2.004 Dynamics and Control II Spring 2008
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1 MIT OpeCourseWare Dyamics ad Cotrol II Sprig 2008 For iformatio about citig these materials or our Terms of Use, visit:
2 Massachusetts Istitute of Techology Departmet of Mechaical Egieerig Dyamics ad Cotrol II Sprig Term 2008 Lecture 33 Readig: Nise: 0. Class Hadout: Siusoidal Frequecy Respose Bode Plots (cotiued. Logarithmic Amplitude ad Frequecy Scales:.. Logarithmic Amplitude Scale: The Decibel Bode magitude plots are frequetly plotted usig the decibel logarithmic scale to display the fuctio H(jω). The Bel, amed after Alexader Graham Bell, is defied as the logarithm to base 0 of the ratio of two power levels. I practice the Bel is too large a uit, ad the decibel (abbreviated db), defied to be oe teth of a Bel, has become the stadard uit of logarithmic power ratio. The power flow P ito ay elemet i a system, may be expressed i terms of a logarithmic ratio Q to a referece power level P ref : ( ) ( ) P P Q = log 0 Bel or Q = 0 log 0 db. () copyright c D.Rowell 2008 P ref Because the power dissipated i a D type elemet is proportioal to the square of the amplitude of a system variable applied to it, whe the ratio of across or through variables is computed the defiitio becomes ( ) 2 ( ) A A Q = 0 log 0 = 20 log 0 db. (2) A ref A ref where A ad A ref are amplitudes of variables. Note: This defiitio is oly strictly correct whe the two amplitude quatities are measured across a commo D type (dissipative) elemet. Through commo usage, however, the decibel has bee effectively redefied to be simply a coveiet logarithmic measure of amplitude ratio of ay two variables. This practice is widespread i texts ad refereces o system dyamics ad cotrol system theory. The table below expresses some commoly used decibel values i terms of the power ad amplitude ratios. 33 P ref
3 Decibels Power Ratio Amplitude Ratio The magitude of the frequecy respose fuctio H (jω) is defied as the ratio of the amplitude of a siusoidal output variable to the amplitude of a siusoidal iput variable. This ratio is expressed i decibels, that is Y (jω) 20 log 0 H(jω) = 20 log 0 db. U(jω) As oted this usage is ot strictly correct because the frequecy respose fuctio does ot defie a power ratio, ad the decibel is a dimesioless uit whereas H (jω) may have physical uits. Example A amplifier has a gai of 28. Express this gai i decibels. We ote that 28 = The gai i db is therefore 20 log log log 0 2, or Gai(dB) = = 29 db. The advatages of a logarithmic amplitude scale iclude: Compressio of a large dyamic rage. Cascaded subsectios may be hadled by additio istead of multiplicatio, that is log( H (jω)h 2 (jω)h 3 (jω) ) = log( H (jω) ) + log( H 2 (jω) ) + log( H 3 (jω) ) which is the basis for the sketchig rules. High ad low frequecy asymptotes become straight lies whe log( H(jω) ) is plotted agaist log(ω). 33 2
4 ..2 Logarithmic Frequecy Scales I the Bode plots the frequecy axis is plotted o a logarithmic scale. Two logarithmic uits of frequecy ratio are commoly used: the octave which is defied to be a frequecy ratio of 2:, ad the decade which is a ratio of 0:.? J = L A? A! # J A M? = J > A I D M C M M Give two frequecies ω ad ω 2 the frequecy ratio W = (ω /ω 2 ) betwee them may be expressed logarithmically i uits of decades or octaves by the relatioships W = log 2 (ω /ω 2 ) octaves = log 0 (ω /ω 2 ) decades. The terms above ad below are commoly used to express the positive ad egative values of logarithmic values of W. A frequecy of 00 rad/s is said to be two octaves (a factor of 2 2 ) above 25 rad/s, while it is three decades (a factor of 0 3 ) below 00,000 rad/s..2 Asymptotic Bode Plots of Low-Order Trasfer Fuctios The Bode plots cosist of () a plot of the logarithmic magitude (gai) fuctio, ad (2) a separate liear plot of the phase shift, both plotted o a logarithmic frequecy scale. I this sectio we develop the plots for first ad secod-order terms i the trasfer fuctio. The approximate sketchig methods described here are based o the fact that a approximate log log magitude plot ca be derived from a set of simple straight lie asymptotic plots that ca be easily combied graphically. The system trasfer fuctio i terms of factored umerator ad deomiator polyomials is H(s) = K (s z )(s z 2 )... (s z m )(s z m ) (s p )(s p 2 )... (s p )(s p ), (3) where the z i, for i =,..., m, are the system zeros, ad the p i, for i =,...,, are the system poles. I geeral a system may have complex cojugate pole ad zero pairs, real poles ad zeros, ad possibly poles or zeros at the origi of the s-plae. Bode plots are costructed from a rearraged form of Eq. (??), i which complex cojugate poles ad zeros are combied ito secod-order terms with real coefficiets. For example a pair of complex cojugate poles s i, s i+ = σ i ± jω i is writte ( ) (s (σ i + jω i )) (s (σ i jω i )) = (4) ω 2 ( (ω/ω ) 2 ) + j2ζω/ω s=jω 33 3
5 M 2 ad described by parameters ω ad ζ. The costat terms /ω is absorbed ito a redefiitio of the gai costat K. I the followig sectios Bode plots are developed for the first ad secod-order umerator ad deomiator terms:.2. Costat Gai Term: The simplest trasfer fuctio is a costat gai, that is H(s) = K. H(jω) = K ad H(jω) = 0, ad covertig to the logarithmic decibel scale 20 log 0 H(jω) = 20 log 0 K ad H(jω) = 0 db. The Bode magitude plot is a horizotal lie at the appropriate gai ad the phase plot is idetically zero for all frequecies..2.2 A Pole at the Origi of the s-plae: A sigle pole at the origi of the s-plae, that is H(s) = /s, has a frequecy respose H(jω) = ad H(jω) = π/2. ω The value of the magitude fuctio i logarithmic uits is or usig the decibel scale log H(jω) = log(ω) 20 log 0 H(jω) = 20 log 0 (ω) db. The decibel based Bode magitude plot is therefore a straight lie with a slope of -20 db/decade ad passig through the 0 db lie ( H(jω) = ) at a frequecy of rad/s. The phase plot is a costat value of π/2 rad, or 90, at all frequecies. The magitude Bode plot for this system is show below. C 0 M! I F A? A!!!!! ) C K = H B H A G K A? O H I A? 33 4
6 M.2.3 A Sigle Zero at the Origi: A sigle zero at the origi of the s-plae, that is H(s) = s, has a frequecy respose H(jω) with magitude ad phase H(jω) = ω The logarithmic magitude fuctio is therefore or i decibels ad H(jω) = π/2. log H(jω) = log(ω) 20 log 0 H(jω) = 20 log 0 (ω) db. The Bode magitude plot is a straight lie with a slope of +20 db/decade. This curve also has a gai of 0 db (uity gai) at a frequecy of rad/s. The phase plot is a costat of π/2 radias, or +90, at all frequecies. The magitude plot is show i below. C 0 M! I F A? A!!!!! ) C K = H B H A G K A? O H I A?.2.4 A Sigle Real Pole The frequecy respose of a uity-gai sigle real pole factor is ad the frequecy respose is: H(jω) = (ωτ) 2 + H(s) =, τs + ad H(jω) = ta ( ωτ). The logarithmic magitude fuctio is log H(jω) = 0.5 log ( (ωτ) 2 + ), or as a decibel fuctio ( 20 log 0 H(jω) = 0 log 0 (ωτ) 2 + ) db. 33 5
7 Whe ωτ, the first term may be igored ad the magitude may be approximated by a low-frequecy asymptote lim 20 log 0 H(jω) = 0 log 0 () = 0 db ωτ 0 which is a horizotal lie o the plot at 0dB (uity) gai. At high frequecies, for which ωτ, the uity term i the magitude expressio may be igored ad the magitude fuctio is approximated by a high-frequecy asymptote 20 log 0 H(jω) 0 log 0 ((ωτ ) 2 ) = 20 log 0 (ω) 20 log 0 (τ) db. which is a straight lie whe plotted agaist log(ω), with a slope of -20 db/decade. The high ad low frequecy asymptotes itersect o the plot o the 0 db lie at a corer or break frequecy of ω = /τ. We ote that whe ω = /τ the magitude is H(jω) = / 2 or -3 db. The complete asymptotic Bode magitude plot as defied by these two lie segmets is show i (a) below usig a ormalized frequecy axis. The exact respose is also show i the figure; at the break frequecy ω = /τ the actual respose is 20 log 0 H(jω) = 0 log 0 (2) = 3 db. C 0 M I F A? A! M J!!!! H = E = C K = H B H A G K A? O 0 M! # $ % & ' M J!!!! H = E = C K = H B H A G K A? O 33 6
8 The phase characteristic is also plotted agaist a ormalized frequecy scale i (a). At low frequecies the phase shift approaches 0 radias. It passes through a phase shift of π/4 radias at the break frequecy ω = /τ, ad asymptotically approaches a maximum phase lag of π/2 radias as the frequecy becomes very large. A piece-wise liear approximatio may be made by assumig that the curve has a phase shift of 0 radias at frequecies more tha oe decade below the break frequecy, a phase shift of π/2 radias at frequecies more tha a decade above the break frequecy, ad a liear trasitio i phase betwee these two frequecies o the logarithmic frequecy scale. This approximatio is withi 0. radias of the exact value at all frequecies..2.5 A Sigle Real Zero A umerator term, correspodig to a sigle real zero, writte i the form H(s) = τs + (where τ is ot strictly a time costat), is hadled i a maer similar to a real pole. I this case H(jω) = jωτ + ad the magitude ad phase resposes are H(jω) = + (ωτ) 2 ad H(jω) = ta (ωτ ) respectively. I decibels the magitude expressio is 20 log 0 H(jω) = 0 log 0 ( + (ωτ) 2 ) db. The low frequecy asymptote is foud by assumig that ωτ i which case lim 20 log 0 H(jω) = 0 log 0 () = 0 db, ωτ 0 The high frequecy asymptote is foud by assumig that ωτ, 20 log 0 H(jω) 20 log 0 (ωτ) = 20 log 0 (ω) 20 log 0 (τ) db whe ω /τ which is a straight lie o the log-log plot, with a slope of +20 db/decade. The break frequecy, defied by the itersectio of these two asymptotes is at a frequecy ω = /τ, ad at this frequecy the exact value of H(jω) is 2 or +3 db. The complete asymptotic Bode magitude plot usig a ormalized frequecy scale is show below. The phase characteristic asymptotically approaches 0 radias at low frequecies ad approaches a maximum phase lead of π/2 radias at frequecies much greater tha the break frequecy. At the break frequecy the phase shift is π/4 radias. A piece-wise liear approximatio, similar to that described for a real pole, is also show below. 33 7
9 C 0 M! I F A? A M J!!!! 0 M H = E = C K = H B H A G K A? O ' & % $ #! M J!!!! H = E = C K = H B H A G K A? O.2.6 Complex Cojugate Pole Pair: The classical secod-order system, H(s) = ω 2 s 2 + 2ζω s + ω2 has a frequecy respose H(jω) = ( (ω/ω ) 2 ) + (2ζ(ω/ω )) ad H(jω) = ta 2ζ (ω/ω ) ( (ω/ω ) 2). 2 2 I logarithmic uits the magitude respose is [ ( ) 2) 2 20 log 0 H(jω) = 0 log 0 (ω/ω + (2ζ(ω/ω )) 2] The Bode forms of the magitude ad phase resposes are plotted i below, with the dampig ratio ζ as a parameter. 33 8
10 The low-frequecy asymptote is foud by assumig that ω/ω so that lim (ω/ω ) 0 (20 log 0 H(jω) ) = 0 log 0 () = 0 db. The high frequecy respose ca be foud by retaiig oly the domiat term whe ω/ω : 20 log 0 H(jω) 0 log 0 [ (ω/ω ) 4] = 40 log 0 (ω) + 40 log 0 (ω ) db whe ω ω, which is a liear fuctio of log 0 ω with a slope of -40 db/decade. The two asymptotes itersect at a break frequecy of ω = ω as show below. The straight lie asymptotic form does ot accout i ay way for the dampig ratio. C 0 M # # $ I F A? A & M M!!!! H = E = C K = H B H A G K A? O 0 M $ # & # # M M!!!! H = E = C K = H B H A G K A? O The phase characteristic asymptotically approaches 0 radias at low frequecies, has a phase lag of π/2 at the break frequecy ω, ad approaches π radias at high frequecies. The steepess of the trasitio is a fuctio of the dampig ratio ζ ad so must be sketched usig the iformatio cotaied above. The resoace peak (for values of ζ < 0.707) must be sketched i after the asymptotes have bee draw. The figure below plots the logarithmic magitude correctio ad frequecy of the resoat peak as a fuctio of ζ; it is a simple matter to sketch i the resoat peak from these values. 33 9
11 C 0 M F M M F! & $! # $ %, = F E C H = J E =! # $ %, = F E C H = J E >.2.7 Complex Cojugate Zero Pair Bode plots for a pair of complex cojugate zeros ca be derived i a maer similar to the cojugate pole pair described above. I this case the block is assumed to have a trasfer fuctio H(s) = ( ) s 2 + 2ζω ω 2 s + ω 2 ad a frequecy respose H(jω) = ( (ω/ω ) 2 ) + (2ζ(ω/ω )) ad H(jω) = ta 2ζ (ω/ω ) ( (ω/ω ) 2). 2 2 The logarithmic magitude respose is [ ( ) 2) 2 20 log 0 H(jω) = 0 log 0 (ω/ω + (2ζ(ω/ω )) 2] db The asymptotic resposes are derived i a similar maer to the complex pole pair; the low frequecy asymptote is lim (ω/ω ) 0 ad the high frequecy asymptote is (20 log 0 H(jω) ) = 0 log 0 () = 0 db, 20 log 0 H(jω) 0 log 0 [ (ω/ω ) 4] = 40 log 0 (ω) 40 log 0 (ω ) db for ω ω. The exact form of the magitude respose is plotted below. This is effectively a iverse of the characteristic of complex-cojugate pole pair described above. There is a otch i the respose i the regio of the frequecy ω, ad the depth is a fuctio of the parameter ζ. The plot has a low frequecy asymptote of 0 db, a break frequecy of ω = ω, ad a 33 0
12 high-frequecy asymptote is a straight lie with a slope of +40 db/decade. The phase characteristic is also a flipped versio of that of a pair of complex cojugate poles; it approaches 0 radias at low frequecies, passes through π/2 at the break frequecy, ad shows a maximum phase lead of π radias at high frequecies. As above, the slope of the curve i the trasitio regio is depedet o the value of ζ. & $ C 0 M I F A? A M M!!!! H = E = C K = H B H A G K A? O 0 M & $ # # & $ # # M M!!!! H = E = C K = H B H A G K A? O #.2.8 Summary The essetial features of the asymptotic forms of the seve compoets of the magitude plot are summarized below. 33
13 Descriptio Trasfer Fuctio Break Frequecy Costat gai Pole at the origi Zero at the origi Real pole Real zero Cojugate poles Cojugate zeros ω 2 K s s τs + (τs + ) ω 2 s 2 + 2ζω s + ω 2 (s 2 + 2ζω s + ω 2 ) (radias/sec.) /τ /τ ω ω High Frequecy Slope (db/decade)
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