CONTROL SYSTEMS. Chapter 7 : Bode Plot. 40dB/dec 1.0. db/dec so resultant slope will be 20 db/dec and this is due to the factor s
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1 CONTROL SYSTEMS Chapter 7 : Bode Plot GATE Objective & Numerical Type Solutio Quetio 6 [Practice Book] [GATE EE 999 IIT-Bombay : 5 Mark] The aymptotic Bode plot of the miimum phae ope-loop trafer fuctio GH () () i a how i the figure. Obtai the trafer fuctio GH () (). GH () () i db 4dB/dec db/decade. (rad/ec) (log cale) 4dB/decade Sol. For the give Bode magitude plot, there are two corer frequecie : ωadω. The iitial lope i 4 db/dec ad thi correpod to a factor fuctio. At ω, the lope chage by i the deomiator of the trafer db/dec o reultat lope will be db/dec ad thi i due to the factor i the umerator of the trafer fuctio. ω At ω, the lope chage by db/dec o reultat lope will be 4 db/dec ad thi i due to the factor i the deomiator of the trafer fuctio. ω Calculatio of : loglog log Calculatio of : log log Calculatio of K : log K 4log. log K K.. rad/ec rad/ec
2 The overall trafer fuctio ca be writte a, K. GH () ().. (.) GH () () ( ) Quetio 3 [Work Book] A. [GATE EE 3 IIT-Madra : Mark] K The aymptotic Bode plot of the trafer fuctio i give i below figure. The error i phae agle a ad db gai at a frequecy of.5a are repectively : G,dB log K, deg db/decade.a a (rad/ec) (rad/ec) a 45 /decade A. Sol. (A) 4.9,.97dB (B) 5.7, 3dB (C) (A) The aymptotic Bode plot i how below. G,dB log K 4.9, 3dB (D) 5.7,.97dB, deg db/decade.a a (rad/ec) (rad/ec) a 45 /decade K From the trafer fuctio the magitude ad phae are give i equatio (i) ad (ii). a ω G( jω) log K log a.5a At.5 a, G( jω) log K log a G( jω) log K.97 db.. (i) ω.5a G( jω) ta ta (ii) a a From the give Bode plot magitude ad phae are how i equatio (iii) ad (iv). At.5 a, G(ω) j logk.. (iii)
3 Error i magitude ca be calculated a, Error = log K.97 log K.97 db G(ω) j y.. (iv) y 45 log.5a log.a y 3.45 Error i phae ca be calculated a, Error = 6.57 ( 3.45 ) 4.89 Hece, the correct optio i (A). Quetio 8 [Practice Book] [GATE EC 5 (Set-) IIT-Kapur : Mark] The trafer fuctio of a ma-prig-damper ytem i give by G (). The frequecy M BK repoe data for the ytem are give i the followig table. i rad/ G( j ) i db arg G( j ) i deg The uit tep repoe of the ytem approache a teady tate value of. A.. Sol. Give : G () M BK U() G () Y() Y () GU () () Y () ( M BK) Steady tate value i give by, y y( ) K Now, at. rad/ec, G( j) 8.5 ( ) lim Y( ) lim ( M B K ) db log G( j) 8.5 log 8.5 K 3
4 Alteratively : From Bode plot cocept : Iitial value of G( j) i cotat. log dc gai 8.5 log 8.5 K 8.5 log K 8.5 y( ).88 K y( ). Hece, the correct awer i.. Quetio 3 [Work Book] [GATE EC 5 (Set-3) IIT-Kapur : Mark] Coider the Bode plot how i the figure. Aume that all the pole ad zero are real-valued. The value of fh fl (i Hz) i. G( j), db 4 db 4 db/dec db f L 3 4 db/dec 9 f H Freq. (Hz) A. 897 Sol. The give Bode plot i how below. G( j), db 4 db 4 db/dec Calculatio of f L : Calculatio of f H : db 4 4 log (3) log ( ) 4 f L 3 log f L 3 f L f 3 Hz.. (i) L 4 4 log log (9) f H f H log 9 f 9 9 (ii) H 4 db/dec fh fl Hz Hece, the correct awer i 897 Hz. f L 3 9 f H Freq. (Hz)
5 Quetio 9 [Practice Book] [GATE EE 5 (Set-) IIT-Kapur : Mark] A Bode magitude plot for the trafer fuctio G() of a plat i how i the figure. Which oe of the followig trafer fuctio bet decribe the plat? log G( j f) ( ) (A) (B) ( ) ( ) (C) ( ) ( ) (D) ( ) A. (D) Sol. For the give Bode magitude plot, there are two corer frequecie : ω ad ω At ω rad/ec, the lope chage by db/dec ad reultat lope will be db, thi i due to the factor i the deomiator of the trafer fuctio. At ω. k k k rad/ec, the lope chage by db/dec ad reultat lope will be db/dec, thi i due to the factor i the umerator of the trafer fuctio. log G( j f) db f (Hz) Slope db/dec log db k (p) (z) Calculatio of K : log K K The overall trafer fuctio ca be writte a, K G () f (Hz) ( ) ( ) G () ( ) ( ) Hece, the correct optio i (D). 5
6 IES Objective Solutio Quetio [Practice Book] [IES EC 995] What i the lope chage at of the magitude veru frequecy characteritic of a uity feedback ytem with the followig ope-loop trafer fuctio? 5( j. ) G( j) j( j.5 ) j.6 j 5 5 (A) 4 db/dec to db/dec (B) 4 db/dec to db/dec (C) db/dec to 4 db/dec (D) 4 db/dec to db/dec A. (A) 5( j. ) Sol. Give : G( j) j(.6) j j( j.5 ) 5 5 (Oe pole) db/dec rad/ec(oe fiite pole) db/dec 3 rad/ec (oe complex zero ) db/dec 4 5rad/ec (oe complex pole ) 4dB/dec Bode plot of give trafer fuctio i how i figure. G( j), db db/dec 5 (rad/ec) 4dB/dec db/dec 6dB/dec Slope chage at rad/ec db/dec to 4dB/dec Slope chage at rad/ec 4dB/dec to db/dec Slope chage at 5 rad/ec db/dec to 6dB/dec Hece, the correct optio i (A). Quetio 3 [Practice Book] [IES EC 6] Coider the followig tatemet regardig the aymptotic Bode plot ued for frequecy repoe aalyi. The deviatio of the actual magitude repoe for a zero o real axi i 3 db at the corer frequecy.. The phae agle for a pair of complex cojugate pole at u-damped frequecy deped upo the value of dampig ratio. Which of the tatemet give above i/are correct? 6
7 A. Sol. (A) Oly (B) Oly (C) Both ad (D) Neither ad (A) (i) For GH () () T Magitude ca be writte a, G( j) H( j) T I db, log G( j) H( j) log log T Error i log-magitude for i give by, T log ( T ) log. Therefore, error at corer frequecy i log( ) log 3 db. T So, the deviatio of the actual magitude repoe for a zero o real axi i 3 db at the corer frequecy. G( j), db Exact plot Corer frequecy T (rad/ec) db/decade (ii) Geeral ecod order ytem G () Put j, G () ( j) ( j) j j G () j I db, magitude ca be writte a, j M M log G( j) log j log 4 Suppoe u 7
8 log G( j) M log ( u ) 4 u Coider the two cae :. u i.e. M log db. u i.e. M log ( u ) log u 4log u So, it i a traight lie havig lope of 4 db/decade ad paig through the poit u. Therefore, the aymptotic plot coit of (i) M u (ii) M 4log u u Phae agle ca be writte a, u G( j) ta u The phae agle i a fuctio of both ad. At, frequecy, the phae agle i (i) For mall value of u, u i mall ta (ii) For large value of ta u u, u u 9 regardle of. the phae agle equal. At the corer (iii) Whe u ta 9 Hece, the correct optio i (A). Quetio 8 [Practice Book] [IES EC 8] The gai-phae plot of a liear cotrol ytem i how i the below figure. db Gai (db) Phae Degree A. db What are the gai-margi (G.M.) ad the phae-margi (P.M.) of the ytem? (A) G.M. > db ad P.M. > degree (B) G.M. > db ad P.M. < degree (C) G.M. < db ad P.M. > degree (D) G.M. < db ad P.M. < degree (D) 8
9 Sol. Sice, gc pc ytem i utable. Gai (db) db P.M. 7 gc pc 8 G.M. 9 Phae (Degree) db At gc, G.M. At gc, P.M. 8 G( jgc) H( j gc) G( jgc ) H( jgc ) 8 P.M. Hece, the correct optio i (D). Quetio 37 [Practice Book] [IES EE ] The trafer fuctio of a liear cotrol ytem i give by, ( 5) G () ( 4)( ) I it bode diagram, the value of gai for.rad/ec i, (A) db (B) 4 db (C) 6 db (D) 8 db A. (B) Sol. Give : ( 5) G () ( 4)( ) Subtitutig j, ( j 5) Gj ( ) j ( j 4)( j ) (5 j.) G(.) j j.( j.4)( j.). 5 G(.) j. (. 4 )(. ) log(4.97) 4 db Hece the mot approximate optio i (B)
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