LESSON 15: BODE PLOTS OF TRANSFER FUNCTIONS

Transcription

1 10/8/015 1 LESSON 15: BODE PLOTS OF TRANSFER FUNCTIONS ET 438a Automatc Control Systems Technology Learnng Objectves After ths presentaton you wll be able to: Compute the magntude of a transfer functon for a gven radan frequency. Compute the phase shft of a transfer functon for a gven radan frequency. Construct a Bode plot that shows both magntude and phase shft as functons of transfer functon nput frequency Use MatLAB nstructons t produce Bode plots of transfer functons. 1

2 10/8/015 Constructon of Bode Plots 3 Bode plots consst of two ndvdual graphs: a) a semlog plot of gan vs frequency b) a semlog plot of phase shft vs frequency. Frequency s the logarthmc axs on both plots. Bode plots of transfer functons gve the frequency response of a control system To compute the ponts for a Bode Plot: 1) Replace Laplace varable, s, n transfer functon wth jw ) Select frequences of nterest n rad/sec (w=pf) 3) Compute magntude and phase angle of the resultng complex expresson. Constructon of Bode Plots 4 Bode plot calculatons- magntude/phase X(jw) Transfer Functon Y(jw) Gan: Y(jw) G(jw) X(jw) db 0log( G(jw) ) Where for a gven frequency Y y o X x Phase: o To fnd the magntude and phase shft of a complex number n rectangular form gven: z a jb z a b b tan 1 a

3 10/8/015 Computng Transfer Functon Values 5 Example 15-1: A self-regulatng tank has a transfer functon of the form shown below. H(s) G Q(s) 1 s The tank has a tme constant, =1590 seconds and a gan, G=000 s/m. Determne the ampltude and phase shft of the system to a snusodal flow nput of Hz Soluton: Substtute values of G,, and jw nto transfer functon and compute the gan magntude and phase shft. Example 15-1 Soluton (1) 6 G 000 s/m 1590 s f Hz H(jw) 000 Q(jw) jw Hz 0.001rad/s w pf w p Place radan frequency nto transfer functon and compute complex number H( j0.001) j901 Q( j0.001) j j.590 Convert the result to polar form to fnd magntude and phase shft 3

4 10/8/015 Example 15-1 Soluton () 7 Complex gan Magntude Calculaton a G(j0.001) G(j0.001) G(j0.001) j j j.590 b -901 G(jw) db 0log(1065) 60.5 db a b Ans tan Phase shft -1 b a -1 b tan tan 58 a Ans At rad/sec, the system has a gan of 60.5 db and the output changes n heght lag the flow changes by 58 degrees 8 MatLAB has control system toolbox functons for defnng Lnear Tme-nvarant systems (LTI) and constructng the Bode plots. Use tf and bode functons to create LTI and plot. Introducng zpk functon sys = zpk(z,p,k) Turns arrays of zeros, poles and gans nto LTI called sys Where z = array of transfer functon zeros p = array of transfer functon poles k = array of transfer functon gans 4

5 Magntude (db) Phase (deg) 10/8/015 9 Example 15-: Construct the Bode plot for the gven transfer functon shown n factored form usng MatLAB control toolbox functons. V o(s) V (s) 0.005s 0.001s s 1 Soluton: Transfer functon has one zero at s=0 and two poles at s=-1/0.001= Dvdng the transfer functon denomnator and numerator by places t nto standard form V o(s) V (s) s s s s s s s s 1000s MatLAB code to produce Bode plot of gven transfer functon Enter at command prompt of nto m-fle Magntude plot k= [5] p=[ ] z=[0] sys=zpk(z,p,k) bode(sys) w=1000 Bode Dagram Phase plot Frequency (rad/s) 5

6 Magntude (db) Phase (deg) 10/8/ The bode(sys) functon can plot more than one transfer functon on the same fgure axs. The fgure produced by the bode(sys) functon can be coped and pasted nto wordprocessors and other programs. To plot more than one transfer functon use the followng syntax: bode(sys1,sys, ). Example 15-3: Compare the Bode plots of the transfer functon gven n Example 15- to the functon gven below. Use MatLAB to generate the Bode plots on a sngle set of axs. Note: the only dfference s the gan s ncreased by a factor of 100. V o(s) V (s) 500s s 1000s Enter at command prompt of nto m-fle sys k= [5] p=[ ] z=[0] sys1=zpk(z,p,k) k1=[500] sys=zpk(z,p,k1) bode(sys1, ro-,sys, b- ) Bode Dagram sys Ths changes plot styles (blue sold lne) Frequency (rad/s) sys1 & sys 6

7 10/8/ ET 438a Automatc Control Systems Technology End Lesson 15: Bode Plots of Transfer Functons 7