NODIA AND COMPANY. GATE SOLVED PAPER Electronics & Communication Control System. Copyright By NODIA & COMPANY

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1 No art of thi ublication may be reroduced ditributed in any fm any mean, electronic, mechanical, hotocoying, otherwie without the ri ermiion of the auth. ATE OLVED PAPER Electronic & Communication Control ytem Coyright By NODIA & COMPANY Infmation contained in thi book ha been obtained by auth, from ource believe to be reliable. However, neither Nodia n it auth guarantee the accuracy comletene of any infmation herein, and Nodia n it auth hall be reonible f any err, omiion, damage ariing out of ue of thi infmation. Thi book i ublihed with the undertanding that Nodia and it auth are ulying infmation but are not attemting to render engineering other rofeional ervice. NODIA AND COMPANY B8, Dhanhree Tower It, Central ine, Vidyadhar Nagar, Jaiur 9 Ph : enquiry@nodia.co.in

2 ONE MAR Q. The Bode lot of a tranfer function ^h i hown in the figure below. The gain _ log ^ h i i db and 8dB at rad/ and rad/ reectively. The hae i negative f all w. Then ^h i (A) 9. 8 (B) 9. 8 (C) (D) TO MAR Y ^ h Q. The ignal flow grah f a ytem i given below. The tranfer function f U ^ h thi ytem i (A) (C) (B) (D) tatement f Linked Anwer Quetion and 4: The tate diagram of a ytem i hown below. A ytem i decribed by the tatevariable equation X o AX+ Bu ; y CX + D u

3 Q. The tatevariable equation of the ytem hown in the figure above are Xo X u (A) > H + > H Xo (B) > X u H + > H y X + u y X + u Xo X u (C) > H + > H Xo (D) > X u H + > H y X u y X u Q. 4 The tate tranition matri e At of the ytem hown in the figure above i (A) e t t e > t th (B) > t th te e te e (C) e t > t th (D) e t te t > t H e e e Q. 5 The oenloo tranfer function of a dc mot i given a Va ^h +. hen connected in feedback a hown below, the aroimate value of a that will reduce the time contant of the cloed loo ytem by one hundred time a comared to that of the oenloo ytem i (A) (B) 5 (C) (D) w^ h ONE MAR ( + 9)( + ) Q. 6 A ytem with tranfer function () ( + )( + )( + 4) i ecited by in( w t). The teadytate outut of the ytem i zero at (A) w rad/ (B) w rad/ (C) w rad/ (D) w 4 rad/ TO MAR Q. 7 The feedback ytem hown below ocillate at rad/ when

4 (A) and a.75 (B) and a.75 (C) 4 and a.5 (D) and a.5 Q. 8 The tate variable decrition of an LTI ytem i given by Jo N J a NJ N JN J N O O O O O o a + u y _ i o O a O O O O L P L PL P L P L P where y i the outut and u i the inut. The ytem i controllable f (A) a!, a, a! (B) a, a!, a! (C) a, a!, a (D) a!, a!, a ONE MAR Q. 9 The root locu lot f a ytem i given below. The oen loo tranfer function creonding to thi lot i given by ( + ) ( + ) (A) H ^ h ^ h k (B) H k ( + )( + ) ^ h ^ h ( + )( + ) (C) H k ( + ) ^ h ^ h (D) H k ( )( + )( + ) ^ h ^ h ( + )( + ) Q. F the tranfer function j ( w) 5 + jw, the creonding Nyquit lot f oitive frequency ha the fm

5 TO MAR Q. The block diagram of a ytem with one inut u and two outut y and y i given below. A tate ace model of the above ytem in term of the tate vect and the outut vect y [ y y] T i (A) o [] + [] u; y [ ] (B) o [ ] + [ ] u; y >H (C) o > + u; y H > H 8 B (D) o > + u; y H > H > H Common Data F Q. and The inutoutut tranfer function of a lant H (). ( + ) The lant i laced in a unity negative feedback configuration a hown in the figure below. Q. The gain margin of the ytem under cloed loo unity negative feedback i (A) db (B) db (C) 6 db (D) 46 db Q. The ignal flow grah that DOE NOT model the lant tranfer function H () i

6 ONE MAR Q. 4 The tranfer function Y ()/ R () of the ytem hown i (A) (B) + (C) (D) + + Y () Q. 5 A ytem with tranfer function ha an outut yt () co t X () + a k f the inut ignal t () co t a k. Then, the ytem arameter i (A) (B) / (C) (D) / Q. 6 F the aymtotic Bode magnitude lot hown below, the ytem tranfer function can be (A) +. + (C) (B) (D) TO MAR Common Data F Q. 7 and 8 The ignal flow grah of a ytem i hown below: Q. 7 The tate variable rereentation of the ytem can be (A) o > u H + > H o (B) > u H + > H yo [. 5] yo 8. 5B

7 o u (C) > H + > H yo B Q. 8 The tranfer function of the ytem i (A) + + (C) o u (D) > H + > H yo B (B) + (D) + + Q. 9 A unity negative feedback cloed loo ytem ha a lant with the tranfer function () + + and a controller () c in the feed fward ath. F a unit et inut, the tranfer function of the controller that give minimum teady tate err i (A) () c + (B) () + c + + ( )( 4) (C) c () + + ( + )( + ) (D) () c ONE MAR Q. The magnitude lot of a rational tranfer function () with real coefficient i hown below. hich of the following comenat ha uch a magnitude lot? (A) Lead comenat (C) PID comenat Q. Conider the ytem d dt (B) Lag comenat (D) Leadlag comenat A + Bu with A and B q where and q are arbitrary real number. hich of the following tatement about the controllability of the ytem i true? (A) The ytem i comletely tate controllable f any nonzero value of and q (B) Only and q reult in controllability (C) The ytem i uncontrollable f all value of and q (D) e cannot conclude about controllability from the given data 9 TO MAR Q. The feedback configuration and the olezero location of () are hown below. The root locu f negative value of k, i.e. f < k <, ha

8 breakaway/breakin oint and angle of dearture at ole P (with reect to the oitive real ai) equal to (A)! and c (B)! and 45c (C)! and c (D)! and 45c Q. The unit te reone of an underdamed econd der ytem ha teady tate value of. hich one of the following tranfer function ha thee roertie? (A) 4. (B) (C) Common Data F Q. 4 and 5 : (D) The Nyquit lot of a table tranfer function () i hown in the figure are intereted in the tability of the cloed loo ytem in the feedback configuration hown. Q. 4 hich of the following tatement i true? (A) () i an alla filter (B) () ha a zero in the righthalf lane (C) () i the imedance of a aive netwk (D) () i marginally table Q. 5 The gain and hae margin of () f cloed loo tability are (A) 6 db and 8c (B) db and 8c (C) 6 db and 9c (D) db and 9c 8 ONE MAR Q. 6 te reone of a et of three econdder underdamed ytem all have the ame ercentage overhoot. hich of the following diagram rereent the ole of the three ytem?

9 ATE OLVED PAPER EC Q. 7 The olezero given below creond to a (A) Law a filter (B) High a filter (C) Band filter (D) Notch filter 8 TO MAR Q. 8 A ignal flow grah of a ytem i given below The et of equalitie that creond to thi ignal flow grah i (A) dt d u u b g a g a b + J L J L e N P O O O N P O O O o R T R T V X V X (B) dt d u u a a b g g b + J L J L e N P O O O N P O O O o R T R T V X V X (C) dt d u u a b a b g g + J L J L e N P O O O N P O O O o R T R T V X V X (D) dt d u u a g b b a a + J L J L e N P O O O N P O O O o R T R T V X V X

10 Q. 9 rou I lit a et of four tranfer function. rou II give a lit of oible te reone yt. () Match the te reone with the creonding tranfer function. (A) P, Q, R4, (B) P, Q, R4, (C) P, Q, R4, (D) P, Q4, R, Q. A certain ytem ha tranfer function () a 4 where a i a arameter. Conider the tandard negative unity feedback configuration a hown below hich of the following tatement i true? (A) The cloed loo ytem i never table f any value of a (B) F ome oitive value of a, the cloed loo ytem i table, but not f all oitive value. (C) F all oitive value of a, the cloed loo ytem i table. (D) The cloed loo ytem table f all value of a, both oitive and negative. Q. The number of oen right half lane of () i (A) (B) (C) (D)

11 Q. The magnitude of frequency reone of an underdamed econd der ytem i 5 at rad/ec and eak to at 5 rad/ec. The tranfer function of the ytem i (A) 5 (B) (C) 7 (D) Q. rou I give two oible choice f the imedance Z in the diagram. The circuit element in Z atify the condition RC> RC. The tranfer function V rereent a kind of controller. V i Match the imedance in rou I with the tye of controller in rou II (A) Q, R (B) Q, R (C) Q, R (D) Q, R 7 ONE MAR Q. 4 If the cloedloo tranfer function of a control ytem i given a T () 5 ( + )( + ), then It i (A) an untable ytem (B) an uncontrollable ytem (C) a minimum hae ytem (D) a nonminimum hae ytem 7 TO MAR Q. 5 A control ytem with PD controller i hown in the figure. If the velocity err contant V and the daming ratio z 5., then the value of P and D are (A),.9 (B),.9 P D (C),.9 (D),.9 P D P P D D

12 Q. 6 The tranfer function of a lant i T () 5 ( + 5)( + + ) The econdder aroimation of T () uing dominant ole concet i (A) (B) 5 ( + 5)( + ) ( + 5)( + ) (C) 5 (D) Q. 7 The oenloo tranfer function of a lant i given a (). If the lant i oerated in a unity feedback configuration, then the lead comenat that an tabilize thi control ytem i ( ) ( + 4) (A) (B) + + ( + ) ( + ) (C) (D) + + Q. 8 A unity feedback control ytem ha an oenloo tranfer function () ( + 7+ ) The gain f which + j will lie on the root locu of thi ytem i (A) 4 (B) 5.5 (C) 6.5 (D) Q. 9 The aymtotic Bode lot of a tranfer function i a hown in the figure. The tranfer function () creonding to thi Bode lot i (A) ( + )( + ) (C) ( + )( + ) (B) ( + )( + ) (D) ( + )( +. 5) Q. 4 The tate ace rereentation of a earately ecited DC ervo mot dynamic i given a dw dt w > dio H u + i dt a where w i the eed of the mot, i a i the armature current and u i the w() armature voltage. The tranfer function of the mot i U () (A) (B) (C) + (D)

13 tatement f linked Anwer Quetion 4 and 4 : Conider a linear ytem whoe tate ace rereentation i () t A() t. If the initial tate vect of the ytem i (), then the ytem reone i e t () > th. If the itial tate vect of the ytem change to () e, then t e the ytem reone become t () > th e Q. 4 The eigenvalue and eigenvect air ( l ivi) f the ytem are (A) e o and e o (B) e, o and e, o (C) e, o and e, o (D) e o and e, o Q. 4 The ytem matri A i (A) (B) (C) (D) 6 ONE MAR Q. 4 The oenloo function of a unitygain feedback control ytem i given by () ( + )( + ) The gain margin of the ytem in db i given by (A) (B) (C) (D) 6 TO MAR Q. 44 Conider two tranfer function () and (). + a + b + a + b The db bandwidth of their frequency reone are, reectively (A) a 4b, a + 4b (B) a + 4b, a 4b (C) a 4b, a 4b (D) a + 4b, a + 4b Q. 45 The Nyquit lot of j ( w) Hj ( w) f a cloed loo control ytem, ae through (, j) oint in the H lane. The gain margin of the ytem in db i equal to (A) infinite (B) greater than zero (C) le than zero (D) zero Q. 46 The oitive value of and a o that the ytem hown in the figure below ocillate at a frequency of rad/ec reectively are (A),.75 (B),.75 (C), (D),

14 Q. 47 The tranfer function of a hae lead comenat i given by () T c + + T where T > The maimum hae hift rovide by uch a comenat i (A) (B) (C) 4 (D) 6 Q. 48 A linear ytem i decribed by the following tate equation Xt o () AX() t + BU(), t A The tate tranition matri of the ytem i co (A) t in t in t co t (B) co t in t in t co t co t in t (C) in t co t (D) co t in t co t in t tatement f Linked Anwer Quetion 49 and 5: Conider a unity gain feedback control ytem whoe oen loo tranfer function i : () a + Q. 49 The value of a o that the ytem ha a hae margin equal to i aroimately 4 equal to (A).4 (B).4 (C).84 (D).74 Q. 5 ith the value of a et f a hae margin of, the value of unit imule 4 reone of the oen loo ytem at t econd i equal to (A).4 (B).4 (C).84 (D).74 5 ONE MAR Q. 5 hich one of the following olar diagram creond to a lag netwk? Q. 5 A linear ytem i equivalently rereented by two et of tate equation : X o AX + BU and o C+ DU

15 The eigenvalue of the rereentation are alo comuted a [ l ] and [ m ]. hich one of the following tatement i true? (A) [ l] [ m] and X (B) [ l] [ m] and X! (C) [ l]! [ m ] and X (D) [ l] [ m] and X! Q. 5 Deite the reence of negative feedback, control ytem till have roblem of intability becaue the (A) Comonent ued have non linearitie (B) Dynamic equation of the ubytem are not known eactly. (C) Mathematical analyi involve aroimation. (D) ytem ha large negative hae angle at high frequencie. 5 TO MAR Q. 54 The olar diagram of a conditionally table ytem f oen loo gain i hown in the figure. The oen loo tranfer function of the ytem i known to be table. The cloed loo ytem i table f (A) < 5 and < < (B) < and < < (C) < and 5 < (D) > and 5 > 8 8 Q. 55 In the derivation of ereion f eak ercent overhoot M ee # % o hich one of the following condition i NOT required? (A) ytem i linear and time invariant (B) The ytem tranfer function ha a air of comle conjugate ole and no zeroe. (C) There i no trantation delay in the ytem. (D) The ytem ha zero initial condition. Q. 56 A ram inut alied to an unity feedback ytem reult in 5% teady tate err. The tye number and zero frequency gain of the ytem are reectively (A) and (B) and (C) and (D) and Q. 57 A double integrat lant () /, H () i to be comenated to achieve the daming ratio z 5. and an undamed natural frequency, w 5 rad/ec n

16 which one of the following comenat e () will be uitable? (A) + (B) (C) 6 (D) ( ) Q. 58 An unity feedback ytem i given a (). Indicate the crect root ( + ) locu diagram. tatement f Linked Anwer Quetion 59 and 6 The oen loo tranfer function of a unity feedback ytem i given by () e ( + ) Q. 59 The gain and hae croover frequencie in rad/ec are, reectively (A).6 and.6 (B).6 and.485 (C).485 and.6 (D).6 and.6 Q. 6 Baed on the above reult, the gain and hae margin of the ytem will be (A) 7.9 db and 87. 5c (B) 79. db and 87. 5c (C) 79. db and 87. 5c (D) 79. and 87. 5c 4 ONE MAR Q. 6 The gain margin f the ytem with oenloo tranfer function ( + ) H () (), i (A) (B) (C) (D) Q. 6 iven H () () ( + )( +.The oint of interection of the aymtote of the root loci with the real ai ) i (A) 4 (B). (C). (D) 4

17 4 TO MAR Q. 6 Conider the Bode magnitude lot hown in the fig. The tranfer function H () i ( + ) (A) ( + )( + ) ( + ) (C) ( + )( + ) ( + ) (B) ( + )( + ) ( + ) (D) ( + )( + ) Q. 64 A caual ytem having the tranfer function H ( ) /( + ) i ecited with ut (). The time at which the outut reache 99% of it teady tate value i (A).7 ec (B).5 ec (C). ec (D). ec Q. 65 A ytem ha ole at. Hz, Hz and 8 Hz; zero at 5 Hz, Hz and Hz. The aroimate hae of the ytem reone at Hz i (A) 9c (B) c (C) 9c (D) 8c Q. 66 Conider the ignal flow grah hown in Fig. The gain 5 i (A) ( be + cf + dg) abcd (C) abcd ( be + cf + dg) + bedg Q. 67 If A, then in At i bedg (B) ( be + cf + dg) ( be + cf + dg) + bedg (D) abcd ( ) ( ) (A) in 4t + in t in( 4t) + in( t) in( 4t) + in( t) in( 4t) + in( t) in( t) in( t) (B) in() t in( t) ( ) ( ) (C) in 4t + in t in( 4t)in(t) in( 4t) + in( t) in( 4t) + in( t) ( ) ( ) (D) co t + co t co( 4t) + co( t) co( 4t) + co( t) co( 4t) + co( t) Q. 68 The oenloo tranfer function of a unity feedback ytem i () ( + + )( + ) The range of f which the ytem i table i (A) > > (B) > > 4

18 (C) < < (D) 6 < < 4 4 Q. 69 F the olynomial P () the number of root which lie in the right half of the lane i (A) 4 (B) (C) (D) Q. 7 The tate variable equation of a ytem are : o u, o and y + u. The ytem i (A) controllable but not obervable (B) obervable but not controllable (C) neither controllable n obervable (D) controllable and obervable Q. 7 iven A, the tate tranition matri e At i given by t e (A) > e t H (B) e t t e (C) e t t e > th (D) e e t ONE MAR Q. 7 Fig. how the Nyquit lot of the oenloo tranfer function H () () of a ytem. If H () () ha one righthand ole, the cloedloo ytem i (A) alway table (B) untable with one cloedloo right hand ole (C) untable with two cloedloo right hand ole (D) untable with three cloedloo right hand ole Q. 7 A PD controller i ued to comenate a ytem. Comared to the uncomenated ytem, the comenated ytem ha (A) a higher tye number (B) reduced daming (C) higher noie amlification (D) larger tranient overhoot TO MAR Q. 74 The ignal flow grah of a ytem i hown in Fig. below. The tranfer function C ()/ R () of the ytem i

19 (A) (B) ( + ) ( + 7) (C) (D) Q. 75 The root locu of ytem H () () ha the breakaway oint ( + )( + ) located at (A) ( 5., ) (B) (. 548, ) (C) ( 4, ) (D) (. 784, ) Q. 76 The aroimate Bode magnitude lot of a minimum hae ytem i hown in Fig. below. The tranfer function of the ytem i 8 ( +. ) 7 ( +. ) (A) (B) ( + ) ( + ) ( + )( + ) ( +. ) ( +. ) (C) (D) ( ) + ( + ) ( )( ) + + Q. 77 A econdder ytem ha the tranfer function C () 4 R () ith rt () a the unitte function, the reone ct () of the ytem i rereented by Q. 78 The gain margin and the hae margin of feedback ytem with H () () 8 are ( + )

20 (A) db,c (B), (C),c (D) db, Q. 79 The zeroinut reone of a ytem given by the tateace equation o () o and () i t (A) te (B) e t t (C) e t t t (D) te te t t ONE MAR Q. 8 Conider a ytem with tranfer function () + 6. It daming ratio will be.5 when the value of k i k (A) (B) 6 (C) 6 (D) 6 Q. 8 hich of the following oint i NOT on the root locu of a ytem with the oenloo tranfer function H () () k ( + )( + ) (A) j (B).5 (C) (D) Q. 8 The hae margin of a ytem with the oen loo tranfer function H () () ( ) ( + )( + ) (A) c (B) 6. 4c (C) 9c (D) Q. 8 The tranfer function Y ()/ U () of ytem decribed by the tate equation t o () t () + ut () and yt () 5. t () i (A) 5. (B) ( ) ( ) (C) 5. (D) ( + ) ( + ) TO MAR Q. 84 The ytem hown in the figure remain table when (A) k < (B) < k < (C) < k < (D) k > Q. 85 The tranfer function of a ytem i () ( + )( +. F a unit te inut to the ytem the aroimate ettling time f % criterion ) i (A) ec (C) ec (B) 4 ec (D). ec

21 Q. 86 The characteritic olynomial of a ytem i 5 4 q () The ytem i (A) table (B) marginally table (C) untable (D) ocillaty Q. 87 The ytem with the oen loo tranfer function H () () ha a gain margin of ( + + ) (A) 6 db (B) db (C) 5 db (D) 6 db ONE MAR Q. 88 The Nyquit lot f the oenloo tranfer function () of a unity negative feedback ytem i hown in the figure, if () ha no ole in the righthalf of lane, the number of root of the ytem characteritic equation in the righthalf of lane i (A) (B) (C) (D) Q. 89 The equivalent of the block diagram in the figure i given i Q. 9 The rootlocu diagram f a cloedloo feedback ytem i hown in the figure. The ytem i overdamed.

22 (A) only if # k # (B) only if < k < 5 (C) only if k > 5 (D) if # k < k > 5 Q. 9 If the characteritic equation of a cloed loo ytem i + +, then the ytem i (A) overdamed (B) critically damed (C) underdamed (D) undamed TO MAR Q. 9 An electrical ytem and it ignalflow grah rereentation are hown the figure (A) and (B) reectively. The value of and H, reectively are Z() Z() (A), Z() + Z() + Z4() Z() + Z() Z() Z() (C), Z () + Z () + Z () Z () + Z () 4 Z() Z() (B), Z() Z() + Z4() Z() + Z() Z() Z() (D), Z () Z () + Z () Z () + Z () 4 Q. 9 The oenloo DC gain of a unity negative feedback ytem with cloedloo tranfer function + 4 i (A) 4 (B) 4 9 (C) 4 (D) Q. 94 The feedback control ytem in the figure i table (A) f all $ (B) only if $ (C) only if # < (D) only if # # ONE MAR Q. 95 An amlifier with reitive negative feedback ha tow left half lane ole in it oenloo tranfer function. The amlifier (A) will alway be untable at high frequency (B) will be table f all frequency (C) may be untable, deending on the feedback fact (D) will ocillate at low frequency.

23 TO MAR Q. 96 A ytem decribed by the tranfer function H () The contraint on a and k are. a k i table. (A) a >, a k < (B) a >, ak > (C) a <, a k > (D) a >, ak < 999 ONE MAR Q. 97 F a econd der ytem with the cloedloo tranfer function T () the ettling time f ercent band, in econd, i (A).5 (B). (C). (D) 4. Q. 98 The gain margin (in db) of a ytem a having the loo tranfer function H () () ( + ) (A) (B) (C) 6 (D) Q. 99 The ytem modeled decribed by the tate equation i X > + u H > H Y 8B (A) controllable and obervable (C) obervable, but not controllable (B) controllable, but not obervable (D) neither controllable n obervable Q. The hae margin (in degree) of a ytem having the loo tranfer function H () () ( + ) i (A) 45c (B) c (C) 6c (D) c 999 TO MAR Q. An amlifier i aumed to have a ingleole highfrequency tranfer function. The rie time of it outut reone to a te function inut i 5 n ec. The uer db frequency (in MHz) f the amlifier to a inuoidal inut i aroimately at (A) 4.55 (B) (C) (D) 8.6 Q. If the cloed loo tranfer function T () of a unity negative feedback ytem i given by T () an an + n n + a an + an then the teady tate err f a unit ram inut i

24 (A) a a n n (B) a a n n (C) an (D) zero a n Q. Conider the oint + j4 and j in the lane. Then, f a ytem with the oenloo tranfer function H () () ( + ) 4 (A) i on the root locu, but not (B) i on the root locu, but not (C) both and are on the root locu (D) neither n i on the root locu Q. 4 F the ytem decribed by the tate equation R V R V o + u 5. T X T X If the control ignal u i given by u [5. 5] + v, then the eigen value of the cloedloo ytem will be (A),, (B),, (C),, (D),, 998 ONE MAR Q. 5 The number of root of in the left half of the lane i (A) zero (B) one (C) two (D) three Q. 6 The tranfer function of a tachometer i of the fm (A) (B) (C) ( + ) (D) ( + ) Q. 7 Conider a unity feedback control ytem with oenloo tranfer function () ( + ). The teady tate err of the ytem due to unit te inut i (A) zero (B) (C) / (D) infinite Q. 8 The tranfer function of a zeroderhold ytem i T (A) ( /)( + e ) (B) ( /)( e T ) (C) (/) e T (D) + (/) e Q. 9 In the Bodelot of a unity feedback control ytem, the value of hae of jw ( ) at the gain cro over frequency i 5c. The hae margin of the ytem i (A) 5c (B) 55c (C) 55c (D) 5c T

25 Q. Conider a feedback control ytem with loo tranfer function ( +. 5) H () () ( + )( + ) The tye of the cloed loo ytem i (A) zero (B) one (C) two (D) three Q. The tranfer function of a hae lead controller i of hae rovided by thi controller i (A) 9c (B) 6c (C) 45c (D) c + T. The maimum value + T Q. The Nyquit lot of a hae tranfer function gj ( w) Hj ( w ) of a ytem encloe the (, ) oint. The gain margin of the ytem i (A) le than zero (B) zero (C) greater than zero (D) infinity Q. The tranfer function of a ytem i ( ). The characteritic equation + ( + ) of the ytem i (A) (B) ( ) + ( + ) (C) ( + ) ( + ) (D) ( + ) ( + ) Q. 4 n In a ynchro err detect, the outut voltage i rotional to [ w( t)], where w ( t) i the rot velocity and n equal (A) (B) (C) (D) 997 ONE MAR Q. 5 In the ignal flow grah of the figure i y / equal (A) (B) 5 (C) (D) None of the above Q. 6 A certain linear time invariant ytem ha the tate and the outut equation given below Xo X > Xo H > u H> + X H > H y X dy 8B: X D, If X(), X(), u(), then i dt t (A) (B) (C) (D) None of the above ***********

26 OLUTION ol. ol. Otion (B) i crect. From the given lot, we obtain the loe a log log loe log w log w From the figure log 8dB log db and w rad/ w rad/ o, the loe i loe 8 log log 4 db/ decade Therefe, the tranfer function can be given a ^ h k at w jw ^ h k k w In decibel, log j ^ wh log k, k 9. 8 Hence, the Tranfer function i. ^ h k 9 8 Otion (A) i crect. F the given F, we have two fward ath P k ^h^ h^ h^h P k ^h^ h^h^h ince, all the loo are touching to the ath P k and P k o, D k Dk Now, we have D (um of individual loo) + (um of roduct of nontouching loo) Here, the loo are L ^ 4h^h4 L ^ 4h^ h 4 L ^ h^ h^ h L 4 ^ h^ h^h A all the loo L, L, L and L 4 are touching to each other o, D ^L+ L+ L+ L4h 4 4 ^ h

27 ol From Maon gain fmulae Y ^ h P k D k U ^ h D Otion (A) i crect. F the hown tate diagram we can denote the tate, a below ol. 4 o, from the tate diagram, we obtain o u o + ^h^h^h^ hu+ ^h^h^ h o + + u and y ^ h^h+ ^h^h^ h+ ^h^h^h^h^hu + u Hence, in matri fm we can write the tate variable equation o > o H > u H> + H > H and y 8 B > + u H which can be written in me general fm a X o > X + H > H y 8 BX+ u Otion (A) i crect. From the obtained tatevariable equation e have A > H o, I A > + + H and ^I Ah + ^ + h > + H R V + ^ + h + T X Hence, the tate tranition matri i obtained a e At L ^IAh ZR V_ ] + b L e [ ` > t th te e ] ^ + h + b \ T Xa

28 ol. 5 ol. 6 ol. 7 Otion (C) i crect. iven, oen loo tranfer function ^ h a a + + By taking invere Lalace tranfm, we have gt ^ h e t Comaring with tandard fm of tranfer function, Ae t/ t, we get the oen loo time contant, t ol Now, we obtain the cloed loo tranfer function f the given ytem a ^ h H a ^ h a + ^ h + + a + ^ a + h Taking invere Lalace tranfm, we get ^ka + t h ht ^ h ka. e o, the time contant of cloed loo ytem i obtained a t cl ka +, t cl (aroimately) ka Now, given that k a reduce oen loo time contant by a fact of. i.e., t cl t ol, ka Hence, k a Otion (C) i crect. ( + 9)( + ) () ( + )( + )( + 4) ( w + 9)( jw+ ) ( jw+ )( jw+ )( jw+ 4) The teady tate outut will be zero if jw ( ) w + 9 & w rad/ Otion (A) i crect. ( + ) Y () [ R ( ) Y ( )] + a + + ( + ) ( + ) Y (); + + a + + E R () + a + + Y ()[ + a+ ( + k) + ( + k)] ( + ) R ( ) Y () ( + ) Tranfer Function, H () R () + a + ( + k) + ( + k) Routh Table :

29 ol. 8 ol. 9 ol. ol. F ocillation, a( + ) ( + ) a a + + Auiliary equation A () a + ( k + ) k + k + a ( k ) ( k + ) ( k + ) + j k+ jw j k+ w k + (Ocillation frequency) k and a Otion (D) i crect. eneral fm of tate equation are given a o A+ Bu yo C+ Du F the given roblem R a V R V A a, B a TR a X VR V R VT X AB a a a TR XT aa X VT R X V R aa V AB aa aa T XT X T X F controllability it i neceary that following matri ha a tank of n. R aa V U 6 B : AB : A B@ a o, a! T X aa! & a! a may be zero not. Otion (B) i crect. F given lot root locu eit from to, o there mut be odd number of ole and zero. There i a double ole at Now ole,,, zero Thu tranfer function H () () Otion (A) i crect. e have jw ( ) 5 + jw k ( + ) ( + )( + ) Here 5. Thu jw ( ) i a traight line arallel to jw ai. Otion (B) i crect. Here y and o dy d

30 y y > y H > H >H Now y u + y ( + ) u yo + y u o + u Drawing F a hown below o + u o [ ] + [] u ol. ol. Thu o [ ] + [ ] u y ; y y y > y H > H Here Otion (C) i crect. e have H () () ( + ) Now j ( w) Hj ( w ) jw( jw+ ) If w i hae cro over frequency + j ( w) Hj ( w ) 8c Thu 8c tan tan tan w a k 8c 9 tan (. w ) 45c tan (. w ) tan 45c.w w rad/ e Now j ( w) Hj ( w ) ww ( + ) At w w j ( w) Hj ( w ) ( + ) ain Margin log j ( w) Hj ( w) log b l 6 db Otion (D) i crect. From otion (D) TF H ()! ( + ) ( + )

31 ol. 4 Otion (B) i crect. From the given block diagram ol. 5 H () Y ()E $ + E () R ()H () R () Y () + E ( + ) E () : + D R ()Y E() R () Y ()...() ( + ) E () Y ()...() + From () and () Y() R ()Y ( + ) Y( ) R () Tranfer function Y () R () + Otion (B) i crect. Tranfer function i given a Y () H () X () + jw Hjw ( ) jw + Amlitude Reone Hjw ( ) w w + Phae Reone qh ( w ) 9c tan Inut t () w a k co t a k Outut yt () Hj ( w) t ( q h ) co t a k Hjw ( ) w w +, ( w rad/ ec) & 4 / Alternative : q h 9 a kc 6

32 ol. 6 ol. 7 ol. 8 o, tan w 6 tan w a k a k 6 w tan a k /, ( w rad/ ec) Otion (A) i crect. Initial loe i zero, o At cner frequency w.5 rad/ ec, loe increae by + db/decade, o there i a zero in the tranfer function at w At cner frequency w rad/ ec, loe decreae by db/decade and become zero, o there i a ole in tranfer function at w a + w k Tranfer function H () + a w k a +. k ( + ) + ( +. ) a. k Otion (D) i crect. Aign outut of each integrat by a tate variable o + o + u y tate variable rereentation o > + u H > H yo [. 5 5.] Otion (C) i crect. By maon gain fmula Tranfer function Y () P H () D / U () D

33 ol. 9 Fward ath given P ( abcdef ) 5 # # # # 5 # #. P ( abcdef ). Loo gain L ( cdc) L ( bcdb) # # D [ L+ L] : D + + D, D Y () o, H () P P D + D U () D : + : ( + ) + + ( + + ) Otion (D) i crect. teady tate err i given a R() e lim " + () C () R () (unit te unit) e lim " + () C () lim " C () e will be minimum if lim C () i maimum " In otion (D) lim C () lim + + " " o, e lim (minimum) " ol. ol. Otion (C) i crect. Thi comenat i roughly equivalent to combining lead and lad comenat in the ame deign and it i referred alo a PID comenat. Otion (C) i crect. Here A and B q AB q q q 8 B ABB q q q ince i ingular, ytem i comletely uncontrollable f all value of and q.

34 ol. Otion (B) i crect. The characteritic equation i + H () () ( + ) ( + ) F break away & break in oint differentiating above w.r.t. we have d ( + )( + ) ( + + )( ) d ( + ) Thu ( + )( + ) ( + + )( )! Let q d be the angle of dearture at ole P, then ol. ol. 4 ol. 5 ol. 6 qd q+ qz+ qz 8c q d 8c ( q+ qz+ q) 8c ( 9c+ 8 45c) 45c Otion (B) i crect. F underdamed econd der reone T () k n w where < + wn + wn Thu (A) (B) may be crect F otion (A) w. and w. 59. n n " F otion (B) w 9. and w n n " Otion (B) i crect. The lot ha one encirclement of igin in clockwie direction. Thu () ha a zero i in RHP. Otion (C) i crect. The Nyzuit lot interect the real ai ate.5. Thu. M. log log db And it hae margin i 9c. Otion (C) i crect. Tranfer function f the given ole zero lot i: ( + Z)( + Z) ( + P)( + P) From the lot Re (P and P )>(Z and Z ) o, thee are two lead comenat. Hence both high a filter and the ytem i high a filter.

35 ol. 7 ol. 8 Otion (C) i crect. Percent overhoot deend only on daming ratio,. e M If M i ame then i alo ame and we get co q Thu q contant The otion (C) only have ame angle. Otion (C) i crect. e labeled the given F a below : ol. 9 From thi F we have o g+ b+ m o g+ a o b a+ u R V R VR V R V g b u Thu g a + e u o b a T X T XT X T X Otion (D) i crect. P 5 w n, " Undamed rah + 5 Q 6 w, > n " Overdamed rah 4 R 6 w, n " Critically rah 7 w n 7, < " underdamed rah ol. ol. Otion (C) i crect. The characteritic equation of cloed lo tranfer function i + H () () a 4 + a ( a + ) + 4 Thi will be table if ( a+ ) > " a >. Thu ytem i table f all oitive value of a. Otion (C) i crect. The characteritic equation i + ()

36 ubtituting z we have z 5 + 5z 4 + 6z + z + z+ The routh table i hown below. A there are two ign change in firt column, there are two RH ole. z 5 6 z 4 5 z z 4 z 7 4 z ol. ol. Otion (A) i crect. F underdamed econd der ytem the tranfer function i T () n w + wn + wn It eak at reonant frequency. Therefe Reonant frequency w w and eak at thi frequency m r r n 5 e have w r 5, and m r. Only otion (A) atify thee value. w n, where w r ` 5 4 j and m r 5 4 Hence atified Otion (B) i crect. The given circuit i a inverting amlifier and tranfer function i Vo Z ZCR ( ) + V R R F Q, F R, i Z Vo Vi Z Vo V i CR+ ( CR+ ) C ( CR+ ) ( CR+ ) # C R R ( CR+ ) R ( CR+ ) # ( C R + ) R ince RC> RC, it i lag comenat. PID Controller ol. 4 Otion (D) i crect. In a minimum hae ytem, all the ole a well a zero are on the left half of

37 the lane. In given ytem a there i right half zero ( 5), the ytem i a nonminimum hae ytem. ol. 5 ol. 6 ol. 7 Otion (B) i crect. e have v lim () H() " ( + D) lim " ( + ) Now characteritic equation i + H () () ( + D) lim " ( + ) Now characteritic equation i + H () () ( + D ) + ( + ) D 4 + ( + ) + Comaring with + wn+ wn we get w + D n D 9. Otion (D) i crect. e have T () 5 ( + 5)( + + ) 5 5 ` + ( + + ) 5j + + In given tranfer function denominat i ( + 5)[( +. 5) + 4 ]. e can ee eaily that ole at 5.! j i dominant then ole at 5. Thu we have aroimated it. Otion (A) i crect. () ( + )( ) The lead comenat C () hould firt tabilize the lant i.e. remove ( ) term. From only otion (A), C () can remove thi term ol. 8 Thu C () () Otion (D) i crect. F ufb ytem the characteritic equation i + () + ( + 7+ ) ( + 7+ ) + ( ) # ( + )( ) ( + ) Only otion (A) atifie. ( + )( + ) Point + j lie on root locu if it atify above equation i.e ( + j)[( + j) + 7( + j) + ) + ]

38 ol. 9 ol. 4 ol. 4 + Otion (D) i crect. At every cner frequency there i change of db/decade in loe which indicate ole at every cner frequency. Thu () ( + ) ` + j Bode lot i in ( + T) fm log 6 db w w. Thu 5 Hence () Otion (A) i crect. ( + )( +. 5) dw dt e have > dia H w u dt i + n dw w + i dt n...() and di a w i a + u...() dt Taking Lalace tranfm (i) we get w () w() I () a ( + ) w( ) Ia ()...() Taking Lalace tranfm (ii) we get Ia () w() Ia () + U() w () ( I ) () + U () a ( )( + ) w() + U() From () w () [ + + ] w() + U() ( + + ) w( ) U() w() U () ( + + ) Otion (A) i crect. e have t o () A() t Let A q r F initial tate vect () Thu d t dt e > d t ( e ) H dt t the ytem reone i t () q r t e > th e () e q > () H 4e r q 4 r e get q and r 4...(i)

39 F initial tate vect () Thu d t dt e > d t ( e ) H dt t the ytem reone i t () q r t e > th e () e q > () H e r q r e get q and r...() olving () and () et of equation we get q r The characteritic equation li A l l + ll+ ( ) + l, Thu Eigen value are and Eigen vect f l ( l I A) X l l + + e have only one indeendent equation. Let, then, the Eigen vect will be Now Eigen vect f l ( li A) X l l + + e have only one indeendent equation. Let, then, the Eigen vect will be

40 ol. 4 ol. 4 ol. 44 ol. 45 ol. 46 Otion (D) i crect. A hown in reviou olution the ytem matri i A Otion (D) i crect. iven ytem i nd der and f nd der ytem.m. i infinite. Otion (D) i crect. Otion (D) i crect. If the Nyquit olt of j ( w) Hj ( w ) f a cloed loo ytem a through (, j) oint, the gain margin i and in db M log db Otion (B) i crect. The characteritic equation i + H () () ( + ) + + a a + ( + ) + + The Routh Table i hown below. F ytem to be ocillaty table a( + ) ( + ) a a + + Then we have a () At rad/ec we have jw " w 4, Thu 4a+ +...() olving (i) and (ii) we get and a 75.. ol a + ( + a ) ( + ) a + Otion (D) i crect. The tranfer function of given comenat i c () + T + T Comaring with c () + at we get a + T The maimum hae ift i T >

41 ol. 48 ol. 49 ol. 5 ol. 5 f tan ma f ma 6 a a tan Otion (A) i crect. ( I A) ( I A) tan + + > H co t in t At f () t e L [( IA)] in t co t Otion (C) i crect. e have () a + + j ( w ) tan ( wa) ince PM i i.e. 45c, thu 4 + +( jw ) 4 g w g " ain cro over Frequency + tan ( w a) 4 g tan ( w a) 4 g a g w At gain croover frequency j ( w ) Thu + a w w g g + wg g g (a aw ) w () 4 Otion (C) i crect. F a 84. we have () Due to ufb ytem H () and due to unit imule reone R (), thu C () R () () () Taking invere Lalace tranfm ct () ( t+84. ) u( t) At t, c( ec) Otion (D) i crect. The tranfer function of a lag netwk i T () + T + bt Tjw ( ) + w T + wbt g b > ;T >

42 ol. 5 ol. 5 ol. 54 ol. 55 ol. 56 ol. 57 ol. 58 and + Tj ( w ) tan ( wt) tan ( wbt) At w, Tjw ( ) At w, + Tj ( w ) tan At w, Tjw ( ) b At w, + Tj ( w ) Otion (C) i crect. e have X o AX + BU where l i et of Eigen value and o C + DU where m i et of Eigen value If a liner ytem i equivalently rereented by two et of tate equation, then f both et, tate will be ame but their et of Eigne value will not be ame i.e. X but l! m Otion (A) i crect. Deite the reence of negative feedback, control ytem till have roblem of intability becaue comonent ued have nonlinearity. There are alway ome variation a comared to ideal characteritic. Otion (B) i crect. Otion (C) i crect. The eak ercent overhoot i determined f LTI econd der cloed loo ytem with zero initial condition. It tranfer function i n T () w + wn + wn Tranfer function ha a air of comle conjugate ole and zeroe. Otion (A) i crect. F ram inut we have R () Now e lim E() But " k v R () lim lim " + () " + () e lim 5% " () lim e () " k v i finite f tye ytem having ram inut. Otion (A) i crect. Finite Otion (C) i crect. Any oint on real ai of i art of root locu if number of OL ole and zero to right of that oint i even. Thu (B) and (C) are oible otion. The characteritic equation i + H () () ( ) + ( + )

43 ol. 59 ol. 6 + F break away & break in oint d ( )( + ) + + d + + which give, Here mut be the break away oint and mut be the break in oint. Otion (D) i crect. () e ( + ) jw jw ( ) e jw( jw+ ) jw ( ) w w + 4 Let at frequency w g the gain i. Thu w ( w + 4) g g g 4 g g w + 4w 9 w. 66 w g 6. rad/ec Now + j ( w ) w tan w Let at frequency w f we have + H 8c w wf f tan wf wf + tan wf wf wf + c ` j m 5wf wf 4 5w f. w f 6. rad Otion (D) i crect. The gain at hae croover frequency w f i j ( w g ) wf ( wf + 4). 6( ) j ( w g ) 7..M. log j ( wg) log db ince.m. i negative ytem i untable. The hae at gain cro over frequency i g + j ( w g ) w w g tan

44 ol. 6 ol. 6 ol. 6.6 tan 6. # 465. rad 66. 5c PM 8c + + j ( wg) 8c 66. 5c 86. 5c Otion (D) i crect. The oen loo tranfer function i ( + ) H () () ubtituting jw we have ( + jw) j ( w) Hj ( w ) w + j ( w) Hj ( w ) 8c + tan w The frequency at which hae become 8c, i called hae croover frequency. Thu 8 8c + tan wf tan w f w f The gain at w f i j ( w) Hj ( w ) + w w Thu gain margin i and in db thi i. Otion (C) i crect. Centroid i the oint where all aymtote interect. Real of Oen Loo Pole Real Part of Oen Loo Pole No.of Oen Loo Pole No.of Oen Loo zero. Otion (C) i crect. The given bode lot i hown below...() ol. 64 At w change in loe i + db " zero at w At w change in loe i db " ole at w At w change in loe i db " ole at w ( + ) Thu T () ( + )( + ) Now log ".. ( + ) ( + ) Thu T () ( + )( + ) ( + )( + ) Otion (C) i crect.

45 ol. 65 ol. 66 ol. 67 e have rt () u() t R () Now H () + C () H () $ R () $ + ( + ) C () ct () 5[ e t ] The teady tate value of ct () i 5. It will reach 99% of teady tate value reache at t, where 5[ e t ] 99. # 5 e t 99. e t. t ln. t. ec Otion (A) i crect. Aroimate (comarable to 9c) hae hift are Due to ole at. Hz " 9c Due to ole at 8 Hz " 9c Due to ole at 8 Hz " Due to zero at 5 Hz " 9c Due to zero at Hz " Due to zero at Hz " Thu aroimate total 9c hae hift i rovided. Otion (C) i crect. Maon ain Fmula T () k k In given F there i only one fward ath and oible loo. abcd (um of indivudual loo) (um of two non touching loo) ( L + L + L ) + ( L L ) Non touching loo are L and L where LL bedg C () Thu R () ( be + cf + dg) + bedg abcd ( be + cf + dg) + bedg Otion (A) i crect. e have A Characteritic equation i

46 [ li A] l + l + ( l+ )( l+ ) l + 5l+ 4 Thu l 4 and l Eigen value are 4 and. Eigen vect f l 4 ( li A) X l + l + + e have only one indeendent equation. Let, then, the Eigen vect will be Now Eigen vect f l ( li A) X l + l + e have only one indeendent equation Let, then. Thu Eigen vect will be Digonalizing matri M Now M ` j Now Diagonal matri of in At i D where in( lt) in( 4t) D in( l t) in( t) Now matri B in At MDM l in( 4t) `j in( t) in(4 t) in( t) in( 4t)in(t) `j in( 4t) + in( t) in(4t)in(t) in(4t)in(t) in( 4t)in(t) `j in( 4t)in(t) in( 4t) + in( t)

47 in( 4t) + in( t) in( 4t) + in( t) ` j in( 4t+ in( t) in( 4t) + in( t) ol. 68 ol. 69 Otion (A) i crect. F ufb ytem the characteritic equation i + () + () + ( + + )( + ) The routh table i hown below. F ytem to be table, ( 4) < and < 7 / Thi give < < / Otion (B) i crect. 5 4 e have P () The routh table i hown below. If e " + then e + e 5e 4 44 i oitive and e e+ i negative. Thu there are two ign change in firt column. Hence ytem ha root on RH of lane e e + e 5 5e 4e44 e+ ol. 7 Otion (D) i crect. e have u + and Y [ ] u + Here A, B and C [ ] The controllability matri i Q C [ B AB]

48 ol. 7 det Q C! Thu controllable The obervability matri i T T T Q [ C A C ]! det Q! Thu obervable Otion (B) i crect. ( I A) ( I A) ( ) ( ) ( ) > H ol. 7 ol. 7 ol. 74 ol. 75 e At L [( IA)] t e t e Otion (A) i crect. Z PN N " Net encirclement of ( + j) by Nyquit lot, P " Number of oen loo ole in right hand ide of lane Z " Number of cloed loo ole in right hand ide of lane Here N and P Thu Z Hence there are no root on RH of lane and ytem i alway table. Otion (C) i crect. PD Controller may accentuate noie at higher frequency. It doe not effect the tye of ytem and it increae the daming. It alo reduce the maimum overhoot. Otion (D) i crect. Maon ain Fmula T () k k In given F there i only fward ath and oible loo L, L 4 and L where L and L are nontouching C () Thi R () ( loo gain) + air of non touching loo ^ h ^ h ^ h ( + 7) Otion (D) i crect. e have

49 + H () () + ( + )( + ) ( ) d ( + + 6) d which give! 7.784, The location of ole on lane i ol. 76 ol. 77 ol. 78 ol. 79 ince breakoint mut lie on root locu o. 748 i oible. Otion (A) i crect. The given bode lot i hown below At w. change in loe i + 6 db " zeroe at w. At w change in loe i 4 db " ole at w At w change in loe i db " ole at w Thu (. + ) T () ( + ) ( + ) Now log Thu T () (. + ) ( + ) ( + ) Otion (B) i crect. The characteritic equation i ( +. ) ( + ) ( + ) Comaring with + wn+ wn we get w n 4 and w n 4 Thu Critically damed t 4 4 w n # Otion (B) i crect. Otion (C) i crect. e have

50 ol. 8 ol. 8 ol. 8 ol. 8 o () o and () A ( I A) ( I A) ( ) ( ) > + ( ) H > + ( ) t At e L [( I A) ] e t t te e t t e At e t () e # [ ( t)] t t t te e te Otion (C) i crect. The characteritic equation i k Comaring with + wn+ wn we have we get w n and w 6 n #. 5# 6w iven 5. 6 & 6 Otion (B) i crect. Any oint on real ai lie on the root locu if total number of ole and zero to the right of that oint i odd. Here 5. doe not lie on real ai becaue there are total two ole and zero ( and ) to the right of 5.. Otion (D) i crect. From the ereion of OLTF it may be eaily ee that the maimum magnitude i.5 and doe not become at any frequency. Thu gain cro over frequency doe not eit. hen gain cro over frequency doe not eit, the hae margin i infinite. Otion (D) i crect. e have t o () t () + ut ()...(i) Taking Lalace tranfm we get X() X () + U () ( + ) X( ) U() U () X () ( + ) Now yt () 5t. ( ) Y () 5X. ( ) 5. # U ( ) Y () + Y () U () ( + ) H

51 ol. 84 ol. 85 ol. 86 Otion (D) i crect. From Maon gain fmula we can write tranfer function a Y () R () ( + ) ( ) F ytem to be table ( ) < i.e. > Otion (B) i crect. The characteritic equation i ( + )( + ) + + Comaring with + wn+ wn we get w n and w n Thu Overdamed F overdamed ytem ettling time can be determined by the dominant ole of the cloed loo ytem. In given ytem dominant ole conideration i at. Thu and T T 4 4 ec T Otion (B) i crect. Routh table i hown below. Here all element in rd row are zero, o ytem i marginal table ol. 87 Otion (B) i crect. The oen loo tranfer function i H () () ( + + ) ubtituting jw we have j ( w) Hj ( w ) jw( w + jw+ ) + j ( w) Hj ( w ) tan w ( w ) The frequency at which hae become 8c, i called hae croover frequency. wf Thu 8 9 tan w f wf 9 tan w w f f

52 ol. 88 ol. 89 ol. 9 ol. 9 ol. 9 w f rad/ec The gain margin at thi frequency w f i M log j ( w ) Hj ( w ) f f log ( w ( w ) + w f f f log Otion (A) i crect. Z PN N " Net encirclement of ( + j) by Nyquit lot, P " Number of oen loo ole in right had ide of lane Z " Number of cloed loo ole in right hand ide of lane Here N ( encirclement in C direction and other in CC) and P Thu Z Hence there are no root on RH of lane. Otion (D) i crect. Take off oint i moved after a hown below Otion (D) i crect. If root of characteritic equation lie on negative ai at different oition (i.e. unequal), then ytem reone i over damed. From the root locu diagram we ee that f < <, the root are on imaginary ai and f < < 5 root are on comle lain. F > 5 root are again on imaginary ai. Thu ytem i over damed f # < and > 5. Otion (C) i crect. The characteritic equation i + + Comaring with + wn+ wn we get w n and w n w n and ince < thu ytem i under damed Otion (C) i crect. From F we have I () V i() + HI()...() I () I ()...() V () I ()...() Now alying VL in given block diagram we have Vi () I() Z() + [ I() I()] Z()...(4)

53 ol. 9 ol. 94 [ I() I()] Z() + I() Z() + I() Z4()...(5) From (4) we have Vi () I()[ Z() + Z( )] I() Z( ) () I () V Z i + I Z () Z () + Z () + Z ()...(6) From (5) we have I() Z ( ) I()[ Z() + Z() + Z4()]...(7) I() Z() I () Z() + Z() + Z4() Comaring () and (7) we have Z() Z() + Z() + Z4() Comaring () and (6) we have Z() H Z() + Z() Otion (B) i crect. F unity negative feedback ytem the cloed loo tranfer function i () CLTF + 4, + () ()" OLain + () + 7+ () () () F DC gain, thu Thu () 4 9 Otion (C) i crect. From the Block diagram tranfer function i () T () + H () () here ( ) () ( + ) and H () ( ) The Characteritic equation i + H () () ( ) + ( ) ( + ) ( + ) + ( ) ( + ) + 4( ) Routh Table i hown below. F ytem to be table + k >, and 4+ 4k > and 4 4k >. Thi give < < A er quetion f # <

54 + k 4+ 4k 4 4k 4+ 4k ol. 95 ol. 96 Otion (B) i crect. It i table at all frequencie becaue f reitive netwk feedback fact i alway le than unity. Hence overall gain decreae. Otion (B) i crect. The characteritic equation i + a + k+ The Routh Table i hown below F ytem to be table a > and a > a Thu a > and a > ol. 97 ol. 98 a a a Otion (B) i crect. Cloed loo tranfer function i given a T () by comaring with tandard fm we get natural freq. w A 9 w n w n 4 Daming fact 4 / # F econd der ytem the etting time f ercent band i given by t 4 4 w n / 4 # Otion (D) i crect. iven loo tranfer function i H () () ( + ) j ( w) Hj ( w ) jw( jw+ ) Phae cro over frequency can be calculated a fw ( ) at 8c w w o here fw ( ) 9c tan ( w) 9c tan ( w) 8c tan ( ) 9c w w

55 ol. 99 ol. ol. ol. ain margin log at w j ( w) Hj ( w) w M.. log e j ( w) Hj ( w) o j ( w) Hj ( w ) w w + o M.. log b l Otion (A) i crect. Here A, B and C [ ] The controllability matri i Q C [ B AB] det Q C! Thu controllable The obervability matri i T T T Q [ C A C ]! det Q! Thu obervable Otion (D) i crect. we have H () () ( + ) j ( w) Hj ( w ) jw( jw+ ) ain cro over frequency j ( w) Hj ( w) at w wg w w + w ( w + ) 4 w + w ( w + 4)( w ) w and w 4 which give w, w! w g fw ( ) at 9 tan ( wg) w w g 9 tan Phae margin 8 +fw ( ) Otion (B) i crect. at w wg 8 5 c Otion (C) i crect. Cloedloo tranfer function i given by T () an an + n n + a a + a n n

56 ol. ol. 4 ol. 5 ol. 6 ol. 7 an+ an n n a... an an+ an n n + a +... a n Thu H () () an an + n n + a +... an F unity feed back H () Thu () an an + n n + a +... an teady tate err i given by E () lim R () " + H () () f unity feed back H () Here inut R () ( unit Ram) o E () lim " + () n n lim a... an " n a n a n an a Otion (B) i crect. Otion (A) i crect. Otion (A) i crect. Alying Routh criteria # n There i no ign change in the firt column. Thu there i no root lying in the lefthalf lane. Otion (A) i crect. Techometer act like a differentiat o it tranfer function i of the fm k. Otion (A) i crect. Oen loo tranfer function i () ( + ) teady tate err R() E () lim " + H () () here R () inut H () (unity feedback) R ()

57 ol. 8 ( ) o E () lim + lim " " ( + ) Otion (B) i crect. Fig given below how a unit imule inut given to a zeroder hold circuit which hold the inut ignal f a duration T & therefe, the outut i a unit te function till duration T. ol. 9 ol. ol. ht () ut ()ut ( T) Taking Lalace tranfm we have H () e T T 6 Otion (C) i crect. Phae margin 8c + qg where q g frequency. Here q 5c g o P.M 8c 5c 55c value of hae at gain croover Otion (B) i crect. Oen loo tranfer function i given by ( +. 5) H () () ( + )( + ) Cloe looed ytem i of tye. It mut be noted that tye of the ytem i defined a no. of ole lie lying in OLTF. Otion (D) i crect. Tranfer function of the hae lead controller i TF. T ( Tw) j ( Tw) j Phae i fw ( ) tan ( Tw) tan ( Tw) fw ( ) tan Tw Tw ; + T w E fw ( ) tan Tw ; + T w E F maimum value of hae dfw ( ) dw T w Tw at igin

58 ol. ol. ol. 4 ol. 5 ol. 6 o maimum hae i f ma tan Tw ; + T w E at Tw tan tan > H c + # ; E Otion (A) i crect. j ( w) Hj ( w ) encloe the (,) oint o here j ( w ) Hj ( w ) > w Phae cro over frequency ain Margin log o gain margin will be le than zero. j ( w ) Hj ( w ) Otion (B) i crect. The denominat of Tranfer function i called the characteritic equation of the ytem. o here characteritic equation i ( + ) ( + ) Otion (C) i crect. In ynchro err detect, outut voltage i rotional to [ w ( t)], where w () t i the rot velocity o here n Otion (C) i crect. By maon gain fmulae y P / D k k D Fward ath gain P 5# # D ( # ) D o gain y # 5 Otion (C) i crect. By given matri equation we can have Xo d + dt Xo d + + m dt dy dt y [ ] > + H dy dt dy dt t d + d dt dt + m () + () m + ***********