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Electric Machines And Drives

Gordon R. Slemon

Chapter 10

Induction Motor Drives - all with Video Answers

Educators

Chapter Questions

Problem 1

A $460-\mathrm{V}, 1770-\mathrm{r} / \mathrm{min}$ wound rotor induction motor has the following parameters, all referred to the stator number of turns:
$\begin{array}{llrl}R_s=0.225 & \Omega & L_{t r}=1.89 & \mathrm{mH} \\ R_r=0.45 & \square & L_{t r}=1.89 & \mathrm{mH} \\ L_m=73.7 & \mathrm{mH} & N_{s r} / N_{r e}=1.88 & \end{array}$
(a) Develop a transient $\Gamma$-model equivalent circuit for the machine with the rotor short-circuited.
(b) Suppose a balanced three-phase resistor with $0.3 \Omega /$ phase is connected to the rotor terminals. Develop an appropriate transient $\Gamma$ model.

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Problem 2

For the machine of Problem 10.1, the rotor circuit is open circuited. A three-phase, $60 \cdot \mathrm{Hz}$ sinusoidal source of $460 \mathrm{~V}$ rms (L-L) is applied to the stator terminals at the instant that the voltage on phase $a$ is maximum positive.
(a) Derive an expression for the stator current space vector.
(b) Plot the three stator phase current waveforms for at least one cycle.

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Problem 3

The induction machine for which data are given in Problem 10.1 has its rotor windings short-circuited. The $460-\mathrm{V}, 60-\mathrm{Hz}$ supply is applied to the stator terminals at time zero. Estimate the peak transient torque produced by the machine and plot this torque for the first few cycles. The effect of the magnetizing inductance may be ignored.

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Problem 4

A three-phase, four-pole, $60-\mathrm{Hz}$ wound-rotor induction machine is to be used as a phase shifter. The leakage reactance is $0.16 \Omega /$ phase and the magnetizing reactance is $2.1 \Omega /$ phase. The winding resistance may be ignored. The effective stator-to-rotor ratio (including the ratio $\gamma$ ) is $1: 0.8$. The stator is connected to a $230-\mathrm{V}$ (L-L), three-phase supply. The rotor is displaced in the direction of field rotation through 15 mechanical degrees from alignment with the corresponding stator phase windings. Determine the internal voltage $(\mathrm{L}-\mathrm{N})$ phasor and the internal impedance/ phase of a single-phase Thevenin equivalent circuit as seen from the rotor terminals.

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Problem 5

A three-phase wound-rotor induction machine can be operated as an induction regulator to provide a source of variable three-phase voltage at line frequency. The connection of the windings is as shown in Fig. 10.36. All six terminals of the three stator phase windings must be available for this connection. Consider a machine with the following parameters, referred to the stator turns: $L_L=5.3 \mathrm{mH}, L_M=40 \mathrm{mH}, R_R$ and $R_s$ negligible, $\gamma N_{s e} / N_{r e}=1.2$. The rotor side (or left side) of the system in Fig. 10.36 is connected to a $220-\mathrm{V}$ (L-L), three-phase, $60-\mathrm{Hz}$ source.
(a) Determine the maximum and minimum values of the rms output voltage per phase on open circuit.
(b) Determine the internal impedance/phase of the regulator.

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Problem 6

A two-pole, $60-\mathrm{Hz}$ wound-rotor motor has a leakage reactance of 0.2 $\Omega /$ phase and a magnetizing reactance of $2.0 \Omega /$ phase. The resistances of the windings are negligible and the effective turns ratio $\gamma N_{s e}: N_{r e}$ is unity. The machine is employed as a variable three-phase inductive load by connecting its stationary windings as shown in Fig. 10.37.
(a) Derive an expression for the impedance per phase of the machine as a function of the rotor angle $\beta$.
(b) What should be the angle $\beta$ if the load is to take $25 \mathrm{kVA}$ from a three-phase, $115-\mathrm{V}$ (L-L), $60-\mathrm{Hz}$ source?

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Problem 7

A three-phase, six-pole, wound-rotor motor is to be employed as a variable-frequency source. For this purpose, the stator is connected to a $60-\mathrm{Hz}, 440-\mathrm{V}$ (L-L), three-phase supply and the machine is driven at a variable speed by a prime mover. When the rotor is stationary, the opencircuit, line-to-line voltage at the rotor terminals is $220 \mathrm{~V}$ (L-L).
(a) Determine the speed range required to give a frequency range of 20 to $150 \mathrm{~Hz}$.
(b) Determine the corresponding range of open-circuit output voltage (L-L).
(c) Assuming an ideal machine, determine the power supplied or absorbed at both of the frequency limits of (a) by (i) the stator source and (ii) the driving machine when the power delivered to the rotor circuit is $20 \mathrm{~kW}$.

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Problem 8

A three-phase, six-pole, $440-\mathrm{V}, 60-\mathrm{Hz}$ induction motor has the following per-phase equivalent-circuit parameters all referred to the stator:
$$
\begin{aligned}
& R_s=0 \\
& L_L=6.27 \quad \mathrm{mH} \\
& R_R=0.318 \\
& L_M=103 \quad \mathrm{mH} \\
&
\end{aligned}
$$

Rotational losses are negligible, and the rotor resistance may be regarded as independent of frequency. When connected to a source of rated voltage and frequency, determine:
(a) the starting torque
(b) the maximum or breakdown torque
(c) the speed at which breakdown torque occurs
(d) the torque per unit of speed change for operation near synchronous speed
(e) the approximate speed at which the motor will drive a load requiring a torque of $2 \mathrm{~N} \cdot \mathrm{m}$ per rad/s of speed
(f) the rotor frequency for the condition of (e).

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Problem 9

A three-phase, six-pole, $220-\mathrm{V}, 60-\mathrm{Hz}$ wound-rotor induction motor has an effective stator-to-rotor ratio $\gamma N_{s e}: N_{r e}$ of 1.25 . The machine may be regarded as ideal. A balanced wye-connected load of $3 \Omega$ resistance in parallel with $2200 \mu \mathrm{F}$ capacitance in each phase is connected to the rotor terminals. When the machine is rotating at $350 \mathrm{r} / \mathrm{min}$, with rated stator voltage and frequency, determine:
(a) the effective impedance per phase as seen from the stator terminals
(b) the total power delivered by the supply
(c) the power delivered to the rotor circuit
(d) the mechanical power
(c) the shaft torque.

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Problem 10

A water-wheel turbine that develops a mechanical power of $1 \mathrm{MW}$ at a speed between 600 and $625 \mathrm{r} / \mathrm{min}$ drives a three-phase, 12 -pole induction generator. The stator windings of the generator are connected to a 4000 $\mathrm{V}$ (L-L), $60 \cdot \mathrm{Hz}$, three-phase distribution system. The magnetizing reactance of the induction machine is $33 \Omega /$ phase, the leakage reactance is $3 \Omega /$ phase, and the rotor resistance as seen from the stator is $0.5 \Omega /$ phase. The resistance of the stator windings is negligibly small, and other losses in the generator may be ignored.
(a) Determine the speed of the turbine when the generator is delivering $1 \mathrm{MW}$ to the electrical system.
(b) What value of capacitance per phase connected in wyeat the stator terminals would be required to bring the terminal power factor to unity?

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Problem 11

For comparative purposes, the parameters of an induction machine are usually best expressed in per unit of the ratings of the machine. $\mathrm{A} 60-\mathrm{Hz}$, six-pole, $440-\mathrm{V}(\mathrm{L}-\mathrm{L}), 12.1-\mathrm{A}, 1170-\mathrm{r} / \mathrm{min}, 7.5-\mathrm{kW}$ induction machine has a stator resistance of $0.15 \Omega$, a leakage reactance of $4.3 \Omega$, a magnetizing reactance of $48 \Omega$, and an effective rotor resistance of $0.4 \Omega$.
(a) Determine the base values of the voltage (L-L), phase current, impedance, frequency, synchronous speed, power per phase, total apparent power, and torque.
(b) Determine the per unit values of the equivalent circuit parameters.
(c) Determine the per unit values of the speed and the output torque under rated load conditions.
(d) What determines the ratio of the rated torque to the base torque?

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Problem 12

A variable-speed drive consists of a squirrel-cage induction motor supplied with three-phase, near-sinusoidal voltage of controllable magnitude at variable frequency from an electronic inverter. The motor has a rating of $460 \mathrm{~V}(\mathrm{~L}-\mathrm{L}), 60 \mathrm{~Hz}, 100 \mathrm{~kW}, 1770 \mathrm{r} / \mathrm{min}, 140 \mathrm{~A}$. The mechanical load requires a torque of $3 \mathrm{~N} \cdot \mathrm{m}$ per rad/s of speed. Stator resistance, leakage inductance, and losses may be ignored.
(a) To drive the load at a speed of $1200 \mathrm{t} / \mathrm{min}$, what should be rms voltage (L-L) and the frequency of the inverter output?
(b) If the magnetizing inductance of the motor is $2.5 \mathrm{pu}$, what will be the source power factor for the condition of (a)?

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Problem 13

A three-phase, four-pole, $220-\mathrm{V}, 60-\mathrm{Hz}$ squirrel-cage induction motor has the following equivalent circuit parameters:
$$
\begin{array}{llll}
R_t=0 & & X_L=0.6 & \Omega \\
R_R=0.3 & \Omega & X_M=15 & \Omega
\end{array}
$$

The motor is supplied from an inverter that provides near-sinusoidal, variable-phase voltages with variable frequency. The ratio of source voitage to source frequency is kept constant at a value appropriate for the machine ratings.
(a) Sketch speed-torque curves for the system at inverter frequencies of 60,20 , and $3 \mathrm{~Hz}$.
(b) Estimate the inverter frequency to drive a load requiring a constant torque of $150 \mathrm{~N} \cdot \mathrm{m}$ at a speed of $100 \mathrm{r} / \mathrm{min}$.
(c) Determine the required source voltage for the operating condition of (b).

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Problem 14

An induction motor is rated at $2300 \mathrm{~V}$ (L-L), three-phase, $60 \mathrm{~Hz}, 1770$ $\mathrm{r} / \mathrm{min}, 150 \mathrm{~A}$. Its no-load current is $50 \mathrm{~A}$. Stator resistance, leakage inductance, and rotational losses may be ignored.
(a) Draw an equivalent circuit for the motor and give the values of its parameters.
(b) Suppose the motor is supplied from a simple voltage source inverter, a controlled rectifier, and a $2300-\mathrm{V}(\mathrm{L}-\mathrm{L}), 60-\mathrm{Hz}$, three-phase supply.
Find the maximum no-load speed at which the motor can be operated with rated stator flux linkage.

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Problem 15

An induction motor has the following ratings: $4.6 \mathrm{kV}(\mathrm{L}-\mathrm{L}), 60 \mathrm{~Hz}$, six pole, $10 \mathrm{MW}, 1192 \mathrm{r} / \mathrm{min}$. It is to drive a compressor requiring a torque of $75 \mathrm{kN} \cdot \mathrm{m}$ at a speed of $1000 \mathrm{r} / \mathrm{min}$. A simple voltage-source inverter supplied from a three-phase, phase-controlled rectifier is to be used. The available utility supply is at $4.6 \mathrm{kV}$ (L-L), three-phase, and $60 \mathrm{~Hz}$. Making any reasonable assumptions, estimate:
(a) the required frequency of the inverter
(b) the required direct-link voltage
(c) the delay angle of the rectifier.

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Problem 16

A refrigerator compressor is to be driven at variable speed over a range of 500 to $3000 \mathrm{r} / \mathrm{min}$. The mechanical power required is $2 \mathrm{~kW}$ at maximum speed. A $230-\mathrm{V}, 60-\mathrm{Hz}$ single-phase utility supply is available. It is proposed to use a controlled bridge rectifier and a simple voltage inverter to supply a three-phase, four-pole squirrel-cage induction motor. At rated load, the rotor frequency of the motor may be assumed to be $5 \%$ of the stator frequency. The stator resistance, leakage inductance, and rotational losses may be ignored.
(a) Specify an appropriate value of rated voltage for the motor.
(b) Estimate the rated phase current of the motor assuming the magnetizing current to be $0.5 \mathrm{pu}$.
(c) Determine the maximum frequency of the inverter.
(d) Assuming the compressor torque to be independent of speed, determine the delay angle of the rectifier for the minimum speed condition.

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Problem 17

A variable-speed drive consists of a $460-\mathrm{V}$ (L-L), four-pole, $60-\mathrm{Hz}, 50-$ $\mathrm{kW}, 1770-\mathrm{r} / \mathrm{min}$ induction motor supplied from a controlled rectifier and a simple voltage-source inverter. The electrical supply is at $460 \mathrm{~V}$ (L-L), three phase, and $60 \mathrm{~Hz}$.
(a) The no-load current of the motor with rated stator voltage and frequency is $24 \mathrm{~A}$. All losses except rotor losses can be ignored and the leakage inductance can be neglected. Estimate the fundamental stator current and power factor when this motor is delivering rated torque.
(b) The starting current for this motor is listed by the manufacturer as $600 \mathrm{~A}$. Estimate the fifth and seventh harmonic stator currents when the motor is providing rated torque at $1200 \mathrm{r} / \mathrm{min}$.
(c) By what factor should the motor torque be derated because of the fifth and seventh harmonics in the stator current?
(d) Find the sixth harmonic torque due to the interaction of rated fundamental flux linkage with the fifth harmonic current found in (b).

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Problem 18

A $460-\mathrm{V}$ ( $\mathrm{L}-\mathrm{L}), 60-\mathrm{Hz}, 1770-\mathrm{r} / \mathrm{min}$ induction motor has the following parameters for a $\mathrm{T}$-form equivalent circuit, all referred to the stator turns:
$\begin{array}{llll}R_s=0.18 & \Omega & L_{t s}=1.89 & \mathrm{mH} \\ R_r=0.22 & \Omega & L_{t r}=1.89 & \mathrm{mH} \\ & & L_{i m}=33.7 & \mathrm{mH}\end{array}$
(a) Develop a transient equivalent circuit for the motor of the inverse$\Gamma$ form (as in Fig. 10.20).
(b) Suppose this motor is supplied from a current source inverter using the system shown in Fig. 10.22. If the stator current is $50 \mathrm{~A}$ and the rotor frequency is held at its value for rated operation of the motor, determine the torque produced.
(c) Describe why the torque in (b) is independent of the speed.
(Section 10.3)

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Problem 19

A Class-D, four-pole, three-phase squirrel-cage induction motor is rated at $415 \mathrm{~V}$ (L-L), $50 \mathrm{~Hz}, 20 \mathrm{~A}$. It is supplied from a three-phase, $415-\mathrm{V}, 50$ $\mathrm{Hz}$ source through a variable-voltage controller. The motor drives a fan for which the power is proportional to the cube of the speed. The motor parameters in per unit are $L_M=2.5$ and $R_R=0.1$. The parameters $L_l$ and $R_s$ as well as the rotational losses may be ignored.
(a) With rated voltage applied to the motor, the steady-state phase current is $15 \mathrm{~A}$. Estimate the speed of the fan for this condition.
(b) Determine the rms value of the component of fundamental voltage (L-L) required to operate the fan at a speed of $1200 \mathrm{r} / \mathrm{min}$.
(c) For the condition of (b), is the motor operating within its stator current rating?

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Problem 20

A fan requires a power of $20 \mathrm{~kW}$ at its rated speed of $1100 \mathrm{r} / \mathrm{min}$. It is to be driven by a wound-rotor induction motor fitted for rotor-power recovery using the system shown in Fig. 10.25. The available utility supply is at $460 \mathrm{~V}(\mathrm{~L}-\mathrm{L}), 60 \mathrm{~Hz}$, three phase. The motor may be considered as ideal.
(a) Choose an appropriate number of poles for the motor.
(b) If the fan is to have a minimum speed of $750 . \pi / \mathrm{min}$, determine the appropriate value for the effective turns ratio $N_{a r} / N_{r r}$. The firing angle of the inverter should not exceed $170^{\circ}$.
(c) To operate the fan at its rated speed, what should be the firing angle of the inverter?
(d) At rated speed, determine the value of the direct link current.
(e) If the magnetizing current of the motor is assumed to be 0.4 pu based on its rating, estimate the stator current of the motor when driving the fan at rated speed.

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Problem 21

A three-phase, $220-\mathrm{V}$ (L-L), $400-\mathrm{Hz}$, two-pole induction machine has a leakage inductance of $8 \mathrm{mH}$ and a rotor resistance of $4 \Omega /$ phase. Its stator resistance and magnetizing current may be ignored. Suppose a singlephase, $220-\mathrm{V}, 400-\mathrm{Hz}$ supply is connected between two stator terminals leaving the third open circuited.
(a) Sketch an equivalent circuit showing the appropriate interconnection of the sequence networks.
(b) Derive an expression for the source current as a function of motor speed. Evaluate the current at standstill.
(c) Derive an expression for the motor torque as a function of speed and use it to evaluate the torque at half synchronous speed.

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01:31

Problem 22

A three-phase, six-pole machine stator has 36 slots. The coils each have ten turns. Each coil is short pitched by one slot.
(a) Determine the effective number of turns per phase.
(b) Evaluate the winding factor for the fifth space harmonic.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:50

Problem 23

A linear induction motor has a wavelength of $480 \mathrm{~mm}$ and has five poles. It can be modeled by an equivalent current with the following parameters:
$$
\begin{aligned}
R_s & =\text { negligible } & & L_L=1.2 \quad \mathrm{mH} \\
R_R & =0.28 \quad \therefore & L_M & =6 \quad \mathrm{mH}
\end{aligned}
$$
(a) Determine the synchronous velocity of the motor when operating on a $20-\mathrm{Hz}$ supply.
(b) Determine the thrust, the mechanical power, the input current, and power factor when the motor is operated on a $300-\mathrm{V}$ (L-L), $40-\mathrm{Hz}$ supply and the speed is $45 \mathrm{~km} / \mathrm{h}$.

Narayan Hari
Narayan Hari
Numerade Educator