Download sample questions - Department of Electrical Engineering, IIT

Department of Electrical Engineering
PhD. Admission Test
Full Marks: 90
Time 90 minutes
Date: 02.12.2014
NAME: ____________________________ Appl. No: ____________________________
Write your answer on the question paper ONLY. All questions carry equal marks.
PART A.
Answer ALL 18 questions.
I. Basic Electrical Engineering, Mathematics, Signals and Network
݀‫)ݐ(ݔ‬
1. Obtain the solution of the differential equation ‫ ݐ݀ ݐ‬+ ‫ ݐ = )ݐ(ݔ‬with initial condition
‫(ݔ‬1) = 0.5.
Answer: ____________________________
ഏ
మ
2. Compute the value of the integral ‫׬‬଴ ܿ‫ ݏ݋‬ଷ ‫ݔ݀ ݔ‬.
Answer: ____________________________
3. The characteristic equation of a matrix ‫ܣ‬ଷ௫ଷ is given as ݂(ߣ) = det(ߣ‫ ܫ‬− ‫ߣ = )ܣ‬ଷ + ߣଶ + 2ߣ +
1 = 0. Compute the inverse of the matrix ‫ ܣ‬as a function of matrix polynomial in ‫ܣ‬.
Answer: ____________________________
4. In a tank containing 10 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 2ܸ with a slide wire of
length 800 mm. A standard cell of ݂݁݉ 1.18ܸ 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 ሼ4, 5, 6ሽ and feedback
coefficients ሼ2, 3ሽ. (i) Find the transfer function of the IIR filter. (ii) Is this IIR filter stable?
Answer: (i)____________________________ (ii)____________________________
1
7. Obtain the ࢘࢓࢙ value of the periodic signal shown in fig. Q7.
Fig.Q7
Answer: _______________
8. A single phase load is connected between ܴ & ܻ terminals of a 415ܸ, balanced 3∅, 4 wire
system with phase sequence ܴܻ‫ܤ‬. 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.
Fig.Q8
Answer: _______________
9. A CRO probe has an impedance of 500݇ࢹ in parallel with a capacitance of 10‫ܨ݌‬. 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.
Fig.Q9
Answer: _______________
5 Ohm
10. Determine the steady state current through the 5ࢹ resistor of the circuit shown in fig.Q10.
Fig.Q10
Answer: ____________________________
2
11. Find the ‫ ܦܥܤܣ‬parameter of the circuit shown in fig.Q11.
i1
Z
b
i2
+
V1
+
V2
Ya
-
-
Fig.Q11
Answer: A = ______________________
B= ______________________
C = ______________________
D =______________________
12. The system shown in fig.Q12 is excited by a signal ‫ݐ݊݅ݏ = )ݐ(ݑ‬. Obtain the steady state
response of the system output y(t).
Fig.Q12
Answer:____________________________
13. 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.
Answer: ____________________________
14. A dc shunt motor operating at an armature terminal voltage of 125V is observed to be operating
at a speed of 1000 rpm at ideal no load condition. By inserting a 5Ω resistance in series with
the shunt field, the motor speed becomes 1050 rpm at the same ideal no load condition.
Calculate the resistance of the shunt field.
Answer: ____________________________
15. A square signal ݂(‫ )ݐ‬is shown in fig.Q15 and it is approximated with a real sinusoidal
signal ‫ݐ݊݅ݏܣ = )ݐ(ݔ‬. It is assumed that both the signals having the same time period ܶ. Find the
amplitude ‫ ܣ‬of 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.
Fig.Q15
Answer:____________________________
3
16. In fig. Q16, the switch S1 has been closed in position “1” for a very long time and then opens
at t=0. What is the value of current through 40 Ω resistor at t= 0+
10 µ F
60 Ω
40 Ω
10 mA
S1
1
t=0
Fig.Q16
Answer: ____________________________
17. In the circuit shown in Fig. Q17, calculate the current through the 10Ω resistance.
5Ω
V z=5.6V
V be
25V
1kΩ
Vbe=0.6V
β=100
10Ω
Fig.Q17
Answer: ____________________________
18. In the operational amplifier circuit shown in Fig. Q18, calculate the output voltage V0.
+1V
10kΩ
50kΩ
+15V
V0
10kΩ
+1V
-15V
20kΩ
Fig.Q18
Answer: ____________________________
END
4
PART B: Section I
Machine Drives & Power Electronics (MDPE)
1. A 3-phase, 415V, 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 100
Nm. The motor is operating from a 415V, 50Hz bus,
(i) Calculate the current drawn from the supply and the power factor,
(ii) if the excitation is reduced by 10% supplying the same load, find the current drawn from the supply
and the power factor.
(4+3)+(4+4)=15
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 10%. 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;
LP=80µH; LS=5µH; C=470µF; R=25Ω; D=0.75; TS=10µSec. LP is the self-inductance of the primary coil
and LS is the self-inductance of the secondary coil and the coils are perfectly coupled. Find the steady
state output voltage Vo, and the peak current through the diode
. Assume that all the circuit
components are ideal.
(6)
Do
Vo
LP
LS
Vg
G
S
Q
C
R
VGS
0
DTs
Ts
t
3. (b) The dc-dc converter shown in figure below has the following circuit parameters: Vg=400V;
Vo=40V; NP: NS=4:1; Lf=0.25 mH; Cf=1000µF; R=4Ω; FS=100kHz (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 VG1S1, VG2S2,
,
and
; (use same time scale to plot all the
waveforms in a single page, give the numerical values and corresponding time durations ).
(9)
Lf
Ig
G1
G2
Q1
Q2
S2
S1
Vg
G3
S3
ILf
D2
Q4
Vo
Cf
Ip
G4
Q3
D1
Np Ns
D3
R
D4
S4
4. A 3-ph two level converter is operated with 180o mode and feeding a three phase passive load. The
configuration is shown in the figure below.
Idc
IL
400V
Vdc
A
O
B
C
C
A
N
B
(a) Draw the phase voltage (vAN, vBN, vCN) and line voltage (vAB, vBC, vCA) waveforms.
(3)
(b) Find the rms value of the fundamental component of line voltage(vAB, vBC, vCA) and line current
(IL).Note that the load is drawing 10kVA 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 (Idc) if the load is drawing 10kVA 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 10kVA 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.
A
A
B
B
C
PASSIVE
LOAD
C
Ic
L
a
b
Vdc
C
c
vL
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.
(10)
(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. vax, Ic, vL, vax). 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 (Vsr) =440V, Rated frequency (fr)=50Hz, No. of
poles = 6, R1=0.1Ω, R2’=0.12Ω, X1=0.3Ω, X2’=0.3Ω, Xm=15Ω.
The profile of the load torque is:
TL  Ar2
Vdc
Vr
f*
V/f profile
V
*
3-ph
inverter
ωr
Vo
fo
fr
(a) Find out the droop voltage (Vo). Calculate the slope of the v/f profile considering fo=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]
PART B: Section II
CONTROL SYSTEM ENGINEERING
1. a) A plant G ( s ) with a variable parameter  is compensated using H ( s ) in the feedback path.
Define:
SG  (G / G )
, ST  ( T / T )
( /  )
(  /  )
where T is the closed loop transfer function, and  is the incremental variation operator. Relate
[6]
ST to SG .
(b) Suppose amplifier units of specification 10  10% are only available. Show how an amplifier
of specification 10  2.5% can be derived using multiple units of the same and appropriate
feedback.
[9]
2. Consider the feedback connection in Fig. 1
u
y
G(s)
_
 (.)
Fig. 1
with G ( s) 
y3
s
,
where  is a positive constant. Using the describing

(
y
)

3
s 2  s  1
function method find frequency and amplitude of periodic solution that exists in the feedback
connection.
[15]
3. Consider an LTI system described by the following state-space model
     1  2  x   2
 x (0)  1
1
 x 1   
   u ,  1    ,



 x 2   1 0   x 2  1
 x 2 (0) 0
x 
y  1 1  1 .
 x2 
(a) Write above state-space model in the modal form.
[10]
(b) From the derived modal form, comment on the state controllability and state observability
of the system.
[5]
4. (a) The loop transfer function of a unity-feedback control system is given by,
Gs H s  
K
ss  3 s 2  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
0
2.5
10.0
25
Magnitude
log  rad/sec
+20 dB/decade
12 dB
20 dB/decade
Fig. 2
5. (a) Consider a system with the transfer function:
G( s) 
( s  2)
( s  1)( s 2  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
function of a continuous time plant G(s)=1/[s(s+10)] for sampling time T=0.1 sec.
transfer
[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]
PART B: Section III
POWER AND ENERGY SYSTEMS
Q 1.
A generator supplies a load over a transmission line having transformer at both ends as shown in
the one-line diagram in Figure 1. Transformers T1 and T2 ratings are shown in Figure 1.
Vs=230 0° V
Xline= 3 Ω
T1
T2
25 kVA
240/480 V
Xeq=0.2 p.u
15 kVA
460/120 V
Xeq=0.2 p.u
Zload=1+j0.3 Ω
Figure 1: 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 YBUS. Neglect
the shunt admittances at the buses and mutual couplings between the lines. The line series impedances
are given in Table-1.
Fig. Q2A
Table-1
2
3
z23
Line(bus to bus)
Impedance
pu values
1-2
z12
0.30+j1.2
1-4
z14
0.25+j1.0
2-3
z23
0.15+j0.6
3-4
z34
0.20+j0.8
1
z12
1
z34
z14
4
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 1-2 and the line series impedance of
(0.6+ j0.2) pu between bus 1-3 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.
Fig. Q2B
2
3
z23
1
ya
yb
z12
1
z34
z13
z14
4
Q 3. A 20 km long three phase line has four no. (4/0) wires of 1.1 cm dia spaced horizontally 1.4 m apart
in a plane. If the currents in the lines be
Ia= -20 + j 40 A, Ib = -25 +j 60 A and Ic = 45 – j 100 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, 13.2 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 0.1 pu
respectively. A single line to ground fault occurs in phase-b of the unloaded alternator.
Determine Ib and Vbc magnitudes at the terminal.
Q 5. For the following system, obtain one iteration for load flow solution by Gauss Seidel or NewtonRaphson method. For system data refer table and the Ybus matrix below.
1
2
3
Figure for load flow problem
Table Operating parameters in pu
Bus
Type
|V| Pg
Qg
PL
QL
1
slack
1.0
--
--
0
0
2
PV
1.1
5.32 --
0
0
3
PQ
--
0
3.64 0.54
0
  j15 j10 j 5

Ybus=  j10  j15 j 5
 j 5
j 5  j10


 pu

Q 6. A 100 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.
PART B: Section IV
INSTRUMENTATION AND SIGNAL PROCESSING
1.a. Match the following (flow meters and the characteristics)
A.
B.
C.
D.
E.
F.
Flowmeter Technology
Venturimeter
Vena Contracta Orifice meter
Clamp-on Ultrasonic
Weir
Electromagnetic
Hot Wire Anemometer
1.
2.
3.
4.
5.
6.
7.
(6)
Characteristics
High permanent pressure loss
Variable Area Principle
Open channel Measurement
Constant Temperature Operation
Nonintrusive Measurement
High Physical Strength
Axisymmetric Flows
1.b. A cylindrical float (volume: 500 mm3, diameter: 15 mm) is used with a tapered glass tube to estimate
the volumetic flow rate of water (density: 1000 kg/m3) in a pipe. Tube dimensions: minimum diameter =
18 mm, included angle of taper = 5°, vertical height = 250 mm. The float is made from aluminium of
relative density 2.7. Assume CD=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 300oC. The
e.m.f values are given as E100 =5.268 mV, E200= 10.777mV., E300= 16.325mV
(a) Find the non linearity at 100oC and 200oC 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 10oC and 30oC the thermocouple e.m.f. is given by the expression
Calculate a1 and a2 (write the units).
(5)
o
(c) The emf is 12.5mV relative to a reference junction of 20 C and the corresponding reference junction
circuit voltage is 1mV. 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 10 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 10 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 = 100pF; Natural frequency = 37
kHz; Damping ratio = 0.01
 
 j 0    
 j     0
4(a) An orthogonal phase shifting filter has a frequency response given as H e j  
Plot the magnitude and phase response of this filter?
(5)
 
4b. Find the unit impulse response hn  of this orthogonal phase shifting filter H e j ?
(10)
5a. Let X 0 be the zero frequency component of the DFT of a N-point sequence xn  and also the mean
 of the n-point sequence is known. Can you find the length N  of the sequence?
(10)
5b. Find the DFT and one sided Z-transform of the following N-point sequence xn   1,2,3,4,5

6a. You are provided with a N-point sequence
hn  1,2, 3,4,5 . Find the convolution

marked on the resulting sequence?
x  hn and
(5)
xn   1,2,3,4,5 and the system response

the correlation
x  hnwith
the zero-point
(5)
6b. Can you prove that the N-point DFT of two real-valued sequences x1 n  and x2 n  can be computed
using only one N-point DFT operation?
(10)