Single Phase Induction Motors, Special Purpose Machines and Electromechanical Energy Conversion System

Question 1
A 4-pole induction motor with inertia of 0.1 kg-m^2 drives a constant load torque of 2 Nm. The speed of the motor is increased linearly from 1000 rpm to 1500 rpm in 4 seconds as shown in the figure below. Neglect losses in the motor. The energy, in joules, consumed by the motor during the speed change is ____________. (round off to nearest integer)

A
1732
B
2534
C
1245
D
3251
GATE EE 2022   Electrical Machines
Question 1 Explanation: 
We have
\begin{aligned} J\frac{d\omega }{dt}&=\tau _e-\tau _L\\ J\omega \frac{d\omega }{dt}&=P_e-P_L \;\;(\because P=\tau \omega )\\ \therefore P_e&=J\omega \frac{d\omega }{dt}+P_L\\ E&=\int P_edt\\ R&=J\int_{1000}^{1500}\omega d\omega +\int_{1000}^{1500}P_Ldt\\ &=\frac{J}{2}\left ( \frac{2 \pi}{60} \right )^2[1500^2-1000^2]+\frac{2 \pi \tau }{60}\int_{1000}^{1500}Ndt\\ &=\frac{0.1}{2}\left ( \frac{2 \pi}{60} \right )^2\times 125 \times 10^4+\frac{2 \pi}{60} \times 2\times [1000 \times 4+\frac{1}{2} \times 500 \times 4]\\ &=1732.586J \end{aligned}
Question 2
The type of single-phase induction motor, expected to have the maximum power factor during steady state running condition, is
A
split phase (resistance start).
B
shaded pole.
C
capacitor start.
D
capacitor start, capacitor run.
GATE EE 2022   Electrical Machines
Question 3
A three-phase, 50 Hz, 4-pole induction motor runs at no-load with a slip of 1%. With full load, the slip increases to 5 %. The % speed regulation of the motor (rounded off to 2 decimal places) is _________ .
A
4.21
B
16.45
C
2.21
D
8.44
GATE EE 2020   Electrical Machines
Question 3 Explanation: 
\begin{aligned} \text{Synchronous speed, }\\ N_s&=\frac{120f}{P}\\ N_s&=\frac{120 \times 50}{4}rpm\\ N_s&=1500 \; rpm\\ \text{Speed at no-load,}\\ &=N_s(1-S_{nL})\\ &=1500(1-0.01)\\ &=1485 \; rpm\\ \text{Speed at full-load,}\\ &=N_s(1-S_{fL})\\ &=1500(1-0.05)\\ &=1425 \; rpm\\ \% \text{ speed regulation}&=\frac{N_{NL}-N_{FL}}{N_{FL}} \times 100\\ &=\frac{1485-1425}{1425} \times 100\\ &=4.21\% \end{aligned}
Question 4
The magnetic circuit shown below has uniform cross-sectional area and air gap of 0.2 cm. The mean path length of the core is 40 cm. Assume that leakage and fringing fluxes are negligible. When the core relative permeability is assumed to be infinite, the magnetic flux density computed in the air gap is 1 tesla. With same Ampere-turns, if the core relative permeability is assumed to be 1000 (linear), the flux density in tesla(round off to three decimal places) calculated in the air gap is __________
A
0.257
B
0.456
C
0.833
D
0.658
GATE EE 2019   Electrical Machines
Question 4 Explanation: 
\begin{aligned} l_{ag} &=0.2 cm \\ l_m&= 40cm\\ B_0 &= 1 \text{Tesla}\\ \phi &= \frac{mmf}{\Re }\\ &\text{For the same mmf,} \\ \phi &\propto \frac{1}{\Re } \\ \text{In case-1:} &\\ \Re _1&=\frac{l_m}{\mu _0A} \\ &= \frac{l_{ag}}{\mu _0 A}+\frac{l_m}{\mu _0 \mu _r A}\\ \text{as} \;\;\mu _0=0& \\ \therefore \;\;\Re _1&= \frac{0.2 \times 10^{-2}}{\mu _0 A}\\ \text{In case-2:} &\\ \Re _2&= \frac{l_{ag}}{\mu _0 A}+\frac{l_m}{\mu _0 \mu _r A}\\ &=\frac{0.2 \times 10^{-2}}{\mu _0 A}+\frac{40 \times 10^{-2}}{1000\mu _0 A}\\ &=\frac{0.24\times 10^{-2}}{\mu _0 A} \end{aligned}
As flux \phi \propto B for uniform cross section area.
\begin{aligned} \therefore \; B_2&=\frac{B_1 \times \Re _1}{\Re _2} \\ B_2&= \frac{1 \times 0.2 \times 10^{-2}/\mu _0 A}{0.24 \times 10^{-2}/\mu _0A}\\ &= 0.833T \end{aligned}
Question 5
The equivalent circuit of a single phase induction motor is shown in the figure, where the parameters are R_{1}=R'_{2}=X_{i1}=X'_{i2}=12\Omega , X_{M}=240\Omega and s is the slip. At no-load, the motor speed can be approximated to be the synchronous speed. The no-load lagging power factor of the motor is___________ (up to 3 decimal places).
A
0.182
B
0.412
C
0.214
D
0.106
GATE EE 2018   Electrical Machines
Question 5 Explanation: 


Simplifying the above circuit into a simple R-L circuit,

\begin{aligned} Z_{eq} &=\frac{(3+j6)(j120)}{(3+j126)}+(12+j32) \\ &=138.563\angle \\ &\text{Current drawn by motor is } \\ I&= \frac{V\angle 0}{Z\angle \theta } \end{aligned}
(\therefore \; \theta : impedance angle will be p.f. angle)
\therefore No laod lagging p.f. of motor is (\cos\theta ),
\cos(83.9)=0.106 lagging power factor
Question 6
A transformer with toroidal core of permeability \mu is shown in the figure. Assuming uniform flux density across the circular core cross-section of radius r\lt \lt R, and neglecting any leakage flux, the best estimate for the mean radius R is
A
\frac{\mu Vr^{2}{N^{2}}_{P}\omega }{I}
B
\frac{\mu VIr^{2}N_{P}N_{S}\omega }{V}
C
\frac{\mu Vr^{2}{N^{2}}_{P}\omega }{2I}
D
\frac{\mu Ir^{2}{N^{2}}_{P}\omega }{2V}
GATE EE 2018   Electrical Machines
Question 6 Explanation: 


Since we know reluctance of core:
\Re =\frac{l_c}{\mu a}
Here, l_c= mean core length, a= area of core reluctance,
\begin{aligned} \Re &=\frac{2 \pi R}{\mu \times \pi r^2}\\ &=\frac{2R}{\mu r^2}\\ [\text{where} R &= \text{mean radius of core}] \\ \text{or}\;\; R&=\frac{\Re \mu r^2}{2} \\ \text{and}\;\; \Re &= \frac{mmf}{flux}=\frac{N_pI_p}{\phi }\\ \text{Here,}\;\; \phi &=-\frac{1}{N_p}\int_{0}^{t}e\;dt \\ [\text{where}\; e &\text{is primary generated voltage}] \\ \text{or}\;\; \phi &=-\frac{1}{N_p}\int_{0}^{t}-(V_p)\;dt \\ \phi &= \frac{1}{N_p}\int_{0}^{t}V \cos \omega t \;dt\\ &= \frac{V}{\omega N_p}\sin \omega t\\ &\text{Now reluctance} \\ \Re &= \frac{N_p I \sin \omega t}{\frac{V}{\omega N_p} \sin \omega t}\\ &= \frac{N_p^2\omega I}{V}\\ & \text{So mean radius,}\\ R&=\frac{N_p^2 \times \omega I \times \mu r^2}{2V} \end{aligned}
Question 7
A 375W, 230 V, 50 Hz capacitor start single-phase induction motor has the following constants for the main and auxiliary windings (at starting): Z_{m}=(24.50+j12.75)\Omega (main winding), Z_{n}=(24.50+j12.75)\Omega (auxiliary winding). Neglecting the magnetizing branch the value of the capacitance (in \muF ) to be added in series with the auxiliary winding to obtain maximum torque at starting is _______.
A
38.43
B
200
C
150
D
50
GATE EE 2017-SET-1   Electrical Machines
Question 7 Explanation: 
Auxiliary winding impedance =(24.5+j12.75)\Omega
Let X_c be the reactance of the capacitor connected in auxiliary winding
\begin{aligned} Z_a &= (24.5+j12.75-jX_c)\Omega \\ X_a &=(X_c-12.75),\; R_a=24.5\Omega \\ \therefore \;\; \phi _a &=\tan^{-1}\frac{X_c-12.75}{24.5}\;...(i) \\ Z_m&= 12.5+j15.75\\ &= 20.10\angle 51.56^{\circ}\Omega \\ \phi _m&= 51.56^{\circ} \end{aligned}
For maximum torque at starting, condition on \phi _a and \phi _m is given by \left [ \phi _a=\frac{90^{\circ}-\phi _m}{2} \right ]
\phi _a=\frac{90-51.56}{2}=19.22^{\circ}}\;...(ii)
Comparing equation (i) and (ii),
\begin{aligned} \tan(19.22^{\circ}) &=\frac{X_c-12.75}{24.5} \\ 0.3486 &= \frac{X_c-12.75}{24.5}\\ X_c &=8.541+12.75 \\ \text{Capacitive reactance,}& \\ X_c &= 21.291\Omega \\ \frac{1}{2 \pi fc}&= 21.291\\ \Rightarrow \;C&= \frac{1}{2\pi \times 50 \times 21.291}\\ &= 149.51\mu F \end{aligned}
Question 8
The flux linkage (\lambda) and current (i) relation for an electromagnetic system is \lambda = (\sqrt{i})/g. When i=2A and g (air-gap length)=10cm, the magnitude of mechanical force on the moving part, in N, is ________.
A
128.52
B
165.65
C
188.56
D
214.25
GATE EE 2016-SET-2   Electrical machines
Question 8 Explanation: 
Energy in magnetic system,
\begin{aligned} W_f &= \int_{0}^{\lambda } i(\lambda )\; d\lambda \\ \because \;\; i &=\lambda ^2g^2 \\ W_f &= \int_{0}^{\lambda } \lambda ^2g^2 \; d\lambda \\ &=g^2\frac{\lambda ^3}{2} \end{aligned}
Now, mechanical force,
\begin{aligned} F_F &=-\frac{\partial W_f (\lambda ,g)}{\partial g} \\ &=-\frac{\partial }{\partial g} \left ( \frac{g^2\lambda ^3}{3} \right )\\ &= -\frac{2}{3}\lambda ^3 g\\ \because \;\; i=2, \;g&=10cm ,\; \lambda =10\sqrt{2} \\ \text{Hence,}\;\;|F_F|&=\frac{2}{3} \times (10\sqrt{2})^3 \times 0.1\\ &=188.56N \end{aligned}
Question 9
The direction of rotation of a single-phase capacitor run induction motor is reversed by
A
interchanging the terminals of the AC supply
B
interchanging the terminals of the capacitor
C
interchanging the terminals of the auxiliary winding
D
interchanging the terminals of both the windings
GATE EE 2016-SET-2   Electrical Machines
Question 9 Explanation: 
Inter changing the terminals of the auxiliary winding.
Question 10
A 220 V, 3-phase, 4-pole, 50 Hz inductor motor of wound rotor type is supplied at rated voltage and frequency. The stator resistance, magnetizing reactance, and core loss are negligible. The maximum torque produced by the rotor is 225 % of full load torque and it occurs at 15% slip. The actual rotor resistance is 0.03 \Omega/phase. The value of external resistance (in Ohm) which must be inserted in a rotor phase if the maximum torque is to occur at start is ______.
A
0
B
0.18
C
0.24
D
0.28
GATE EE 2015-SET-2   Electrical Machines
There are 10 questions to complete.