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Essentials of Electrical and Computer Engineering

David V. Kerns, Jr., J. David Irwin

Chapter 16

AC Polyphase Machines - all with Video Answers

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Chapter Questions

02:41

Problem 2

A three-phase $4 \mathrm{hp}$ induction motor is rated at $120 \mathrm{~V}, 400 \mathrm{~Hz}$, and $3840 \mathrm{rpm}$. Find the number of poles, the slip, and the frequency of the rotor currents.

Narayan Hari
Narayan Hari
Numerade Educator
01:49

Problem 3

A $50 \mathrm{~Hz}$ induction motor has a full-load speed of $460 \mathrm{rpm}$. If the no-load speed is $476 \mathrm{rpm}$, determine the slip at full load.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 4

A $3 \phi, 440 \mathrm{~V}, 60 \mathrm{~Hz}$, eight-pole, $Y$-connected induction motor has the following parameters: $R_s=0.29 \Omega, X_s=1.25 \Omega, X_r^{\prime}=1.25 \Omega$, $R_r^{\prime}=0.1 \Omega$, and $X_m=18.5 \Omega$. Find the terminal current for a slip of $10 \%$.

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02:41

Problem 5

An often-used simplified equivalent circuit for an induction motor is shown in Figure P16.5, where the shunt branch has been moved to the voltage source side.
Figure P16.5 can't copy
Using this simplified equivalent circuit, determine the rotor speed and torque of a $40 \mathrm{hp}, 440 \mathrm{~V}$, three-phase, $60 \mathrm{~Hz}$, four-pole, $Y$-connected induction motor with the following parameters: $R_r^{\prime}=0.13 \Omega, R_i=0.12 \Omega$, $X_i=0.7 \Omega, X_m=\infty$, and a slip of $4 \%$.

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 6

A three-phase $20 \mathrm{hp}, 220 \mathrm{~V}, 60 \mathrm{~Hz}$, six-pole, $Y$-connected induction motor has the following parameters: $R_1=0.15 \Omega, R_2=0.1 \Omega$, $X_1=0.2 \Omega, X_2=0.3 \Omega$, and $X_m=j 10 \Omega$. Using the approximate equivalent circuit in problem 16.5, determine the input line current and the input power factor if the slip is $3 \%$.

Narayan Hari
Narayan Hari
Numerade Educator
02:41

Problem 7

A $3 \phi$, six-pole, $60 \mathrm{~Hz}, Y$-connected induction motor has the following parameters: $R_S=$ $0.21 \Omega, X_s=1.01 \Omega, X_r^{\prime}=1.01 \Omega, R_r^{\prime}=0.11 \Omega$ and $X_m=\infty$. The motor is used to drive a fan. The fan must turn at $1145 \mathrm{rpm}$ and $10 \mathrm{hp}$ of shaft power is required to run the fan at this speed. Find the magnitude of the terminal voltage required to achieve this operating condition.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 8

A $3 \phi$, four-pole, $60 \mathrm{~Hz}, Y$-connected induction motor rated at $440 \mathrm{~V}$ and $1750 \mathrm{rpm}$ has the following per-phase model parameters: $R_S=$ $0.35 \Omega, X_s=1.1 \Omega . X_r^{\prime}=1.1 \Omega, R_r^{\prime}=0.15 \Omega$, and $X_m>>\left|R_r^{\prime} / s+j X_r^{\prime}\right|$ and can, therefore, be neglected. If the motor is operated at rated voltage and shaft speed, find the magnitude of the line current if the motor is used in a hoist that requires a torque of $146.29 \mathrm{~N}-\mathrm{m}$ to drive a load.

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02:41

Problem 9

A three-phase, four-pole, $230 \mathrm{~V}, 60 \mathrm{~Hz}$ induction motor has the following parameters: $R_1=0.2 \Omega, R_2=0.1 \Omega,\left(X_1+X_2\right)=0.3 \Omega$, and $X_m$ is infinite. Find the output power and torque for a slip of $4 \%$.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 10

A $3 \phi$, four-pole, $440 \mathrm{~V}, 60 \mathrm{~Hz}, Y$-connected induction motor has the following parameters: $R_s=0.22 \Omega, X_s=1.3 \Omega, X_r^{\prime}=1.0 \Omega, R_r^{\prime}=$ $0.1 \Omega$, and it is assumed that $X_m=\infty$. Find the torque developed at starting if rated voltage is applied to the machine terminals.

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Problem 11

A $3 \phi$, four-pole, $60 \mathrm{~Hz}, 550 \mathrm{~V}, Y$-connected induction motor has the following parameters: $R_s=1.0 \Omega, X_s=3.0 \Omega, X_r^{\prime}=4.5 \Omega$, $R_r^{\prime}=0.9 \Omega$, and $X_m=35 \Omega$. When the motor is running at rated speed, the frequency of the rotor currents is measured to be $5 \mathrm{~Hz}$. (a) Find the rated speed. (b) Find the torque developed.

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02:41

Problem 12

A three-phase, $50 \mathrm{hp}, 440 \mathrm{~V}$, four-pole, $60 \mathrm{~Hz}$ induction motor has a stator resistance of $0.05 \Omega$, a rotor resistance of $0.072 \Omega$, and a combined $\left(X_1+X_2\right)$ reactance of $0.06 \Omega$. Find the horsepower and output torque for a speed of $1746 \mathrm{rpm}$.

Narayan Hari
Narayan Hari
Numerade Educator
04:52

Problem 13

A $3 \phi$, two-pole, $60 \mathrm{~Hz}, 550 \mathrm{~V}, Y$-connected induction motor has the following parameters: $R_s=0.3 \Omega, X_s=0.75 \Omega, X_r^{\prime}=0.75 \Omega, R_r^{\prime}=$ $0.17 \Omega$, and $X_m=\infty$. The motor is used to drive a centrifugal pump. When the pump is primed and running, the motor slip is 0.1 . Neglecting rotational and stray losses, find the efficiency of the motor at this slip.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 14

A $3 \phi$, eight-pole, $60 \mathrm{~Hz}, 440 \mathrm{~V}, Y$-connected induction machine has the following parameters: $R_s=0.5 \Omega, X_s=1.0 \Omega, X_r^{\prime}=1.0 \Omega$, $R_r^{\prime}=0.4 \Omega$, and $X_m=\infty$. If the motor draws a line current of $29.09 \mathrm{~A}$, find the shaft speed.

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06:15

Problem 15

A local power company charges $$\$ 0.08$$ per $\mathrm{kW}$-hr of energy used and pays $$\$ 0.02 \mathrm{~kW}-\mathrm{hr}$$ that a customer supplies back to the power company's system. A $3 \phi$, four-pole, $60 \mathrm{~Hz}$, $440 \mathrm{~V}, Y$-connected motor with the following parameters: $R_s=0.8 \Omega, X_s=1.9 \Omega$, $X_r^{\prime}=1.9 \Omega, R_r^{\prime}=0.45 \Omega$, and $X_m=\infty$, is employed at a local construction site to raise and lower a hoist. The motor speed in the "raise" mode is $1600 \mathrm{rpm}$ and the speed in the "lower" mode is $2000 \mathrm{rpm}$, and the shaft always turns in the same direction because of a mechanical gearbox. If it takes 10 seconds to raise the loaded hoist and 5 seconds to lower the hoist, find the total cost to operate the hoist through one complete raise-and-lower cycle. Neglect rotational losses.

RZ
Rubeena Zulfiqar
Numerade Educator
01:38

Problem 16

A three-phase, $220 \mathrm{~V}, 60 \mathrm{~Hz}$, six-pole induction motor has the following parameters: $R_1=$ $0.18 \Omega, R_2=0.1 \Omega, X_1=0.4 \Omega, X_2=0.24 \Omega$, and $X_m=j 20 \Omega$. Find the input power and output power and the efficiency if the fixed losses are $300 \mathrm{~W}$ and the slip is $2 \%$.

Anand Jangid
Anand Jangid
Numerade Educator

Problem 17

In parts of South America, the frequency of the power system is $50 \mathrm{~Hz}$. If a three-phase, eight-pole synchronous machine is used as a generator, determine the speed of the rotor.

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02:07

Problem 18

A $1000 \mathrm{kVA}$, three-phase, $Y$-connected, 4160 $\mathrm{V}, 60 \mathrm{~Hz}$ synchronous gererator has an armature resistance and synchronous reactance of $0.14 \Omega$ and $2.5 \Omega$ per phase, respectively. If the power factor is 0.77 lagging, determine the full load generated voltage per phase.

Kajal Gautam
Kajal Gautam
Numerade Educator
02:45

Problem 19

A $960 \mathrm{hp}$ synchronous motor is fed by a 2300 $\mathrm{V}$, three-phase supply at $0.75 \mathrm{pf}$ lagging. If the rotational losses are known to be $20 \mathrm{~kW}$, the armature resistance is negligible and the synchronous reactance is $2.2 \Omega$, determine the excitation voltage at full load.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 20

A $3 \phi$, six-pole, $Y$-connected synchronous machine is operating as a motor. The machine is rated at 2080 volts and $X_s=4.0 \Omega$. If the magnitude of the induced voltage $\mathbf{E}_f$ is 1500 volts, find the maximum three-phase power that the motor can develop without losing synchronism.

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Problem 21

A three-phase, $4160 \mathrm{~V}$ synchronous motor has a $X_s$ value of $6 \Omega$. If the motor supplies 600 $\mathrm{kW}$ of shaft power and 360 kvars of reactive power, determine the induced voltage $E_f$.

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02:45

Problem 22

A $3 \phi$, four-pole, $Y$-connected synchronous motor is operating at a rated voltage of $550 \mathrm{~V}$ and the input current is $\mathbf{I}_s=45 /-25^{\circ} \mathrm{A}$. Find the three-phase power developed by the machine.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 23

A $10,000 \mathrm{kVA}$, three-phase, $Y$-connected, 60 $\mathrm{Hz}$ synchronous generator is fed by a $13.8 \mathrm{kV}$ line. If the armature resistance and synchronous reactance are $0.8 \Omega$ and $2 \Omega$, respectively, find the full-load generated voltage if the $\mathrm{PF}$ is (a) 0.85 lagging, and (b) 0.85 leading.

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Problem 24

A $3 \phi$, six-pole, $Y$-connected synchronous generator is rated at 550 volts and has a synchronous reactance of $X_d=2.0 \Omega$. If the generator supplies $50 \mathrm{kVA}$ at a rated voltage and a power factor of 0.95 lagging, find $\mathbf{I}_s$ and $\mathbf{E}_f$ and sketch the phasor diagram for $\mathbf{V}_s, \mathbf{I}_s$, and $\mathbf{E}_f$.

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04:38

Problem 25

A $3 \phi$, six-pole, $Y$-connected synchronous generator with a synchronous reactance of $X_d=12 \Omega$ is rated at $4160 \mathrm{~V}$. If the dc field current is adjusted to produce an induced voltage of $5000 \mathrm{~V}$ and the rotor angle delta is known to be $35^{\circ}$, find the three-phase complex power output at the generator terminals.

Hemraj Kumawat
Hemraj Kumawat
Numerade Educator
01:55

Problem 26

For the machine in problem 16.25 , if the threephase real power output does not change, find the new rotor angle if the field current is adjusted to produce an $E_f=4000 \mathrm{~V}$ while the terminal voltage is held constant.

Narayan Hari
Narayan Hari
Numerade Educator
04:03

Problem 27

A $3 \phi$, eight-pole, $4160 \mathrm{~V}, Y$-connected synchronous generator is operated at a rated terminal voltage to supply a terminal current of 100 A at 0.9 power factor lagging. If $X_d=12.0 \Omega$ and the dc field current in the rotor is related to $E_f$ by the equation $I_f=0.15 E_f$, find the dc field current required to operate the generator.

Narayan Hari
Narayan Hari
Numerade Educator
07:43

Problem 28

Repeat problem 16.27 if the machine is operated as a motor and draws a current of $75.0 \angle 15^{\circ} \mathrm{A}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 29

A $3 \phi$, six-pole, $2080 \mathrm{~V}, Y$-connected synchronous machine is operated as a generator. The generator delivers $300 \mathrm{kVA}$ at a power factor of 0.75 lagging. Find the induced voltage, $E_f$, and rotor angle, $\delta$, if the per-phase stator impedance is $\mathbf{Z}_s=0.1+j 8 \Omega$. Note that in this case we are simply including the effects of stator winding resistance.

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02:41

Problem 61

A three-phase 6 hp induction motor is rated at $220 \mathrm{~V}, 50 \mathrm{~Hz}$, and $1410 \mathrm{rpm}$. Find the number of poles, the slip, and the frequency of the rotor currents.

Narayan Hari
Narayan Hari
Numerade Educator