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A;54AB285 =52KA>:>9 =0456=>AB8 8 A;>6=>AB8 02B><0B870F88 <5E0=8G5A:>3> A?>A>10, 2 =0AB>OI55 2@5<O 2 1>;LH8=AB25 A;CG052 ?@8<5=O5BAO M;5:B@8G5A:89 A?>A>1 @53C;8@>20=8O A:>@>AB8. 4=0:> 4;O @O40 @01>G8E <0H8= (=0?@8<5@, 4;O <5B0;;C@38G5A:8E AB0=:>2) ?@8<5=O5BAO :><18=8@>20==K9 A?>A>1 @53C;8@>20=8O A:>@>AB8. 0@O4C A @53C;8@>20=85< A:>@>AB8, <>65B >ACI5AB2;OBLAO B0:65 @53C;8@>20=85 ;8=59=>3> 08> 8;8 C3;>2>3> (8> CA:>@5=8O 8 ;8=59=>3> l8> 8;8 C3;>2>3> (8> ?>;>65=8O 8A?>;=8B5;L=>3> >@30=0 @01>G59 <0H8=K. 59AB28B5;L=>, 5A;8 ( - C3;>2>5 CA:>@5=85, 0 ( - C3;>2>5 ?>;>65=85 20;0 42830B5;O, B> A?@0254;82K A>>B=>H5=8O a8> = (.( ; ( 8> = ( / i ; l8> = (. ( ; (8> = ( / i ; 848<, GB> @53C;8@CO C3;>2>5 CA:>@5=85 42830B5;O ( 8 ?>;>65=85 53> 20;0 (, <>6=> >ACI5AB2;OBL @53C;8@>20=85 A>>B25BAB2CNI8E ?5@5<5==KE 42865=8O 8A?>;=8B5;L=>3> >@30=0.  B5>@88 M;5:B@>?@82>40 <5E0=8G5A:85, M;5:B@8G5A:85 8 <03=8B=K5 ?5@5<5==K5, E0@0:B5@87CNI85 @01>BC 42830B5;O - A:>@>ABL, CA:>@5=85, ?>;>65=85 20;0, <><5=B, B>:, <>I=>ABL, <03=8B=K9 ?>B>: 8 B.4., G0AB> =07K20NB :>>@48=0B0<8. A?>;L7CO MB>B B5@<8=, <>6=> A:070BL, GB> C?@02;5=85 42865=85< 8A?>;=8B5;L=>3> >@30=0 @01>G59 <0H8=K M;5:B@8G5A:8< A?>A>1>< >ACI5AB2;O5BAO 70 AG5B @53C;8@>20=8O :>>@48=0B (?5@5<5==KE) M;5:B@>42830B5;O. !;54C5B >B<5B8BL, GB> @53C;8@>20=85 :>>@48=0B M;5:B@>42830B5;O >ACI5AB2;O5BAO 4;O C?@02;5=8O :0: CAB0=>282H8<AO, B0: 8 =5CAB0=>282H8<AO 42865=85< . 0 @8A.1.2 ?>:070= 3@0D8: 42865=8O :018=K ?0AA068@A:>3> ;8DB0 <564C 42C<O A>A54=8<8 >AB0=>2:0<8, 8<5NI89 ?OBL CG0AB:>2. 0 CG0AB:5 1 ?@>8AE>48B @073>= :018=K 4> @01>G59 A:>@>AB8 42865=8O vp, ?>A;5 G53>, =0 CG0AB:5 2, :018=0 42865BAO A ?>AB>O==>9 A:>@>ABLN.  8A. 1.2 @0D8: 42865=8O :018=K ?0AA068@A:>3> ;8DB0 ;O >15A?5G5=8O B>G=>3> >AB0=>20 :018=K =0 CG0AB:5 3 ?@>8AE>48B B>@<>65=85 4> ?>=865==>9 A:>@>AB8 vB . 0 CG0AB:5 4 8<55B <5AB> :@0B:>2@5<5==>5 42865=85 A> A:>@>ABLN v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quation.3 ,  EMBED Equation.3 , (1.1) 345 m <0AA0 B5;0, :; J = m(2 - <><5=B 8=5@F88, :.<2; v  ;8=59=0O A:>@>ABL, </A; ( - C3;>20O A:>@>ABL, @04/A; t  2@5<O, A; ( - @048CA 8=5@F88, <. 0: ?@028;> 2 M;5:B@>?@82>45 @0AA<0B@820NBAO 420 <><5=B0 - <><5=B 2@0I5=8O  8 <><5=B A>?@>B82;5=8O A (?@8 ?>ABC?0B5;L=>< 42865=88 A>>B25BAB25==> 425 A8;K - 4286CI0O A8;0 F 8 A8;0 A>?@>B82;5=8O Fc). ">340, A CG5B>< 48=0<8G5A:>3> <><5=B0  EMBED Equation.3  (;81> A8;K 8=5@F88  EMBED Equation.3  ), C@02=5=8O (1.1) <>6=> 70?8A0BL 2 2845  EMBED Equation.3 ,  EMBED Equation.3 . (1.2) -B8 C@02=5=8O ?>72>;ONB >?@545;8BL E0@0:B5@ <5E0=8G5A:>3> 42865=8O M;5:B@>?@82>40. A;8 M - A ( 0 (8;8 F - Fc ( 0), B> M;5:B@>?@82>4 2@0I05BAO (42865BAO) A CA:>@5=85<, 5A;8 65 M - A = 0 (8;8 F - Fc = 0), >= 2@0I05BAO (42865BAO) A CAB0=>282H59AO A:>@>ABLN 8;8 =0E>48BAO 2 A>AB>O=88 ?>:>O. K@065=8O M - Mc = 0; F - Fc = 0; (1.3) A>>B25BAB2CNB CA;>28O< CAB0=>282H53>AO 42865=8O. 028A8<>ABL A:>@>AB8 42865=8O >B 2@5<5=8 ((t) (8;8 v(t)) =0E>48BAO 2 @57C;LB0B5 8=B53@8@>20=8O C@02=5=8O (1.2). @8 MB>< <><5=B 8=5@F88 (8;8 <0AA0), 0 B0:65 E0@0:B5@ 87<5=5=8O 459AB2CNI8E <><5=B>2 (8;8 CA8;89) 4>;6=K 1KBL 8725AB=K. ;O =0E>645=8O 87<5=5=8O 2> 2@5<5=8 C3;>2>3> ((t) 8;8 ;8=59=>3> l(t) ?>;>65=8O B5;0 >ACI5AB2;O5BAO 8=B53@8@>20=85< 48DD5@5=F80;L=KE C@02=5=89 ( = d(/dt v = dl/dt (1.4) -;5<5=BK <5E0=8G5A:>9 G0AB8 M;5:B@>?@82>40 A2O70=K <564C A>1>9, >1@07CO 548=CN :8=5<0B8G5A:CN F5?L, A2O7K20NICN 42830B5;L 8 8A?>;=8B5;L=K9 >@30=. 064K9 87 M;5<5=B>2 :8=5<0B8G5A:>9 F5?8 >1;0405B C?@C3>ABLN, 0 2 A>548=5=8OE M;5<5=B>2 8<5NBAO 2>74CH=K5 707>@K. A;8 CG8BK20BL MB8 D0:B>@K, B> @0AG5B=0O AE5<0 <5E0=8G5A:>9 G0AB8 ?@82>40 1C45B ?@54AB02;OBL <=>3><0AA>2CN A8AB5<C A C?@C38<8 A2O7O<8 8 707>@0<8, @0AG5B 48=0<8:8 :>B>@>9 ?@54AB02;O5B 1>;LH85 B@C4=>AB8. 0 ?@0:B8:5 2 8=65=5@=KE @0AG5B0E >1KG=> ?@5=51@530NB 707>@0<8 8 C?@C3>ABLN, ?@8=8<0O <5E0=8G5A:85 A2O78 01A>;NB=> 65AB:8<8.  MB>< A;CG05 42865=85 >4=>3> M;5<5=B0 405B ?>;=CN 8=D>@<0F8N > 42865=88 2A5E >AB0;L=KE M;5<5=B>2 8 42865=85 M;5:B@>?@82>40 <>6=> @0AA<0B@820BL =0 :0:><-;81> >4=>< <5E0=8G5A:>< 725=5. 1KG=> 2 :0G5AB25 B0:>3> M;5<5=B0 ?@8=8<0NB 20; M;5:B@>42830B5;O. 0AG5B=CN AE5<C <5E0=8G5A:>9 G0AB8 ?@82>40 A2>4OB : >4=><C >1>1I5==><C 65AB:><C <5E0=8G5A:><C 725=C, 8<5NI5<C M:2820;5=B=K9 <><5=B 8=5@F88 J, =0 :>B>@K9 2>7459AB2C5B M;5:B@><03=8B=K9 <><5=B 42830B5;O  8 AC<<0@=K9, ?@82545==K9 : 20;C 42830B5;O, <><5=B A>?@>B82;5=8O A, 2:;NG0NI89 2A5 <5E0=8G5A:85 ?>B5@8 2 A8AB5<5, 2 B>< G8A;5 8 <5E0=8G5A:85 ?>B5@8 2 42830B5;5. @8 0=0;875 <5E0=8G5A:>9 G0AB8 M;5:B@>?@82>40 A;54C5B 8<5BL 2 284C, GB> <><5=B A>?@>B82;5=8O <5E0=87<0 A>AB>8B 87 42CE A;0305<KE  ?>;57=>9 @01>BK 8 @01>BK A8; B@5=8O. >;57=0O @01>B0, A2O70=0 A 2K?>;=5=85< ?@>872>4AB25==K< <5E0=87<>< A>>B25BAB2CNI59 B5E=>;>38G5A:>9 >?5@0F88. @0D8: ?>;57=>9 @01>BK <>65B 1KBL ?>AB@>5= =0 >A=>20=88 @0AG5B>2 8;8 ?> >?KB=K< 40==K<. 01>B0 A8; B@5=8O >1KG=> CG8BK205BAO  <5E0=8G5A:8E A2O759 ?@82>40. @82545=85 <><5=B0 =03@C7:8 >ACI5AB2;O5< 8AE>4O 87 @025=AB20 <5E0=8G5A:>9 <>I=>AB8 =03@C7:8 42830B5;O 2 @50;L=>9 8 ?@82545==>9 AE5<0E. @8 MB>< ?>B5@8 <>I=>AB8 2 ?@><56CB>G=KE ?5@540G0E CG8BK205< 22545=85< 2 @0AG5BK  ?5@540G8 (?. !CI5AB2> >?5@0F88 ?@82545=8O @0AA<>B@8< =0 ?@8<5@5 :8=5<0B8G5A:>9 AE5<K M;5:B@>?@82>40 ;5154:8, ?>:070==>9 =0 @8A.1.3. 5E0=8G5A:0O <>I=>ABL ?@8 2@0I0B5;L=>< 42865=88 @02=0 ?@>872545=8N <><5=B0  =0 C3;>2CN A:>@>ABL  (?@8 ?>ABC?0B5;L=>< - ?@>872545=8N A8;K =0 A:>@>ABL) P = M*( 8  = F*V. (1.5)  8A. 1.3 !E5<K <5E0=8G5A:>9 G0AB8 M;5:B@>?@82>40 ;5154:8: 0) @50;L=0O; 1) ?@82545==0O (@0AG5B=0O) 1>7=0G0O G5@57 (4 8 A C3;>2CN A:>@>ABL 8 <><5=B 42830B5;O 8 G5@57 (< 8 A< C3;>2CN A:>@>ABL 8 <><5=B ?@>872>4AB25==>3> <5E0=87<0, 87 @025=AB20 <>I=>AB59 ?>;CG8<: A<.(</ (? = A.(4 >B:C40 A = A< / ( i.(? ), (1.6) 345 i = (4 / (< - ?5@540B>G=>5 G8A;>.@82545=85 A8; A>?@>B82;5=8O ?@>872>48BAO 0=0;>38G=>: FA<.v</ (? = A.(4 >B:C40 FA< = A.v / ( (4.(? ), (1.7) @82545=85 <><5=B>2 8=5@F88 : >4=>9 >A8 2@0I5=8O >A=>20=> =0 B><, GB> AC<<0@=K9 70?0A :8=5B8G5A:>9 M=5@388 >AB05BAO =587<5==K<. @8 =0;8G88 2@0I0NI8EAO G0AB59, >1;040NI8E <><5=B0<8 8=5@F88 J4, J1, J2 & Jn (B0: A?;>H=>3> F8;8=4@0 @048CA0 R (<) 8 <0AA>9 m (:3) <><5=B 8=5@F88 (:3<2) J = m.R2/2) 8 C3;>2K<8 A:>@>ABO<8 (4, (1, (2 & . (n, >1I89 70?0A :8=5B8G5A:>9 M=5@388 J(42/ 2 = J4(42/ 2 + J1(12/ 2 + J2(22/ 2 + & .. + Jn(n2/ 2, 8 @57C;LB8@CNI89 <><5=B 8=5@F88, ?@82545==K9 : 20;C 42830B5;O ( 745AL i1=(4/(1, i2=(4/(2& ) J = J4 + J1 / i12 + J2 / i22 + & + Jn / in2. (1.8) @82545=85 <0AA, 4286CI8EAO ?>ABC?0B5;L=>, B0:65 >ACI5AB2;O5BAO =0 >A=>20=88 @025=AB20 70?0A>2 :8=5B8G5A:>9 M=5@388 m.v2 / 2 = J. (42 / 2 >B:C40 J = m..(v / (4)2 (1.9) A;8 <5E0=87< 8<55B 8 2@0I0NI85AO 8 ?>ABC?0B5;L=> 4286CI85AO M;5<5=BK, B> AC<<0@=K9 ?@82545==K9 : 20;C 42830B5;O <><5=B 8=5@F88 >?@545;O5BAO =0 >A=>20=88 (1.8) 8 (1.9) J = J4 + J1 /i12 + J2 /i22 + & + m.(v / (4)2. (1.10) 2865=85 M;5:B@>?@82>40 2 >1I5< A;CG05 <>65B ?@>8AE>48BL 2 42CE @568<0E  CAB0=>282H5<AO, ?@8 =587<5==>9 A:>@>AB8 42865=8O ( 2 B.G. = 0) 8 ?5@5E>4=>< (48=0<8G5A:><). #A;>285< CAB0=>282H53>AO 42865=8O O2;O5BAO @025=AB2> <><5=B>2 2@0I5=8O (42830B5;O) 8 A>?@>B82;5=8O (=03@C7:8)  = A. @>25@:0 2K?>;=5=8O MB>3> CA;>28O >1KG=> >ACI5AB2;O5BAO 3@0D8G5A:8, A ?><>ILN <5E0=8G5A:8E E0@0:B5@8AB8: 42830B5;O 8 8A?>;=8B5;L=>3> >@30=0 (?@>872>4AB25==>3> <5E0=87<0). ><5=BK A>?@>B82;5=8O <>6=> @0745;8BL =0 425 :0B53>@88  @50:B82=K5, 2> 2A5E A;CG0OE ?@5?OBAB2CNI85 42865=8N ?@82>40 (>B 45D>@<0F88 =5C?@C38E B5;, @570=8O, B@5=8O 8 B.?.), 87<5=ONI85 A2>9 7=0: ?@8 87<5=5=88 =0?@02;5=8O 2@0I5=8O 8 0:B82=K5 8;8 ?>B5=F80;L=K5 ( A8;K BO65AB8, A8;K >B @0ABO65=8O, A60B8O 8 A:@CG820=8O C?@C38E B5;), A>E@0=ONI85 A2>9 7=0: ?@8 87<5=5=88 =0?@02;5=8O 2@0I5=8O ?@82>40. >B5=F80;L=K5 <><5=BK <>3CB B>@<>78BL 42865=85 ?@82>40 8;8, =0>1>@>B, A?>A>1AB2>20BL 53> 42865=8N.  5E0=8G5A:>9 E0@0:B5@8AB8:>9 42830B5;O =07K205BAO 7028A8<>ABL 53> C3;>2>9 A:>@>AB8 >B @0728205<>3> 8< <><5=B0 ((). 07;8G0NB 5AB5AB25==CN 8 8A:CAAB25==K5 E0@0:B5@8AB8:8 42830B5;59. AB5AB25==>9 =07K205BAO <5E0=8G5A:0O E0@0:B5@8AB8:0 42830B5;O, :>B>@0O A>>B25BAB2C5B >A=>2=>9 AE5<5 2:;NG5=8O 42830B5;O, =><8=0;L=K< ?0@0<5B@0< ?8B0NI53> =0?@O65=8O 8 >BACBAB28N 2 M;5:B@8G5A:8E F5?OE 42830B5;O 4>?>;=8B5;L=KE M;5<5=B>2.  8A.1.4 5E0=8G5A:85 E0@0:B5@8AB8:8 42830B5;59 (0) 8 8A?>;=8B5;L=KE >@30=>2 (1) 0 @8A. 1.4. 0 ?>:070=K 5AB5AB25==K5 <5E0=8G5A:85 E0@0:B5@8AB8:8 =081>;55 @0A?@>AB@0=5==KE 42830B5;59: 1  A8=E@>==>3> 42830B5;O, 2  42830B5;O ?>AB>O==>3> B>:0 =57028A8<>3> 2>71C645=8O, 3  42830B5;O ?>AB>O==>3> B>:0 ?>A;54>20B5;L=>3> 2>71C645=8O, 4  0A8=E@>==>3> 42830B5;O. %0@0:B5@8AB8:0 5  ?@8<5@ 01A>;NB=> <O3:>9 <5E0=8G5A:>9 E0@0:B5@8AB8:8. 0 @8A.1.4 1 ?>:070=K ?@82545==K5 : 20;C 42830B5;O <5E0=8G5A:85 E0@0:B5@8AB8:8 ((A) =5:>B>@KE 8A?>;=8B5;L=KE >@30=>2. %0@0:B5@8AB8:0 1, 2 2845 ?@O<>9 25@B8:0;L=>9 ;8=88, A>>B25BAB2C5B @07;8G=K< ?>4J5<=K< <5E0=87<0<. 5 >B;8G8B5;L=>9 >A>15==>ABLN O2;O5BAO =587<5==>5 =0?@02;5=85 <><5=B0 =03@C7:8 A , =5 7028AOI55 >B 7=0:0 A:>@>AB8 (0:B82=K9 <><5=B). %0@0:B5@8AB8:0 2 2845 ;><0==>9 ;8=88 2 A>>B25BAB2C5B 8A?>;=8B5;L=><C >@30=C, A>?@>B82;5=85 :>B>@>3> A>7405BAO 3;02=K< >1@07>< A8;0<8 B@5=8O. 5 B0:65 =07K20NB E0@0:B5@8AB8:>9 ACE>3> B@5=8O. "0:>9 (8;8 1;87:>9 : =59) E0@0:B5@8AB8:>9 >1;040NB <5E0=87<K ?>40G8 AB0=:>2, 3>@87>=B0;L=K5 :>=2595@K 8 B@0=A?>@B5@K, <5E0=87<K ?5@542865=8O ?>4J5<=KE :@0=>2. ><5=B =03@C7:8 MB>3> 2840 =0?@02;5= =02AB@5GC 42865=8N 8 O2;O5BAO @50:B82=K<. @820O 3 E0@0:B5@87C5B <><5=B =03@C7:8 25=B8;OB>@>2, F5=B@>156=KE :><?@5AA>@>2, 4K<>A>A>2, :>B>@K9 >1KG=> ?@>?>@F8>=0;5= :204@0BC A:>@>AB8. %0@0:B5@8AB8:8 2840 3 =07K20NB 25=B8;OB>@=K<8. %0@0:B5@8AB8:>9 2840 4, 1;87:>9 : 38?5@1>;8G5A:>9 7028A8<>AB8, >1;040NB <5E0=87<K 3;02=>3> 42865=8O B>:0@=KE 8 D@575@=KE AB0=:>2, @07;8G=K5 =0<0BK20NI85 CAB@>9AB20. %0@0:B5@8AB8:8 ?>:070==K5 =0 @8A.1.4 ?@54AB02;ONB A>1>9 8450;878@>20==K5, B5>@5B8G5A:85 :@82K5. 50;L=K9 <><5=B =03@C7:8 >1KG=> >?@545;O5BAO >4=>2@5<5==> =5A:>;L:8<8 A>AB02;ONI8<8 8 <5E0=8G5A:0O E0@0:B5@8AB8:0 8A?>;=8B5;L=>3> >@30=0 8<55B 1>;55 A;>6=K9 284. ;O >F5=:8 <5E0=8G5A:>9 E0@0:B5@8AB8:8 (:0: 42830B5;59, B0: 8 <5E0=87<>2) 22>48BAO ?>=OB85 65AB:>AB8, :0: <>4C;O ?@>872>4=>9 <><5=B0 ?> A:>@>AB8 ( =| dM/d(| (|(M/((| (1.11) 5E0=8G5A:85 E0@0:B5@8AB8:8 <>6=> @0745;8BL =0 G5BK@5 >A=>2=K5 :0B53>@88: 01A>;NB=> 65AB:85 (( = "), 65AB:85 (A 1>;LH>9 65AB:>ABLN), <O3:85 (A <0;>9 65AB:>ABLN) 8 01A>;NB=> <O3:85 (( = 0). "0: E0@0:B5@8AB8:C A8=E@>==>3> 42830B5;O (:@820O 1 =0 @8A. 1.40) <>6=> AG8B0BL 01A>;NB=> 65AB:>9, E0@0:B5@8AB8:C 42830B5;O ?>AB>O==>3> B>:0 A =57028A8<K< 2>71C645=85< (:@820O 2 =0 @8A. 1.40)  65AB:>9, 0 E0@0:B5@8AB8:C 42830B5;O ?>AB>O==>3> B>:0 A ?>A;54>20B5;L=K< 2>71C645=85< (3 =0 @8A. 1.40)  <O3:>9. %0@0:B5@8AB8:0 0A8=E@>==>3> 42830B5;O 4 8<55B ?5@5<5==CN 65AB:>ABL. 0 @01>G5< CG0AB:5 (0b) 55 65AB:>ABL >B@8F0B5;L=0 (=> 4>2>;L=> 25;8:0 ?> <>4C;N), 2 B>G:5 :@8B8G5A:>3> <><5=B0 b >=0 @02=0 0, 0 ?@8 <5=LH8E A:>@>ABOE  ?>;>68B5;L=0 8 =525;8:0. 1A>;NB=> <O3:0O <5E0=8G5A:0O E0@0:B5@8AB8:0 (( = 0)  MB> E0@0:B5@8AB8:0, ?@8 :>B>@>9 <><5=B 42830B5;O A 87<5=5=85< C3;>2>9 A:>@>AB8 >AB05BAO =587<5==K< (:@820O 5 =0 @8A.1.40). "0:>9 E0@0:B5@8AB8:>9 >1;0405B, =0?@8<5@, 42830B5;L ?>AB>O==>3> B>:0 =57028A8<>3> 2>71C645=8O ?@8 ?8B0=88 >B 8AB>G=8:0 B>:0 (8;8 ?@8 @01>B5 2 70<:=CBKE A8AB5<0E M;5:B@>?@82>40 2 @568<5 AB018;870F88 B>:0 O:>@O).  8A.1.5  >?@545;5=8N ?0@0<5B@>2 CAB0=>282H53>AO 42865=8O 0 @8A. 1.5 ?>:070=K <5E0=8G5A:85 E0@0:B5@8AB8:8 42830B5;O ?>AB>O==>3> B>:0 A =57028A8<K< 2>71C645=85< (:@820O 1) 8 25=B8;OB>@0 (:@820O 2). ">G:0  O2;O5BAO B>G:>9 CAB0=>282H53>AO 42865=8O, 0 55 :>>@48=0BK (CAB 8 CAB  :>>@48=0B0<8 CAB0=>282H53>AO 42865=8O. <5O <5E0=8G5A:CN E0@0:B5@8AB8:C 42830B5;O (() 8 ?@82545==CN E0@0:B5@8AB8:C 8A?>;=8B5;L=>3> >@30=0 ((A) <>6=> =09B8 B>G:C CAB0=>282H53>AO @568<0 @01>BK  = A. ;O MB>3> 87>1@078< MB8 425 E0@0:B5@8AB8:8 =0 >4=>< 3@0D8:5. $0:B ?5@5A5G5=8O MB8E E0@0:B5@8AB8: 3>2>@8B > 2>7<>6=>AB8 A>2<5AB=>9 @01>BK 42830B5;O 8 @01>G59 <0H8=K, 0 B>G:0 8E ?5@5A5G5=8O O2;O5BAO B>G:>9 CAB0=>282H53>AO 42865=8O. ;O ?>;=>3> 0=0;870 @568<0 CAB0=>282H53>AO 42865=8O (CAB>9G82>9 @01>BK) =5>1E>48<> >?@545;8BL, O2;O5BAO ;8 42865=85 CAB>9G82K<. !B0B8G5A:8 CAB>9G82K< 1C45B B0:>5 CAB0=>282H55AO 42865=85, :>B>@>5, 1C4CG8 2K2545==K< 87 CAB0=>282H53>AO @568<0 :0:8<-;81> 2=5H=8< 2>7<CI5=85<, 2>72@0I05BAO 2 MB>B @568< ?>A;5 8AG57=>25=8O MB>3> 2>7<CI5=8O.  ?@>B82=>< A;CG05 @568< 1C45B =5CAB>9G82K<. @8 =5CAB>9G82>< 42865=88 ;N1>5, 4065 A0<>5 <0;>5, >B:;>=5=85 A:>@>AB8 >B CAB0=>282H53>AO 7=0G5=8O ?@82>48B : 87<5=5=8N A>AB>O=8O ?@82>40  >= =5 2>72@0I05BAO 2 B>G:C CAB0=>282H53>AO @568<0. @82>4 AB0B8G5A:8 CAB>9G82, 5A;8 2 B>G:5 CAB0=>282H53>AO @568<0 2K?>;=O5BAO CA;>285 (@8A.1.5)  EMBED Equation.3  8;8 ( - (A < 0 (1.12) #A;>285 (1.12) >7=0G05B, GB> ?@82>4 AB0B8G5A:8 CAB>9G82, 5A;8 ?@8 ?>;>68B5;L=>< ?@8@0I5=88 C3;>2>9 A:>@>AB8 <><5=B 42830B5;O >:065BAO <5=LH5 <><5=B0 A>?@>B82;5=8O 8 ?@82>4 2A;54AB285 MB>3> 70B>@<>78BAO 4> ?@56=53> 7=0G5=8O A:>@>AB8. @8 >B@8F0B5;L=>< ?@8@0I5=88 C3;>2>9 A:>@>AB8 <><5=B 42830B5;O >:065BAO 1>;LH5 <><5=B0 A>?@>B82;5=8O 8 ?@82>4 2A;54AB285 MB>3> @073>=8BAO 4> ?@56=53> 7=0G5=8O A:>@>AB8.  8A.1.6  ?>=OB8N CAB>9G82>AB8 @8 ?>AB>O==>< <><5=B5 =03@C7:8 A=  EMBED Equation.3  (?@O<0O 1 =0 @8A. 1.41) 8 AB0B8G5A:0O CAB>9G82>ABL >?@545;O5BAO B>;L:> 65AB:>ABLN <5E0=8G5A:>9 E0@0:B5@8AB8:8 42830B5;O. A;8 >=0 >B@8F0B5;L=0, B> @01>B0 2 CAB0=>282H5<AO @568<5 CAB>9G820, 8 =0>1>@>B. "0:, 5A;8 0A8=E@>==K9 42830B5;L (4 =0 @8A. 1.40) =03@C60BL ?>AB>O==K< <><5=B><, B> =0 8=B5@20;5 0  1  EMBED Equation.3  8 @568< 1C45B CAB>9G82K<, 0 =0 8=B5@20;5 1  2  EMBED Equation.3  8 @568< 1C45B =5CAB>9G82K<. 53:> ?>:070BL, GB> ?@8 @01>B5 B>3> 65 42830B5;O =0 25=B8;OB>@=CN =03@C7:C, 53> @01>B0 1C45B CAB>9G82>9 2> 2A5E B>G:0E. =25@B>@K A8AB5<0E M;5:B@>?@82>40 ?5@5<5==>3> B>:0 2.1. !>2@5<5==>5 A>AB>O=85 A8;>2KE ?>;C?@>2>4=8:>2KE M;5<5=B>2 A=>2=K<8 ?@81>@0<8 A8;>2>9 M;5:B@>=8:8 2 =0AB>OI55 2@5<O O2;ONBAO: 18?>;O@=K5 B@0=78AB>@K ( "), ?>;52K5 B@0=78AB>@K A 87>;8@>20==K< 70B2>@>< (MOSFET), 18?>;O@=K5 B@0=78AB>@K A 87>;8@>20==K< 70B2>@>< (IGBT), B8@8AB>@K (Thyristors, SCR) 8 70?8@05<K5 B8@8AB>@K (GTO), 0 B0:65 48>4K (Diodes), A8;>2K5 8=B53@0;L=K5 AE5<K (Power IC) 8 8=B5;;5:BC0;L=K5 A8;>2K5 8=B53@0;L=K5 AE5<K (Smart Power IC). !0<CN 7=0G8B5;L=CN G0ABL ?@81>@>2 2 480?07>=5 4> 50  A>AB02;ONB ?>;52K5 B@0=78AB>@K A 87>;8@>20==K< 70B2>@>< MOSFET. 1;040O <0;K<8 AB0B8G5A:8<8 8 48=0<8G5A:8<8 ?>B5@O<8, <8=8<0;L=K<8 70B@0B0<8 =0 C?@02;5=85, :@09=5 <0;K< 2@5<5=5< ?5@5:;NG5=8O (A?>A>1=K @01>B0BL =0 G0AB>B0E 4> 1 F) 8 2KA>:>9 AB>9:>ABLN : ?5@53@C7:0<, >=8 ?@0:B8G5A:8 ?>;=>ABLN 2KB5A=8;8 87 =87:>2>;LB=KE ?@5>1@07>20B5;L=KE CAB@>9AB2 (4> 200 ) 2A5 >AB0;L=K5 B8?K A8;>2KE ?>;C?@>2>4=8:>2KE ?@81>@>2.  @07@01>B:8 D8@<K Simens 2 >1;0AB8 A>740=8O 2KA>:>2>;LB=KE (600  1000 ) MOSFET B@0=78AB>@>2 @0AH8@ONB >1;0ABL 8E ?@8<5=5=8O 4> 10 :B, 8 ?>72>;8;8 ?>B5A=8BL A8;>2K5 18?>;O@=K5 B@0=78AB>@K ( "). "@0=78AB>@K MOSFET H8@>:> 8A?>;L7CNBAO 2 A8;>2KE 8=B53@0;L=KE AE5<0E (Power IC) 8 8=B5;;5:BC0;L=KE A8;>2KE 8=B53@0;L=KE AE5<0E (Smart Power IC). !8;>2K5 18?>;O@=K5 B@0=78AB>@K ( "), 2 B>:>2>< 480?07>=5 4> 50 , =0E>4OB ?@8<5=5=85 2 >A=>2=>< 2 <0AA>2>< 8 45H52>< 1KB>2>< 8 ?@><KH;5==>< >1>@C4>20=88.  >1;0AB8 A@54=8E =0?@O65=89 (500  600 8 2KH5) =081>;55 ?@54?>GB8B5;L=K<8 4;O ?@8<5=5=8O O2;ONBAO 18?>;O@=K5 B@0=78AB>@K A 87>;8@>20==K< 70B2>@>< (IGBT- Insulated Gate Bipolar Transistor). >O2;5=85 MB>3> ?@81>@0 ?>2;5:;> 70 A>1>9 AB>;L @048:0;L=K5 87<5=5=8O 2 A8;>2>9 M;5:B@>=8:5, GB> <>6=> 3>2>@8BL > 2B>@>9 @52>;NF88, ?5@568205<>9 MB8< =0CG=>  B5E=8G5A:8< =0?@02;5=85<.  =0AB>OI55 2@5<O IGBT B@0=78AB>@K >15A?5G820NB :><<CB0F8N B>:>2 4> 1800  ?@8 =0?@O65=88 4> 4,5 :. @8 MB>< 2@5<O ?5@5:;NG5=8O ;568B 2 480?07>=5 200  400 =A5:. -B> ?@825;> : 2KB5A=5=8N 70?8@05<KE B8@8AB>@>2 (GTO) ?@8 <>I=>ABOE 4> 1 B (U d" 3,5 :). !0<8 GTO 2 ?>A;54=85 3>4K 1K;8 <>45@=878@>20=K (, Mitsubishi), 8 ?>O28;AO =>2K9 :;0AA ?@81>@>2  B8@8AB>@, :><<CB8@C5<K9 ?> 70B2>@C (GCT  Gate Thyristor 8;8 IGCT  Integrated Gate Commutated Thyristor). 081>;55 ?5@A?5:B82=K<8 87 =8E, 4;O AE5< ?@5>1@07>20B5;59 <>I=>ABLN >B 548=8F 20BB 4> 548=8F <53020BB, O2;ONBAO B@0=78AB>@K MOSFET 8 IGBT. @8=F8? 459AB28O A8;>2KE 8=25@B>@>2 !8;>2K5 <>4C;8 >1KG=> 8A?>;L7CNBAO 2 F5?OE ?@5>1@07>20B5;59 ?5@5<5==>3> =0?@O65=8O 2 ?>AB>O==>5 (:>=25@B>@K 8;8 2K?@O<8B5;8) 8 ?>AB>O==>3> =0?@O65=8O 2 ?5@5<5==>5 - (8=25@B>@K).  A>2@5<5==KE A8AB5<0E M;5:B@>?@82>40 ?5@5<5==>3> B>:0 ?@0:B8G5A:8 ?>2A5<5AB=> 2 :0G5AB25 A8;>2KE @53C;OB>@>2 8A?>;L7CNBAO 8=25@B>@K A H5ABLN A8;>2K<8 <>4C;O<8 (B@0=78AB>@=K<8 :;NG0<8 - ").  SHAPE \* MERGEFORMAT  8A.2.1 "@5ED07=K9 8=25@B>@ A ( !E5<0 B@5ED07=>3> 8=25@B>@0 =0 IGBT A8;>2KE <>4C;OE, 2:;NG0NI53> B0:65 2K?@O<8B5;L=K9 <>4C;L, ?>40NI89 =0 8=25@B>@ =0?@O65=85 U?, 8 ?@54=07=0G5==>3> 4;O ?8B0=8O 42830B5;O ?5@5<5==>3> B>:0, ?>:070=0 =0 @8A. 2.1. !B0B>@=K5 >1<>B:8 <0H8=K (>1KG=> MB> 0A8=E@>==0O <0H8=0 A :>@>B:>70<:=CBK< @>B>@><) ?@8 ?8B0=88 >B B0:>3> 8=25@B>@0 2:;NG0NBAO 2 72574C 8;8 B@5C3>;L=8:. 0AA<>B@8< ?@8=F8? ?>;CG5=8O @53C;8@C5<>9 G0AB>BK A ?><>ILN C?@02;O5<>3> 8=25@B>@0. @>AB59H8< A?>A>1>< C?@02;5=8O B@0=78AB>@=K<8 :;NG0<8, O2;O5BAO A?>A>1 A  = 180 (@8A. 2.2).  8A.2.2 ;3>@8B< C?@02;5=8O 8=25@B>@>< A  = 180 45AL 2 B5G5=85 1/6 ?5@8>40 2KE>4=>3> =0?@O65=8O 2:;NG5=K B@8 :;NG0. >A;54>20B5;L=>ABL C?@02;5=8O :;NG0<8 A;54CNI0O: 123, 234, 345, 456, 561 8 612.  8A. 2.3 50;870F8O ( =0 1075 8=25@B>@0: 0) A8=CA>840;L=K9 8 ?8;>>1@07=K9 A83=0;K @02=>9 0<?;8BC4K; 1) =0?@O65=85 =0 =03@C7:5 >3CB B0:65 @50;87>2K20BLAO A?>A>1K A  = 120 8  = 150. @8  = 120 ?>A;54>20B5;L=>ABL C?@02;5=8O B@0=78AB>@=K<8 :;NG0<8: 12, 23, 34, 45, 56, 61; ?@8  = 150 B@0=78AB>@=K5 :;NG8 ?5@5:;NG0NBAO 2 B0:>9 ?>A;54>20B5;L=>AB8: 12, 123, 23, 234, 34, 345, 45, 456, 56, 561,61.  =0AB>OI55 2@5<O, 2 A2O78 A> 7=0G8B5;L=K<8 4>AB865=8O<8 B5E=>;>388 873>B>2;5=8O A8;>2KE B@0=78AB>@>2, ?@0:B8G5A:8 ?>2A5<5AB=> 8A?>;L7CNBAO 0;3>@8B<K A A8=CA>840;L=>9 (, ?@8=F8? @01>BK :>B>@KE 8;;NAB@8@C5B @8A. 2.3. !E5<0 70<5I5=8O, E0@0:B5@8AB8:8 8 @568<K @01>BK  3.1. !E5<0 70<5I5=8O "  >1@07=0O AE5<0 70<5I5=8O B@5ED07=>9 (m1=3) 0A8=E@>==>9 <0H8=K AB@>8BAO 4;O >4=>9 D07K, 8, ?> ACI5AB2C, 0=0;>38G=0 AE5<5 70<5I5=8O B@0=AD>@<0B>@0 (A<. @8A. 3.1).  8A. 3.1 !E5<0 70<5I5=8O 0A8=E@>==>9 <0H8=K 45AL: U1 - D07=>5 =0?@O65=85, ; R1 8 R2  0:B82=K5 A>?@>B82;5=8O ?5@28G=>9 8 2B>@8G=>9 >1<>B>:; X1 = 1.L1 8 X2 = 1.L2   8=4C:B82=K5 A>?@>B82;5=8O @0AA5O=8O ?5@28G=>9 8 2B>@8G=>9 >1<>B>:, 345 1 = 2..f1  F8:;8G5A:0O G0AB>B0 A5B8, 0 f1  G0AB>B0 A5B8; Xm 8 Rm  8=4C:B82=>5 A>?@>B82;5=85 =0<03=8G820NI59 25B28, >1CA;>2;5==>5 >A=>2=K< <03=8B=K< ?>B>:><, 8 0:B82=>5 A>?@>B82;5=85, 8A:CAAB25==> 2=>A8<>5 2 AE5<C 70<5I5=8O 4;O CG5B0 <03=8B=KE ?>B5@L 2 D5@@><03=8B=KE G0ABOE <03=8B>?@>2>40; I1, I2  D07=K5 B>:8 2 >1<>B:0E AB0B>@0 8 @>B>@0; Im - B>: =0<03=8G820=8O, A>740NI89 >A=>2=>9 <03=8B=K9 ?>B>: $m; 45AL 8=45:A HB@8E ( ), >B=>AOI89AO :> 2B>@8G=>9 >1<>B:5, >1>7=0G05B, GB> 40==0O 25;8G8=0 ?@82545=0 : ?5@28G=>9 >1<>B:5. !:>;L65=85 s = (0  ) / 0 = (n0  n)/ n0 (3.1) ?@54AB02;O5B A>1>9 >B=>A8B5;L=>5 2@0I5=85 @>B>@0 >B=>A8B5;L=> ?>;O AB0B>@0, 345 0 = 2.(.f/p  C3;>20O A:>@>ABL 2@0I5=8O <03=8B=>3> ?>;O, 0   C3;>20O A:>@>ABL @>B>@0 ( @  G8A;> ?0@ ?>;NA>2 0A8=E@>==>9 <0H8=K). A?>;L7CO D>@<C;C 4;O A:>;L65=8O <>6=> 70?8A0BL 2K@065=8O 4;O C3;>2>9 A:>@>AB8 @>B>@0 2 2845  = 0 ( 1- s ) (3.2) 5B@C4=> ?>:070BL, GB> G0AB>B0 -! 8 B>:0 2 @>B>@5 f2 7028A8B >B A:>;L65=8O: ?@8 s = 0 (@568< 8450;L=>3> E>;>AB>3> E>40) f2 = 0, 0 ?@8 s = 1 (@>B>@ =5?>42865=) f2 = f1 .  >1I5< A;CG05 f2 = f1.s , >B:C40 ?>;CG05< 4@C3>5 >?@545;5=85 A:>;L65=8O: s = f2 / f1 (3.3) A?>;L7CO 70:>=K 8@E3>D0 4;O C7;0 8 42CE :>=BC@>2 AE5<K 70<5I5=8O, <>6=> 70?8A0BL A;54CNI85 C@02=5=8O 4;O A>>B25BAB2CNI8E 25:B>@>2 (25:B>@K >1>7=0G5=K 2K45;5=85<): U1 = - E1 + I1.(R1+j.X1), E2 = I2 .(R2 /s+j.X2 ), (3.4) I1 = Im + (-I2 ), 345 E1 = E2 = ."2.$m W1.Kw1  -! =02>48<0O 2 ?5@28G=>9 8 2B>@8G=>9 (?@82545==>9) >1<>B:0E >A=>2=K< <03=8B=K< ?>B>:>< $m, 0 W1.Kw1  MDD5:B82=>5 G8A;> 28B:>2 2 D075 ?5@28G=>9 >1<>B:8. #G8BK20O, GB>, :0: ?@028;>, A>?@>B82;5=85 25B28 =0<03=8G820=8O ( Zm=Rm + jXm) =0<=>3> 1>;LH5, G5< A>?@>B82;5=85 ?>A;54>20B5;L=>9 25B28 AE5<K 70<5I5=8O [(R1 + R2 /s) + j(X1 + X2 )], <>6=> 70?8A0BL ?@81;865==>5 2K@065=85 4;O ?@82545==>3> B>:0 @>B>@0 (157 CG5B0 Zm) 2 2845  EMBED Equation.3 ; (3.5) 345 X: = X1 + X2 = 1(Ls+L r)  8=4C:B82=>5 A>?@>B82;5=85 :>@>B:>3> 70<K:0=8O. 568<K @01>BK 8 M=5@35B8G5A:0O 4803@0<<0   7028A8<>AB8 >B 25;8G8=K 8 7=0:0 A:>;L65=8O, 0A8=E@>==0O <0H8=0 <>65B @01>B0BL 2 42830B5;L=>< (0 < s <1), 35=5@0B>@=>< (s < 0) 8;8 B>@<>7=>< (s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m1.U1.I1.cos(, 345 ( - D07>2K9 C3>; <564C =0?@O65=85< 8 B>:><. >I=>ABL @0AE>4CNI0OAO =0 M;5:B@8G5A:85 ?>B5@8 2 >1<>B:0E AB0B>@0 8 @>B>@0 ( M;1 = m1.I12.R1, ( M;2 = m1.I2 2.R2 . (3.6) >I=>ABL <03=8B=KE ?>B5@L ( <3 , A2O70==0O A ?>B5@O<8 =0 38AB5@578A 8 =0 28E@52K5 B>:8, 2K45;O5BAO 2 >A=>2=>< 2 <03=8B>?@>2>45 AB0B>@0  745AL <03=8B=>5 ?>;5 87<5=O5BAO A G0AB>B>9 A5B8 (50 8;8 60 F).  @01>G5< @568<5 0A8=E@>==>9 <0H8=K G0AB>B0 B>:0 2 @>B>@5 =525;8:0, A>AB02;OO >1KG=> =5A:>;L:> 35@F, 8 ?>MB><C <03=8B=K<8 ?>B5@O<8 2 <03=8B>?@>2>45 @>B>@0 ?@5=51@530NB. >B5@8 8  0A8=E@>==>3> 42830B5;O  =0AB>OI55 2@5<O M;5:B@>B5E=8G5A:85 A8AB5<K A 0A8=E@>==K<8 42830B5;O<8 O2;ONBAO >A=>2=K< ?>B@518B5;5< M;5:B@8G5A:>9 M=5@388. >MB><C >?@545;5=85 >A=>2=KE M=5@35B8G5A:8E ?>:070B5;59 @01>BK 0A8=E@>==>3> 42830B5;O 8 ?>8A: A?>A>1>2 8E C;CGH5=8O 8<55B 1>;LH>5 ?@0:B8G5A:>5 7=0G5=85.  >A=>2=K< M=5@35B8G5A:8< ?>:070B5;O< @01>BK 42830B5;O >B=>AOBAO ?>B5@8 <>I=>AB8 8 M=5@388,  8 :>MDD8F85=B <>I=>AB8 (cos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s = ( M;2 (3.9) 7 (9) A;54C5B, GB> A:>;L65=85 @02=> >B=>H5=8N M;5:B@8G5A:8E ?>B5@L 2 @>B>@5 (( M;2) : M;5:B@><03=8B=>9 <>I=>AB8 ( M<):. s = ( M;2/ M<, >B:C40 ?>;CG05< (8A?>;L7CO B0:65 (5)) D>@<C;C 4;O M;5:B@><03=8B=>3> <><5=B0 0A8=E@>==>9 <0H8=K: M< = ( M;2 / (s.(0) = m1.I2 2.R2 /(s.(0) =  EMBED Equation.3  (3.10) $>@<C;0 (3.10) O2;O5BAO 107>2>9 ?@8 0=0;875 M;5:B@><5E0=8G5A:8E A2>9AB2 0A8=E@>==>3> M;5:B@>?@82>40. 84=>, GB> <><5=B B@5ED07=>9 (m1 = 3) O2;O5BAO DC=:F859 B@5E ?5@5<5==KE  =0?@O65=8O (U1) 8 G0AB>BK A5B8 (f), 0 B0:65 A:>;L65=8O s. @><5 B>3>, =0 <><5=B 2;8ONB ?0@0<5B@K AE5<K 70<5I5=8O (R1, R2 8 Lk).  EMBED Mathcad  EMBED Mathcad  0) 1) 8A. 3.4 %0@0:B5@8AB8:8 0A8=E@>==>9 <0H8=K: 0) 028A8<>AB8 (() 8 ((I2 ) 4;O B>@<>7=>3> (( < 0), 42830B5;L=>3> (0 < ( < (o) 8 35=5@0B>@=>3> ((o < () @568<>2; 1) 028A8<>AB8 s(M), s(I2 ), 8 s(I2a ) 4;O 42830B5;L=>3> @568<0. @820O 0:B82=>9 A>AB02;ONI59 @>B>@=>3> B>:0 ?>:070=0 HB@8E>2K<8 ;8=8O<8. 0 @8A.3.4 40=K 7028A8<>AB8 ?@82545==>3> B>:0 @>B>@0 8 M;5:B@><03=8B=>3> <><5=B0 2 DC=:F88 C3;>2>9 A:>@>AB8 (0 < (< ), ?>AB@>5==K5 4;O  B8?0 41124, 8<5NI53> A;54CNI85 =><8=0;L=K5 40==K5: G8A;> D07 m1 = 3; G8A;> ?0@ ?>;NA>2 @ = 2 (G5BK@5E?>;NA=0O <0H8=0); <>I=>ABL (8<55BAO 2 284C >B40205<0O <>I=>ABL) = = 5.5 :B; =><8=0;L=K9 <><5=B = = 36.3 <; D07=K5 =0?@O65=85 8 B>: U= =220 , I= = 11.5; G0AB>B0 A5B8 f= = 50 F; 0:B82=K5 A>?@>B82;5=8O >1<>B>: AB0B>@0 8 @>B>@0 (?@82545==>5) R1 = 1.24 < 8 R2 = 0.935 <, 0 B0:65 Xk = 3.98 <. %0@0:B5@=K5 B>G:8 <5E0=8G5A:>9 E0@0:B5@8AB8:8 s(M): B>G:0 8450;L=>3> E>;>AB>3> E>40 (s = 0, ( = (>,  = 0), B>G:0 :>@>B:>3> 70<K:0=8O (s = 1, ( = 0,  = ?) 8 B>G:0 :@8B8G5A:>3> <><5=B0 (M:AB@5<C<0) (s = s: , ( = (:,  = :). 06=>9 >A>15==>ABLN <5E0=8G5A:>9 E0@0:B5@8AB8:8 (() 0A8=E@>==>3> 42830B5;O O2;O5BAO =0;8G85 M:AB@5<C<0 (?5@53810) 2 :@8B8G5A:>9 B>G:5 :, (: 8 sk.  =0H5< ?@8<5@5 (@8A.3.4) :@8B8G5A:85 <><5=B 8 A:>@>ABL (2 >B=>A8B5;L=KE 548=8F0E) @02=K: :>=2.15 8 (:= 122 8 sk = 0.22. @8 C<5=LH5=88 A:>@>AB8 =865 :@8B8G5A:>9 <><5=B ?0405B, =5A<>B@O =0 @>AB B>:0. -B> >1JOA=O5BAO B5<, GB> <><5=B A>7405BAO =5 2A5< B>:>< @>B>@0, 0 B>;L:> 53> 0:B82=>9 A>AB02;ONI59, :>B>@0O C<5=LH05BAO ?@8 ?>2KH5=88 A:>;L65=8O 8 G0AB>BK 2 @>B>@5 (A<. @8A.3.4). 865 ?@82545=K D>@<C;K 4;O @0AG5B0 <0E 8 sk:  EMBED Equation.3   EMBED Equation.3  (3.11) ?@8G5< 745AL 7=0: + >B=>A8BAO : 42830B5;L=><C @568<C, 0 7=0: - - : 35=5@0B>@=><C. 53C;8@>20=85 A:>@>AB8, B>:0 8 <><5=B0 , A8AB5<0 '- 4.1 53C;8@>20=85 A:>@>AB8  A ?><>ILN @578AB>@>2 2 F5?8 @>B>@0 0A?@>AB@0=5==K< O2;O5BAO A?>A>1 @53C;8@>20=8O A:>@>AB8, B>:0 8 <><5=B0  A D07=K< @>B>@>< ?@8 22545=88 (8 87<5=5=88) 4>102>G=KE A>?@>B82;5=89 2 F5?L @>B>@0. 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athcad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athcad  8A. 4.4 53C;8@>20=85 A:>@>AB8  87<5=5=85< =0?@O65=8O =0 AB0B>@5: 0) <5E0=8G5A:85 E0@0:B5@8AB8:8, 1) AE5<0 >7<>6=>ABL @53C;8@>20=8O A:>@>AB8  ?@8 87<5=5=88 =0?@O65=8O A;54C5B 87 D>@<C;K (4.11), 2 A>>B25BAB288 A :>B>@>9 A 87<5=5=85< U <>6=> 87<5=OBL 25;8G8=C :@8B8G5A:>3> (<0:A8<0;L=>3>) <><5=B0, ?>;CG0O B5< A0<K< 8A:CAAB25==K5 E0@0:B5@8AB8:8. @8B8G5A:>5 A:>;L65=85 8 A8=E@>==0O A:>@>ABL  >B =0?@O65=8O =5 7028AOB, 8 ?@8 B0:>< A?>A>15 @53C;8@>20=8O >AB0NBAO =587<5==K<8. 0: 284=> 87 3@0D8:>2, ?>;CG05<K5 2 40==>9 @07><:=CB>9 AE5<5 8A:CAAB25==K5 E0@0:B5@8AB8:8 >:07K20NBAO <0;> ?@83>4=K<8 4;O @53C;8@>20=8O A:>@>AB8, B0: :0: A C<5=LH5=85< =0?@O65=8O @57:> A=860NBAO :@8B8G5A:89 <><5=B 8 ?5@53@C7>G=0O A?>A>1=>ABL, 0 480?07>= @53C;8@>20=8O A:>@>AB8 >:07K205BAO >G5=L <0;5=L:8<. 5A:>;L:> ;CGH85 @57C;LB0BK <>6=> ?>;CG8BL, 8A?>;L7CO A8AB5<C - A >B@8F0B5;L=>9 >1@0B=>9 A2O7LN ?> A:>@>AB8. >3CB 1KBL ?>;CG5=> >B=>A8B5;L=> 65AB:85 <5E0=8G5A:85 E0@0:B5@8AB8:8 ?@8 C4>2;5B2>@8B5;L=>9 ?5@53@C7>G=>9 A?>A>1=>AB8 8 480?07>=5 @53C;8@>20=8O A:>@>AB8 ?>@O4:0 10. 53C;8@>20=85 ?;02=>5, 2=87 >B 5AB5AB25==>9 <5E0=8G5A:>9 E0@0:B5@8AB8:8. <5AB5 A B5< MB0 A8AB5<0 @53C;8@>20=8O A:>@>AB8 A2O70=0 A 1>;LH8<8 ?>B5@O<8 2 >1<>B:5 @>B>@0, >A>15==> ?@8 @01>B5 =0 =87:8E A:>@>ABOE 2 >1;0AB8 1>;LH8E A:>;L65=89. @5>1@07>20B5;L G0AB>BK  0A8=E@>==K9 42830B5;L '0AB>B=K9 A?>A>1 @53C;8@>20=8O O2;O5BAO =081>;55 ?5@A?5:B82=K< 8 H8@>:> 8A?>;L7C5<K< 2 =0AB>OI55 2@5<O A?>A>1>< @53C;8@>20=8O A:>@>AB8 . @8=F8? 53> 70:;NG05BAO 2 B><, GB> 87<5=OO G0AB>BC ?8B0NI53> =0?@O65=8O f1, <>6=>, 2 A>>B25BAB288 A 2K@065=85< 0 = 2.(.f1/p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const. A;8 ?@5=51@5GL 0:B82=K< A>?@>B82;5=85< AB0B>@0, B> 87 (9) ?>;CG8<  EMBED Equation.3  (4.12) 345   ?>AB>O==0O, =5 7028AOI0O >B f1. 7 (10) A;54C5B, GB> 4;O 42CE ;N1KE G0AB>B f1i 8 f1k  EMBED Equation.3 , (4.13) 345 Ai 8 Ak  <><5=BK =03@C7:8 ?@8 A:>@>ABOE, A>>B25BAB2CNI8E G0AB>B0< f1i 8 f1k. BAN40 ?>;CG05< 70:>= 87<5=5=8O =0?@O65=8O ?@8 G0AB>B=>< A?>A>15 @53C;8@>20=8O A:>@>AB8:  EMBED Equation.3  (4.14) ! ?><>ILN (4.12) <>3CB 1KBL ?>;CG5=K G0AB=K5 70:>=K 87<5=5=8O =0?@O65=8O 8 G0AB>BK ?@8 @07;8G=KE 7028A8<>ABOE <><5=B0 =03@C7:8 A >B A:>@>AB8. "0: ?@8 ?>AB>O==>< <><5=B5 =03@C7:8 A = const: U1 / f1 = const, (4.15) B> 5ABL =0?@O65=85 4>;6=> 87<5=OBLAO ?@>?>@F8>=0;L=> G0AB>B5. 0 @8A. 4.9 ?@82545=K <5E0=8G5A:85 E0@0:B5@8AB8:8  ?@8 G0AB>B=>< @53C;8@>20=88, >ACI5AB2;5==>< 2 A>>B25BAB288 A CA;>285< U1 / f1 = const. 4=0:>, :0: 284=> 87 3@0D8:>2, ?5@53@C7>G=0O A?>A>1=>ABL =5 >AB0;0AL =587<5==>9.  459AB28B5;L=>AB8, ?@8 >4=>2@5<5==>< C<5=LH5=88 G0AB>BK 8 =0?@O65=8O, ?5@53@C7>G=0O A?>A>1=>ABL ?0405B - ?@5645 2A53>, 87-70 2;8O=8O 0:B82=>3> A>?@>B82;5=8O AB0B>@0 R1, 2K7K20NI53> C<5=LH5=85 -! 8 <03=8B=>3> ?>B>:0 <0H8=K.   8A. 4.9 !E5<0 (0) 8 <5E0=8G5A:85 E0@0:B5@8AB8:8 (1) ?@8 G0AB>B=>< C?@02;5=88 . 45AL f1=fn/2, U1=Un/2, f2=2.fn, U2=2.Un ;O :><?5=A0F88 MB>3> 2;8O=8O A;54C5B, A C<5=LH5=85< G0AB>BK, A=860BL =0?@O65=85 2 <5=LH59 AB5?5=8, G5< MB> A;54C5B 87 (4.15). !;54C5B 70<5B8BL, GB> @0AA<>B@5==K5 70:>=K 87<5=5=8O =0?@O65=8O U1 ?@8 87<5=5=88 G0AB>BK f1 =5 2 ?>;=>9 <5@5 >B@060NB A?5F8D8:C @01>BK  ?@8 G0AB>B=>< @53C;8@>20=88. 0: A;54C5B 87 (4.4), <03=8B=K9 ?>B>: 0A8=E@>==>9 <0H8=K,  EMBED Equation.3  , (4.18) 7028A8B =5 B>;L:> >B =0?@O65=8O U1 8 G0AB>BK f1, => 8 >B B>:0 <0H8=K. 0:A8<0;L=K9 <03=8B=K9 ?>B>: (mo 8<55B <5AB> 2 @568<5 E>;>AB>3> E>40, :>340 <>6=> A 4>AB0B>G=>9 B>G=>ABLN ?@8=OBL U1 ( -1 8  EMBED Equation.3  (4.19) =0G5=85 ?>B>:0 E>;>AB>3> E>40, A>>B25BAB2CNI55 =><8=0;L=><C =0?@O65=8N 8 =><8=0;L=>9 G0AB>B5, O2;O5BAO ?@545;L=K< 4;O 0A8=E@>==>9 <0H8=K. @8 ;N1>< 4@C3>< @568<5 B5:CI55 7=0G5=85 ?>B>:0 =5 4>;65= 53> ?@52KH0BL 1>;55 G5< =0 10  15 %. '0AB>B=>5 @53C;8@>20=85 2 A>>B25BAB288 A 70:>=>< U1 / f1 = const, :>340 >4=>2@5<5==> 87<5=O5BAO 8 G0AB>B0 8 =0?@O65=85, A>>B25BAB2C5B MB><C B@51>20=8N. 59AB28B5;L=>, 2 D>@<C;5 (4.18) >4=>2@5<5==> @0ABCB (8;8 ?040NB) 8 G8A;8B5;L 8 7=0<5=0B5;L 8 ?>B>: >AB05BAO ?@0:B8G5A:8 =587<5==K<. !>2A5< 4@C3>5 45;> 70:>= U1 / f12 = const (4.16). = B@51C5B 1>;55 1KAB@>3> C25;8G5=8O =0?@O65=8O, G5< G0AB>BK, 8 745AL =5>1E>48<> ?@>25@OBL, =5 ?@52>AE>48B ;8 25;8G8=0 <03=8B=>3> ?>B>:0 4>?CAB8<>5 7=0G5=85 ((mo ( (m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~1500 >1/<8=), 25;8G8=0 2@0I0NI53> <><5=B0 4>;6=0 1KBL C<5=LH5=0, 5A;8 B@51C5BAO A:>@>ABL 1>;LH5 G5< =><8=0;L=0O (1500 >1/<8=). 0 @8A.4.10 <>6=> 2K45;8BL B@8 @01>G85 >1;0AB8: ?>AB>O==>3> <><5=B0 =const, ?>AB>O==>9 <>I=>AB8 P=const, 345 2@0I0NI89 <><5=B O2;O5BAO DC=:F859 >1@0B=>9 A:>@>AB8, (8 B0:8< >1@07>< >15A?5G8205B ?>AB>O=AB2> <>I=>AB8 P = M.) 8 >1;0ABL <>I=>ABL * A:>@>ABL = const, 345 =><8=0;L=K9 <><5=B O2;O5BAO DC=:F859 >1@0B=> ?@>?>@F8>=0;L=>9 :204@0BC A:>@>AB8. 'B>1K 2KOA=OBL @07;8G85 <564C 42C<O ?>A;54=8<8 7>=0<8 @0AA<>B@8< E0@0:B5@ 87<5=5=8O <0:A8<0;L=>3> <><5=B0 4;O CAB0=>282H53>AO @568<0 @01>BK. !>3;0A=> (4.11) <0:A8<0;L=K9 2@0I0NI89 <><5=B <>65B 1KBL ?@81;878B5;L=> =0945= A 8A?>;L7>20=85< A;54CNI59 D>@<C;K:  EMBED Equation.3  (4.20) 345 Ls 8 L r - A>>B25BAB25==> 8=4C:B82=>AB8 @0AA5O=8O AB0B>@0 8 @>B>@0, p - G8A;> ?0@ ?>;NA>2, 0 1=2f1 .  ?5@2>9 7>=5, DC=:F8O <0:A8<0;L=>3> 2@0I0NI53> <><5=B0 ?>AB>O==0, B0: :0: U (D07=>5 =0?@O65=85) C25;8G8205BAO ;8=59=> A> A:>@>ABLN.  7>=5 2KH5 =><8=0;L=>9 A:>@>AB8, =0?@O65=85 ?>445@68205BAO ?>AB>O==K< 8 @02=K< 53> =><8=0;L=><C 7=0G5=8N. @8 MB>< <0:A8<0;L=K9 2@0I0NI89 <><5=B ?@54AB02;O5B A>1>9 DC=:F8N >1@0B=CN :204@0BC A:>@>AB8.  8A. 4.11 0:A8<0;L=K9 8 =><8=0;L=K9 <><5=BK 2 DC=:F88 A:>@>AB8 7 8A. 4.11 284=>, GB> :@820O =><8=0;L=>3> 2@0I0NI53> <><5=B0 ?5@5A5:05B :@82CN <0:A8<0;L=>3> 2@0I0NI53> <><5=B0. -B0 B>G:0 ?5@5A5G5=8O  3@0=8F0, >B45;ONI0O 7>=C ?>AB>O==>9 <>I=>AB8 >B 7>=K <>I=>ABL*A:>@>ABL = const. B<5B8<, GB> :@820O =><8=0;L=>3> 2@0I0NI53> <><5=B0 ?5@5A5:05B AB0B8G5A:85 <5E0=8G5A:85 E0@0:B5@8AB8:8 2 B>G:0E CAB>9G82>9 @01>BK (=><8=0;L=K9 2@0I0NI89 <><5=B <5=LH5 G5< <0:A8<0;L=K9 2@0I0NI89 <><5=B) 2?;>BL 4> MB>9 3@0=8G=>9 B>G:8. 0: B>;L:> MB0 B>G:0 4>AB83=CB0, =><8=0;L=K9 2@0I0NI89 <><5=B ?@8=8<05BAO @02=K< <0:A8<0;L=><C 2@0I0NI5<C <><5=BC, 0 <>I=>ABL AB0=>28BAO DC=:F859 >1@0B=>9 A:>@>AB8. ! ?><>ILN 2KH5?@82545==KE D>@<C; 8 ?0@0<5B@>2 42830B5;O <>6=> =09B8 MBC B>G:C.  @0AA<0B@8205<>< ?@8<5@5 B>G:0 ?5@5A5G5=8O 8<55B <5AB> ?@8 A:>@>AB8 @02=>9 1.8 =><8=0;L=>9 (1.8*1500 = 2700>1/<8=). 0<=>3> 1>;LH89 480?07>= @53C;8@>20=8O A:>@>AB8 <>65B 1KBL 4>AB83=CB 4;O 42830B5;59 A 1>;55 2KA>:8< >B=>H5=85< <0:A8<0;L=>3> <><5=B0 : =><8=0;L=><C, GB> A428305B B>G:C ?5@5A5G5=8O 2?@02>. 3@0=8G5=8O<8, ?@5?OBAB2CNI8<8 @0AH8@5=8N 480?07>=0 @53C;8@>20=8O A:>@>AB8, O2;ONBAO, 2>-?5@2KE, D07=>5 =0?@O65=85 8, 2>-2B>@KE, D07=K9 B>:. #G8BK20O, GB> E>BO D07=>5 =0?@O65=85 4>;6=> 2>7@0AB0BL A C25;8G5=85< A:>@>AB8, => 53> 7=0G5=85 =5 4>;6=> ?@52KA8BL =><8=0;L=>5 7=0G5=85, 25;8G8=0 ?>B>:0 4>;6=0 1KBL C<5=LH5=0 4> 7=0G5=8O, :>B>@>5 A>>B25BAB2C5B ?>445@60=8N =><8=0;L=>3> D07=>3> =0?@O65=8O ?@8 B@51C5<>9 A:>@>AB8.  8A. 4.12 3@0=8G5=8O @53C;8@>20=8O A:>@>AB8 >A;01;5=85< ?>;O "0: :0: D07=K9 B>: C25;8G8205BAO A @>AB>< =03@C7:8, B> <0:A8<0;L=K9 2@0I0NI89 <><5=B A>?@>B82;5=8O ?@82>40 ?@8 C25;8G5=88 A:>@>AB=>3> 480?07>=0, 4>;65= 1KBL B0:8<, GB>1K D07=K5 B>:8 =5 ?@52>AE>48;8 =><8=0;L=>5 7=0G5=85. 0:A8<0;L=K9 <><5=B A>?@>B82;5=8O 4>;65= C<5=LH0BLAO 2 DC=:F8O A:>@>AB8. 0 @8AC=:5 4.12 8 <0:A8<0;L=>5 =0?@O65=85 D07K 8 ?>B>: ?>:070=K :0: 4;O =>@<0;L=>3>, B0: 8 4;O @0AH8@5==>3> 480?07>=>2 A:>@>AB8. ">@<>7=K5 @568<K @01>BK @8 2:;NG5=88 0A8=E@>==>3> 42830B5;O ?> >A=>2=>9 AE5<5, <>65B 1KBL @50;87>20=> B>@<>65=85 ?@>B82>2:;NG5=85< 8;8 @5:C?5@0B82=>5 B>@<>65=85. ;O ?5@52>40  2 @568< B>@<>65=8O ?@>B82>2:;NG5=85< =5>1E>48<> 87<5=8BL =0?@02;5=85 2@0I5=8O <03=8B=>3> ?>;O. ;O MB>3> 4>AB0B>G=> 87<5=8BL ?>@O4>: G5@54>20=8O D07 =0 7068<0E AB0B>@=>9 >1<>B:8, 4;O G53> 4>AB0B>G=> ?><5=OBL <5AB0<8 420 2K2>40. 0?@8<5@ ?@8 ?>4:;NG5=88  ( !1,  ( !2 8 ! ( !3 4>AB0B>G=> ?5@5:;NG8BL  ( !2,  ( !1 8 ! ( !3. CA:09  @01>B05B =0 <5E0=8G5A:>9 E0@0:B5@8AB8:5 1 (@8A. 4.13) 2 B>G:5 0 =0 ?5@5A5G5=88 <5E0=8G5A:8E E0@0:B5@8AB8: 42830B5;O (1) 8 =03@C7:8 (4) ?@8 G5@54>20=88 D07 !. @8 ?5@5:;NG5=88 42CE D07, =0?@8<5@  8 , 42830B5;L ?5@5945B =0 CG0AB>: 2A <5E0=8G5A:>9 E0@0:B5@8AB8:8 2 , A>>B25BAB2CNI89 B>@<>65=8N ?@>B82>2:;NG5=85<. B<5B8<, GB> ?@8 @50;870F88 B>@<>7=KE @568<>2, :0: ?@028;>, B@51C5BAO >3@0=8G8BL B>: 8 <><5=B , 4;O G53> 2 F5?L @>B>@0 8;8 AB0B>@0 22>4OBAO 4>102>G=K5 A>?@>B82;5=8O.   EMBED Mathcad  8A. 4.13 ">@<>65=85  ?@8 >A=>2=>9 AE5<5 2:;NG5=8O @8 0:B82=>< E0@0:B5@5 =03@C7:8 8 =0;8G88  A D07=K< @>B>@>< <>65B 1KBL 8A?>;L7>20= 4@C3>9 <5B>4 B>@<>65=8O. >?CAB8<, GB> B@51C5BAO >ACI5AB28BL A?CA: 3@C70, >ACI5AB2;OO 53> B>@<>65=85 A ?><>ILN 42830B5;O. ;O MB>3>, ?CB5< 2:;NG5=8O 2 F5?L @>B>@0 4>102>G=>3> A>?@>B82;5=8O, 42830B5;L ?5@52>48BAO =0 8A:CAAB25==CN E0@0:B5@8AB8:C (:@820O 3 =0 @8A. 4.13). 7-70 ?@52KH5=8O <><5=B>< =03@C7:8 <><5=B0  8 53> 0:B82=>3> E0@0:B5@0, 3@C7 =0G=5B >?CA:0BLAO A CAB0=>282H59AO A:>@>ABLN -(CAB. 5:C?5@0B82=>5 B>@<>65=85 >ACI5AB2;O5BAO 2 B>< A;CG05, 5A;8 A:>@>ABL @>B>@0 ?@52KH05B A8=E@>==CN. "0:>9 @568< <>65B 2>7=8:=CBL ?@8 B>@<>65=88  ?5@5E>45 42CEA:>@>AB=>3>  A 2KA>:>9 A:>@>AB8 =0 =87:CN. -B>B 65 284 B>@<>65=8O <>65B 1KBL @50;87>20= 2 A8AB5<5 ?@5>1@07>20B5;L G0AB>BK  42830B5;L ?@8 >AB0=>2:5  8;8 ?5@5E>45 53> =0 4@C3CN E0@0:B5@8AB8:C. ;O >ACI5AB2;5=8O @568<0 48=0<8G5A:>3> B>@<>65=8O >1<>B:C AB0B>@0  >B:;NG0NB >B A5B8 ?5@5<5==>3> B>:0 8 ?>4:;NG0NB : 8AB>G=8:C ?>AB>O==>3> B>:0, :0: MB> ?>:070=> =0 @8A.4.14. I4 = U/(1.5 Rs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s) ) @0AA<0B@8205BAO :0: <3=>25==>5 7=0G5=85 =5:>B>@>3> ?5@5<5==>3> B>:0, A>740NI53> B0:CN 65 ! IM:2, ?@8G5< IM:2 = I4 /(2. >4@>1=K9 0=0;87 [1, AB@. 256 - 260] ?>72>;O5B ?>;CG8BL 2K@065=8O 4;O M;5:B@><5E0=8G5A:8E E0@0:B5@8AB8:  ?@8 48=0<8G5A:>< B>@<>65=88:  EMBED Equation.3 ,  EMBED Equation.3 . (4.21) 345 A:>;L65=85 ?@8 48=0<8G5A:>< B>@<>65=88 s = ( /(o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atlab, 3@0D8:8 7028A8<>AB8 A:>@>AB8 ((t) 8 <><5=B0 (t) >B 2@5<5=8 ?@8 ?@O<>< ?CA:5 0A8=E@>==>3> :>@>B:>70<:=CB>3> 42830B5;O.  8A. 5.1 @0D8:8 7028A8<>AB8 A:>@>AB8 (t) 8 <><5=B0 (t) >B 2@5<5=8  8A. 5.2 028A8<>ABL <><5=B0 >B C3;>2>9 A:>@>AB8 ().  ?@>F5AA5 <>45;8@>20=8O 2=0G0;5 2K?>;=O;AO ?@O<>9 ?CA: , 0 70B5<  =01@>A (A:0G:>>1@07=>5 C25;8G5=85) =03@C7:8 4> =><8=0;L=>9. 0 3@0D8:0E E>@>H> 284=K (2 A>>B25BAB2CNI89 <><5=B 2@5<5=8 t = 1) C<5=LH5=85 A:>@>AB8 8 @>AB <><5=B0 ! ?><>ILN 7028A8<>AB59 ((t) 8 (t) <>6=> ?>AB@>8BL B0: =07K205<CN 48=0<8G5A:CN <5E0=8G5A:CN E0@0:B5@8AB8:C  (() (@8A.5.2), ?@54AB02;ONICN A>1>9 D07>2CN B@05:B>@8N ?5@5E>4=>3> ?@>F5AA0 ?CA:0 . !@02=820O MBC E0@0:B5@8AB8:C A> AB0B8G5A:>9 <5E0=8G5A:>9 E0@0:B5@8AB8:>9, <>6=> 70<5B8BL 25AL<0 ACI5AB25==K5 >B;8G8O.  ?5@2CN >G5@54L >=8 A2O70=K A :>;510=8O<8 48=0<8G5A:>3> <><5=B0, ?@8G5< <0:A8<0;L=K5 7=0G5=8O ?5@5E>4=>3> <><5=B0 25AL<0 7=0G8B5;L=K, 8 <>3CB 2 =5A:>;L:> @07 ?@52>AE>48BL <0:A8<0;L=K9 <><5=B AB0B8G5A:>3> @568<0.  =5:>B>@K5 <><5=BK 2@5<5=8 48=0<8G5A:89 <><5=B <>65B 4065 AB0BL >B@8F0B5;L=K< (B>@<>7=K<), GB> <>6=> 2845BL =0 =0G0;L=>< CG0AB:5 :@82>9 <><5=B0. @C3>5 ACI5AB25==>5 >B;8G85 48=0<8G5A:>9 <5E0=8G5A:>9 E0@0:B5@8AB8:8 >B AB0B8G5A:>9 70:;NG05BAO 2 B><, GB> >=0 >?@545;O5BAO =5 B>;L:> ?0@0<5B@0<8 A0<>9 <0H8=K, => 8 <><5=B>< =03@C7:8 8 <><5=B>< 8=5@F88 M;5:B@>?@82>40. , @07C<55BAO, E0@0:B5@ :@82KE 7028A8B >B 2840 ?5@5E>4=>3> ?@>F5AA0 (?CA:, =01@>A =03@C7:8, @525@A, B>@<>65=85). !@02=5=85 3@0D8:>2 @07;8G=KE ?5@5E>4=KE ?@>F5AA>2 ?>72>;O5B >B<5B8BL A;54CNICN 70:>=><5@=>ABL: ?@8 C25;8G5=88 :0: <><5=B0 8=5@F88, B0: 8 <><5=B0 =03@C7:8 C25;8G8205BAO 4;8B5;L=>ABL :0: <5E0=8G5A:>3>, B0: 8 M;5:B@><5E0=8G5A:>3> ?5@5E>4=>3> ?@>F5AA0. ;O ?5@5E>4=KE ?@>F5AA>2 2 A8AB5<5 '   E0@0:B5@=K ACI5AB25==> <5=LH85 1@>A:8 B>:0 8 <><5=B0, G5< 2 A;CG05 ?@O<>3> ?CA:0 42830B5;O. 04020O 65;05<K9 70:>= 87<5=5=8O G0AB>BK 8 =0?@O65=8O, <>6=> @50;87>20BL B@51C5<K5 3@0D8:8 87<5=5=8O A:>@>AB8  2 ?5@5E>4=>< ?@>F5AA5. 6. !8;>2K5 ?@5>1@07>20B5;8 M;5:B@>?@82>40 ?>AB>O==>3> B>:0  @53C;8@C5<KE M;5:B@>?@82>40E ?>AB>O==>3> B>:0 A8;>2K5 25=B8;L=K5 ?@5>1@07>20B5;8 ?@8<5=ONBAO =5 B>;L:> 4;O 2K?@O<;5=8O ?5@5<5==>3> B>:0 2 ?>AB>O==K9, :0: 2 >1KG=KE 2K?@O<8B5;OE. 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:@82>9 2.  :>=F5 ?>;C?5@8>40 ( t = ) 2 <5=O5B 7=0:, 8 B8@8AB>@ VS1 70?8@05BAO. 0B5<, G5@57 8=B5@20; t =  C?@02;ONI89 8<?C;LA ?>405BAO C65 =0 B8@8AB>@ VS2 8 >= 2ABC?05B 2 @01>BC. 0;55 ?@>F5AAK ?>2B>@ONBAO. @8 @01>B5 2K?@O<8B5;O =0 0:B82=CN =03@C7:C D>@<0 :@82>9 B>:0 ?>;=>ABLN ?>2B>@O5B D>@<C :@82>9 =0?@O65=8O (A<. 3@0D8: B>:0 =0 @8A. 6.21). A;8 87<5=OBL C3>; C?@02;5=8O , B> 1C45B 87<5=OBLAO 2@5<O >B:@KB>3> A>AB>O=8O B8@8AB>@>2 8, A>>B25BAB25==>, A@54=OO 25;8G8=0 2K?@O<;5==>9 -! Ed =  EMBED Equation.3  d0 (1+cos )/2, (6.1) 345: Ed0 = 0.9.E2D.  =081>;LH55 A@54=55 7=0G5=85 2K?@O<;5==>3> =0?@O65=8O ?@8 (=0). 028A8<>ABL Ed() =07K205BAO @53C;8@>2>G=>9 E0@0:B5@8AB8:>9. ;O 0:B82=>9 =03@C7:8 >=0 ?@54AB02;5=0 =0 @8A.6.30 (R). @8 @01>B5 C?@02;O5<>3> 2K?@O<8B5;O =0 0:B82=>-8=4C:B82=CN =03@C7:C 8=4C:B82=>ABL L= A >4=>9 AB>@>=K ?@5?OBAB2C5B =0@0AB0=8N B>:0 (=5?>A@54AB25==> ?>A;5 2:;NG5=8O B8@8AB>@0), 0 A 4@C3>9  ?>445@68205B 53> 4065 ?>A;5 87<5=5=8O 7=0:0 2. "5?5@L B8@8AB>@ 70:@K205BAO =5 2 <><5=B t = , 0 2 <><5=B t = +, 345   4;8B5;L=>ABL ?@>2>4OI53> A>AB>O=8O 25=B8;59 (?@>B5:0=8O B>:0), :>B>@0O 1>;LH5, G5< -. -B> ?@82>48B : B><C, GB> A@54=55 7=0G5=85 2KE>4=>9 -! A=8605BAO.  8A. 6.3 53C;8@>2>G=K5 (0) 8 2=5H=85 (1) E0@0:B5@8AB8:8 >4=>D07=>3> C?@02;O5<>3> 2K?@O<8B5;O >7<>6=K 420 @568<0 @01>BK C?@02;O5<>3> 2K?@O<8B5;O: @568< ?@5@K28AB>3> B>:0 8 @568< =5?@5@K2=>3> B>:0.  @568<5 ?@5@K28AB>3> B>:0 ( > :@) <564C 8=B5@20;0<8 ?@>2>48<>AB8 25=B8;59 8<5NBAO 15AB>:>2K5 ?0C7K. @8 C25;8G5=88 8=4C:B82=>AB8 8;8 C<5=LH5=88 R= 4;8B5;L=>ABL 15AB>:>2>9 ?0C7K C<5=LH05BAO 8 ?@8  >  8 Id > Id:@ 2K?@O<8B5;L ?5@5E>48B 2 @568< =5?@5@K2=>3> B>:0. 568< =5?@5@K2=>3> B>:0 O2;O5BAO =081>;55 E0@0:B5@=K< 4;O <>I=KE 2K?@O<8B5;59, 4;O :>B>@KE >1KG=> L= >>R=.  MB>< @568<5 4;8B5;L=>ABL ?@>B5:0=8O B>:0 G5@57 25=B8;L @02=0 , B.5. 2 ;N1>9 <><5=B 2@5<5=8 =03@C7:0 ?>4:;NG5=0 : >4=>9 87 ?>;C>1<>B>: B@0=AD>@<0B>@0. !@54=55 7=0G5=85 -! 2K?@O<8B5;O 2 MB>< @568<5 Ed =  EMBED Equation.3  d0 cos , (6.2) 53C;8@>2>G=0O E0@0:B5@8AB8:0 2K?@O<8B5;O 2 @568<5 =5?@5@K2=>3> B>:0 ?@54AB02;5=0 =0 @8A.6.30 L , 0 2=5H=85 ?@8 Id > Id:@  =0 @8A. 6.31. 4=>D07=0O <>AB>20O AE5<0 (@8A.6.12) 8<55B C42>5==>5 G8A;> B8@8AB>@>2, >4=0:> =5 B@51C5B ?@8<5=5=8O A?5F80;L=>3> B@0=AD>@<0B>@0 A 2K2545==>9 A@54=59 B>G:>9. 5 A2>9AB20 1;87:8 >4=>D07=>9 =C;52>9 AE5<5, 2 G0AB=>AB8, >=0 B0:65 8<55B 2KA>:89 C@>25=L ?C;LA0F88 2K?@O<;5==>3> =0?@O65=8O q (A<. B01;.6.1) ;O <>I=KE CAB0=>2>: ?@8<5=ONBAO B@5ED07=K5 AE5<K 2K?@O<;5=8O, >15A?5G820NI85 7=0G8B5;L=>5 A=865=85 ?C;LA0F89. 081>;55 ?@>AB>9 O2;O5BAO AE5<0 B@5ED07=>3> =C;52>3> ?@5>1@07>20B5;O (@8A.6.11), 8<5NI0O <8=8<0;L=> G8A;> B8@8AB>@>2 (B@8) 8 >1KG=K9 42CE>1<>B>G=K9 B@5ED07=K9 B@0=AD>@<0B>@. 45AL :@ = 30 8 ?@8 0 <  < 30 2K?@O<;5==K9 B>: =5?@5@K2=K9 8 7028A8<>ABL 2K?@O<;5==>3> =0?@O65=8O >B C3;0 C?@02;5=8O >?@545;O5BAO 2K@065=85< (6.2). @8 0:B82=>9 =03@C7:5 >1;0ABL  > 30 A>>B25BAB2C5B @568<C ?@5@K28AB>3> B>:0: Ud = Ud0 [1+cos ( 30 + )]/ "2. (6.3) @545;L=K< C3;>< @53C;8@>20=8O ?@8 @01>B5 MB8E ?@5>1@07>20B5;59 =0 0:B82=CN =03@C7:C O2;O5BAO C3>;  = 150. 54>AB0B:>< B0:>9 AE5<K O2;O5BAO A@02=8B5;L=> 2KA>:89 C@>25=L ?C;LA0F89 (q) 2K?@O<;5==>3> =0?@O65=8O (A<. B01;.6.1).  B@5ED07=>< <>AB>2>< C?@02;O5<>< ?@5>1@07>20B5;5 (@8A.6.1, 3), >4=>2@5<5==> @01>B0NB 420 B8@8AB>@0; >48= 87 :0B>4=>9, 4@C3>9 87 0=>4=>9 3@C??K. @5>1@07>20B5;L 8<55B C42>5==>5 G8A;> 25=B8;59 (?> A@02=5=8N A B@5ED07=>9 =C;52>9 AE5<>9) 8 4>AB0B>G=> =87:89 C@>25=L ?C;LA0F89 2K?@O<;5==>3> =0?@O65=8O (A<. B01;.6.1). 45AL :@ = 60 8 2K?@O<;5==K9 B>: 1C45B =5?@5@K2=K< ?@8 0 <  < 60 . 028A8<>ABL 2K?@O<;5==>3> =0?@O65=8O >B C3;0 C?@02;5=8O >?8AK205BAO 2K@065=85< (6.1). @8  > 60 8 0:B82=>9 =03@C7:5, :@82K5 2K?@O<;5==>3> B>:0 8 =0?@O65=8O AB0=>2OBAO ?@5@K28ABK<8 8 2K?@O<;5==>5 =0?@O65=85 Ud = Ud0 [1+cos ( 60 + )]. (6.4) @545;L=K< C3;>< @53C;8@>20=8O ?@8 @01>B5 MB8E ?@5>1@07>20B5;59 =0 0:B82=CN =03@C7:C O2;O5BAO C3>;  = 120. !@02=5=85 >A=>2=KE ?0@0<5B@>2, E0@0:B5@87CNI8E @01>BC @07;8G=KE AE5< ?@5>1@07>20=8O ?@8 8E @01>B5 =0 G8AB> 0:B82=CN =03@C7:C (L =0), ?@82545=K 2 B01;.6.1. "01;8F0 6.1 !@02=5=85 ?0@0<5B@>2 @07;8G=KE AE5< ?@5>1@07>20B5;59 ?@8 L =0 !E5<0 do/2DI2/Id>1@</doq4=>D07=0O =C;520O0.90.7852.140.674=>D07=0O <>AB>20O0.91.111.570.67"@5ED07=0O =C;520O1.170.5852.090.25"@5ED07=0O <>AB>20O2.340.8171.050.05745AL q = d1m / d  :>MDD8F85=B ?C;LA0F88, @02=K9 >B=>H5=8N 0<?;8BC4K <0:A8<0;L=>9 30@<>=8:8 : A@54=59 -! =0 2KE>45 2K?@O<8B5;O ?@8 =0, 0 >1@<  <0:A8<0;L=0O >1@0B=0O -!, ?@8;>65==0O : B8@8AB>@C. 7. !E5<K, E0@0:B5@8AB8:8 8 @568<K @01>BK " !E5<0 2:;NG5=8O 42830B5;O ?>AB>O==>3> B>:0 (") A =57028A8<K< 2>71C645=85< ?@82545=0 =0 @8A.7.10 8A. 7.1 !E5<0 2:;NG5=8O (0) 8 <5E0=8G5A:85 E0@0:B5@8AB8:8 " 45AL: Ia 8 If  B>:8 O:>@O 8 >1<>B:8 2>71C645=8O; 0  -! >1<>B:8 O:>@O; (  C3;>20O A:>@>ABL 42830B5;O; 2  <><5=B =0 20;C 42830B5;O;R2 8 R4  4>102>G=>5 A>?@>B82;5=85 2 F5?8 O:>@O (>=> <>65B >BACBAB2>20BL); R0 = r0 + r4? + rI  ?>;=>5 0:B82=>5 A>?@>B82;5=85 O:>@=>9 F5?8, A>AB>OI55 87 A>?@>B82;5=8O >1<>B:8 O:>@O, 4>102>G=KE ?>;NA>2 8 I5B>G=>3> :>=B0:B0; La - 8=4C:B82=>ABL O:>@=>9 F5?8; Rf 8 Lf 0:B82=>5 A>?@>B82;5=85 8 8=4C:B82=>ABL >1<>B:8 2>71C645=8O #@02=5=8O, :>B>@K5 >?8AK20NB M;5:B@><03=8B=K5 8 M;5:B@><5E0=8G5A:85 ?@>F5AAK 2 MB>< 42830B5;5, 8<5NB 284:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (7.1)  EMBED Equation.DSMT4 . 45AL:  EMBED Equation.DSMT4  - -! 42830B5;O;  EMBED Equation.DSMT4  - M;5:B@><03=8B=K9 <><5=B;  EMBED Equation.DSMT4 , ! =  EMBED Equation.DSMT4   ?>AB>O==0O 42830B5;O, p  G8A;> ?0@ ?>;NA>2, N  G8A;> 0:B82=KE ?@>2>4=8:>2 2 O:>@5, 20  G8A;> ?0@0;;5;L=KE 25B259 >1<>B:8 O:>@O, $  <03=8B=K9 ?>B>:, Laf  8=4C:B82=>ABL 2708<>8=4C:F88 >1<>B>: O:>@O 8 2>71C645=8O. =45:A>< 0 >B<5G5=K ?5@5<5==K5 8 ?0@0<5B@K, >B=>AOI85AO : >1<>B:5 O:>@O, 0 8=45:A>< f - >B=>AOI85AO : >1<>B:5 2>71C645=8O. @8 @0AA<>B@5=88 AB0B8G5A:8E @568<>2 @01>BK, 48=0<8G5A:8<8 A>AB02;ONI8<8 2 C@02=5=8OE M;5:B@8G5A:>3> 8 <5E0=8G5A:>3> @02=>25A8O (  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 8 EMBED Equation.DSMT4 ) ?@5=51@530NB, ?>;030O 8E @02=K<8 =C;N. >;CG05< 2<5AB> (7.1): U0 = E0+ Ia.(R0+Rd) Uf = If.Rf (7.2)  = !Ia = A >4AB02;OO 2K@065=85 4;O E0 C@02=5=85 M;5:B@8G5A:>3> @02=>25A8O O:>@=>9 F5?8, ?>;CG05< D>@<C;C 4;O G0AB>BK 2@0I5=8O " - C@02=5=85 M;5:B@><5E0=8G5A:>9 E0@0:B5@8AB8:8 ((I): ( = U/(C$)  Ia.(Ra+Rd)/(C$) (7.3) $>@<C;C 4;O <5E0=8G5A:>9 E0@0:B5@8AB8:8 (() ?>;CG05<, ?>4AB02;OO 2 (7.3) 2K@065=85 4;O B>:0, ?>;CG5==>5 87 D>@<C;K <><5=B0 ( = C.$.Ia): ( = U/(C$)  .(Ra+Rd)/(C$)2 (7.4) "0: :0: <><5=B =0 20;C 42830B5;O 2 =5A:>;L:> <5=LH5 (>1KG=> =0 4  6 %), G5< M;5:B@><03=8B=K9 <><5=B , ?>;CG5==0O D>@<C;0 O2;ONBAO 2 =5:>B>@>9 AB5?5=8 ?@81;865==>9. 7 (7.3) 8 (7.4) A;54CNB B@8 A?>A>10 @53C;8@>20=8O A:>@>AB8 42830B5;O - 87<5=5=85< =0?@O65=8O U, <03=8B=>3> ?>B>:0 $ 8 4>102>G=>3> A>?@>B82;5=8O 2 O:>@=>9 F5?8 R4. K@065=8O (7.3) 8 (7.4) <>6=> ?@54AB028BL 2 2845 ( = (0 - ((, 345 (0 - A:>@>ABL 8450;L=>3> E>;>AB>3> E>40 (0 = U/(C$), 0 (( - 87<5=5=85 A:>@>AB8 >B=>A8B5;L=> A:>@>AB8 8450;L=>3> E>;>AB>3> E>40 (0. (( = (0 - ( = Ia.(Ra+Rd) / (C$) = .(Ra+Rd) / (C$)2 (7.5) 2>4O ?>=OB85 >B=>A8B5;L=>3> 87<5=5=8O A:>@>AB8 " (0=0;>3 A:>;L65=8O 2 0A8=E@>==KE 42830B5;OE) ( = ((0 - ()/(0, ?>;CG05<: ( = ((0 - ()/(0 = Ia.(Ra+Rd)/U. (7.6) K@065=85 (7.6) <>6=> AD>@<C;8@>20BL B0:: >B=>A8B5;L=>5 87<5=5=85 A:>@>AB8 " @02=> >B=>A8B5;L=><C ?045=8N =0?@O65=8O 2 F5?8 O:>@O. 0 @8A. 7.1, 1 ?>:070=K <5E0=8G5A:85 E0@0:B5@8AB8:8 ((=f()) ?@8 @07=KE 7=0G5=8OE =0?@O65=8O ?@8;>65==>3> : O:>@N. -;5:B@><5E0=8G5A:85 E0@0:B5@8AB8:8 ((=f(Ia)) 8<5NB 0=0;>38G=K9 284. 5E0=8G5A:0O (8 A>>B25BAB2CNI0O M;5:B@><5E0=8G5A:0O) E0@0:B5@8AB8:0 " A =57028A8<K< 2>71C645=85< ?@8 U = 0 (A@54=OO ?@O<0O =0 @8A. 7.1, 1) =07K205BAO E0@0:B5@8AB8:>9 48=0<8G5A:>3> B>@<>65=8O. -;5:B@8G5A:CN AE5<C 4;O A=OB8O E0@0:B5@8AB8:8 48=0<8G5A:>3> B>@<>65=8O (@8A.7.2) ?>;CG05<, 70<K:0O O:>@=CN F5?L 42830B5;O =0 4>102>G=>5 A>?@>B82;5=85 (8;8 =0:>@>B:>). 568<K @01>BK " 7028AOB >B <5E0=8G5A:8E (M. 8 () 8 M;5:B@8G5A:8E (E. 8 I) :>>@48=0B, >?@545;ONI8E 53> <5E0=8G5A:CN P2 = M.( 8 M;5:B@><03=8B=CN PM< = E.Ia <>I=>AB8.  8A. 7.2 !E5<0 48=0<8G5A:>3> B>@<>65=8O " 568< E>;>AB>3> E>40 8<55B <5AB> 2 B>G:5 0. 45AL  ( U ( C.$.(0, M = 0, Ia ( 0, ( = (0. 2830B5;L =5 ?>;CG05B M=5@388 - =8 87 M;5:B@8G5A:>9 A5B8 (70 8A:;NG5=85< <>I=>AB8 2>71C645=8O 8 ?>B5@L E>;>AB>3> E>40) =8 A 20;0; 2830B5;L=K9 @568< 8<55B <5AB> =0 CG0AB:5 1 (01) 2 ?5@2>< :204@0=B5 ?@8 0 < ( < (0. 45AL ( 8 M A>2?040NB ?> =0?@02;5=8N, -! <5=LH5 =0?@O65=8O ( < U), B>: Ia A>2?0405B ?> =0?@02;5=8N A U 8 =0?@02;5= =02AB@5GC -!; 5=5@0B>@=K9 @568< 8<55B <5AB> 2> 2B>@>< :204@0=B5 ?@8 ( > (0 . 45AL A:>@>ABL ( 8 <><5=B M =0?@02;5=K 2AB@5G=>, -! 1>;LH5 =0?@O65=8O  > U, B>: Ia A>2?0405B ?> =0?@02;5=8N A -! 8 =0?@02;5= =02AB@5GC =0?@O65=8N U; 568< :>@>B:>3> 70<K:0=8O 8<55B <5AB> 2 B>G:5 1 ?@8 ( = 0 8  = 0.  MB>< @568<5 B>: Ia = U / (Ra+Rd), 8 2AO ?>ABC?0NI0O 87 A5B8 M=5@38O @0AA58205BAO 2 42830B5;5 8 2 R4 , 0 2 = 0; 568< B>@<>65=8O ?@>B82>2:;NG5=85< 8<55B <5AB> ?@8 ( < 0. 7 70 87<5=5=8O =0?@02;5=8O A:>@>AB8 87<5=O5BAO =0?@02;5=85 -!, :>B>@0O B5?5@L A>2?0405B ?> =0?@02;5=8N A =0?@O65=85< A5B8. ">: 2 O:>@5 I0 = (U + E ) / (Ra+Rd). AO M=5@38O - :0: ?>ABC?0NI0O 87 A5B8, B0: 8 ?>ABC?0NI0O A 20;0, @0AA58205BAO 2 42830B5;5 8 4>102>G=>< A>?@>B82;5=88 R4, ?>MB><C @0AA<0B@8205<K9 @568< 2 B5?;>2>< >B=>H5=88 O2;O5BAO 25AL<0 =0?@O65==K<;  @568<5 48=0<8G5A:>3> B>@<>65=8O B>: A>2?0405B ?> =0?@02;5=8N A -!, 0 <>I=>ABL, ?>ABC?0NI0O A 20;0 42830B5;O, @0AA58205BAO 2 2845 B5?;0 =0 A>?@>B82;5=8OE O:>@=>9 F5?8. 53C;8@>20=85 A:>@>AB8, B>:0 8 <><5=B0 " 8.1 53C;8@>20=85 A:>@>AB8, B>:0 8 <><5=B0 A ?><>ILN @578AB>@>2 2 F5?8 O:>@O -B>B A?>A>1, G0AB> =07K205<K9 @5>AB0B=K<, ?@>AB 2 @50;870F88 8 H8@>:> 8A?>;L7C5BAO 4;O @53C;8@>20=8O A:>@>AB8, 0 B0:65 4;O @53C;8@>20=8O (>3@0=8G5=8O B>:0) 8 <><5=B0 " =57028A8<>3> 2>71C645=8O. !5<59AB2> 8A:CAAB25==KE (@53C;8@>2>G=KE) E0@0:B5@8AB8:, ?>;CG5==KE 4;O @07;8G=KE 7=0G5=89 A>?@>B82;5=8O O:>@=>9 F5?8 R (70 AG5B 87<5=5=8O 4>102>G=>3> A>?@>B82;5=8O R4) ?@54AB02;5=> =0 @8A.8.1.  8A.8.1 5>AB0B=K5 E0@0:B5@8AB8:8 " =57028A8<>3> 2>71C645=8O. "CB R42 > R41 7 C@02=5=8O <5E0=8G5A:>9 E0@0:B5@8AB8:8 (8.4) 284=>, GB> 25;8G8=0 A:>@>AB8 E>;>AB>3> E>40 (0 =5 7028A8B >B Ra+Rd. >MB><C 2A5 E0@0:B5@8AB8:8  :0: 5AB5AB25==0O (R4 = 0), B0: 8 8A:CAAB25==K5 (R4 ( 0) 8AE>4OB 87 >4=>9 B>G:8 (( = (0,  = 0). 0:;>= E0@0:B5@8AB8:, >?@545;O5<K9 87<5=5=85< A:>@>AB8 (( (A<.7.7), ?@>?>@F8>=0;5= ?>;=><C A>?@>B82;5=8N O:>@=>9 F5?8 (Ra+Rd), 8 A @>AB>< R4 C25;8G8205BAO, >=8 AB0=>2OBAO 1>;55 <O3:8<8. F5=8< MB>B A?>A>1 @53C;8@>20=8O: 80?07>= @53C;8@>20=8O A:>@>AB8 (D) =51>;LH>9, >1KG=> =5 ?@52>AE>48B 2(3; 0?@02;5=85 @53C;8@>20=8O  2=87 >B=>A8B5;L=> 5AB5AB25==>9 E0@0:B5@8AB8:8; ;02=>ABL @53C;8@>20=8O >?@545;O5BAO ?;02=>ABLN 87<5=5=8O R4 (>1KG=> =525;8:0); !B018;L=>ABL A:>@>AB8 A=8605BAO ?> <5@5 C25;8G5=8O 480?07>=0 @53C;8@>20=8O 8 2 A@54=5< =52KA>:0; -:>=><8G=>ABL @53C;8@>20=8O >F5=8<, A>?>AB0282 70B@0BK =0 @50;870F8N 40==>3> A?>A>10 8 AB>8<>ABL ?>B5@L <>I=>AB8 ?@8 @53C;8@>20=88. !B>8<>ABL 8A?>;L7C5<KE @578AB>@>2 >1KG=> <0;0, >4=0:> AB>8<>ABL ?>B5@L <>I=>AB8 <>65B 1KBL 25AL<0 7=0G8B5;L=>9. 0AA<>B@8<, :0: 87<5=ONBAO ?>B5@8 <>I=>AB8 ?@8 C<5=LH5=88 G0AB>BK 2@0I5=8O 42830B5;O. >B5@8 <>I=>AB8 (  @02=K @07=>AB8 ?>B@51;O5<>9 <>I=>AB8 1 = U.I 8 ?>;57=>9 <>I=>AB8 2 = .(. >;030O U H" =C.$.( =094Q< (  = U.I - .( = C.$.(0.I - C.$.I( = C.$.(0.I.( = 1.( (8.1) 7 (8.1) A;54C5B, GB> ?>B5@8 <>I=>AB8 2 42830B5;5 ?@O<> ?@>?>@F8>=0;L=K >B=>A8B5;L=><C 87<5=5=8N A:>@>AB8 (. "0:, =0?@8<5@, ?@8 A=865=88 A:>@>AB8 2 2 @070 ?> A@02=5=8N A> A:>@>ABLN 8450;L=>3> E>;>AB>3> E>40 ((=0,5), ?>;>28=0 ?>B@51;O5<>9 <>I=>AB8 845B =0 ?>B5@8, 8  ?@82>40 =5 <>65B ?@52>AE>48BL 50%. >?CAB8<0O =03@C7:0 A>>B25BAB2C5B =><8=0;L=><C <><5=BC 42830B5;O, 5A;8 ?@8 A=865=88 A:>@>AB8 =5 CEC4H0NBAO CA;>28O >E;0645=8O, =0?@8<5@, ?@8 =57028A8<>9 2=5H=59 25=B8;OF88.  A;CG05 65 ?@8<5=5=8O 42830B5;O A A0<>25=B8;OF859, ?@8 A=865=88 A:>@>AB8 >E;0645=85 CEC4H05BAO, GB> B@51C5B A>>B25BAB2CNI53> A=865=8O =03@C7:8. 5A<>B@O =0 =5 >G5=L 2KA>:85 B5E=8:>-M:>=><8G5A:85 ?>:070B5;8, @5>AB0B=>5 @53C;8@>20=85 A:>@>AB8 87-70 ?@>AB>BK @50;870F88 8A?>;L7C5BAO 4>AB0B>G=> H8@>:> - B>340, :>340 B@51C5BAO =51>;LH>9 480?07>= @53C;8@>20=8O 8;8 :>340 @01>B0 =0 ?>=865==KE A:>@>ABOE 8<55B :@0B:>2@5<5==K9 E0@0:B5@: 2 M;5:B@>?@82>40E ;8DB>2, ?>4J5<=KE :@0=>2, =5:>B>@KE AB0=:>2 8 4@. >;CG5==K5 8A:CAAB25==K5 E0@0:B5@8AB8:8 H8@>:> 8A?>;L7CNB 4;O >3@0=8G5=8O B>:0 8 <><5=B0 2 ?5@5E>4=KE @568<0E (?CA:, B>@<>65=85), :>340 B>: 8 <><5=B <>3CB AB0BL =54>?CAB8<> 1>;LH8<8.  ?5@2K9 <><5=B ?CA:0 42830B5;L =0E>48BAO 2 @568<5 :>@>B:>3> 70<K:0=8O (( = 0,  = 0, U = U=) 8 B>:, ?@8 53> ?CA:5 ?> 5AB5AB25==>9 E0@0:B5@8AB8:5 (R4=0), I? = U / R0. 7-70 <0;>AB8 R0 ?CA:>2>9 B>: ?@8 ?@O<>< ?CA:5 <>65B ?@52>AE>48BL =><8=0;L=K9 2 10(50 @07, 2 B> 2@5<O :0: 4>?CAB8<K9 B>: I4>? = (2(2.5)I=. 3@0=8G5=85 B>:0 ?@8 ?CA:5 4>AB8305BAO 22545=85< 2 O:>@=CN F5?L ?CA:>2>3> A>?@>B82;5=8O R4. "0:0O 65 =5>1E>48<>ABL 2>7=8:05B ?@8 B>@<>65=88 8 @525@A5 42830B5;O. ;O >3@0=8G5=8O ?CA:>2>3> B>:0 2 ?@>AB59H5< A;CG05 <>65B 1KBL 8A?>;L7>20=0 8A:CAAB25==0O M;5:B@><5E0=8G5A:0O E0@0:B5@8AB8:0, =0?@8<5@ E0@0:B5@8AB8:0 1 =0 @8A.8.2. >@O4>: ?CA:0 42830B5;O A;54CNI89: >= =0G8=05B @01>B0BL, ?@8 =0;8G88 2 F5?8 O:>@O 4>102>G=>3> A>?@>B82;5=8O R41, A <0:A8<0;L=> 4>?CAB8<>3> ?CA:>2>3> B>:0 Imax = (2 2,5).In 8 40;55 ?> E0@0:B5@8AB8:5 1. 0B5<, ?@8 B>:5 Imin H" 1.2 In 8 A:>@>AB8 (1, A>?@>B82;5=85 R41 HC=B8@C5BAO, 8 42830B5;L ?5@5E>48B =0 5AB5AB25==CN E0@0:B5@8AB8:C. !>?@>B82;5=85 R41 2K18@05BAO 87 CA;>28O >15A?5G5=8O <0:A8<0;L=>3> 4>?CAB8<>3> ?CA:>2>3> B>:0 Imax R41 = U / Imax  R0 (8.2)  8A. 8.2 AB5AB25==0O 8 8A:CAAB25==K5 E0@0:B5@8AB8:8 42830B5;O ?>AB>O==>3> B>:0  1>;LH8=AB25 A;CG0OE ?@8 ?CA:5 8A?>;L7C5BAO =5 >4=0, 0 =5A:>;L:> ABC?5=59 ?CA:>2>3> @5>AB0B0 8, A>>B25BAB25==>, =5A:>;L:> 8A:CAAB25==KE E0@0:B5@8AB8:. E :>;8G5AB2> >?@545;O5BAO <><5=B>< =03@C7:8 42830B5;O ?@8 ?CA:5 8 B@51>20=8O<8 : ?;02=>AB8 ?CA:0. 8=0<8G5A:>5 B>@<>65=85 (E0@0:B5@8AB8:0 2 =0 @8A.8.2) >ACI5AB2;O5BAO >B:;NG5=85< O:>@O >B A5B8 8 53> 70<K:0=85< =0 A>?@>B82;5=85 R42. 5;8G8=0 R41 >?@545;O5BAO ?> 4>?CAB8<><C 1@>A:C B>:0 2 <><5=B ?5@5:;NG5=8O A CG5B>< A>>B=>H5=8O (U R42 =  / I4>?  R0 ( U / I4>?  R0. (8.3) 525@A 8;8 B>@<>65=85 ?@>B82>2:;NG5=85< (E0@0:B5@8AB8:0 3) >ACI5AB2;O5BAO 87<5=5=85< ?>;O@=>AB8 ?@8;>65==>3> : O:>@N =0?@O65=8O, A >4=>2@5<5==K< 2:;NG5=85< 2 F5?L O:>@O @578AB>@0 R43.  MB>< @568<5 -! 8 =0?@O65=85 A>2?040NB ?> =0?@02;5=8N 8  R43 = (U +) / I4>?  RO ( 2.U / I4>?  RO. (8.4)  8A. 8.3  @0AG5BC @53C;8@>2>G=KE A>?@>B82;5=89 2 F5?8 O:>@O 0AA<>B@8< ?>4@>1=55 <5B>48:C @0AG5B0 @53C;8@>2>G=KE @578AB>@>2. 040G0 D>@<C;8@C5BAO A;54CNI8< >1@07><: 8725AB=K B5E=8G5A:85 40==K5 42830B5;O (RO, (0) 8 5AB5AB25==K5 (<5E0=8G5A:0O 8 M;5:B@><5E0=8G5A:0O) E0@0:B5@8AB8:8. > CA;>28O< @53C;8@>20=8O A:>@>AB8, B>:0 8;8 <><5=B0 7040=0 8A:CAAB25==0O E0@0:B5@8AB8:0, ?@>E>4OI0O G5@57 >?@545;5==CN B>G:C (I?, (1 =0 @8A.8.3). 5;8G8=0 4>102>G=>3> A>?@>B82;5=8O R4, ?@8 2:;NG5=88 :>B>@>3> 2 F5?L O:>@O >=0 1C45B ?>;CG5=0: R4 = U.(1 / I?  R0, (8.5) 345 (1 = ((0 - (1)/(0 - >B=>A8B5;L=>5 87<5=5=85 A:>@>AB8 2 B>G:5 I?, (1.  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'B> B0:>5 5AB5AB25==0O (8A:CAAB25==0O)<5E0=8G5A:0O E0@0:B5@8AB8:0 "? 0: >?@545;8BL :>>@48=0BK B>G:8 CAB0=>282H53>AO @568<0? >G5<C 2 <><5=B ?CA:0 " 2>7=8:05B 1>;LH>9 B>:, 8 GB> 45;0NB 4;O 53> C<5=LH5=8O? 0@8AC9B5 E0@0:B5@8AB8:8, ?>;CG05<K5 ?@8 @5>AB0B=>< A?>A>15 @53C;8@>20=8O A:>@>AB8 "? 0@8AC9B5 E0@0:B5@8AB8:8, ?>;CG05<K5 ?@8 @53C;8@>20=88 A:>@>AB8 " 87<5=5=85< <03=8B=>3> ?>B>:0? 0@8AC9B5 E0@0:B5@8AB8:8, ?>;CG05<K5 ?@8 @53C;8@>20=88 A:>@>AB8 " 87<5=5=85< =0?@O65=8O? >G5<C ?CA: " ?@>872>4OB ?@8 =><8=0;L=>< <03=8B=>< ?>B>:5? 0: >?@545;8BL CAB>9G82>ABL ?@82>40? 0:85 A?>A>1K B>@<>65=8O " ?@8<5=ONBAO =0 ?@0:B8:5? 0: >?@545;8BL 65AB:>ABL <5E0=8G5A:>9 E0@0:B5@8AB8:8 "? 0:85 K 7=05B5 A?>A>1K @53C;8@>20=8O G0AB>BK 2@0I5=8O "? 5@5G8A;8B5 >A=>2=K5 4>AB>8=AB20 8 =54>AB0B:8 @07;8G=KE A?>A>1>2 @53C;8@>20=8O A:>@>AB8 ". 0?8H8B5 C@02=5=85 M;5:B@8G5A:>3> @02=>25A8O 8 A>>B25BAB2CNICN 25:B>@=CN 4803@0<<C =5O2=>?>;NA=>3> A8=E@>==>3> 42830B5;O. 0: 87<5=8BAO 25:B>@=0O 4803@0<< ! ?@8: 0) 87<5=5=88 B>:0 2>71C645=8O, 1) ?@8 87<5=5=88 <><5=B0 =03@C7:8, 2) ?@8 87<5=5=88 =0?@O65=8O? 0@8AC9B5 8 ?>OA=8B5 7028A8<>ABL M;5:B@><03=8B=>3> <><5=B0 ! >B C3;0 =03@C7:8 (. >OA=8B5 E0@0:B5@ 87<5=5=8O U  >1@07=KE E0@0:B5@8AB8: !. '5< >1JOA=O5BAO A?>A>1=>ABL ! @01>B0BL 2 :0G5AB25 :><?5=A0B>@0 @50:B82=>9 <>I=>AB8? >OA=8B5 ?@8=F8? 459AB28O 8 :>=AB@C:B82=K5 >A>15==>AB8 H03>2KE 42830B5;59. >OA=8B5 ?@8=F8? 459AB28O 8 :>=AB@C:B82=K5 >A>15==>AB8 25=B8;L=KE 8=4C:B>@=KE 42830B5;59. 0:85 D0:B>@K =5>1E>48<> CG8BK20BL ?@8 2K1>@5 42830B5;O 4;O @53C;8@C5<>3> M;5:B@>?@82>40? 0: >?@545;O5BAO M:2820;5=B=K9 <><5=B ?@8 F8:;8G5A:>< E0@0:B5@5 87<5=5=8O =03@C7:8? 0: >ACI5AB2;O5BAO ?@>25@:0 2K1@0==>3> 42830B5;O ?> ?5@53@C7>G=>9 A?>A>1=>AB8? 0?8H8B5 8 ?>OA=8B5 C@02=5=85 B5?;>2>3> 10;0=A0 M;5:B@>42830B5;O. 0?8H8B5 48DD5@5=F80;L=>5 C@02=5=85 (8 53> @5H5=85), >?8AK20NI55 =5AB0F8>=0@=K5 B5?;>2K5 ?@>F5AAK M;5:B@>42830B5;O. >G5<C >B;8G0NBAO ?> 25;8G8=5 (8 :0:0O 1>;LH5) B5?;>2K5 ?>AB>O==K5 2@5<5=8 =03@520 "= 8 >E;0645=8O "> M;5:B@>42830B5;O? 09B5 >?8A0=85 AB0=40@B=KE @568<>2 @01>BK 42830B5;O (4;8B5;L=>3>, :@0B:>2@5<5==>3> 8 ?>2B>@=>  :@0B:>2@5<5==>3>). 0: ?@>25@8BL ?> =03@52C 42830B5;L, @01>B0NI89 2 :@0B:>2@5<5==>< @568<5? 0: ?@>25@8BL ?> =03@52C 42830B5;L, @01>B0NI89 2 ?>2B>@=> - :@0B:>2@5<5==>< @568<5? 040G8 5E0=8G5A:0O G0ABL M;5:B@>?@82>40 ;5154:8 A>AB>8B 87 M;5:B@>42830B5;O, @54C:B>@0 8 10@010=0 A B@>A><, =0 :>B>@>< 28A8B 3@C7. 0AAG8B0BL ?@82545==K9 : 20;C 42830B5;O <><5=B0 8=5@F88, 0 B0:65 <><5=B A>?@>B82;5=8O A, <>I=>ABL 42830B5;O 8 A:>@>ABL ?>4J5<0 3@C70 ?@8 A;54CNI8E CA;>28OE: <><5=B 8=5@F88 @>B>@0 42830B5;O J4 = 0.025 :3.<2;480<5B@ 8 <><5=B 8=5@F88 10@010=0 Db = 0.3, J1 = 1.8 :3.<2; ?5@540B>G=>5 >B=>H5=85 8  @54C:B>@0 i@ = 24, @=0.92; A:>@>ABL 42830B5;O 4 = 150 @04 /A; <0AA0 3@C70 (2<5AB5 A :@N:><) m3 = 850 :3. "@5ED07=K9 G5BK@5E?>;NA=K9 (2@=4) 0A8=E@>==K9 42830B5;L 8<55B A;54CNI85 ?0@0<5B@K: U==220 , I1= = 5,6 , I2 = 5,6 , = = 2,8 :B, f = 50 F, (= = 0.84, R1=1,8 <, R2 =2,9 <, X1=2,9 <, X2 =3,6 <. 09B8 =><8=0;L=K5 A:>@>ABL 8 <><5=B. 0AAG8B0BL ?CA:>2K5 B>: 8 <><5=B (40==K5  2 ?.1). 09B8 B>: =0<03=8G820=8O 8 <03=8B=K5 ?>B5@8 (40==K5 2  ?.1). 0AAG8B0BL :@8B8G5A:>5 A:>;L65=85 8 <0:A8<0;L=K9 <><5=B (40==K5  2 ?.1). 0AAG8B0BL 25;8G8=C ?CA:>2KE A>?@>B82;5=89 R4>1, :>B>@K5 =C6=> 225AB8 2 F5?L @>B>@0, GB>1K ?>;CG8BL <0:A8<0;L=K9 ?CA:>2>9 <><5=B (40==K5  2 ?.1). 0: 87<5=8BAO <0:A8<0;L=K9 8 ?CA:>2>9 <><5=BK 42830B5;O, 0 B0:65 :@8B8G5A:>5 A:>;L65=85, 5A;8 53> ?>4:;NG8BL : A5B8 A G0AB>B>9 f = 60 F (40==K5  2 ?.1)? > 8725AB=K< =><8=0;L=K< 40==K< 0A8=E@>==>9 <0H8=K @0AAG8B0BL ?0@0<5B@K "->1@07=>9 AE5<K 70<5I5=8O, ?@8=O2: Xm*H"2.5,  <3/( M;+ <3) H" 0.2, R1/Rk H" 0.55, Pmx H" 1.5.109. n2.h2 8 Km=1.25. 0AAG8B09B5 8 ?>AB@>9B5 5AB5AB25==K5 M;5:B@><5E0=8G5A:CN (<5E0=8G5A:CN) E0@0:B5@8AB8:C " =57028A8<>3> 2>71C645=8O. > 7040==K< ?0@0<5B@0< =03@C7>G=>3> F8:;0 ( i, ti) =09B8 B@51C5<CN <>I=>ABL 42830B5;O. -;5:B@>42830B5;L, 8<5NI89 B5?;>2CN ?>AB>O==CN =03@520 "= = 250A. 8 ?@54=07=0G5==K9 4;O 4;8B5;L=>9 @01>BK, 8A?>;L7C5BAO 2 :@0B:>2@5<5==>< @568<5 A t@ = 100A. ?@545;8BL :>MDD8F85=B 4>?CAB8<>9 ?5@53@C7:8 ?> <>I=>AB8 (@ = :/ 4). " "#  >A:0;5=:> .. 2B><0B878@>20==K9 M;5:B@>?@82>4  .: -=5@3>0B><8740B; 1986.-416 A.: '8;8:8= .., !0=4;5@ .!. 1I89 :C@A M;5:B@>?@82>40  .: -=5@3>8740B ; 1981.-576 A.: . C;30:>2 .. '0AB>B=>5 C?@02;5=85 0A8=E@>==K<8 42830B5;O<8 -.: -=5@3>8740B, 1982.- 216 A., 8;. 5@<0=-0;:8= !.. ><?LNB5@=>5 <>45;8@>20=85 ?>;C?@>2>4=8:>2KE A8AB5< 2 MATLAB 6.0 - !.:  -?@8=B, 2001.-320 A. 5B>48G5A:85 C:070=8O : ;01>@0B>@=K< @01>B0< ?> :C@AC -;5<5=BK 8 A8AB5<K M;5:B@>?@82>40, A>AB. ..!;>1>4=8G5=:>, %, 2004.           "01;8F0 1 2830B5;8 ?>AB>O==>3> B>:0 A5@88 2 Ua = Uf =220  =0G5=8O A>?@>B82;5=89 40=K 4;O 75(!. n/n , :Bn,>1/<8=(, %Ra, <Rf, <La, <Lf, 2112 J = 0.018 :<2 10.68000.60513.8287584220.87500.6409.8029052.54230.859500.6409.56287444041.2510000.6806.10250343951,515000.704.1022319.53862.215000.752.6207143772.522000.761.811929.33683.422000.781.301546.22993.630000.790.91594.830105.330000.800.641193.5222132 J = 0.048 :<2 111.67500.6854.6016538.633121.97500.733.4117028.334132.510000.7352.8016522.933143.010000.7552.2017018.134154.015000.791.351651133165.515000.810.92125722177.022400.830.621374.626188.522400.840.471103.5241910.530000.840.381372.826201431500.860.25941.8232160 J = 0.1 :<2 2137500.771.9015820.2282248000.791.4014414.727234.510000.7951.1513310.526246.310000.8150.801088.723257.515000.840.53101521261115000.8650.28803.119n/n , :Bn,>1/<8=(, %Ra, <Rf, <La, <Lf, 271321200.8600.24762.218281623600.8750.15611.416291831500.880.1265117302431500.890.1610.8162180 J = 0.23 :<2 315.67500.790.909210.924327.17500.8050.708092133810600.830.55926.124341010000.830.50905.623351515000.8560.25682.72036 18.515000.880.18802218372622400.890.10601.2122200 J = 0.3 :<2 388.58000.820.47766.41939118000.850.35684.617401311200.860.29763.619411610000.870.23683.213422215000.880.14651.612433015000.900.09521.1510443622000.900.075570.9111>7=0G5=8O: - >B40205<0O <>I=>ABL, :B; n A:>@>ABL, >1/<8=; ( - :>MDD8F85=B ?>;57=>3> 459AB28O, %; Ra , Rf 0:B82=K5 A>?@>B82;5=8O O:>@=>9 F5?8 8 F5?8 2>71C645=8O ?@8 75!, <; La  8=4C:B82=>ABL O:>@=>9 F5?8, <; Lf 8=4C:B82=>ABL F5?8 2>71C645=8O, .; 0AH8D@>2:0 >1>7=0G5=8O B8?0 42830B5;O - 2-h: 2  <0H8=0 ?>AB>O==>3> B>:0 2B>@>9 A5@88, 70I8I5==>3> 8A?>;=5=8O A A0<>25=B8;OF859 (IP22); h  2KA>B0 >A8 2@0I5=8O, << (90, 112, 132, 160, 180, 200, 225, 250, 280, 315). >71C645=85 - =57028A8<>5. 5E0=8G5A:85 ?>B5@8  <5E = 15+(.h/2000)2, 345  - G0AB>B0 2@0I5=8O, @04/A. "01;8F0 2 @5>1@07>20B5;8 G0AB>BK n/n"8?>I=>ABL 42830B5;O :B><. B>: n/n"8?>I=>ABL 42830B5;O :B><. B>: 1ECO-37/3-10.371.3511ECO-1500/3-515302ECO-55/3-20,551.9512ECO-1850/3-518.5373ECO-75/3-20,752.513ECO-2200/3-52243.54ECO-110/3-31,13.114ECO-3000/3-630585ECO-220/3-32,25.815ECO-3700/3-637716ECO-300/3-437.716ECO-4500/3-645847ECO-400/3-4410.217ECO-5500/3-7551028ECO-550/3-45,513.218ECO-7500/3-7751389ECO-750/3-47,51719ECO-9000/3-79016810ECO-1100/3-41123.520ECO-11000/3-8110210  >1>7=0G5=88 ?@5>1@07>20B5;O 40=K: B8? (A>), <>I=>ABL (B/10), G8A;> D07 (3) 8 B8?>@07<5@. "01;8F0 3 A8=E@>==K5 :>@>B:>70<:=CBK5 42830B5;8 A5@88 4 n/n"8? = (cos(=?<Es=I?I=J4--:B--<>.5.>.5.%>.5.:<2146340,370,680,692,622,2941,20,0014247140,550,710,703,7822,27,34,51,70,0013347140,750,720,735,1622,27,54,52,10,0014448041,10,750817,422,25,452,70,0032548041,50,770,8310,122,35,353,50,00326490L42,20,800,8314,822,45,1650,003374A100S430,820,83201,92,44,466,70,005684A100L440,840,8426,71,82,44,668,60,008794A112M45,50,850,8536,31,82,23,6611,50,017104A132S47,50,870,8649,21,62,62,96,515,20,028114A132M4110,880,87721,62,62,86,521,70,04124A160S4150,880,88981,42,32,3729,30,1134A160M418.50,890,881201,42,32,2735,80,13144A180S4220,870,901431,42,32,06,542,50,19154A180M4300,910,891951,42,21,96,556,10,23164A200M4370,910,902401,42,41,7768,4037174A200L4450,920,902911,42,41,6782,30,45184A225M4550,930,903551,32,41,471000,64194A250S4750,930,904831,22,31,271361204A250M4900,930,915801,22,31,371611,2214A280S41100,930,907171,221,572002,3 =  <>I=>ABL, ( - , cos( - :>MDD8F85=B <>I=>AB8, =, ? 8 <E  <><5=BK (=><8=0;L=K9, ?CA:>2>9 8 <0:A8<0;L=K9), s=  A:>;L65=85, I? 8 I=  B>:8 (?CA:>2>9 8 =><8=0;L=K9), J4 - <><5=B 8=5@F88. "01;8F0 3. @>4>;65=85 0@0<5B@K "->1@07=>9 AE5<K 70<5I5=8O  2 >B=>A8B5;L=KE 548=8F0E n/n"8? = :BXR1X1RkXk:B>.5.>.5.>.5.>.5.>.5.146340,371,40,170,0860,300,23247140,551,60,130,0860,240,23347140,751,50,110,0840,230,22448041,11,70,120,0780,200,14548041,51,90,120,0780,190,156490L42,22,10,0980,0760,160,1474A100S432,20,0780,0790,140,1484A100L442,40,0670,0790,130,1594A112M45,52,80,0540,0780,110,14104A132S47,53,00,0480,0850,0880,15114A132M4113,20,0430,0850,0820,15124A160S4154,00,0470,0860,0850,15134A160M418.54,00,0420,0850,0790,14144A180S4224,00,0410,080,0760,15154A180M4303,90,0640,0680,0640,13164A200M4374,40,0390,0860,0740,16174A200L4454,60,0340,0820,0590,16184A225M4554,20,0270,0860,0580,15194A250S4754,40,0250,0890,0520,15204A250M4905,00,0240,0930,0540,15214A280S41104,90,0230,1220,0710,21n/n"8? = :BXR1X1RkXk     PAGE  PAGE 2 v 8>, a 8>, l 8>,  (, (, ( (8>, (8>, (8>,    i, ( 1 2 3 4 5 V  V" V t 1) 0)  1 (,M,J4 J, Mc  , M  (1, J1, (1 2 m 1 1 ( ( 2 3 5 4 3 4 2 A  1) 0) (  CAB 0  0  1 2 (CAB 0) 1) 1) 0) Ed  = 0  = 60 Id:@ 180  90 R L Id Ed  = 30 ~ !$# L 2   I _ + y VS2 R= Ed 2 (t= (t= t1  2b 2a (t= VS1 3) 2) 1) 0) ( RH Ud RH Ud ( U2 ( U1 Ud RH U2 Ud RH (U1  "1 "3 "5 "4 "6 "2 ~ "2 "5 "3 "4 "6 "1 2 1 3 6 4 5 t Ua U? /3 -2U? /3 - - + + I24 I23 1    ,  ~ 1) 0) I(M) ?  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