Diagrammi di Bode
² i³u¬ t¡ ´ © ³ull¡ ² i³po³t¡ armo® ic¡ £ © ¤ icono c¨e W (jω 0 ) ³©
° uo' i® ´e² p² ta² coe ¬'"aº io®e" ¤e¬ ³i³´em¡ all¡ pa² ´e ¤ ©
ing²sso ¤ © ° ulsaº io®e ω 0 . ² ta® to, a¬ va² ia² ¤ © ω, W (jω)
¤esc² i¶ coe i¬ ³i³´em¡ a§ is£e ³ul¬e co po®e® ´ © al¬e va² ©e
° ulsaº io® © (f²± µenºe) ¤e¬ ³ gna¬e ¤ © ing²sso.
i³u¬ t¡ °e² ta® to i® ´e²ssa® ´e r¡ ° p²³ ® ta² gra¦ icae® ´e
± µe³t¡ informaº io®e os³i¡ for® i² un¡ r¡ ° p²³ ® taº io®e
gra¦ ic¡ ¤ © W (jω). µe³t¡ e' un¡ ¦ unº io®e complessa
¤ell¡ va² ia¢i¬e reale ω e ± µ in¤ © l¡ ³u¡ r¡ ° p²³ ® taº io®e
² ic¨ ©e¤e²b¢ u® gra¦ ico t² i¤ ien³iona¬e.
³© ¤e³i¤er¡ e¶© ta² ± µe³t¡ (poco pr¡ ´ ic¡) pos³i¢i¬ © t¡', i¬
modo ° © µ' n¡ ´ µra¬e e inform¡ ´ ivo ¤ © r¡ ° p²³ ® ta² W (jω) e' ± µello ¤ © ² icor²² ¡ due gra¦ i£ © °er ¬e ¤ µe ¦ unº io® © (²a¬ ©
¤ © va² ia¢i¬e ²a¬e) |W (jω)| e Arg[W (jω)].
r ¯´ ´e®e² gra¦ i£ © ° © µ' ¬eg§ i¢i¬ © e trac£ ia¢i¬ © i® modo ° © µ'
a§vo¬e ³© ² icor² a¤ al£ u® © accor§ ie® ´ ©:
1. P¯ic¨e' W (−jω) = W ((jω) ∗ ) = [W (jω)] ∗ , ¤ © modo c¨e
|W (−jω)| = |W (jω)|, ∀ω (0.1a)
Arg[W (−jω)] = Arg[W (jω)], ∀ω (0.1b)
³© possono trac£ ia² © gra¦ i£ © °er © so¬ © valo² © po³© ´ i¶©
¤ © ω (inf¡ ´ ´ © ¬e informaº io® © ²l¡ ´ i¶ ¡ © valo² © ®eg¡ ´ i¶©
¤ © ω ³© ² icavano d¡ © gra¦ i£ © ²l¡ ´ i¶© ¡ ω > 0 tra © ´e ¬e
(0.1)).
2. r a¶ ² un¡ ig¬ io² ² iso¬ uº io®e ³µ u®'a ° i¡ gamm¡
¤ © f²± µenºe (o ° ulsaº io® ©), ³© trac£ iano © gra¦ i£ © ad¯´- tando i® as£ iss¡ un¡ coor¤ in¡ t¡ loga² © t ic¡: ° © µ' p²-
£ isae® ´e © gra¦ i£ © ¶ ngono trac£ i¡ ´ © i® ¦ unº io®e ¤ ©
log 10 (ω) in¶ £e c¨e i® ¦ unº io®e ¤ © ω (³© n¯´ © c¨e ³´ ©- amo con³i¤erando solo valo² © po³© ´ i¶© ¤ © ω e ± µ in¤ © i¬
loga² © tmo e' ¢ ® ¤e¦ i® © to).
3. I¬ mo¤ ulo ³© r¡ ° p²³ ® t¡ i® decibel . I® a¬ t² paro¬e
no® ³© r¡ ° p²³ ® t¡ i¬ gra¦ ico ¤ © |W (jω)| m¡ ± µello ¤ ©
|W (jω)| dB := 20 log 10 |W (jω)|. (0.2) Os³ r¶iamo ³u¢© to c¨e i¬ mo¤ ulo e³p²sso i® ¤e£ i¢ ¬ ° uo' as³ue² ³i¡ valo² © po³© ´ i¶© ³i¡ valo² © ®eg¡ ´ i¶© anº ©:
|W (jω)| dB > 0 ⇔ |W (jω)| > 1
|W (jω)| dB < 0 ⇔ |W (jω)| < 1
|W (jω)| dB = 0 ⇔ |W (jω)| = 1
io' ³ig® i¦ ic¡ c¨e:
°er valo² © po³© ´ i¶© ¤ © |W (jω)| dB i¬ ³i³´em¡ a p¬ i¦ ic¡
l¡ co po®e® ´e ¤el¬'ing²sso all¡ ° ulsaº io®e ω e °er
°er valo² © ®eg¡ ´ i¶© ¤ © |W (jω)| dB i¬ ³i³´em¡ ¡ ´ ´e® u¡ l¡
co po®e® ´e ¤el¬'ing²sso all¡ ° ulsaº io®e ω.
I® conc¬ u³io®e, ¤evo r¡ ° p²³ ® ta² © gra¦ i£ © ¤el¬e ¤ µe ¦ u®- º io® ©
|W (jω)| dB := 20 log 10 |W (jω)| e Arg[W (jω)]
i® ¦ unº io®e ¤ © log 10 (ω) (ω > 0). Ta¬ © gra¦ i£ © ³© c¨ iamano diagrammi di Bode e °er trac£ iar¬ © ² i³u¬ t¡ con¶ ® ©e® ´e
sc² i¶ ² l¡ ¦ unº io®e ¤ © tras¦ ² ie® to W (s) i® un¡ pa² ´ ico-
la² form¡ (¤e ´ t¡ form¡ ¤ © Bo¤e). ± µe³to sc¯po sc² i¶iamo d¡ ° p² im¡ W (s) e ´ ´endo®e i® e¶i¤enz¡ po¬ © e ºe² © e i®- co² porando i® u® ´er i®e ¤ © ³ condo grado ¬e c¯° ° ©e ¤ ©
po¬ © e ºe² © co p¬es³© co® © ug¡ ´ ©:
W (s) = K E 1 s ν
ˆ r
Y
i=1
(s − z i )
ˆ c
Y
i=1
(s 2 + 2 ˆ ξ i ω ˆ i s + ˆ ω i 2 )
r
Y
i=1
(s − p i )
c
Y
i=1
(s 2 + 2ξ i ω i s + ω i 2 )
(0.3)
L¡ (0.3) e¶i¤enº i¡:
• l¡ co³ta® ´e K E ¤e ´ t¡ Guadagno di Evans;
• ¬'e¶ ® ´ ua¬e polo ®el¬'o² i§ i®e ¤ © mo¬ ´e p¬ i£ © t¡' ν (o, ³ ν < 0 , lo ºero ®el¬'o² i§ i®e ¤ © mo¬ ´e p¬ i£ © t¡' −ν);
• © po¬ © ²a¬ © p i e i¬ loro ® uero r;
• g¬ © ºe² © ²a¬ © z i e i¬ loro ® uero ˆr;
• ¬e c c¯° ° ©e ¤ © po¬ © co p¬es³© co® © ug¡ ´ © (do¶ g¬ © e¬ee® ´ ©
¤ell¡ i−e³im¡ c¯° ° i¡ hanno mo¤ ulo ω i e pa² ´e ²a¬e
−ξ i ω i , co³icc¨e' ω i > 0 e −1 < ξ i < 1 ).
• ¬e ˆc c¯° ° ©e ¤ © ºe² © co p¬es³© co® © ug¡ ´ © (do¶ g¬ © e¬ee® ´ ©
¤ell¡ i−e³im¡ c¯° ° i¡ hanno mo¤ ulo ˆω i e pa² ´e ²a¬e
− ˆ ξ i ω ˆ i , co³icc¨e' ˆω i > 0 e −1 < ˆξ i < 1 ).
© n¯´ © c¨e ³ i¬ grado ²l¡ ´ ivo ¤ © W (s) e' ºero, allor¡
K E = W (∞) e' i¬ § uadagno i® a¬ t¡ f²± µenz¡ ¤ © W (s).
± µe³to ° u® to ¤e¦ i® iamo τ i := − 1
p i τ ˆ i := − 1
z i (0.4)
e os³ r¶iamo c¨e
s − p i = −p i (1 + sτ i ), s − z i = −z i (1 + sˆ τ i ) (0.5) e
s 2 + 2ξ i ω i s + ω i 2 = ω i 2 ( s 2
ω i 2 + 2 ξ i
ω i s + 1) (0.6)
¤ © modo c¨e pos³iamo sc² i¶ ² W (s) ®ell¡ form¡:
W (s) = K B 1 s ν
ˆ r
Y
i=1
(1 + sˆ τ i )
ˆ c
Y
i=1
( s 2 ˆ
ω i 2 + 2 ξ ˆ i
ˆ
ω i s + 1)
r
Y
i=1
(1 + sτ i )
c
Y
i=1
( s 2
ω i 2 + 2 ξ i
ω i s + 1)
(0.7)
do¶
K B := K E
ˆ r
Y
i=1
(−z i )
ˆ c
Y
i=1
ˆ ω i 2
r
Y
i=1
(−p i )
c
Y
i=1
ω i 2
(0.8)
Definizione. L¡ (0.7) ³© ¤ i£e forma di Bode ¤ © W (s) e K B
³© ¤ i£e guadagno di Bode ¤ © W (s).
© n¯´ © c¨e ³ ν = 0, allor¡ K B = W (0) e' i¬ § uadagno i®
co® ´ i® u¡ ¤ © W (s).
Tracciamento dei diagrammi di Bode
D¯po a¶ r esso W (s) i® form¡ ¤ © Bo¤e ³iamo pro® ´ © °er trac£ iar®e © ¤ iagram © ¤ © Bo¤e. ± µe³to sc¯po ² icor¤ iamo c¨e, d¡ ´ © ¤ µe ® ue² © co p¬es³© a e b, ³© h¡:
Arg(a · b) = Arg(a) + Arg(b), (0.9)
Arg(a/b) = Arg(a) − Arg(b), (0.10)
20 · log[|a · b|] = 20 · log[|a|] + 20 · log[|b|], (0.11)
e
20 · log[|a/b|] = 20 · log[|a|] − 20 · log[|b|], (0.12)
² ta® to © ¤ iagram © ¤ © Bo¤e ³i¡ ¤e¬ mo¤ ulo ³i¡ ¤ell¡
fa³ ¤ © W (jω) ³© possono ¯´ ´e®e² coe somm¡ al§b² ic¡
(mo¬ to a§vo¬e d¡ ef¦ ´ ´ ua² gra¦ icae® ´e) ¤e © ¤ iagram ©
¤ © ¦ unº io® © e¬ee® ta² © ¤ © ± u¡ ´ tro ´ © ° ©.
1. ´er i®e co³ta® ´e (§ uadagno ¤ © Bo¤e): T 1 (jω) = K B ; 2. ´er i®e mono io (ºero o polo i® ºero): T 2 (jω) = (jω) ν ; 3. ´er i®e ¢ino io (ºe² © o po¬ © ²a¬ ©): T 3 (jω) = 1 + jωτ ; 4. ´er i®e t² ino io (ºe² © o po¬ © co p¬es³©):
T 4 (jω) = 1 + j2 ξ
ω n ω − ω 2 ω 2 n .
Un¡ vo¬ t¡ trac£ i¡ ´ © © ¤ iagram © ¤ © Bo¤e ¤ © £ ias£ uno ¤e ©
´er i® © c¨e ¡ ° pa² ®ell¡ form¡ ¤ © Bo¤e ¤ © W (s), ³© so- mano gra¦ icae® ´e © co® t² i¢µ ´ © ¤e © ´er i® © ¡ ® uer¡ to²
e ³© s¯´ traggono ± µel¬ © ¤e © ´er i® © ¡ ¤eno in¡ to².
© ³iamo co³©' ² id¯´ ´ © a¬ prob¬em¡ ¤ © trac£ ia² © ¤ iagram ©
¤ © Bo¤e ¤e © ´er i® © T 1 (jω) , T 2 (jω) , T 3 (jω) e T 4 (jω) .
Termine costante T 1 (jω) = K B .
© h¡:
|T 1 (jω)| dB = 20 · log |K B | (0.13a)
Arg[T 1 (jω)] =
0 (mod 2π) se K > 0
−π (mod 2π) se K < 0
(0.13b)
e ± µ in¤ © ³i¡ i¬ ¤ iagramm¡ ¤ © Bo¤e ¤e¬ mo¤ ulo ³i¡ ± µello
¤ell¡ fa³ sono ² ´ ´e o² izzo® ta¬ © (co³ta® ´ © ¡ ° ° u® to) mo¬ to
fa£ i¬ © d¡ trac£ ia².
Termine monomio T 2 (jω) = (jω) ν .
r ± ua® to ² i§ uard¡ i¬ mo¤ ulo, ³© h¡:
|T 2 (jω)| dB = 20 · log |(jω) ν | = 20 · ν · log(ω) (0.14) c¨e, i® ¦ unº io®e ¤ © log(ω), un¡ ² ´ t¡ passa® ´e °er ¬'o² i§ i®e
e ¤ © °en¤enz¡ pa² © ¡ 20 · ν dB/decade (² icor¤ iamo c¨e
un¡ decade e' ¬'i® ´ervallo co p²so fr¡ ω 0 e 10ω 0 e c¨e,
i® coor¤ in¡ ´e loga² © t ic¨e, l¡ ¬ ung¨ezz¡ ¤ © ta¬e i® ´er- vallo e' in¤ © °en¤e® ´e d¡ ω 0 ; °er e³ ° io, l¡ ¬ ung¨ezz¡
¤el¬'i® ´ervallo fr¡ 1 e 10 e± µ iva¬e ¡ ± µell¡ ¤el¬'i® ´ervallo
fr¡ 10 e 100).
© ² icord¡ c¨e ± µe³to ´er i®e ³© trov¡ ¡ ¤eno in¡ to² ¤ ©
W (s) e ± µ in¤ © i¬ ³uo co® t² i¢µ to v¡ s¯´ tr¡ ´ to da¬ t¯ta¬e.
r ± ua® to ² i§ uard¡ l¡ fa³ ,
Arg[T 2 (jω)] = ν · π/2 (mod 2π) (0.15) c¨e cor² i³pon¤e a¤ un¡ ² ´ t¡ o² izzo® ta¬e. Anc¨e °er ¬e fa³©,
i¬ co® t² i¢µ to v¡ s¯´ tr¡ ´ to da¬ t¯ta¬e ¶i³to c¨e ± µe³to ´er i®e
³© trov¡ ¡ ¤eno in¡ to².
L¡ ¦ i§ ur¡ ³ § µe® ´e ² © po² t¡ © ¤ iagram © ¤ © Bo¤e ¤ © (jω) ν
®e¬ caso ¤ © ν = 2.
10 -2 10 -1 10 0 10 1 10 2
-100 -50 0 50 100
Pulsazione (rad/sec)
Modulo (dB)
10 -2 10 -1 10 0 10 1 10 2
179 179.5 180 180.5 181
Pulsazione (rad/sec)
Fase (gradi)
Figure 1: Diagrammi di Bode di (jω) 2 .
Termine binomio T 3 (jω) = 1 + jωτ
r ana¬ izza² ¬'andae® to ¤ © mo¤ ulo e fa³ ¤ © T 3 (jω) ³
®e va¬ µ tano d¡ ° p² im¡ © gra¦ i£ © °er
ω 1
|τ | . (0.16)
I® ta¬ caso ³© h¡ ω|τ| 1 e ± µ in¤ ©:
|1 + jωτ | dB ' 20 · log(1) = 0 dB (0.17a)
Arg(1 + jωτ ) ' 0, (0.17b)
e ± µ in¤ © i¬ co® t² i¢µ to ¤ © T 3 (jω) e' tras£ ura¢i¬e °er valo² ©
¤ © ω mo¬ to all¡ ³i® i³tr¡ ¤ © |τ | 1 .
Con³i¤e² iamo or¡ valo² © ¤ © ω °er £ µ ©:
ω 1
|τ | . (0.18)
I® ta¬ caso ³© h¡:
|1 + jωτ | dB ' 20 · log(ω|τ |) dB = 20 · log(ω) + 20 · log(|τ |) dB Arg(1 + jωτ ) ' sgn(τ ) · π/2,
£ i¯ ', °er valo² © ¤ © ω ³uf¦ i£ ©e® ´ee® ´e gran¤ © ² i³°e ´ to ¡ |τ | 1 :
• i¬ co® t² i¢µ to ¤ © T 3 (jω) a¬ mo¤ ulo e' ¡ ° pros³ima¢i¬e co®
un¡ ² ´ t¡ ¤ © °en¤enz¡ pa² © ¡ 20 dB/decade c¨e i® ´er³ c¡
¬'as³ ¤el¬e or¤ in¡ ´e ®e¬ ° u® to 20·log[|τ|] dB e ¬'as³ ¤el¬e
as£ is³ ®e¬ ° u® to ω = |τ | 1 ;
• i¬ co® t² i¢µ to ¤ © T 3 (jω) all¡ fa³ e' ¤ © £ irc¡ π/2 ³ τ > 0 e ¤ © £ irc¡ −π/2 ³ τ < 0.
A¬ so¬ © to, ta¬ © co® t² i¢µ ´ © vanno s¯´ tr¡ ´ ´ © ³ i¬ ´er i®e ³©
trov¡ ¡ ¤eno in¡ to².
Ad¯´ tando ¬e ¡ ° pros³imaº io® © cor² i³pon¤e® ´ © ¡ ω << |τ | 1
°er ´ µ ´ ´ © g¬ © ω ≤ |τ | 1 e ¬e ¡ ° pros³imaº io® © cor² i³pon¤e® ´ ©
¡ ω >> |τ | 1 °er ´ µ ´ ´ © g¬ © ω ≥ |τ | 1 , ³© ¯´ ´engono © co³id¤e ´ ´ ©
diagrammi asintotici ¤ © Bo¤e. I¬ ¤ iagramm¡ es¡ ´ to ¤e¬
mo¤ ulo e' un¡ £ urv¡ mon¯ton¡ c²s£e® ´e c¨e ² ima®e ³ p²
a¬ ¤ © s¯pr¡ ¤e¬ ¤ iagramm¡ a³i® t¯´ ico ¤ isco³tando³ ®e ¤ © u®
mas³imo ¤ © £ irc¡ 3 d i® cor² i³pon¤enz¡ a¤ ω = 1/|τ|, ¤e ´ to punto di spezzamento ¤e¬ ¤ iagramm¡ a³i® t¯´ ico.
I¬ ¤ iagramm¡ a³i® t¯´ ico ¤ell¡ fa³ p²³ ® t¡ u® sa¬ to ¤ ©
π/2 (i® ° © µ' o i® eno ¡ ³ cond¡ ¤e¬ ³ gno ¤ © τ) i® cor-
² i³pon¤enz¡ ¡ ω = 1/|τ|. ¬ ¤ iagramm¡ es¡ ´ to ¤ell¡ fa³ ,
in¶ £e, l¡ va² iaº io®e ¤ © fa³ av¶©e®e co® co® ´ i® µ © t¡' e coe
³© ¶ ¤e dall¡ ¦ i§ ur¡ i¬ ²l¡ ´ ivo co® t² i¢µ to ³© ° uo tras£ ura²
¦ ino a¤ un¡ ¤eca¤e p² im¡ ¤e¬ ° u® to ¤ © ³°ezzae® to e¤
e' £ irc¡ ¡ p² © ¡ π/2 ¡ pa² ´ i² d¡ un¡ ¤eca¤e d¯po ¤e¬ ° u® to
¤ © ³°ezzae® to.
ll¡ i§ ur¡ ³ § µe® ´e © ¤ iagram © es¡ ´ ´ © e ± µel¬ © a³i® t¯´ i£ ©
¤ © 1 + jωτ (°er τ = 1) sono r¡ ° p²³ ® t¡ ´ © i® rosso e b¬ µ,
² i³°e ´ ´ ivae® ´e.
© os³ r¶© c¨e:
• °er valo² © ¤ © τ ¤ i¶ r³© d¡ 1, m¡ ³ p² po³© ´ i¶©, ©
¤ i¡gra © con³ rvano l¡ e¤e³im¡ form¡ e ¶ ngono
³ p¬ i£ee® ´e trasl¡ ´ © ¡ ¤e³tr¡ o ¡ ³i® i³tr¡ ¡ ³ cond¡
c¨e τ ³i¡ ino² o mag§ io² ¤ © 1;
• i¬ ¤ i¡gram¡ ¤e¬ mo¤ ulo no® ¤ © °en¤e da¬ ³ gno ¤ © τ,
e® t² ± µello ¤ell¡ fa³ h¡ °er τ < 0 valo² © ¯° po³´ © ¡
± µel¬ © ¤e¬ ¤ iagramm¡ ²l¡ ´ ivo ¡ −τ.
0 5 10 15 20 25
Modulo (dB)
10-1 100 101
0 45 90
Fase (o argomento) (deg)
Bode Diagram
Pulsazione in coordinate logaritmiche (rad/s)
