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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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Äëÿ íàõîæäåíèÿ ÷èñëåííîãî ðåøåíèÿ âûðîæäåííîãî óðàâíåíèÿ íå òèïà Êîøè-Êîâàëåâñêîé íåâîçìîæíî ïðèìåíèòü õîðîøî èçâåñòíûå ðàçíîñòíûå ìåòîäû (ñì., íàïðè- ìåð, [3,4]) èç-çà íåóñòîé÷èâîñòè ðàçíîñòíîé ñõåìû. Ïîýòîìó ðàçðàáîòàí íîâûé ÷èñëåííûé ìåòîä íà- õîæäåíèÿ ðåøåíèÿ. Âñå ðåçóëüòàòû ñòàòüè ÿâëÿþò- ñÿ íîâûìè. 2. Îïèñàíèå ìàòåìàòè÷åñêîé ìîäåëè Ðàññìîòðèì öèëèíäðè÷åñêèé âîëíîâîä, èçîáðà- æåííûé íà ðèñ. 1. Ðèñ. 1. Öèëèíäðè÷åñêèé âîëíîâîä Âíóòðåííîñòü âîëíîâîäà çàïîëíåíà íåîäíîðîäíîé ñëîèñòîé ñðåäîé ñ ïðîñòðàíñòâåííîé äèñïåðñèåé. Áîêîâàÿ ïîâåðõíîñòü  – èäåàëüíûé ìåòàëë, ñâîáîäíûõ çàðÿäîâ è ñâîáîäíûõ èñòî÷íèêîâ íåò, ïîïåðå÷íîå ñå÷åíèå S – ïðîñòàÿ ñâÿçíàÿ îáëàñòü ñ ãëàäêîé ãðàíèöåé S  . Õîðîøî èçâåñòíî (ñì., íà- ïðèìåð, [5, ãëàâà IX], [6, ÷àñòü IV]), ÷òî ýâîëþöèÿ ýëåêòðîìàãíèòíîãî ïîëÿ â âîëíîâîäå îïèñûâàåòñÿ óðàâíåíèÿìè Ìàêñâåëëà â äèôôåðåíöèàëüíîé ôîðìå 0. E n H n B D B E D H              | ] , [ , 0 | ) , ( , 0 div , 0 div ; t rot , t rot (1) Çäåñü E – íàïðÿæåííîñòü ýëåêòðè÷åñêîãî ïîëÿ; H – íàïðÿæåííîñòü ìàãíèòíîãî ïîëÿ; D – ýëåêòðè- ÷åñêàÿ èíäóêöèÿ; B – ìàãíèòíàÿ èíäóêöèÿ; n – íîðìàëü ê ïîâåðõíîñòè. Ñèñòåìà óðàâíåíèé (1) íå îïðåäåëÿåò îäíîçíà÷íûì îáðàçîì ýëåêòðîìàãíèò- íûå ïîëÿ. Îíà äîïîëíÿåòñÿ ìàòåðèàëüíûìè óðàâ- íåíèÿìè, óñòàíàâëèâàþùèìè ñâÿçè ìåæäó âåêòî- ðàìè èíäóêöèè D , B è âåêòîðàìè íàïðÿæåííîñòè ïîëåé E , H . Ìàòåðèàëüíûå óðàâíåíèÿ îïðåäåëÿ- þòñÿ ñâîéñòâàìè ìàòåðèàëüíîé ñðåäû. Êëàññè÷åñ- êèå ìàòåðèàëüíûå óðàâíåíèÿ â èçîòðîïíûõ ñðåäàõ èìåþò âèä . , H B E D     (2) Ñîîòíîøåíèÿ (2) íàðóøàþòñÿ â àíèçîòðîïíûõ ñðåäàõ, êîãäà ïîÿâëÿåòñÿ äèñïåðñèÿ [5, § 58, 60, 83], [6, ÷àñòü IV, ãëàâà 1, § 6], [1, ðàçäåë 1].  ñëó÷àå ñðåäû ñ ïðîñòðàíñòâåííîé äèñïåðñèåé âåêòîð èíäóêöèè D ìîæåò çàâèñåòü íå òîëüêî îò ïîëÿ E , íî è îò åãî ïðîñòðàíñòâåííûõ ïðîèçâîäíûõ [5, § 83], [1, ðàçäåë 1].  äàííîé ñòàòüå ìû ïðåäïîëàãàåì, ÷òî ìàòåðè- àëüíûå óðàâíåíèÿ èìåþò âèä , , z b z H B E E D          (3) ãäå   , – ïîëîæèòåëüíûå ôóíêöèè îò z , ôóíêöèÿ ) z ( b ìîæåò ïðèíèìàòü îòðèöàòåëüíûå çíà÷åíèÿ. Ìàòåðèàëüíûå óðàâíåíèÿ (3) îòâå÷àþò ñëîèñòîé ñðåäå ñ ïðîñòðàíñòâåííîé äèñïåðñèåé. ×àñòíûé ñëó÷àé ìàòåðèàëüíûõ óðàâíåíèé (3) áûë ðàññìîò- ðåí â [1, ðàçäåë 1], ñîîòíîøåíèÿ (3) îáùåãî âèäà áûëè ââåäåíû â [2, ðàçäåë 7].  ñëó÷àå èçîòðîïíîé è îäíîðîäíîé ñðåäû, õàðàêòå- ðèçóåìîé ìàòåðèàëüíûìè óðàâíåíèÿìè (2) ñ ïîñòî- ÿííûìè   , , èññëåäóþò ýëåêòðîìàãíèòíûå ïîëÿ âèäà ) z , y , x ( ˆ e ), z , y , x ( ˆ e t i t i H H E E       , (4) ãäå  – ñîáñòâåííàÿ ÷àñòîòà ýëåêòðîìàãíèòíûõ êîëåáàíèé. Âåêòîð-ôóíêöèè H E ˆ , ˆ ïðåäñòàâëÿþò â âèäå ñóïåðïîçèöèè òàê íàçûâàåìûõ ìîäîâûõ  E è  H âîëí, êîòîðûå îáðàçóþò ïîëíóþ ñèñòåìó. Ìîäîâûå  E âîëíû èìåþò ÷èñòî ïîïåðå÷íîå ìàã- íèòíîå ïîëå ( 0 H ˆ z  ), ìîäîâûå  H âîëíû èìåþò ÷èñòî ïîïåðå÷íîå ýëåêòðè÷åñêîå ïîëå ( 0 E ˆ z  ). Ñèñòåìó ìîäîâûõ  E è  H âîëí ïîëó÷àþò ïóòåì ðàçäåëåíèÿ ïåðåìåííûõ y , x îò z (ñì., íàïðèìåð, [5, ãëàâà X, § 71]). Äëÿ íåîäíîðîäíîé ñðåäû èëè ñðåäû ñ äèñïåðñèåé (â ÷àñòíîñòè, äëÿ ñðåäû, õàðàêòåðèçóåìîé ìàòåðè- àëüíûìè óðàâíåíèÿìè (3), â êîòîðûõ õîòÿ áû îäíà èç ôóíêöèé b , ,   íå ÿâëÿåòñÿ ïîñòîÿííîé, ò.å. çàâèñèò îò z ) ñòðîèòü ýëåêòðîìàãíèòíûå ïîëÿ â âèäå (4) è íàõîäèòü âåêòîð-ôóíêöèè H E ˆ , ˆ ïóòåì ðàçäåëåíèÿ ïåðåìåííûõ y , x îò z íå ïðåäñòàâëÿåòñÿ âîçìîæíûì. Ïîýòîìó â ñëó÷àå ìàòåðèàëüíûõ óðàâ- íåíèé (3) â ðàáîòàõ [1, ðàçäåë 2], [2, ðàçäåë 2] áûëî ïðåäëîæåíî ðàçäåëÿòü ïåðåìåííûå y , x îò ïåðåìåí- íûõ t , z . Îáîáùàÿ âèä ìîäîâîé  H âîëíû â âîë- íîâîäå, çàïîëíåííîì èçîòðîïíîé è îäíîðîäíîé ñðåäîé, ýëåêòðîìàãíèòíîå ïîëå â ñëó÷àå ìàòåðèàëü- íûõ óðàâíåíèé (3) èùóò â âèäå . ) t , z ( f ) y , x ( F ) t , z ( h ) y , x ( , 0 ) t , z ( e ) y , x (               Γ H Θ E (5) Çäåñü F , f , h , e – äîñòàòî÷íî ãëàäêèå ñêàëÿðíûå ôóíêöèè; Γ Θ, – äîñòàòî÷íî ãëàäêèå äâóìåðíûå âåêòîð-ôóíêöèè. Åñëè â ìàòåðèàëüíûõ óðàâíåíèÿõ (3) b , ,   ÿâëÿþòñÿ ïîñòîÿííûìè, òî ìîæíî ïîñò- ðîèòü ìîäîâûå  H âîëíû âèäà (4), ðàçäåëèâ ïåðå- ìåííûå y , x îò z [1, ðàçäåë 5].  ýòîì ñëó÷àå, êðîìå ìîäîâûõ  H âîëí, ñóùåñòâóþò è äðóãèå ýëåêòðî- ìàãíèòíûå ïîëÿ îáùåãî âèäà (5). Ïðåäïîëàãàåì, ÷òî â ïîïåðå÷íûõ ñå÷åíèÿõ 0 z  è   z âîëíîâîä îãðàíè÷èâàþò ìåòàëëè÷åñêèå ïëà- ñòèíû. 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Z<-Sܷ$[=zߑ;+{ ?]ӶlsϨuN<6- ̾>I-\MzC?_,A6A!:\b^ D{`(Ó죹ᱼH oބEAm@{'(恥)OY7X9Cr!Br[vYXUwé׿x< B.W26¬+p@yWf :0oƲP/TAWqKa{y\)"w @uK]r2amYh<:$~oGKJrInݺ9J"h~-L&}qg #-F?~K3F}YX>nzA#!ܫQ=c(q@r-rƱWߋ|#dsS 9GUUλT_0_~[XU/e6 *:ȿ,&l+KK5ifƂbx32x48Wgy7g2&sS|EL{+tIۧ<)Fp_,,{ ÐÈ, 2005, ¹ 1 16 Ýòè óñëîâèÿ ýêâèâàëåíòíû ñëåäóþùèì: . 0 ) t , ( f ) t , 0 ( f    (6) Ìåòîä ðàçäåëåíèÿ ïåðåìåííûõ, îïèñàííûé â [2, ðàçäåë 2], ïîêàçûâàåò, ÷òî ôóíêöèè F , f , h , e è âåêòîð-ôóíêöèè Γ Θ, íàõîäÿòñÿ ñëåäóþùèì îáðà- çîì. Ôóíêöèÿ ) y , x ( F ÿâëÿåòñÿ ðåøåíèåì çàäà÷è Íåéìàíà â äâóìåðíîé îáëàñòè: . 0 S F , S ) y , x ( , 0 ) y , x ( F ) y , x ( F | 2          n (7) Çàäà÷à (7) èìååò ñ÷åòíîå ìíîæåñòâî ñîáñòâåííûõ ôóíêöèé ) y , x ( F ) y , x ( F n  , îòâå÷àþùèõ âåùåñòâåí- íûì ñîáñòâåííûì ÷èñëàì      2 n 2 (   n ). Âûáåðåì 2 n 2    . Ïóñòü q , p – êîíñòàíòû òàêèå, ÷òî 2 pq    . Âåêòîð-ôóíêöèè ) y , x ( q ), y , x ( q Θ Γ íàõî- äèì èç ñîîòíîøåíèé ]. ), y , x ( F [ ) y , x ( q ), y , x ( F ) y , x ( q 0 z Θ Γ     (8) Ôóíêöèÿ ) t , z ( f ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ 0 f z 1 z t f z b z 2 2 2                                (9) ñ êðàåâûìè óñëîâèÿìè (6). Äëÿ îäíîçíà÷íîãî íàõîæäåíèÿ ðåøåíèÿ ) t , z ( f çàäà÷è (9),(6) ðàññìîò- ðèì íà÷àëüíûå óñëîâèÿ âèäà ). z ( g ) 0 , z ( t f ), z ( g ) 0 , z ( f 1 0     (10) Ôóíêöèè ) t , z ( ph ), t , z ( pe îïðåäåëÿþòñÿ èç ñîîòíî- øåíèé   . f z 1 ) t , z ( ph ), t , z ( z f ) t , z ( pe          Òàêèì îáðàçîì, äëÿ íàõîæäåíèÿ ìîäîâîãî ïîëÿ â âèäå (5) íàì íåîáõîäèìî óñòàíîâèòü, ÷òî ñìåøàí- íàÿ çàäà÷à (9),(6),(10) èìååò åäèíñòâåííîå ðåøåíèå ) t , z ( f , ãäå 2 n 2    – ñîáñòâåííîå ÷èñëî çàäà÷è Íåéìàíà (7), êîòîðîìó îòâå÷àåò ñîáñòâåííàÿ ôóí- êöèÿ ) y , x ( F n . Òîãäà, ïîäñòàâëÿÿ â (5) ôóíêöèþ ) y , x ( F ) y , x ( F n  , ðåøåíèå ) t , z ( f ñìåøàííîé çàäà÷è (9), (6), (10), à òàêæå îïðåäåëÿåìûå ñ ïîìîùüþ ñîîòíîøåíèé (8) âåêòîð-ôóíêöèè   ) t , z ( ph ) y , x ( q 1 ) t , z ( h ) y , x ( 2 Γ Γ   è   ) t , z ( pe ) y , x ( q 1 ) t , z ( e ) y , x ( 2 Θ Θ   , ìû îäíîçíà÷íî íàõîäèì ýëåêòðîìàãíèòíîå ïîëå. 3. Àáñòðàêòíîå ïðåäñòàâëåíèå ñìåøàííîé çàäà÷è Óðàâíåíèå (9) íå ÿâëÿåòñÿ óðàâíåíèåì òèïà Êîøè- Êîâàëåâñêîé. Ïîä óðàâíåíèåì òèïà Êîøè-Êîâà- ëåâñêîé ïîäðàçóìåâàþò óðàâíåíèå, ðàçðåøåííîå îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé ïî âðåìåíè. Ðÿä ïðèìåðîâ óðàâíåíèé è ñèñòåì óðàâíåíèé íå òèïà Êîøè-Êîâàëåâñêîé ðàññìîòðåí â [8, ñ. 57-58, 78, 133]. Äëÿ ðàçðåøèìîñòè òàêèõ óðàâíåíèé íåëüçÿ ïðèìåíèòü õîðîøî èçâåñòíûå êëàññè÷åñêèå òåîðå- ìû (ñì., íàïðèìåð, [9, ãëàâà 4, §2]). Íåêîòîðûå çàäà÷è äëÿ óðàâíåíèé íå òèïà Êîøè-Êîâàëåâñêîé (òàêèå óðàâíåíèÿ òàêæå íàçûâàþò óðàâíåíèÿìè òèïà Ñ.Ë. Ñîáîëåâà) èññëåäîâàëèñü â [10-13,2].  äàííîé ñòàòüå ïðåäëàãàåòñÿ ìåòîä, îñíîâàííûé íà ïåðåõîäå îò ñìåøàííîé çàäà÷è (9),(6),(10) ê åå àáñòðàêòíîé ôîðìå – çàäà÷å Êîøè â àáñòðàêòíîì ïðîñòðàíñòâå; äîêàçûâàåòñÿ îäíîçíà÷íàÿ ðàçðåøè- ìîñòü àáñòðàêòíîé çàäà÷è Êîøè, à çàòåì îñóùåñòâ- ëÿåòñÿ îáðàòíûé ïåðåõîä ê ñìåøàííîé çàäà÷å. Îáîçíà÷èì ÷åðåç ] , 0 [ L L 2 2   ïðîñòðàíñòâî èíòåã- ðèðóåìûõ ñ êâàäðàòîì íà ] , 0 [  ôóíêöèé. Ââåäåì ïðîñòðàíñòâî Ñîáîëåâà ] , 0 [ W W 2 2 2 2   ôóíêöèé ) z ( u òàêèõ, ÷òî ) z ( u è åå îáîáùåííûå ïðîèçâîäíûå ) z ( ' ' u ). z ( ' u ïðèíàäëåæàò 2 L . Òàêæå ðàññìîòðèì ïðîñòðàíñòâî }. 0 ) ( u ) 0 ( u , W ) z ( u { ] , 0 [ W W 2 2 2 2 2 2           ïðîñòðàíñòâå 2 L ñìåøàííàÿ çàäà÷à (9), (6), (10) äîïóñêàåò ïðåäñòàâëåíèå â âèäå àáñòðàêòíîãî óðàâ- íåíèÿ 0 t , 0 ) t ( Bu ) t ( ' ' Au    (11) ñ íà÷àëüíûìè óñëîâèÿìè . u ) 0 ( ' u , u ) 0 ( u 1 0   (12) Îïåðàòîðû B , A ïîðîæäàþòñÿ äèôôåðåíöèàëüíû- ìè âûðàæåíèÿìè ) u ( dz d 1 dz d ) z ( u Bu ), u ( dz d b dz d ) z ( u Au 2          è èìåþò îáëàñòü îïðåäåëåíèÿ  2 2 B A W D D   , ) t , z ( f ) z )( t ( u  , ) z ( g ) z ( u 0 0  , ) z ( g ) z ( u 1 1  . Ëþáóþ ôóíêöèþ ) t , z ( v t , z : v  áóäåì òàêæå ðàññìàòðè- âàòü êàê ôóíêöèþ îò t ñî çíà÷åíèÿìè â ïðîñòðàí- ñòâå ôóíêöèé îò z è çàïèñûâàòü êàê ) z )( t ( v . Îïèñàíèå ïðîñòðàíñòâ ôóíêöèé îò t è z ìîæíî íàéòè â [8, ðàçäåë 1.2].  ðàáîòå [2] ñìåøàííàÿ çàäà÷à (9), (6), (10) è àáñòðàêòíàÿ íà÷àëüíàÿ çàäà÷à (11), (12) èññëåäîâà- ëèñü â êîìïëåêñíîì ïðîñòðàíñòâå. Çäåñü ìû áóäåì ðàññìàòðèâàòü ñëó÷àé âåùåñòâåííîãî ïðîñòðàíñòâà. Äëÿ ðàçðåøèìîñòè ñìåøàííîé çàäà÷è (9), (6), (10) â ðàáîòå [2] íàëàãàëèñü ñïåöèàëüíûå îãðàíè÷åíèÿ íà ôóíêöèè b , ,   (ñì. ïðåäëîæåíèå 7.1). Ýòî áûëî ñâÿçàíî ñ ìåòîäîì äîêàçàòåëüñòâà.  äàííîé ñòàòüå ìû ïðèìåíèì äðóãîé ìåòîä äîêàçàòåëüñòâà ðàçðå- øèìîñòè ñìåøàííîé çàäà÷è è îòêàæåìñÿ îò ýòèõ îãðàíè÷åíèé íà b , ,   , íî ïðåäïîëîæèì èõ ïîñòî- ÿííîñòü. Ïðåäëîæåííûé â äàííîé ñòàòüå ìåòîä äîêàçàòåëüñòâà òàêæå ìîæíî ïðèìåíèòü äëÿ êîìï- ëåêñíûõ ïðîñòðàíñòâ. 4. 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G9i]E]gY,mJ$?>eqJrZh^ |䄾;S|U#dP(ʇT*x#B'}zxx m/t:\?21.`r2Xd¢w |Ty4op63*`F/ۘlgN!uD/*7 KҠx[.1ڡ]z3Ze8aޗëSR^߭w`czHeiZ1#I#X|/ڏ;>@+% F1% +X9zB:`1V7}.Nʍ0BS'5\GD1Wĝq3W˹nvTe~k>{(K=G^}a΂ɸ"my ]+ʌW7SI29%˖=C[a}@ $![3f)M'w0?a ?.²yJ4yҠeC'靍iX$X9e=Ⴏ{b VbkX,yUEb6kIx<$D| G_utneY3y^&{Α=#krR$s1{Z&944(pO`}]܎,"3k*}sL|QϜmYV(X s?C>ڜ~K~.׿ cڕ~Hcc@i54lj> <" ~6FKɀZ#>E|9-6uld<Ι^5|M2@']6R=eZ$/u6/VW0ޭF nLLD-ͼ"9=saJ# VoYց\^xoUYɕq::cWkϏMK~5?Y;oJo5q\q+#\u_OywcʯV̮0{xomLͭS#[[h R-Ⱥѣ/q h|OSίgׇSymzbL8ySV=+Y_h={dž]5gNއwcC{_ؤ2bpQ}NXl;p$ C(_m +w^yÅHߊd,,e ÐÈ, 2005, ¹ 1 17 ñòðàíñòâî Y , èìåþò îáëàñòè îïðåäåëåíèÿ B A D , D ñîîòâåòñòâåííî. Ñèìâîë ) X ), , 0 ([ C n  îáîçíà÷àåò ïðîñòðàíñòâî ôóíêöèé X ) , 0 [ : u   , êîòîðûå n ðàç íåïðåðûâíî äèôôåðåíöèðóåìû â íîðìå X ïðè âñåõ 0 t  . Ôóíêöèþ ) X ), , 0 ([ C ) t ( u 2   íàçîâåì ð åøåíèåì óðàâíåíèÿ (11), åñëè ) Y ), , 0 ([ C ) t ( Au 2   è ôóíêöèÿ ) t ( u óäîâëåòâîðÿåò óðàâíåíèþ (11). Èç îïðåäåëåíèÿ ðåøåíèÿ ) t ( u ñëåäóåò, ÷òî B A D D D ) t ( u    äëÿ âñåõ 0 t  , ) Y ), , 0 ([ C ) t ( Bu   . Ðåøåíèåì çàäà÷è Êîøè (11),(12) íàçûâàåòñÿ ðåøå- íèå óðàâíåíèÿ (11), êîòîðîå óäîâëåòâîðÿåò íà÷àëü- íûì óñëîâèÿì (12). Âñþäó â äàëüíåéøåì ìû áóäåì òðåáîâàòü } 0 { D  . Ïóñòü Y ~ , X ~ – êîìïëåêñíûå îáîëî÷êè ïðîñòðàíñòâ Y , X è B ~ , A ~ – êîìïëåêñíûå ðàñøèðåíèÿ îïåðàòî- ðîâ B , A (ñì. [7, c. 25-26]). Ðàññìîòðèì ïó÷îê îïåðàòîðîâ B ~ A ~   , îïðåäåëåííûé íà B ~ A ~ D D D ~   . ×åðåç ) Y ~ , X ~ ( L îáîçíà÷àåòñÿ ïðîñòðàíñòâî îãðàíè- ÷åííûõ ëèíåéíûõ îïåðàòîðîâ èç X ~ â Y ~ , ) X ~ , X ~ ( L ) X ~ ( L  . Àíàëîãè÷íî áóäåì ðàññìàòðèâàòü ïðîñòðàíñòâà îãðàíè÷åííûõ ëèíåéíûõ îïåðàòîðîâ ) Y , X ( L è ) X ( L . Ïðåäïîëîæèì, ÷òî âûïîëíåíî ñëåäóþùåå óñëîâèå: (H) Äëÿ âñåõ êîìïëåêñíûõ ÷èñåë  òàêèõ, ÷òî 1 C | |   ( 0 C 1  ), ïó÷îê îïåðàòîðîâ B ~ A ~   èìååò ðåçîëüâåí- òó ) X ~ , Y ~ ( L ) B ~ A ~ ( ) ( R 1       è ñïðàâåäëèâà îöåíêà 2 C || ) ( R ||   ( 0 C 2  ). Åñëè âûïîëíåíî óñëîâèå (H), òî èç [14, ðàçäåë 6] ñëåäóåò, ÷òî ñóùåñòâóþò ïðÿìûå ðàçëîæåíèÿ 2 1 D D D    , 2 1 Y Y Y    , AD Y 1  , D KerA D 2   , 2 2 BD Y  , 1 1 1 Y ) B A ( D     . Çäåñü KerA – ÿäðî îïåðàòîðà A . Ïðè îïðåäåëåíèè 1 D ðàññìàòðèâàþòñÿ òå êîìïëåêñíûå ÷èñëà  , äëÿ êîòîðûõ ñóùåñòâóåò îãðàíè÷åííûé îáðàòíûé îïå- ðàòîð. Ëèíåàë 1 D íå çàâèñèò îò âûáîðà òàêèõ òî÷åê  . Íàïðèìåð, äîñòàòî÷íî âûáðàòü 2 C   . Ïóñòü 2 1 P , P – âçàèìíî äîïîëíèòåëüíûå ïðîåêòîðû íà 2 1 D , D ñîîòâåòñòâåííî, 2 1 Q , Q – âçàèìíî äîïîëíè- òåëüíûå ïðîåêòîðû íà 2 1 Y , Y ñîîòâåòñòâåííî. Îïå- ðàòîð B Q A Q BP AP G 2 1 2 1     îïðåäåëåí íà D è èìååò îãðàíè÷åííûé îáðàòíûé ) X , Y ( L G 1   .  [14] óñòàíîâëåíû ñâîéñòâà îïåðàòîðà G . Óñëîâèå (H) èñïîëüçîâàëîñü ïðè äîêàçàòåëüñòâå òåîðåì ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè äëÿ âû- ðîæäåííûõ óðàâíåíèé ïåðâîãî ïîðÿäêà (ñì., íà- ïðèìåð, [14]).  äàííîé ñòàòüå ìû ïðèìåíèì ýòî óñëîâèå, ÷òîáû óñòàíîâèòü îäíîçíà÷íóþ ðàçðåøè- ìîñòü âûðîæäåííîãî óðàâíåíèÿ âòîðîãî ïîðÿäêà (11), à òàêæå ÷òîáû îïèñàòü ìíîæåñòâî äîïóñòèìûõ íà÷àëüíûõ âåêòîðîâ 1 0 u , u (12). Èç òåîðåìû 1 ñëå- äóåò, ÷òî íå ëþáûå âåêòîðû D u , u 1 0  ìîãóò áûòü íà÷àëüíûìè äëÿ ðåøåíèÿ çàäà÷è Êîøè (11),(12); íà÷àëüíûå âåêòîðû îáÿçàòåëüíî äîëæíû óäîâëåò- âîðÿòü ñïåöèàëüíûì óñëîâèÿì ñîãëàñîâàíèÿ (13). Íàëè÷èå ýòèõ óñëîâèé îáóñëîâëåíî âûðîæäåííîñ- òüþ óðàâíåíèÿ (11). Äëÿ íåâûðîæäåííîãî óðàâíå- íèÿ 0 P 2  è íåîáõîäèìîñòü â óñëîâèÿõ (13) îòïà- äàåò. Òåîðåìà 1. Ïóñòü âûïîëíåíî óñëîâèå (H). Òîãäà äëÿ ëþáûõ âåêòîðîâ D u , u 1 0  òàêèõ è òîëüêî òàêèõ, ÷òî âûïîëíåíû óñëîâèÿ , 0 u P , 0 u P 1 2 0 2   (13) ñóùåñòâóåò åäèíñòâåííîå ðåøåíèå ) t ( u çàäà÷è Êîøè (11),(12), ïðè÷åì 0 ) t ( u P 2  ïðè âñåõ 0 t  . Äîêàçàòåëüñòâî. Ëþáîå ðåøåíèå óðàâíåíèÿ (11) ïðåäñòàâèìî â âèäå ) t ( u P ) t ( u P ) t ( u 2 1   . Ïðèìåíÿÿ ê (11) ïðîåêòîðû 2 1 Q , Q , óáåæäàåìñÿ â òîì, ÷òî óðàâíåíèå (11) ýêâèâàëåíòíî ñèñòåìå äâóõ óðàâíå- íèé , 0 )] t ( u P [ B , 0 ) t ( v BG Q ) t ( ' ' v 2 1 1     (14) ãäå ) t ( u GP ) t ( v 1  . Ó÷èòûâàÿ, ÷òî 2 2 1 P BP G   , ïîëó- ÷àåì 0 ) t ( u P 2  äëÿ âñåõ 0 t  . Ïåðâîå óðàâíåíèå â (14) ðàññìàòðèâàåòñÿ â ïðîñòðàíñòâå 1 Y . Òàê êàê îïåðàòîð 1 1 BG Q T   îãðàíè÷åí â 1 Y , òî ýòî óðàâ- íåíèå èìååò åäèíñòâåííîå ðåøåíèå äëÿ ëþáûõ íà÷àëüíûõ äàííûõ 1 1 1 0 Y v ) 0 ( ' v , Y v ) 0 ( v     : , v ) t T sin T ( v ) t T (cos ) t ( v 1 2 / 1 2 / 1 0 2 / 1    ãäå îïåðàòîð-ôóíêöèè t T cos 2 / 1 è t T sin T 2 / 1 2 / 1  îïðåäåëÿþòñÿ ñõîäÿùèìèñÿ ïî íîðìå îïåðàòîðîâ ðÿäàìè              0 k 1 k 2 k k 2 / 1 2 / 1 0 k k 2 k k 2 / 1 )! 1 k 2 ( t T ) 1 ( t T sin T , )! k 2 ( t T ) 1 ( t T cos [7, c. 113-114]. Îòñþäà ñëåäóåò, ÷òî äëÿ ëþáûõ íà÷àëüíûõ âåêòîðîâ D u , u 1 0  , óäîâëåòâîðÿþùèõ ñîîòíîøåíèÿì (13), ñóùåñòâóåò åäèíñòâåííîå ðå- øåíèå çàäà÷è Êîøè (11),(12), îïðåäåëÿåìîå ïî ôîðìóëå ]. Au ) t T sin T ( Au ) t T [(cos G ) t ( u 1 2 / 1 2 / 1 0 2 / 1 1     (15) Ïîíÿòíî, ÷òî óñëîâèÿ (13) ÿâëÿþòñÿ òàêæå íåîáõî- äèìûìè äëÿ ðàçðåøèìîñòè çàäà÷è Êîøè (11), (12). Òåîðåìà 1 äîêàçàíà. 5. Ïðèëîæåíèå ðåçóëüòàòîâ ðàçäåëà 4 ê èññëåäîâàíèþ ðàçðåøèìîñòè ñìåøàííîé çàäà÷è Âåðíåìñÿ ê ñìåøàííîé çàäà÷å (9), (6), (10), îïèñàí- íîé â ðàçäåëå 2. Ýòà çàäà÷à äîïóñêàåò ïðåäñòàâëåíèå â àáñòðàêòíîé ôîðìå (11), (12). Çäåñü 2 L Y X   . Äëÿ ïðîñòîòû ðàññìîòðèì ñëó÷àé ïîñòîÿííûõ êî- Rjx xUՕ׹!$Q,*$(jV: >FdT*PjXD(j(D*A4ˡJ/ȺY}oR('?,@_ ^^OyS]jlmlB3~9Jw98гUڣgT2o10QzƋwʦ"aK1Mby[?g"~H5ދѳ0w9ڻ)Y[g˥;o l[dq^C|GЩ줺6nƎ7cڮS׾gGK[X rorGˇsn0BJsSRUP65JZK24U\;$ @^yWc GKG/vv`Oi^; wu8}Ǻ!ss{^{6Ȍg<Ȝ b=1:wʹ Ac8W3g>My]8v;yB[y~ YH[_S3X,F. 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Âûâîäû Îïèñàíà ìàòåìàòè÷åñêàÿ ìîäåëü ýâîëþöèè ýëåêò- ðîìàãíèòíîãî ïîëÿ â êîíå÷íîì âîëíîâîäå ñî ñëî- èñòîé äèñïåðñíîé ñðåäîé. Äëÿ îïèñàíèÿ ýëåêòðî- ìàãíèòíîãî ïîëÿ íåîáõîäèìî íàõîäèòü ðåøåíèÿ óðàâíåíèÿ â ÷àñòíûõ ïðîèçâîäíûõ íå òèïà Êîøè- Êîâàëåâñêîé. Ýòî óðàâíåíèå íå ÿâëÿåòñÿ ðàçðåøà- åìûì îòíîñèòåëüíî ñòàðøåé ïðîèçâîäíîé ïî âðå- ìåíè, ÷òî ñóùåñòâåííî óñëîæíÿåò èññëåäîâàíèå ìîäåëè. Óñòàíîâëåíà íîâàÿ òåîðåìà ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿ íà÷àëüíî-êðàåâîé çàäà- ÷è äëÿ óðàâíåíèÿ íå òèïà Êîøè-Êîâàëåâñêîé. Ðåçóëüòàòû ñòàòüè ìîãóò íàéòè ïðàêòè÷åñêîå ïðè- ìåíåíèå ïðè èññëåäîâàíèè ýëåêòðîìàãíèòíûõ ïî- ëåé â âîëíîâîäàõ. Äëÿ ïðàêòè÷åñêîãî ïðèìåíåíèÿ ðåçóëüòàòîâ äàííîé ñòàòüè ðàçðàáîòàí íîâûé ÷èñ- ëåííûé ìåòîä íàõîæäåíèÿ ðåøåíèÿ ñìåøàííîé çàäà÷è äëÿ óðàâíåíèÿ íå òèïà Êîøè-Êîâàëåâñêîé. Ëèòåðàòóðà: 1. Ðóòêàñ À.Ã. Ìîäîâûå ïîëÿ â âîëíîâîäå ñî ñëîèñòîé äèñïåðãèðóþùåé ñðåäîé. Õàðüêîâ: Ïðå- ïðèíò N 360, ÀÍ ÓÑÑÐ. Èí-ò ðàäèîôèçèêè è ýëåêòðî- íèêè, 1987. 34c. 2. Rutkas A.G., Vlasenko L.A. Implicit operator differential equations and applications to electrodynamics // Mathematical Methods in the Applied Sciences. 2000. Vol. 23, N 1. P. 1-15. 3. Ãîäóíîâ Ñ.Ê., Ðÿáåíüêèé Â.Ñ. Ðàçíîñòíûå ñõåìû. Ì.: Íàóêà, 1977. 440 ñ. 4. Ðÿáåíüêèé Â.Ñ., Ôèëèïïîâ À.Ô. Îá óñòîé÷èâîñòè ðàçíîñòíûõ óðàâíåíèé. Ì.: Ãîñòåõèçäàò, 1956. 171 ñ. 5. Ëàíäàó Ë.Ä., Ëèâøèö Å.Ë. Ýëåêòðîäèíàìèêà ñïëîø- íûõ ñðåä. Ì.: Ôèçìàòãèç, 1959. 532 ñ. 6. Ëåâè÷ Â.Ã. Êóðñ òåîðåòè÷åñêîé ôèçèêè. Ò.1. Ì. : Íàóêà, 1969. 910 ñ. 7. Äàëåöêèé Þ.Ë., Êðåéí Ì.Ã. Óñòîé÷èâîñòü ðåøåíèé äèôôåðåíöèàëüíûõ óðàâíåíèé â áàíàõîâîì ïðîñòðàí- ñòâå. Ì.: Íàóêà,1970. 536 ñ. 8. Ëèîíñ Æ.Ë. Íåêîòîðûå ìåòîäû ðåøåíèÿ íåëèíåéíûõ êðàåâûõ çàäà÷. Ì.: Ìèð, 1972. 588 ñ. 9. Ìèçîõàòà Ñ. Òåîðèÿ óðàâíåíèé ñ ÷àñòíû- ìè ïðîèçâîäíûìè. Ì.: Ìèð, 1977. 504 ñ. 10. Ñîáîëåâ Ñ.Ë. Îá îäíîé çàäà÷å ìàòåìàòè÷åñêîé ôèçèêè // Èçâ. ÀÍ ÑÑÑÐ. Ñåð. Ìàòåìàòèêà. 1954. Ò. 18, N 1. C. 3-50. 11. Ãàëüïåðí Ñ.À. Çàäà÷à Êîøè äëÿ ñèñòåìû ëèíåéíûõ óðàâíåíèé ñ ÷àñòíûìè ïðîèçâîäíûìè // Äîêë. ÀÍ ÑÑÑÐ. 1958. N 4. C. 640-643. 12. Êîñòþ÷åíêî À.Ã., Ýñêèí Ã.È. Çàäà÷à Êîøè äëÿ óðàâíåíèé Ñîáîëåâà-Ãàëüïåðíà // Òðóäû Ìîñêîâñêîãî ìàòåìàòè÷åñêîãî îáùåñòâà. 1961. Ò. 10. Ñ. 273-284. 13. Âëàñåíêî Ë.À., Ðóòêàñ À.Ã. Ðàçðåøèìîñòü è ïîëíîòà äëÿ ýëåêòðîäèíàìè÷åñêîé ñèñòåìû íå òèïà Êîâàëåâñêîé // Ìàòåìàòè÷åñêèå çà- ìåòêè. 1993. Ò. 53. Âûï. 1. Ñ.138-141. 14. Rutkas A.G., Vlasenko L.A. Existence, uniqueness and continuous dependence for implicit semilinear functional differential equations // Nonlinear Analysis. TMA. 2003. V. 55, N 1- 2. P. 125-139. Ïîñòóïèëà â ðåäêîëëåãèþ 31.01.2005 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Ðóòêàñ À.Ã. Âëàñåíêî Ëàðèñà Àíäðååâíà, êàíä. ôèç.-ìàò. íàóê, äîöåíò êàôåäðû ìîäåëèðîâàíèÿ è ìàò. îáåñïå÷åíèÿ ÝÂÌ Õàðüêîâñêîãî íàöèîíàëüíîãî óíèâåðñèòåòà èì. Â.Í.Êàðàçèíà. Íàó÷íûå èíòåðåñû: ìîäåëèðîâàíèå, äèôôåðåíöèàëüíûå óðàâíåíèÿ. Àäðåñ: Óêðàèíà, 61001, Õàðüêîâ, óë. Ïëåõàíîâñêàÿ, 2/5, êâ. 29, äîì. òåë.: (057) 732-28-35. ÓÄÊ 621.317 ÊÐÈÒÅÐÈÈ ÎÖÅÍÊÈ ÏÎÑÒÎßÍÑÒÂÀ ÂÛÁÎÐÎÊ ÄËß ÀËÃÎÐÈÒÌΠÎÁÐÀÁÎÒÊÈ ÑÈÃÍÀËΠ ÖÈÔÐÎÂÛÕ ÎÑÖÈËËÎÃÐÀÔÀÕ ÁÀÐÀÍΠÍ.Ã., ÊËÞ×ÍÈÊ È.È., ËÎÄÛÃÈÍ Ì.À. Ïðèâîäÿòñÿ ðåçóëüòàòû ñîïîñòàâèòåëüíîãî àíàëèçà êðèòåðèåâ îöåíêè âûáîðîê âèçóàëèçèðóåìûõ ñèãíà- ëîâ. Ïîêàçûâàåòñÿ, ÷òî ïî ñîâîêóïíîñòè ïàðàìåíòðîâ íàèáîëåå îïòèìàëüíûì ÿâëÿåòñÿ êðèòåðèé, ïðåäëî- æåííûé àâòîðàìè. Ââåäåíèå. Ðàíåå [1] áûë ïðåäëîæåí ñïîñîá îïðåäå- ëåíèÿ ïîñòîÿíñòâà ïîëó÷àåìûõ âûáîðîê àíàëîãîâî- ãî ñèãíàëà íà îñíîâå ðàñ÷¸òà òàê íàçûâàåìîãî êðè- òåðèÿ Ñ. Ýòîò êðèòåðèé ïðåäíàçíà÷åí äëÿ áûñòðîãî ñðàâíåíèÿ ïîëó÷àåìûõ âûáîðîê è îïðåäåëåíèÿ ñòå- ïåíè èõ èäåíòè÷íîñòè, ïðè ýòîì ñðàâíåíèå òåêóùåé è ïðåäûäóùåé âûáîðîê ïðîèçâîäèòñÿ òàêèì îáðà- çîì, ÷òî ïîÿâëåíèå ðåçóëüòàòà îáåñïå÷èâàåòñÿ â ìîìåíò ïîëó÷åíèÿ ïîñëåäíåé âûáîðêè. Àêòóàëüíîé ÿâëÿåòñÿ ïðîáëåìà ñîïîñòàâèìîñòè äàííîãî êðèòå- ðèÿ ñ óæå èñïîëüçóåìûìè àíàëîãè÷íûìè êðèòåðè- ÿìè è â ïåðâóþ î÷åðåäü ñ íàèáîëåå áëèçêèìè è øèðîêî èñïîëüçóåìûìè â òåõíèêå. Öåëüþ ïðîâåä¸ííûõ èññëåäîâàíèé ÿâëÿåòñÿ îáî- ñíîâàíèå âûáîðà íàèáîëåå ïåðñïåêòèâíîãî â àëãî- ðèòìàõ îáðàáîòêè ñèãíàëîâ öèôðîâûõ îñöèëëîãðà- ôîâ êðèòåðèÿ îöåíêè ïîñòîÿíñòâà âûáîðîê ñèãíà- ëîâ íà îñíîâå ñîïîñòàâèòåëüíîãî àíàëèçà êðèòåðè- åâ, èñïîëüçóåìûõ ïî óêàçàííîìó íàçíà÷åíèþ. Çàäà÷è íàñòîÿùåé ðàáîòû: íàõîæäåíèå íàèáîëåå áëèçêîãî êðèòåðèÿ-àíàëîãà, ïðîâåäåíèå èõ ñðàâíè- òåëüíîãî àíàëèçà, îïðåäåëåíèå ïðåèìóùåñòâ ïðåä- ëîæåííîãî êðèòåðèÿ íàä àíàëîãîì ïðèìåíèòåëüíî ê îáðàáîòêå àíàëîãîâûõ ñèãíàëîâ. Ñóòü èññëåäîâàíèÿ. Ïðîöåññ ñðàâíåíèÿ âûáîðîê ñèãíàëà, ïðîèçâîäèìûé ñ èñïîëüçîâàíèåì êðèòå- ðèÿ Ñ, èìååò ìíîãî îáùåãî ñ çàäà÷åé ñòàòèñòè÷åñ- êîãî êîíòðîëÿ êà÷åñòâà [2], ðåøàåìîé äëÿ ïðîâåðêè âîçìóùåíèÿ â õîäå íåêîòîðîãî ïðîöåññà.  ýòèõ ñëó÷àÿõ, êàê ïðàâèëî, èñïîëüçóåòñÿ ðÿä ñòàòèñòè- ÷åñêèõ êðèòåðèåâ, ñ ïîìîùüþ êîòîðûõ ïðîâåðÿþò- ñÿ ãèïîòåçû î ïîñòîÿíñòâå äèñïåðñèé êîíòðîëèðó- åìîãî ïîêàçàòåëÿ èëè ãèïîòåçû î ðàâåíñòâå ýòîãî ïîêàçàòåëÿ íîìèíàëüíîìó çíà÷åíèþ. 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Äëÿ ñðàâíåíèÿ íà ðèñ.3 ïðèâîäèòñÿ àíàëîãè÷íûé ãðàôèê, íî çàâè- ñèìîñòè êðèòåðèÿ Ñ îò ðàçìåðà âûáîðêè. Êàê âèäíî èç ïðåäñòàâëåííûõ ãðàôèêîâ, íàáëþäà- åòñÿ êîðåëëÿöèÿ ìåæäó ñðàâíèâàåìûìè êðèòåðèÿ- ìè, è õîòÿ ÷èñëåííûå çíà÷åíèÿ ñóùåñòâåííî îòëè- ÷àþòñÿ, “ õàðàêòåðíûå òî÷êè ” ÿñíî ðàçëè÷èìû è ìîãóò áûòü èäåíòèôèöèðîâàíû àëãîðèòìè÷åñêèìè ìåòîäàìè. Ýòîò ôàêò ìîæíî ñ÷èòàòü ñâèäåòåëü- ñòâîì òîãî, ÷òî êðèòåðèé Ñ, ÿâëÿåòñÿ àíàëîãîì êðèòåðèÿ Êîõðåíà â ïðèìåíåíèè ê àíàëèçó ñèãíà- ëîâ. Ñðàâíèâàÿ àëãîðèòì ïîëó÷åíèÿ ÷èñëîâûõ çíà- ÷åíèé äëÿ êàæäîãî èç êðèòåðèåâ, ìîæíî çàìåòèòü, ÷òî äëÿ ðàñ÷¸òà êðèòåðèÿ Ñ èñïîëüçóåòñÿ àëãîðèòì, â ïðîöåññå ðàáîòû êîòîðîãî ðåçóëüòàò ïîÿâëÿåòñÿ íå òîëüêî îäíîâðåìåííî ñ ïîñëåäíèì çíà÷åíèåì âûáîðêè, íî è ïîëó÷àåìûå çíà÷åíèÿ íàõîäÿòñÿ â îáëàñòè öåëûõ ÷èñåë, ÷òî âûãîäíî îòëè÷àåò åãî îò äðóãèõ êðèòåðèåâ è â ÷àñòíîñòè îò êðèòåðèÿ Êîõ- ðåíà. Ýòî äà¸ò âîçìîæíîñòü ñóùåñòâåííî óïðîñòèòü âû÷èñëåíèÿ è ïîâûñèòü èõ ñêîðîñòü. Êðèòåðèé Êîõðåíà èñïîëüçóåò çíà÷èòåëüíî áîëåå ñëîæíûå âû÷èñëåíèÿ, ÷àñòü êîòîðûõ, ê òîìó æå, äîëæíà áûòü ïðîèçâåäåíà ïîñëå ïîëó÷åíèÿ ñðàâíèâàåìîé âûáîðêè. Ðèñ. 1. Àíàëèçèðóåìûé òåñòîâûé ñèãíàë Ðèñ. 2. Çíà÷åíèÿ êðèòåðèÿ Êîõðåíà äëÿ äâóõ âûáîðîê, ïðè àíàëèçå òåñòîâîãî ñèãíàëà, â çàâèñè- ìîñòè îò ðàçìåðà âûáîðêè (âðåìåíè, âûäåëÿåìîãî íà âûáîðêó) Ðèñ. 3. Çíà÷åíèÿ êðèòåðèÿ Ñ äëÿ äâóõ âûáîðîê, ïðè àíàëèçå òåñòîâîãî ñèãíàëà, â çàâèñèìîñòè îò ðàçìåðà âûáîðêè (âðåìåíè, âûäåëÿåìîãî íà âûáîðêó) Êðîìå òîãî, êðèòåðèé Êîõðåíà ïðåäïîëàãàåò, ÷òî åãî çíà÷åíèÿ ìîãóò áûòü â îáëàñòè ÷èñåë ñ ïëàâàþùåé òî÷êîé, ÷òî òðåáóåò çíà÷èòåëüíî áîëåå ïðîèçâîäè- òåëüíîãî àïïàðàòíîãî îáåñïå÷åíèÿ äëÿ ðàáîòû. Öèôðîâûå îñöèëëîãðàôû â íàñòîÿùåå âðåìÿ ðàáî- òàþò â ðåæèìå ñëó÷àéíûõ âûáîðîê ïî ïðè÷èíå òîãî, ÷òî êëàññè÷åñêèé ðåæèì ïîñëåäîâàòåëüíûõ âûáîðîê íå â ñîñòîÿíèè îáåñïå÷èòü ñòàáèëüíóþ ðàáîòó ïðèáîðà ïðè íå èäåàëüíî óñòîé÷èâîì ñèãíà- ëå. Êðèòåðèè æå, êîòîðûìè ìîæíî âîñïîëüçîâàòñÿ äëÿ îïðåäåëåíèÿ òîãî, ÷òî àíàëèçèðóåìûé ñèãíàë ïåðèîäè÷åñêèé, î÷åíü ñëîæíû äëÿ àïïàðàòíîé ðåëàëèçàöèè è íå ðàáîòàþò â ðåæèìå ðåàëüíîãî âðåìåíè. Íàéäåííûé êðèòåðèé Ñ ëèø¸í ýòèõ íåäîñòàòêîâ è ñ óñïåõîì ïðèìåí¸í â îñöèëëîãðà- ôàõ, ðàçðàáàòûâàåìûõ â ÕÍÓÐÝ, äëÿ çíà÷èòåëü- íîãî óâåëè÷åíèÿ ïîëîñû ïðîïóñêàíèÿ áåç óñëîæíå- íèÿ ïðè¸ìíîé ÷àñòè è, â ÷àñòíîñòè, ÀÖÏ. Îöåíêó òî÷íîñòè ñîîòâåòñòâèÿ ïîëó÷àåìûõ îñöèë- ëîãðàìì ðåàëüíîìó ñèãíàëó ìîæíî ïðîèçâåñòè ïîëó÷èâ çíà÷åíèå âîçìîæíîé îøèáêè, âîçíèêàþ- ùåé â ïðîöåññàõ ïðåîáðàçîâàíèÿ è ñðàâíåíèÿ. Ïîãðåøíîñòü ñîñòîèò èç òð¸õ îñíîâíûõ ÷àñòåé – îøèáêà, ïîëó÷àåìàÿ âñëåäñòâèå íåñîîòâåòñòâèÿ ïå- ðèîäîâ äðóã äðóãó, îøèáêà àíàëîãî-öèôðîâîãî ïðåîáðàçîâàíèÿ è îøèáêà àïïðîêñèìàöèè. Ìàêñè- ìàëüíîå îòêëîíåíèå ìãíîâåííûõ çíà÷åíèé ñðàâíè- âàåìûõ âûáîðîê îïðåäåëÿåòñÿ ïàðàìåòðîì “ ÷óâ- ñòâèòåëüíîñòü íóëÿ ” [1]. Êàæäàÿ åäèíèöà ýòîãî ïàðàìåòðà îïðåäåëÿåò îòêëîíåíèå, ÷èñëåííî ðàâíîå îäíîé ñòóïåíè äèñêðåòèçàöèè. Òàêèì îáðàçîì, ìàêñèìàëüíîå ñðåäíåå çíà÷åíèå ðàñõîæäåíèÿ ïåðè- îäîâ ñèãíàëà ìîæíî ðàññ÷èòàòü êàê: d Vref * Zs   , (1) ãäå s – ìàêñèìàëüíîå ñðåäíåå çíà÷åíèå îøèáêè, Â; Zs – âåëè÷èíà «÷óâñòâèòåëüíîñòè íóëÿ»; Vref – çíà÷åíèå îïîðíîãî íàïðÿæåíèÿ ÀÖÏ, Â; d – ÷èñëî ñòóïåíåé äèñêðåòèçàöèè ÀÖÏ. Îøèáêó äèñêðåòèçàöèè àíàëîãî-öèôðîâîãî ïðåîá- ðàçîâàíèÿ ìîæíî ïðåäñòàâèòü ñëåäóþùèì îáðàçîì: d=0,75*Vref/d. (2) Îøèáêà àïïðîêñèìàöèè ìåòîäà Ëàãðàíæà ðàññ÷è- òûâàåòñÿ ïî ôîðìóëå [4]:     N 1 k k N ) x (x max * N! D δ , (3) ãäå D N – ìàêñèìàëüíîå çíà÷åíèå ôóíêöèè, â ðàññìàòðèâàåìîì ñëó÷àå ìîæíî ñ÷èòàòü ðàâíûì Vref; N – ÷èñëî ïîëþñîâ; (x-x k )– âðåìåííîé èíòðåâàë ìåæäó òî÷êîé àïïðîêñèìàöèè è áëèæàé- øåé òî÷êîé ïîëþñà. Ñôîðìóëèðîâàòü àíàëèòè÷åñêèå çàâèñèìîñòè êðè- òåðèåâ Êîõðåíà è êðèòåðèÿ Ñ îò ïîãðåøíîñòè îêàçàëîñü ñëîæíîé çàäà÷åé. Ïîýòîìó áûëî ïðîèç- âåäåíî êîìïüþòåðíîå ìîäåëèðîâàíèå è ðàñ÷¸ò ýòîé çàâèñèìîñòè ÷èñëåííûìè ìåòîäàìè. 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Ïðåäñòàâëåííûå çàâèñèìîñòè ïîëó÷åíû ïðè ðàçìå- ðå âûáîðêè 0,1ñ – ïåðâûé ìèíèìóì íà ãðàôèêàõ ðèñ. 2 è 3. Ðèñ. 4. Çàâèñèìîñòü ïîãðåøíîñòè îò êðèòåðèÿ Êîõðåíà Ðèñ. 5. Çàâèñèñìîñòü ïîãðåøíîñòè îò êðèòåðèÿ Ñ Âûâîäû. Òàêèì îáðàçîì, ïðàêòè÷åñêèì ðåçóëüòà- òîì èñïîëüçîâàíèÿ ïðåäëîæåííîãî êðèòåðèÿ ÿâëÿ- åòñÿ ðàáîòà ïðèáîðîâ, îòëè÷àþùàÿñÿ áîëüøåé ðà- öèîíàëüíîñòüþ èñïîëüçîâàíèÿ äîñòóïíûõ ðåñóð- ñîâ ëèáî èñïîëüçîâàíèåì ìåíüøåãî êîëè÷åñòâà ðåñóðñîâ äëÿ ïîëó÷åíèÿ àíàëîãè÷íûõ ðåçóëüòàòîâ. Ýòî äîñòèãàåòñÿ ïóòåì ðàñïàðàëëåëèâàíèÿ ïðîöåñ- ñîâ ïîëó÷åíèÿ äàííûõ è èõ îáðàáîòêè. Óêàçàííûå ïðåèìóùåñòâà ïîçâîëÿþò ñíèçèòü ñåáåñòîèìîñòü èçìåðèòåëüíûõ ïðèáîðîâ â íåñêîëüêî ðàç.  öåëîì, ïî ñîâîêóïíîñòè ñâîéñòâ öèôðîâûå îñöèëëîãðàôû, ôóíêöèîíèðóþùèå íà îñíîâå ïðåäëîæåííîãî ñïî- ñîáà, çíà÷èòåëüíî ïðåâûøàþò ñâîéñòâà ñóùåñòâó- þùèõ ïðèáîðîâ. Íàó÷íàÿ íîâèçíà äàííîé ðàáîòû ïîêàçàíà â ñðàâíå- íèè ðåçóëüòàòîâ, ïîëó÷àåìûõ ñ èñïîëüçîâàíèåì êðèòåðèÿ Ñ è êðèòåðèÿ Êîõðåíà. Ïðåèìóùåñòâîì êðèòåðèÿ Ñ ÿâëåòñÿ çíà÷èòåëüíî áîëüøàÿ ìîùíîñòü, ÷òî îáåñïå÷èâàåò áîëåå òî÷íûé ðåçóëüòàò ïðè êîëè- ÷åñòâå âûáîðîê 2. Ïðè ýòîì ÿâíî ïîêàçàí èçâåñòíûé íåäîñòàòîê êðèòåðèÿ Êîõðåíà- åãî ìàëàÿ ìîùíîñòü. Ëèòåðàòóðà: 1. Áàðàíîâ Í.Ã., Êëþ÷íèê È.È., Ëîäûãèí Ì.À. Öèôðîâîé îñöèëëîãðàô íà îñíîâå óñòðîéñòâ ïðî- ãðàììèðóåìîé ëîãèêè // Ðàäèîýëåêòðîíèêà è èíôîð- ìàòèêà. 2004. ¹ 2. Ñ.31-33. 2. Ìèòòàã Õ.-É., Ðèííå Õ. Ñòàòèñòè÷åñêèå ìåòîäû îáåñïå÷åíèÿ êà÷åñòâà. Ì.: Ìàøèíîñòðîåíèå. 1995. 600 ñ. 3. Cochran W.G. The distribution of the largest of a set of estimated variances as a fraction of their total // Ann. of eugenics. 1941. 11. P. 47- 52. 4. Ñïðàâî÷íèê ïî ìàòåìàòèêå äëÿ èíæåíåðîâ è ó÷àùèõñÿ âòóçîâ / Áðîíøòåéí È.Í., Ñåìåíèöåâ Ê.À. Ì.:Íàóêà, 1981. 704ñ. Ïîñòóïèëà â ðåäêîëëåãèþ 25.11.2004 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Èëþøêî Â.Ì. Áàðàíîâ Íèêîëàé Ãàâðèëîâè÷, ñòàðøèé ïðåïîäàâà- òåëü êàôåäðû ÏÝÝÀ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: öèô- ðîâàÿ îñöèëëîãðàôèÿ. Àäðåñ: Óêðàèíà, 61146, Õàðü- êîâ, óë. Àêàäåìèêà Ïàâëîâà, 148à , êâ. 17, òåë. +380-572- 654425. Êëþ÷íèê Èãîðü Èâàíîâè÷, êàíä. òåõí. íàóê, ïðîôåññîð êàôåäðû ÏÝÝÀ ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: òåõíèêà ÑÂ×, àâòîìàòèçàöèÿ ïðîåêòèðîâàíèÿ, ìåäèöèíà, ïñè- õîëîãèÿ. Óâëå÷åíèÿ: îõîòà, ôèëàòåëèÿ. Àäðåñ: Óêðàè- íà, 61166, Õàðüêîâ, ïð. Ëåíèíà, 14, òåë.+380-57-7021- 440; +380-57-7021-448;+380-57-7021-494. Ëîäûãèí Ìèõàèë Àëåêñàíäðîâè÷, àñïèðàíò ÕÍÓÐÝ. Íàó÷íûå èíòåðåñû: èçìåðèòåëüíûå ïðèáîðû, öèôðî- âàÿ òåõíèêà. Àäðåñ: Óêðàèíà, 61045, Õàðüêîâ, óë.Øåê- ñïèðà, 12, êâ.39, òåë. +380-572-321974. ÓÄÊ 621.385.6 ÝÂÎËÞÖÈß ÑÏÅÊÒÐΠÑÈÃÍÀËΠÏÐÈ ÏÅÐÅÕÎÄÍÛÕ ÏÐÎÖÅÑÑÀÕ Â ÃÅÍÅÐÀÒÎÐÀÕ È ÓÑÈËÈÒÅËßÕ Ì-ÒÈÏÀ ÂÀÑßÍÎÂÈ× À.Â., ÃÐÈÖÓÍΠÀ.Â., ËÅÁÅÄÅ  Î.Ã., ×ÓÐÞÌΠÃ.È. Ñ ïîìîùüþ ñïåêòðàëüíîãî ïîäõîäà èññëåäóåòñÿ âðå- ìåííàÿ ýâîëþöèÿ ñïåêòðîâ âûõîäíîãî ñèãíàëà â íå- ñòàöèîíàðíûõ ðåæèìàõ ìàãíåòðîíîâ è àìïëèòðîíîâ. Óñòàíàâëèâàåòñÿ, ÷òî ïðè âêëþ÷åíèè ýòèõ ïðèáîðîâ ìåæäó ïåðèîäîì ñàìîãðóïïèðîâêè âòóëêè è ïåðèîäîì åå ñèíõðîíèçàöèè Â×-ïîëåì ýëåêòðîäèíàìè÷åñêîé ñèñòåìû (ÝÑ) íàõîäèòñÿ ïåðèîä ïîäâîçáóæäåíèÿ ñîá- ñòâåííîãî êîëåáàíèÿ ýëåêòðîííîãî îáëàêà ïîëåì ÝÑ, ò.å. â ñïåêòðå ñèãíàëà ïðèñóòñòâóþò äâå ãàðìîíèêè: ñ ÷àñòîòîé ñîáñòâåííûõ êîëåáàíèé âòóëêè è ÷àñòîòîé âîçáóæäàåìîé íîðìàëüíîé ìîäû ÝÑ. Ïðè èçìåíåíèè âèäà êîëåáàíèé ìàãíåòðîííîãî àâòîãåíåðàòîðà ýëåêò- ðîííûé ïîòîê â òå÷åíèå íåêîòîðîãî âðåìåíè èìååò ñëîæíóþ êîíôèãóðàöèþ, îáðàçîâàííóþ âîçäåéñòâèåì íà íåãî Â×-ïîëåé ïðåäûäóùåé è ïîñëåäóþùåé íîð- ìàëüíûõ ìîä ÝÑ ñ ñîïîñòàâèìûìè àìïëèòóäàìè.  ðåçóëüòàòå ñïåêòð âûõîäíîãî ñèãíàëà îáîãàùàåòñÿ ìíî- ãî÷èñëåííûìè êîìáèíàöèîííûìè ñîñòàâëÿþùèìè. 1. Ââåäåíèå  ñâÿçè ñ ðàçâèòèåì ñèñòåì ìîáèëüíûõ òåëåêîììó- íèêàöèé è ðàñøèðåíèåì òåõíîëîãè÷åñêèõ ïðèìå- íåíèé ÑÂ×-ýíåðãèè ïðîáëåìà ýëåêòðîìàãíèòíîé ñîâìåñòèìîñòè (ÝÌÑ) ðàäèîýëåêòðîííûõ ñðåäñòâ ïðèîáðåòàåò â ïîñëåäíèå ãîäû îñîáóþ àêòóàëü- íîñòü. Áîëåå âñåãî îíà çàòðàãèâàåò ìîùíûå ãåíåðà- òîðû è óñèëèòåëè ÑÂ×-êîëåáàíèé, êàêèìè ÿâëÿ- þòñÿ, â ÷àñòíîñòè, ïðèáîðû Ì-òèïà. Äàæå íåáîëü- øîé â îòíîñèòåëüíîì âûðàæåíèè óðîâåíü ïîáî÷- íûõ êîìïîíåíò â ñïåêòðå âûõîäíîãî ñèãíàëà òàêèõ ïðèáîðîâ â àáñîëþòíûõ åäèíèöàõ ìîæåò äîñòèãàòü íåïðèåìëåìûõ âåëè÷èí. 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Îäíàêî âñå îíè âûïîëíåíû, âî-ïåðâûõ, â ìíîãî- ÷àñòîòíîì ïðèáëèæåíèè, ñóùåñòâåííî îãðàíè÷èâà- þùåì äîñòîâåðíîñòü ðåçóëüòàòîâ è âîçìîæíîñòè àíàëèçà ïîëó÷åííûõ ñïåêòðîâ, âî-âòîðûõ – ðàñ- ñìàòðèâàþò òîëüêî ñòàöèîíàðíûå ðåæèìû ðàáîòû èññëåäóåìûõ ïðèáîðîâ.  òî æå âðåìÿ ñïåêòðàëü- íûå õàðàêòåðèñòèêè, ïîëó÷åííûå â ñòàöèîíàðíîì ñîñòîÿíèè, ÿâëÿþòñÿ íåäîñòàòî÷íî ïîëíûìè ñ òî÷- êè çðåíèÿ ïðîáëåì ÝÌÑ ðàäèîòåõíè÷åñêèõ è òåõíîëîãè÷åñêèõ óñòðîéñòâ, ðàáîòàþùèõ â èìïóëü- ñíîì ðåæèìå, òàê êàê íà ôðîíòàõ èìïóëüñîâ âûõîäíîé ñèãíàë îáû÷íî îáîãàùàåòñÿ äîïîëíè- òåëüíûìè ñïåêòðàëüíûìè ñîñòàâëÿþùèìè. Êîí- êóðåíöèÿ âèäîâ êîëåáàíèé òàêæå ÷àùå âñåãî ïðî- ÿâëÿåòñÿ âî âðåìÿ âêëþ÷åíèÿ ïðèáîðà (íà ôðîíòå èìïóëüñà). Íåó÷åò óêàçàííûõ ÿâëåíèé ìîæåò ñóùå- ñòâåííî îáåñöåíèòü ðåçóëüòàòû èññëåäîâàíèé ïî îòäåëüíîñòè êàê ñïåêòðîâ ñèãíàëîâ, òàê è ïåðåõîä- íûõ ïðîöåññîâ â ÑÂ×-ïðèáîðàõ [2], êîòîðûå ñâÿ- çàíû ìåæäó ñîáîé íàñòîëüêî òåñíî, ÷òî èìååò ñìûñë òîëüêî êîìïëåêñíîå (ñîâìåñòíîå) èõ èçó÷åíèå. Òàêàÿ âîçìîæíîñòü ïîÿâèëàñü ëèøü ïîñëå ðåàëèçà- öèè ñïåêòðàëüíîãî ïîäõîäà ê ìîäåëèðîâàíèþ ÑÂ×- ïðèáîðîâ [3]. Îí çàêëþ÷àåòñÿ â òðàíñôîðìàöèè ïóòåì îáðàòíîãî ïðåîáðàçîâàíèÿ Ôóðüå çàäàííîãî â ÷àñòîòíîé îáëàñòè âõîäíîãî Â×-ñèãíàëà âî âðå- ìåííóþ îáëàñòü è èñïîëüçîâàíèè åãî çàòåì êàê ãðàíè÷íîãî óñëîâèÿ íåñòàöèîíàðíîé ìîäåëè. Ïîñëå ìîäåëèðîâàíèÿ ïðîõîæäåíèÿ ñèíòåçèðîâàííîé òà- êèì îáðàçîì íåãàðìîíè÷åñêîé ýëåêòðîìàãíèòíîé âîëíû ÷åðåç ÝÑ ñ ó÷åòîì åå íåëèíåéíîãî âçàèìî- äåéñòâèÿ ñ ýëåêòðîííûì ïîòîêîì âðåìåííàÿ âû- áîðêà Â×-ñèãíàëà íà âûõîäå ïðèáîðà, íàðÿäó ñ äðóãèìè âûõîäíûìè ïàðàìåòðàìè (íàïðèìåð, òî- êàìè ýëåêòðîäîâ), ïîäâåðãàåòñÿ ñïåêòðàëüíîìó àíà- ëèçó äëÿ ïåðåâîäà åå îáðàòíî â ÷àñòîòíóþ îáëàñòü. Öåëüþ äàííîé ðàáîòû ÿâëÿåòñÿ êîìïëåêñíîå èçó÷å- íèå ÿâëåíèé â ñêðåùåííûõ ïîëÿõ íà îñíîâå ñïåê- òðàëüíîãî ïîäõîäà, âêëþ÷àþùåå àíàëèç ñïåêòðîâ ñèãíàëîâ ïðè ïåðåõîäíûõ ïðîöåññàõ è èõ ïîñëåäó- þùóþ ôèçè÷åñêóþ èíòåðïðåòàöèþ. Îñíîâíûå çàäà÷è èññëåäîâàíèÿ: – ïðèìåíåíèå ñïåêòðàëüíîãî ïîäõîäà ê ìîäåëèðî- âàíèþ ìàãíåòðîííûõ àâòîãåíåðàòîðîâ (ìàãíåòðî- íîâ) è ìàãíåòðîííûõ óñèëèòåëåé îáðàòíîé âîëíû (àìïëèòðîíîâ) ïðè èõ âêëþ÷åíèè èç õîëîäíîãî ñîñòîÿíèÿ; – èñïîëüçîâàíèå óêàçàííîãî ïîäõîäà äëÿ èññëåäî- âàíèÿ ñïåêòðîâ âûõîäíîãî ñèãíàëà ìàãíåòðîííûõ ãåíåðàòîðîâ ïðè èçìåíåíèè âèäà êîëåáàíèé â íèõ. Ïîñêîëüêó â áîëüøèíñòâå ìàãíåòðîíîâ, ñ îäíîé ñòîðîíû, è àìïëèòðîíîâ – ñ äðóãîé íàáëþäàþòñÿ àíàëîãè÷íûå ÿâëåíèÿ, íåò ñìûñëà ðàññìàòðèâàòü èõ äëÿ êàæäîãî êîíêðåòíîãî ïðèáîðà.  ðåçóëüòàòå âû÷èñëèòåëüíûõ ýêñïåðèìåíòîâ ñî ìíîãèìè êîí- ñòðóêöèÿìè âûáðàíû äâå èç íèõ, íàèáîëåå íàãëÿä- íî èëëþñòðèðóþùèå òèïè÷íûå ÿâëåíèÿ â èññëåäó- åìûõ êëàññàõ ïðèáîðîâ Ì-òèïà. Ýòî ìàãíåòðîí ñðåäíåé ìîùíîñòè Ì-156, ïðåäíàçíà÷åííûé äëÿ áûòîâûõ ÑÂ×-ïå÷åé, è àìïëèòðîí ñðåäíåé ìîùíî- ñòè ÌÈÓ-90 [4], èñïîëüçóåìûé â ÐËÑ ãðàæäàíñ- êîãî íàçíà÷åíèÿ. Âñå ïîñëåäóþùåå èçëîæåíèå áà- çèðóåòñÿ íà àíàëèçå ôèçè÷åñêèõ ïðîöåññîâ èìåííî â ýòèõ ïðèáîðàõ. Îñíîâíûå ïàðàìåòðû ìàãíåòðîíà Ì-156: ðàäèóñ êàòîäà 2,3 ìì; ðàäèóñ àíîäà (ïî òîðöàì ëàìåëåé) 4,45 ìì; øèðèíà ïðèáîðà 10,0 ìì; ÷èñëî ëàìåëåé N = 10; ñîáñòâåííàÿ ÷àñòîòà N/2 ìîäû ÝÑ 2,45 ÃÃö, åå íàãðóæåííàÿ äîáðîòíîñòü 127,0; àíàëîãè÷íî äëÿ N/2–1 ìîäû 4,4 ÃÃö è 286,0 ñîîòâåòñòâåííî; èíäóêöèÿ âíåøíåãî ìàãíèòíîãî ïîëÿ 0,157 Òë; àíîäíîå íàïðÿæåíèå 3,05 êÂ. Ïàðàìåòðû àìïëèòðîíà ÌÈÓ-90: ðàäèóñ êàòîäà 5,5 ìì; ðàäèóñ àíîäà 10,75 ìì; øèðèíà ïðèáîðà 36,0 ìì; ÷èñëî ëàìåëåé N = 11; íîìåð ðàáî÷åé íîðìàëü- íîé ìîäû ÝÑ 4; ÷àñòîòà ðàáî÷åé íîðìàëüíîé ìîäû 1,25 ÃÃö; åå ñîïðîòèâëåíèå âçàèìîäåéñòâèÿ [5] 175 Îì; èíäóêöèÿ âíåøíåãî ìàãíèòíîãî ïîëÿ 0,12 Òë; àíîäíîå íàïðÿæåíèå 9,5 êÂ; âõîäíàÿ ìîùíîñòü 1,0 êÂò (âõîäíîé ñèãíàë ïðåäïîëàãàåòñÿ ãàðìîíè÷åñ- êèì). 2. Ýâîëþöèÿ âûõîäíîãî ñïåêòðà ìàãíåòðîíà ïðè âêëþ÷åíèè ïðèáîðà Îáùàÿ êàðòèíà ïåðåõîäíîãî ïðîöåññà ïðè âêëþ÷å- íèè àâòîãåíåðàòîðîâ è óñèëèòåëåé îáðàòíîé âîëíû Ì-òèïà ñ êàòîäîì â ïðîñòðàíñòâå âçàèìîäåéñòâèÿ è çàìêíóòûì ýëåêòðîííûì ïîòîêîì îïèñàíà â ðàáîòå [2].  ÷àñòíîñòè, óñòàíîâëåíî, ÷òî äàííûé ïðîöåññ ìîæíî ðàçäåëèòü íà òðè ýòàïà: – ïðåäãåíåðàöèîííûé, âî âðåìÿ êîòîðîãî ñïèöû îáúåìíîãî çàðÿäà åùå íå îáðàçîâàëèñü; – ýòàï ëèíåéíîãî âçàèìîäåéñòâèÿ ïîòîêà ñ âîëíîé, êîãäà íåëèíåéíûìè ýôôåêòàìè ìîæíî ïðåíåá- ðå÷ü.  ýòî âðåìÿ ïðîèñõîäèò ôîðìèðîâàíèå ñïèö; – ýòàï íåëèíåéíîãî âçàèìîäåéñòâèÿ, õàðàêòåðèçó- þùèéñÿ íàðàñòàíèåì íåëèíåéíîñòè ïðîöåññîâ â ïðèáîðå èç-çà äîñòèæåíèÿ ñïèöàìè àíîäà è ïîñòå- ïåííûì óñòàíîâëåíèåì ñòàöèîíàðíîãî ðåæèìà. Ñïåêòð âûõîäíîãî ñèãíàëà ìàãíåòðîíà â íà÷àëå ïðåäãåíåðàöèîííîãî ýòàïà èìååò êâàçèøóìîâîé (ñ íåñêîëüêèìè øèðîêèìè ñïåêòðàëüíûìè êîìïî- íåíòàìè) õàðàêòåð. Îäíàêî ÷àñòîòû äîìèíèðóþ- ùèõ êîìïîíåíò â íåì ïîêà ñîîòâåòñòâóþò íå ÷àñòîòàì ñîáñòâåííûõ êîëåáàíèé ýëåêòðîííîãî îáëàêà, à ñîáñòâåííûì ÷àñòîòàì íîðìàëüíûõ ìîä ÝÑ. Ýòî òèïè÷íî ïðè âîçáóæäåíèè ìíîãîìîäîâîé êîëåáàòåëüíîé ñèñòåìû ñ âûñîêîäîáðîòíûìè ìîäà- ìè øèðîêîïîëîñíûì âíåøíèì âîçäåéñòâèåì. Ñàì ñèãíàë ñóùåñòâåííî õàîòè÷åí, íî âåëè÷èíà åãî äîâîëüíî ìàëà, íà íåñêîëüêî äåñÿòêîâ äåöèáåë ìåíüøå, ÷åì â ðåæèìå ãåíåðàöèè. 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Ò{e y#3?9bXjZߢg`Ox= Q`gݟ}fagw ߯=Ye~OW){)&/K2{fg?}9gǻ?p{YpړcWfƮٳPgl)goans[֞?ۻV6sȫG=ghip3{ #ҏmRr,z&km)>Zy2|>.H#\grحW6k,0n7_@L{:aӆC?}k dY*`E#~8ϝ;~8bքly?X?܆?: CǶ>aeN~5/D#|8^Ff܃o3.?Vn}t>ʃpV.'%C{+"E.{Kѿ2W ~䱆 1 tNݪ|Vy#&f_ 6_z_J7Ԇ {%c%ܹsl}\X?ݙ'-Rx }\q=woj_gK ϋгqqq {2}* S/\ݚ%2}doP;'UC2}܏jƶy^E}ܟYqݲ,9E留#'Ҟmy?5BGz2)}7E^}PwN]d}g#X_ 7}u } aK|x(//&H?8Z~+- ++y|Q3mth5_l+mB&ia>6X#5ޝKv1jF:4[",~ ~e͡{2<ːFp09"(w?,ͫ*Ga[Y(;}}f܄VU%thGٚLҗ/3y!% hmZ{ܹ|8c 7~W*`2Qmtl،i;|<:y-!xX'ひwjC =(ݡit]_Ǿ&oMތ/6t;XX/cOv4o+KrϚ+`xgFD&+7yfilʊ#F8o)&{o7O 2B4o<+q3fEژ;xot9pGH\Yb-c OV%qoO;wB{*ܽMK]|`c(V΀F9o e,,6>gj~r+P`ֆyr<{Xuw6-&(-/ac0Fog'76%!yZ4 'o|#z,Ƈ!-y{\UN 4"qHLv)' ~(#D~G%ZE|2mVCniqѓĆGf/FӞ)NH>m:SLuh˔YcFJk=<:y!6y6p[݀pO yʖyO#pE.x@*eŷ?љC<.ye@N2r-^y8& |^4k w:Ew.sbM. ?а)'<[—wܺw;~"}~Gj,,ÐÈ, 2005, ¹ 1 24 Ðèñ.1. Ñïåêòð âîçáóæäàþùåãî òîêà â ìàãíåòðîíå âî âòîðîé ïîëîâèíå ïðåäãåíåðàöèîííîãî ýòàïà Ðèñ. 2. Ñïåêòð âûõîäíîãî ñèãíàëà ìàãíåòðîíà âî âòîðîé ïîëîâèíå ïðåäãåíåðàöèîííîãî ýòàïà íå ðàâíîé åé (ñì. ðèñ. 1, ãäå âèäíà äîìèíèðóþùàÿ ãàðìîíèêà òîêà ñ ÷àñòîòîé 0,91 tube f ). Çäåñü tube f – íåêàÿ óñëîâíàÿ ÷àñòîòà ïðèáîðà, ôèãóðèðóþùàÿ â óðàâíåíèÿõ ìîäåëè è ðàâíàÿ â äàííîì ñëó÷àå 2,45 ÃÃö. Àìïëèòóäà óêàçàííîé êîìïîíåíòû äîñòàòî÷íî âåëèêà, ïîýòîìó ñîñòàâëÿþùàÿ ñ åå ÷àñòîòîé ïîÿâ- ëÿåòñÿ â âûõîäíîì ñèãíàëå ïðèáîðà íàðÿäó ñ ñîñòàâëÿþùåé íà ÷àñòîòå ðàáî÷åãî âèäà, îáû÷íî ïðåâûøàÿ åå ïî àìïëèòóäå (ðèñ. 2). Ê äàííîìó ìîìåíòó âäîëü ïåðèìåòðà ïðîìîäóëèðîâàííîé ãðà- íèöû ýëåêòðîííîé âòóëêè óæå íàñ÷èòûâàåòñÿ N/2 âîëí, íî ñêîðîñòü èõ âðàùåíèÿ âîêðóã êàòîäà åùå íå ðàâíà ôàçîâîé ñêîðîñòè N/2 âèäà. Çàòåì àìïëè- òóäà N/2 ìîäû íàðàñòàåò, è åå Â×-ïîëå ñèíõðîíè- çèðóåò ýëåêòðîííûé ïîòîê. Íà ñïåêòðîãðàììå âèä- íî, êàê ÷àñòîòà «íåñèíõðîííîé» êîìïîíåíòû ïîñòå- ïåííî êàê áû «ïîäòÿãèâàåòñÿ» ê ÷àñòîòå N/2 ìîäû ÝÑ, è äàííàÿ êîìïîíåíòà èñ÷åçàåò, ñëèâàÿñü ñ «ñèíõðîííîé» ñîñòàâëÿþùåé íà ÷àñòîòå ðàáî÷åãî âèäà. Ýòîò ìîìåíò õàðàêòåðèçóåò îêîí÷àíèå ïðåä- ãåíåðàöèîííîãî ýòàïà è ïåðåõîä ê ýòàïó ëèíåéíîãî âçàèìîäåéñòâèÿ. Îòñþäà ñëåäóåò âûâîä, ÷òî íà ïðåäãåíåðàöèîííîì ýòàïå, ìåæäó ïåðèîäîì ñàìîãðóïïèðîâêè âòóëêè, ñ îäíîé ñòîðîíû, è ïåðèîäîì åå ñèíõðîíèçàöèè ïîëåì ðàáî÷åé íîðìàëüíîé ìîäû ÝÑ – ñ äðóãîé, ëåæèò ïåðèîä «ïîäâîçáóæäåíèÿ» ñîáñòâåííîãî êî- ëåáàíèÿ ýëåêòðîííîãî îáëàêà Â×-ïîëåì ÝÑ (ñð. ñ ðàáîòîé [6]). Ýâîëþöèÿ ñïåêòðà âûõîäíîãî ñèãíàëà ìàãíåòðîíà íà ïðîòÿæåíèè òðåòüåãî ýòàïà ïåðåõîäíîãî ïðîöåñ- ñà ïîêàçàíà íà ðèñ. 3.  îòëè÷èå îò ïðåäûäóùèõ ñïåêòðîãðàìì, ãäå äëÿ ñïåêòðàëüíîãî îöåíèâàíèÿ ïðèìåíÿëñÿ ìîäèôèöèðîâàííûé êîâàðèàöèîííûé ìåòîä, çäåñü èñïîëüçîâàíî ãàðìîíè÷åñêîå ðàçëîæå- íèå ñèãíàëà ïî íåçàòóõàþùèì êîìïëåêñíûì ýêñ- ïîíåíòàì [7]. Âèäíî, â ÷àñòíîñòè, ïîñòåïåííîå ñóæåíèå øèðèíû ïîëîñû ðàáî÷åãî âèäà (äëÿ íà- ãëÿäíîñòè óñëîâíàÿ ÷àñòîòà ïðèáîðà tube f çäåñü âûáðàíà ðàâíîé 2,3 ÃÃö). Ê ìîìåíòó ïðåêðàùåíèÿ íàðàñòàíèÿ êîëåáàíèé ñïåêòð âûõîäíîãî ñèãíàëà ñîñòîèò èç äîìèíèðóþùåé ãàðìîíè÷åñêîé êîìïî- íåíòû íà ÷àñòîòå N/2 âèäà ÝÑ 1,065 tube f (åå àìïëèòóäà ñîâïàäàåò ñ àìïëèòóäîé âûõîäíîãî ñèã- íàëà ìàãíåòðîíà â óñòàíîâèâøåìñÿ ðåæèìå) è íåñêîëüêèõ ñîñòàâëÿþùèõ ñ íåçíà÷èòåëüíîé àìï- ëèòóäîé âáëèçè ñëåãêà âîçáóæäåííîé N/2–1 íîð- ìàëüíîé ìîäû ÝÑ íà ÷àñòîòå 1,913 tube f . 3. Ýâîëþöèÿ âûõîäíîãî ñïåêòðà àìïëèòðîíà ïðè âêëþ÷åíèè ïðèáîðà  îòëè÷èå îò ìàãíåòðîíà, â àìïëèòðîíå ãðóïïèðîâ- êà âòóëêè óæå â ñàìîì íà÷àëå ïðåäãåíåðàöèîííîãî ýòàïà îïðåäåëÿåòñÿ âîçäåéñòâèåì ïîëÿ âõîäíîãî ñèãíàëà. Ýòî ïðîÿâëÿåòñÿ â íåçàâèñèìîñòè ÷èñëà âîëí, óêëàäûâàþùèõñÿ âäîëü åå ïåðèìåòðà, îò èíäóêöèè âíåøíåãî ìàãíèòíîãî ïîëÿ è àíîäíîãî íàïðÿæåíèÿ, è ðàâåíñòâå ýòîãî ÷èñëà íîìåðó ðàáî- ÷åé íîðìàëüíîé ìîäû ÝÑ. Êàê ñëåäñòâèå, ïî÷òè ñðàçó æå âîçíèêàåò îïèñàííîå âûøå ÿâëåíèå ïîä- âîçáóæäåíèÿ ñîáñòâåííîãî êîëåáàíèÿ ýëåêòðîííî- ãî îáëàêà Â×-ïîëåì âõîäíîãî ñèãíàëà â ÝÑ. Ñïåêòð âûõîäíîãî ñèãíàëà àìï- ëèòðîíà ëèøü â òå÷åíèå êîðîòêîãî âðåìåíè èìååò êâàçèøóìîâîé õàðàêòåð. Çàòåì â íåì îñòàþòñÿ äâå ñðàâíèòåëüíî óçêîïîëîñ- íûå êîìïîíåíòû (ðèñ. 4): ñ ÷àñòîòîé âõîäíîãî ñèã- íàëà tube f è ñ ÷àñòîòîé íå- ñèíõðîíèçèðîâàííîãî, íî ïîäâîçáóæäåííîãî ñîá- ñòâåííîãî êîëåáàíèÿ âòóëêè 0,88 tube f . Áèåíèÿ ýòèõ êîëåáàíèé âûçûâàþò õàðàêòåðíûå ïóëüñàöèè àìïëèòóäû è ôàçû âûõîäíîãî ñèãíàëà, êîòîðûå õîðîøî âèäíû íà ðèñ. 2 è 3 ðàáîòû [2] ñîîòâåòñòâåííî. 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Ó÷è- òûâàÿ ïðàêòè÷åñêóþ âàæíîñòü äàííîãî âîïðîñà ñ òî÷êè çðåíèÿ ïðîáëåì ÝÌÑ, ïðèâåäåì ðåçóëüòàòû ÷èñëåííîãî àíàëèçà óêàçàííîãî ÿâëåíèÿ ñ ïîìî- ùüþ ñïåêòðàëüíîé ìîäåëè.  êà÷åñòâå èñõîäíîãî âûáåðåì ñòàöèîíàðíûé ðå- æèì ðàáîòû ìàãíåòðîíà Ì-156 íà N/2 âèäå. Àíîä- íîå íàïðÿæåíèå ðàâíî 3,9 êÂ. Ðàñïðåäåëåíèå îáúåì- íîãî çàðÿäà â ïðèáîðå äëÿ äàííîãî ðåæèìà ïîêàçàíî íà ðèñ. 3, a ðàáîòû [9]. Ñïåêòð ãàðìîíèê âûõîäíîãî ñèãíàëà èçîáðàæåí íà âåðõíåì ãðàôèêå ðèñ. 5 (óñëîâíàÿ ÷àñòîòà ïðèáîðà tube f çäåñü ðàâíà 2,3 ÃÃö).  íåêèé ìîìåíò âðåìåíè àíîäíîå íàïðÿæåíèå ñêà÷êîì óâåëè÷èâàåòñÿ äî 4,2 êÂ. Íåïîñðåäñòâåííî âñëåä çà ýòèì íà÷èíàåòñÿ ïîñòåïåííîå ðàçðóøåíèå ñïèö è óìåíüøåíèå àìïëèòóäû N/2 ìîäû. Îäíî- âðåìåííî âîçðàñòàåò àìïëèòóäà N/2–1 âèäà, ñëåãêà âîçáóæäåííîãî è ðàíåå (ïðè ðàáîòå ïðèáîðà íà N/ 2 ìîäå). Âðåìåííàÿ çàâèñèìîñòü ýòîé àìïëèòóäû â îáùèõ ÷åðòàõ àíàëîãè÷íà íàðàñòàíèþ àìïëèòóäû ðàáî÷åãî âèäà ïðè âêëþ÷åíèè ìàãíåòðîíà (ðèñ. 1 ðàáîòû [2]). Îäíàêî èìåþòñÿ ñóùåñòâåííûå ðàçëè- ÷èÿ â êîíôèãóðàöèè ýëåêòðîííîãî ïîòîêà, îáóñ- ëîâëåííûå îäíîâðåìåííûì ñóùåñòâîâàíèåì íà ïðîòÿæåíèè äëèòåëüíîãî âðåìåíè îáîèõ âèäîâ êîëåáàíèé ñîïîñòàâèìîé àìïëèòóäû.  òå÷åíèå äàííîãî ïåðèîäà ýëåêòðîííàÿ âòóëêà ïîäâåðæåíà âîçäåéñòâèþ Â×-ïîëåé äâóõ íîðìàëüíûõ ìîä ÝÑ ñ ðàçëè÷íûìè ÷àñòîòàìè è ïîñòîÿííûìè ðàñïðîñ- òðàíåíèÿ. Êàê ñëåäñòâèå, ñòðóêòóðà ýëåêòðîííîãî îáëàêà ñòàíîâèòñÿ íåðåãóëÿðíîé. Ñ äðóãîé ñòîðî- íû, ãðóïïèðîâêà âòóëêè äîñòàòî÷íî âåëèêà, ÷òîáû âûçâàòü íåëèíåéíûå ýôôåêòû. Îòñþäà ñëåäóåò âåðîÿòíîñòü íàëè÷èÿ áîëüøîãî ÷èñëà êîìáèíàöè- îííûõ ñîñòàâëÿþùèõ ÷àñòîò îáåèõ íîðìàëüíûõ ìîä ÝÑ è èõ ãàðìîíèê, à òàêæå êîìïîíåíò, ñâÿçàííûõ ñ ñîáñòâåííûìè êîëåáàíèÿìè ýëåêòðîííîãî îáëà- êà, â ñïåêòðå âûõîäíîãî ñèãíàëà ïðèáîðà. Ãàðìî- íè÷åñêîå ðàçëîæåíèå ñèãíàëà ïîäòâåðæäàåò ýòî ïðåäïîëîæåíèå. Íà ñðåäíåì ãðàôèêå ðèñ. 5 ïîêàçàí ýòîò ñïåêòð ñïóñòÿ 100 ïåðèîäîâ ñèãíàëà ñ ÷àñòîòîé tube f ïîñëå óâåëè÷åíèÿ àíîäíîãî íàïðÿæåíèÿ. Çàìåòíî ðàñøèðåíèå ïîëîñû ñîñòàâëÿþùèõ îáåèõ íîðìàëüíûõ ìîä (ñ ÷àñòîòàìè 1,065 tube f è 1,913 tube f ). Êðîìå òîãî, íàáëþäàåòñÿ îáøèðíàÿ ãðóïïà ñïåêò- ðàëüíûõ êîìïîíåíò â îêðåñòíîñòè ÷àñòîòû 1,5 tube f . Ñîñòàâëÿþùàÿ ñ äàííîé ÷àñòîòîé ÷åòêî ïðîñëåæè- âàåòñÿ è â ñïåêòðå íàâåäåííîãî òîêà çîíäà (ñì. [9]). Çà ïðåäåëàìè èçîáðàæåííîãî íà ðèñ. 5 äèàïàçîíà ïîÿâëÿþòñÿ òàêæå äðóãèå êîìïîíåíòû. Ñïóñòÿ 200 … 250 ïåðèîäîâ ñèãíàëà ñ ÷àñòîòîé tube f ïåðåõîäíûé ïðîöåññ ïðàêòè÷åñêè çàêàí÷èâàåòñÿ. Ê ýòîìó ìîìåíòó ôîðìèðóåòñÿ óñòîé÷èâàÿ êîíôèãó- ðàöèÿ ýëåêòðîííîãî îáëàêà ñ øåñòüþ ñïèöàìè (ìèíóñ ïåðâàÿ ïðîñòðàíñòâåííàÿ ãàðìîíèêà N/2– 1 íîðìàëüíîé ìîäû ÝÑ), êàê íà ðèñ. 3, b ðàáîòû [9]. Ñïåêòð âûõîäíîãî ñèãíàëà «î÷èùàåòñÿ» (íèæ- íèé ãðàôèê íà ðèñ. 5). Ïîìèìî äîìèíèðóþùåé ñîñòàâëÿþùåé ñ ÷àñòîòîé N/2–1 ìîäû, â íåì ïðèñóòñòâóåò íåîæèäàííî áîëüøàÿ ãàðìîíèêà N/2 âèäà ñ àìïëèòóäîé âñåãî îêîëî –5 … –7 äÁ îò àìïëè- òóäû âûñîêîâîëüòíîé ìîäû. Ïðè÷èíà ñòîëü ñèëü- íîãî âîçáóæäåíèÿ N/2 ìîäû ýëåêòðîííûì ïîòî- êîì, êîíôèãóðàöèÿ êîòîðîãî íà ïåðâûé âçãëÿä îðòîãîíàëüíà ïðîñòðàíñòâåííîé ñòðóêòóðå ýòîé ìîäû, îñòàåòñÿ ïîêà íå âûÿñíåííîé. 5. Âûõîäíîé ñïåêòð ìàãíåòðîíà ïðè ïðîïóñêå èìïóëüñà Êàê îòìå÷åíî â ðàáîòå [2], îñîáåííîñòüþ âêëþ÷å- íèÿ ïðèáîðîâ Ì-òèïà, óñòàíîâëåííîé â õîäå âû- ÷èñëèòåëüíûõ ýêñïåðèìåíòîâ, ÿâëÿåòñÿ òî, ÷òî ïåðåõîä îò ïðåäãåíåðàöèîííîãî ñîñòîÿíèÿ ìàãíåò- ðîíà ê ãåíåðèðóþùåìó îáû÷íî ïðîèñõîäèò ëèøü â óçêîì èíòåðâàëå àíîäíûõ íàïðÿæåíèé, â ïðåäåëàõ ïîðÿäêà 1,0 … 1,05 îò ïîðîãîâîãî íàïðÿæåíèÿ. Çà âåðõíåé ãðàíèöåé äàííîãî äèàïàçîíà ïðåäãåíåðà- öèîííûé ýòàï èíîãäà ñóùåñòâåííî óäëèíÿåòñÿ, ÷àñòî âîîáùå íå ïåðåõîäÿ â óñòîé÷èâóþ ãåíåðàöèþ (ýòî âûçûâàåò ïðîïóñê èìïóëüñà ìàãíåòðîííîãî ãåíåðàòîðà). Âíà÷àëå, ïðè áîëüøèõ àíîäíûõ íàïðÿ- æåíèÿõ, ãðóïïèðîâêà ýëåêòðîííîãî ïîòîêà íàðàñòà- åò çíà÷èòåëüíî áûñòðåå, ÷åì ïðè íàïðÿæåíèÿõ, áëèçêèõ ê ïîðîãîâîìó, íî çàòåì îíà íà÷èíàåò îñëàáåâàòü è ïîñòåïåííî ðàçðóøàåòñÿ.  òî æå âðåìÿ ãåíåðèðóþùèé ïðèáîð ôóíêöèîíèðóåò â çíà÷èòåëü- íî áîëüøåì èíòåðâàëå àíîäíûõ íàïðÿæåíèé, ïîðÿä- êà 1,0 … 1,2 îò íàïðÿæåíèÿ Õàðòðè è âûøå, áåç ñðûâà êîëåáàíèé. Òàêèì îáðàçîì, «ìÿãêèé» ðåæèì âêëþ- ÷åíèÿ ìàãíåòðîíà ðåàëèçóåòñÿ òîëüêî âáëèçè ïîðî- ãîâîãî íàïðÿæåíèÿ. Åñëè èìååò ìåñòî ïðî- ïóñê èìïóëüñà, ò.å. ïåðåõîäíûé ïðîöåññ â ìàãíåòðîíå íå çàêàí- ÷èâàåòñÿ óñòàíîâëåíè- åì êàêîãî-ëèáî èç âîç- ìîæíûõ âèäîâ êîëåáà- íèé (N/2, N/2–1 è ò.ä.), ñî âðåìåíåì ôîð- ìèðóåòñÿ îäíî èç ñî- ñòîÿíèé ýëåêòðîííîãî ïîòîêà, õàðàêòåðíûõ äëÿ ñãðóïïèðîâàííîé âòóëêè â ìàãíåòðîííîì Ðèñ. 4. 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Ýêñïåðèìåíòàëüíî îï- ðåäåëÿþòñÿ ðåæèìû óäâîåíèÿ è ïðåîáðàçîâàíèÿ ÷àñ- òîòû ââåðõ äî 120  ÃÃö. Ââåäåíèå  ñîâðåìåííûõ óñòðîéñòâàõ ñèñòåì ñâÿçè øèðîêîå ïðèìåíåíèå íàõîäÿò òâåðäîòåëüíûå ïðèáîðû, ÷òî îáóñëîâëåíî æåñòêèìè òðåáîâàíèÿìè ê ãàáàðèò- íûì ðàçìåðàì ôóíêöèîíàëüíûõ óçëîâ. Îñâîåíèå ìèëëèìåòðîâîãî äèàïàçîíà âîëí òðåáóåò ñîçäàíèÿ òâåðäîòåëüíûõ ãåíåðàòîðîâ, óñèëèòåëåé, ïðåîáðà- çîâàòåëåé ÷àñòîòû ýòîãî äèàïàçîíà è ò.ä. Ïðåîáðà- çîâàòåëè ÷àñòîòû ââåðõ ÿâëÿþòñÿ âòîðè÷íûìè èñ- òî÷íèêàìè ÑÂ× êîëåáàíèé è ê íèì ïðåäúÿâëÿþòñÿ ïîâûøåííûå òðåáîâàíèÿ ïî óðîâíþ âûõîäíîé ìîùíîñòè, êîýôôèöèåíòó ïðåîáðàçîâàíèÿ, ïîëîñå ðàáî÷èõ ÷àñòîò, äèàïàçîíó ïåðåñòðîéêè. Ê íàñòîÿùåìó âðåìåíè íàêîïëåí íåêîòîðûé ýêñ- ïåðèìåíòàëüíûé ìàòåðèàë ïî èñïîëüçîâàíèþ â ïðåîáðàçîâàòåëÿõ ÷àñòîòû ââåðõ íåëèíåéíûõ ñâîéñòâ ïîëóïðîâîäíèêîâûõ ïðèáîðîâ òèïà äèîäîâ ñ áàðü- åðîì Øîòêè (ÄÁØ), âàðàêòîðíûõ äèîäîâ, à òàêæå ëàâèííî-ïðîëåòíûõ äèîäîâ, ïðè÷åì áîëüøàÿ ÷àñòü ëèòåðàòóðíûõ èñòî÷íèêîâ ïîñâÿùåíà èçó÷åíèþ ïðåîáðàçîâàòåëåé ÷àñòîòû â ñðàâíèòåëüíî äëèííî- âîëíîâîì äèàïàçîíå. Öåëüþ íàñòîÿùåé ðàáîòû ÿâëÿåòñÿ èññëåäîâàíèå ïðèíöèïèàëüíîé âîçìîæíîñòè èñïîëüçîâàíèÿ ñõåì ïàðàìåòðè÷åñêèõ óñèëèòåëåé íà âñòðå÷íûõ âîëíî- âîäàõ [1-3] äëÿ ïðåîáðàçîâàíèÿ ÷àñòîòû ââåðõ êîðîòêîâîëíîâîé ÷àñòè ìèëëèìåòðîâîãî äèàïàçîíà âîëí.  ñâÿçè ñ ýòèì â ðàáîòå ðåøåíû çàäà÷è ïîâûøåíèÿ êîýôôèöèåíòà ïðåîáðàçîâàíèÿ â ðå- æèìå óäâîåíèÿ ÷àñòîòû è ðàñøèðåíèÿ ÷àñòîòíîãî äèàïàçîíà ðàáîòû ïðåîáðàçîâàòåëÿ â ðåæèìå ñìåñè- òåëÿ ñ âåðõíåé áîêîâîé ïîëîñîé. 1. Îáùèå ñâîéñòâà äèîäíûõ ïîâûøàþùèõ ïðåîáðàçîâàòåëåé  ñõåìå äèîäíîãî ïðåîáðàçîâàòåëÿ ÷àñòîòû ââåðõ ïðèñóòñòâóåò ãåíåðàòîð ÑÂ× ìîùíîñòè, êîòîðûé âîçäåéñòâóåò íà çàïåðòûé p-n ïåðåõîä åìêîñòíîãî äèîäà ñ íåëèíåéíîé çàâèñèìîñòüþ çàðÿäà îò ïðè- ëîæåííîãî íàïðÿæåíèÿ, âûçûâàÿ ïåðèîäè÷åñêîå èçìåíåíèå åìêîñòè ïåðåõîäà.  ñëó÷àå ìàëîãî âõîäíîãî ñèãíàëà ïðåîáðàçîâàòåëè õàðàêòåðèçóþò- ñÿ ïåðåìåííûì âî âðåìåíè ïàðàìåòðîì – åìêîñòüþ p-n ïåðåõîäà, áëàãîäàðÿ ÷åìó îíè ïîëó÷èëè íàçâà- íèå ïàðàìåòðè÷åñêèõ. Ïðè îäíîâðåìåííîì âîçäåéñòâèè íà íåëèíåéíûé ýëåìåíò âõîäíîãî ñèãíàëà è ñèãíàëà ãåòåðîäèíà íà âûõîäå ïðåîáðàçîâàòåëÿ ÷àñòîòû âîçíèêàþò êîëå- áàíèÿ ñ ÷àñòîòàìè âèäà mf ñ  nf Ã, , ãäå f c – ÷àñòîòà âõîäíîãî ñèãíàëà, f à – ÷àñòîòà ãåòåðîäèíà, m è n – öåëûå ÷èñëà.  ïîâûøàþùèõ ïðåîáðàçîâàòåëÿõ ÷àñòîòû îñóùåñòâëÿåòñÿ ïåðåíîñ ÷àñòîòû âõîäíîãî ñèãíàëà f c â áîëåå âûñîêî÷àñòîòíûé äèàïàçîí, ïðè ýòîì ÷àñòîòà âûõîäíîãî ñèãíàëà f âûõ =f ñ +f à (ñìåñè- òåëü ñ âåðõíåé áîêîâîé ïîëîñîé) èëè f âûõ =f ñ -f à (ñìåñèòåëü ñ íèæíåé áîêîâîé ïîëîñîé). Åñëè èí- ôîðìàöèîííûé ñèãíàë f ñ áîëåå âûñîêîé ÷àñòîòû, ÷åì f à , òàêèå ïðåîáðàçîâàòåëè íàçûâàþò ñìåñèòåëÿ- ìè ñäâèãà [4]. Ýôôåêòèâíîñòü ïðåîáðàçîâàòåëÿ öåëåñîîáðàçíî îöåíèâàòü êîýôôèöèåíòîì Ê ïð =Ð âûõ /Ð ñ , ñðàâíèâàÿ âûõîäíóþ ìîùíîñòü ïðå- îáðàçîâàòåëÿ Ð âûõ ñ ìîùíîñòüþ ñèãíàëà Ð ñ , ïîäàâà- åìîé íà ïðåîáðàçîâàòåëü [5].  ïîâûøàþùèõ ïðåîáðàçîâàòåëÿõ ÷àñòîòû îáû÷íî èñïîëüçóþòñÿ âàðàêòîðû è ÄÁØ, èìåþùèå íåëè- íåéíóþ çàâèñèìîñòü çàðÿäà íà ïåðåõîäå îò íàïðÿ- æåíèÿ. Îñíîâíîå ïðåèìóùåñòâî íåëèíåéíîé ðåàê- òèâíîñòè ïåðåä íåëèíåéíûì ñîïðîòèâëåíèåì çàê- ëþ÷àåòñÿ â òîì, ÷òî â íåé ïðàêòè÷åñêè îòñóòñòâóþò ïîòåðè ýíåðãèè. Àíàëèç ýêñïåðèìåíòàëüíûõ ðàáîò ïîêàçûâàåò, ÷òî â ìèëëèìåòðîâîì äèàïàçîíå âîëí âàðàêòîðû è ÄÁØ îáåñïå÷èâàþò Ê ïð =617 äÁ è ïðè îäèíàêîâîé âûõîäíîé ìîùíîñòè òðåáóþò ïðèìåð- íî íà ïîðÿäîê ìåíüøåé âõîäíîé ìîùíîñòè, ÷åì ïðåîáðàçîâàòåëè íà ïàññèâíûõ ýëåìåíòàõ [4]. Ïî- ýòîìó ïðèìåíåíèå àêòèâíûõ íåëèíåéíûõ ýëåìåí- òîâ ÿâëÿåòñÿ äîñòàòî÷íî ïåðñïåêòèâíûì. 2. Êîíñòðóêöèÿ ïðåîáðàçîâàòåëÿ è ìåòîäèêà ýêñïåðèìåíòà  ìèëëèìåòðîâîì äèàïàçîíå âîëí ïðåäïî÷òèòåëü- íû êîíñòðóêöèè ïðåîáðàçîâàòåëåé ÷àñòîòû, àíàëî- ãè÷íûå êîíñòðóêöèÿì óñèëèòåëåé òîãî æå äèàïàçî- íà [4]. Íàèáîëåå ïîëíî ýêñïåðèìåíòàëüíî èññëåäî- âàíû êîíñòðóêöèè ïðåîáðàçîâàòåëåé îòðàæàòåëü- íîãî è ïðîõîäíîãî òèïîâ.  îòðàæàòåëüíûõ ïðåîá- ðàçîâàòåëÿõ äëÿ ðàçâÿçêè ãåòåðîäèííîãî è âûõîä- íîãî âûñîêî÷àñòîòíûõ òðàêòîâ èñïîëüçóåòñÿ öèð- êóëÿòîð. Ïðåîáðàçîâàòåëü ïðîõîäíîé êîíñòðóêöèè íå òðåáóåò öèðêóëÿòîðà, ëåã÷å íàñòðàèâàåòñÿ íà ìàêñèìóì âûõîäíîé ìîùíîñòè è ìîæåò îáåñïå÷èòü áîëåå ãëàäêóþ ÷àñòîòíóþ õàðàêòåðèñòèêó è ëó÷øåå ïîäàâëåíèå íåæåëàòåëüíûõ ÷àñòîòíûõ êîìïîíåíò â âûõîäíîì ñèãíàëå, ÷åì ïðåîáðàçîâàòåëü ÷àñòîòû îòðàæàòåëüíîãî òèïà. Íà ðèñ. 1 ñõåìàòè÷åñêè ïîêàçàíà êîíñòðóêöèÿ ïî- âûøàþùåãî ïðåîáðàçîâàòåëÿ ïðîõîäíîãî òèïà. Âõîäíîé ñèãíàë ÷åðåç ïëàâíûé ïåðåõîä ïîñòóïàåò íà ñèãíàëüíûé âîëíîâîä 1, à ïðåîáðàçîâàííûé ñíèìàåòñÿ ñ âûõîäíîãî âîëíîâîäà 3, ÿâëÿþùåãîñÿ çàïðåäåëüíûì äëÿ âõîäíîãî ñèãíàëà. 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Ywe,,UɛZ6ŎNol̔雑<0|/q`3ȯ(2v•ldž2CsWujQWgx6}Kf8 =yaܽՃAi:Gft&/Ɋic:,a^v1}c7{gPOnyiܧX[7Vqb,m>6#vCӕbsvG5*NiYGsҞzH#p) ߊ㿎lX +[9/mwyosWǷN7{fe`S ׵gk/23 /xq c9<BwLrw>*P?O޲Ů +Cg>WG1>FuU}=iקI'zjNH#K#_;4,,#ÐÈ, 2005, ¹ 1 31 ðèòü äèàïàçîí ðàáî÷èõ ÷àñòîò ïðåäëîæåííîãî óñò- ðîéñòâà âïëîòü äî ñóáìèëëèìåòðîâîãî äèàïàçîíà âîëí. Ëèòåðàòóðà: 1. Sukhoruchko  O.N. Balanced parametric amplifier of the mm range // “ International Journal of Infrared and Millimeter Waves ” . 2001. Vol. 22, ¹ 7. P. 1047- 1053. 2.  Sukhoruchko  O.N., Bulgakov  B.M., Fisun  A.I., and Belous  O.I.. Parametric Amplification of Signals in the Short-Wave Part of MM Wave Band // Telecommunication and Radio Engineering ’ . 2002. Vol.  58, ¹  7-8. Ð. 64-73. 3.  Ñóõîðó÷êî  Î.Í., Ïëàêñèé  Â.Ò. Ïàðàìåòðè÷åñêèé óñè- ëèòåëü 5-ìì äèàïàçîíà // ³ñíèê Õàðê³âñüêîãî Íàö³î- íàëüíîãî Óí³âåðñèòåòó. Cåð. Ðàä³îô³çèêà òà åëåêòðî- í³êà. 2002. ¹ 2 (570). Ñ. 94-96. 4.  Ïîëóïðîâîäíèêîâûå ïàðàìåòðè÷åñêèå óñèëèòåëè è ïðåîáðàçîâàòåëè ÑÂ× / Ïîä ðåä. Â.Ñ.  Ýòêèíà. // Ì.: Ðàäèî è ñâÿçü, 1983. 333 ñ. 5.  Äüÿ÷åíêî  Ñ.Ì., Çèí÷åíêî  Ñ.À., Êîöåðæèíñêèé Á.À., Òàðàíåíêî Â.Ï. Èñïîëüçîâàíèå íåëèíåéíûõ ñâîéñòâ ËÏÄ â óñòðîéñòâàõ ìèëëèìåòðîâîãî äèàïàçîíà // Ðà- äèîýëåêòðîíèêà. 1983. Ò. 26, ¹ 10. Ñ. 4-18. Ïîñòóïèëà â ðåäêîëëåãèþ 25.12.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Áóëãàêîâ  Á.Ì. Ñóõîðó÷êî Îëåã Íèêîëàåâè÷, êàíä. ôèç.-ìàò. íàóê, íàó÷í. ñîòð. ÈÐÝ ÍÀÍ Óêðàèíû. Íàó÷íûå èíòåðåñû: ôèçèêà ïðèáîðîâ, ýëåìåíòîâ è ñèñòåì, òâåðäîòåëüíàÿ ýëåêòðîíèêà ìèëëèìåòðîâîãî äèàïàçîíà âîëí. Àäðåñ: Óêðàèíà, 61085, Õàðüêîâ, óë. Àê. Ïðîñêóðû, 12, òåë. 744-83-08, e-mail: oniks@ire.kharkov.ua. Ôèñóí Àíàòîëèé Èâàíîâè÷, ä-ð ôèç.-ìàò. íàóê, cò. íàó÷í. ñîòð, âåäóùèé íàó÷í. ñîòð. ÈÐÝ ÍÀÍ Óêðàèíû. Íàó÷íûå èíòåðåñû: ýëåêòðîäèíàìèêà îòêðûòûõ ðåçî- íàíñíûõ ñèñòåì, êâàçèîïòè÷åñêèå òâåðäîòåëüíûå èñ- òî÷íèêè ýëåêòðîìàãíèòíûõ âîëí ìèëëèìåòðîâîãî äèàïàçîíà âîëí. Àäðåñ: Óêðàèíà, 61085, Õàðüêîâ, óë. Àê. Ïðîñêóðû, 12, òåë. 744-83-08, e-mail: afis@ire.kharkov.ua. Áåëîóñ Îëåã Èãîðåâè÷, êàíä. ôèç.-ìàò. íàóê, ñò. íàó÷í. ñîòð. ÈÐÝ ÍÀÍ Óêðàèíû. Íàó÷íûå èíòåðåñû: ýëåêò- ðîäèíàìèêà îòêðûòûõ ðåçîíàíñíûõ ñèñòåì ìèëëè- ìåòðîâîãî äèàïàçîíà âîëí. Àäðåñ: Óêðàèíà, 61085, Õàðüêîâ, óë. Àê.Ïðîñêóðû, 12, òåë. 744-83-08, e-mail: obel@ire.kharkov.ua. ÓÄÊ 621.396 ÌÅÒÎÄ ÂÂÎÄÀ ÊÎÍÒÐÎËÜÍÎÃÎ ÑÈÃÍÀËÀ  ÌÎÍÎÈÌÏÓËÜÑÍÎÌ ÏÅËÅÍÃÀÒÎÐÅ ÇÀÉ×ÅÍÊÎ À.Í., ÑÓÌÊÈÍ Ã.Ï., ÂÅÐÅÙÀÊ À.Ï., ÄÀÍÈËÈÍ À.Á., ÊÎËÅÑÍÈÊΠÞ.À., ÒÎËÊÀ×Å Â.È. Ïðåäëàãàåòñÿ ìåòîä ââîäà êîíòðîëüíîãî ñèãíàëà äëÿ êàëèáðîâêè àìïëèòóäíî-ôàçîâûõ õàðàêòåðèñòèê ïðè- åìíûõ è àíòåííî-ôèäåðíûõ òðàêòîâ ìîíîèìïóëüñíîé ïåëåíãàöèîííîé ñèñòåìû íà áàçå ïàðàáîëè÷åñêîé àí- òåííû ñ àìïëèòóäíûì ñóììàðíî-ðàçíîñòíûì ìîíî- èìïóëüñíûì óãëîâûì äèñêðèìèíàòîðîì. 1. Ïîñòàíîâêà ïðîáëåìû â îáùåì âèäå Çàäà÷à ñëåäÿùåãî ïåëåíãîâàíèÿ, ò.å. çàäà÷à îáíàðó- æåíèÿ èçëó÷àåìûõ ñèãíàëîâ, íàâåäåíèÿ àíòåíí ñèñòåìû è ñîïðîâîæäåíèÿ îáúåêòà â òå÷åíèå ñåàíñà ñâÿçè âîçíèêàåò â øèðîêîì êëàññå ðàäèîòåõíè÷åñ- êèõ ñèñòåì (ðàäèîýëåêòðîííûå è íàâèãàöèîííûå ñèñòåìû, ñèñòåìû òðàåêòîðíûõ èçìåðåíèé è äð.). Êàê ïîêàçàíî â [1], íàèáîëåå øèðîêî â ñîâðåìåí- íûõ ñèñòåìàõ èñïîëüçóþòñÿ ïåëåíãàòîðû ñ àìïëè- òóäíûì ñóììàðíî-ðàçíîñòíûì ìîíîèìïóëüñíûì óãëîâûì äèñêðèìèíàòîðîì, êàê ìåíåå òðåáîâàòåëü- íûì ê èäåíòè÷íîñòè õàðàêòåðèñòèê ïðèåìíûõ êàíàëîâ.  òàêèõ ïåëåíãàòîðàõ ñèãíàëû ñ âûõîäà àíòåííû ïîñòóïàþò â ñóììàðíî-ðàçíîñòíûé ïðåîá- ðàçîâàòåëü, íà âûõîäå êîòîðîãî ôîðìèðóþòñÿ ñóì- ìàðíûé è ðàçíîñòíûå (â îáåèõ ïëîñêîñòÿõ) âûñî- êî÷àñòîòíûå ñèãíàëû. Àìïëèòóäà ðàçíîñòíîãî ñèã- íàëà îïðåäåëÿåò çíà÷åíèå óãëîâîé îøèáêè, à ðàç- íîñòü ôàç ìåæäó ñóììàðíûì è ðàçíîñòíûì ñèãíà- ëàìè – çíàê óãëîâîé îøèáêè, ò.å. íàïðàâëåíèå îòêëîíåíèÿ èñòî÷íèêà èçëó÷åíèÿ îò ðàâíîñèãíàëü- íîãî íàïðàâëåíèÿ. Ñóùåñòâåííîå âëèÿíèå íà êà÷åñòâî èçìåðåíèÿ óã- ëîâûõ êîîðäèíàò ìîíîèìïóëüñíûì ìåòîäîì îêà- çûâàåò íåèäåíòè÷íîñòü àìïëèòóäíî-ôàçîâûõ õà- ðàêòåðèñòèê ïðèåìíûõ è àíòåííî-ôèäåðíûõ òðàê- òîâ ñèñòåìû. Ýòîò âîïðîñ ðàññìîòðåí â ðÿäå ðàáîò, íàïðèìåð [1,2]. Ìåòîäû óìåíüøåíèÿ îøèáîê ïå- ëåíãîâàíèÿ, îáóñëîâëåííûõ íåèäåíòè÷íîñòüþ àì- ïëèòóäíî-ôàçîâûõ õàðàêòåðèñòèê, äîñòàòî÷íî ïîë- íî è ïîäðîáíî îïèñàíû â [1]. Ìîæíî âûäåëèòü äâà ïîäõîäà ê ðåøåíèþ ýòîé çàäà÷è. Êîíñòðóêöèîí- íûé, îõâàòûâàþùèé ðàçëè÷íûå ñïîñîáû îáúåäèíå- íèÿ ïðèåìíûõ êàíàëîâ â îäèí, è ýêñïëóàòàöèîííî- òåõíîëîãè÷åñêèé, ïðè êîòîðîì àìïëèòóäíî-ôàçî- âûå õàðàêòåðèñòèêè ïðèåìíûõ òðàêòîâ âûðàâíèâà- þòñÿ ïî êîíòðîëüíîìó ñèãíàëó ñ ïîìîùüþ ïðåäóñ- ìîòðåííûõ äëÿ ýòîãî ðåãóëèðóþùèõ ýëåìåíòîâ. Áåçóñëîâíî, âñå îíè èìåþò ñâîè äîñòîèíñòâà è íåäîñòàòêè. Ê ñîæàëåíèþ, â ðÿäå ïðàêòè÷åñêè âàæíûõ ñëó÷àåâ èçâåñòíûå ðåøåíèÿ ëèáî íå ïîçâî- ëÿþò ðåøèòü ïîñòàâëåííóþ çàäà÷ó, ëèáî èõ èñïîëü- çîâàíèå èìååò íåïðèåìëåìûå îãðàíè÷åíèÿ. Öåëü èññëåäîâàíèÿ – ðàçðàáîòêà ìåòîäà ââîäà êîíò- ðîëüíîãî ñèãíàëà äëÿ êàëèáðîâêè àìïëèòóäíî- ôàçîâûõ õàðàêòåðèñòèê ïðèåìíûõ è àíòåííî-ôè- äåðíûõ òðàêòîâ ìîíîèìïóëüñíîé ïåëåíãàöèîííîé ñèñòåìû. 2. Àíàëèç èçâåñòíûõ ðåøåíèé Âîïðîñó êàëèáðîâêè è êîìïåíñàöèè íåèäåíòè÷íî- ñòè àìïëèòóäíî-ôàçîâûõ õàðàêòåðèñòèê ïðèåìíûõ òðàêòîâ óäåëÿåòñÿ áîëüøîå âíèìàíèå. Ýòà ïðîáëåìà ðàññìàòðèâàåòñÿ âî ìíîãèõ ðàáîòàõ, íàïðèìåð â [1- 4]. Ìîæíî âûäåëèòü íåñêîëüêî ïîäõîäîâ ê åå ðåøåíèþ. Ðàññìîòðèì ñíà÷àëà âîçìîæíûå ðåøåíèÿ â ðàìêàõ êîíñòðóêöèîííîãî ïîäõîäà.  [1] ïðåäëîæåíî íå- ñêîëüêî âîçìîæíûõ âàðèàíòîâ îáúåäèíåíèÿ ïðè- åìíûõ êàíàëîâ. 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Îáúåäèíåíèå íà ïðîìåæóòî÷íîé ÷àñòîòå è ðàçíåñåíèå ïî ÷àñòîòå ïîäðàçóìåâàåò íàëè÷èå â êàæäîì êàíàëå íåêîòîðîé ÷àñòè ïðèåìíîãî óñòðîé- ñòâà, ÷òî íå ïîçâîëÿåò èñêëþ÷èòü âñå èìåþùèåñÿ íåèäåíòè÷íîñòè. Êîììóòàöèÿ ïðèâîäèò ê ýíåðãå- òè÷åñêèì ïîòåðÿì, â áîëüøèíñòâå ïðàêòè÷åñêèõ ñëó÷àåâ ýòî íåïðèåìëåìî. Èñïîëüçîâàíèå ôàçîâûõ ñäâèãîâ îãðàíè÷åíî âîçìîæíîñòüþ îáúåäèíåíèÿ òîëüêî äâóõ ñèãíàëîâ [5]. Ïðèìåíåíèå âðåìåííîé çàäåðæêè çàòðóäíåíî ïîëó÷åíèåì ýëåìåíòîâ çàäåð- æêè ñ ìàëûìè ðàçáðîñàìè. Íàèáîëåå ïðèåìëåìûìè äëÿ ïðàêòè÷åñêîé ðåàëèçàöèè ÿâëÿþòñÿ êîäèðîâà- íèå è íèçêî÷àñòîòíàÿ ìîäóëÿöèÿ íà âûñîêîé ÷àñ- òîòå. Èñïîëüçîâàíèå êîäèðîâàíèÿ ðàññìîòðåíî â [6,7]. Çäåñü ñèãíàë ñ êàæäîãî ýëåìåíòà àíòåííû ìîäóëè- ðóåòñÿ îðòîãîíàëüíûì ïñåâäîñëó÷àéíûì ñèãíàëîì, âñå ñèãíàëû ñêëàäûâàþòñÿ, ïðîõîäÿò ÷åðåç îäíîêà- íàëüíûé ïðèåìíèê, îöèôðîâûâàþòñÿ, à çàòåì â öèôðîâîé ôîðìå äåìîäóëèðóþòñÿ è ïðîèçâîäèòñÿ äàëüíåéøàÿ îáðàáîòêà. Òàêîå ðåøåíèå ïîçâîëÿåò îáúåäèíèòü â îäèí êàíàë ëþáîå êîëè÷åñòâî ñèãíà- ëîâ, çàòåì âîññòàíîâèòü è îáðàáîòàòü èõ. Îäíàêî àïïàðàòóðà, ðåàëèçóþùàÿ òàêóþ îáðàáîòêó, ïîëó- ÷àåòñÿ áîëåå ñëîæíîé, òàê êàê ïîñëå êîäèðîâàíèÿ ñïåêòð ñóììàðíîãî ñèãíàëà ñîîòâåòñòâåííî ðàñøè- ðÿåòñÿ, è äëÿ åãî óñèëåíèÿ è ïðåîáðàçîâàíèÿ òðåáóåòñÿ áîëåå øèðîêîïîëîñíûé ïðèåìíèê è óñ- òðîéñòâî öèôðîâîé îáðàáîòêè, à äëÿ êà÷åñòâåííîãî âîññòàíîâëåíèÿ ñèãíàëîâ íåîáõîäèìà æåñòêàÿ ñèí- õðîíèçàöèÿ ïî âðåìåíè ìîäóëèðóþùèõ è äåìîäó- ëèðóþùèõ êîäîâûõ ïîñëåäîâàòåëüíîñòåé.  [1] ðàññìîòðåí âîïðîñ èñïîëüçîâàíèÿ íèçêî÷àñ- òîòíîé ìîäóëÿöèè ñèãíàëîâ. Ñòðóêòóðà ñèñòåìû àíàëîãè÷íà îïèñàííîé âûøå äëÿ êîäèðîâàíèÿ, íî, ïðè ýòîì, äîñòàòî÷íî ïðîñòà â íàñòðîéêå è ñòàáèëü- íà â ðàçëè÷íûõ óñëîâèÿõ ýêñïëóàòàöèè. Îäíàêî, ñîãëàñíî [1], ââåäåíèå íèçêî÷àñòîòíîé ìîäóëÿöèè ïðè îáúåäèíåíèè êàíàëîâ äåëàåò ñèñòåìó ïåëåíãà- öèè ÷óâñòâèòåëüíîé ê àìïëèòóäíûì ôëóêòóàöèÿì ïðèíèìàåìûõ ñèãíàëîâ, à òàêæå óÿçâèìîé ñî ñòî- ðîíû îðãàíèçîâàííûõ àìïëèòóäíî-ìîäóëèðîâàí- íûõ ïîìåõ.  ðàìêàõ ýêñïëóàòàöèîííî-òåõíîëîãè÷åñêîãî ïîä- õîäà âîçìîæíû òàêæå ðàçëè÷íûå âàðèàíòû ðåøå- íèÿ çàäà÷è, îõâàòûâàþùèå ðàçíûå ñïîñîáû ïîäà÷è êîíòðîëüíîãî ñèãíàëà, èçìåðåíèÿ ïàðàìåòðîâ ïðè- åìíûõ òðàêòîâ è ââåäåíèÿ êîìïåíñèðóþùèõ âîç- äåéñòâèé. Íàèáîëåå ïîëíî îíè îïèñàíû â [4]. Ðàçëè÷èå â âàðèàíòàõ êàëèáðîâêè è êîìïåíñàöèè íåèäåíòè÷íîñòè àìïëèòóäíî-ôàçîâûõ õàðàêòåðèñ- òèê çàêëþ÷àåòñÿ â ñïîñîáå ïîäà÷è êîíòðîëüíîãî ñèãíàëà – ââåäåíèå â ïðèåìíûé òðàêò ÷åðåç êîì- ìóòàòîð, îòâåòâèòåëü è ò.ï., èëè ïîäà÷à íà âõîä àíòåííîé ñèñòåìû ÷åðåç âûíåñåííûé íà þñòèðî- âî÷íóþ ìà÷òó èçëó÷àòåëü; â ñïîñîáå èçìåðåíèÿ õàðàêòåðèñòèê ïðèåìíîãî òðàêòà – ââåäåíèå ñïåöè- àëüíîãî èçìåðèòåëüíîãî òðàêòà, âêëþ÷àþùåãî àì- ïëèòóäíûå è ôàçîâûå äåòåêòîðû, èëè èñïîëüçîâà- íèå òðàêòà îñíîâíîé îáðàáîòêè ñèãíàëîâ; â ñïîñîáå îñóùåñòâëåíèÿ êîìïåíñàöèè íåèäåíòè÷íîñòè – ïðèìåíåíèå ðåãóëèðóåìûõ ôàçîâðàùàòåëåé è àòòå- íþàòîðîâ, êîòîðûå ìîãóò áûòü âûïîëíåíû êàê â àíàëîãîâîì, òàê è â öèôðîâîì âèäå, èëè ó÷åò â àëãîðèòìàõ îñíîâíîé îáðàáîòêè ñèãíàëîâ. Óñòðîéñòâà, â êîòîðûõ êîíòðîëüíûé ñèãíàë ââî- äèòñÿ â ïðèåìíûé òðàêò ÷åðåç êîììóòàòîð, îòâåò- âèòåëü è ò.ï., øèðîêî ðàñïðîñòðàíåíû, íàïðèìåð [8-14]. Ýòîò ñïîñîá äîñòàòî÷íî ýôôåêòèâíî ïîçâî- ëÿåò êàëèáðîâàòü è êîìïåíñèðîâàòü àìïëèòóäíóþ íåèäåíòè÷íîñòü òðàêòîâ, à êîìïåíñàöèÿ ôàçîâîé õàðàêòåðèñòèêè âîçìîæíà ëèøü â ñëó÷àÿõ, êîãäà ãàðàíòèðóåòñÿ ñèíôàçíîñòü ââîäèìûõ âîçäåéñòâèé èëè äëèíà âîëíû íàìíîãî áîëüøå ýëåêòðè÷åñêèõ äëèí ïîäâîäÿùèõ öåïåé – óäîâëåòâîðèòü ýòèì òðåáîâàíèÿì âî ìíîãèõ ïðàêòè÷åñêèõ ñëó÷àÿõ î÷åíü ñëîæíî. Êðîìå òîãî, íèêàê íå ó÷èòûâàþòñÿ öåïè àíòåííî-ôèäåðíîãî òðàêòà. Óñòðîéñòâà, â êîòîðûõ êîíòðîëüíûé ñèãíàë ïîäà- åòñÿ íà âõîä àíòåííîé ñèñòåìû ÷åðåç âûíåñåííûé èçëó÷àòåëü, ñâîáîäíû îò ýòèõ íåäîñòàòêîâ, íàïðè- ìåð [15]. Îäíàêî âñå îíè îðèåíòèðîâàíû íà ïîñò- ðîåíèå ñèñòåì, â êîòîðûõ âûíåñåííûé èçëó÷àòåëü ðàñïîëàãàåòñÿ â äàëüíåé çîíå íà þñòèðîâî÷íîé ìà÷òå.  ðÿäå ïðàêòè÷åñêèõ ñëó÷àåâ ýòî íå ïðèåì- ëåìî êàê ïî òåõíè÷åñêèì, òàê è ïî ýêñïëóàòàöèîí- íî-òåõíîëîãè÷åñêèì ïðè÷èíàì. 3. Àíòåííà ïåëåíãàòîðà è ìåòîä ââîäà êîíòðîëüíîãî ñèãíàëà Ñîãëàñíî òðåáîâàíèÿì òåõíè÷åñêîãî çàäàíèÿ, â ðàìêàõ êîòîðîãî ïðîâîäèëàñü ðàáîòà, ïåëåíãàòîð äîëæåí èìåòü øèðèíó äèàãðàììû àíòåííîãî óñò- ðîéñòâà ïî óðîâíþ –3 äÁ â S-äèàïàçîíå îêîëî 5°. Êàê èçâåñòíî, íàèáîëåå òåõíîëîãè÷åñêè ïðîñòîé àíòåííîé ñ øèðèíîé äèàãðàììû íàïðàâëåííîñòè ìåíåå 10° ÿâëÿåòñÿ ïàðàáîëè÷åñêàÿ àíòåííà. Ñî- ãëàñíî ðàñ÷åòó, äèàìåòð àíòåííû ñîñòàâëÿåò 1800 ìì.  êà÷åñòâå îáëó÷àòåëÿ àíòåííû ïðèìåíåí äâóõìîäîâûé âîëíîâîäíûé îáëó÷àòåëü, îáåñïå÷è- âàþùèé ôîðìèðîâàíèå ñóììàðíîãî è äâóõ ðàçíî- ñòíûõ (ïî àçèìóòó è óãëó ìåñòà) êàíàëîâ. Äëÿ äîñòèæåíèÿ âûñîêîé ÷óâñòâèòåëüíîñòè ïåëåíãàöè- îííîãî óñòðîéñòâà êàæäûé êàíàë èìååò ñâîé ïðè- åìíûé òðàêò ñ ìàëîøóìÿùèì óñèëèòåëåì íà âõîäå, îäíî èç îñíîâíûõ òðåáîâàíèé ê êîòîðîìó – âûñî- êàÿ èäåíòè÷íîñòü è ñòàáèëüíîñòü àìïëèòóäíûõ è ôàçîâûõ õàðàêòåðèñòèê. Äëÿ óäîâëåòâîðåíèÿ ýòèì òðåáîâàíèÿì íåîáõîäèìî èñïîëüçîâàíèå äîðîãîñ- òîÿùèõ âûñîêîñòàáèëüíûõ ïðèåìíûõ óñòðîéñòâ è îáåñïå÷åíèå ðàâíûõ ýëåêòðè÷åñêèõ äëèí òðàêòîâ è èäåíòè÷íîãî òåïëîâîãî ðåæèìà êàê äëÿ ýëåêòðîí- íûõ êîìïîíåíòîâ, òàê è äëÿ àíòåííî-ôèäåðíûõ òðàêòîâ. Êàê ïîêàçûâàåò îïûò, òàêèå ðåøåíèÿ ñëîæíû è äîðîãîñòîÿùè. Ïðåäëîæåíî èçìåðÿòü àìïëèòóäíî-ôàçîâûå õàðàêòåðèñòèêè ïðèåìíûõ òðàêòîâ è àíòåííî-ôèäåðíûõ óñòðîéñòâ, à ðåçóëü- òàòû èçìåðåíèé ó÷èòûâàòü â âèäå ïîïðàâîê ïðè îáðàáîòêå ñèãíàëîâ. Ñ ýòîé öåëüþ â ðàñêðûâå àíòåííû ðàçìåùåí èçëó÷àòåëü êîíòðîëüíîãî ñèãíàëà. 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Íà ðèñ.1 ñõåìàòè÷- íî ïîêàçàíî âçàèìíîå ðàñïîëîæåíèå çåðêàëà àíòåí- íû 1, èçëó÷àòåëÿ êîíòðîëüíîãî ñèãíàëà 2 è ìîíî- èìïóëüñíîãî îáëó÷àòåëÿ 3. 1 2 3 à 1 2 3  á Ðèñ. 1. Ñõåìà âçàèìíîãî ðàñïîëîæåíèÿ çåðêàëà àíòåííû 1, èçëó÷àòåëÿ êîíòðîëüíîãî ñèãíàëà 2 è ìîíîèìïóëüñíîãî îáëó÷àòåëÿ 3: à – âèä ñïåðåäè; á – âèä ñáîêó Ðàññìîòðèì ôîðìèðîâàíèå óêàçàííûõ ñèãíàëîâ â îäíîé ïëîñêîñòè, äëÿ ïðèìåðà â âåðòèêàëüíîé. Íà ðèñ.2 óïðîùåííî èçîáðàæåíû äâà êàíàëà îáëó÷àòå- ëÿ è ïàäàþùàÿ âîëíà êîíòðîëüíîãî ñèãíàëà.  d/2 d/2 1 2 Ðèñ. 2. Óïðîùåííàÿ èëëþñòðàöèÿ îáëó÷àòåëÿ è ïàäàþùåé âîëíû êîíòðîëüíîãî ñèãíàëà Íàïðàâëåíèå ïàäàþùåé âîëíû èç-çà ñìåùåíèÿ èçëó÷àòåëÿ êîíòðîëüíîãî ñèãíàëà ñ ôîêàëüíîé îñè ïàðàáîëîèäà ïðîèñõîäèò ïîä óãëîì  ê ïëîñêîñòè ðàñêðûâà îáëó÷àòåëÿ. Ñèãíàëû íà âûõîäàõ îáëó÷à- òåëÿ 1 è 2 ìîæíî ïðåäñòàâèòü ñëåäóþùèìè âûðà- æåíèÿìè: ) sin / d t sin( A U       0 1 1 , (1) ) sin / d t sin( A U       0 2 2 . (2) Óñòðîéñòâî ôîðìèðîâàíèÿ ñóììàðíîãî è ðàçíî- ñòíîãî ñèãíàëîâ âûäåëÿåò ñèãíàëû: ñóììàðíûé ) t sin( A U        , (3) ãäå )] sin / d ( tg A A A A [ arctg         0 2 1 2 1 , (4) è ðàçíîñòíûé ) t sin( A U        , (5) ãäå )] sin / d ( tg A A A A [ arctg         0 2 1 2 1 . (6) Ñ ó÷åòîì òîãî, ÷òî 2 1 A A  , ïîëó÷àåì 2 /           . (7) Ââèäó óñëîâèé ðàçìåùåíèÿ íà êîíå÷íîì ðàññòîÿ- íèè èçëó÷àòåëÿ êîíòðîëüíîãî ñèãíàëà è îáëó÷àòåëÿ àíòåííû (íåñêîëüêî äëèí âîëí) àìïëèòóäû À 1 è À 2 ñèãíàëîâ U 1 è U 2 íå îäèíàêîâû. Îäíàêî ÷èñëåí- íûé àíàëèç ïîêàçàë, ÷òî âåëè÷èíà îòëè÷èÿ íå ïðåâûøàåò 5%, ïðè ýòîì   îòëè÷àåòñÿ îò 2 /  íå áîëåå ÷åì íà 3°. Ýòî ïîäòâåðæäåíî ðåçóëüòàòàìè ýêñïåðèìåíòàëüíûõ èññëåäîâàíèé. Ðàâåíñòâî àìïëèòóä  A è  A ñèãíàëîâ  U è  U îáåñïå÷èâàåòñÿ âûïîëíåíèåì óñëîâèÿ 2 2 0 / sin / d      . (8) Îáðàáîòêà êîíòðîëüíîãî ñèãíàëà ïðåäóñìàòðèâàåò ïðîöåäóðó äåìîäóëÿöèè è ôîðìèðîâàíèÿ êîìïëåê- ñíûõ îòñ÷åòîâ ñèãíàëîâ, ñíèìàåìûõ ñ âûõîäîâ ñóììàðíîãî è ðàçíîñòíûõ êàíàëîâ ÀÓ, êîòîðàÿ ñâîäèòñÿ ê ñâåðòêå ïðèíèìàåìûõ ñèãíàëîâ ñ åãî êîïèåé íà èíòåðâàëå íàáëþäåíèÿ: dt ) t ( ) t ( U Y * k k T k          , (9) dt ) t ( ) t ( U Y * k k T k          , (10) ãäå T k – âðåìÿ íàêîïëåíèÿ êîíòðîëüíîãî ñèãíàëà; k  – êîïèÿ êîíòðîëüíîãî ñèãíàëà. Îïðåäåëåíèå íåèäåíòè÷íîñòè ôàçîâûõ õàðàêòåðè- ñòèê ïðèåìíûõ êàíàëîâ: 2 / } Y Y arg{ * k k           , (11) îïðåäåëåíèå íåèäåíòè÷íîñòè àìïëèòóäíûõ õàðàê- òåðèñòèê ïðèåìíûõ êàíàëîâ:  2x{pTg< F~@EHY+6e3 ֶPgԎ:;mtXԊ*`x?""yT%j Kws#̮s3{y|s# bDU`riNɢ_4mWS|-nP136&iGepfl@$l&VK';H" lY͖)Nޞ-Q'[y~RedEf,ʲ}\N|_rɑhȡ s_ Ibm@He){|i8 :-gFt2sMHpmxɗ~蝎Ll%l ^fʍڨ2ٵ}=6>q'Q!K1yƮǐѮSM6cfpj>ʥPF[˾XU.@y)%޻d8tW0~Kk8rv62)ܔR&22Cn'KނS-`aN35By'D}e{܂o yov{5 Լ> 05ۭ g;1bq,b)b89g=NFG7WELОd2oZ8375;z^kGNr\.Qg:niȵ<N{*.ҵNN}<9.z}Gw]y E.~ꊍ_M'6>֖ƹ^'6~jKcU?Q/gQ΅)q>+@B aEdNٛw&y9t[[Y䩭&K dIߘMHV‘SW)|R9C#PKl]y@Robq^\ێ,_ɓu=On@g 2лDAzbkC٢ޭlb9&O{u]|;֭:9ZòsnmY>$K֟f{kkd19.myl;Zrvw=i1?])Ŭ>(vދYS;h%Y=Pm*Q [u-YǓ:αuE֚bǶ>$1yz>wIzvl[=ZԱjz׎/U_33Q_h5Z{clk]Ǫ{[QmgoLΪzlK6Ϗ?~#G@|{wɏ?~b Microsoft Office Document Image Writer 11.03.1897.00 2005:11:12 12:46:27  TP -{DPLP(16l2m/ZP1 c{xVuE1hPHKTf&DbRPARDqf$TSڔI hsEPԋ^~#Jd=ދ?rk]k8};,[.zQAӽl})>;"OyJudCY3%I>B(rxݭzpӿ)ȎalV(]]墋?Х9޼ZA1FG t[G啶uJmeAthgH2C!q\ktAdLeoե~O-Vv@"OlLnp?5T6^yY6*zQkw5jڇc5e7z^|➯ Ye-^]6F+bvg`glJC՘eدC,:Ҙ>rn齕^5=֓T*1L`c cLӤu1 Uӻ/y kw4Ou1=KcɽqZK=՗yu}1I*H a_WZi#QeÞtܸ<ْ\^cV^c!^#~_{%Ykqv}۹domQsЫ07y~=lk*9{d3/:jI\e]dwx'^3d۷/b'T++NxeIYs+KJd'g'dYʝ}"N%fkO{,B|ts$$NCK5cxCȦM%qe*DL1Jzʤ~Ѕ_ʻ~gr⹮QKw^/6֕K>ԕd]u/zGcww1vÈɊٍŧŘ;#˱ -Pwh|&x*W~Ew_hS1j 4 yWN)&+s7)yh4s oח0wuz5f~٭~mgR<'*ܢ99pj1΂om_B=|_ɯ\Xﹰگmc ;*霞f?]WnFRn诜Oj֤oy7F w__71?_7O>w.?ʯS/gj̵.H;j5w덤/굌W]Wx=GÅw?qX2:_9)<?fet 2y>?7j1׎A`J4'R/\kyֲs^ {3ě&BN"bP:]ƨqor(r(?e4m4n=s^G;JoEwJ%4=g^iqč$6?urq6)9hi۩Iҏ< wxP:Ɛ7LvԁgI]}@h!:# Kׂb!^ηIr-BUaFng#*Z@wK*׭:/U!gŏd[!x&0 q7sg#E]V}:QVj=Gl Cģ^K%{E/!j#b/FoL  5ku XU*3ZgF!I}ae s0gϞH'=?% saQ0IoӺ3Ll.:+t-VXUX9M+45mPcg@5^ b}C9xqt>?qDjqG)/Lz8:OChZOGuqU+|Iqtbe/+&_∳5aW|k6ۨ~VۺD*uĘ'cLcesseH(5)c셵 7WZXv2ggb/\.7BOzK]҅-<~)s9弧޴u'9؃]kղ61Oήa-QI/NwvZWqU+lzkOgcIJ=( t)5W/|g+n[<;QN? ^ݧb vo"O,a xz Dy,##5&y<=ny\7ޟ)G=Xv^G~1}FMs[XZb%bx ='6w/ƒ^ԃ._~O4t[~eǙ⡣.F",%D } >3 i`2)1\9Bu!Ȏ_ƯW`ab]FY\}sK1Gj'b|x(ۈ X^syUK0;~(cpM"W οyx֫8z.u89a#þNy*`lF<>6x?Ɔ9)!.t 3ȹOellcSvئkͶ쭀Z̻MK?N`nMI}u1;"u];=oG\>q}crQ;nl?S tl+{Eyٵp-/h|+ cwn nlgl!ۉʁq o{UXQzMRs{N1?_>-^8P<ׇbڭ^5 blG7jv}C#`愭Һ3} . 5<䒑w9ƌ:-w]{.K nJ>!l0ヴ޺)^=m Ws(Ǹs=8ggM>[Rl`-`q2`貤i⡣n?;`OOXysg-t߯U{x<k6ț=)wmg]bՄؑ/(5N3Ω` D߹&Sx\Y'}M2c9mRy|+0Gڍ-:gpAsS8>#ั4=Qnܛn#J1Gv!`iyǮ eO&.]=$e.WCyX\Fbud8s+Btn;JX"Tf+-?\SlGa3k(y(L:!zi]R?"` R:OnM2?d'~"Έ`˰R!>rR"Ef9|Q= gW;k>-~A|Ʒ1^5+O8c1ϻ!QgU}c?3Ǣ1͸Ȱ[Qu3X?)2|n!OeqڭeBQͽut5>! ~cؔ>ƋVɚҁ0v> $ ׅn|E ¹a4f`cX3lrlBFCj <6S1u1|h@_}I%+RTf,&1Gif /)݌C"xB`͘:Aʻuf7s>PʷeNM7|7뉽o7\6Wi~}yW4yp&v$iBT-w[䍝sK㳩[VpO31e_b+Ï{׃q 9&Лh25Ax):CK|A'r(U-EoG 9on1H;J)m dAuq1OczxЏ ?G=J]\<#<7Md;Q=,^\Sd0-q|ۼj;go6qO/Oݣh OLl!ybeur⯳CQDZ5"9Ѧ; Q*Zb>0x|#Wd3x}>Rm?/o2vdSg D]sl岋(~1XًDZm11ᛰXE~Sz[/e{;bj@%frUcZ\vq'Ї5{BIΙ=Jvbȶj^OA8;\WxdCc@׬fc:B8J1씜7\FeSo_ã]'cO]t#CWC#c0|NZ!Dqdrrg36>(zT9<&N!gϪvp۩v er}2vx(μTw|KoƙԱ#b}vuX_%dթn8U`g}.q=syX"6O.2F2}/≸m\C]f];pyuSd쨟ur5} ڦ ,ƊdžOE`~]u>VG|v's6g5f/{EJRc+eR|.-D9#^a]v"vks\綼6S[-c!eLtdO(_S;hPx=Hzge3x7Ye\==I 0ё"]kk׳/J-c;V닆dS ܬTw\?ʝ'vV2U,ɦ;6BT;hEP2ڠH]|F.Jm ,J˘!C?6-Êx׸&V>Opd?߯wrƼBocT+l3Xmg/B?kCvdg^3&Nwy;OivU__p\OJIs%n|gҏib̍gwxu.i}@zvߜl|v}X|s“+gg5)>E7:]z 17<2&`J-pþT5V %{H+tgveO+H}gp|Dh{Rw{L 6WV}?۔~hRX m{Fu"o{' . 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(12) Ïîëó÷åííûå îöåíêè ìîãóò áûòü èñïîëüçîâàíû â àëãîðèòìàõ îáðàáîòêè îñíîâíîãî ñèãíàëà äëÿ êîì- ïåíñàöèè íåèäåíòè÷íîñòè àìïëèòóäíî-ôàçîâûõ õàðàêòåðèñòèê òðàêòîâ. 4. Âûâîäû Íàó÷íûé ðåçóëüòàò ðàáîòû – ïîëó÷åí ìåòîäà ââîäà êîíòðîëüíîãî ñèãíàëà è êàëèáðîâêè íåèäåíòè÷íî- ñòè àìïëèòóäíî-ôàçîâûõ õàðàêòåðèñòèê àíòåííî- ôèäåðíûõ è ïðèåìíûõ òðàêòîâ ìîíîèìïóëüñíîé ïåëåíãàöèîííîé ñèñòåìû. Ïðàêòè÷åñêàÿ çíà÷èìîñòü ïîëó÷åííîãî ðåçóëüòàòà: èñïîëüçîâàíèå îïèñàííîãî âûøå ìåòîäà ââîäà êîí- òðîëüíîãî ñèãíàëà ïîçâîëÿåò ïîñòðîèòü ìîíîèì- ïóëüñíóþ ïåëåíãàöèîííóþ ñèñòåìó ñ ðàçäåëüíûìè ïðèåìíûìè òðàêòàìè, â êîòîðîé èñêëþ÷åíî âëèÿ- íèå íåèäåíòè÷íîñòè àìïëèòóäíî-ôàçîâûõ õàðàêòå- ðèñòèê ïðèåìíûõ òðàêòîâ è àíòåííî-ôèäåðíûõ óñòðîéñòâ, ê ïðèåìíûì òðàêòàì íå ïðåäúÿâëÿþòñÿ òðåáîâàíèÿ ïî äîëãîâðåìåííîé ñòàáèëüíîñòè ïàðà- ìåòðîâ.  ðÿäå ñëó÷àåâ â ñèëó êîíñòðóêöèîííûõ èëè ñõåìíî-òåõíè÷åñêèõ îñîáåííîñòåé ðåàëèçàöèè èìå- åòñÿ íåîáõîäèìîñòü êîíòðîëèðîâàòü àìïëèòóäíî- ôàçîâûå õàðàêòåðèñòèêè ïðèåìíûõ òðàêòîâ ïîñòî- ÿííî. Îðòîãîíàëüíîñòü êîíòðîëüíîãî è ïåëåíãóå- ìîãî ñèãíàëîâ â ðàçíîñòíûõ êàíàëàõ ñóùåñòâåííî îáëåã÷àåò ðåøåíèå òàêîé çàäà÷è. Îäíàêî ýòîò âîïðîñ âûõîäèò çà ðàìêè äàííîé ðàáîòû è ìîæåò áûòü ïðåäìåòîì äàëüíåéøèõ èññëåäîâàíèé. Ëèòåðàòóðà: 1. Ëåîíîâ À.È., Ôîìè÷åâ Ê.È. Ìîíîèì- ïóëüñíàÿ ðàäèîëîêàöèÿ. Ì.: Ðàäèî è ñâÿçü, 1984. 312 ñ. 2. Ïðîáëåìû àíòåííîé òåõíèêè /Ïîä ðåä. Ë.Ä.Áàõèðå- âà, Ä.È.Âîñêðåñåíñêîãî. Ì.: Ðàäèî è ñâÿçü, 1989. 368 ñ. 3. Ñïðàâî÷íèê ïî ðàäèîëîêàöèè /Ïîä. ðåä. Ì. Ñêîëíè- êà. Íüþ-Éîðê, 1970: Ïåð. ñ àíãë. (â ÷åòûðåõ òîìàõ) ïîä îáùåé ðåä. Ê.Í.Òðîôèìîâà. Òîì. 2. Ðàäèîëîêàöèîí- íûå àíòåííû è óñòðîéñòâà /Ïîä ðåä. Ï.È. Äóäíèêà. Ì.: Ñîâ.ðàäèî, 1977. 408 ñ. 4. Øèøîâ Þ.À., Ãîëèê À.Ì. è äð. Àäàïòàöèÿ ÔÀÐ ïî ðåçóëüòàòàì âñòðîåííîãî êîíòðî- ëÿ // Çàðóáåæíàÿ ðàäèîýëåêòðîíèêà. 1990. ¹ 9. Ñ. 69- 89. 5. Ïàòåíò ÐÔ 2087005. Ïðåîáðàçîâàòåëü ñèãíàëîâ äëÿ ìîíîèìïóëüñíûõ ðàäèîëîêàòîðîâ. Ùóð Þ.È., Îñóëü È.Ê., 1997. 6. Èíôîðìàöèîííûå òåõíîëîãèè â ðàäèîòåõíè÷åñêèõ ñèñòåìàõ: Ó÷åáíîå ïîñîáèå / Â.À. Âàñèí, È.Á. Âëàñîâ, Þ.Ì. Åãîðîâ è äð./Ïîä ðåä. È.Á. Ôåäîðîâà. Ì.: Èçä-âî ÌÃÒÓ èì. Í.Ý. Áàóìàíà, 2003. 672 ñ. 7. Patent US 6384784. Direction finder system using spread spectrum techniques. Linley F. Gumm; 2002. 8. Patent US 5315304. Digital monopulse. Sam Ghaleb, Michael Stokes; 1994. 9. Patent US 4394695. Monopulse radar with pilot signal generator. Bernard Gellekink; 1983. 10. Patent US 4994810. Monopulse processor digital correction circuit. Allen I.Sinsky; 1991. 11. Patent US 5051752. Angle measurement compensation technique for amplitude comparison monopulse receiver. Richard L.Woolley; 1991. 12. Patent US 5808578. Guided missile calibration method. Peter F. Barbella; Malcolm F. and others. 1998. 13. Patent US 6100841. Radio frequency receiving circuit. John Toth; Michael J. Delcheccolo and others. 2000. 14. Patent US 6498582. Radio frequency receiving circuit having a passive monopulse comparator. Anthony Sweeney; Mark A. Hebeisen and others. 2002. 15. Patent US 6144333. Method for estimating gain and phase imbalance using self-calibrating monopulse angle discriminates in a monopulse radar system. Kwang M.Cho; 2000. Ïîñòóïèëà â ðåäêîëëåãèþ 20.12.2004 Ðåöåíçåíò: ä-ð òåõí. íàóê, ïðîô. Ãîìîçîâ Â.È. Çàé÷åíêî Àëåêñàíäð Íèêîëàåâè÷, êàíä. òåõí. íàóê, íà÷àëüíèê ñåêòîðà ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëü- ñêèé èíñòèòóò ðàäèîòåõíè÷åñêèõ èçìåðåíèé ” . Íàó÷- íûå èíòåðåñû: èçìåðèòåëüíûå ðàäèîñèñòåìû, öèôðî- âàÿ îáðàáîòêà ñèãíàëîâ. Àäðåñ: Óêðàèíà, 61054, Õàðü- êîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. òåë. (057) 738-20-93. E- mail: zaychenko@niiri.kharkov.com Ñóìêèí Ãåðìàí Ïàâëîâè÷, ñòàðøèé íàó÷íûé ñîòðóä- íèê ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëüñêèé èíñòèòóò ðàäèîòåõíè÷åñêèõ èçìåðåíèé ” . Íàó÷íûå èíòåðåñû: àí- òåííûå óñòðîéñòâà ðàäèîòåõíè÷åñêèõ ñèñòåì. Àäðåñ: Óêðàèíà, 61054, Õàðüêîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. Âåðåùàê Àëåêñàíäð Ïåòðîâè÷, êàíä. òåõí. íàóê, äî- öåíò, ãåíåðàëüíûé êîíñòðóêòîð, äèðåêòîð ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëüñêèé èíñòèòóò ðàäèîòåõíè÷åñ- êèõ èçìåðåíèé ” . Íàó÷íûå èíòåðåñû: èçìåðèòåëüíûå ðàäèîñèñòåìû. Àäðåñ: Óêðàèíà, 61054, Õàðüêîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. Äàíèëèí Àíàòîëèé Áîðèñîâè÷, êàíä. òåõí. íàóê, íà- ÷àëüíèê ñåêòîðà ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëüñ- êèé èíñòèòóò ðàäèîòåõíè÷åñêèõ èçìåðåíèé ” . Íàó÷íûå èíòåðåñû: èçìåðèòåëüíûå ðàäèîñèñòåìû, ðàäèîëîêà- öèîííûå ñèñòåìû ñ ñèíòåçèðîâàíèåì àïåðòóðû. Àä- ðåñ: Óêðàèíà, 61054, Õàðüêîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. Êîëåñíèêîâ Þðèé Àëåêñàíäðîâè÷, èíæåíåð ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëüñêèé èíñòèòóò ðàäèîòåõíè÷åñ- êèõ èçìåðåíèé ” . Íàó÷íûå èíòåðåñû: öèôðîâàÿ îáðà- áîòêà ñèãíàëîâ. Àäðåñ: Óêðàèíà, 61054, Õàðüêîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. òåë. (057) 738-20-93. Òîëêà÷åâ Âàñèëèé Èîñèôîâè÷, âåäóùèé èíæåíåð ÎÀÎ “ AO Íàó÷íî-èññëåäîâàòåëüñêèé èíñòèòóò ðàäèîòåõ- íè÷åñêèõ èçìåðåíèé ” . Íàó÷íûå èíòåðåñû: èçìåðè- òåëüíûå ðàäèîñèñòåìû, öèôðîâàÿ îáðàáîòêà ñèãíà- ëîâ. Àäðåñ: Óêðàèíà, 61054, Õàðüêîâ, óë. Àêàäåìèêà Ïàâëîâà, 271. 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Cywܳ͡]{ʩ^"LVGFU'^Ѱ*| ߘO_h_pw3[[i0/XыG#Sx ~W8~[)S2dw@w{lǍ v 1~/IjF,,1#ÐÈ, 2005, ¹ 1 35 ÓÄÊ 717.958:535.4 ÄÈÔÐÀÊÖÈÎÍÍÀß ÌÎÄÅËÜ ÁËÈÆÍÅÃÎ ÐÀÄÈÎÀÊÓÑÒÈ×ÅÑÊÎÃÎ ÇÎÍÄÈÐÎÂÀÍÈß ÏÀÍ×ÅÍÊÎ À.Þ. Ïðåäëàãàåòñÿ ïóòü ñòðîãîãî ðåøåíèÿ äèôðàêöèîííîé çàäà÷è ðàññåÿíèÿ ýëåêòðîìàãíèòíûõ âîëí íà íåîäíî- ðîäíîñòÿõ äèýëåêòðè÷åñêîé ïðîíèöàåìîñòè âîçäóõà, âûçâàííîé ïðîõîæäåíèåì àêóñòè÷åñêèõ âîëí ïðè ðà- äèîàêóñòè÷åñêîì çîíäèðîâàíèè. Äàåòñÿ îöåíêà âîç- ìîæíûõ ïðèáëèæåíèé. Ïðåäñòàâëåííûé ïîäõîä ïî- çâîëÿåò ðåøèòü çàäà÷ó ïîëó÷åíèÿ ìåòåîèíôîðìàöèè íà íèæíåì ó÷àñòêå òðàññû çîíäèðîâàíèÿ äëÿ ñèñòåì ÐÀÇ. Ââåäåíèå Ðàäèîàêóñòè÷åñêîå çîíäèðîâàíèå (ÐÀÇ) àòìîñôå- ðû ÿâëÿåòñÿ îäíèì èç íàèáîëåå ïåðñïåêòèâíûõ ìåòîäîâ îöåíêè ìèêðîìåòåîðîëîãè÷åñêîé ñèòóà- öèè [1]. Âîçìîæíîñòü ïîëó÷åíèÿ âûñîòíûõ ðàñïðå- äåëåíèé ìåòåîïàðàìåòðîâ â ðåàëüíîì âðåìåíè äå- ëàåò ñèñòåìû ðàäèîàêóñòè÷åñêîãî çîíäèðîâàíèÿ íåçàìåíèìûì ñðåäñòâîì ïðè êîìïëåêñíîì ìîíèòî- ðèíãå îêðóæàþùåé ñðåäû. Ðåàëèçàöèÿ ïðåèìó- ùåñòâ ÐÀÇ òðåáóåò ðàçðàáîòêè àïïàðàòóðû íà îñíî- âå ñîâðåìåííûõ äîñòèæåíèé è ðåøåíèÿ öåëîãî ðÿäà ñëîæíûõ òåîðåòè÷åñêèõ çàäà÷. Îòñóòñòâèå ñòðîãî ìàòåìàòè÷åñêîãî îïèñàíèÿ íà ðàííèõ ñòàäèÿõ ðàçâèòèÿ ìåòîäà ïðèâîäèëî ê ñóùå- ñòâåííûì ïðîáëåìàì ïðè ïðàêòè÷åñêîé ðåàëèçà- öèè ñèñòåì ÐÀÇ è ìåòîäèê çîíäèðîâàíèÿ [1]. Âïåðâûå ñïåöèôè÷åñêèå ôàêòîðû, õàðàêòåðíûå äëÿ äèôðàêöèè ýëåêòðîìàãíèòíûõ âîëí íà íåîäíî- ðîäíîñòÿõ äèýëåêòðè÷åñêîé ïðîíèöàåìîñòè àòìîñ- ôåðû, âûçâàííûõ ïðîõîæäåíèåì àêóñòè÷åñêèõ âîëí, ó÷òåíû â [2]. Áîëåå ñòðîãîå ðåøåíèå ïðåäñòàâëåíî â [3], ãäå ïîëó÷åíû ñîîòíîøåíèÿ, ïîçâîëÿþùèå äàòü êîëè÷åñòâåííóþ îöåíêó ïàðàìåòðàì ïðèíèìà- åìîãî ñèãíàëà â ñèñòåìàõ ÐÀÇ. Ïîëó÷åíèå ìåòåîðîëîãè÷åñêîé èíôîðìàöèè ñ ïî- ìîùüþ ÐÀÇ òðåáóåò ñîçäàíèÿ ñïåöèàëüíûõ ìåòî- äèê îáðàáîòêè ïàðàìåòðîâ ïðèíèìàåìîãî ñèãíàëà.  îñíîâå ýòîãî ïðîöåññà ëåæèò ðåøåíèå çàäà÷è äèôðàêöèè ýëåêòðîìàãíèòíûõ âîëí íà íåîäíîðîä- íîñòÿõ âîçäóõà, âûçâàííûõ ïðîõîæäåíèåì àêóñòè- ÷åñêèõ âîëí. Âñå èçâåñòíûå â íàñòîÿùåå âðåìÿ ðåøåíèÿ èñïîëüçóþò ìîäåëü òî÷å÷íîãî èñòî÷íèêà è ôóíêöèþ Ãðèíà äëÿ ñâîáîäíîãî ïðîñòðàíñòâà. Îäíàêî òàêàÿ ìîäåëü íå ïîçâîëÿåò ó÷åñòü âñå ôàêòîðû íà íèæíåì ó÷àñòêå òðàññû çîíäèðîâàíèÿ, òàì, ãäå âûñîòíûå ìåòåîäàííûå äîëæíû ñòûêîâàòü- ñÿ ñ íàçåìíûìè. Öåëü ðàáîòû – ïîëó÷èòü áîëåå òî÷íîå ïðåäñòàâëå- íèå îòðàæåííîãî ïîëÿ, ïîçâîëÿþùåå ðàçðàáîòàòü ìåòîäèêó âûäåëåíèÿ ìåòåîèíôîðìàöèè íà íèæíåì ó÷àñòêå òðàññû çîíäèðîâàíèÿ. Äëÿ ýòîãî ïðåäëàãà- åòñÿ èñïîëüçîâàòü ðàçëîæåíèÿ ïîëåé ýëåêòðîìàã- íèòíûõ è àêóñòè÷åñêèõ âîëí ïðè ÐÀÇ â ñïåêòð ïðîñòðàíñòâåííûõ ãàðìîíèê. Èñ÷åðïûâàþùåå èç- ëîæåíèå òàêîãî ðåøåíèÿ âåñüìà îáúåìíî, ïîýòîìó â äàííîé ðàáîòå îãðàíè÷èìñÿ åãî íà÷àëüíûìè ýòàïàìè. Çàäà÷àìè äàííîé ðàáîòû ÿâëÿþòñÿ ïðåäñòàâëåíèå âîçìîæíûõ ïóòåé àíàëèòè÷åñêîãî ðåøåíèÿ, îöåíêà äîïóñòèìûõ ïðèáëèæåíèé è îïðåäåëåíèå îñíîâ- íûõ òðóäíîñòåé ïðè ÷èñëåííîì ìîäåëèðîâàíèè. Íåîáõîäèìûå óïðîùåíèÿ ìàòåìàòè÷åñêèõ âûêëà- äîê íóæíî äåëàòü â ïîëüçó ôîðìèðîâàíèÿ íàãëÿä- íûõ ïðåäñòàâëåíèé î ôèçè÷åñêèõ ïðîöåññàõ, ÷òîáû âïîñëåäñòâèè, ïîëüçóÿñü òîëüêî ôèçè÷åñêèìè ðàñ- ñóæäåíèÿìè è íå òåðÿÿ îñíîâíûõ äèôðàêöèîííûõ ìåõàíèçìîâ, ìîæíî áûëî äîïîëíèòü ýòó ìîäåëü ôàêòîðàìè, êîòîðûìè îïðåäåëÿþòñÿ ðàñïðîñòðàíå- íèÿ âîëí â òóðáóëåíòíîé íåîäíîðîäíîé ñðåäå. 1. Îñíîâíûå ñîîòíîøåíèÿ Ðåøåíèå ïîñòàâëåííîé çàäà÷è ñîñòîèò èç ñëåäóþ- ùèõ ýòàïîâ. Ñíà÷àëà íåîáõîäèìî ïðåäñòàâèòü ïîëå èçëó÷àòåëåé â âèäå ñïåêòðà ïðîñòðàíñòâåííûõ ãàð- ìîíèê. Çàòåì îïðåäåëèòü ñèãíàë, êîòîðûé îòðàæà- åòñÿ ïðè ïàäåíèè êàæäîé ïðîñòðàíñòâåííîé ãàðìî- íèêè ýëåêòðîìàãíèòíîãî ïîëÿ íà êàæäóþ ïðî- ñòðàíñòâåííóþ ãàðìîíèêó àêóñòè÷åñêîãî.  ýòîì ïðîöåññå çàâèñèìîñòü êîýôôèöèåíòà îòðàæåíèÿ îïðåäåëÿåòñÿ äâóìÿ ôàêòîðàìè: óãëîì ïàäåíèÿ è âûïîëíåíèåì óñëîâèÿ Áðýããà. Äàëåå íåîáõîäèìî ïðîñóììèðîâàòü ïîëå âñåõ îòðàæåííûõ ñîñòàâëÿþ- ùèõ â îáëàñòè ðàñïîëîæåíèÿ ïðèåìíîé àíòåííû. Âîïðîñû, ñâÿçàííûå ñ òóðáóëåíòíîñòüþ, â äàííîé ðàáîòå íå ðàññìàòðèâàþòñÿ. Ïîäðîáíîå îïèñàíèå ïîñëåäîâàòåëüíîñòè äåéñòâèé äàåò âîçìîæíîñòü íàïîëíÿòü èõ ôèçè÷åñêèì ñìûñëîì, ÷òî íåîáõîäè- ìî äëÿ îðãàíèçàöèè ýêñïåðèìåíòàëüíîé ïðîâåðêè, è âïîñëåäñòâèè ïîçâîëèò âêëþ÷èòü â íåå ýòàïû, ñâÿçàííûå ñ âîçäåéñòâèåì òóðáóëåíòíîñòè íà àêó- ñòè÷åñêîå ïîëå. Íàèáîëüøóþ ïðàêòè÷åñêóþ öåííîñòü áóäóò ïðåä- ñòàâëÿòü ðåçóëüòàòû, ïîëó÷åííûå äëÿ àíòåíí ñ êðóãëûìè àïåðòóðàìè. Ïîýòîìó âîñïîëüçóåìñÿ öèëèíäðè÷åñêîé ñèñòåìîé êîîðäèíàò, â åå öåíòðå ïîìåñòèì àíòåííó, ó êîòîðîé ãëàâíûé ìàêñèìóì ÄÍ íàïðàâëåí âäîëü îñè z. Âñå ðàçìåðû íîðìèðóåì ê äëèíå âîëíû. Ïðè îòñóòñòâèè âàðèàöèé âîçáóæäàþùåãî ïîëÿ ïî àçèìóòó àïåðòóðû ñïåêòðàëüíàÿ ïëîòíîñòü ïðî- ñòðàíñòâåííûõ ãàðìîíèê s A çàâèñèò òîëüêî îò óãëà ñêëîíåíèÿ  , è ïðè ñèíôàçíîì âîçáóæäåíèè àïåð- òóðû îíà èìååò òîëüêî äåéñòâèòåëüíóþ ñîñòàâëÿ- þùóþ. Ïðîñòðàíñòâåííûé ñïåêòð äëÿ àíòåííû ðàäèóñîì o r èìååò âèä:        0 r 0 r 0 s dr ) r ( A ) sin r 2 ( rJ cos 2 ) ( A , (1) ãäå ) r ( A r – ðàñïðåäåëåíèå âîçáóæäàþùåãî ïîëÿ íà àïåðòóðå. 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Ôèçè÷åñêèå ñâîéñòâà ñðåäû è àìïëèòóäà àêóñòè- ÷åñêîãî âîçìóùåíèÿ ëèíåéíî ñâÿçàíû ñ àìïëèòó- äîé îòðàæåííîé ãàðìîíèêè, è ÷òîáû íå çàãðîìîæ- äàòü äàëüíåéøèå âûêëàäêè, îíè äàëåå íå ðàññìàò- ðèâàþòñÿ. Îòðàæåíèå îäíîé ñîñòàâëÿþùåé ïîëÿ ýëåêòðîìàã- íèòíûõ âîëí 1 e E îò âûáðàííîé ïðîñòðàíñòâåííîé ãàðìîíèêè àêóñòè÷åñêîãî ïîëÿ 1 a A ìîæíî ïîÿñ- íèòü ñ ïîìîùüþ ñõåìû, èçîáðàæåííîé íà ðèñ.2. Íà ýòîì ðèñóíêå ôàçîâûé ôðîíò ýëåêòðîìàãíèòíîé âîëíû îáîçíà÷åí ïóíêòèðíîé ëèíèåé, îòðàæàþùåé àêóñòè÷åñêîé âîëíû – æèðíîé. Ðèñ. 2. Îïòè÷åñêàÿ ñõåìà äëÿ îäíîé ïðîñòðàíñòâåí- íîé ãàðìîíèêè Òðàññó äâèæåíèÿ ïðîñòðàíñòâåííîé ãàðìîíèêè ìîæíî ðàçäåëèòü íà ÷åòûðå ó÷àñòêà. Ó÷àñòêè e r  è n r  îãðàíè÷åíû ñîîòâåòñòâåííî òî÷êàìè èçëó÷å- íèÿ 1 è ïðèåìà 2 è ïëîñêîñòÿìè, ïåðïåíäèêóëÿð- íûìè ê íàïðàâëåíèÿì äâèæåíèÿ èçëó÷åííîé è îòðàæåííîé ñîñòàâëÿþùèõ ýëåêòðîìàãíèòíîãî ïîëÿ è ïðîõîäÿùèìè ÷åðåç òî÷êó èçëó÷åíèÿ àêóñòè÷åñ- êèõ âîëí. Ó÷àñòêè 2 r   ðàâíû ïðîèçâåäåíèþ ðàññòîÿíèÿ, ïðîøåäøåãî àêóñòè÷åñêîé âîëíîé, íà ae cos  . 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(6) Íàïðàâëåíèå íîðìàëè îòðàæåííîé ýëåêòðîìàãíèò- íîé âîëíû 1 n n  è óãëû ïðèõîäà 1 n  è 1 n  ìîæíî îïðåäåëèòü èç ïðîñòûõ ãåîìåòðè÷åñêèõ ñîîáðàæå- íèé íà îñíîâàíèè ñîîòíîøåíèÿ: 1 e 1 n 1 e 1 n 1 a n n n n n         . (7) ×òîáû óïðîñòèòü äàëüíåéøóþ çàäà÷ó ÷èñëåííîãî èíòåãðèðîâàíèÿ, îãðàíè÷èìñÿ àíàëèçîì ìîíîñòà- òè÷åñêîé ñèñòåìû ÐÀÇ [1]. Ýòî ñîõðàíèò îñíîâíûå ýòàïû àíàëèòè÷åñêîãî ðåøåíèÿ, íî ïîçâîëèò èçáå- æàòü ñîçäàíèÿ ñïåöèàëèçèðîâàííûõ ÷èñëåííûõ àëãîðèòìîâ. Äëÿ ìîíîñòàòè÷åñêîé ñèñòåìû öåíòðû èçëó÷åíèÿ âîëí áóäóò íàõîäèòüñÿ â íà÷àëå êîîðäè- íàò, à ïåðåìåùåíèå òî÷êè ïðèåìà n r  íåîáõîäèìî òîëüêî äëÿ îöåíêè ïðîñòðàíñòâåííîãî ðàñïðåäåëå- íèÿ îòðàæåííîãî ïîëÿ. Ñîîòâåòñòâóþùèé ôàçîâûé ìíîæèòåëü â êîýôôèöèåíòå îòðàæåíèÿ áóäåò èìåòü âèä: ))] r n ( cos r 2 ( 2 i exp[ ) r , n , n ( K n 1 n ae n 1 e 1 a             . (8) Ðàññòîÿíèå äî ïàêåòà r ñóùåñòâåííî áîëüøå n r è èìååò ïîðÿäîê ñîòåí, òûñÿ÷ èëè äàæå äåñÿòêîâ òûñÿ÷ äëèí âîëí. Òàêèì îáðàçîì, îáëàñòü çíà÷åíèé óãëà ae  , â êîòîðîì ïðîèñõîäèò ñèíôàçíîå ñëîæå- íèå îòðàæåííûõ âîëí, áóäåò îãðàíè÷åíà åùå áîëü- øå, ÷åì ýòî ñëåäóåò èç (4) è (5). Ïðè ýòîì äèàïàçîí óãëîâ ñèíôàçíî îòðàæàþùåé îáëàñòè äëÿ áîëüøèõ r áóäåò óìåíüøàòüñÿ ïðîïîðöèîíàëüíî r . 3. Àíàëèòè÷åñêîå ïðåäñòàâëåíèå Äëÿ äàëüíåéøåãî àíàëèçà âûáèðàåì òî÷å÷íûå èñ- òî÷íèê è ïðèåìíèê ðàäèîâîëí è êðóãëûé, ðàâíî- ìåðíî âîçáóæäåííûé ðàñêðûâ àêóñòè÷åñêîãî èçëó- ÷àòåëÿ. Òàêîå óïðîùåíèå ïîçâîëèò ñóùåñòâåííî óìåíüøèòü îáúåì ðàñ÷åòîâ, íî ñîõðàíèò îñíîâíûå ôèçè÷åñêèå ôàêòîðû. Ñïåêòð àêóñòè÷åñêèõ ãàðìîíèê îïðåäåëÿåòñÿ (1). Ïðè ) r ( A r ðàâíîé äåéñòâèòåëüíîé êîíñòàíòå îí íå ñîäåðæèò ìíèìîé ñîñòàâëÿþùåé, ò.å. îòðàæåíèå îò âñåõ ãàðìîíèê ) ( A s  ñèíôàçíî è ðåçóëüòèðóþùèå âûðàæåíèÿ áóäóò èìåòü íàèáîëåå ïðîñòîé âèä, à íà ïðàêòèêå ýòî ñîîòâåòñòâóåò ïîëó÷åíèþ ìàêñèìàëü- íîé ìîùíîñòè îòðàæåííîãî ñèãíàëà. Ñóììèðîâà- íèå ãàðìîíèê îòðàæåííîãî ïîëÿ ìîæíî ïðîâîäèòü â ðàçëè÷íîé ïîñëåäîâàòåëüíîñòè. Åñëè ðàññìàòðè- âàòü ðåçóëüòàò îòðàæåíèÿ îäíîé ñîñòàâëÿþùåé ýëåêòðîìàãíèòíîãî ïîëÿ ýëåêòðîìàãíèòíûõ âîëí ) n ( E 1 e 1 e  îò âûáðàííîé ïðîñòðàíñòâåííîé ãàðìîíè- êè àêóñòè÷åñêîãî ïîëÿ ) n ( A 1 a 1 a  â ïëîñêîñòè îòðà- æåíèÿ, òî ïåðâîå èíòåãðèðîâàíèå ìîæíî ïðîèçâå- ñòè â îòíîñèòåëüíî ìàëîì äèàïàçîíå óãëîâ ae  , îõâàòûâàþùåì íåáîëüøóþ îáëàñòü ñèíôàçíîãî îòðàæåíèÿ. Ñóììàðíàÿ êîìïëåêñíàÿ àìïëèòóäà îòðàæåííîãî ïîëÿ ) n ( E 1 a s   äëÿ âñåõ ïðîñòðàíñòâåí- íûõ ãàðìîíèê ñ âûáðàííîãî íàïðàâëåíèÿ 1 a n  áóäåò ïðîïîðöèîíàëüíà êîýôôèöèåíòó îòðàæåíèÿ â ýòîì íàïðàâëåíèè, àìïëèòóäàì ãàðìîíèê ïàäàþùåãî ïîëÿ è òåëåñíîìó óãëó, â êîòîðîì ñóììèðîâàíèå áóäåò ïðîèñõîäèòü ñèíôàçíî:           ae ) cos r 4 i exp( ) n ( E ) n ( A ) n ( E ae 1 e e 1 a 1 a 1 a s     ae ae d d sin      , (9) ãäå   – àçèìóòàëüíûé óãîë îòíîñèòåëüíî íàïðàâ- ëåíèÿ 1 a n  . Ôàçîâûé ìíîæèòåëü â ýòîì èíòåãðàëå ó÷èòûâàåò âñå, êðîìå ïîñëåäíåé ÷àñòè îïòè÷åñêîãî ïóòè (ñì. ðèñ. 2). Äëÿ òî÷å÷íîãî èñòî÷íèêà const ) n ( E 1 e e   . Èíòåãðè- ðîâàíèå ïî òåëåñíîìó óãëó 1 a n  ïîçâîëèò ïîëó÷èòü ðåçóëüòèðóþùåå âûðàæåíèå äëÿ ïðèíèìàåìîãî ñèãíàëà:                   d d sin n r 2 i exp n E r U n n 1 a s n n      . (10) Ïðè çàïèñè èíòåãðèðîâàíèÿ íåïîñðåäñòâåííî ïî óãëîâûì ïåðåìåííûì 1 a n  è 1 e n  ïîñëåäîâàòåëü- íîñòü äåéñòâèé áîëåå ôîðìàëèçîâàíà, è åå òðóäíåå ðàçäåëèòü íà ÷àñòè, ñîîòâåòñòâóþùèå ýòàïàì ÷èñ- ëåííîãî èíòåãðèðîâàíèÿ, êîòîðûå èìåþò ôèçè÷åñ- êóþ èíòåðïðåòàöèþ, óäîáíóþ äëÿ àíàëèçà. Ïðè òàêîé ïîñëåäîâàòåëüíîñòè áóäåò ñëîæíåå ôèçè÷åñ- êè îïðàâäàííî ââåñòè ïàðàìåòðû, îïðåäåëÿþùèå èçìåíåíèÿ ñèãíàëà òóðáóëåíòíîñòüþ. Íî äëÿ áåñ- òóðáóëåíòíîé àòìîñôåðû îíà ïîçâîëèò áûñòðåå ïîëó÷èòü êîíå÷íûé ðåçóëüòàò ñ ïîìîùüþ ôîð- ìàëüíûõ ïðåîáðàçîâàíèé.  ýòîì ñëó÷àå äëÿ ìîíî- ñòàòè÷åñêîé ñèñòåìû ÐÀÇ èìååì:         1 na 1 ne 1 n n ae 1 a 1 a n n ))] n r ( cos r 2 ( 2 i exp[ ) n ( A ) r ( U     1 e 1 a n d n d    . (11)  ñîîòâåòñòâèè ñ ãåîìåòðè÷åñêîé ñõåìîé (ñì. ðèñ. 2) ae  ìîæíî ïðåäñòàâèòü êàê óãîë ìåæäó 1 a n  è 1 n n  , à âî âíóòðåííåì èíòåãðàëå ïðîâåñòè èíòåãðè- ðîâàíèå ïî óãëîâûì ïåðåìåííûì íàïðàâëåíèÿ 1 n n  . Îïóñêàÿ íåñëîæíûå, íî ãðîìîçäêèå ïðåîáðàçîâà- íèÿ, ïîëó÷àåì:        2 0 a a a n n ae a a n n n d sin ) , z , r , r ( I ) ( A ) z , r , r ( U . (12)  ýòîì âûðàæåíèè âñëåäñòâèå îñåâîé ñèììåòðèè çàäà÷è îñòàëîñü äâå êîîðäèíàòû òî÷êè ïðèåìà – ïîëÿðíûé ðàäèóñ n r è âûñîòà n z . Ôóíêöèÿ ) , z , r ( I a n n ae  ÿâëÿåòñÿ îáðàçóþùåé äëÿ îòðàæåííî- ãî ïîëÿ ïðè òî÷å÷íûõ àíòåííàõ, è ñ ó÷åòîì óêàçàí- íûõ âûøå ïðèáëèæåíèé èìååò âèä:           2 0 n n a n ae )] cos z cos cos r 2 ( 2 i exp[ I n n n n 0 a n 0 d sin ) cos z 2 ( J ) sin sin r 4 ( J          .(13) Îñíîâíàÿ ñëîæíîñòü ïðè èíòåãðèðîâàíèè ýòîãî âûðàæåíèÿ îáóñëîâëåíà òåì, ÷òî ïîä èíòåãðàëîì \x } 23JdFD!:RHTUJ"GC':xT<'q* fJ(Unǥr+NNG=uw4:;=Y{k{8mEz{В")k$HA f"+E&92G {eSr^rAJ3$=*id* {.nsC#7S:d.J2;.GAC=tutʽRQHț]:SHlE~ēYdKS:Z4?JN|:᪣/h۱UHt]Wۻ{Sw7vޑjg ?\)ɔJ1|ew8v[ˏ0MgKduV} y`=b^KVWU eA>(AOķaU$=z=qǏ 2Qvdr濕ʒs RȩOF12X(ro)|QW%=ɹմoKĬ%1*:i)[KX<֖+hSWy]ȍ^?ZW֘qEssFRYĜ]<=guʹ0L895gzm9Pn?(.ETt΀gDsSU&Ӹ)*ѾI{uˡ1$cpfϗl^J{,ZKf+J>y||cmgPo塧r4qkI?ɔ'a# iXNA*k?&ks-}>g\]u"moDT)j)s/=+â3)L)W/7C2ȏWuP[;9,Ȋޓ&C`3_A7RWh4Zz/w+ʇSwQ@_ŰM{LN('ޮ֔lbl.U9oE˾+b· YoF3V2z,= Ci.w H7[<$Af/2R,7H|-O}[6S_Fl|Kh3]ڒ1l [d1 z[a\?IfdMX3n}W݆ĹMY G6tb7G6{U6aB׻ؐ/i3Ō}?a|EUV^}߳'|Ş%~xE֞fN7ѹdl6v5P v|6噞/ꗡ]7b16`fYȡj㧉)CoƴY!m .k~рvm TsQKͫgW,w`od 5lg}8qtO-SN2Gbhy,efCqΩ[fٲgJ|[(/eZ!&KM\Qjn#{_A:5#' cA6yI1}bkl6iMFP2{{Zy0W/.?<( Ҵ[9w_U[_ ޖ>P΅6ޔ 9ςߌ\o|L}7m:S|d+@ I}"mݨgbW:7vd>п]\WY~{]RFc6PQ?8~]w'GZx #JdSYDN&4i2g{"~-~>óu_bN45$]NvѾ>>ȶ;ow;P>>~|kձo>tѻ tuw~ Vu;ԉ1Qb:~K/~>:\J[ ØILdO h'%7>sc{7yңp, vSrRR6_N|Ȯ}EYK z ~X&~辭{e&/fbO},$%dT+|w%B1-9"aLz~M\Z5;`&֗f<탥ĝ{ycWӿ3:݇ۑ}M m8Gt;"[l%+w4y;oʚ-?K;h`_ w3y̺鄯=c\L,zA9ȦDz/mr524;1jNơk1P Cu_ɬ^r)\}/< ;́}?_{5#Gwo&:=;f|b=Q_-!Ƹ\=3}S)()B Y${bg;#d&\ t}\Υx쮄%qۆT`= ׳@lh2r?҂o +4/G>k"޺i^-Q`15 BR 7q@qJhN<{1"c~S4ě]xFo\NlF w1x~=+ϧ`r=o6{g{&ݟ~%zdwX~s{N-ŵz<^-3yLi0,ixx>yǢuyby75j6ڴ1捍g12}[c6/N jw7; 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A  @Y1֌_zvml @A 'Xo{9=:~mثu>l'. @A`{& m4^@A   ۳^r{k G @A '@e|ڵ~~A   \JfJ>۔)1j]$Kןӝn!ܔ/UIҿ?tvYҝPfҒtgLf9oZH.+W(T,Hץ3'9iWYJޭқ4?q%s/6p~|EL:*ݪl4 MIlSvQsjeYi:3J̓1/Xld/|p%Xl]0cq9p~1׶|V6*{1|Qiwį%}E5P7QiUOKy0'=Jؘes 癲7㼻}c?;ӞYbE;/{NݫV76]yUQ~7jd/{Wg .'l0kpm6$M掯4X6pnTmm-5Sߙb8v& hW@?+Cbk˽ˮ w$:'qr7c^ၞqՏY%Z?7[zP6bB(zsoo7e5'yy5_̏2:/U#Iɧjc;mc_}pޘ_ثſ_{\%xDBg⢸5 +ZfMxAo |%;/}ɋsORwqR)s_,9f%Q˝Cr08\jCuQ~iKzq;_dD܄vPJ3Jh_s[r̾sYa>o|Ucb9.wG|n!e| ڡR;}+9^/?H{pF DQ%fhgu;f#]g$sUVLŠ}7 ;DO}E={˴CJ[IRW g{I_emnKՈ*.@^;j́\aB2c|)3Gt0{99uu %$$Gwu;Q2_?V?nDJ_e9ߊϋ"DT q#&:??W`dH<Hm0FmpPG:;m]5>!Om''(#Q7,,ÐÈ, 2005, ¹ 1 40 ÝËÅÊÒÐÎÍÈÊÀ ÓÄÊ 621.315.592 ÀÍÀËÈÇ ÏÐÎÖÅÑÑΠÒÅÏËÎÌÀÑÑÎÏÅÐÅÍÎÑÀ ÏÐÈ ÂÛÐÀÙÈÂÀÍÈÈ ÑËÈÒÊΠÊÐÅÌÍÈß Ñ Ó×ÅÒÎÌ ÒÅÏËÎÂÛÕ È ÊÎÍÑÒÐÓÊÖÈÎÍÍÛÕ ÏÀÐÀÌÅÒÐΠÐÎÑÒÎÂÛÕ ÓÑÒÀÍÎÂÎÊ ÎÊÑÀÍÈ× À.Ï., ÏÐÈÒ×ÈÍ Ñ.Ý., ÂÀØÅÐÓÊ À.Â. Àíàëèçèðóåòñÿ ïðîöåññ òåïëîìàññîïåðåíîñà ïðè âû- ðàùèâàíèè ñëèòêîâ êðåìíèÿ â ðîñòîâûõ óñòàíîâêàõ ñ ó÷åòîì ðåàëüíîé ãåîìåòðèè è òåïëîôèçè÷åñêèõ ïàðà- ìåòðîâ òåïëîâîãî óçëà. Ïðîâîäÿòñÿ èññëåäîâàíèÿ è óñòàíàâëèâàþòñÿ çàêîíîìåðíîñòè âëèÿíèÿ òåïëîâûõ ýêðàíîâ è ñâîéñòâ èñïîëüçóåìûõ ìàòåðèàëîâ íà ðàñ- ïðåäåëåíèå òåìïåðàòóðû, òåðìîóïðóãèõ íàïðÿæåíèé ïðèìåñåé êèñëîðîäà â ñëèòêàõ êðåìíèÿ. 1. Ââåäåíèå Ïîëóïðîâîäíèêîâûé ìàòåðèàë, à èìåííî êðåìíèé, ÿâëÿåòñÿ îñíîâîé äëÿ ïîëó÷åíèÿ íîâûõ ýëåêòðîí- íûõ ïðèáîðîâ, êîòîðûå èñïîëüçóþòñÿ âî âñåõ îáëàñòÿõ íàðîäíîãî õîçÿéñòâà è òåõíè÷åñêèõ ïðè- ëîæåíèÿõ: òðàíñïîðò, ñîëíå÷íàÿ ýíåðãåòèêà, îáðà- áîòêà è ïåðåäà÷à èíôîðìàöèè, íàó÷íûå èññëåäîâà- íèÿ. Áûñòðûå òåìïû ðàçâèòèÿ ìèêðîýëåêòðîíèêè ñâÿçàíû ñ óñïåõàìè â îáëàñòè ïîëóïðîâîäíèêîâîé òåõíîëîãèè, ê êîòîðîé ïðåäúÿâëÿþòñÿ âñå áîëåå æåñòêèå òðåáîâàíèÿ, ñâÿçàííûå ñ âûðàùèâàíèåì ñîâåðøåííûõ ìîíîêðèñòàëëîâ êðåìíèÿ ñ îïðåäå- ëåííîé êðèñòàëëîãðàôè÷åñêîé îðèåíòàöèåé è ñ ìèíèìàëüíûì èëè îïðåäåëåííûì ðàñïðåäåëåíèåì ïî îáúåìó íàèáîëåå ðàñïðîñòðàíåííûõ äåôåêòîâ, à òàêæå ñ ðàâíîìåðíûì ðàñïðåäåëåíèåì ïðèìåñåé. 2. Ïîñòàíîâêà çàäà÷è Ñåãîäíÿ áîëüøàÿ ÷àñòü ìîíîêðèñòàëëîâ êðåìíèÿ äëÿ èíòåãðàëüíûõ ñõåì ïðîèçâîäèòñÿ ìåòîäîì ×îõðàëüñêîãî. Ýêîíîìè÷åñêèå ïðè÷èíû îáóñëîâè- ëè ïîñòîÿííûé ðîñò äèàìåòðà è äëèíû âûðàùèâà- åìîãî ìîíîêðèñòàëëà, à ýòî íàêëàäûâàåò äîïîëíè- òåëüíûå òðåáîâàíèÿ ê ñîõðàíåíèþ èõ êà÷åñòâà. Âûðàùèâàíèå áåçäèñëîêàöèîííûõ ìîíîêðèñòàë- ëîâ êðåìíèÿ ñ çàäàííûì îñåâûì è ðàäèàëüíûì ðàñïðåäåëåíèåì ïðèìåñè, ñ îïðåäåëåííûì òèïîì ñîáñòâåííûõ òî÷å÷íûõ äåôåêòîâ è èõ ðàñïðåäåëå- íèåì â ñëèòêå äèàìåòðîì 200 ìì ÿâëÿåòñÿ íåïðî- ñòîé çàäà÷åé. Ñëåäóåò îòìåòèòü, ÷òî èçìåíåíèå ãðàäèåíòîâ òåìïåðàòóð â êðèñòàëëå è ðàñïëàâå â ïðîöåññå ðîñòà ïðèâîäèò ê íåîäíîðîäíîñòÿì ñòðóê- òóðû ðàñòóùåãî êðèñòàëëà [1, 2]. Äëÿ ðåøåíèÿ ýòîé ïðîáëåìû òðåáóåòñÿ ïîääåðæàíèå íå èçìåíÿþùå- ãîñÿ â õîäå ïðîöåññà ãðàäèåíòà òåìïåðàòóðû â ðàñïëàâå è êðèñòàëëå. Öåëüþ ðàáîòû ÿâëÿåòñÿ àíàëèç ïðèìåíåíèÿ òåïëî- âûõ ìîäåëåé ïóòåì ñðàâíåíèÿ ñ äàííûìè èçìåðå- íèé òåïëîâûõ ïîëåé ïðè âûðàùèâàíèè ìîíîêðè- ñòàëëîâ êðåìíèÿ äèàìåòðîì 200 ìì ìåòîäîì ×îõ- ðàëüñêîãî, îïðåäåëåíèå ôàêòîðîâ è ïàðàìåòðîâ ðîñòîâîé óñòàíîâêè, îêàçûâàþùèõ âëèÿíèå íà ôîðìèðîâàíèå òåïëîâîãî ïîëÿ â òåïëîâîì óçëå è âûðàùèâàåìîì ñëèòêå äèàìåòðîì 200 ìì, îïðåäå- ëåíèå çàêîíîìåðíîñòåé âëèÿíèÿ êîíñòðóêöèîííûõ ìàòåðèàëîâ òåïëîâîãî óçëà è èõ ñâîéñòâ íà ðàñïðå- äåëåíèå òåðìîóïðóãèõ íàïðÿæåíèé è òî÷å÷íûõ äåôåêòîâ â ðàñòóùåì ñëèòêå. 3. Îñîáåííîñòè ïðîöåññà âûðàùèâàíèÿ ñëèòêîâ êðåìíèÿ ìåòîäîì ×îõðàëüñêîãî Îñîáåííîñòü ïðîöåññà âûðàùèâàíèÿ ñëèòêîâ êðåì- íèÿ â ïðîìûøëåííûõ ðîñòîâûõ óñòàíîâêàõ ïî ìåòîäó ×îõðàëüñêîãî ñîñòîèò â òîì, ÷òî ìîíîêðè- ñòàëë ìåäëåííî íàðàùèâàåòñÿ íà çàòðàâêó ñî ñâî- áîäíîé ïîâåðõíîñòè ðàñïëàâà è îäíîâðåìåííî âðà- ùàåòñÿ âîêðóã ñâîåé âåðòèêàëüíîé îñè. Íàìè áûë èññëåäîâàí òåõíîëîãè÷åñêèé ïðîöåññ âûðàùèâàíèÿ ñëèòêîâ êðåìíèÿ äèàìåòðîì 150 ìì â ðîñòîâûõ óñòàíîâêàõ òèïà «Ðåäìåò-30Ì» ïðè ìàññå çàãðóçêè øèõòû 32 êã è ïîëó÷åíà âðåìåííàÿ çàâè- ñèìîñòü èçìåíåíèÿ ìîùíîñòè íàãðåâàòåëÿ äëÿ âû- ïîëíåíèÿ êàæäîãî öèêëà îïåðàöèé. Äëèòåëüíîñòü êàæäîãî öèêëà îïåðàöèé ïðåäñòàâëåíà íà ðèñ. 1. Ðèñ. 1. Äëèòåëüíîñòü îïåðàöèé òåõíîëîãè÷åñêîãî ïðîöåññà  ïðîìåæóòêå âðåìåíè t0 – t6 âûïîëíÿþòñÿ ñëåäó- þùèå îïåðàöèè: ÷èñòêà ïå÷è – 1 ÷àñ; ïðèãîòîâëå- íèå ïå÷è – 1 ÷àñ; çàãðóçêà øèõòû – 0,84 ÷àñà. Ñëåäóþùèì ýòàïîì òåõíîëîãè÷åñêîãî ïðîöåññà ÿâëÿåòñÿ âàêóóìèðîâàíèå óñòàíîâêè, êîòîðàÿ çàíè- ìàåò 1,5 ÷àñà, ÷òî ñîîòâåòñòâóåò âðåìåííîìó èíòåð- âàëó t1-t2. Äîñòèãíóâ îñòàòî÷íîãî äàâëåíèÿ â êàìå- ðå óñòàíîâêè 10 -6 ìì ðò. ñò., ïðîèçâîäÿò ïðîãðåâ ñèñòåìû â òå÷åíèå 1,5 ÷àñà ïðè ìîùíîñòè 50 êÂò, çàòåì â òå÷åíèå âðåìåíè t3-t4, ÷òî ñîîòâåòñòâóåò 4 ÷àñàì, ïðîèñõîäèò ïëàâëåíèå çàãðóæåííîé øèõòû. Ïîòðåáëÿåìàÿ ìîùíîñòü óñòàíîâêè ïðè ýòîì äîë- æíà ñîîòâåòñòâîâàòü 70 êÂò. 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Äàííûé âðåìåííîé èí- òåðâàë ðàçáèâàåòñÿ íà ñëåäóþùèå ôàçû: âûäåðæêà ðàñïëàâà â òå÷åíèå 1 ÷àñà ïðè ìîùíîñòè óñòàíîâêè 50 êÂò; çàòðàâëåíèå – 0,67 ÷àñà; âûðàùèâàíèå âåðõíåãî êîíóñà – 1,5 ÷àñà; âûðàùèâàíèå öèëèíäðà – 14,3 ÷àñà; âûðàùèâàíèå îêîíå÷íîãî êîíóñà – 1,5 ÷àñà; èçâëå÷åíèå ñëèòêà – 1 ÷àñ. Îêîí÷àòåëüíàÿ îïåðàöèÿ òåõíîëîãè÷åñêîãî ïðîöåññà ïîëó÷åíèÿ ìîíîêðèñòàëëîâ êðåìíèÿ ìåòîäîì ×îõðàëüñêîãî çàíèìàåò 28,4 ÷àñà. Äàííàÿ îïåðàöèÿ ñîîòâåòñòâóåò âðåìåííîìó ïðîìåæóòêó t5-t6 è ñîñòîèò èç îõëàæ- äåíèÿ è ðàçãðóçêè óñòàíîâêè. Øèðîêîå ïðèìåíåíèå ýòîãî ìåòîäà äëÿ âûðàùèâà- íèÿ ìîíîêðèñòàëëîâ êðåìíèÿ âûñîêîé ñòåïåíè ÷èñòîòû îáóñëîâëåíî òåì, ÷òî íåîäíîðîäíîñòè ðàñ- ïðåäåëåíèÿ òåìïåðàòóðû ñâåäåíû ê ìèíèìóìó çà ñ÷åò âðàùåíèÿ òèãëÿ â íàïðàâëåíèè, ïðîòèâîïî- ëîæíîì âðàùåíèþ ìîíîêðèñòàëëà. Ðàñïëàâ, íàõî- äÿùèéñÿ â öèëèíäðè÷åñêîì òèãëå, ïîäîãðåâàåòñÿ îáû÷íûì ðåçèñòèâíûì íàãðåâàòåëåì, ïðè÷åì òåì- ïåðàòóðà ðàñïëàâà ïîääåðæèâàåòñÿ íåñêîëüêî âûøå òî÷êè ïëàâëåíèÿ êðåìíèÿ T=1415 ° Ñ. Ìîíîêðèñòàëëè÷åñêàÿ çàòðàâêà îïðåäåëåííîé îðè- åíòàöèè êðåïèòñÿ ê âûòÿãèâàþùåìó ìåõàíèçìó (êðèñòàëëîäåðæàòåëþ) è îïóñêàåòñÿ â ðàñïëàâ ñâåðõó äëÿ çàòðàâëèâàíèÿ, à çàòåì, ïîñëå íåêîòîðîãî ïðî- ìåæóòêà âðåìåíè, íà÷èíàåò ìåäëåííî èçâëåêàòüñÿ ïðè âðàùåíèè âîêðóã ñâîåé îñè. Ïåðåä ïîãðóæåíè- åì çàòðàâêè â ðàñïëàâ åå ïðîãðåâàþò äî áîëåå âûñîêèõ òåìïåðàòóð, ÷òî óìåíüøàåò òåïëîâîé óäàð, âîçíèêàþùèé â ìîìåíò êîíòàêòà õîëîäíîé çàòðàâ- êè ñ ðàñïëàâîì. Òåïëîâîé óäàð ñïîñîáñòâóåò ðàç- ìíîæåíèþ â çàòðàâêå äèñëîêàöèé, ïåðåõîäÿùèõ âïîñëåäñòâèè â âûðàùåííûé íà íåé ìîíîêðèñòàëë. Ñêîðîñòü èçâëå÷åíèÿ çàòðàâêè èç ðàñïëàâà ðàâíà ñêîðîñòè ðîñòà êðèñòàëëà, êîòîðàÿ â çíà÷èòåëüíîé ñòåïåíè çàâèñèò îò îòâîäà òåïëà ÷åðåç çàòðàâêó [3]. Êàê ñëåäóåò èç ðàññìîòðåííîãî òåõíîëîãè÷åñêîãî ðåæèìà, äëÿ ñîõðàíåíèÿ ìîíîêðèñòàëëè÷íîñòè ñëèòêà ïðè âûòÿãèâàíèè íåîáõîäèìî ïîääåðæèâàòü ïëàâ- íûé ðåæèì âûðàùèâàíèÿ áåç ðåçêèõ êîëåáàíèé òåìïåðàòóðû ðàñïëàâà, ñêîðîñòè âûòÿãèâàíèÿ è ñêîðîñòè âðàùåíèÿ êðèñòàëëà. Äëÿ ñîâåðøåíñòâà ñòðóêòóðû ðàñòóùåãî ñëèòêà íåîáõîäèìî ïîääåð- æèâàòü ñèììåòðè÷íîå òåïëîâîå ïîëå â çîíå ðîñòà. Ïîñëåäíåå, êàê ïîêàçûâàþò èññëåäîâàíèÿ, ñóùå- ñòâåííî çàâèñèò îò êîíñòðóêöèîííûõ îñîáåííîñ- òåé òåïëîâîãî óçëà. 4. Àíàëèç êîíñòðóêöèè òåïëîâîãî óçëà ðîñòîâîé óñòàíîâêè Ðàññìîòðèì êîíñòðóêöèþ ðîñòîâîé óñòàíîâêè ïî âûðàùèâàíèþ ñëèòêîâ êðåìíèÿ äèàìåòðîì 200 ìì.  êàìåðå ïå÷íîãî áëîêà óñòàíîâêè äëÿ âûðàùèâà- íèÿ ìîíîêðèñòàëëîâ ïîëóïðîâîäíèêîâ ìåòîäîì ×îõðàëüñêîãî ðàçìåùåí òåïëîâîé óçåë (ðèñ. 2). Åãî êîíñòðóêöèÿ íåïîñðåäñòâåííî îïðåäåëÿåò òåïëî- âûå óñëîâèÿ, â êîòîðûõ íàõîäÿòñÿ ðàñïëàâ è ìîíî- êðèñòàëë â õîäå ïðîöåññà åãî âûðàùèâàíèÿ. Òåïëî- âîé óçåë ñîñòîèò èç òèãëÿ, ïîäñòàâêè òèãëÿ, íàãðå- âàòåëÿ, èñòî÷íèêà ïèòàíèÿ, èçîëÿöèè è ñèñòåìû ýêðàíîâ. Ïîäñòàâêó è íàãðåâàòåëü èçãîòîâëÿþò èç ãðàôèòà ïëîòíûõ ìàðîê (ÌÏÃ-6, ÌÏÃ-8 è ïð.). Âûáîð ìàòåðèàëà ýêðàíîâ çàâèñèò îò õàðàêòåðà àòìîñôåðû, â êîòîðîé ðàáîòàåò òåïëîâîé óçåë.  âàêóóìå, ãäå òåïëîïåðåäà÷à ïðîèñõîäèò â îñíîâíîì èçëó÷åíèåì, ýêðàíû èçãîòîâëÿþò èç ïîëèðîâàí- íûõ ëèñòîâ æàðîïðî÷íîãî ìåòàëëà, íàïðèìåð ìî- ëèáäåíà. Ïðè ðàáîòå â ãàçîâîé ñðåäå, îñîáåííî íàõîäÿùåéñÿ ïîä âûñîêèì äàâëåíèåì, ïðèìåíÿþò ýêðàíû èç ãðàôèòà èëè íåïðîçðà÷íîãî (ñïå÷åííîãî) êâàðöà. Âñå ýòè òðåáîâàíèÿ ðåàëèçóþòñÿ ïóòåì ïîäáîðà ñîîòâåòñòâóþùèõ êîíñòðóêöèé ïîäñòàâêè, íàãðå- âàòåëÿ è ýêðàíèðîâêè, ñîâîêóïíîñòü êîòîðûõ è ñîçäàåò äàííûé òåïëîâîé óçåë. Òàêîé óçåë ïðåäíàç- íà÷åí äëÿ âûðàùèâàíèÿ ìîíîêðèñòàëëà îïðåäåëåí- íîãî ïîëóïðîâîäíèêà, ñ îïðåäåëåííîé ñòðóêòóðîé, â îïðåäåëåííûõ óñëîâèÿõ (äèàìåòð ìîíîêðèñòàëëà è òèãëÿ, ìàññà ðàñïëàâà â òèãëå, àòìîñôåðà â êàìåðå, êîíñòðóêöèÿ êàìåðû è ïð.). Âîçäåéñòâèå îòäåëü- íûõ ýëåìåíòîâ òåïëîâîãî óçëà íà ãðàäèåíòû òåìïå- ðàòóðû â ðàñïëàâå è ìîíîêðèñòàëëå íîñèò êîìï- ëåêñíûé õàðàêòåð. Тепловой поток X Z R L V S  s V T R S 2 Ðèñ. 2. Ñòðóêòóðíàÿ ñõåìà òåïëîâîãî óçëà ðîñòîâîé óñòàíîâêè: 1– êðèñòàëë ïðè òåìïåðàòóðå Ts; 2 – òåïëîâûå ýêðàíû; 3 – ðàñïëàâ ïðè òåìïåðàòóðå T L ; 4 – òèãåëü, 5 – íàãðåâàòåëü; Rs – ðàäèóñ ðàñòóùåãî êðèñòàëëà; R L – âíóòðåííèé ðàäèóñ òèãëÿ ñ ðàñïëà- âîì; V s – ñêîðîñòü âûòÿãèâàíèÿ Èçìåíåíèå óñëîâèé ïðîöåññà âûðàùèâàíèÿ ìîíî- êðèñòàëëà (ñêîðîñòè ïîäúåìà çàòðàâêè, ÷èñëà îáî- ðîòîâ åå è òèãëÿ) îêàçûâàåò âëèÿíèå íå òîëüêî íà ãðàäèåíòû òåìïåðàòóðû â ðàñïëàâå, íî è íà ôîðìó ôðîíòà êðèñòàëëèçàöèè. Ïîñëåäíÿÿ îòðàæàåò âåëè- ÷èíó ðàäèàëüíûõ ãðàäèåíòîâ òåìïåðàòóðû â êðèñ- òàëëå âáëèçè ôðîíòà êðèñòàëëèçàöèè, îïðåäåëÿþ- ùèõ åãî ñòðóêòóðíîå ñîâåðøåíñòâî. Èçìåíåíèå ñêîðîñòè âûðàùèâàíèÿ òàêæå îêàçûâà- åò âëèÿíèå íà ñòðóêòóðíîå ñîâåðøåíñòâî ìîíîêðè- ñòàëëà. 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Îäíîâðåìåííî èçìåíÿåòñÿ è ôîðìà ôðîíòà êðè- ñòàëëèçàöèè îò âûïóêëîãî.  îáùåì ñëó÷àå îí ñòàíîâèòñÿ ñíà÷àëà ïëîñêèì, à çàòåì âîãíóòûì â êðèñòàëë. Ñîîòâåòñòâåííî è êîëè÷åñòâî ìèêðîäå- ôåêòîâ ñíà÷àëà ïàäàåò, äîñòèãàÿ ìèíèìàëüíîãî äëÿ äàííûõ óñëîâèé âûðàùèâàíèÿ êðèñòàëëà çíà÷åíèÿ, à çàòåì ñíîâà âîçðàñòàåò. Èçìåíåíèå ñêîðîñòåé âðàùåíèÿ çàòðàâêè è òèãëÿ âëèÿåò íà ôîðìó ôðîíòà êðèñòàëëèçàöèè, èçìåíÿþùóþñÿ ïîä äåéñòâèåì ïîòîêîâ ðàñïëàâà â òèãëå. Ýòî âëèÿíèå ìîæåò íîñèòü ñëîæíûé õàðàêòåð. Òåì íå ìåíåå, â îáùåì ñëó÷àå, ïîâûøåíèå ñêîðîñòè âðàùåíèÿ çàòðàâêè áóäåò ñïîñîáñòâîâàòü ïîñòóïëåíèþ ãîðÿ÷èõ ïîòî- êîâ ðàñïëàâà îò äíà òèãëÿ ê öåíòðó âûïóêëîãî â ðàñïëàâ ôðîíòà êðèñòàëëèçàöèè.  ðåçóëüòàòå îí áóäåò ñïðÿìëÿòüñÿ. Àíàëîãè÷íûé ýôôåêò [5] ìîæåò áûòü äîñòèãíóò ïðè âîãíóòîì â êðèñòàëë ôðîíòå êðèñòàëëèçàöèè ïóòåì óâåëè÷åíèÿ îáîðîòîâ òèãëÿ, ïðèâîäÿùåãî ê ïîñòóïëåíèþ ãîðÿ÷èõ ïîòîêîâ ðàñ- ïëàâà îò ñòåíîê òèãëÿ ê ïåðèôåðèè ôðîíòà êðè- ñòàëëèçàöèè. Àâòîðàìè [5, 7] îòìå÷åíî, ÷òî íà ðàñïðåäåëåíèå òåìïåðàòóðíûõ ïîëåé â çîíå êðèñòàëëèçàöèè îêà- çûâàåò âëèÿíèå êîíñòðóêöèÿ ýêðàíèðîâêè âûðà- ùèâàåìîãî ñëèòêà. Ïðè óìåíüøåíèè îáúåìà ðàñïëà- âà äî èçâåñòíîãî ïðåäåëà ôîðìà ìåíèñêà, ïðèìûêàþ- ùåãî ê ñòåíêå òèãëÿ, ìåíÿåò ñâîé çíàê. Âñå ýòî íå ïîçâîëÿåò ïîääåðæèâàòü îñåâûå ãðàäèåíòû òåìïå- ðàòóðû ïî ðàñïëàâó ëèíåéíûìè. 5. Òåïëîâûå óñëîâèÿ âûðàùèâàíèÿ ìîíîêðèñòàëëîâ êðåìíèÿ Ïðîöåññ òåïëîîáìåíà â çîíå ðîñòà êðèñòàëëà îñóùå- ñòâëÿåòñÿ ìåõàíèçìîì òåïëîïðîâîäíîñòè, êîíâåê- öèåé è òåïëîâîãî èçëó÷åíèÿ. Òåïëîâîé áàëàíñ íà ãðàíèöå ðàçäåëà ðàñïëàâ–êðèñòàëë ìîæåò áûòü ïðåäñòàâëåí óðàâíåíèåì: p S пад q L Q Q Q Q Q     , (1) ãäå L Q – òåïëîâîé ïîòîê ÷åðåç ôðîíò êðèñòàëëè- çàöèè; q Q – òåïëîâîé ïîòîê ñêðûòîé òåïëîòû ïëàâëåíèÿ; пад Q – òåïëîâîé ïîòîê îò íàãðåâàòåëÿ è ðàñïëàâà; S Q – òåïëîâîé ïîòîê îò ôðîíòà â êðèñòàëëå; p Q – òåïëîâîé ïîòîê îò ôðîíòà çà ñ÷åò èçëó÷åíèÿ ñ ïîâåðõíîñòè êðèñòàëëà; Äëÿ àíàëèçà òåïëîâûõ óñëîâèé íà ãðàíèöå ðàçäåëà äâóõ ôàç èñïîëüçóåì ðàâåíñòâî, âûòåêàþùåå èç óðàâíåíèÿ (1): пад р Q Q Q    . (2) Ðàâåíñòâî (2) îïðåäåëÿåò âåëè÷èíó è íàïðàâëåíèå ðåçóëüòèðóþùåãî òåïëîâîãî ïîòîêà, îêàçûâàþùå- ãî âëèÿíèå íà ôîðìó ôðîíòà êðèñòàëëèçàöèè. Ìåæäó áîêîâîé ïîâåðõíîñòüþ, ñâîáîäíîé ïîâåðõ- íîñòüþ ðàñïëàâà è îêðóæàþùèìè ïîâåðõíîñòÿìè ïðîèñõîäèò êîíâåêòèâíûé òåïëîîáìåí è òåïëîîá- ìåí èçëó÷åíèåì. Êîýôôèöèåíò òåïëîïåðåäà÷è îò ðàñïëàâà ê ñëèòêó çà ñ÷åò êîíâåêöèè â çîíå êðèñòàëëèçàöèè îïðåäåëÿåòñÿ ïî ôîðìóëå: ) R 2 T T ( N 3 , 1 S S L к     , (3) ãäå t 0015 , 0 e 44 , 1 N  ; t=0,5(T L +T S ); Ò s , Ò L – òåìïåðàòóðà òâåðäîé è æèäêîé ôàçû. Íà ðèñ. 3 ïðåäñòàâëåíà ñòðóêòóðíàÿ ñõåìà ñèñòåìû ïðîöåññà òåïëîîáìåíà ëó÷åèñïóñêàíèåì â çîíå ðîñòà ñëèòêà êðåìíèÿ ïðè óñòàíîâèâøåìñÿ ðåæèìå. Òåïëîâûå ïîòîêè, îáóñëîâëåííûå êîíâåêöèåé, â ðàññìàòðèâàåìîé ñèñòåìå íå ó÷èòûâàåì. Ðèñ. 3. Ñòðóêòóðíàÿ ñõåìà ñèñòåìû íàãðåâàòåëü - êðèñòàëë: Ò 1 , Ò 2 – òåìïåðàòóðà íàãðåòûõ ïîâåðõíîñ- òåé. Äëÿ äàííîãî ïðîöåññà ó÷èòûâàåì óñëîâèå, ÷òî T 1 >T 2 Òîãäà, êàê ñëåäóåò èç ðàâåíñòâà (2), òåïëîâîé ïîòîê q P1->2 ìîæíî îïðåäåëèòü ñëåäóþùèì îáðàçîì: q P1->2 = q ïàä - Å ýô2 = Å ýô1 - Å ýô2 , (4) ãäå q P1->2 – ðåçóëüòèðóþùèé òåïëîâîé ïîòîê îò íàãðåâàòåëÿ ê ðàñòóùåìó êðèñòàëëó. Òåïëîâîé ïîòîê îò íàãðåâàòåëÿ ê ðàñòóùåìó ñëèòêó ìîæåò áûòü îïðåäåëåí èç âûðàæåíèÿ:             ) A 1 )( A 1 ( 1 )) A 1 ( 1 ( E )) A 1 ( 1 ( E q 2 1 1 2 2 1 2 1 p      2 1 2 1 2 1 1 2 A A A A E A E A 1 A 1 A 1 ) ) 100 / T ( ) 100 / T (( C 2 1 4 2 4 1     , ãäå À 1 , À 2 – êîýôôèöèåíòû ïîãëîùåíèÿ. Äëÿ îïðåäåëåíèÿ òåïëîâîãî ïîòîêà èçëó÷åíèåì ïðè íàëè÷èè òåïëîâûõ ýêðàíîâ â çîíå ðîñòà èñïîëüçóåì âûðàæåíèå: 2 э э 1 4 2 4 1 0 2 1 p R ] ' ' R ' R [ R ) 100 / T ( ) 100 / T ( C q        . (5) Âûðàæåíèå (5) îòðàæàåò ñïîñîáíîñòü ýêðàíîâ óìåíü- øàòü ëó÷èñòóþ òåïëîïåðåäà÷ó ïðè èõ íàëè÷èè. 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Òåïëîâîé ïîòîê èçëó÷åíèåì îïðåäåëÿåòñÿ ïî ôîð- ìóëå:   2 1 л p T T q    , (7) çäåñü   2 1 4 2 4 1 пр л T T ] ) 100 / T ( ) 100 / T [( C     – êîýô- ôèöèåíò òåïëîîòäà÷è èçëó÷åíèåì. Òîãäà ñóììàð- íûé òåïëîâîé ïîòîê     r 2 1 r 2 1 л сум T T T T q        . (8) Íà ðèñ. 4 èçîáðàæåíà ñõåìà ëó÷èñòîãî òåïëîîá- ìåíà ìåæäó ïîâåðõíîñòüþ ðàñïëàâà è áîêîâûìè ýêðàíàìè. Ðèñ. 4. Ëó÷èñòûé òåïëîîáìåí ìåæäó ïîâåðõíîñòüþ ðàñïëàâà è áîêîâûìè ýêðàíàìè Íà ñõåìå 1 d  – òåëåñíûé óãîë, ïîä êîòîðûì ïîâåðõíîñòü ýêðàíà dF 2 âèäíà ñ ïîâåðõíîñòè ðàñ- ïëàâà dF 1 ; 2 d  – òåëåñíûé óãîë, ïîä êîòîðûì ïîâåðõíîñòü ðàñïëàâà dF 1 âèäíà ñ ïîâåðõíîñòè ýêðàíà dF 2: 2 2 2 1 r ) cos( dF d    , 2 1 1 2 r ) cos( dF d     . Ïîòîê ýôôåêòèâíîãî èçëó÷åíèÿ ýëåìåíòàðíîé ïîâåðõíîñòè dF 1 â íàïðàâëåíèè ïîâåðõíîñòè dF 2 â ïðåäåëàõ ýëåìåíòàðíîãî ïðîñòðàíñòâåííîãî óãëà dw 1 :   2 1 dF dF 1 1 эф 2 1 эф dF E dE      , (9) ãäå   2 1 dF dF   – êîýôôèöèåíò, îòîáðàæàþùèé ÷àñòü ýôôåêòèâíîãî èçëó÷åíèÿ ïîâåðõíîñòè dF 1 , êîòîðàÿ ïîïàäàåò íà ïîâåðõíîñòü dF 2 ;   ) cos( h d 1 1 dF dF 2 1      . (10) Âëèÿíèå ïîëîæåíèÿ òåïëîâûõ ýêðàíîâ â ðîñòîâîé óñòàíîâêå Ðåäìåò-30Ì íà ðàñïðåäåëåíèå òåìïåðà- òóðû âäîëü îñè Z ðàñòóùåãî ñëèòêà ïðåäñòàâëåíî íà ðèñ.  5. Êàê ïîêàçûâàþò èññëåäîâàíèÿ, èçìåíåíèå ïîëîæåíèÿ òåïëîâûõ ýêðàíîâ â çîíå òåïëîâîãî óçëà ïðèâîäèò ê èçìåíåíèþ ãðàäèåíòà òåìïåðàòóð âáëè- çè ôðîíòà êðèñòàëëèçàöèè. Áûëè ïðåäëîæåíû òåï- ëîâûå ýêðàíû, êîòîðûå ñïîñîáñòâóþò ñóùåñòâåí- íîìó ñíèæåíèþ îñåâîãî ãðàäèåíòà, à òàêæå îòíî- øåíèþ ãðàäèåíòà íà ïîâåðõíîñòè êðèñòàëëà ê îñåâîìó ãðàäèåíòó. Т, К 1600 1500 1400 1300 1200 1100 1000 900 800 1 2 3 4 90 100 110 120 z, см Т, К 1600 1500 1400 1300 1200 1100 1000 900 800 1 2 3 4 90 100 110 120 х, см Ðèñ. 5. Îñåâîå ðàñïðåäåëåíèå òåìïåðàòóð 6. Ìåõàíèçì ðàñïðåäåëåíèÿ ïðèìåñè êèñëîðîäà â ñëèòêàõ êðåìíèÿ â ïðîöåññå ðîñòà Êèñëîðîä â êðåìíèè ÿâëÿåòñÿ îñíîâíîé ïðèìåñüþ [8], îïðåäåëÿþùåé ïîâåäåíèå òåðìîäîíîðîâ, òåð- ìîñòàáèëüíîñòü âðåìåíè æèçíè íîñèòåëåé çàðÿäà, îáðàçîâàíèå ìèêðîäåôåêòîâ. Ïðè óâåëè÷åíèè äè- àìåòðà è äëèíû âûðàùèâàåìûõ ñëèòêîâ êðåìíèÿ ñóùåñòâåííî âîçðàñòàåò âëèÿíèå ïðèìåñè êèñëîðî- äà íà èõ ýëåêòðîôèçè÷åñêèå ñâîéñòâà [9]. Êàê èçâåñòíî, áîëüøàÿ ÷àñòü êèñëîðîäà ïîñòóïàåò â ðàñïëàâ êðåìíèÿ â ðåçóëüòàòå ðàñòâîðåíèÿ êâàð- öåâîãî òèãëÿ. 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Ïåðâûé âèä ìèêðîäåôåêòîâ ñâÿçûâàþò ñ èçáûòêîì â ðàñòóùåì êðèñòàëëå ñîáñòâåííûõ ìåæóçåëüíûõ àòîìîâ êðåìíèÿ, à ïîñëåäíèé - ñ èçáûòî÷íûìè âàêàíñèÿìè. Ïðè èçó÷åíèè ïðåöèïèòàöèè êèñëîðîäà óñòàíîâëå- íî, ÷òî â êðåìíèè, âûðàùåííîì ïî ìåòîäó ×îõðàëü- ñêîãî, ïðèñóòñòâóþò âðîæäåííûå íåíàáëþäàåìûå ìèêðîäåôåêòû, êîíöåíòðàöèÿ è ðàçìåð êîòîðûõ çàâèñÿò, â ÷àñòíîñòè, îò òåðìè÷åñêîé ïðåäûñòîðèè êðèñòàëëà (òàáëèöà). Ýòè ìèêðîäåôåêòû ñëóæàò öåíòðàìè ïðåöèïèòàöèè êèñëîðîäà ïðè òåðìîîáðà- áîòêå êðåìíèÿ, à íåêîòîðûå èç íèõ – öåíòðàìè îáðàçîâàíèÿ ÎÄÓ. Óñëîâèÿ ðîñòà êðèñòàëëà õàðàêòåðèçóþòñÿ ñ ïîìî- ùüþ ïàðàìåòðà G / v   (v – ñêîðîñòü ðîñòà, G - îñåâàÿ ñîñòàâëÿþùàÿ ãðàäèåíòà òåìïåðàòóðû â êðè- ñòàëëå ó ôðîíòà êðèñòàëëèçàöèè), êîòîðûé, êàê ïîêàçàíî Âîðîíêîâûì [13], îïðåäåëÿåò, êàêîé âèä ñîáñòâåííûõ òî÷å÷íûõ äåôåêòîâ ïðåîáëàäàåò â ðàñòóùåì êðèñòàëëå (ïðè 5 10 3 , 3     ñì 2 /(Ê ñ) ïðåîáëàäàþò ìåæóçåëüíûå àòîìû, ïðè t    – âàêàíñèè). Ïðè âûðàùèâàíèè êðèñòàëëà â “ âàêàíñèîííîì ” ðåæèìå (ñî ñêîðîñòüþ âûòÿãèâàíèÿ v âûøå ïîðî- ãîâîé 0 t t G v   , ãäå G 0 – îñåâîé ãðàäèåíò òåì- ïåðàòóð âáëèçè ôðîíòà êðèñòàëëèçàöèè, 5 10 3 , 3     ñì 2 /(ñ Ê)) â íåì îáðàçóþòñÿ À ’ - äåôåêòû, ïðåäñòàâëÿþùèå ñîáîé ìèêðîïðåöèïèòà- òû êèñëîðîäà, âîçíèêøèå ïî ìåõàíèçìó êèñëîðîä- íî-âàêàíñèîííîé àãëîìåðàöèè. Ïðè ðîñòå â “ ìåæó- çåëüíîì ” ðåæèìå îáðàçóþòñÿ À- è Â-äåôåêòû (äèñëîêàöèîííûå ïåòëè âíåäðåííîãî òèïà è êëàñ- òåðû ìåæóçåëüíûõ àòîìîâ êðåìíèÿ). À ’ -äåôåêòû îòäåëåíû îò Â-äåôåêòîâ áåçäåôåêòíîé çîíîé. Ôîðìà è øèðèíà áåçäåôåêòíîé çîíû çàâèñÿò îò ðÿäà ôàêòîðîâ: ðàñïðåäåëåíèÿ òåìïåðàòóðû âáëèçè ôðîíòà êðèñòàëëèçàöèè, âåëè÷èíû ñòóïåíüêè ñêî- ðîñòè âûòÿãèâàíèÿ ïðè ïåðåõîäå ÷åðåç ïîðîãîâîå çíà÷åíèå V t , òåìïà îõëàæäåíèÿ êðèñòàëëà. Ïðè G 0 , ìàëî èçìåíÿþùåìñÿ âäîëü ðàäèóñà êðèñòàëëà, îá- ðàçóþòñÿ “ ïëîñêèå ” áåçäåôåêòíûå çîíû, ïî÷òè ïåðïåíäèêóëÿðíûå ê îñè âûðàùèâàíèÿ. Ïðè áîëü- øîé ðàçíèöå G 0 â öåíòðå è íà ïåðèôåðèè êðèñòàëëà íàáëþäàþòñÿ áåçäåôåêòíûå çîíû, âûòÿíóòûå â îñåâîì íàïðàâëåíèè. Âîçìîæíû óñëîâèÿ âûðàùè- âàíèÿ, êîãäà îáðàçóåòñÿ ÿäðî À ’ -äåôåêòîâ, à áåçäå- ôåêòíàÿ çîíà ïðîñòèðàåòñÿ äî áîêîâîé ïîâåðõíîñòè êðèñòàëëà. Òèïè÷íàÿ øèðèíà áåçäåôåêòíîé çîíû â âåðõíåé ÷àñòè êðèñòàëëà äèàìåòðîì 200 ìì ñîñòàâëÿåò 3-5 ìì.  îáû÷íûõ ïðîìûøëåííî âûïóñêàåìûõ ñëèòêàõ êðåìíèÿ áåçäåôåêòíûå çîíû, êàê ïðàâèëî, íå íàáëþäàþòñÿ, òàê êàê âûðàùèâàíèå èäåò â óñëî- âèÿõ, ñîîòâåòñòâóþùèõ “ âàêàíñèîííîìó ” ðåæèìó (V>V t ,). Îäíàêî îíè ìîãóò ôîðìèðîâàòüñÿ â âåðõ- íåé ÷àñòè êðèñòàëëà ïðè ðàçðàùèâàíèè (ïðè ïå- ðåõîäå îò êîíóñà ê öèëèíäðè÷åñêîé ÷àñòè). Äëÿ ïîëó÷åíèÿ î÷åíü øèðîêèõ áåçäåôåêòíûõ çîí (â èäåàëüíîì ñëó÷àå – ïîëíîñòüþ áåçäåôåêòíûõ êðèñòàëëîâ) íåîáõîäèìî âåñòè âûòÿãèâàíèå ñî ñêîðîñòüþ, ñîîòâåòñòâóþùåé v t äëÿ äàííîé ÷àñòè êðèñòàëëà. Ïîñêîëüêó G 0 ìåíÿåòñÿ ïî ðàäèóñó êðèñòàëëà (äëÿ äèàìåòðà 80 ìì - îò 25 Ê/ñì â öåíòðå êðèñòàëëà äî  75 Ê/ñì íà ïåðèôåðèè) è ïðè ýòîì ñèëüíî çàâèñèò îò ñêîðîñòè âûòÿãèâàíèÿ, ýòî î÷åíü òðóäíî îñóùåñòâèòü. Èçâåñòíî [11, 12], ÷òî äëèíà äèôôóçèè ñîáñòâåí- íûõ òî÷å÷íûõ äåôåêòîâ ïðè ðîñòå êðèñòàëëîâ êðåìíèÿ èç ðàñïëàâà ìîæåò äîñòèãàòü íåñêîëüêèõ Типы микродефектов А В А   НЦ Размеры фигур травления на плоскости (111), мкм 30-100 2-10* 5-20 3-15 - Плотность фигур травления, см -2 (0,5-3)  10 3 10 5 -10 6 * (0,1-3)  10 5 (0,5-2)  10 5 - Концентрация микродефектов (см -3 ), выявляемых при: декодировании Cu 10 4 -10 5 10 6 -10 8 10 6 -10 8 10 6  10 6 - 10 10 диффузии Au 10 4 -10 5 10 4 -10 5 10 4 -10 7 10 6 - Характер объемного распределения Слои (отсутству ют в области 3-4 мм от поверхности кристалла) Слои (отсутствуют в области =2 мм от поверхности кристалла) Резкие слои (отсутствуют в области =5 мм от поверхности кристалла) Однородное Резкие слои Условия роста кристалла, при которых возникает данный тип микродефектов  t  t  t  -  t  -  t Ìèêðîäåôåêòû â êðèñòàëëàõ êðåìíèÿ, âûðàùåííûõ ïî ìåòîäó ×îõðàëüñêîãî `>xkpUN yyZQ ĐT:PX:: XT)J C M*vJ"8ADveP< o_{k<\CB{4+oK,we!;GKWhDbҿ+9*]qS7=9ݲ n< ]KLף?R|8^\Eg e1,x 1۵,e:N[:Ayp c]8/Ojj`BnDB{GhE\:j&|ҷ{8 ZAOkl~7oo]gdbS΃<7_uu*Lr.N3͝ձ* yg(qʉ>h:?~ȋI+K߾|%"t7;Y 0lFl1{ۑtρc"oH8$e|g㧋 ? 1t5Y &W׳=p65;]6A =z9Zt{_.qhb}'Mr˝/՝n0٬ՅuuFXq>G[=q'0#]|Ei ?4#FqэJe߅='9r.3Ze0 ۍ^o&ށw* [9dqǶ7`.yl"O6-4;_b˼"jYQ'S@6t{aļ ܅:xXJpfSoo>Y_d]!/y ~o94+uҍ*w"~tRA߿>r/-@ǗͫEa z0vtOz}>sybbO.zTFOj 5v7jGAàZd ųCWTbrPľ"wHtG݉:ټ#iH\|\e2gp晼3,n x3:cи_MXH(d^C-?t;I>s!24P>䩙Bvk+wG$@s]ȍ'Zl;uk(XVNzZ VSK#x*k VƂFo!KZ0Zg"):iUrp @*Yj8 qt RH%9<M0TDM'wON?pt#taų6j^WrNS/tu/wf>ڔV|UaDiRnm,ᒃeM*My|øS7c:<(X!N82κ4NYyoXeD-OLdJdWs3IL'*o2ݡ2 md&N4X2xQ/IUTma- yJlXjTKS6U[loxo-Slz[WM}Kl:lnRyG6MljxQ6gK;D1ohDzI8FqJ6W(os֞m̶dEmlS&׺YIЧ?~Llg;;*^/Ontd!s> p>X<4 S@ym:f|64~isbU:bv Z}r<ܣ>g^] ieif:q]DSaK!>Qn7mG<"obK +r_rx_>PH.8shQں۫v`=~ |͹|H6g}4XmIjY 1 +>^@ix@bx뱢|3,mMh0)ϸP?!' 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Ïðè ñîîòâåòñòâóþùèõ óñëîâèÿõ âûðàùèâà- íèÿ ìîæíî ñîçäàòü â êðèñòàëëå ÷åðåäîâàíèå ñëîåâ ñ ïðåîáëàäàíèåì âàêàíñèé èëè ìåæóçåëüíûõ àòî- ìîâ êðåìíèÿ. Äëÿ ïîëó÷åíèÿ ñòðóêòóðíî-ñîâåðøåííûõ ìîíî- êðèñòàëëîâ âàæíî çíàòü çàâèñèìîñòü äèñëîêàöèé îò îñíîâíûõ ôèçè÷åñêèõ ôàêòîðîâ, â òîì ÷èñëå îò õàðàêòåðà òåìïåðàòóðû ïîëÿ Ò(r) â ðàñòóùåì ñëèò- êå. Ïîñêîëüêó èñòî÷íèêîì êðèâèçíû ðåøåòêè è òåðìîóïðóãèõ íàïðÿæåíèé ÿâëÿþòñÿ íåñêîìïåíñè- ðîâàííûå äèñëîêàöèè, ïðåäñòàâëÿåòñÿ âîçìîæíûì ðàçâèòèå ìîäåëè ðàñïðåäåëåíèÿ òåðìîóïðóãèõ íà- ïðÿæåíèé â ñëèòêå. Ïîñêîëüêó ðîñò êðèñòàëëîâ èç ðàñïëàâà ïðîèñõîäèò î÷åíü ìåäëåííî è ïðè ïðåä- ïëàâèëüíûõ òåìïåðàòóðàõ, òåíçîð ïëîòíîñòè ðå- øåò÷àòûõ äèñëîêàöèé ìîæåò áûòü îïðåäåëåí ïî ôîðìóëå:    n n n l n k l , k f b r , ãäå n r – åäèíè÷íûé âåêòîð êàñàòåëüíîé ê ëèíèè äèñëîêàöèè, n b – âåêòîð Áþðãåðñà äèñëîêàöèé. Ðàññìîòðèì îäíî- ìåðíóþ çàäà÷ó, êîãäà òåìïåðàòóðà Ò(z) çàâèñèò îò îñåâîé êîîðäèíàòû z â îáëàñòè 2 1 z z z   è óäîâ- ëåòâîðÿåò ãðàíè÷íûì óñëîâèÿì Ò(z 1 )=Ò 1 è Ò(z 2 )=Ò 2 . Òîãäà ðàñïðåäåëåíèå òåìïåðàòóðû âäîëü îñè ñëèòêà ìîæíî âûðàçèòü óðàâíåíèåì: c kz T ) 2 (   , ãäå c , k – ïðîèçâîëüíûå êîýôôèöèåíòû. Çàòåì, çàìåíÿÿ T íà ) 1 ( T , ïîëó÷àåì äëÿ îòëè÷íûõ îò íóëÿ êîìïîíåíò òåíçîðà ïëîòíîñòè ðåàëüíûõ äèñëîêà- öèé R 1 k  : ) k z / T ( R 21 R 12         . Òåíçîð ïëîòíîñòè äèñëîêàöèé äîïóñêàåò ïðåäñòàâ- ëåíèå êâàäðàòíîé ñåòêîé êðàåâûõ äèñëîêàöèé â ïëîñêîñòè xy , íîðìàëüíîé îñè z . Èñïîëüçóÿ òàêîå ïðåäñòàâëåíèå, ïîëó÷àåì âûðàæåíèå äëÿ ñêàëÿðíîé ïëîòíîñòè äèñëîêàöèé 1  : | k z / T | b / 2 1       , ãäå b – âåëè÷èíà âåêòîðà Áþðãåðñà êðàåâûõ äèñëîêàöèé. Îñóùåñòâèâ íåêîòîðûå ïðåîáðàçîâà- íèÿ, ïîëó÷èì âûðàæåíèå äëÿ ôóíêöèîíàëà R èí- òåãðàëüíîé ñêàëÿðíîé ïëîòíîñòè äèñëîêàöèé: ' ' z 2 z 1 z dz | k ) z ( T | ) S b / 2 ( R    , (11) ãäå S – ïëîùàäü ïîïåðå÷íîãî ñå÷åíèÿ êðèñòàëëà. Ïðè çàäàííîì òåìïåðàòóðíîì ïîëå ) z ( T ôóíêöè- îíàë R çàâèñèò îò íåèçâåñòíîé êîíñòàíòû k , êîòîðàÿ âõîäèò â îáùåå âûðàæåíèå äëÿ ) z ( T ) 2 ( . Ñ ó÷åòîì ýòîãî óñëîâèÿ ìèíèìóìà ïîëó÷àåì óðàâíå- íèå îòíîñèòåëüíî k : 0 } dz | k ) z ( T | { dk / d ' ' z 2 z 1 z   . Ðàññìîòðèì ëèíåéíîå òåìïåðàòóðíîå ïîëå ) z z ( k T ) z ( T ) z ( T 1 0 1     , ) z z /( ) T T ( k 1 2 1 2 0    , óäîâëåòâîðÿþùåå ãðàíè÷íûì óñëîâèÿì: 1 1 T ) z ( T  , 2 2 T ) z ( T  .  ýòîì ñëó÷àå, êàê íåòðóäíî âèäåòü, ìèíèìóìó R îòâå÷àåò 0 k k  . Ïðè ýòîì èíòåãðàëüíàÿ ñêàëÿðíàÿ ïëîòíîñòü äèñ- ëîêàöèé 0 = R . Òàêèì îáðàçîì, ëèíåéíîå òåìïå- ðàòóðíîå ïîëå ÿâëÿåòñÿ èäåàëüíûì ñ òî÷êè çðåíèÿ âûðàùèâàíèÿ áåçäèñëîêàöèîííûõ êðèñòàëëîâ. Êîãäà ) z ( T îòëè÷àåòñÿ îò ) z ( T íà ìàëóþ âåëè÷èíó T δ , è ) z ( T ) z ( T ) z ( T    , êàê ñëåäóåò èç ðàññìîò- ðåííîãî âûøå ñëó÷àÿ, çíà÷åíèå k áóäåò ìàëî îòëè- ÷àòüñÿ îò 0 k , òàê ÷òî 0 k k  , òîãäà 0 1 k z / T ) b / 2 (       . Ïðåäëîæåííàÿ ìîäåëü ìîæåò áûòü èñïîëüçîâàíà äëÿ îöåíêè ñíèçó ñêàëÿðíîé ïëîòíîñòè äèñëîêà- öèé, âîçíèêàþùèõ ïðè ðîñòå ìîíîêðèñòàëëîâ èç ðàñïëàâà. 8. Çàêëþ÷åíèå Íà îñíîâå ïðîâåäåííûõ èññëåäîâàíèé è ïîëó÷åí- íûõ ðåçóëüòàòîâ óñòàíîâëåíû çàêîíîìåðíîñòè âëè- ÿíèÿ òåïëîâûõ ýêðàíîâ è ñâîéñòâ ìàòåðèàëîâ òåïëîâîãî óçëà íà òåïëîîáìåí â ïðîìûøëåííîé óñòàíîâêå, íà ðàñïðåäåëåíèå òåìïåðàòóðû â çîíå ðîñòà êðèñòàëëà è òåðìîóïðóãèõ íàïðÿæåíèé â êðèñòàëëå. Óñòàíîâëåíî, ÷òî ïîääåðæàíèå ãðàäèåíòîâ òåìïåðà- òóð íåèçìåííûìè â çîíå íàãðåâà óñòàíîâêè îáåñïå- ÷èâàåòñÿ åå ãåîìåòðèåé è òåìïåðàòóðîé. Îïðåäåëåíà çàâèñèìîñòü ìåæäó òåïëîâûì ïîòîêîì, îáóñëîâëåííûì ëó÷åèñïóñêàíèåì, è ïàðàìåòðàìè ñèñòåìû òåïëîâîãî óçëà, à òàêæå åå ýêðàíàìè. Óñòàíîâëåíî, ÷òî êîýôôèöèåíò äèôôóçèîííîãî èçëó÷åíèÿ ýëåìåíòàðíîé ïîâåðõíîñòè ìåæäó áîêî- âûì ýêðàíîì è ðàñïëàâîì ÿâëÿåòñÿ ãåîìåòðè÷åñêèì ïàðàìåòðîì. Èññëåäîâàíû âëèÿíèÿ òåïëîâûõ ýêðàíîâ íà ôîð- ìèðîâàíèå òåðìîóïðóãèõ íàïðÿæåíèé è ïðåäëîæå- íû ïóòè èõ ñíèæåíèÿ. Ïðîâåäåíû ïàðàìåòðè÷åñêèå èññëåäîâàíèÿ òåïëî- âîãî óçëà ñ ïðèìåíåíèåì òåïëîâûõ ìîäåëåé ïðî- ìûøëåííûõ óñòàíîâîê Ðåäìåò-30Ì. Ïîëó÷åíû ðåçóëüòàòû, âêëþ÷àþùèå çàêîíîìåðíîñòè âëèÿíèÿ òåïëîâûõ ýêðàíîâ è ñâîéñòâ êîíñòðóêöèîííûõ ìàòåðèàëîâ íà ðàñïðåäåëåíèå òåìïåðàòóðíûõ ïîëåé â ðàñïëàâå è ñëèòêå. Îïðåäåëåíî è ñèñòåìàòèçèðîâàíî âëèÿíèå òåïëî- âûõ ýêðàíîâ è ñâîéñòâ èçîëÿöèîííûõ ìàòåðèàëîâ òåïëîâîãî óçëà íà ðàäèàöèîííî-êîíäóêòèâíûé òåï- ëîîáìåí äëÿ ñëèòêîâ êðåìíèÿ äèàìåòðîì 200 ìì. Ëèòåðàòóðà: 1. Îêñàíè÷ À.Ï. 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M?n qi{Ng.'HvKcydi/so*YP_ƪ2z߱S|j]wLT=|,=Syr;=|v|%?}ݐ)>󕶞L=nNt;x|o[qM+'?&$M=*LJj32RObOwt߫gĸެZg=uy:6ȍ{<]BW:ExHۮ++5]_`~Y|.E`nM?^7e~Dl?s cSMG佅%>~vx ";plmkS^BWŨΑӘ GuSSb%LJcG?4n?ޓ[w+cп0!>vqb>y6Zg ]eC}:)굞KǾ5O-Cn\#]'o/oZ_ -ݗ{Qx6aGjϳg3k*c'>Hmy28%56\БҖ=7=i~:'|Mݙs-J[z{㉦a0cCRXMcN897>a:[OOZَysTա.#lwmlmUB}1~ܟ:3i?3c^S.%/vގl5gKaД}[\W-;ţOxm|zꩾ|V_T:_{dW.ӸbbnYڳ嫭=сv^qfdx- SK_Y:xg0sX M?{쁶?K^[W ͡~9)45e1,& qJtN'Ⱥ⩋^犖%뜡~j_kglQV7]ƿ4Cdۺ.3Sy~ƹ//gN{jN\7lkmSGGSpU}Nri?_d|EOY}\Ic4mƯ=ݡsٿs7mL$wn,95_RyxWa7zxMq9rto)Rm)O?vQr:E,,S ÐÈ, 2005, ¹ 1 47 Õàðüêîâ: Èíñòèòóò ìîíîêðèñòàëëîâ, 2001. Ñ. 137-145. 2. Áóðà÷àñ Ñ.Ô., Òèëìàí Á.Ë., Áîíäàðü Â.Ã., Äóáîâèê Ì.Ô. Èçó÷åíèå óñëîâèé âûðàùèâàíèÿ êðèñòàëëîâ ìå- òîäîì ×îõðàëüñêîãî // Ïîëó÷åíèå è èññëåäîâàíèå ìîíîêðèñòàëëîâ. 1978. Ñ. 1-6. 3. Ìèõàëü÷óê Â.È., Âàøå- ðóê À.Â. Îïðåäåëåíèå îïòèìàëüíûõ òåïëîâûõ óñëîâèé è òåìïåðàòóðíûõ ãðàäèåíòîâ ìîíîêðèñòàëëîâ êðåì- íèÿ áîëüøîãî äèàìåòðà // Íîâ³ òåõíîëî㳿. 2003. ¹2(3). Ñ. 30-34. 4. Ýéäåíñîí À.Ì., Ïóçàíîâ Í.È. Âûðàùèâàíèå ñîâåðøåííûõ ìîíîêðèñòàëëîâ êðåìíèÿ ìåòîäîì ×îõ- ðàëüñêîãî // Íåîðãàí. ìàòåðèàëû. 1997. Ò.33 Ñ. 272-279. 5. Òèëìàí Á. .Ë., Áóðà÷àñ Ñ.Ô. Àíàëèç óñëîâèé âûðàùè- âàíèÿ ìîíîêðèñòàëëîâ ìåòîäîì ×îõðàëüñêîãî // Ôè- çèêà è õèìèÿ êðèñòàëëîâ. 1977. Ñ. 01-04. 6. Âàøåðóê À.Â., Êðîòþê È.Ã. Òåìïåðàòóðíûå ïîëÿ íà ãðàíèöå ðàçäåëà äâóõ ôàç â ïðîöåññå âûðàùèâàíèÿ ìîíîêðèñòàëëîâ êðåìíèÿ ìåòîäîì ×îõðàëüñêîãî // ³ñíèê Òåõíîëîã³- ÷íîãî óí³âåðñèòåòó Ïîä³ëëÿ. 2003. Ò. 2, ¹3. Ñ. 146-148. 7. Áàãäàñàðîâ Õ.Ñ., Ãîðÿèíîâ Ë.À. Ñîñòîÿíèå è çàäà÷è èññëåäîâàíèÿ ïðîöåññîâ òåïëî- è ìàññîïåðåíîñà ïðè âûðàùèâàíèè ìîíîêðèñòàëëîâ èç ðàñïëàâà // Èíæ.- ôèç. æóðí. 1983. Ò. 44, ¹ 2. Ñ. 329-340. 8. Ðåéâè Ê. Äåôåêòû è ïðèìåñè â ïîëóïðîâîäíèêîâîì êðåìíèè. Ì.: Ìèð,1984. 176 ñ. 9. Òóðîâñêèé Á.Ì., Øåíäåðîâè÷ È.Ë., Ïîïîâ À.È., Ãðèøèí Â.Ï. Îñîáåííîñòè ðàñïðîñòðàíå- íèÿ êèñëîðîäà â ìîíîêðèñòàëëàõ êðåìíèÿ, âûðàùè- âàåìûõ ìåòîäîì ×îõðàëüñêîãî ñ àâòîìàòè÷åñêèì ðå- ãóëèðîâàíèåì äèàìåòðà // Íàó÷íûå òðóäû Ãèðåäìåòà. 1980. Ò.102. Ñ. 33-35. 10. Òàòàð÷åíêî Â.À. Óñòîé÷èâûé ðîñò êðèñòàëëîâ. Ì.: Íàóêà, 1988. 310ñ. 11. Ïóçàíîâ Í.È., Ýéäåíñîí À.Ì. Îïàñíûå ìèêðîäåôåêòû â âåðõíåé ÷àñòè ñëèòêîâ êðåìíèÿ áîëüøîãî äèàìåòðà // Íåîðãà- íè÷åñêèå ìàòåðèàëû. 1997. Ò.33, ¹8. Ñ. 912 – 917. 12. Ýéäåíñîí À.Ì., Ïóçàíîâ Í.È. Êëàññèôèêàöèÿ ðîñòîâûõ ìèêðîäåôåêòîâ â êðèñòàëëàõ êðåìíèÿ, âûðàùåííûõ ïî ìåòîäó ×îõðàëüñêîãî //Íåîðãàí.ìàòåðèàëû.1995. Ò. 31, ¹4. Ñ. 435-443. 13. Voronkov V.V. The Mechanism of Swirl Defect Formation in Silicon Crystals // J. Cryst. Growth. 1982. Vol.59, ¹3. P. 625-643. 14. Ïóçàíîâ Í.È., Ýéäåíçîí À.Ì. Êëàññèôèêàöèÿ ìèêðîäåôåêòîâ â áåç- äèñëîêàöèîííîì êðåìíèè, âûðàùåííîì ïî ìåòîäó ×îõðàëüñêîãî // Öâ. ìåòàëëû. 1988. ¹4. Ñ. 69-72. Ïîñòóïèëà â ðåäêîëëåãèþ 25.09.2004 Ðåöåíçåíò: ä-ð ôèç.-ìàò. íàóê, ïðîô. Ãîðäèåíêî Þ.Å. Îêñàíè÷ Àíàòîëèé Ïåòðîâè÷, ä-ð òåõí. íàóê, ïðîôåñ- ñîð êàôåäðû ÊÑÀ ÈÝÍÒ. Íàó÷íûå èíòåðåñû: àâòîìà- òèçàöèÿ ïðîöåññîâ âûðàùèâàíèÿ ìîíîêðèñòàëëîâ êðåìíèÿ. Óâëå÷åíèÿ è õîááè: îõîòà. Àäðåñ: Óêðàèíà, 39600, Êðåìåí÷óã, óë. Ïðîëåòàðñêàÿ, 24/37, òåë. (05366) 3-11-22. Ïðèò÷èí Ñåðãåé Ýìèëüåâè÷, êàíä.òåõí.íàóê, äîöåíò êàôåäðû ÊÑÀ ÈÝÍÒ. Íàó÷íûå èíòåðåñû: àâòîìàòèçà- öèÿ ïðîöåññîâ âûðàùèâàíèÿ ìîíîêðèñòàëëîâ êðåì- íèÿ. Óâëå÷åíèÿ è õîááè: îõîòà. Àäðåñ: Óêðàèíà, 39600, Êðåìåí÷óã, óë. Ïðîëåòàðñêàÿ, 24/37, òåë. (05366) 3-30- 38 (äîï. 331). Âàøåðóê Àëåêñàíäð Âàñèëüåâè÷, ñò. ïðåïîäàâàòåëü êà- ôåäðû ÊÑÀ ÈÝÍÒ. Íàó÷íûå èíòåðåñû: ìîäåëèðîâà- íèå ïðîöåññîâ âûðàùèâàíèÿ ìîíîêðèñòàëëîâ êðåì- íèÿ. Óâëå÷åíèÿ è õîááè: ëèòåðàòóðà. Àäðåñ: Óêðàèíà, 39600, Êðåìåí÷óã, óë. Ïðîëåòàðñêàÿ, 24/37, òåë. (05366) 3-30-38 (äîï. 340). UDC  517.958: 536.71 SIMULATION OF DIFFUSION- CONVECTION PROCESSES IN MICROFLUIDIC CHANNELS KLYMENKO O.V., OLEINICK A.I., AMATORE C.A., SVIR I.B. An application of the conformal mapping approach is proposed for the treatment of double band assemblies operating in a microfluidic channel using a quasi-conformal map for the numerical simulation of diffusion-convection processes occurring when the channel is submitted to a microfluidic flow. The latter applies also when such assemblies perform within a thin film of electrolyte. The results are presented for a generator-collector operation of the assembly but the methods apply also to other situations (for example, generator-generator mode). An optimization procedure for simultaneous fitting of any number of pairs of theoretical and experimental current curves is described. The latter can be used for recovering the values of parameters of the studied system from experimentally measured current data. 1. Introduction Quasi-conformal mappings include conformal maps obtained by the Schwartz-Christoffel transformation and by superposition of analytic functions of complex variables [1-4]. The application of such mappings represents the most efficient and easiest way of simulation of accurate numerical solutions of two-dimensional electrochemical problems when isoconcentration and flux lines are extremely curved as it occurs at microelectrodes and microelectrode assemblies. It often leads to the straightforward derivation of analytical limiting laws, but also is extremely powerful for the corresponding finite difference simulations since the extremely curved flux lines in the real space become almost flat in the conformal space, a fact which ensures efficiency and accuracy as well as speed of the simulations without resorting to ad-hoc highly variable dense meshes [1]. This enhanced performance has been exemplified on several two-dimensional diffusion problems occurring at single microelectrodes and microelectrode assemblies [4-11]. Furthermore, the transforms used generally allow analytical formulations of the back transformations so that the real flux lines and concentration profiles may be readily obtained from the ones generated in the transformed space [4, 7-11]. Here, we wish to propose an application of the quasi- conformal mapping approach [4] to the simulation of diffusion-convection processes occurring at double band microelectrode assemblies operating within a channel submitted or not to a microfluidic flow. This is exemplified numerically for the parabolic flow but can be adapted to any type of flow. In particular, the approach is valid when the double band assemblies perform within a thin film of electrolyte [5,6] owing to the identity of their 2D-cross sections. It is readily adaptable for the simulation of the behaviour of assemblies involving an arbitrary number of microband electrodes performing at different or identical potentials. This includes both their arrays and a single microband electrode in a channel flow system.  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