CHAPTER NINE 9. Itroductio It as poited earlier that i practice the performace of a feedback cotrol system is more preferably measured by its time - domai respose characteristics. This is i cotrast to the aalysis ad desig of systems i the commuicatio field, here the frequecy respose is of more importace, sice i this case most of the sigals to be processed are either siusoidal or periodic i ature. Hoever, aalytically, the time respose of a cotrol system is usually difficult to determie, especially i the case of high - order systems. I the desig aspects, there are o uified ays of arrivig at a desig system give the time - domai specificatios, such as peak overshoot, rise time, delay time ad settig time. O the other had, there is a ealth of graphical methods available i the frequecy - domai aalysis, all suitable for the aalysis ad desig of liear feedback cotrol system. Oce the aalysis ad desig are carried out i the frequecy - domai, the time - domai behavior of the system ca be iterpreted based o the relatioships that exist betee the time - domai ad the frequecy - domai properties. Therefore, might cosider that the mai purpose of coductig cotrol systems aalysis ad desig i the frequecy domai is merely to use the techiques as a coveiet vehicle toard the same objectives as ith time - domai method. Frequecy respose methods ere developed i 930s ad 940s by Nyquist, Bode Nichols, ad may others. The frequecy respose methods are most poerful i covetioal cotrol theory. They are also idispesable to robust cotrol theory. The priciple of frequecy respose testig is based i comparig the iput ad output of the elemets uder test over a ide rage of frequecies, he subjected to a siusoidal iput sigal. The Nyquist stability criterio eables to ivestigate both the absolute ad relative stabilities of liear closed loop systems from koledge of their ope loop frequecy respose characteristics. A advatage of the frequecy respose approach is that frequecy respose tests are; i geeral, simple ad ca be made accurately by use of readily available siusoidal sigal geerators ad precise measuremet equipmet. Ofte the trasfer fuctios of complicated compoets ca be determied experimetally by frequecy respose tests. I additio, the frequecy respose approach has the advatages that a system may be desiged so that the effects of udesirable oise are egligible ad that such aalysis ad desig ca be exteded to certai oliear cotrol systems. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
Although the frequecy respose of a cotrol system presets a qualitative picture of the trasiet respose, the correlatio betee frequecy ad trasiet resposes is idirect, except for the case of secod- order systems. I desigig a closed - loop system, e adjust the frequecy respose characteristic of the ope - loop trasfer fuctio by usig several desig criteria i order to obtai acceptable trasiet respose characteristics of the system. 9. Frequecy respose coverage the folloig plots 9.. Polar plot Polar plot is a graph of Im [GH] versus Re [GH] o the [GH(j)] - plae for - < <. = o Figure 9. Polar Plot 9.. Magitude - Phase plot The magitude i decibel versus the phase o rectagular coordiates ith as a variable parameter is called Magitude versus Phase Plot. Figure 9. Magitude versus Phase Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
9..3 Bode plot The plot of the magitude i decibel versus (or log ) i semilog (or rectagular) coordiate called Bode plot (corer plot). Figure 9.3 Bode Plot ----------------------------------------------------------------------------------------------------- 9.3 Frequecy Respose Characteristic For closed loop cotrol system Gs ( ) M T. F. GH ( s ) let s j M ( j ) G ( j ) G ( j ) H ( j ) G ( j ) G ( j ) H ( j ) M ( j ) magitude phase agle M(j) may be regarded as the magificatio of the feedback cotrol system. A typical magificatio curve of a feedback cotrol system is sho i the folloig figure: Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-3
Mp - Resoace Peak: is the maximum value of the magitude of the closed loop frequecy respose. p - Resoace frequecy: is the frequecy at hich Mp occurs. B.W - Bad Width; is the rage of frequecies (of the iput) over hich the system ill respod satisfactorily. Satisfactorily ill be at value equal to 0.707 of magificatio. c - Cut off frequecy: it is occurs he the magitude ratio ted to 0.707 of its value. ------------------------------------------------------------------------------------ 9.4 Polar Plots (Nyquist Plot) The polar plot of a siusoidal trasfer fuctio G(j) is a plot of the magitude of G(j) versus the phase agle of G(j) o polar coordiates as is varied from zero to ifiity. Thus, the polar plot is the locus of vectors G ( j ) G ( j ) as is varied from zero to ifiity. Note that i polar plots a positive (egative) phase agle is measured couter clockise (clockise) from positive real axis. The polar plot is ofte called Nyquist plot. A example of such a plot is sho i Figure 9.4. Each poit o the polar plot of G(j) represets the termial poit of a vector at a particular value of. i the polar plot, it is importat to sho the frequecy graduatio of the locus. The projectios of G(j) o the real ad imagiary axes is real ad imagiary compoets. A advatage i usig a polar plot is that it depicts the frequecy respose characteristics of a system over the etire frequecy rage i a sigle plot. Oe disadvatage is that the plot does ot clearly idicate the cotributios of each idividual factor of the ope loop trasfer fuctio. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-4
Itegrated ad Derivative Factors (j) ± Figure 9.4 Polar Plot The polar plot of G(j) = / j is the egative imagiary axis sice G ( j ) j 90 j The polar plot of G(j) = j is the positive imagiary axis. First order Factors ( + j T) ± For the siusoidal trasfer fuctio G ( j ) ta j T T T The values of G(j) at = 0 ad = /T are respectively G ( j 0) 0 ad G ( j ) 45 T If approaches ifiity, the magitude of G(j) approaches zero ad the phase agle approaches -90 o. The polar plot of this trasfer fuctio is a semicircle as the frequecy is varied from zero to ifiity. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-5
Quadratic Factors [ ( j / ) ( j / ) ] The lo ad high frequecy portios of the polar plot the folloig siusoidal trasfer fuctio G ( j ) for 0 ( j ) ( j ) are give respectively by lim G ( j ) 0 ad lim G ( j ) 0 80 The polar plot of this siusoidal trasfer fuctio starts at 0 ad eds at 0 80 as icreases from zero to ifiity. Thus, the high frequecy portio of G(j) is taget to the egative real axis. ------------------------------------------------------- Geeral Shapes of Polar Plots The polar plots of a trasfer fuctio of the form K ( jta)( jtb)... G ( j ) ( j ) ( jt)( jt )... Where > m or the degree of the deomiator polyomial is greater tha that of the umerator, ill have the folloig geeral shapes:. Type zero cotrol systems, λ = 0: The startig poit of the polar plot (hich correspod to = 0) is fiite ad is o the positive real axis. The taget to the polar plot at = 0 is perpedicular to the real axis. The termial poit, hich correspods to =, is at the origi, ad the curve is taget to oe of the axis. Example: K G ( j ) ( jt )( jt ) at = 0, G ( j ) K 0, G ( j ) 0 80, ta 90 Each factor of deomiator cotributes a agle of -90 o or 90 o i clockise directio. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-6
. Type Cotrol System λ=: The j term i the deomiator cotributes - 90 to the total phase agle of G(j) for 0. At = 0, the magitude of G(j) is ifiity, ad the phase agle becomes -90. At lo frequecies, the polar plot is asymptotic to a lie parallel to the egative imagiary axis. At =, the magitude becomes zero, ad the curve coverges to the origi ad is taget to oe of the axes. Example: K G ( j ) ( j )( jt)( jt )( jt3) at 0, G ( j ) 90, G ( j ) 0 360 Vx is asymptotes s approaches to zero, G(j) approaches to ifiity alog to asymptotes Vx, ad Vx is foud from: Vx lim Re[ G ( j )] 0 ad is deter mied from x Im [ G ( j )] 0 Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-7
3. Type Cotrol System λ=: The (j) term i the deomiator cotributes -80 to the total phase agle of G(j) for 0. At = 0, the magitude of G(j) is ifiity, ad the phase agle is equal to - 80. At lo frequecies, the polar plot is asymptotic to a lie parallel to the egative real axis. At =, the magitude becomes zero, ad the curve is taget to oe of the axes. K Example: G ( j ) ( j ) ( jt )( jt ) at 0, G ( j ) 80, G ( j ) 0 360 Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-8
Rough Sketch of the Polar Plot The sketchig of the polar plot is facilitated by the folloig iformatio:.the behavior of the magitude ad the phase at = 0 ad at =..The poit of itersectio of the polar plot ith the real ad imagiary axes, ad the value of at the itersectios. ------------------------------------------------------------- Absolute ad Relative Stability The simplified Nyquist criterio for system stability may be stated as follos: If GH(j) does ot have poles i the right half s - plae, the closed loop system is stable, if ad oly if the - pit lies to the left of the polar plot he movig alog this plot i the directio of icreasig. That is, the polar plot passes o the right side of -. --------------------------------------------------------------- Gai Margi (GM) GM is defied as the reciprocal of the magitude of the ope loop trasfer fuctio he the phase shift is 80. This is, therefore the factor by hich the gai must be icreased i order to produce istability. Phase Margi (ΦPM) Φ PM is defied as 80 mius the phase agle of the ope loop trasfer fuctio [GH] of the frequecy he the gai is uity. This is therefore the amout of the phase agle, ould have to be icreased to make the system ustable. ------------------------------------------------------------------------------------------------------------- 9.5 Bode Diagrams (or Logarithmic Plots) A Bode diagram (plot) cosists of to graphs: oce is a plot of the logarithm of the magitude of a siusoidal trasfer fuctio; the other is a plot of the phase agle; both are plotted agaist the frequecy o a logarithmic scale. The stadard represetatio of the logarithmic magitude of G(j) is 0log G(j), here the base of the logarithm is 0. The uit used i this represetatio of the magitude is the decibel, usually abbreviated db. I the logarithmic represetatio, the curves are dra o semi log paper, usig the log scale for frequecy ad the liear scale for either magitude (but i decibels) or phase agle (i degrees). The frequecy rage of iterest determies the umber of logarithmic cycles required o the abscissa. The mai advatage of usig Bode diagram is that multiplicatio of the magitudes ca be coverted ito additio. Further more; a simple method for sketchig a approximate logmagitude curve is available. It is based o asymptotic approximatios. Such approximatio Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-9
by straight-lie asymptotes is sufficiet if oly rough iformatio o the frequecy - respose characteristics is eeded. Should the exact curve be desired, correctios ca be made easily to these basic asymptotic plots. Expadig the lo - frequecy rage by use of a logarithmic scale for the frequecy is highly advatageous sice characteristics at lo frequecies are most importat impractical systems. Although it is ot possible to plot the curves right do to zero frequecy because of the logarithmic frequecy (log 0 = - ). Note that the experimetal determiatio of a trasfer fuctio ca be made simple if frequecy - respose data are preseted i the form of a Bode diagram. Basic Factors of G(j)H(j) The basic factors that vary frequetly occur i arbitrary trasfer fuctio G(j)H(j) are:. Gai K.. Itegrated ad derivative factors (j) ±. 3. First - order factors ( + j T) ±. 4. Quadratic factors [ ( / ) ( / ) ] j j. Oce e become familiar ith the logarithmic plots of these basic factors, it is possible to utilize them i costructig a composite logarithmic plot for ay geeral form of G(j)H(j) by sketchig for each factor ad addig idividual curves graphically; because addig the logarithms of the gais correspods to multiplyig them together. The Gai K A umber greater tha uity has a positive value i decibels, hile a umber smaller tha uity has a egative value. The log - magitude curve for a costat gai K is a horizotal straight lie at the magitude of 0 log K decibels. The phase agle of the gai K is zero. The effect of varyig the gai K i the trasfer fuctio is that it raises or loers the log - magitude curve of the trasfer fuctio by the correspodig costat amout, but it has o effect o the phase curve. db Φ 0 log K Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-0
Itegral factor j The logarithmic magitude of /j i decibel is 0 log 0 log db j This is a straight lie of slop - 0 db/decade. The phase agle of /j is costat ad equal to - 90 o. Derivative factor j The logarithmic magitude of j i decibel is 0 log j 0 log db This is a straight lie of slop 0 db/decade. The phase agle of j is costat ad equal to 90 o. First order factor a. Factor j T Log magitude is 0 log 0 log T db j T For lo frequecies such that «/ T, the log magitude may be approximated by 0 log T 0 log 0 Thus, the log magitude curve at lo frequecies is the costat 0 db lie. For high frequecies, such that» / T, 0 log T 0 logt db This is a approximate expressio for the high frequecy rage. At = /T, the log magitude equals 0 db; at = 0/T, the log magitude is -0 db, the value of -0 log T Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
db decreases by 0 db for every decade of. For»/T, the log magitude curve is thus a straight lie ith a slope of -0 db/ decade. The frequecy at hich the to asymptotes meet is called the corer frequecy or break frequecy. For the factor /( + j T), the frequecy = /T is the corer frequecy sice at = /T the to asymptotes have the same value. The corer frequecy divides the frequecy respose curve ito to regios; a curve for the lo frequecy regio ad a curve for high frequecy regio. The corer frequecy is very importat i sketchig logarithmic frequecy respose curves. The exact phase agle Φ of the factor /( + j T) is ta T At zero frequecy ( = 0), the phase agle is 0 o. At corer frequecy ( = /T), the phase agle is ta T / T ta 45. At ifiity ( ), the phase agle becomes - 90 o. Sice the phase agle is give by a iverse taget fuctio, the phase agle is ske symmetric about the iflectio poit at Φ = -45 o. b. Factor + j T Log magitude is 0 log j T 0 log T db For lo frequecies such that «/ T, the log magitude may be approximated by 0 log T 0 log 0 Thus, the log magitude curve at lo frequecies is the costat 0 db lie. For high frequecies, such that» / T, 0 log 0 log T T db The exact phase agle Φ of the factor ( + j T) is ta T At zero frequecy ( = 0), the phase agle is 0 o. At corer frequecy ( = /T), the phase agle is ta T / T ta 45. At ifiity ( ), the phase agle becomes 90 o. Sice the phase agle is give by a iverse taget fuctio, the phase agle is ske symmetric about the iflectio poit at Φ = 45 o. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9 -
Quadratic Factors a. The factor j ( j ) If ζ >, this quadratic factor ca be expressed as a product of to first order factors ith real poles. If 0 < ζ <, this quadratic factor is the product of to complex - cojugate factors. Asymptotic approximatios to the frequecy - respose curves are ot accurate for a factor ith lo values of ζ. This is because the magitude ad phase of the quadratic factor deped o both the corer frequecy ad the dampig ratio ζ. The asymptotic frequecy-respose curve may be obtaied as the follos. Sice 0 log 0 log ( ) ( ) j ( j ) For lo frequecies such that «, the log magitude becomes -0 log = 0 db The lo-frequecy asymptote is thus a horizotal lie at 0 db. For high frequecies such that», the log magitude becomes 0 log 40 log db The equatio for the high-frequecy asymptote is a straight lie havig the slope - 40 db/decade sice 0 40 log 40 40 log The high frequecy asymptote itersects the lo-frequecy oe at = sice at this frequecy 40 log 40 log 0 db This frequecy,, is the corer frequecy for the quadratic factor cosidered. The phase agle of the quadratic factor j ( j ) ta ( j ) ( j ) ( ) The phase agle is a fuctio of both ad ζ. At = 0, the phase agle equals 0 o. At the corer frequecy =, the phase agle is - 90 o regardless of ζ, sice is Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-3
ta ( ) ta 90 0 At =, the phase agle becomes - 80 o. The phase agle curve is ske symmetric about the iflectio poit (the poit here Φ = - 90 o ). 0 Bode Diagram j ( j ) Magitude (db) 0-0 -40-60 corer frequecy asymptote slope - 40 db/decade -80 0 Phase (deg) -45-90 -35-80 0-0 0 0 0 Frequecy (rad/sec) 80 Bode Diagram j ( j ) 60 Magitude (db) 40 0 0 corer frequecy asymptote slope 40 db/decade -0 80 Phase (deg) 35 90 45 0 0-0 0 0 0 Frequecy (rad/sec) Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-4
9.6 The Resoat Frequecy, r, ad the Resoat Peak Value, Mr The magitude of G ( j ) is G ( j ) j ( ) ( ) ( ) j If G(j) has a peak value at some frequecy, this frequecy is called the resoat frequecy. Sice the umerator of G(j) is costat, a peak value of G(j) ill occur he g ( j ) ( ) ( ) is a miimum, ad this equatio ca e ritte as ( ) ( ) 4 ( ) g j is miimum value of g() occur at Thus the resoat frequecy r is r for 0 0.707 As the dampig ratio ζ approaches zero, the resoat frequecy approaches. For 0 0.707, the resoat frequecy r, is less tha the damped atural frequecy d, hich is exhibited i the trasiet respose. From the equatio of r above it ca be see that for ζ > 0.707, there is o resoat peak. The magitude G(j) decrease mootoically ith icreasig frequecy. This magitude is less tha 0 db for all values of > 0. The magitude of the resoat peak M, ca be foud by substitutig equatio of r ito equatio of G(j) for 0 0.707, M r G ( j ) max G ( j r) For ζ > 0.707 M r =. As ζ approaches zero, M r approaches ifiity. This meas that if the udamped system is excited at its atural frequecy, the magitude of G(j) becomes ifiity. The phase agle of G(j) at the frequecy here the resoat peak occurs ca be obtaied by G ( j ) ta 90 si o --------------------------------------------------------------------------------------------------------------------------------e Notes: a. If gai crossover frequecy is less tha the phase crossover frequecy, the system is stable. b. If the phase crossover frequecy is less tha the gai crossover frequecy, the system is ustable. Dr. Fadhil Aula Salahaddi Uiversity /College of Egieerig/ Electrical Egieerig Dept. 06 Chapter 9-5