# GATE EE 2003

 Question 1
Figure Shows the waveform of the current passing through an inductor of resistance $1\Omega$ and inductance 2 H. The energy absorbed by the inductor in the first four seconds is
 A 144J B 98J C 132J D 168J
Electric Circuits   Basics
Question 1 Explanation:
For $0 \lt t \lt 2s$ current varies linearly with time and given as $i(t)=3t$ and for $2s \lt t \lt 4s$ current is constant, $i(t)=6A$.
The energy absorbed by the inductor (Resistance neglected) in the first 2 sec,
$E_L=\int_{0}^{T}Li\frac{di}{dt}=E_{L_1}+E_{L_2}$
$E_{L_1}=\int_{0}^{2}Li \left (\frac{di}{dt} \right )dt$
$\;\;=\int_{0}^{2} 2 \times 3t \times 3\; dt$
$\;\;=18\int_{0}^{2}t\; dt$
$\;\;=18 \times \left.\begin{matrix} \frac{t^2}{2} \end{matrix}\right|^2_0$
$\;\;= 18 \times \left [ \frac{4}{2}-0 \right ]=36J$
The energy absorbed by the inductor in ($2\rightarrow 4$) second
$E_{L_2}=\int_{2}^{4}Li\left ( \frac{di}{dt}dt \right )$
$\;\;=\frac{2}{4}2 \times 6 \times 0 \;dt=0 J$
A pure indictor does not dissipate enegy but only stores it. Due to resistance, some energy is dissipated in the resistor. Therefore, total energy stored in the inductor and the energy dissipated in the resistor.
The energy dissipated by the resistance in 4 sec.
$E_R=\int_{0}^{T}i^2 R\;dt$
$\;\;=\int_{0}^{2}(3t)^2 \times 1 \;dt+\int_{2}^{4}6^2 \times 1 \; dt$
$\;\;=9\int_{0}^{2}t^2\;dt +36\int_{2}^{4}1 \; dt$
$\;\;=9 \times \left.\begin{matrix} \frac{t^3}{3} \end{matrix}\right|_0^2+36t|_2^4$
$\;\;=9 \times \left ( \frac{8}{3} \right )+36 \times 2$
$\;\;=24+72=96J$
The total energy absorbed by the inductor in 4 sec
$=96 J +36 J =132 J$
 Question 2
A segment of a circuit is shown in figure. $V_{R}$ = 5V, $V_{C}$ = $4 \sin 2t$. The voltage $V_{L}$ is given by
 A $3-8 \cos 2t$ B $32 \sin 2t$ C $16 \sin 2t$ D $16 \cos 2t$
Electric Circuits   Basics
Question 2 Explanation:
By KCL,
$I_P+I_Q+I_C+I_L=0$
$2+2+I_C+I_L=0$
But, $I_C=C \times \frac{dv}{dt}$
$\;\; =1 \times \frac{d}{dt}(4 \sin 2t)$
$\;\;=(8 \cos 2t)$
$\therefore \;\; I_L=-(2+1+8 \cos 2t)$
$\;\; \;\;=-3-8 \cos 2t$
$\therefore \;\; V_L=L\left ( \frac{di}{dt} \right )$
$\;\;\;=2 \times 2 \times 8 \sin 2t$
$\;\;\;=32 \sin 2t$
NOTE: KCL is based on the law of conservation of charges.
 Question 3
In the figure,
$Z_{1}=10\angle -60 ^{\circ} , Z_{2}=10\angle 60^{\circ} , Z_{3}=50\angle 53.13^{\circ}$.
The venin impedance seen form X-Y is
 A $56.66\angle 45^{\circ}$ B $60\angle 30^{\circ}$ C $70\angle 30^{\circ}$ D $34.4\angle 65^{\circ}$
Electric Circuits   Network Theorems
Question 3 Explanation:
By Thevenin's theorem

$Z_{Th}=Z_{X-Y}=Z_1||Z_2+Z_3$
$\;\;=\frac{Z_1 \times Z_2}{Z_1+Z_2}+Z_3$
$\;\;=\frac{10\angle -60 \times 10\angle 60}{10\angle -60 + 10\angle 60}+50\angle 53.13$
$\;\;=56.66\angle 45^{\circ}\Omega$
 Question 4
Two conductors are carrying forward and return current of +I and -I as shown in figure. The magnetic field intensity H at point P is
 A $\frac{I}{\pi d}\vec{Y}$ B $\frac{I}{\pi d}\vec{X}$ C $\frac{I}{2 \pi d}\vec{Y}$ D $\frac{I}{2 \pi d}\vec{X}$
Electromagnetic Fields   Magnetostatic Fields
Question 4 Explanation:
$\bar{H}=\bar{H_1}+\bar{H_2}=\frac{I}{2 \pi d}\vec{y}+\frac{I}{2 \pi d}\vec{y}=\frac{I}{ \pi d}\vec{y}$

 Question 5
Two infinite strips of width w m in x -direction as shown in figure, are carrying forward and return currents of +I and -I in the z - direction. The strips are separated by distance of x m. The inductance per unit length of the configuration is measured to be L H/m. If the distance of separation between the strips in snow reduced to x/2 m, the inductance per unit length of the configuration is
 A 2L H/m B L/4 H/m C L/2 H/m D 4L H/m
Electromagnetic Fields   Magnetostatic Fields
Question 5 Explanation:
Inductance is proportional to separation between the strips so when separation is reduced to x/2 then inductance will become L/2 H/m.
 Question 6
A sinple phase transformer has a maximum efficiency of 90% at full load and unity power factor. Efficiency at half load at the same power factor is
 A 86.70% B 88.26% C 88.90% D 87.80%
Electrical Machines   Transformers
Question 6 Explanation:
\begin{aligned} \text{Efficiency} &=\frac{\text{Output}}{\text{Input}} \\ &= \frac{\text{Output}}{\text{Output}+\text{Losses}}\\ \text{Output}&= x S \cos \phi \\ \text{Where,}\;\;S &=\text{Rating of the machine} \\ x&= \% \text{of the full load}\\ \cos \phi &= \text{Power factor} \\ P_i &= \text{Iron or core loss}\\ P_{cu} &=\text{Full load copper loss} \\ &\text{For maximum frequency}\\ P_i &=x^2P_{cu} \\ \text{Efficiency} =\eta &= \frac{xS \cos \phi }{xS \cos \phi +P_i+x^2 P_{cu}} \\ \text{Max efficiency}=\eta _{max}&=\frac{xS \cos \phi }{xS \cos \phi +2P_i}\\ \Rightarrow \;\;0.9&=\frac{1 \times S \times 1}{1 \times S \times 1+2P_i}\\ P_i&=0.055S \end{aligned}
Full load copper loss $= 1^2 P_{cu}=P_i$
$P_{cu}=0.055S$
$x=\frac{1}{2}$
$\eta =\frac{\frac{1}{2} \times S \times 1 }{\frac{1}{2} \times S \times 1 +0.055S+\left ( \frac{1}{2} \right )^2 \times 0.055S} \times 100$
$\;\;\;\approx 87.8\%$
 Question 7
Group-I lists different applications and Group-II lists the motors for these applications. Match the application with the most suitable motor and choose the right combination among the choices given thereafter
 A P-3 Q-6 R-4 S-5 B P-1 Q-3 R-2 S-4 C P-3 Q-1 R-2 S-4 D P-3 Q-2 R-1 S-4
Electrical Machines   Single Phase Induction Motors, Special Purpose Machines and Electromechanical Energy Conversion System
Question 7 Explanation:
Universal motors are used where light weight is important., as in vacuum cleaners and portable tools e.g. food mixer which usually operate at high speed (1500-15000 rpm). Domestic water pump are usually of low rating. So single phase induction motor can be used for such application. Three phase induction motor is suitable for escalator as it has high starting torque.
 Question 8
A stand alone engine driven synchronous generator is feeding a partly inductive load. A capacitor is now connected across the load to completely nullify the inductive current. For this operating condition.
 A the field current and fuel input have to be reduced B the field current and fuel input have to be increased C the field current has to be increased and fuel input left unaltered D the field current has to be reduced and fuel input left unaltered
Electrical Machines   Synchronous Machines
Question 8 Explanation:

Assuming resistance of the armature to be zero. In first case, the generator is feeding a partly inductive load. It means that generator is supplying lagging power. The generator supplies a lagging power factor current when it is overexcited which is represented by $E_{f_1}$.
In second case, a capacitor is connected across the load to completely nullifythe inductive current. It means the generator supplies no reactive power and unity power factor current is drawn from the generator. The excitation ($E_{f_2}$) corresponding to unity power factor is known as normal excitation.

From the phasor diagram $E_{f_1} \gt E_{f_2}$ and as the field current approximately directly proportional to excitation, the field current has to be reduced. From phasor diagram $E_{f_1} \sin \delta _1=E_{f_2} \sin \delta _2=\text{constant}$
$P_e(\text{ power delivered})=\frac{E_fV_t}{X_s}\sin \delta$
Fuel input remains unaltered.
 Question 9
Curves X and Y in figure denote open circuit and full-load zero power factor(zpf) characteristics of a synchronous generator. Q is a point on the zpf characteristics at 1.0 p.u. voltage. The vertical distance PQ in figure gives the voltage drop across
 A Synchronous reactance B Magnetizing reactance C Potier reactance D Leakage reactance
Electrical Machines   Synchronous Machines
Question 9 Explanation:

The vertical distance PQ between O.C. (curve X) and ZPF characteristic (curve Y) in above characteristic is leakage reactance.
 Question 10
No-load test on a 3-phase induction motor was conducted at different supply voltage and a plot of input power versus voltage was drawn. This curve was extrapolated to intersect the y-axis. The intersection point yields
 A Core loss B Stator copper loss C Stray load loss D Friction and windage loss
Electrical Machines   Three Phase Induction Machines
Question 10 Explanation:
Power input at no-load $(P_0)$ provided losses only as the shaft output is zero. These losses comprise $P_i$(Iron/core loss) and $P_{wf}$(windage and friction loss).

As the voltage is reduced below the rated value, the core-loss decreases as the square of voltage. Since the slip does not increase significantly, the windage and friction loss remains almost constant. The plot of $P_0$ versus V is extrapolated at V=0 which gives, $P_{wf}$ as $P_{i}=0$ art zero voltage.
There are 10 questions to complete.