Lecture 3 Graphical represetatio of the frequecy respose Luca Ferrarii - Basic Automatic Cotrol
Graphical represetatio of the frequecy respose Polar plot G Bode plot ( j), G Im 3 Re of the magitude G ( j), of the phase argg j, ( ) ( ), G j G Luca Ferrarii - Basic Automatic Cotrol
Bode magitude plot: covetio Ordiate i db G ( j) db = = log G ( j) Abscissa i logarithmic scale log log = log Luca Ferrarii - Basic Automatic Cotrol 3
Decibel x db log x x db x db 4 db -db. db Luca Ferrarii - Basic Automatic Cotrol 4
Specificatios of liear ad logarithmic scales LINEAR Scale + arithmetic mea 3 4 4 3 the distace betwee ad is - LOGARITHMIC Scale (Attetio: the frequecies have liear values) geometric mea 3 4 log ( 4 / 3 ) the distace betwee ad is log log, that is log ( / ) Luca Ferrarii - Basic Automatic Cotrol 5
Plots i semi logarithmic scale - Decades Bode Plots Magitude (db) 8 6 4 - -4-6 -8 8 decade =. =. =.3=.6 = decade 9 Phase (deg) -9-8 - - Frequecy (rad/sec) Luca Ferrarii - Basic Automatic Cotrol 6
Bode magitude plot: drawig G () s = μ g s i ( + st ) ( + sτi ) i i G ( j) = μ j g i ( + jt ) ( + jτi ) i i G g ( j) = log μ log j + db gai poles or zeros i the origi + i log+ j T i + zeros (complex cojugate & real) i log+ jτ i poles (complex cojugate & real) Luca Ferrarii - Basic Automatic Cotrol 7
Gai log μ costat lie μ=± μ=± μ=±. Luca Ferrarii - Basic Automatic Cotrol 8
Poles or zeros i the origi log j = g 8 g log Covetio: the slope is idicated with -g lie with slope -g db/decade crossig the db axis at = g= 6 4 + + g= db - -4 g= -6-8 - - g= Luca Ferrarii - (rad/s) Basic Automatic Cotrol 9
Real zero log T + jt = log + real T The drawig of this fuctio of is ot simple. We ca fid a reasoable approximatio cosiderig the behavior at high ad low frequecies. Low if T <<, that is << log + T T High if T >>, that is >> T log + T log T log + logt Luca Ferrarii - Basic Automatic Cotrol
The Low for High for << >> T T costat lie at db lie with slope + ad maximum error of 3 db crossig the db axis at =/ T (3 db is log ) at = T Asymptotic approximatio Luca Ferrarii - Basic Automatic Cotrol
Example Real zero 4 G ( s) =+ s db true plot + asymptotic plot -.. (rad/s) 3 db Luca Ferrarii - Basic Automatic Cotrol
Example p Real zeros 3 G( s) = ( + s) p = Bode Diagramma plot - magitude di Bode - Modulo 4 3 8 db 6 4 true plot asymptotic plot - - frequecy pulsazioe Luca Ferrarii - Basic Automatic Cotrol 3
Luca Ferrarii - Basic Automatic Cotrol 4 T j T j + + + log log complex T () ( )( ) ~ s s st st s G + ξ + = + + = Remember that Complex cojugate zeros ( ) ( ) j j j j G ξ + = + ξ + = ~
The magitude (i db) is ~ G ( j) = log + 4ξ Fid the asymptotic approximatio (suppose that ξ=): Low for << costat lie at db High for >> ~ G = ( j) 4log lie with slope + crossig the db axis at log = = 4log 4log = Luca Ferrarii - Basic Automatic Cotrol 5
Example Complex cojugate zeros G( s) 5 = + ξs + s = 4. 3.3 db +.5.7.9 - - - rad/s asymptotic plot Luca Ferrarii - Basic Automatic Cotrol 6
Example (particular case) purely imagiary zeros G( s) = + s = 4 - -4-6 -8 - - - Luca Ferrarii - Basic Automatic Cotrol 7
The error at = depeds o ξ E = log + 4ξ = = log 4ξ = log ( ξ ) if E = log 6 db if ξ E The same results apply to the poles, except by the sig Luca Ferrarii - Basic Automatic Cotrol 8
Example Real pole Diagramma Bode magitude di Bode - plot Modulo G( s) = + s -5 - true plot asymptotic plot -5 - db -5-3 -35-4 -45 - - frequecy pulsazioe Luca Ferrarii - Basic Automatic Cotrol 9
Example Complex cojugate poles G( s) = = + ξs + s asymptotic plot..3 db - - -.5.7.9-3 -4-5 - rad/s Luca Ferrarii - Basic Automatic Cotrol
Example (particular case) Purely imagiary poles G( s) = = = + ξ s 8 6 4 - -4 - Luca Ferrarii - Basic Automatic Cotrol
Asymptotic Bode magitude plot: rules for drawig. the iitial slope is -g. the iitial segmet (or its extesio) crosses the μ at = db rad/s 3. slope chages accordig to poles ( ) ad zeros (+) Observatio The fial slope (per ) is give by : zeros - poles (= oly if G(s) is ot strictly proper) Luca Ferrarii - Basic Automatic Cotrol
Example g = μ = G( s) = μ db s( + = 4dB ( + s) s)( +.6s + ξ + s + zero : T = T = =. T real pole : τ = τ = =.5 τ complex poles : =.8 slope - s s ) slope + slope - Luca Ferrarii - Basic Automatic Cotrol 3
Bode magitude plot 5 - - db -3-5 - - - Luca Ferrarii - Basic Automatic Cotrol 4
Bode phase plot: covetio 9 Ordiate i degrees G( j) Abscissa i logarithmic scale log log = log degrees -9-8 -7 3 4 (rad/s) Luca Ferrarii - Basic Automatic Cotrol 5
Argumet or phase of a complex umber Im ϑ Phase calculatio if a x λ = a + Re jb λ = ata Covetios: b a 8 λ < 8 The phase of a egative real umber is 8 ( 9 λ + 9 ) o if a < b > b λ = ata + 8 a ( + 9 < λ < + 8 ) if a < b b λ = ata 8 a Luca Ferrarii - Basic Automatic Cotrol 6 ( 8 λ < 9 )
Bode phase plots: drawig G () s = μ g s i ( + st ) ( + sτi ) i i gai poles or zeros i the origi g ( j) = μ ( j) + ( + jt ) i ( + jτi ) G i zeros (complex cojugate & real) i poles (complex cojugate & real) Luca Ferrarii - Basic Automatic Cotrol 7
Gai μ = if μ > -8 if μ < costat lie 9 μ > degrees -9-8 μ < -7 3 4 (rad/s) Luca Ferrarii - Basic Automatic Cotrol 8
Poles ad zeros i the origi g o ( j) = g ( j) = g9 7 costat lie 8 g= degrees 9-9 g= g= -8 g= -7 3 4 (rad/s) Luca Ferrarii - Basic Automatic Cotrol 9
Real zero ( + jt ) = ata( T ) real T The drawig of this fuctio of is simple, yet we will try to fid a approximatio for low ad high frequecies. Low for ata( T ) High for ata( T ) Asymptotic approximatio + 9 if T > (zero o the left) 9 if T < (zero o the right) Luca Ferrarii - Basic Automatic Cotrol 3
Example Real zero 8 Asymptotic plot (T>) here is exactly 45 degrees 9 T> (zero o the left) G(s)=+s Real plot -9-8./T./T./T /T /T /T /T (rad/s) Asymptotic plot (T<) T< (zero o the right) G(s)=-s Luca Ferrarii - Basic Automatic Cotrol 3
Luca Ferrarii - Basic Automatic Cotrol 3 complex T ( ) ( ) T j T j + + + Complex cojugate zeros () ( )( ) ~ s s st st s G + ξ + = + + = Remember that ( ) ( ) j j j j G ξ + = + ξ + = ~
~ G ( j) = + jξ Low for High for ~ G ~ G ( j ) = = ( j) = lim a ta ± 8 ξ (+8 if ξ>) (-8 if ξ<) Luca Ferrarii - Basic Automatic Cotrol 33
But lim a ξ ta = The G ~ ( j) = for for + 8 if ξ > (zeros o the left) 8 if ξ < (zeros o the right) Luca Ferrarii - Basic Automatic Cotrol 34
For ξ= (purely imagiary zeros) ~ G ( j) = + ( j) = Real fuctio If G ~ ( j) is a real umber, the the phase is whe positive ad 8 whe egative. G ~ ( j)= for for < > 8 Luca Ferrarii - Basic Automatic Cotrol 35
Example Complex cojugate zeros G( s) 8 6 Asymptotic plot = + ξs + s = Zeros o the left 4 8 6..3.5.7.9 4 - Luca Ferrarii - Basic Automatic Cotrol 36
G( s) = + ξs + s = - -4-6 -8 - - -4 Asymptotic plot Zeros o the right..3.5.7.9-6 -8 - The same results apply to the poles, except the sig Luca Ferrarii - Basic Automatic Cotrol 37
Example Real pole 8 6 4 Real plot Asymptotic plot Pole o the right τ< G( s) = s - -4-6 -8 Real plot Asymptotic plot - - - Pole o the left τ> G( s) = + s Luca Ferrarii - Basic Automatic Cotrol 38
Example Complex cojugate poles 8 6 4 8 6 4 Real plot Asymptotic plot G( s) = = + ξs + s Poles o the right..3.5.7.9 - Luca Ferrarii - Basic Automatic Cotrol 39
Example Complex cojugate poles G( s) = = + ξs + s - -4 Poles o the left -6. -8 - - -4 Real plot Asymptotic plot.3.5.7.9-6 -8 - Luca Ferrarii - Basic Automatic Cotrol 4
Asymptotic Bode phase plot: rules for drawig. iitial value μ g9. value chages accordig to poles ad zeros left semi-plae right semi-plae poles 9 +9 zeros +9 9 Luca Ferrarii - Basic Automatic Cotrol 4
Miimum phase systems gai μ> poles ad zeros have egative or zero real part It is possible to obtai the phase plot from the magitude oe. pole magitude plot slope phase 9 zero magitude plot slope + phase +9 Luca Ferrarii - Basic Automatic Cotrol 4
Polar plot It is the graphical represetatio of G j, i the complex plae. G ( ) G( j ) Im G( j ) G G( j ) Re G( j ) Luca Ferrarii - Basic Automatic Cotrol 43
Saliet poits ad curves of the complex plae G = 9 +j Im G = = db G = ± 8 G = - Re G = = db -j G = 9 Luca Ferrarii - Basic Automatic Cotrol 44
Example G( s) = ( + s) 3 G( j) = ( + j) 3 5-5 -3 - -5 - - - - -3 - - Luca Ferrarii - Basic Automatic Cotrol 45
8 6 Nyquist plot Nyquist plot Γ 4 The Nyquist plot is the polar plot ad its symmetric diagram with respect to the real axis put together. = ± = - -4 polar plot (cotiuous lie) -6-8 -4-4 6 8 Luca Ferrarii - Basic Automatic Cotrol 46
Example G () s = ( + s)( + s) μ = = db g = τ = = τ = =..5 Luca Ferrarii - Basic Automatic Cotrol 47
4 db - -4-6 - - 9 (rad/s) gra degrees -9-8 -7 - - (rad/s) Luca Ferrarii - Basic Automatic Cotrol 48
6 Nyquist Diagram 4 Imagiary Axis = = - -4.3-6 - 4 6 8 Real Axis Luca Ferrarii - Basic Automatic Cotrol 49
Example G () s = + ξ s + s μ = G ( ) = g = ξ variable geeric Luca Ferrarii - Basic Automatic Cotrol 5
- db -4-6 -8.. 9 (rad/s)..3.5.7.9 grad degrees -9-8 -7.. (rad/s) Luca Ferrarii - Basic Automatic Cotrol 5
6 Nyquist Diagram 4. Imagiary Axis.3.5.7 - ξ decreasig.9-4 -6-3 - - 3 Real Axis Luca Ferrarii - Basic Automatic Cotrol 5
if G = s + () s G ( j) = for < G(j)> for > G(j)< real fuctio Im + = = = = Re Luca Ferrarii - Basic Automatic Cotrol 53
Example G () s = s s + ( s + ) = ( +.s) s( s + ) =, τ =, τ =. = μ = = db g = for G( j) o G( j) = 9 the magitude plot starts with slope - ad the phase plot with -9 There is a vertical asymptote i the polar plot. The asymptote positio is give by lim Re( G( j )) =... = 9 Luca Ferrarii - Basic Automatic Cotrol 54
Bode Diagram 5 Magitude (db) -5 - -9 Phase (deg) - -5 - - 3 Frequecy (rad/sec) Luca Ferrarii - Basic Automatic Cotrol 55
5 Nyquist Diagram db 5 Imagiary Axis -5 - -5 - -9-8 -7-6 -5-4 -3 - - Real Axis Luca Ferrarii - Basic Automatic Cotrol 56