• Non ci sono risultati.

20 dB K

N/A
N/A
Protected

Academic year: 2021

Condividi "20 dB K"

Copied!
21
0
0

Testo completo

(1)

798;:.<=8?>@8BACA8;:EDGFH<JILK;:AMINAODGFQPR8TS#I1:.FE8VU F

WRXZY\[^]`_baMcedf_hgMijYlkM[m]`_b])nceiogqpN_r[^cCgdscutGY\[mv kM[^w _bY\[^cxaE_yn`dzg^]vzcedf_rijce[OnY|{~}f€`ƒ‚g

v kM[^w _bY\[^c„aE_ydf_b]†…Y])n%g‡gdˆijY\[_btCg‰aMcep!]`_b])nceiogupN_r[^cCgdsc‹Šc‡{~} ŒMŽ€`

W‘…M…MpN_bte’_hgMijY“_r[”_r[^•\dscG]–]–Y—gMp]`_b])nceiog˜pN_r[^cCgdsc˜kM[™]–cG•\[gMp,clšJ}›€xtGY\[™]†…c†nn`dsY

œ

} ŒMŽ€`Ž‚Yx]†…c†nn`dsYžR} ŒMŽ€LaMcep'pŸg df_b]†…Y])n%g‹¡¢}›€L]`_!Ynn _bce[^c£tGY\ijc¤

R} ŒMŽ€¦¥§{~} ŒMŽ€

œ

} ŒMŽ€

WR¨¦pN_!]†…c†nn`df_!aE_BgMi‹…_bcGwwOg£covg^]–c£aMcep'pŸg df_b]†…Y])n%g‰¡¢}›€9df_b]†kMp©n%gM[^YuªBk_r[^aE_«¤

¬

|} ŒMŽ€

¬b­®

¥ ¬

{~} ŒMŽ€

¬b­®¯°¬

œ

} ŒMŽ€

¬b­®

±

|} ŒMŽ€—¥

±

{~} ŒMŽ€

¯ ± œ

} ŒMŽ€

Wl‚Y—]†…c†nn`dsY—|} ŒMŽ€aMcep'pŸg“df_b]†…Y])n%g²Šc™aBkM[^ªBk^c³kM[g™´ ceds]`_bY\[^cµ„gMp©ncedzgOn%g™a!gMp'pŸg

v kM[^w _bY\[^c‡aE_;df_b]†…Y])n%gxgdˆijY\[_btCg~{~} ŒMŽ€ µZaMcep'p,Y]†…c†nn`dsY

œ

} ŒMŽ€¶aMcep.]–cG•\[gMp,c·aE_

_r[^•\dscG]–]–YB¹¸y_!aE_btGc„te’^c‡_rp!]`_b])nceiogžpN_r[^cCgdsc»º½¼#¾f¿0À _rpÁ]–cG•\[gMp,c„aE_@_r[^•\dscG]–]–YB

Wl‚c·aE_,ÂQcedsce[^wc~[^cep'p,Yƒ]†…c†nn`dsY~cGªBk_ôegMp,•Y\[^Y|gxaE_,ÂQcedsce[^wc~[^cep'pŸgƒdf_b]†…Y])n%gžncei‹…Y Ä

dzgMp,c”Å«p;]`_b])nceiogÆpN_r[^cCgdscÇdf_rk^]–t_rdˆŠg|gl_r[^]–cG•\k_rdscƒcG]gOnn%gMijce[OncÈprÉ'gM[^a!gMijce[OnYÆaMcep

]–cG•\[gMp,c£aE_@_r[^•\dscG]–]–YEµ.Y´C´ cedsYx¡¢}›€¥§Ê˚J}›€»ÌÊ ÍÏÎÐ µB]–c£c£]–Y\p,Yu]–c¤

¬

{~} ŒMŽ€

¬

¥ÑÊ Ò ±

{~} ŒMŽ€¥ÏÎ

Wl‚cuaE_,ÂQcedsce[^wc‡n`dzgÈpŸg|df_b]†…Y])n%gm¡¢}›€Óc_rpy]–cG•\[gMp,cxaE_Ô_r[^•\dscG]–]–YȚJ}›€9]–Y\[^Yqn%gM[OnY

iog^••(_bYMdf_(ªBkgM[OnYq…_«Šk

¬

{~} ŒMŽ€

¬;Õ

¥ÑÊ c ±

{~} ŒMŽ€

Õ

¥ÏÎ\

WlÖEced;ijY\p©n _^]`_b])ncei‰_pN_r[^cCgdf_ cG]`_b])nc‘]–Y\p,YžkM[„_r[Onced+´egMp'p,YoaE_Á…MkMp,]g^w _bY\[_C?…ced¢_rp^ªBkgMp,c

´egMp,c·pŸgudscepŸg^w _bY\[^c

¬

{~} ŒMŽ€

¬!×

Ê c ±

{~} ŒMŽ€

×

Î\¹Øjk^cG])nY~_r[Onced+´egMp'p,YžaE_M´egMp,YMdf_

aE_MÙ]`_(te’_hgMiogqÚ`ÀMÛ¢Ü(À‰ÝyÀÞ`ÞOÀMÛZ¾ß 

WlÖEced(tGY\[O´ ce[^w _bY\[^c9pŸg9àgM[^a!gj…g^]–]gM[Onc¶Ì–[^cep'pŸgÓiog^••(_bYMdV…gd+nc¹aMc_tCg^]`_aE_ _r[OncedscG]–]–c

…dzgOn _btGYÁжŠc£a!gOn%g‹a!g_Á´egMp,YMdf_(aE_Má…ced_(ªBkgMpN_

¬

{~} ŒMŽ€

¬b­®˜â

Ê ã“ä

­®



Wƒå‘_r…_btCgMijce[OncƒpŸg|àgM[^a!gDžg^]–]gM[Onc~Šc a!gOn%g~a!gRkM[R_r[Onced+´egMp'p,YÈaE_V´egMp,YMdf_QaMcep@n _r…Y

æƒçèéçèê?tGY\[„æ~cjê?aMc†nncjÝ¢ë.¼ìÞOÀeíGîïÛVîŽÜMî½¾ ÀðÁ¼«îï 

(2)

Wq¸y_b])nceiog£aE_!n _r…Y‡ÝyÀÞ`ÞOÀÇÚ`ÀÞ`ÞCï ÔgM[^a!g …g^]–]gM[Onc£ÎxçٍÑçٍ ê 

w j dB

G ( ) |

| w

20 dB K

-20 dB

K-3 db

banda passante

Passa basso

w H

3 dB

w j dB

G ( ) |

| w

20 dB K

-20 dB

K-3 db

banda passante

Passa basso

w H

3 dB

Wq¸y_b])nceiog£aE_!n _r…Y‡ÝyÀÞ`ÞOÀRÀÁ¼#¾ï ÔgM[^a!g …g^]–]gM[Oncj æ çèéçá

w j dB

G ( ) |

| w

20 dB

K

-20 dB

K-3 db

banda passante Passa alto

w L

3 dB

w j dB

G ( ) |

| w

20 dB

K

-20 dB

K-3 db

banda passante Passa alto

w L

3 dB

(3)

Wq¸y_b])nceiog£aE_!n _r…Y‡ÝyÀÞ`ÞOÀÇÚ`ÀMÛ¢Ü(À  ÔgM[^a!g …g^]–]gM[OncӍ æ çٍÑçٍ ê 

w j dB

G ( ) |

| w

20 dB

K

-20 dB

K-3 db

banda passante

Passa banda

w L w

H

3 dB

w j dB

G ( ) |

| w

20 dB

K

-20 dB

K-3 db

banda passante

Passa banda

w L w

H

3 dB

Wq¸y_b])nceiog£aE_!n _r…YÆßC¼«î  î3Û¢ÀRÚ`ÀMÛ¢Ü(À 

ÔgM[^a!gx…g^]–]gM[Onc„Îuçèéçè

æ

cj

ê

çٍÑç á

w j dB

G ( ) |

| w

20 dB K

-20 dB

K-3 db

banda passante

Elimina banda

banda passante

w L w

H

3 dB 3 dB

w j dB

G ( ) |

| w

20 dB K

-20 dB

K-3 db

banda passante

Elimina banda

banda passante

w L w

H

3 dB 3 dB

(4)

WRXZY\[^]`_baMcedf_hgMijYƒkM[‹]`_b])nceiog„aMcep@…df_rijYuYMdsaE_r[^c„tGY\[·•\kg^a!g^•\[^Y ])n%gOn _btGYkM[_Ãn%gdf_bYB¤

{~}f€¥





¯



WlÅe]`_b])ncei‰_OaMcepÁ…df_rijYoYMdsaE_r[^c¶]–Y\[^Y£aE_Cn _r…Y‡…g^]–]g£àg^]–]–Yoc pŸg„p,YMdsYžàgM[^a!g„…g^]–]gM[Onc

Šc„Îxçèéçٍê tGY\[!¤

 ê ¥ 



W‘…M…MpN_bte’_hgMijYÆ_r[È_r[^•\dscG]–]–YRgMpy]`_b])nceiog{~}f€‰kM[|_ri‹…MkMp,]–Yldsc†nn%gM[^•Y\pŸgdscxšJ}›€9aE_

aBkMdzgOn%gțžcÈgMi‹…_bcGwwOg



›ÌˆªBk_r[^aE_JgdscCg kM[_Ãn%gdf_hgeÐ`²¸y_hgÇg^a™cG]–cei‹…_bYǛƒ¥

  

Î\¤

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Impulso rettangolare

Wl‚g£n`dzg^]vzYMdˆiogOn%g„aE_Q‚gM…MpŸg^tGc

œ

}f€LaMcep!]–cG•\[gMp,c£aE_@_r[^•\dscG]–]–YxšJ}›€»df_b]†kMp©n%g!¤

œ

}f€¦¥



›V



ãTÒ

Wl‚Yu]†…c†nn`dsY

œ

} ŒMŽ€LaMcep!]–cG•\[gMp,c£aE_y_r[^•\dscG]–]–YušJ}›€»df_b]†kMp©n%g‹ªBk_r[^aE_«¤

œ

} ŒMŽ€¦¥



›ÁŒM



ã?Ò

(5)

Wl‚Yu]†…c†nn`dsY

œ

} ŒMŽ€LaMcep!]–cG•\[gMp,c£aE_y_r[^•\dscG]–]–YušJ}›€»df_b]†kMp©n%g!¤

œ

} ŒMŽ€¦¥



›ÁŒM



ã?Ò

WlÖEcedԛL¥

  

Î~p,Yu]†…c†nn`dsY

œ

} ŒMŽ€9df_b]†kMp©n%g!¤

10 0 10 1 10 2

−40

−35

−30

−25

−20

−15

−10

−5 0 5

ω

|F( ω )| [dB]

Spettro delle Ampiezze

‚YÆ]†…c†nn`dsYÆ]`_ gM[M[MkMp'pŸg …ced½ ¥ £



tGY\[q

 ¥

 

› 

³ÖEcedL Í äC



p,Y

]†…c†nn`dsYmaMcep'p,cžgMi‹…_bcGwwc‹aMcepB]–cG•\[gMp,cuŠcm_r[Cvzcedf_bYMdscug

 

Îqdf_b]†…c†nnYƒgMp'p,Y~]†…c†nn`dsY

gMp'p,c‡àg^]–]–cž…MkMp,]g^w _bY\[_«

(6)

Wq¸y_hgo ê ¥ÏäeΉ¥ÏäC  Ž‚Yx]†…c†nn`dsYžR} ŒMŽ€½aMcep'pŸgxdf_b]†…Y])n%g‹¡¢}›€9df_b]†kMp©n%g!¤

10 0 10 1 10 2

−40

−35

−30

−25

−20

−15

−10

−5 0 5

ω

|X( ω )|(c), |G(j ω )|(r), |Y( ω )|(b) [dB]

Spettro delle Ampiezze

Wl‚(É'gM[^a!gMijce[OnY‡ncei‹…YMdzgMp,coaMcep'pŸgxdf_b]†…Y])n%g [^cepMncei‹…Ydf_b]†kMp©n%g!¤

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Risposta(r) all’impulso(b)

Wl‚cDždf_r[^t_r…gMpN_ÔtGY\i‹…Y\[^ce[On _=]†…c†nn`dzgMpN_J̖…cedL    `Ðu…g^]–]gM[^Y…dzgOn _btCgMijce[Onc

_r[O´egdf_hgOnc·gMp'prÉ k^]–t_Ãn%gžaMcepE]`_b])nceiogmpN_r[^cCgdsc„Å«[O´ cGtGcmp,c‰tGY\i‹…Y\[^ce[On _.]†…c†nn`dzgMpN_Q…ced

 ͧ

ê

]–Y\[^YqgOnnce[MkgOnc·a!gMpB]`_b])nceiog‡aMcep¢…df_rijY~YMdsaE_r[^c‰Øjk_r[^aE_;pŸgƒdf_b]†…Y])n%g

¡¢}›€[^Y\[„…MkÁŠY£cG]–]–cedsc ´ cep,Y!tGc½tGY\ijc9_rpC]–cG•\[gMp,c‘aE__r[^•\dscG]–]–YošJ}›€ µ\iogŽ]`_MiogM[On _bce[^c

kM[g àMk^Y\[g·]`_ri‰_rpN_Ãn`k^aE_r[^cov dzgu_(aBk^c„]–cG•\[gMpN_«

(7)

Wq¸y_hgo ê ¥ 

Ή¥?  ¶‚Yu]†…c†nn`dsYu|} ŒMŽ€½aMcep'pŸg df_b]†…Y])n%g‰¡¢}›€9df_b]†kMp©n%g!¤

10 0 10 1 10 2

−40

−35

−30

−25

−20

−15

−10

−5 0 5

ω

|X( ω )|(c), |G(j ω )|(r), |Y( ω )|(b) [dB]

Spettro delle Ampiezze

Wl‚(É'gM[^a!gMijce[OnY‡ncei‹…YMdzgMp,coaMcep'pŸgxdf_b]†…Y])n%g [^cepMncei‹…Ydf_b]†kMp©n%g!¤

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Risposta(r) all’impulso(b)

WÔ_b]†…c†nnYÇgMpZtCg^]–Y™…dscGtGcGaMce[OncƒgM[^te’^c|kM[gl…gd+ncmaMcep'p,c|…df_r[^t_r…gMpN_;tGY\i‹…Y\[^ce[On _

]†…c†nn`dzgMpN_(̖…cedy  ?     ÐLŠc gOnnce[MkgOn%gja!gMp]`_b])nceiogÓaMcep\…df_rijY‹YMdsaE_r[^cŽÅ«[

|} ŒMŽ€¦]`_E…cedsaMc‹àMk^Y\[gž…gd+ncjaMceptGY\[Once[MkOnYm_r[‰gMp©n%gjv dscGªBk^ce[^wOg£aMcep]–cG•\[gMp,cÓaE_

_r[^•\dscG]–]–YuªBk_r[^aE_QpŸgxdf_b]†…Y])n%g‹¡¢}›€ŽŠcž…_rkOnnY])nYqp,ce[On%g!

(8)

Wq¸y_hgo ê ¥„¥? 



¶‚Yu]†…c†nn`dsYuR} ŒMŽ€LaMcep'pŸg df_b]†…Y])n%g‰¡¢}›€9df_b]†kMp©n%g!¤

10 0 10 1 10 2

−40

−35

−30

−25

−20

−15

−10

−5 0 5

ω

|X( ω )|(c), |G(j ω )|(r), |Y( ω )|(b) [dB]

Spettro delle Ampiezze

Wl‚(É'gM[^a!gMijce[OnY‡ncei‹…YMdzgMp,coaMcep'pŸgxdf_b]†…Y])n%g [^cepMncei‹…Ydf_b]†kMp©n%g!¤

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Risposta(r) all’impulso(b)

WlÅ«[oªBk^cG])nY·tCg^]–Y‡gM[^te’^c‹p,cÓtGY\i‹…Y\[^ce[On _M]†…c†nn`dzgMpN_Mg‡àg^]–]gjv dscGªBk^ce[^wOg£]–Y\[^YugOnnc`Ä

[MkgOncoa!gMpM]`_b])nceiog£aMcep.…df_rijYžYMdsaE_r[^c£c ªBk^cep'p,c‡_r[·gMp©n%g£v dscGªBk^ce[^wOg‰]–Y\[^Yu]–Y])n%gM[GÄ

w _hgMp'ijce[Onc¶]–tGY\i‹…gds]–c½‚g‹df_b]†…Y])n%g ¡¢}›€¹Šc‰ijY\p©nYxp,ce[On%gjc‰[^Y\[‡df_bcG]–tGcjgj]–cG•\k_rdsc

…ced¦[MkMp'pŸg prÉ«_r[^•\dscG]–]–YxšJ}›€`

(9)

W‘…M…MpN_bte’_hgMijY˜_r[ _r[^•\dscG]–]–Y gMp ]`_b])nceiogl{~}f€ kM[ _ri‹…MkMp,]–Yln`df_hgM[^•Y\pŸgdscȚJ}›€‹aE_

aBkMdzgOn%g



›¶c‰gMi‹…_bcGwwOg



›oÌ gdscCg kM[_Ãn%gdf_hgeÐ`¹¸y_hg‡g^a cG]–cei‹…_bY‡›L¥

  

Î\

Wq¸y_hgoê³¥ÏäeÎ\Ž‚Yx]†…c†nn`dsYžR} ŒMŽ€LaMcep'pŸg df_b]†…Y])n%g‹¡¢}›€9df_b]†kMp©n%g!¤

10 0 10 1 10 2

−40

−35

−30

−25

−20

−15

−10

−5 0 5

ω

|X( ω )|(c), |G(j ω )|(r), |Y( ω )|(b) [dB]

Spettro delle Ampiezze

Wl‚(É'gM[^a!gMijce[OnY‡ncei‹…YMdzgMp,coaMcep'pŸgxdf_b]†…Y])n%g [^cepMncei‹…Ydf_b]†kMp©n%g!¤

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Risposta(r) all’impulso(b) triangolare

Wl‚Y£]†…c†nn`dsYoaMcep'p,c9gMi‹…_bcGwwc½aE_R} ŒMŽ€Šc»]–Y])n%gM[^w _hgMp'ijce[Onc9_baMce[On _btGY‰gMp'p,Y£]†…c†nn`dsY

aE_

œ

} ŒMŽ€‘g‹tGY]gžŠc£aMY´ÁkOnY~gMp'p,YMdzgx_rp@df_Ãn%gdsaMYuaE_(¡¢}›€9df_b]†…c†nnY g‰šJ}›€

(10)

    ! " "#%$'&()&+* " *,"$'&+-.*0/,*,%&132#* 4%54 6

Wl‚Yu]†…c†nn`dsYuaMcep'p,covg^]`_!aMcep'pŸg df_b]†…Y])n%g‹¡¢}›€»df_b]†kMp©n%g!¤

10 0 10 1 10 2

−250

−200

−150

−100

−50 0 50 100 150 200

ω

arg X( ω )(c), G(j ω )(r), Y( ω )(b) [deg]

Spettro delle Fasi

W·ce[On`dsc„p,Y·]†…c†nn`dsY‹aMcep'p,c gMi‹…_bcGwwc»aE_M|} ŒMŽ€LŠcj]–Y])n%gM[^w _hgMp'ijce[Onco_baMce[On _btGYžgMp'p,Y

]†…c†nn`dsYqaE_

œ

} ŒMŽ€ µ_QaBk^cx]†…c†nn`df_VaMcep'p,cuvg^]`_VaE_,ÂQcedf_b]–tGY\[^Yqv dzgRp,YMdsYEµ gM[^te’^cq…ced

…MkMp,]g^w _bY\[_EdscepŸgOn _ôegMijce[Onc£…_btGtGY\p,cŽÅ«[‡…gd+n _btGY\pŸgdsc‹p,Y‹]†…c†nn`dsY‹aE_vg^]–cÓaE_M|} ŒMŽ€

…dscG]–ce[On%g‡gM[^•Y\pN_yi‰_r[^YMdf_ df_b]†…c†nnY gMp'p,Yu]†…c†nn`dsYžaE_(vg^]–coaE_

œ

} ŒMŽ€`

Wq¸y_…gdˆpŸg9_r[½ªBk^cG])nY9tCg^]–Y»aE_ÞOÀÞOÀ  ßOÛZ¾ïî3Û ¿îf¾ ÀM¿0Ü\ï Y9aE_¿îf¾ ÀM¿0Ü\ïÈÜMîOÀÞCß aMcep'pŸg

vg^]–coaMcep!]–cG•\[gMp,c£aE_Vk^]–t_Ãn%gxdf_b]†…c†nnY g‰ªBk^cep'pŸg·aMcep!]–cG•\[gMp,c£aE_y_r[^•\dscG]–]–YB

WÈØjkgMpN_Ãn%gOn _ôegMijce[Onc»kM[„df_Ãn%gdsaMY aE_Cvg^]–c9_r[On`dsY!aMYnnYoa!g£kM[»]`_b])nceiog pN_r[^cCgdscÓ_r[^aBk^tGc

kM[ df_Ãn%gdsaMYuv dzgx_rpÁ]–cG•\[gMp,c£aE_y_r[^•\dscG]–]–Yuc‡_rp!]–cG•\[gMp,c£aE_Vk^]–t_Ãn%g!

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Segnale

Risposta(r) all’impulso(b) triangolare

(11)

Wq¸y_!tGY\[^]`_baMcedf_Bg^a cG]–cei‹…_bY_rpÁ]`_b])nceiog‹g‰vg^]–c‡[^Y\[mi‰_r[_riog!¤

{G}f€¦¥

 ã?

 ¯ 

gO´ ce[Onc|ijY!aBkMp,Y kM[_Ãn%gdf_bYlgn`kOnnc|p,c|…MkMp,]g^w _bY\[_̈ªBk_r[^aE_àgM[^a!gl_r[V[_Ãn%geÐ µ½iog

te’^c‡_r[On`dsY!aBk^tGcukM[mdf_Ãn%gdsaMYuaE_!vg^]–c a!gxg£Ä



Î †

WR¨¦pN_ ]†…c†nn`df_ aMcep'p,c gMi‹…_bcGwwclaE_

œ

} ŒMŽ€~c aE_LR} ŒMŽ€~]–Y\[^YÙ_baMce[On _bt_LcG]–]–ce[^aMY

¬

{G} ŒMŽ€

¬ ¥ 

…ced ¢ Æ‚YR]†…c†nn`dsY|aMcep'p,c vg^]`_ ̖…ced  ¥ 

ÎÐ aMcep'pŸgÇdf_b]†…Y])n%g

¡¢}›€»df_b]†kMp©n%g!¤

10 0 10 1 10 2

−350

−300

−250

−200

−150

−100

−50 0 50 100 150 200

ω arg X( ω )(c), G r (j ω )(r), Y( ω )(b) [deg]

Spettro delle Fasi

WlÅ«pVdf_Ãn%gdsaMYxaE_!vg^]–c _r[On`dsY!aMYnnY~a!gMpE]`_b])nceiogu{G}f€Ž]`_(n`dzg^aBk^tGcx_r[kM[df_Ãn%gdsaMYmaMcep

]–cG•\[gMp,c¡¢}›€ df_b]†…c†nnY»gLšJ}›€ µ^Y\p©n`dsc g^a£kM[ŽgM[^a!gMijce[OnYj_r[ †ïÛZ¾f¿ ï OÀÞCß _r[L]–cG•\k_ÃnY

g_!tCgMi‹à_!aE_(aMcedf_ôegOn%g‹aMcep!]–cG•\[gMp,c£aE_@_r[^•\dscG]–]–YB¤

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Segnale

Risposta(r) all’impulso(b) triangolare

(12)

798;:.<=8?>@8BACA8;:EDGFH<;FyI AMINAODGFQPHILU FÁDU;8@I :y8EDeI

WRXZY\[^]`_baMcedf_hgMijYƒkM[‹]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnYutGY\[mdsc†n`dsYMg^w _bY\[^cžkM[_Ãn%gdf_hg!¤

C(s) G(s) Y(s)

R(s) X(s)

C(s) G(s) Y(s)

R(s) X(s)

Å«p•\kg^a!g^•\[^Y—aE_LgM[^cep'p,Y²aMcep¦]`_b])nceiog {}f€xc˜pŸgHv kM[^w _bY\[^cHaE_n`dzg^]vzcedf_rijce[OnY

_r[^•\dscG]–]–Y Äk^]–t_Ãn%gu{  }f€9df_b]†kMp©n%gM[^YB¤

{ 

}f€¦¥~}f€%{~}f€ { 

}f€¥

|}f€

}f€

¥

~}f€%{~}f€

 ¯

~}f€%{~}f€

¥ {  }f€

 ¯

{ }f€

Wl‚gžpŸgds•\’^cGwwOg„aE_.àgM[^a!g‰aMcepM]`_b])nceiog·dsc†n`dsYMg^w _bY\[gOnY {  }f€¶…MkÁŠYžcG]–]–cedscj])n _riogOn%g

tGY\[^Y]–tGce[^aMYq_rp!•\kg^a!g^•\[^Y a.É'gM[^cep'p,Y{ }f€`

WlÖEced¦p,cž…MkMp,]g^w _bY\[_M n%gMpN_!te’^c

¬ {  } ŒMŽ€

¬



µ\´egMp,c‡prÉ'gM…M…dsY]–]`_riog^w _bY\[^c¤

¬ {  } ŒMŽ€

¬ ¥ ¬{  } ŒMŽ€

¬

¬  ¯

{ } ŒMŽ€

¬ × 

ijce[On`dsc·…ced¦p,cž…MkMp,]g^w _bY\[_Á?n%gMpN_(te’^c

 ¬

{ } ŒMŽ€

¬Bâ

Î(µ.]`_@’g!¤

¬{ O} ŒMŽ€

¬ ¥ ¬

{ } ŒMŽ€

¬

¬  ¯ {  } ŒMŽ€

¬ × ¬

{ } ŒMŽ€

¬  ¬ { C} ŒMŽ€

¬Bâ

Î

Å«[^aE_btCgOn _ôegMijce[Onc9aBkM[^ªBk^c·p,c‹…MkMp,]g^w _bY\[_MaE_n%g^•\pN_bY‹te’^cjaMcepN_ri‰_Ãn%gM[^Y pŸg‡àgM[^a!g£aMcep

]`_b])nceiog™dsc†n`dsYMg^w _bY\[gOnY”{ C}f€‡]–Y\[^Y³_r[OnYMdˆ[^Y”gMp'p,c …MkMp,]g^w _bY\[_  …ced·p,c|ªBkgMpN_

¬ {  } ŒMŽ€

¬!×

 

Wl‚gàgM[^a!gƒ…g^]–]gM[Onc‰aE_¢kM[u]`_b])nceiog _r[ƒdsc†n`dsYMg^w _bY\[^cmkM[_Ãn%gdf_hgžtGYMdˆdf_b]†…Y\[^aMcm_r[^aE_©Ä

tCgOn _ôegMijce[Onc»gMp'p,c£…MkMp,]g^w _bY\[_ è…ced p,c9ªBkgMpN_(_rp!ijY!aBkMp,Y‰aMcepe•\kg^a!g^•\[^Y‹aE_MgM[^cep'p,Y

¬ {  } ŒMŽ€

¬

Šcuiog^••(_bYMdsc£aE_ 

W cepetCg^]–Y‰te’^c‰[^Y\[ot_]`_hgM[^Y …MkMp,]g^w _bY\[_ Ï…ced p,c9ªBkgMpN_(_rp\ijY!aBkMp,Y‰aMcep•\kg^a!g^•\[^Y

aE_=gM[^cep'p,YÇ]`_hg

¬{  } ŒMŽ€

¬ Í 

µLpŸglàgM[^a!gqaMcep¢]`_b])nceiog|_r[ldsc†n`dsYMg^w _bY\[^cRkM[_Ãn%gdf_hg

tGY(_r[^t_baMc„tGY\[‹ªBk^cep'pŸg·aMcep!•\kg^a!g^•\[^Y aE_EgM[^cep'p,YB

(13)

W‘[^te’^cx…ced¹_rp(]`_b])nceiog dsc†n`dsYMg^w _bY\[gOnY{  }f€L´egMp,•Y\[^Y|p,c„])ncG]–]–c„tGY\[^]`_baMcedzg^w _bY\[_

te’^c‡p,cG•MgM[^YƒpŸgx…dsY\[OncGwwOg‹aMcepÁ]`_b])nceiog‰gMp'pŸg‰]†kgxpŸgds•\’^cGwwOg‰aE_yàgM[^a!g!¶ÖEced+n%gM[OnY

…cedJiog^]–]`_ri‰_bwwOgdsc£pŸgo´ cep,Y!t_Ãn`Šg„aE_.df_b]†…Y])n%g„Y!tGtGYMdˆdsc„gMp'pŸgds•Mgdsc‹_rpB…_«Šk …Y]–]`_rà_rp,c·pŸg

àgM[^a!g‹aMcep!]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnY~{  }f€`

W¦gOnYxte’^cž_rp(•\kg^a!g^•\[^Y~aE_BgM[^cep'p,Yƒ{ }f€ ¥ ~}f€%{~}f€LtGY\[On _bce[^c‰tGY\ijc£vgOnnYMdsc

pŸg v kM[^w _bY\[^cmaE_yn`dzg^]vzcedf_rijce[OnYqaMcepVtGY\[On`dsY\p'p,YMdsc ~}f€£Šc c†´M_baMce[Oncxte’^cx•ce[^cedzgMpÄ

ijce[Onc‹tGY\[O´M_bce[^cÌˆtGY\i‹…gOn _rà_rp'ijce[Onc·tGY\[ÈpŸgu])n%gMà_rpN_Ãn`ŠguaMcep.]`_b])nceiogžc‡tGY\[mgMp©n`dsc

]†…cGt_\te’^c„aE_EtGY\[On`dsY\p'p,YÁÐ …dsY•c†nn%gdscu_rp(tGY\[On`dsY\p'p,YMdsc ~}f€½tGY\[‡•\kg^a!g^•\[^YRiog^• Ä

•(_bYMdscƒ…Y]–]`_rà_rp,c~Å«[mªBk^cG])nYlijY!aMYƒ]`_QgMp'pŸgds•MgÈpŸgÈàgM[^a!g|̈cžªBk_r[^aE_¢gMkMijce[On%gqpŸg

…dsY\[OncGwwOgeÐLaMcep!]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnYB

W  ÓÖÔdsY•c†nn%gM[^aMY _r[ÆijY!aMY|Y\…M…YMd+n`kM[^Y _rpQtGY\[On`dsY\p'p,YMdsc ~}f€

Šcƕce[^cedzgMp'ijce[Onc…Y]–]`_rà_rp,cµ£ijcGaE_hgM[OncÇtGY\[On`dsY\p'p,YT_r[”dsc†n`dsYMg^w _bY\[^cµÓgMp'pŸgds•MgdscHg

…_hg^tGcedsc‡pŸg àgM[^a!g·aE_ykM[·]`_b])nceiog·{~}f€`

Å«p…dsY\àMp,ceiog£…dzgOn _btGY„ŠcŽte’^c prÉ'gMp'pŸgds•MgMijce[OnYjaE_MàgM[^a!g9aE_ÁkM[»]`_b])nceiog p,ce[OnY·_ri‹…MpN_©Ä

tCg‡prÉ'gM…M…MpN_btCg^w _bY\[^c aE_M]–cG•\[gMpN_MaE_E_r[^•\dscG]–]–Y‡aE_Mcep,c†´egOn%g„gMi‹…_bcGwwOgjte’^c·…Yn`dsceàMàcedsY

[^Y\[ df_bce[On`dzgdsc‡[^cep'p,co]†…cGt_\te’^coaMcepÁ]`_b])nceiog£tGY\[On`dsY\p'pŸgOnYÌˆcG]–cei‹…_«¤J]gOn`kMdzg^w _bY\[_ìµ

pN_ri‰_Ãn _.ijcGtGtCgM[_bt_!Y \]`_bt_ìµZpN_ri‰_Ãn _!ce[^ceds•c†n _bt_ìµ+##'Ð`

W  !"$# %$&('%$)+*,-#-)/.0."&(,)1%2!"!3.")546&78&7 9., %%$&(,)

{~}f€:746;8<)5,)59.")= +%$)>8)5)5!"!"?.



&@A 9., %%$8B

WRXZY\[^]`_baMcedf_hgMijY g^a cG]–cei‹…_bY~_rp!]–cG•\k^ce[Onc£]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnYB¤

K Y(s)

R(s) X(s)

G(s)

K Y(s)

R(s) X(s)

G(s)

{~}f€¦¥

 ê

 ¯  ê

Å«p\•\kg^a!g^•\[^Y~aE_.gM[^cep'p,Yq{ }f€‘cxpŸg·v kM[^w _bY\[^c‹aE_\n`dzg^]vzcedf_rijce[OnYƒ_r[^•\dscG]–]–Y Äk^]–t_Ãn%g

{ O}f€9df_b]†kMp©n%gM[^YB¤

{  }f€—¥ Ê {~}f€¥

Ê  ê

 ¯

ê

{ C}f€—¥

|}f€

}f€

¥ Ê {~}f€

 ¯ Ê {~}f€

¥ Ê ê

 ¯ }fÊ

¯ 

€ˆ ê

(14)

W(]–cei‹…_bY„aE_MgMp'pŸgds•MgMijce[OnY„aMcep'pŸg‹àgM[^a!gÓn`dzgMi‰_Ãncodsc†n`dsYMg^w _bY\[^c„…ced=kM[o•\kg^a!g^•\[^Y

aE_EgM[^cep'p,YuaE_!n _r…Yƒ…g^]–]gGÄàg^]–]–YB

w 20 dB

-20 dB

w H

banda passante guadagno d’anello

j dB

G ( ) |

| 0 w

banda passante sistema retroazionato

w H0

0 dB

dB a j

G ( ) |

| w

w 20 dB

-20 dB

w H

banda passante guadagno d’anello

j dB

G ( ) |

| 0 w

banda passante sistema retroazionato

w H0

0 dB

dB a j

G ( ) |

| w

W(]–cei‹…_bY„aE_MgMp'pŸgds•MgMijce[OnY„aMcep'pŸg‹àgM[^a!gÓn`dzgMi‰_Ãncodsc†n`dsYMg^w _bY\[^c„…ced=kM[o•\kg^a!g^•\[^Y

aE_EgM[^cep'p,YuaE_!n _r…Yƒ…g^]–]gGÄàgM[^a!g!

w

dB a j

G ( ) |

| w

20 dB w

L

banda passante sistema retroazionato

j dB

G ( ) |

| 0 w

w H

w L0 w

H0

banda passante guadagno d’anello

0 dB w

dB a j

G ( ) |

| w

20 dB w

L

banda passante sistema retroazionato

j dB

G ( ) |

| 0 w

w H

w L0 w

H0

banda passante guadagno d’anello

0 dB

(15)

Wl‚g„v kM[^w _bY\[^c‰aE_Án`dzg^]vzcedf_rijce[OnYžaMcep(]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnYmŠc‹gM[^tGYMdzg‹aMcepy…df_rijY

YMdsaE_r[^c

{  }f€”¥

|}f€

}f€

¥ Ê {~}f€

 ¯ Ê {~}f€

¥ Ê  ê

 ¯ }fÊ

¯ 

€ˆê

ªBk_r[^aE_V…Y\[^ce[^aMY

Ê ¥  ê 

ê

ã 

pŸgxpŸgds•\’^cGwwOg‹aE_yàgM[^a!g·aMcep!]`_b])nceiogždsc†n`dsYMg^w _bY\[gOnYƒ…g^]–]g‹a!g£ ê go ê  

WRXZY\[ƍê ¥ 

Ιcȍê ™¥



ÎeΔ_aE_hg^•\dzgMi‹i‰_JaMcep'p,clgMi‹…_bcGwwcRaE_½{ } ŒMŽ€xc

{  } ŒMŽ€Ódf_b]†kMp©n%gM[^YB¤

10 0 10 1 10 2 10 3

−20

−15

−10

−5 0 5 10 15 20

ω

|G a (j ω )| (b−−), |G 0 (j ω )| (r) [dB]

Diagramma delle Ampiezze con ω H =10 e K=9

(16)

WRXZY\[ƍ ê ¥ 

Ιcȍ ê  ¥ 

ÎeΔ_aE_hg^•\dzgMi‹i‰_JaMcep'p,clgMi‹…_bcGwwcRaE_½{  } ŒMŽ€xc

{ O} ŒMŽ€Ódf_b]†kMp©n%gM[^YB¤

10 0 10 1 10 2 10 3

−20

−15

−10

−5 0 5 10 15 20

ω

|G a (j ω )| (b−−), |G 0 (j ω )| (r) [dB]

Diagramma delle Ampiezze con ω H =10 e K=9

WRXZY\[ƍê ¥ äeΙcȍê ™¥



ÎeΔ_aE_hg^•\dzgMi‹i‰_JaMcep'p,clgMi‹…_bcGwwcRaE_½{ } ŒMŽ€xc

{ O} ŒMŽ€Ódf_b]†kMp©n%gM[^YB¤

10 0 10 1 10 2 10 3

−20

−15

−10

−5 0 5 10 15 20

ω

|G a (j ω )| (b−−), |G 0 (j ω )| (r) [dB]

Diagramma delle Ampiezze con ω H =30 e K=2.33

(17)

WRXZY\[j ê ¥ 

΄c» ê  ¥ 

ÎeÎup,c‰df_b]†…Y])nc gMp'prÉ«_ri‹…MkMp,]–Ymdsc†nn%gM[^•Y\pŸgdscÓaMcep]`_b])nceiog

{~}f€ c£aMcep!]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnY~{



}f€»df_b]†kMp©n%gM[^YB¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

anello aperto(r−−), retroazione e comp.(g)

Confronto risposte all’impulso(b)

WRXZY\[jê³¥Ïäe΄c»ê ½¥



ÎeÎup,c‰df_b]†…Y])nc gMp'prÉ«_ri‹…MkMp,]–Ymdsc†nn%gM[^•Y\pŸgdscÓaMcep]`_b])nceiog

{~}f€ c£aMcep!]`_b])nceiogudsc†n`dsYMg^w _bY\[gOnY~{  }f€»df_b]†kMp©n%gM[^YB¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

anello aperto(r−−), retroazione e comp.(g)

Confronto risposte all’impulso(b)

(18)

WRXZY\[ƍ ê ¥ 

Ιc| ê  ¥ 

ÎeΙ•\pN_‘_r[^•\dscG]–]`_ gMpJ]`_b])nceiog {~}f€ucHgMpJ]`_b])nceiog

dsc†n`dsYMg^w _bY\[gOnYm{ C}f€9df_b]†kMp©n%gM[^YB¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−15

−10

−5 0 5 10 15

Tempo [s]

anello aperto(r−−), retroazione e comp.(g)

Confronto segnali di controllo

WRXZY\[ƍê ¥ äeΙc|ê ™¥



ÎeΙ•\pN_‘_r[^•\dscG]–]`_ gMpJ]`_b])nceiog {~}f€ucHgMpJ]`_b])nceiog

dsc†n`dsYMg^w _bY\[gOnYm{  }f€9df_b]†kMp©n%gM[^YƒÌ gOnnce[^w _bY\[^c‰gMp'pŸg‹]–tCgMpŸg£´ ced+n _btCgMp,c«Ð`¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−3

−2

−1 0 1 2 3 4 5 6

Tempo [s]

anello aperto(r−−), retroazione e comp.(g)

Confronto segnali di controllo

(19)

WRXZY\[u ê ¥ 

Îmc‹ ê  ¥ 

ÎeÎ|p,cdf_b]†…Y])ncxgMp'prÉ«_ri‹…MkMp,]–YÇdsc†nn%gM[^•Y\pŸgdscžaMcepy]`_b])nc`Ä

iogm{~}f€¶c‡aMcep.]`_b])nceiog~dsc†n`dsYMg^w _bY\[gOnY|{



}f€»tGY\[xc‡]–ce[^wOg ¸\Qå V Å 

]†kMp'prÉ«_r[^•\dscG]–]–YB¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Sistema con (−) e senza(−−) saturazione

Confronto risposte all’impulso(b)

WRXZY\[uꧥ äeÎmc‹ê £¥



ÎeÎ|p,cdf_b]†…Y])ncxgMp'prÉ«_ri‹…MkMp,]–YÇdsc†nn%gM[^•Y\pŸgdscžaMcepy]`_b])nc`Ä

iogm{~}f€¶c‡aMcep.]`_b])nceiog~dsc†n`dsYMg^w _bY\[gOnY|{  }f€»tGY\[xc‡]–ce[^wOg ¸\Qå V Å 

]†kMp'prÉ«_r[^•\dscG]–]–YB¤

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Tempo [s]

Sistema con (−) e senza(−−) saturazione

Confronto risposte all’impulso(b)

(20)

    ! " "#%$'&()&+* " *,"$'&+-.*0/,*,%&132#* 4%54

798;:.<=8T>@8BACA8;:EDGF F@FVSyI÷D8˜<JILUINA!> QAODC8

Wl‚g pŸgds•\’^cGwwOg‹aE_VàgM[^a!g‡aE_VkM[·]`_b])nceiog‡Šcž_r[mdscepŸg^w _bY\[^c„tGY\[ _!ncei‹…_(aE_Bg^]–]–cG])n%gGÄ

ijce[OnYžc£aE_(]gMpN_Ãn%g‰te’^c‡_baMce[On _\tCgM[^YqpŸg„´ cep,Y!t_Ãn`Šg‰aE_ydf_b]†…Y])n%g!

WlÖEced£kM[Ç]`_b])nceiogÈaMcep¹…df_rijY YMdsaE_r[^cḦcH…ced9cG])nce[^]`_bY\[^cl…ced£_ ]`_b])ncei‰_=tGY\[ kM[

…Y\p,YÇaMY\i‰_r[gM[Onc^Ї_rpVncei‹…YRaE_Ôg^]–]–cG])n%gMijce[OnY|c|pŸgÆpŸgds•\’^cGwwOgƒaE_JàgM[^a!gq]–Y\[^Y

_r[O´ ceds]gMijce[Onc·…dsY\…YMdsw _bY\[gMpN_«¤

{~}f€¥

 ê

 ¯

ê



¶¥

ä

ê

WlÖEced»kM[ƒ]`_b])nceiog aMcepV]–cGtGY\[^aMYRYMdsaE_r[^cmtGY\[DžY\pN_ZtGY\i‹…Mp,cG]–]`_QcxtGY\[_rk^•MgOn _ ̈cȅced

cG])nce[^]`_bY\[^c …cedL_.]`_b])ncei‰_\tGY\[q…Y\pN_.tGY\i‹…Mp,cG]–]`_Bc‹tGY\[_rk^•MgOn _@aMY\i‰_r[gM[On _÷Ð µ¢gƒ…gdf_

tGY!cžt_bce[Onc~aE_Ô]†ijYMdswOgMijce[OnYEµ _rpZncei‹…YÇaE_ g^]–]–cG])n%gMijce[OnY|cÇpŸglpŸgds•\’^cGwwOgqaE_

àgM[^a!g‹]–Y\[^Yƒ_r[O´ ceds]gMijce[Onc‡…dsY\…YMdsw _bY\[gMpN_«¤

{~}f€¥



 ¯



  ¯   ê ×    ¥ ä



(21)

WÈØjkgMpN_Ãn%gOn _ôegMijce[Onc„_rpncei‹…Y‡aE_!]gMpN_Ãn%g

 

aE_@kM[‹]`_b])nceiog£aE_Mn _r…Y…g^]–]guàg^]–]–Y Šc

_r[O´ ceds]gMijce[Onc·…dsY\…YMdsw _bY\[gMp,c·gMp'pŸgxàgM[^a!gm…g^]–]gM[Onc

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5 0 0.5 1 1.5

diagramma di Nyquist

0.47 0.56

0.68 0.82

1 1.2 1.5 2.2

0.15 0.18 0.22 0.27 0.33 0.39 0.47

0.56 0.68 1

0 10 20 30 40 50 60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

risposta nel tempo

secondi

Øjk^cep'pN_Ždf_r…YMd+n%gOn _½_r[ \•\kMdzg ]–Y\[^YT_¦aE_hg^•\dzgMi‹i‰_ aE_  ªBk_b])nmaMc_¦]–cG•\k^ce[On _¦aBk^c

]`_b])ncei‰_«¤

{ 6 }f€¦¥

 Î

(}f

¯  € }f

¯ 

΀

{ }f€¦¥



(}f

¯  € }f

¯ 

΀

 BgM[^tGYEµ;]–Y\[^Yldf_r…YMd+n%gOn _@•\pN_QgM[^a!gMijce[On _Encei‹…YMdzgMpN_.aMcep'pŸgÈdf_b]†…Y])n%g gMp.•\dzg^aE_r[^Y

aMc_¹tGYMdˆdf_b]†…Y\[^aMce[On _L]`_b])ncei‰_½_r[³dsc†n`dsYMg^w _bY\[^c²kM[_Ãn%gdf_hg! XZY\ijcl]`_¶…MkÁŠY²´ cGaMcedsc

a!g_J•\dzg\t_ìµ [^c_aBk^cÇtCg^]`_L_rp ncei‹…Y aE_]gMpN_Ãn%gÌ

  6 ×



c   × 

]0ÐmŠc



t_rdstCg x_r[O´ ceds]gMijce[Onco…dsY\…YMdsw _bY\[gMp,cogMp'pŸg‹pŸgds•\’^cGwwOg aE_\àgM[^a!goaMc_aBk^cÓ]`_b])ncei‰_

dsc†n`dzg^w _bY\[gOn _LÌ3 ê  6 × Î

 

cȍ ê  × Î  

Оt_bYÁŠcHgMp'p,c™…MkMp,]g^w _bY\[_  ê  aE_

_r[Onceds]–cGw _bY\[^c£aMc_(aE_hg^•\dzgMi‹i‰_ÁaE_ ªBk_b])n‘tGY\[ _rp!tGcedste’_bYƒkM[_Ãn%gdf_bYB

Riferimenti

Documenti correlati

teorema degli zeri).. teorema degli zeri).. teorema

Università degli Studi di Trento Corso di Laurea in

Sia E 3 il 3–spazio euclideo ordinario dotato del riferimento cartesiano standard di coordinate (x,

Infatti in tale topologia due aperti non vuoti hanno almeno il punto p in comune e quindi non possono esistere intorni disgiunti.. Supponiamo che f

[r]

[r]

Successivamente abbiamo ricavato, tramite il teorema di Noether per traslazioni rigide infinitesime, il relativo pseudotensore energia-impulso e dopo averlo sviluppato all’ordine h 2

Si tratta quindi di determinare il piano ortogonale a s passante