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1 Department of Electrical Engineering PhD. Admission Test Full Marks: 90 Time 90 minutes Date: NAME: Appl. No: Write your answer on the question paper ONLY. All questions carry equal marks. PART A. Answer ALL 8 questions. I. Basic Electrical Engineering, Mathematics, Signals and Network. Obtain the solution of the differential equation () ()= Compute the value of the integral. +()= with initial condition 3. The characteristic equation of a matrix is given as ()=det( )= =0. Compute the inverse of the matrix as a function of matrix polynomial in. 4. In a tank containing 0 fishes, there are three yellow and seven black fishes. Select three fishes at random. What is the probability that exactly one yellow fish gets selected? Answer: 5. A dc potentiometer is designed to measure voltage up to 2 with a slide wire of length 800 mm. A standard cell of.8 obtains balance at 600 mm. A test cell is seen to obtain balance at 680 mm. Find the value of the test cell. Answer: 6. Suppose we have an IIR filter with feed-forward coefficients 4, 5, 6 and feedback coefficients 2, 3. (i) Find the transfer function of the IIR filter. (ii) Is this IIR filter stable? Answer: (i) (ii)

2 7. Obtain the value of the periodic signal shown in fig. Q7. Answer: Fig.Q7 8. A single phase load is connected between & terminals of a 45, balanced 3,4 wire system with phase sequence. A wattmeter is connected in the system as shown in fig.q8. Find the reading of the wattmeter when the power factor of the load is 0.8 lagging. Answer: Fig.Q8 9. A CRO probe has an impedance of 500 in parallel with a capacitance of 0. The probe is used to measure the voltage between as shown in fig.q9. Find the rms value of the voltage measured by the CRO. Answer: Fig.Q9 0. Determine the steady state current through the 5 resistor of the circuit shown in fig.q0. 5 Ohm Fig.Q0 2

3 . Find the parameter of the circuit shown in fig.q. Z i b i V V 2 Y a - - Fig.Q Answer: A = C = B= D = 2. The system shown in fig.q2 is excited by a signal ()=. Obtain the steady state response of the system output y(t). Answer: Fig.Q2 3. A source which can be represented by a voltage source of 8V r.m.s. in series with an internal resistance of 2kΩ is connected to a 500Ω load resistance through an ideal transformer. Calculate the value of turns ratio of the ideal transformer for which maximum power is supplied to the load. 4. A dc shunt motor operating at an armature terminal voltage of 25V is observed to be operating at a speed of 000 rpm at ideal no load condition. By inserting a 5Ω resistance in series with the shunt field, the motor speed becomes 050 rpm at the same ideal no load condition. Calculate the resistance of the shunt field. 5. A square signal () is shown in fig.q5 and it is approximated with a real sinusoidal signal ()=. It is assumed that both the signals having the same time period. Find the amplitude of the sinusoidal signal so that the energy of the error signal (i.e. e(t)= (f(t) A sint) over the time period is minimized, that is minimize its size. Answer: Fig.Q5 3

4 6. In fig. Q6, the switch S has been closed in position for a very long time and then opens at t=0. What is the value of current through 40Ω resistor at t= Ω 0µ F 40Ω 0 ma S t=0 Fig.Q6 7. In the circuit shown in Fig. Q7, calculate the current through the 0Ω resistance. 25V V z =5.6V 5Ω V be =0.6V V be β=00 kω 0Ω Fig.Q7 8. In the operational amplifier circuit shown in Fig. Q8, calculate the output voltage V 0. +V 0kΩ 50kΩ +5V +V 0kΩ 20kΩ -5V V 0 Fig.Q8 END 4

5 PART B: Section I Machine Drives & Power Electronics (MDPE). A 3-phase, 45V, 6-pole, 50Hz. Star-connected synchronous motor has an emf of 520V(Line to line). The stator winding has a synchronous reactance of 2 per phase and the motor develops a torque of 00 Nm. The motor is operating from a 45V, 50Hz bus, (i) Calculate the current drawn from the supply and the power factor, (ii) if the excitation is reduced by 0% supplying the same load, find the current drawn from the supply and the power factor. (4+3)+(4+4)=5 2. A single phase full bridge diode rectifier operates from a 220V, 50 Hz single phase supply and delivers power to a 50 Ω resistance connected across the output capacitor. The voltage across the output capacitor may be assumed to be ripple free. i) Find out the value of the filter inductor (with negligible resistance) on the DC side so that the rectifier operates at the boundary between continuous and discontinuous conduction modes. (8) ii) State with proper justification what will happen if the load resistance increases by 0%. Also draw approximately the inductor current waveform in this case. (3+4) 3. (a) The dc-dc converter shown in the figure below has the following circuit parameters: V g =48V; L P =80µH; L S =5µH; C=470µF; R=25Ω; D=0.75; T S =0µSec. L P is the self-inductance of the primary coil and L S is the self-inductance of the secondary coil and the coils are perfectly coupled. Find the steady state output voltage V o, and the peak current through the diode. Assume that all the circuit components are ideal. (6) Do Vo LP LS C R Vg G S Q V GS 0DT s t T s 3. (b) The dc-dc converter shown in figure below has the following circuit parameters: V g =400V; V o =40V; N P : N S =4:; L f =0.25 mh; C f =000µF; R=4Ω; F S =00kHz (switching frequency). The duty ratio of each switch is equal to 0.5. Consider steady state operation and assume that all the components are ideal. Sketch V GS, V G2S2,, and ; (use same time scale to plot all the waveforms in a single page, give the numerical values and corresponding time durations ). (9)

6 Lf Ig G Q G2 Q2 D ILf D2 Vo Vg S S2 Ip Cf R G3 S3 Q3 G4 S4 Q4 Np Ns D3 D4 4. A 3-ph two level converter is operated with 80 o mode and feeding a three phase passive load. The configuration is shown in the figure below. I dc IL A 400V V dc O A B C C N B (a) Draw the phase voltage (v AN, v BN, v CN ) and line voltage (v AB, v BC, v CA ) waveforms. (3) (b) Find the rms value of the fundamental component of line voltage(v AB, v BC, v CA ) and line current (I L ).Note that the load is drawing 0kVA with 0.7 power factor. (4) (c) Find the magnitude of first three dominant harmonics present in the line voltage. (6) (d) Calculate the average DC current (I dc ) if the load is drawing 0kVA with 0.7 power factor. (2) 5. A 3-phase two level converter is connected to a 400V system as shown in the next figure. The load is drawing 0kVA with a lagging power factor of 0.5. The converter supplies only the reactive power to achieve unity power factor at the source side. Answer the following questions.

7 A A C B B PASSIVE LOAD C L I c C Vdc a b c v L Figure: A 3-ph two level converter is connected between supply and load (a) Find the required DC voltage for this application considering 5% reactor and sine-triangle PWM. Consider the maximum modulation index is 0.9. (0) (b) Calculate the voltage rating of the switches without considering any margin. (3) (c) Draw the steady state instantaneous voltage and current waveforms (e.g. v ax, I c, v L, v ax ). Where x is a virtual common point. (2) 6. An open loop v/f controlled induction machine (IM) drive is shown in the Figure below. The rating and parameters of the IM is given below. Rated power (P) =45 kw, Rated voltage at stator side (V sr ) =440V, Rated frequency (f r )=50Hz, No. of poles = 6, R =0.Ω, R 2 =0.2Ω, X =0.3Ω, X 2 =0.3Ω, X m =5Ω. The profile of the load torque is: T L A 2 r V dc V r f * V/f profile V * 3-ph inverter ω r V o f o f r (a) Find out the droop voltage (Vo). Calculate the slope of the v/f profile considering f o =5Hz. [2] (b) If the rated speed is 970rpm, determine the constant A of the load profile. [8] (c) Determine the speed of the machine at f*=40hz. [5]

8 PART B: Section II CONTROL SYSTEM ENGINEERING. a) A plant Gs () with a variable parameter is compensated using H() s in the feedback path. Define: ( G / G) S, ( / ) G S T T ( / ) T ( / ) wheret is the closed loop transfer function, and is the incremental variation operator. Relate S to S. [6] T G (b) Suppose amplifier units of specification 0 0% are only available. Show how an amplifier of specification 0 2.5% can be derived using multiple units of the same and appropriate feedback. [9] 2. Consider the feedback connection in Fig. _ u G(s) y (.) Fig. 3 s y with G( s), ( y) where is a positive constant. Using the describing 2 s s 3 function method find frequency and amplitude of periodic solution that exists in the feedback connection. [5] 3. Consider an LTI system described by the following state-space model 2 2 x x, 0 2 u 2 x x x y. x2 x (0), 2 (0) 0 x (a) Write above state-space model in the modal form. [0] (b) From the derived modal form, comment on the state controllability and state observability of the system. [5]

9 4. (a) The loop transfer function of a unity-feedback control system is given by, G s H s s K 2 s 3s 2s 2 Draw the root locus for K 0. Calculate the value of K for which the system becomes oscillatory. [9] b) Find the transfer function of a plant whose Bode plot is shown in Fig. 2. [6] db Magnitude log rad/sec 2 db 20 db/decade +20 db/decade Fig (a) Consider a system with the transfer function: ( s 2) G( s) 2 ( s )( s 2s 2) Obtain the phase variable canonical form realization of this system. [5] (b)draw the signal flow graph corresponding to this realization. [5] (c) We wish to apply state feedback control to this system to locate the closed loop poles at s 2, s 2 j. Determine the gain vector to achieve this specification. [5] 6. (a) Using the zero-order hold equivalent method, derive the equivalent discrete-time transfer function of a continuous time plant G(s)=/[s(s+0)] for sampling time T=0. sec. [7] (b) For the above continuous time plant (given in 6.(a)), using bilinear transformation, derive discrete-time transfer for the same sampling period and compare discrete-time transfer functions using both the methods. [8]

10 PART B: Section III POWER AND ENERGY SYSTEMS Q. A generator supplies a load over a transmission line having transformer at both ends as shown in the one-line diagram in Figure. Transformers T and T 2 ratings are shown in Figure. Vs=230 0 V T Xline= 3 Ω T2 Zload=+j0.3 Ω 25 kva 240/480 V Xeq=0.2 p.u 5 kva 460/20 V Xeq=0.2 p.u Figure : Single phase circuit Draw the per-unit circuit and determine the per-unit impedances and the per-unit source voltage. Then calculate the load current both in per-unit and in amperes. Transformer winding resistances and shunt admittance branches are neglected. Q 2. A): For the 4-bus power system shown in the Fig.Q2A with on-line diagram, determine the Y BUS. Neglect the shunt admittances at the buses and mutual couplings between the lines. The line series impedances are given in Table-. Fig. Q2A Table- 2 z 23 3 Line(bus to bus) Impedance pu values -2 z j.2-4 z j z j0.6 z 2 z z j0.8 z 4 4

11 B): Referring to the modified one line diagram of Fig.Q2A, as shown in Fig.Q2B. In this modified system, the shunt admittance of -j0.20 pu between bus -2 and the line series impedance of (0.6+ j0.2) pu between bus -3 is also taken into consideration. Determine the modified bus admittance matrix for the power system shown in Fig.Q2B, considering the remaining parameters to be the same. y a z z 34 z 2 3 y b Fig. Q2B 2 3 z 23 z 4 4 Q 3. A 20 km long three phase line has four no. (4/0) wires of. cm dia spaced horizontally.4 m apart in a plane. If the currents in the lines be I a = j 40 A, I b = -25 +j 60 A and I c = 45 j 00 A Fourth neutral wire carries zero current. Find the flux linkages in the neutral wire and determine voltage drop in neutral wire. Q 4. A 25 MVA, 3.2 kv, 50 Hz alternator with solidly grounded neutral has a subtransient reactance of 0.25 pu. The negative and zero sequence reactances are 0.35 and 0. pu respectively. A single line to ground fault occurs in phase-b of the unloaded alternator. Determine I b and V bc magnitudes at the terminal. Q 5. For the following system, obtain one iteration for load flow solution by Gauss Seidel or Newton- Raphson method. For system data refer table and the Y bus matrix below. 2 3 Figure for load flow problem

12 Table Operating parameters in pu Bus Type V P g Q g P L Q L slack PV PQ Y bus = j5 j0 j5 j0 j5 j5 pu j 5 j5 j0 Q 6. A 00 MVA 50 Hz turboalternator operates at no load at 3000 rpm. A load of 25 MW is suddenly applied to the machine and the steam valves to the turbine commence to open after 0.6 secs due to the time-lag in the governor system. Assuming inertia constant H of 4.5 kw-sec per KVA of generator capacity, calculate the frequency to which the generated voltage drops before the steam flow commences to increase to meet the new load.

13 PART B: Section IV INSTRUMENTATION AND SIGNAL PROCESSING.a. Match the following (flow meters and the characteristics) (6) Flowmeter Technology Characteristics A. Venturimeter. High permanent pressure loss B. Vena Contracta Orifice meter 2. Variable Area Principle C. Clamp-on Ultrasonic 3. Open channel Measurement D. Weir 4. Constant Temperature Operation E. Electromagnetic 5. Nonintrusive Measurement F. Hot Wire Anemometer 6. High Physical Strength 7. Axisymmetric Flows.b. A cylindrical float (volume: 500 mm 3, diameter: 5 mm) is used with a tapered glass tube to estimate the volumetic flow rate of water (density: 000 kg/m 3 ) in a pipe. Tube dimensions: minimum diameter = 8 mm, included angle of taper = 5, vertical height = 250 mm. The float is made from aluminium of relative density 2.7. Assume C D =0.8 (i) Determine the range of volume-flow rate (Q) of this setup. (5) (ii) Is the estimated Q dependant on the variations in the density of water? Justify your answer (4) 2. An iron constantan thermocouple is to be used to measure temperatures between 0 and 300 o C. The e.m.f values are given as E 00 =5.268 mv, E 200 = 0.777mV., E 300 = 6.325mV (a) Find the non linearity at 00 o C and 200 o C as a percentage of full scale. If only these data are available then what will be the non-linearity specifications in your sensor data sheet. (4+2) (b) Between 0 o C and 30 o C the thermocouple e.m.f. is given by the expression Calculate a and a 2 (write the units). (5) (c) The emf is 2.5mV relative to a reference junction of 20 o C and the corresponding reference junction circuit voltage is mv. Use the result of (b) to estimate the measured junction temperature. (4) 3. A piezo electric crystal acting as a force sensor is connected by a short cable of negligible capacitance and resistance to a voltage detector of infinite bandwidth and purely resistive impedance of 0 M. a. Draw circuit diagram and use the crystal data below to calculate the system transfer function (you should represent the crystal with its electrical equivalent circuit). (2+3) b. Find the static sensitivity of the system. (2) c. Sketch the approximate frequency response characteristic of the system (both phase and magnitude). (5) d. The time variation in the thrust of an engine is a square wave of period 0 ms. What is the fundamental frequency and harmonics present in the input signal? Explain (without detailed calculation) why the above system is unsuitable for this application. (3) Crystal Data: Charge sensitivity to force: = 2 pc/n; Capacitance = 00pF; Natural frequency = 37 khz; Damping ratio = 0.0

14 j 0 j 0 Plot the magnitude and phase response of this filter? (5) 4(a) An orthogonal phase shifting filter has a frequency response given as He j 4b. Find the unit impulse response n j h of this orthogonal phase shifting filter e H? (0) 5a. Let X 0 be the zero frequency component of the DFT of a N-point sequence x n and also the mean N of the sequence? (0) of the n-point sequence is known. Can you find the length 5b. Find the DFT and one sided Z-transform of the following N-point sequence n,2,3,4, 5 x (5) 6a. You are provided with a N-point sequence xn,2,3,4, 5 h n,2,3,4, 5. Find the convolution x hn and the correlation x hn and the system response with the zero-point marked on the resulting sequence? (5) x 2 can be computed using only one N-point DFT operation? (0) 6b. Can you prove that the N-point DFT of two real-valued sequences x n and n

EE Branch GATE Paper 2010

EE Branch GATE Paper 2010 Q.1 Q.25 carry one mark each 1. The value of the quantity P, where, is equal to 0 1 e 1/e 2. Divergence of the three-dimensional radial vector field is 3 1/r 3. The period of the signal x(t) = 8 is 0.4