéíîãî âîçäåéñòâèÿ ÑÂ× ñèãíàëà ñ àìïëèòóäîé 0 F è ðåçîíàíñíîé ÷àñòîòîé  íà ïîëóïðîâîäíèê. Î÷åâèäíî, ÷òî ïî- äîáíîå ðàññóæäåíèå ìîæíî ðàñïðîñòðàíèòü è íà áîëåå âûñîêèå ãàðìîíèêè èñõîäíîãî ÑÂ× ñèãíàëà. Ïîëó÷åííûå ðåøåíèÿ äëÿ âûñîêèõ ãàðìîíèê ìîãóò áûòü òàêæå èñïîëüçîâàíû äëÿ äàëüíåéøåãî óòî÷íå- íèÿ ñîáñòâåííûõ ðåçîíàíñíûõ ÷àñòîò ñèñòåìû íà îñíîâàíèè (8), ïîëó÷åííûõ ïðè ðåøåíèè (13). 5. Çàêëþ÷åíèå  äàííîé ðàáîòå â êâàçèñòàòè÷åñêîì ïðèáëèæåíèè èçëîæåí àëãîðèòì ðåøåíèÿ çàäà÷è î ðåçîíàíñíûõ ÷àñòîòàõ ðåçîíàòîðà êîàêñèàëüíîãî òèïà, íàãðó- æåííîãî ïîëóïðîâîäíèêîâîé ñòðóêòóðîé ñ ó÷åòîì èñêðèâëåíèÿ ýíåðãåòè÷åñêèõ çîí ó å¸ ïîâåðõíîñòè è íåëèíåéíîãî õàðàêòåðà âçàèìîäåéñòâèÿ íîñèòå- ëåé ñâîáîäíîãî çàðÿäà ñ ÑÂ× ïîëåì. Ýòîò àëãîðèòì äàåò âîçìîæíîñòü ïðîâîäèòü àíàëèç íåëèíåéíîãî âîçäåéñòâèÿ ÑÂ× ñèãíàëà íà ïîëóïðîâîäíèê ñ ó÷åòîì íåðàâíîìåðíîãî ðàñïðåäåëåíèÿ ðàâíîâåñ- íîé ïëîòíîñòè ñâîáîäíîãî çàðÿäà ïóòåì ïîëó÷åíèÿ ñîáñòâåííûõ ðåçîíàíñíûõ ÷àñòîò ñèñòåìû. Ëèòåðàòóðà: 1. Ãîðäèåíêî Þ.Å. Âû÷èñëåíèå êîìïëåêñ- íûõ ðåçîíàíñíûõ ÷àñòîò ÑÂ× ðåçîíàòîðíûõ äàò÷èêîâ àïåðòóðíîãî òèïà // Ðàäèîýëåêòðîíèêà è èíôîðìàòè- êà. 2001. ¹2. Ñ. 4-7. 2. Ñëèï÷åíêî Í.È., Êîñòû÷åâ Þ.Ã., Çîëîòàð¸â Â.À. Îöåíêà ýôôåêòèâíîñòè ìåòîäà Òðåôò- öà ïðè àíàëèçå ýëåêòðîäèíàìè÷åñêèõ ñèñòåì äëÿ ÑÂ× äèàãíîñòèêè ïîëóïðîâîäíèêîâ è äèýëåêòðèêîâ // Ðà- äèîýëåêòðîíèêà è èíôîðìàòèêà. 2001. ¹1. Ñ. 20-24. 3. Ëèñþê Þ.Â. Ïîãëîùåíèå ÑÂ× ìîùíîñòè â ÎÏÇ âûñî- êîîìíûõ ïîëóïðîâîäíèêîâ // Èçâ. âóçîâ. Ôèçèêà. 1981. ¹11. Ñ. 18-22. 4. Íèêîëüñêèé Â.Â. Ýëåêòðîäèíàìèêà è ðàñïðîñòðàíåíèå ðàäèîâîëí. Ì.: Íàóêà, 1973. 5. Áåë- ëìàí Ð., Êàëàáà Ð. Êâàçèëèíåàðèçàöèÿ è íåëèíåéíûå êðàåâûå çàäà÷è. Ì.: Ìèð, 1968. Ïîñòóïèëà â ðåäêîëëåãèþ 12.03.2005 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Ãîðäèåíêî Þ.Å. Ñëèï÷åíêî Íèêîëàé Èâàíîâè÷, êàíä. òåõí. íàóê, ïðî- ôåññîð, ïðîðåêòîð ïî íàó÷íîé ðàáîòå ÕÍÓÐÝ. Íàó÷- íûå èíòåðåñû: òâåðäîòåëüíàÿ ýëåêòðîíèêà, ìèêðî- âîëíîâàÿ ìèêðîñêîïèÿ, ðàäèîôèçèêà. Àäðåñ: Óêðàè- íà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë.: (057) 702-10-20. ÓÄÊ 517.9 ×ÈÑËÅÍÍÎ-ÀÍÀËÈÒÈ×ÅÑÊÀß ÌÎÄÅËÜ ÝÂÎËÞÖÈÈ ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÃÎ ÏÎËß Â ÂÎËÍÎÂÎÄÅ Ñ ÄÈÑÏÅÐÑÍÎÉ ÑÐÅÄÎÉ ÂËÀÑÅÍÊÎ Ë.À. Îïèñûâàåòñÿ ìàòåìàòè÷åñêàÿ ìîäåëü ýâîëþöèè ýëåê- òðîìàãíèòíîãî ïîëÿ â âîëíîâîäå, çàïîëíåííîì ñëîè- ñòîé ñðåäîé ñ ïðîñòðàíñòâåííîé äèñïåðñèåé.  ðåçóëü- òàòå ðàçäåëåíèÿ ïåðåìåííûõ â íà÷àëüíî-êðàåâîé çàäà- ÷å äëÿ óðàâíåíèé Ìàêñâåëëà âîçíèêàåò óðàâíåíèå â ÷àñòíûõ ïðîèçâîäíûõ íå òèïà Êîøè-Êîâàëåâñêîé. Ïîëó÷åíà òåîðåìà ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿ ñîîòâåòñòâóþùåé ñìåøàííîé çàäà÷è. Ïðè- âîäèòñÿ ÷èñëåííûé ìåòîä íàõîæäåíèÿ ðåøåíèÿ. 1. Ââåäåíèå Öåëü ðàáîòû: èññëåäîâàíèå ýâîëþöèè ýëåêòðîìàã- íèòíîãî ïîëÿ â âîëíîâîäå ñ äèñïåðñíîé ñðåäîé.  ðàññìàòðèâàåìîì ñëó÷àå ïîëå óäîâëåòâîðÿåò õîðî- øî èçâåñòíûì óðàâíåíèÿì Ìàêñâåëëà, à êëàññè÷åñ- êèå ìàòåðèàëüíûå óðàâíåíèÿ íàðóøàþòñÿ èç-çà íàëè÷èÿ äèñïåðñèè. Îïèñàíèå ýëåêòðîìàãíèòíîãî ïîëÿ ñâîäèòñÿ ê íàõîæäåíèþ åãî êîìïîíåíò, êîòî- ðûå ïîÿâëÿþòñÿ â ðåçóëüòàòå ïðèìåíåíèÿ ìåòîäà ðàçäåëåíèÿ ïåðåìåííûõ. Òðóäíîñòè âûçûâàåò íà- õîæäåíèå êîìïîíåíòû, êîòîðàÿ óäîâëåòâîðÿåò óðàâ- íåíèþ â ÷àñòíûõ ïðîèçâîäíûõ, íå ðàçðåøåííîìó îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé ïî âðåìåíè (óðàâíåíèþ íå òèïà Êîøè-Êîâàëåâñêîé).  ñòàòüå óñòàíîâëåíû óñëîâèÿ îäíîçíà÷íîé ðàçðåøèìîñòè ñîîòâåòñòâóþùåé ñìåøàííîé çàäà÷è. Çàäà÷è èññëåäîâàíèÿ: ïðåäñòàâèòü ñìåøàííóþ çàäà- ÷ó â âèäå àáñòðàêòíîé çàäà÷è Êîøè; óêàçàòü óñëî- âèÿ îäíîçíà÷íîé ðàçðåøèìîñòè àáñòðàêòíîé çàäà- ÷è; ïðèìåíèòü àáñòðàêòíûå ðåçóëüòàòû ê íà÷àëüíî- êðàåâîé çàäà÷å íå òèïà Êîøè-Êîâàëåâñêîé; îïè- ñàòü ÷èñëåííûé ìåòîä íàõîæäåíèÿ ðåøåíèÿ ñîîò- âåòñòâóþùåé ñìåøàííîé çàäà÷è.  ðàáîòàõ [1,2] òàêæå èññëåäóåòñÿ ýëåêòðîìàãíèòíîå ïîëå â âîëíî- âîäå ñ äèñïåðñíîé ñðåäîé è îäíà èç êîìïîíåíò ïîëÿ îïèñûâàåòñÿ óðàâíåíèåì íå òèïà Êîøè-Êîâàëåâñ- êîé.  ðàáîòå [1] ýòî óðàâíåíèå ÿâëÿåòñÿ íåâûðîæ- äåííûì (îïåðàòîð ïðè ñòàðøåé ïðîèçâîäíîé ïî âðåìåíè îáðàòèì).  [2], êàê è â äàííîé ñòàòüå, óðàâíåíèå íå òèïà Êîøè-Êîâàëåâñêîé ÿâëÿåòñÿ âûðîæäåííûì.  ýòîé ñòàòüå, â îòëè÷èå îò ðàáîòû [2], àáñòðàêòíîå óðàâíåíèå, îòâå÷àþùåå ñîîòâåò- ñòâóþùåé ñìåøàííîé çàäà÷å, èçó÷àåòñÿ â âåùå- ñòâåííîì ïðîñòðàíñòâå, ïîýòîìó ïðèìåíÿþòñÿ äðó- SFx t7}5+hX(XTBZJ[Zˢ.VXPS6cc$HUԈ0" ys2Rvι}[fK"FtlCۓD2a1,&__恎^dG? *Eeo~)I4hc'wCe.%225IufW,3 lö0FH'y̖zJ_3BuY)$8 Gk[dI3Gf4§en784C=x(fӨ6ZBpq`Sĺ%V>Ls1H"ݎ|V*%ʖ5;(cc\=^o31];[1eaCea#<~m9.L^LL6{fzmɴ)9u)ȍ hk:6%CWZaӘGFy.Bb[Ʃu:A@SZ-ʣv/q kZh2W!6'ڦtmo,6e2H>k-Rsh;F~uxsSL=ڜvn>'CiQ+dFeիlꝲ:g۳#O +vL;WYrtt9R*]qzbk\OSmN۞w 8+,%S@ePX/oK<7yY,opO9.Uye|,g'>#Qb]3+}y,F/uɉ{{O42u0:a~#g4@㡳(胾Ї80x/F#S܇t_^߅~6 |R5 J'I"V<$“e|39ck}4Q8޹Si.u^K=T],`A𵬫"Z2 F<͜Ozd@s,k-E{^oo&;_B9s|SZ߽͖xYDL f⻈90?+|tlVc>A~'!&!g7m yJ9=*i?]JHrĹ!n=  &jHcK0/!b Eֽ<>>@fP}q>r5>O>Y'K|dFf<#Px!>ȕL7gº m7[f?3W >)"'q%8j܄(P1qtw$qZ6Nֲ7x1tv $HdְNfՇ½5{En瞆~C"qy0JoK5 ^Tp9x5mQ|OUr>Hc_яT XY߳XJ!kݠj /'|huL&o3-G5u8|`+$s$ 1 nt '~fOdsZOvG7@_V? 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