ME375 Hadouts Bode Diagrams Recall that if m m bs m + bm s + + bs+ b Gs () as + a s + + as+ a The bm( j z)( j z) ( j zm) G( j ) a ( j p )( j p ) ( j p ) bm( s z)( s z) ( s zm) a ( s p )( s p ) ( s p ) bm ( j z) ( j z) ( j zm ) a ( j p ) ( j p ) ( j p ) bm log ( G( j ) ) log log log a KJ + ( j p ) KJ + + ( j p ) + log c ( j z) h+ + log c ( j z) h ad bm( j z)( j z) ( j zm) G( j ) a( j p)( j p) ( j p) ( j z ) + ( j z ) + + ( j z ) ( j p ) ( j p ) ( j p ) F H G I F HG m I F HG I KJ School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 9 Ex: Fid the magitude ad the phase of the followig trasfer fuctio: 3 3s + s + 9s Gs () 3 s + s + 76s+ 8 ( )( ) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 3
ME375 Hadouts Bode Diagram Buildig Blocks st Order Real Poles Trasfer Fuctio: G p ( s) τ s +, τ > Frequecy Respose: ( j ), τ > τ j + R ( j ) τ + T ( j) ata( τ, ) ta τ a f Phase (deg); Magitude (db) -3 - -4-45 Q: By just lookig at the Bode diagram, ca you determie the time costat ad the steady state gai of the system? -9./τ./τ /τ /τ /τ Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 3 st Order Real Poles Trasfer Fuctio: 5 G( s) s + 5 Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 5 5-45 -9 - Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 3
ME375 Hadouts Bode Diagram Buildig Blocks st Order Real Zeros Trasfer Fuctio: Gz( s) τ s+, τ > Frequecy Respose: G ( j ) τ j +, τ > R T z Gz( j) τ + Gz( j) ata( τ, ) ta τ a f Phase (deg); Magitude (db) 4 3 9 45./τ./τ /τ /τ /τ Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 33 st Order Real Zeros Trasfer Fuctio: G ( s) 7. s + 7. Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 5 5 9 45 - Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 34 3
ME375 Hadouts Lead Compesator Trasfer Fuctio: 35s + 35 G( s) s + 5 Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 3-9 -9 - Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 35 st Order Bode Diagram Summary st Order Poles ( s), τ > τ s + Break Frequecy b [ rad/s] τ Mag. Plot Approximatio db from DC to b ad a straight lie with db/decade slope after b. Phase Plot Approximatio deg from DC to. Betwee ad b b b, a straight lie from deg to 9 deg (passig 45 deg at b ). For frequecy higher tha b, straight lie o 9 deg. st Order Zeros G ( ), z s τ s+ τ > Break Frequecy b [ rad/s] τ Mag. Plot Approximatio db from DC to b ad a straight lie with db/decade slope after b. Phase Plot Approximatio deg from DC to. Betwee b b ad b, a straight lie from deg to 9 deg (passig 45 deg at b ). For frequecy higher tha b, straight lie o 9 deg. Note: By lookig at a Bode diagram you should be able to determie the relative order of the system,, its break frequecy,, ad DC (steady-state) state) gai.. This process should also be reversible, i.e. give a trasfer fuctio, be able to plot a straight lie approximated Bode diagram. School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 36 4
ME375 Hadouts Bode Diagram Buildig Blocks Itegrator (Pole at origi) Trasfer Fuctio: ( s) s Frequecy Respose: ( j ) j R ( j ) G ( j ) 9 T p Phase (deg); Magitude (db) - -4-6 -45-9 -35. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 37 Bode Diagram Buildig Blocks Differetiator (Zero at origi) Trasfer Fuctio: G ( z s ) s Frequecy Respose: Gz ( j ) j R Gz ( j ) Gz ( j ) 9 T Phase (deg); Magitude (db) 6 4-35 9 45. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 38 5
ME375 Hadouts Combiatio of Systems Trasfer Fuctio: 35s + 35 G ( s) s( s + 5 ) Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 3-9 -9 - School of Mechaical Egieerig Purdue Uiversity Frequecy (rad/sec) ME375 Frequecy Respose - 39 Combiatio of Systems Trasfer Fuctio: 5 G ( s) s( s + 55s + 5) Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 4-4 -8 - -9-8 -7-3 School of Mechaical Egieerig Purdue Uiversity Frequecy (rad/sec) ME375 Frequecy Respose - 4 6
ME375 Hadouts Bode Diagram Buildig Blocks d Order Complex Poles Trasfer Fuctio: ( s) + ζ +, ζ s s Frequecy Respose: ( j ) ζ j + ( j ) 4ζ + Phase (deg); Magitude (db) - -4-6 -8-9 ζ G ( ) j ta p -8.. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 4 Secod-Order System Trasfer Fuctio: 5 G( s) s + s + 5 Plot the straight lie approximatio of G(s) s s Bode diagram: Phase (deg); Magitude (db) 4-4 -8 - -9-8 3 4 Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 4 7
ME375 Hadouts Bode Diagram Buildig Blocks d Order Complex Zeros Trasfer Fuctio: s + ζ s+ Gz ( s), ζ Frequecy Respose: ζ Gz ( j ) j + ( ) p ( ) G j G j z 4ζ + ( ) p ( ) Gz j G j ζ ta Phase (deg); Magitude (db) 8 6 4-8 9.. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 43 Bode Diagrams of Poles ad Zeros Bode Diagrams of stable complex zeros are the mirror images of the Bode diagrams of the idetical stable complex poles w.r.t. the db lie ad the deg lie, respectively. Let ( s) G ( s) R T R T ( j ) G ( j ) G ( j) G ( j) p log G ( j) log G ( j) G ( j) G ( j) p z z z e p j c z h z Phase (deg) Magitude (db) 4 - -4 8-8. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 44 8
ME375 Hadouts d Order Bode Diagram Summary d Order Complex Poles ( s), ζ> s + ζ s+ Break Frequecy b [ rad/s] Mag. Plot Approximatio db from DC to ad a straight lie with 4 db/decade slope after. Peak value occurs at: r ζ ( j r) MAX ζ ζ Phase Plot Approximatio ζ ( ) ( ) deg from DC to. Betwee 5 5 ad 5 ζ, a straight lie from deg to 8 deg (passig 9 deg at ). For frequecy higher tha 5 ζ, straight lie o 8 deg. ζ d Order Complex Zeros s + ζ s+ Gz( s), ζ> Break Frequecy b [ rad/s] Mag. Plot Approximatio db from DC to ad a straight lie with 4 db/decade slope after. Phase Plot Approximatio ζ deg from DC to ( 5 ). Betwee ( ζ ) 5 ad 5 ζ, a straight lie from deg to 8 deg (passig 9 deg at ). For frequecy higher tha 5 ζ, straight lie o 8 deg. School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 45 d Order System Frequecy Respose A Closer Look: ( s), ζ s + ζ s+ Frequecy Respose Fuctio: ( j ) ( j) + ζ ( j) + ζ j + Magitude: Phase: ( j ) ( ) ( ) j + j ζ + ζ ata, ( ) The maximum value of G(j) occurs at the Peak (Resoat) Frequecy r : r ζ ad ( jr) ζ ζ ζ School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 46 9
ME375 Hadouts d Order System Frequecy Respose 4 Phase (deg); Magitude (db) - -4-6 -45-9 -35-8. Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 47 d Order System Frequecy Respose A Few Observatios: Three differet characteristic frequecies: Natural Frequecy ( ) Damped Natural Frequecy ( d ): d Resoat (Peak) Frequecy ( r ): r d ζ Whe the dampig ratio ζ >.77,, there is o peak i the Bode magitude plot. DO NOT cofuse this with the coditio for over-damped ad uder-damped damped systems: whe ζ < the system is uder-damped damped (has overshoot) ad whe ζ > the system is over-damped (o overshoot). As ζ Æ, r Æ ad G(j) ΜΑΧ icreases; also the phase trasitio from deg to 8 deg becomes sharper. r ζ School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 48
ME375 Hadouts Combiatio of Systems Trasfer Fuctio: G ( s) ( s + s + 5) s( s + )( s + s + 5) Plot the straight lie approximatio of G(s) s s Bode diagram: School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 49 4 Magitude (db) - -4-6 -8 8 9 Phase (deg) -9-8 -7-3 Frequecy (rad/sec) School of Mechaical Egieerig Purdue Uiversity ME375 Frequecy Respose - 5