Vettori spaziali per lo studio dei sistemi trifase 3.1
Transcript
Vettori spaziali per lo studio dei sistemi trifase 3.1
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' %λ 1 λ21 λ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π + 5 " (' )( < π ] 5$ (00 '(#! 5 1 521 5" ,!-00 % * "9( ! -! / !$ "$)/0(&&$ "$! &&( #( 0( (''$ α ( &&( )) 5 ! # $ (''$ β # //#(&(!' -! 3(''$#( )$ -0$ ( +&( 5(!(# " )(!'( 3 # 2 0 !(0 '()/$1 5 0 3(''$#( 5(' ) = 5 α (' ) + <5 β (' ) , (''$ α,β (00 '(#! 5# ! (44( "$!& (# '( (+! 4$!( (0 3(''$#( &/ 4 0( /-= (&&(#( (&/0 " ' ' /(# 0( &-( "$)/$!(!' #( 0( ( )) 5 ! # 00 % * & # " 3 !+'' ; 5 (' ) 5 (' ) 5 (' ) − 2 − " [ ] ( 5(' ) = 5 α (' ) = "9( .- ! $ 3 0( 0 % *1 "9( "$!&(!'( % * % * % * /$##( 5 7 52 5"1 "$## &/$! ( [ ] ( 5(' ) = 5 α (' ) = 5 (' ) / #'( )) 5 ! # 3 0( !3("( [ ] ) 5 (' ) = 5 β (' ) = 5 2 (' ) % * # //#(&(!' 0 *(% &/ 4 0( 5(' ) ( % − 5 " (' ) = [5 2 (' ) − 5 " (' )] 0(55( '# &+$#) 4$!( 00 '(#! 5 %'*1 52%'*1 5"%'* 0 3(''$#( * !( !!$ &(/ # ' )(!'( 0( "$)/$!(!' $#'$5$! 0 (# 0 '# &+$#) 4$!( !3(#& 1 "9( "$!&(!'( / && #( 5(' ) 5 %'*1 52%'*1 5"%'*1 & $&&(#3 !! !4'-''$ "9( /(# & &'() 2 0 !" ' 3 0( 0 % * >- ! ! .-(&'$ " &$ 5 %'* ,0 / #'( #( 0( 5 (# .-(&'$ )$' 3$ 0? &&( α , %/#$ (4$!( &-00? &&( α* (0 3(''$#( 5(' ) 1 "$)( )$&'# 0 (''$ !"9( β 5 (' ) 7 5(' ) 5 " (' ) α 52%'* 5"%'* π/3 5 %'* π/3 5 2 (' ) 5 @ ('(#) ! 4$!( 5 %'*1 52%'*1 5"%'* 3(''$# 5 (' ) 1 5 2 (' ) ( 5 " (' ) "$!& (# /$ 0 3(''$#( 5 2 (' ) = 5(' )( − < π = [5 (' )( −< π + 5 2 (' ) + 5 " (' )( < π ] % * "9( "$## &/$! ( 0 3(''$#( &/ 4 0( 5(' ) #-$' '$ "9( 9 !!$ /$#' '$ 00 % * $# //0 " ' 00 % '#$3 "9( [ ] [ ( 5 2 (' ) = ( 5 (' )( − < π ] = −5 π = 5 2 (' ) + 5 β %'* α %'* # ! &(!&$ $# # $ $! 50 &'(&& / && 55 *1 &()/#( &$''$ 0 "$! 4$!( (00 % *1 & % A* -!.-( 0 52%'* & $'' (!( /#(! (! $ 0 /#$ (4$!( &-00? &&( #( 0( (0 3(''$#( &/ 4 0( 5 2 (' ) $''(!-'$ #-$' ! $ 5(' ) π # ! &(!&$ $# # $ 00$ &'(&&$ )$ $ 0 "$)/$!(!'( 5"%'* & $'' (!( [ ] [ ( 5 " (' ) = ( 5 (' )( − < π ] = −5 00 = 5 " (' ) − 5 β %'* α %'* 5 00-&'# ! +$#) 5# +" 0( #(0 4$!( % (+! '$ !"9( 0 3(''$#( 5 (' ) "$ !" (!'( "$! 5(' ) *1 % % A* ( % B* B* $3(1 /(# (&'(!& $!(1 , 5"%'* β 2 5(' ) 7 5 (' ) 52%'* α, 52%'* 5 %'* " 5 @ ('(#) ! 4$!( 5 %'*1 52%'* ( 5"%'* 0 3(''$#( &/ 4 0( 5(' ) !3("( &+& #( 0 3(''$#( &/ 4 0( & /-= )) 5 ! #( 0 &" #0$ !(00 &- /#$/# /$& 4$!( ( #-$' #( π # 0? &&( &- "- (&(5- #( 0 /#$ (4$!(1 ! &(!&$ !' $# # $ /(# 0 ( -4$!( * ( .- ! π # /(# 5"%'* % (+!(! $ 0? *1 52%'* % (+!(! $ -! !-$3$ &&(1 (''$ "$)( ! 5 (# (&'(!& $!(1 0? &&( α , (''$ !"9( ( "$!& (# 4$! &$/# (&/$&'( )$&'# !$ "9( 0 /#$ (4$!( (0 3(''$#( &/ 4 0( 5(' ) &-50 && 121" #(&' '- &"( (& '' )(!'( 0 3 0$#( (00( 5# ! (44( '# +&( 5 %'*1 52%'* ( 5"%'*1 &(!4 0"-! +''$#( &" 0 (# " =& "( "9( 0 '# &+$#) 4$!( % *, 1 /#$/# ('C"9( ,$''(!-' ! 3 #'D (0 +''$#( "9( // #( !(00 % * &'(&& '# +&( %& "( !"9( "$!& (# $# 0 " &$ * " $,/(# 0 .- 0(; -! '(#! 5# ! (44( 5 $ %'* = 5 %' * + 5 2 %' * + 5 " %' * ≠ % E* 00$# & /-=&"# 3(#(; 5 %'* = 5 F %'* + 5 $ %'* 5 2 % ' * = 5 F2 % ' * + 5 $ % ' * % * 5 " % ' * = 5 F" % ' * + 5 $ % ' * $3(1 /(# 0 % //0 " #( 0 % 5(' ) = E* 0 '(#! 5? %'*1 5?2%'* ( 5?"%'* ,$33 )(!'( 2 0 !" ' * "9( & 3(# +" +" 0)(!'( &$ &+ 00 ; [5 (' ) + 5 2 (' )( < + 5 " (' )( < π π ] = [5 (' ) + 5 F F 2 (' )( < π 00 % + 5 F" (' )( < π * & /-= !"$# ] % * " $,, ! /(! (!'( 0 3 0$#( 5$%'* 5(' ) (+! '$ 00 % * (1 "$!&(5-(!4 0? //0 " 4$!( (00( % *1 % A* ( % B* .- ! #(&' '- &"( &$0$ 0 '(#! 2 0 !" ' 5? %'*1 5?2%'* ( 5?"%'* ! 5(!(# 0( & /$'#C /(#' !'$ ++(#) #( "9( .- 0& & '(#! 5 %'*1 52%'* ( 5"%'* /-= (&&(#( '# &+$#) ' !(00 '(#! 5$%'*1 5α%'*1 5β%'* (+! ' 00(; 5 $ %'* = 5 %'* + 5 2 %'* + 5 " %'* 5 2 (' ) 5 α (' ) = 5 (' ) − 5 β (' ) = [5 2 (' ) − 5 " (' )] − 5 " (' ) % * % * )(!'#( 0 '# &+$#) 4$!( !3(#& # &-0' ; 5 (' ) = [ ] [ ( 5 (' ) + 5 $ % ' * = ( 5 [ ( [5 (' )( 5 2 (' ) = ( 5 (' )( − < π 5 " (' ) = −< π (' )]+ 5 $ % ' * ]+ 5 %' * = ([5 (' )]+ 5 %' * ]+ 5 %' * = ([5 (' )]+ 5 %' * $ 2 $ $ " $ "$! 5(' ) 7 5α%'* <5β%'* # //#(&(!' 4$!( 5# +" 5 +$#! &"( $# 0( "$)/$!(!' '# +&( 5? %'*1 5?2%'* ( 5?"%'*1 )(!'#( 0? !'(#/#(' 4$!( 5# +" (00 '# &+$#) 4$!( 5(!(# 0( % * , )$&'# ' ! 5 ( #(&' '- &"( 0 '(#! '# +&( "$)/0(' "$)/$!(!'( $)$/$0 #( '# &+$#) 4$!( -! '(#! 5# ! (44( '# +&( !(00( "$)/$!(!' (0 3(''$#( &/ 4 0( ( .- ! 0? !' '# &+$#) 4$!( ! -! !-$3 '(#! '# +&( ,-! '("! " )/ (5 ' /(# (0 ) ! #( 0 "$)/$!(!'( $)$/$0 #( β 5 (' ) 7 5(' ) 5 " (' ) 52%'* 5$%'* 5"%'* π/3 α 5 %'* π/3 5 2 (' ) 5 = αβ $ $3( 5 αβ $ % , 0 ) '# "( %"$0$!! * G5α%'*1 5β%'*1 5$%'*H' 5 00 ) '# "(1 # " 3 ' − ! 0$5 )(!'( 0 % % 2" * * αβ $ 2" = *1 '(!(! $ "$!'$ − − % * % * % A* 3(!' = αβ $ → 2" 5 αβ $ "$!1 ! ""$# $ "$! % * , 0 ) '# "( G5 %'*1 52%'*1 5"%'*H' ( 0 − − 2" → αβ $ = 2" * /-= (&&(#( (&/#(&&( ! +$#) 2" → αβ $ 5 2" '# &+$#) 4$!( , (+! ' 5 5 (' ) 1 5 2 (' ) ( 5 " (' ) /(# -! 5 %'*1 52%'*1 5"%'* 3(''$# '(#! !$! 2 0 !" ' '# &+$#) 4$!( % ''# 3(#&$ 0( &(5-(!'( (&/#(&& $!( ) '# " 0( ; "$)/ '' 5 @ ('(#) ! 4$!( % A* ( % B* − ( 5# ! (44( ) '# " 0 &$!$ ! " '( "$! -! & )2$0$ &$''$0 !( '$ $//-#( (&/0 " ' )(!'( "$! / #(!'(& .- #(6"$! 0 & )2$0$ G•H' & !'(! ( 0 ) '# "( '# &/$&' G•H "$!& (# !$ -( 5(!(# "9( '(#!( -! '# +&( 5 /$'(!4 &' !' !( '(!& $! ( -! "$##(!' #(0 ' 3( 00$ &'(&&$ & &'() &&$#2 ' 0 " # "$ '# +&( 3 0( /% ' * = - % ' * % ' * + - 2 % ' * 2 % ' * + - " % ' * " % ' * % * % * % * % * "9( & /-=(&/# )(#( "$! 0 /% ' * = - ' 2" 2" //0 " ! $ 0 % * & '#$3 ' ' /% ' * = - αβ $ αβ $ 2" αβ $ 2" αβ $ "9( (&/0 " ' ' ,; /% ' * = [- α %' * α %'* + - β %' * β %' *]+ - $ %' * $ %' * % * )$&'# "9( 0 '# &+$#) 4$!( ! (& )( !(0 &(!&$ "9( 0 /$'(!4 " 0"$0 ' "$! 0( 5# ! (44( '# &+$#) '( αβ % &&-)(! $1 /(# &()/0 " 'C &"$#&$1 !-00 -! $ 0? 0'# (00( "$)/$!(!' $)$/$0 # * 3 "$##('' "$! 0 +''$#( /(# 3(#( 0 #( 0( /$'(!4 (0 & &'() '# +&( 0( # &-0' '$ & //0 " $5! " 0"$0$ "9( "$ !3$05 0 /#$ $''$ -( '(#!( 5# ! (44( $33(#$ -( 3(''$# &/ 4 0 ! (&&(#( # &"# '' !(00 +$#) /% ' * = I(0 " &$ ! "- -$7 $ $7 1 0 % * /-= !"9( ∨ % ( - %'* %'* $3( - % ' * , 0 3(''$#( &/ 4 0( &&$" '$ 00 '(#! * (00( '(!& $! )(!'#( 0 & )2$0$ ∨ (!$' 0?$/(# 4$!( (0 3(''$#( &/ 4 0( % ' * (00( "$##(!' !$0'#(1 /(# (&'(!& $!( (00( % *(% * +"(! $ "$ !" (#( $5! '(!& $!( ( "$##(!'( -! 5(!(# " "$)-!( 5# ! (44 5%'*1 & # "$!$&"( "9( 5 % ' * + 5 2 % ' * + 5 " % ' * = G5 α % ' * + 5 β % ' *H = 5% ' * % "9( , -!?(&/#(&& $!( "9( #(&' '- &"( #('' )(!'( 0 3 0$#( &' !' !($ (0 )$ -0$ &/ 4 0( / #' #( 3 0$# &' !' !( (00( "$)/$!(!' (00 '(#! '# +&( 2 0 !" ' 5%' * = 5 %'* + 5 2 %'* + 5 " %'* * (0 3(''$#( % A* )(!'#( 0? #5$)(!'$ γ % !5$0$ # &/(''$ 00? &&( α* (0 3(''$#( & # " 3 ! $5! " &$ "$! 0 γ = #5G5 % ' *H = ' !G5 β % ' * 5 α % ' *H % (# 3 ! $ 0 % B* & $'' (!( 0?(&/#(&& $!( (00 3(0$" 'C &/ 4 0( "9( # &-0' / # ; ω= 5β %'* J #$' 4$!( &' !' !( B* (0 3(''$#( 5β %'* 5 α %' * − 5 α %'* J ' ' 5 α %'* + 5 β %'* % E* " (# 0( !-)(#$&( //0 " 4$! !(0 " )/$ (50 4$! )(!' (0(''# " 1 , F !'(#(&&( 3( (#( "$)( 0 3(''$#( &/ 4 0( 5 75α <5β &&$" '$ 00 5(!(# " '(#! %5 1 52 15"* /$&& (&&(#( (&/#(&&$ &("$! $ -! & &'() # +(# )(!'$ $#'$5$! 0( . "$! 3(0$" 'C !5$0 #( ω .%'* # &/(''$ 0 α β "$!& (# '$ !(0 / # 5# +$ /#("( (!'( 5 (# &' !5-(#( 0( 0 3(''$#( &/ 4 0( &("$! $ -! # +(# )(!'$ #$' !'( 1. -( # //#(&(!' 4$! 1 & # //#(&(!' 4$!( !(0 & &'() ! "9(#C "$! 5 αβ 75α <5β7K 5 K(<γαβ 0 # +(# )(!'$ α−β ( "$! 5 . 75 <5.7K 5 K(<γ .1 5 33 )(!'( 3 0( K 5 αβ K7K 5 . K7K 5 K ( γαβ7γ . ϑ . $3( ϑ . , 0 /$& 4$!( &' !' !( & &'() . # &/(''$ 0 & &'() α−β $! 0? -'$ (00 +5-# , ))( '$ .- ! &"# 3(#( (0 5 αβ = 5 ( ( < γ . +ϑ . ) = 5 ( <γ . ( <ϑ . =5 . ( <ϑ . . .-(00 ! % * % * ( -!.-( 5 . = 5 αβ ( − <ϑ . A % * /-= (&&(#( &"# '' )) 5 ! # ; ( ) ( ) 5 "$& ϑ . = 5. −& ! ϑ . +&( ϑ . ! +$#) ( ) ( ) &!ϑ . "$& ϑ . #(0 4$!( ) '# " 0( &(/ # ! $ 0 / #'( #( 0( ( 5α 5α = αβ → . 5β 5β % * % * ' 0( & &'() , ! 5(!(# 0( +-!4$!( (0 '()/$ ( , ' ' ϑ . (' ) = ω . (' ) ' ) '# "( αβ→ . "9( // #( !(00 % 0 '# &/$&' ; . → αβ ( ) ( ) "$& ϑ . − = αβ → . = &!ϑ . * ,$#'$!$#) 0(1 -!.-( 0 &- ( ) ( ) −& ! ϑ . "$& ϑ . !3(#& "$ !" ( "$! % * $)( & 3( #C !(0 &(5- '$1 )$0'$ &/(&&$ & # "$##( .- 0"9( / #' "$0 #( & &'() # +(# )(!'$ %/(# (&()/ $1 L&$0 0(M "$! 0 #$'$#( (0 )$'$#(* /(#"9,.-(&'$ /(#)(''( &"# 3(#( 0( (.- 4$! (00( '(!& $! (50 33$05 )(!' ( (00 "$// / #' "$0 #)(!'( &()/0 " ( ''( ))( '( !'(#/#(' 4$! 3$0'( /-= "" (#( "9( !(00 '# '' 4$!( ! &$'#$/( & "$!3(! (!'( $ !("(&& # $ ) !'(!(#( 0 -! '# &+$#) 4$!( "9( /#(&(#3 ' 0( "$)/$!(!'( ) '# "( '# &+$#) 4$!( 0 & &'() %α1β1$* 00 % *1 "$! 0 "$! 4$!( "$!&(#3 #( 0 &'(&& ( ) ( ) "$& ϑ . .$ → αβ $ = & ! ϑ . 4$! )(!' /(# 0"-!( ) ""9!( (0(''# "9( "$)/$!(!'( $)$/$0 #(1 $'' ! $ -!.-( 0 & &'() % 1.1$* & $'' (!( ! )$ $ "$)/$!(!'( $)$/$0 #(; ( ) ( ) % * ( ) ( ) % A* &!ϑ . "$& ϑ . #(''$ "9( 9 /(# !3(#& 0 ) '# "( ( ) ( ) "$& ϑ . αβ $ → .$ = & ! ϑ . ) '# "( "$)/0(' $&&(#3 ! $ "9( &!ϑ . "$& ϑ . '# &+$#) 4$!( 5 .$ 7 αβ$→ .$ 5αβ$ 7 αβ$→ .$ ( &$&' '-(! $ 0( % A* ( % 0 & &'() % 121"* 0 & &'() #$' !'( % 1.1$* & $'' (!( 2"→αβ$ 5 2" 7 2"→ .$ 5 2" % B* * & $'' (!( B 2" → .$ ( ) ( ) ( "$& ϑ . − &(! ϑ . = ) "$& ϑ . − π − &(! ϑ . − π ( ) ( "$& ϑ . − π − &(! ϑ . − π ( ) ) % E* ( 0 &- !3(#& , .$ → 2" = ( ) "$& ϑ . "$& ϑ . − π "$& ϑ . − π ( ( ( ) − &(! ϑ . − &(! ϑ . − π − &(! ϑ . − π ) ) ( ( ) ) % * # (## !!$ $# /#(& ! (& )( 0"-! (&()/ //0 " 4$!( ( 3(''$# &/ 4 0 ( !$! 3(#& )(!'( ! " '$1 $5! 3(''$#( &/ 4 0( & !'(! (#C(&/#(&&$ !(0 & &'() # +(# )(!'$ α−β. "$!& (# /(# 0 &'#-''-# '# +&( '(!& $! &'(00 '( & !-&$ 0 ; # $ -! & &'() '# +&( & ))('# "$ - %'* = "$&(ω' + ϑ $ ) - 2 %' * = "$& ω' + ϑ$ − - " %' * = "$& ω' + ϑ $ − π % ( <(ω' + ϑ$ ) = * π "$! ω /-0& 4$!( !5$0 #(1 (&/#(&& # &-0' 1 $/$ .- 0"9( / && 55 $ ; - %' * = 5 !# &1 ( ϑ$ +&( ! 4 0( //0 " ! $ 0 ( <ϑ$ ( <ω' (+! 4$!( % % * * ? ''# 2-'$ L& ))('# "$M &' ! " #( "9( 0( '#( '(!& $! &$!$ -5- 0( )/ (44 ( &+& '( #(" /#$" )(!'( π 0 & &'() & ))('# "$ & "( L #(''$M &( /-!' "$## &/$! (!' (00( '#( 5# ! (44(1 /(# (&()/ $ 3 0$# ) && ) 1 & &-""( $!$ !(00?$# !( 121"1 1N 6L !3(#&$M &( 0?$# !( , 1"121 1"121N 0 & &'() '$ 00( % * , #(''$ ! '# "" /(# / && 55 ,0 &(5-(!'( $!(! $ ξ7ω' α$ /(# /# ' " 'C1 & &"# 3(; -= ["$&(ξ) + "$&(ξ − π )( < π G"$&%α β* ( # "$# ! $ 0( +$#)-0( "$&%α*"$&%β*7 9; -= + = "$&(ξ ) + < ("$&(ξ − (& ! (ξ) − & ! (ξ − π π + "$&(ξ − π )( < ] "$&%α β*H 6& !%α*"$&%β*7 ) + "$&(ξ ) + "$&(ξ − ) + & ! (ξ) − & ! (ξ − π π π )) G& !%α β* & !%α β*H & + ) + & ! (ξ) − & ! (ξ)) = "$&(ϑ + α $ ) + < & ! (ϑ + α $ ) E -!.-(1 !(0 & &'() # +(# )(!'$ &' 4$! # $ α1β1 -! 3(''$#( √ ( #$' !'( ! &(!&$ !' $# # $ "$! 3(0$" 'C !5$0 #( ω1 / #' #( 00 /$& 4$!( 5 ( "$)/$!(!' #( 0( ( )) 5 ! # ' 0( 3(''$#( # &-0' !$ ! 4 0( ϑ$ "$)( )$&'# 0 ! '-# 0)(!'(; '# '' "$&(ω' + ϑ $ ) - α %' * = % & ! (ω' + ϑ $ ) - β %'* = β '7 * ω '7 ϑο α 5 @ (''$#( &/ 4 0( -! 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