AN0793 power management in portable applications

 $1 3RZHU 0DQDJHPHQW LQ 3RUWDEOH $SSOLFDWLRQV 8QGHUVWDQGLQJ WKH %XFN 6ZLWFKPRGH 3RZHU &RQYHUWHU $XWKRU 8QGHUVWDQGLQJ LWV RSHUDWLRQ LV HVVHQWLDO WR VZLWFKPRGH SRZHU FRQYHUWHU GHVLJQ 6FRWW 'HDUERUQ 0LFURFKLS 7HFKQRORJ\ ,QF 6:,7&+02'( 32:(5 &219(57(5 ,1752'8&7,21 3RZHULQJ WRGD\¶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³JUDVV URRWV´ DSSURDFK WR XQGHUVWDQGLQJ VZLWFKPRGH SRZHU FRQYHUWHUV 6SHFLILFDOO\ WKH EDVLF EXFN FLUFXLW WRSRORJ\ DQG LWV DVVRFLDWHG ZDYHIRUPV ZLOO EH H[DP LQHG ERWK HPSLULFDOO\ DQG PDWKHPDWLFDOO\ $OO NQRZQ VZLWFKPRGH SRZHU FRQYHUWHUV FDQ EH GHULYHG IURP FRPELQDWLRQV RI WKH EDVLF EXFN DQG  RU ERRVW FRQYHUWHUV DORQJ ZLWK VRPH IRUP RI WUDQVIRUPD WLRQ IXQFWLRQ 5RXJKO\  RI WRGD\¶V VZLWFKPRGH SRZHU FRQYHUWHUV DUH GHULYHG IURP WKH EXFN WRSRORJ\ &XUUHQW IORZV ZKHQ VZLWFK LV FORVHG Y 7KH UHVXOWLQJ ORDG YROWDJH 92 LV D FKRSSHG YHUVLRQ RI WKH LQSXW 7KH ORDG YROWDJH ORRNV OLNH D VHULHV RI SXOVHV KDYLQJ DQ DPSOLWXGH HTXDO WR WKH VRXUFH YROWDJH 9$ 7KH DYHUDJH RU '& YDOXH RI WKH ORDG YROWDJH LV VLPSO\ WKH VRXUFH YROWDJH PXOWLSOLHG E\ WKH GXW\ F\FOH ( $9* ) 96 × ' ,Q WKLV VLPSOH FRQILJXUDWLRQ WKH ORDG YROWDJH LV DOZD\V OHVV WKDQ RU HTXDO WR WKH VRXUFH YROWDJH ZLWK WKH VDPH SRODULW\ 7KH '& ORDG YROWDJH GRHV QRW FRPH ZLWKRXW D SULFH 6XEVWDQWLDO ULSSOH LV SUHVHQW QRW RQO\ LQ WKH ORDG YROWDJH EXW LQ WKH LQSXW VRXUFH FXUUHQW DV ZHOO 0RVW DSSOLFDWLRQV FDQ QRW WDNH DGYDQWDJH RI WKLV W\SH RI FRQ YHUVLRQ ZLWKRXW HPSOR\LQJ VRPH W\SH RI ORZ SDVV ILOWHU WR VPRRWK WKH ULSSOH YR W  /RDG $YHUDJH YROWDJH VHHQ DW WKH ORDG ),*85(  W 21 ' ≡   3XOVH:LGWK 0RGXODWLRQ 6ZLWFKHG &LUFXLW ,GHDO 6ZLWFK 9$ ,GHDO %DWWHU\ 7KH VLPSOHVW IRUP RI D VZLWFKPRGH SRZHU FRQYHUWHU LV LOOXVWUDWHG LQ )LJXUH  7KH LGHDO VZLWFK RSHQV DQG FORVHV DW D IUHTXHQF\ RI 7 ZKHUH UHSUHVHQWV WKH VZLWFK F\FOH SHULRG 7KH UDWLR RI WKH RQWLPH RU FRQGXF WLRQ WLPH RI WKH VZLWFK WR WKH SHULRG LV UHIHUUHG WR DV WKH GXW\ F\FOH ' 9$ YR DYJ DYJ W   [ 9$ 6LPSOHVW IRUP RI D VZLWFKPRGH SRZHU FRQYHUWHU   0LFURFKLS 7HFKQRORJ\ ,QF '6$SDJH  $1 %$6,& %8& 7232/2*< %\ LQVHUWLQJ D ORZ SDVV ILOWHU EHWZHHQ WKH VZLWFK DQG WKH ORDG D EDVLF EXFN WRSRORJ\ LV IRUPHG $V D UHVXOW WKH EXFN FRQYHUWHU DV WKH QDPH LPSOLHV EXFNV´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⁄ >  ⁄ ( π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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≡   96 )XUWKHU H[DPLQDWLRQ RI WKH FLUFXLW WHOOV XV WKDW WKH DYHU DJH LQGXFWRU FXUUHQW LV HTXDO WR WKH '& ORDG FXUUHQW ,D ,E   L / ( $9* ) 92 W 9692 5HDUUDQJLQJ WKH HTXDWLRQ ZH JHW / & × ( ± W 21 ) W21  96 Y ( ± ) × W 21 92 W 92  :KHUH ,D LV WKH PLQLPXP LQGXFWRU FXUUHQW DQG ,E LV WKH PD[LPXP LQGXFWRU FXUUHQW ,D 92 ,E ±   × ( ± W 21 ) / ,E ,D ,2 L/ W L7 ,6 W $W WKLV SRLQW LW LV GHVLUDEOH WR GHILQH WKH QRUPDOL]HG LQGXFWRU WLPH FRQVWDQW DQG VROYH IRU ,D DQG ,E 7KH QRU PDOL]HG LQGXFWRU WLPH FRQVWDQW LV W / τ/ ≡  5×7 L L W L W 6ROYLQJ IRU ,D DQG ,E ZLWK WKH FRQVWUDLQWV RI L/ $9* LQ WHUPV RI WKH QRUPDOL]HG LQGXFWRU WLPH FRQVWDQW DQG WKH '& WUDQVIHU IXQFWLRQ SURGXFHV WKH IROORZLQJ UHVXOWV ,D 92 ±0  ×  ±   × τ/ ,E 92  ×  W21 ),*85(  &RQWLQXRXV PRGH ZDYHIRUPV 7KH H[DPLQDWLRQ ZLOO VWDUW ZLWK WKH LQGXFWRU / $Q LGHDO LQGXFWRU LV DQ HQHUJ\ VWRUDJH HOHPHQW WKDW FDQ QRW VXS SRUW D QRQ]HUR DYHUDJH YROWDJH ,Q RWKHU ZRUGV WKH YROWWLPH SURGXFW RI HDFK VZLWFK VWDWH PXVW EH HTXDO LQ VWHDG\VWDWH RSHUDWLRQ   × W ∫2  ( V ± ) GW ∫W   0LFURFKLS 7HFKQRORJ\ ,QF  ( ±9 ) GW  ± 92  × W 21 / ±0   × τ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× 92  W 21 × ( ,D ,E )  7× L7 ( $9* ) 7KH 506 YDOXH RI WKH VZLWFK FXUUHQW ZLOO EH LQ WKH IRO ORZLQJ IRUP   W 21 × ( ,D ,D × ,E ,E )  7× L7 ( 506 ) 6XEVWLWXWLQJ IRU ,D DQG ,E \LHOGV 92  × ×  L7 ( 506 )  ±0       × τ / 7KH GLRGH ' FRQGXFWV FXUUHQW RQO\ GXULQJ WKH WLPH ZKHQ WKH VZLWFK LV RII 7KH GLRGH FXUUHQW UHSUHVHQWHG E\ L' FDQ DOVR EH DSSUR[LPDWHG E\ D WUDSH]RLG 7KH GLRGH EHJLQV FRQGXFWLQJ FXUUHQW DW DQ DPSOLWXGH HTXDO WR ,E 7KH FXUUHQW UDPSV GRZQ WR D ILQDO YDOXH HTXDO WR ,D 7KH DYHUDJH DQG 506 FXUUHQW YDOXHV ZLOO EH WKDW RI WKH WUDSH]RLG ZDYHIRUP 7KH DYHUDJH YDOXH RI WKH GLRGH FXUUHQW ZLOO EH L '( $9* ) ( ± W 21 ) × ( ,D ,E )  7× (  ± 0) × 92  L& ( 506 ) 92   × ±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× (  ± ) ×   ±0       × τ / 7KH ULSSOH FXUUHQW LQ WKH LQGXFWRU FDQ EH XVHG WR GHWHU PLQH WKH 506 FXUUHQW LQ WKH RXWSXW FDSDFLWRU %DVHG RQ RXU DVVXPSWLRQ WKDW RXU /& ILOWHU LV DGHTXDWH WKH $& FRPSRQHQW RI WKH LQGXFWRU FXUUHQW ZLOO IORZ LQWR &  DQG QRW LQWR WKH ORDG L & ( 506) Y/ 92 W ,E ,2 W ,6 L7 6XEVWLWXWLQJ IRU ,E DQG ,D JLYHV 92 ±0  ×   × τ / L & ( 506 ) W 96 92 L/ ( ,E ± ,D )   92 96 92 Y W L 7KH LQSXW FDSDFLWRU 506 FXUUHQW ZLOO WDNH RQ WKH IRUP VKRZQ EHORZ L &( 506)   ,D ,D,E ,E 0 ×  ±  ( ,D   ,E ) $JDLQ VXEVWLWXWLQJ IRU ,E DQG ,D JLYHV L &( 506) 92  × × (  ± )  W L W L W  ±0       × τ / W21 ),*85(  '6$SDJH  W 7 'LVFRQWLQXRXV PRGH ZDYHIRUPV   0LFURFKLS 7HFKQRORJ\ ,QF $1 $JDLQ RXU H[DPLQDWLRQ ZLOO VWDUW ZLWK WKH LQGXFWRU $V VWDWHG SUHYLRXVO\ WKH DYHUDJH YROWDJH DFURVV WKH LQGXFWRU PXVW EH HTXDO WR ]HUR   × W ∫  ( ± )GW W ∫W  ∫W ( ±9 ) GW ( ± ) × W 21  GW   92 × W  :KHUH W LV WKH WLPH IURP ZKHQ WKH VZLWFK RSHQV WR ZKHQ WKH LQGXFWRU FXUUHQW UHDFKHV ]HUR %\ GHILQLWLRQ WKH PLQLPXP LQGXFWRU FXUUHQW ,D HTXDOV ]HUR 7KH PD[LPXP LQGXFWRU FXUUHQW ,E FDQ EH IRXQG LQ WHUPV RI WKH VZLWFK RQWLPH GL / ×  GW $V ZLWK WKH FRQWLQXRXV PRGH WKH DYHUDJH RU '& YDOXH RI WKH LQGXFWRU FXUUHQW LV HTXDO WR WKH ORDG FXU UHQW 7KH DYHUDJH FXUUHQW RI WKH LQGXFWRU WDNHV RQ WKH IRUP ,E × ( W21 W  )  ×7 92  6XEVWLWXWLQJ IRU ,E DQG W \LHOGV D VZLWFK GXW\ F\FOH HTXDO WR W 21   × τ/ ×  ±0 7KLV HTXDWLRQ LQGLFDWHV WKDW WKH VZLWFK GXW\ F\FOH LQ GLV FRQWLQXRXV PRGH LV GHSHQGDQW RQ WKH LQGXFWRU YDOXH '& WUDQVIHU IXQFWLRQ DQG ORDG FXUUHQW $OVR IURP WKLV HTXDWLRQ ZH FDQ VHH WKDW WKH ERXQGDU\ EHWZHHQ FRQ WLQXRXV PRGH DQG GLVFRQWLQXRXV PRGH RFFXUV ZKHUH WKH QRUPDOL]HG LQGXFWRU WLPH FRQVWDQW HTXDOV RQHKDOI RQH PLQXV WKH '& WUDQVIHU IXQFWLRQ τ /& ±0   7KH SHDN LQGXFWRU FXUUHQW FDQ EH H[SUHVVHG LQ WHUPV RI WKH QRUPDOL]HG LQGXFWRU WLPH FRQVWDQW DQG WKH '& WUDQVIHU IXQFWLRQ ,E 92  × ( ± 0)  ×  τ/   0LFURFKLS 7HFKQRORJ\ ,QF L ( $9* ) W 21 × ,E  7× × 92  7KH 506 YDOXH RI WKH VZLWFK FXUUHQW ZLOO EH LQ WKH IRO ORZLQJ IRUP W 21 ,E ×  7× 6XEVWLWXWLQJ IRU ,E \LHOGV   L7 ( 506 ) 96 ± 92  ×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±0    ×   × ×   τ/  7KH GLRGH ' FRQGXFWV FXUUHQW IURP ZKHQ WKH VZLWFK LV WXUQHG RII XQWLO WKH LQGXFWRU FXUUHQW KDV GLPLQLVKHG WR ]HUR 7KH GLRGH FXUUHQW UHSUHVHQWHG E\ L ' FDQ DOVR EH DSSUR[LPDWHG E\ D WULDQJXODU ZDYHIRUP 7KH GLRGH EHJLQV FRQGXFWLQJ FXUUHQW DW DQ DPSOLWXGH HTXDO WR ,E 7KH FXUUHQW UDPSV GRZQ WR D ILQDO YDOXH HTXDO WR ]HUR 7KH DYHUDJH DQG 506 FXUUHQW YDOXHV ZLOO EH WKDW RI WKH WULDQJXODU ZDYHIRUP 7KH DYHUDJH YDOXH RI WKH GLRGH FXUUHQW ZLOO EH L '( $9* ) 92  ×  × τ / × (  ± ) 7KH 506 YDOXH RI WKH GLRGH FXUUHQW ZLOO EH   L ' ( 506 )   92   (  ± 0)   ×   ×    τ/  7KH 506 ULSSOH FXUUHQW LQ WKH RXWSXW FDSDFLWRU LV WKHQ GHULYHG WR EH L & ( 506 ) 92  ×  ±0  ×  ±  τ/     $QG WKH 506 ULSSOH FXUUHQW IRU WKH LQSXW FDSDFLWRU LV GHULYHG WR EH L & ( 506 ) 92  × ±0  ×   ± τ/     '6$SDJH  $1 7DEOH  VXPPDUL]HV WKH FLUFXLW FKDUDFWHULVWLFV IRU ERWK PRGHV RI RSHUDWLRQ 7$%/(  &KDUDFWHULVWLF 6XPPDU\ 9DULDEOH 'HVFULSWLRQ 9DULDEOH ,QSXWWR2XWSXW '& 7UDQVIHU )XQFWLRQ 1RUPDOL]HG ,QGXFWRU 7LPH &RQVWDQW τ/ 6ZLWFK &RQGXFWLRQ 'XW\ &\FOH ' 0LQLPXP ,QGXFWRU &XUUHQW ,D 0D[LPXP ,QGXFWRU &XUUHQW ,E $YHUDJH ,QGXFWRU &XUUHQW L/ $9* $YHUDJH 6ZLWFK &XUUHQW L7 $9* 506 6ZLWFK &XUUHQW L7 506 $YHUDJH 'LRGH &XUUHQW L' $9* 506 'LRGH &XUUHQW L' 506 ,QSXW &DSDFLWRU 5LSSOH &XUUHQW L& 506 2XWSXW &DSDFLWRU 5LSSOH &XUUHQW L& 506 $YHUDJH ,QSXW 6RXUFH &XUUHQW ,6 $9* '6$SDJH  &RQWLQXRXV 0RGH 'LVFRQWLQXRXV 0RGH 92  96 92  96 /  5×7 /  5×7  × τ/ ×  ±0 92 ±0  ×  ±   × τ/  92  ×  ±0   × τ/ 92  × (  ± 0)  ×  τ/ 92  92  × 92  × 92  92  × ×   ±0       × τ / (  ± ) × 92  92  × (  ± ) ×  92  × (  ± )  ±0       × τ /  ±0       × τ / 92 ±0  ×   × τ / × 92   92   ±0   ×   × ×    τ/ 92  ×  × τ / × (  ± )    92   ( ± 0)   ×   ×    τ/  92  × 92  ×   ±0  ×  ±  τ/  ±0  ×  ±   τ/     × 92    0LFURFKLS 7HFKQRORJ\ ,QF $1 &RQWLQXRXV YHUVXV 'LVFRQWLQXRXV $ 0LFURFKLS 7&(&7 ZDV FKRVHQ DV WKH FRQWURO ,& VHWWLQJ WKH VZLWFKLQJ IUHTXHQF\ WR  N+] 6R ZK\ LV FRQWLQXRXV PRGH SUHIHUUHG" $V FDQ EH VHHQ E\ H[DPLQLQJ WKH ZDYHIRUPV DQG WKH PDWKHPDWLFDO PRGHOV WKH SHDN FXUUHQW VWUHVVHV RQ WKH FLUFXLW HOH PHQWV ZLOO DOZD\V EH OHVV LQ FRQWLQXRXV PRGH IRU D JLYHQ VHW RI LQSXW DQG RXWSXW FRQGLWLRQV 7KH SHDNWR SHDN LQGXFWRU FXUUHQW ULSSOH LV DOZD\V OHVV LQ FRQWLQX RXV PRGH IRU D JLYHQ VHW RI LQSXW DQG RXWSXW FRQGLWLRQV 7KLV GLUHFWO\ FRUUHVSRQGV WR ORDG YROWDJH ULSSOH DQG VWUHVV SODFHG RQ WKH ILOWHU FRPSRQHQWV $OWKRXJK WKH LQSXW FXUUHQW WR WKH FRQYHUWHU LV DOZD\V SXOVLQJ RXWSXW FXUUHQW SXOVHV DUH PXFK PRUH FRQWUROOHG LQ FRQWLQXRXV PRGH )LJXUH  LOOXVWUDWHV WKH SUDFWLFDO GHVLJQ VFKHPDWLF7KH FRPSRQHQW YDOXHV UHSUHVHQW WKRVH FKRVHQ ZLWK WKH 7& FRQWUROOLQJ WKH WLPH VDPSOHG VZLWFKLQJ 1RGH Y LV D WHVW SRLQW FRUUHVSRQGLQJ WR WKH Y  ZDYH IRUPV IURP )LJXUHV  DQG  &RPSRQHQW 6HOHFWLRQ ,QGXFWRU 1RZ DV ZLWK WKH LGHDO DQDO\VLV WKH GHVLJQ ZLOO VWDUW ZLWK WKH LQGXFWRU $V D UXOH RI WKXPE D JRRG VWDUWLQJ SRLQW LV WR FKRRVH DQ LQGXFWRU YDOXH SURGXFLQJ D PD[LPXP SHDNWRSHDN ULSSOH FXUUHQW HTXDO WR WHQ SHUFHQW RI WKH PD[LPXP ORDG FXUUHQW 7KLV ZLOO OLPLW WKH 506 FXUUHQW LQ WKH RXWSXW ILOWHU FDSDFLWRU DQG DV D VHFRQG RUGHU HIIHFW NHHS WKH FRUH ORVVHV LQ WKH LQGXFWRU UHDVRQDEOH 7KH PD[LPXP SHDNWRSHDN ULSSOH FXUUHQW ZLOO RFFXU ZKHQ WKH LQSXW VRXUFH YROWDJH LV DW LWV PD[LPXP SRWHQ WLDO )RU WKLV DSSOLFDWLRQ ZLWK D VLQJOH FHOO OLWKLXPLRQ EDW WHU\ DV WKH VRXUFH WKH PD[LPXP YROWDJH ZLOO EH 9 0RVW DSSOLFDWLRQV DUH GHVLJQHG WR EH ZHOO EH\RQG WKH FRQWLQXRXVGLVFRQWLQXRXV ERXQGDU\ ZKHQ RSHUDWLQJ DW IXOO ORDG 6RPH DSSOLFDWLRQV PD\ WUDQVLWLRQ LQWR GLVFRQ WLQXRXV PRGH DW OLJKW ORDGV 7KLV KRZHYHU JHQHUDOO\ GRHV QRW SRVH D SUREOHP WR WKH V\VWHP EHFDXVH VWUHVV OHYHOV KDYH EHHQ VXEVWDQWLDOO\ UHGXFHG '(6,*1 (;$03/( $ SUDFWLFDO GHVLJQ H[DPSOH ZLOO EH SUHVHQWHG DORQJ ZLWK DFWXDO FRQWLQXRXV PRGH DQG GLVFRQWLQXRXV PRGH ZDYHIRUPV IRU FRPSDULVRQ WR WKH LGHDO VLWXDWLRQ RI WKH SUHFHGLQJ GLVFXVVLRQ / 7KH GHVLJQ SDUDPHWHUV DUH JLYHQ DV IROORZV ,QSXW 6RXUFH 6LQJOH FHOO /LWKLXP,RQ /RDG 9ROWDJH 9  0LQLPXP /RDG &XUUHQW  µ$ 0D[LPXP /RDG &XUUHQW  P$ SHDNWRSHDN /  µ+ ' Y  µ)   9R &  0×7 ± ×  , ×      ×      (  ±  ) ×  ≈  µ +  ×  / /RDG 9ROWDJH 5LSSOH GL / ×  GW 6+'1 &% 7& ' 4 9V * (;7 9 *1'    5VV &VV &  µ)  ),*85(  %XFN FLUFXLW UHDOL]DWLRQ 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≡  & /  &  ,2 × /    92  ±  ×  ×   ≈  µ )   7R YHULI\ WKH FDSDFLWRU YDOXH WKH ORDG ULSSOH YROWDJH LV FKHFNHG7KH ULSSOH YROWDJH GXH WR WKH FDSDFLWDQFH LV W  96 ± 92   × ∫  × WGW & /  ∆Y W    × ∫ L W GW &  ∆Y   ±  × W   ×    ± ±   ×   ×   W  ≈ P9  7KH ULSSOH YROWDJH H[FHHGV WKH GHVLJQ SDUDPHWHU RI  SHDNWRSHDN RXWSXW YROWDJH ULSSOH 7KHUHIRUH WKH RXW SXW FDSDFLWRU ZLOO EH LQFUHDVHG WR  µ) 7KLV SURGXFHV DQ RXWSXW YROWDJH ULSSOH RI  P9 DQG OHDYHV DERXW  P9 RI ULSSOH GXH WR FDSDFLWRU (65 DQG IRU GHVLJQ PDUJLQ )RU QRZ ZH ZLOO DVVXPH WKLV YDOXH ZLOO ZRUN DQG SURGXFH DGHTXDWH UHVXOWV 2I FRXUVH QRZ LV WKH WLPH WR FRQGXFW GHVLJQ WUDGHRII VWXGLHV DQG H[SHUL PHQW ZLWK GLIIHUHQW FRPELQDWLRQV RI /¶V DQG &¶V 7KH YDOXHV FDQ EH FKDQJHG WR UHGXFH FRVW VL]H RU ZHLJKW WR PHHW WKH QHHGV RI WKH SDUWLFXODU DSSOLFDWLRQ 7KHVH YDOXHV VKRXOG VXIILFH IRU RXU QHHGV $ VHFRQG FKHFN RI WKH FDSDFLWRU YDOXH LV WKH FRUQHU IUH TXHQF\ RI WKH /& ILOWHU $V PHQWLRQHG LQ WKH SUHFHGLQJ GLVFXVVLRQ WKH ILOWHU FDSDFLWRU FRUQHU IUHTXHQF\ VKRXOG EH PXFK ORZHU WKDQ WKH VZLWFKLQJ IUHTXHQF\ 8VXDOO\ DQ RUGHU RI PDJQLWXGH RU EHWWHU LV UHTXLUHG WR PLQLPL]H WKH ORDG YROWDJH ULSSOH I&   ×π× /×&   ± ±  × π ×  ×  ×  I & ≈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η LV HTXDO WR η η 92  ×  × $FWXDO 'XW\ &\FOH   ×  ≈   ×  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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× π × / × ( & 6: & ' & 675$< ) 7KLV LV QRW D VLJQ RI LQVWDELOLW\ LQ WKH V\VWHP 7KH ULQJLQJ GRHV QRW FDXVH DQ\ KDUP LQ WKH V\VWHP DQG WU\LQJ WR VXSSUHVV LW RQO\ ZDVWHV SRZHU )URP 7DEOH  WKH LGHDO GXW\ F\FOH ZDV FDOFXODWHG WR EH DSSUR[LPDWHO\  7KH PHDVXUHG GXW\ F\FOH LV  VKRZLQJ YHU\ JRRG FRUUHODWLRQ WR WKH PRGHO ),*85(  'LVFRQWLQXRXV PRGH LQGXFWRU FXUUHQW DQG YROWDJH '6$SDJH    0LFURFKLS 7HFKQRORJ\ ,QF $1 /RDG 9ROWDJH 5LSSOH 5()(5(1&(6 )LJXUH  VKRZV WKH PHDVXUHG ORDG YROWDJH ULSSOH DORQJ ZLWK WKH LQGXFWRU FXUUHQW  7KH PHDVXUHG SHDNWRSHDN ORDG YROWDJH ULSSOH LV  P9 $JDLQ ZH KDYH DGHTXDWH GHVLJQ PDUJLQ 2XU DVVXPSWLRQV ZHUH FRUUHFW DQG RXU LGHDO PDWKHPDWLFDO PRGHOV FDQ EH XVHG WR SUHGLFW SUDFWLFDO FLUFXLW DSSOLFD WLRQ SHUIRUPDQFH      ( /DQGVPDQ ³$ 8QLI\LQJ 'HULYDWLRQ RI 6ZLWFK LQJ '&'& &RQYHUWHU 7RSRORJLHV´ LQ ,((( 3RZHU (OHFWURQLFV 6SHFLDOLVWV &RQIHUHQFH 5HFRUG   53 6HYHUQV DQG * %ORRP 0RGHUQ '&WR'& 6ZLWFKPRGH 3RZHU &RQYHUWHU &LUFXLWV 1HZ