Electrical Engineering: Principles and Applications

Allan R. Hambley

Chapter 17

AC Machines - all with Video Answers

Educators

Chapter Questions

Problem 1

A 60 -Hz induction motor is needed to drive a load at approximately 850 rpm. How many poles should the motor have? What is the slip of this motor for a speed of 850 rpm?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Prepare a table that shows synchronous speeds for three-phase induction motors operating at $50 \mathrm{Hz}$ Consider motors having eight or fewer poles. Repeat for $400-\mathrm{Hz}$ motors.

Narayan Hari

Numerade Educator

Problem 3

Explain why induction motors develop zero torque at synchronous speed.

Narayan Hari

Numerade Educator

Problem 4

A 10 -hp six-pole 60 -Hz three-phase induction motor runs at 1160 rpm under full-load conditions. Determine the slip and the frequency of the rotor currents at full load. Also estimate the speed if the load torque drops in half.

Narayan Hari

Numerade Educator

Problem 5

Consider the induction motor shown in Figure 17.7 on page 817. Redraw the figure showing the current directions in the rotor conductors, the magnetic rotor poles and the direction of the developed torque if a prime mover drives the rotor at a speed higher than synchronous speed. In this case, does the machine operate as a motor or as a generator?

Narayan Hari

Numerade Educator

Problem 6

The magnetic field produced in the air gap of an induction motor by the stator windings is given by $B=B_{m} \cos (\omega t-2 \theta),$ in which $\theta$ is angular displacement in the counterclockwise direction as illustrated in Figure 17.4 on page $814 .$ How many poles does this machine have? Given that the frequency of the source is $50 \mathrm{Hz}$, determine the speed of rotation of the field, Does the field rotate clockwise or counterclockwise? Repeat for a field given by $B=B_{m} \cos (\omega t+3 \theta)$.

Narayan Hari

Numerade Educator

Problem 7

A four-pole induction motor drives a load at 2500 rpm. This is to be accomplished by using an electronic converter to convert a 400 -V dc source into a set of three-phase ac voltages, Find the frequency required for the ac voltages assuming that the slip is 4 percent The load requires 2 hp. If the dc-to-ac converter has a power efficiency of 88 per cent and the motor has a power efficiency of 80 percent, estimate the current taken from the dc source.

Narayan Hari

Numerade Educator

Problem 8

In a proposed design for an electric automobile, the shaft of a four pole three-phase induction motor is connected directly to the drive axle; in other words, there is no gear train. The outside diameter of the tires is 20 inches Instead of a transmission, an electronic converter produces variable-frequency three-phase ac from a $48-\mathrm{V}$ battery. Assuming negligible slip, find the range of frequencies needed for speeds ranging from 5 to 70 mph. The vehicle, including batteries and occupants, has a mass of 1000 kg The power efficiency of the dc-to-ac converter is 85 percent, and the power efficiency of the motor is 89 percent. Find the current taken from the battery as a function of time while accelerating from 0 to 40 mph uniformly (i.e., acceleration is constant) in 10 seconds Neglect wind load and road friction.

Prabhu Ramji

Numerade Educator

Problem 9

Repeat Problem P17.8 assuming that the vehicle is accelerated with constant power.

Narayan Hari

Numerade Educator

Problem 10

It is necessary to reduce the voltage applied to an induction motor as the frequency is reduced from the rated value. Explain why this is so.

Narayan Hari

Numerade Educator

Problem 11

Consider the two-pole two-phase induction motor having two windings displaced $90^{\circ}$ in space shown in Figure $P 17.11$. The fields produced by the windings are given by $B_{a}=$ $K i_{a}(t) \cos (\theta)$ and $B_{b}=K i_{b}(t) \cos \left(\theta-90^{\circ}\right)$. The two-phase source produces currents given by $i_{a}(t)=I_{m} \cos (\omega t)$ and $i_{B}(t)=$ $I_{m} \cos \left(\omega t-90^{\circ}\right) .$ Show that the total field rotates. Determine the speed and direction of rotation. Also find the maximum flux density of the rotating field in terms of $K$ and $I_{m}$.

Narayan Hari

Numerade Educator

Problem 12

Show that if the connections to the $b$ winding for the machine of Problem $P 17.11$ are reversed, the field rotates in the opposite direction.

Prabhu Ramji

Numerade Educator

Problem 13

The air-gap flux density of a two-pole induction motor is given by $$B=B_{m} \cos (\omega t-\theta)$$ where $B_{m}$ is the peak flux density, $\theta$ is the angular displacement around the air gap, and we have assumed clockwise rotation. Give the corresponding expression for the flux density of a four-pole induction motor: of a six-pole induction motor.

Narayan Hari

Numerade Educator

Problem 14

Suppose that we could use superconducting material for the rotor conductors (i.e. rotor conductors with inductance and zero resistance) of an induction motor. Would this improve the performance of the motor? Explain by considering the torque-speed characteristic.

Narayan Hari

Numerade Educator

Problem 15

What are the two basic types of construction used for the rotors of induction motors? Which is the most rugged?

Narayan Hari

Numerade Educator

Problem 16

Sketch the torque-speed characteristic of a delta-connected 220-V-rms 5-hp four-pole $60-\mathrm{Hz}$ three-phase induction motor. Estimate values and label key features for things such as the full-load running speed, the full-load torque, the pull-out torque, and the starting torque, Estimate the full-load line current and the starting line current.

Narayan Hari

Numerade Educator

Problem 17

List five important specifications to be considered (besides cost) in selecting induction motors. Indicate the optimum value or whether a high or low value is desirable for each specification.

Narayan Hari

Numerade Educator

Problem 18

A 60-Hz wound-rotor induction motor operates at 40 percent slip with added rotor resistance to achieve speed control. The stator resistance is negligible compared with the rotor resistance. Neglect rotational losses. Find the efficiency of this motor.

Narayan Hari

Numerade Educator

Problem 19

An eight-pole 60 -Hz ac induction motor produces an output power of 2 hp and has rotational losses of $100 \mathrm{W}$ at a speed of 850 rpm. Determine the slip, the frequency of the stator currents, the frequency of the rotor currents, and the rotor copper loss.

Narayan Hari

Numerade Educator

Problem 20

Sometimes, to reduce starting current to reasonable values, induction motors are started with reduced source voltage. When the motor approaches its operating speed, the voltage is increased to full rated value. Compute the starting line current and torque for the motor of Example 17.2 if it is started with a source voltage of $220 \mathrm{V}$ Compare results with the values found in the example, and comment.

Narayan Hari

Numerade Educator

Problem 21

Another method that is used to limit starting current is to place additional resistance in series with the stator windings during starting The resistance is switched out of the circuit when the motor approaches full speed. Compute the resistance that must be placed in series with each phase of the motor of Examples 17.1 and 17.2 to limit the starting line current to $50 \sqrt{3}$ A. Determine the starting torque with this resistance in place. Compare the starting torque with the value found in Example 17.2 and comment.

Keshav Singh

Numerade Educator

Problem 22

A certain four-pole $240-\mathrm{V}-$rms $60-\mathrm{Hz}$ delta-connected three-phase induction motor has $$\begin{array}{ll}R_{2}=1 \Omega & R_{r}^{\prime}=0.5 \Omega \\
X_{2}=1.5 \Omega & X_{r}^{\prime}=0.8 \Omega \\X_{m}=40 \Omega &\end{array}$$ Under load, the machine operates at 1728 rpm and has rotational losses of 200 W. Find the power factor, output power, copper losses, output torque, and efficiency.

Narayan Hari

Numerade Educator

Problem 23

Neglecting the rotational losses find the no-load speed, line current, and power factor for the motor of Problem $\mathrm{P} 17.22$.

Narayan Hari

Numerade Educator

Problem 24

Neglecting losses, find the starting torque and starting line current for the motor of Problem P17.22.

Narayan Hari

Numerade Educator

Problem 25

A certain six-pole 440-V-rms 60-Hz three-phase delta-connected induction motor has $$\begin{aligned}R_{3} &=0.08 \Omega & & R_{r}^{\prime}=0.06 \Omega \\X_{3} &=0.20 \Omega & & X_{r}^{\prime}=0.15 \Omega \\X_{m} &=7.5 \Omega\end{aligned}$$ Under load, the machine operates with a slip of 4 percent and has rotational losses of $2 \mathrm{kW}$. Determine the power factor, output power, copper losses, output torque, and efficiency.

Narayan Hari

Numerade Educator

Problem 26

Neglecting the rotational losses, find the no-load speed, line current, and power factor for the motor of Problem P17.25.

Narayan Hari

Numerade Educator

Problem 27

Neglecting losses, find the starting torque and starting line current for the motor of Problem P17.25.

Narayan Hari

Numerade Educator

Problem 28

A 2-hp six-pole 60-Hz delta-connected three-phase induction motor is rated for $1140 \mathrm{rpm}, 220 \mathrm{V} \mathrm{rms},$ and $5.72 \mathrm{A}$ (line current) at an 80 percent lagging power factor. Find the full-load efficiency.

Narayan Hari

Numerade Educator

Problem 29

A 440 -V-rms (line-to-line) $60-\mathrm{Hz}$ three phase wye-connected induction motor draws $16.8 \mathrm{A}$ at a power factor of 80 percent lagging The stator copper losses are $350 \mathrm{W}$, the rotor copper loss is $120 \mathrm{W}$, and the rotational losses are 400 W. Find the power crossing the air gap $P_{\mathrm{ag}},$ the developed power $P_{\text {dev }},$ the output power $P_{\text {out }},$ and the efficiency.

Narayan Hari

Numerade Educator

Problem 30

The torque-speed characteristics of a $60-\mathrm{Hz}$ induction motor and a load are shown in Figure $\mathrm{Pl} 7.30 .$ How many poles does the motor have? In steady-state operation, find the speed, the slip, the output power, and the rotor copper loss. Neglect rotational losses.

Narayan Hari

Numerade Educator

Problem 31

The rotational inertia of the motor and load of Problem $\mathrm{P} 17.30$ is $5 \mathrm{kgm}^{2}$. Estimate the time required for the motor to accelerate the load from a standing start to 1000 rpm. (Hint: The difference between the motor output torque and the load torque is approximately $25 \mathrm{Nm}$ in the range of speeds under consideration.

Narayan Hari

Numerade Educator

Problem 32

A two-pole 60 -Hz induction motor produces an output power of 5 hp at a speed of 3500 rpm. With no load, the speed is 3598 rpm. Assume that the rotational torque loss is independent of speed. Find the rotational power loss at 3500 rpm.

Narayan Hari

Numerade Educator

Problem 33

A four-pole $60-\mathrm{Hz}$ 240-V-rms induction motor operates at $1750 \mathrm{rpm}$ and produces 2 hp of output power. The load is a hoist that requires constant torque versus speed. Assume that the motor is operating in the range for which torque is proportional to slip and write an equation for motor torque in terms of slip when operating at rated voltage. Then modify the equation for operation from a 220-V-rms source. Estimate the speed when operating the hoist from a 220-V-rms-source.

Narayan Hari

Numerade Educator

Problem 34

Give two situations for which a synchronous motor would be a better choice than an induction motor in an industrial application.

Manish Kumar

Numerade Educator

Problem 35

List several methods for starting synchronous motors.

Vishal Gupta

Numerade Educator

Problem 36

What is a synchronous capacitor? What is the practical benefit of using one?

Vishal Gupta

Numerade Educator

Problem 37

Sketch the V curve for a synchronous motor. Label the axes. Indicate where the power factor is lagging and where it is leading Draw the phasor diagram corresponding to the minimum point on the V curve.

Carson Merrill

Numerade Educator

Problem 38

A synchronous motor is running at 75 percent of rated load with unity power factor. If the load increases to the rated output power, how do the following quantities change?
a. field current; b. mechanical speed; c. out put torque; d. armature current; e. power factor; f. torque angle.

Bruce Edelman

Numerade Educator

Problem 39

A synchronous motor is running at 100 per cent of rated load with unity power factor. If the field current is increased, how do the following quantities change? a. out-put power; b. mechanical speed; c. output torque; d. armature current; e, power factor; f. torque angle.

Dominador Tan

Numerade Educator

Problem 40

A six-pole 60 -Hz synchronous motor is operating with a developed power of 5 hp and a torque angle of $5^{\circ}$. Find the speed and developed torque. Suppose that the load increases such that the developed torque doubles Find the new torque angle. Find the pull-out torque and maximum developed power for this machine.

Narayan Hari

Numerade Educator

Problem 41

A 10 pole 60-Hz synchronous motor operates with a developed power of 100 hp, which is its rated full load. The torque angle is $20^{\circ}$. Plot the torque-speed characteristic to scale. showing the values for rated torque and for the pull-out torque.

Narayan Hari

Numerade Educator

Problem 42

An eight-pole $240-\mathrm{V}-$rms $60-\mathrm{Hz}$ deltaconnected synchronous motor operates with a constant developed power of 50 hp, unity power factor, and a torque angle of $15^{\circ}$. Then, the field current is increased such that $B_{r}$ increases in magnitude by 20 percent. Find the new torque angle and power factor. Is the new power factor leading or lagging?

Narayan Hari

Numerade Educator

Problem 43

A six-pole $240-\mathrm{V}-\mathrm{rms}$ $60-\mathrm{Hz}$ delta-connected synchronous motor operates with a developed power of 50 hp, unity power factor, and a torque angle of $15^{\circ}$. Find the phase current. Suppose that the load is removed so that the developed power is zero. Find the new values of the current, power factor, and torque angle.

Narayan Hari

Numerade Educator

Problem 44

A certain 480-V-rms delta-connected syn chronous motor operates with zero developed power and draws a phase current of 15 A. which lags the voltage. The synchronous reactance is $5 \Omega$. The field current is 5 A. Assuming that the rotor field magnitude is proportional to field current, what field current is needed to reduce the armature current to zero?

Narayan Hari

Numerade Educator

Problem 45

A $240-\mathrm{V}-\mathrm{rms}$ delta-connected $100-\mathrm{hp}$ $60-\mathrm{Hz}$ six-pole synchronous motor operates with a developed power (including losses) of 50 hp and a power factor of 90 percent leading. The synchronous reactance is $X_{s}=0.5 \Omega$ a. Find the speed and developed torque, b. Determine the values of $\mathbf{I}_{a}, \mathbf{E}_{r},$ and the torque angle. c. Suppose that the excitation remains constant and the load torque increases until the developed power is 100 hp. Determine the new values of $\mathbf{I}_{a}, \mathbf{E}_{r},$ the torque angle, and the power factor.

Narayan Hari

Numerade Educator

Problem 46

A $240-\mathrm{V}-\mathrm{rms}$ $100-\mathrm{hp}$ $60-\mathrm{Hz}$ six-pole delta-connected synchronous motor operates with a developed power (including losses) of 100 hp and a power factor of 85 percent lagging. The synchronous reactance is $X_{j}=$ $0.5 \Omega$. The field current is $l_{f}=10$ A. What must the new ficld current be to produce 100 percent power factor? Assume that magnetic saturation does not occur so that $B_{r}$ is proportional to $I_{f}$.

Narayan Hari

Numerade Educator

Problem 47

a. A 12-pole $60-\mathrm{Hz}$ synchronous motor drives a 10-pole synchronous machine that acts as a generator. What is the frequency of the voltages induced in the armature windings of the generator? b. Suppose that we need to drive a load at exactly 1000 rpm. The power available is 60-Hz three-phase. Diagram a system of synchronous machines to drive the load, specifying the number of poles and frequency of operation for each. (Multiple correct answers exist.)

Narayan Hari

Numerade Educator

Problem 48

Suppose that a synchronous machine is instrumented to measure its armature current, armature voltage, and field current. The field circuit contains a rheostat so that the field current can be adjusted, Discuss how to adjust the field current to obtain unity power factor.

Narayan Hari

Numerade Educator

Problem 49

A $60-\mathrm{Hz}$ $480-\mathrm{V}-\mathrm{rms}$ $200-\mathrm{hp}$ delta-connected
synchronous motor runs under no-load conditions. The field current is adjusted for minimum line current, which turns out to be $16.45 \mathrm{A}$ rms. The per-phase armature impedance is $R_{s}+j X_{r}=0.05+j 1.4 .$ (Until now in this chapter, we have neglected $R_{s}$. However, it is significant in efficiency calculations.) Estimate the efficiency of the machine under full-load conditions operating with 90 percent leading power factor.

Narayan Hari

Numerade Educator

Problem 50

A $1-\mathrm{hp}$ $120-\mathrm{V}-\mathrm{rms}$ $1740-\mathrm{rpm}$ $60-\mathrm{Hz}$ capacitor-start induction-run motor draws a current of $10.2 \mathrm{A}$ rms at full load and has an efficiency of 80 percent. Find the values of a. the power factor and b. the impedance of the motor at full load. c. Determine the number of poles that the motor has.

Narayan Hari

Numerade Educator

Problem 51

Assuming small slip, the output power of a single-phase induction motor can be written as $P_{\text {out }}=K_{1} s-K_{2},$ where $K_{1}$ and $K_{2}$ are constants and $s$ is the slip. A 0.5-hp motor has a full-load speed of 3500 rpm and a no-load speed of 3595 rpm. Determine the speed for $0.2-$ hp output.

Narayan Hari

Numerade Educator

Problem 52

The winding impedances under starting conditions for a $60-\mathrm{Hz}$ 0.5-hp motor are shown in Figure $\mathrm{P} 17.52 .$ Determine the capacitance $C$ needed so that the phase angle between the currents $\mathbf{I}_{a}$ and $\mathbf{L}_{m}$ is $90^{\circ}$.

Narayan Hari

Numerade Educator

Problem 53

A farm house is located at the end of a country road in northern Michigan. The Thevenin impedance seen looking back into the power line from the electrical distribution panel is $0.2+j 0.2 \Omega$. The The voltage is $240 \mathrm{V}$ $\mathrm{rms} 60 \mathrm{Hz}$ ac. A 2-hp 240-V-rms capacitor-start motor is used for pumping water. We want to estimate the voltage drop observed in the house when the motor starts. Typically, such a motor has a power factor of 75 percent and an efficiency of 80 percent at full load. Also, the starting current can be estimated as six times the full-load current. Estimate the worst-case percentage voltage drop observed when the motor starts.

Narayan Hari

Numerade Educator

Problem 54

How could the direction of rotation of a single-phase capacitor-start induction motor be reversed?

Vishal Gupta

Numerade Educator

Problem 55

Which would be more suitable for use in a portable vacuum cleaner, an induction motor or a universal motor? For the fan in a heating system? For the compressor motor in a refrigerator? For a variable-speed handheld drill? Give the reasons for your answer in each case,

Narayan Hari

Numerade Educator

Problem 56

Sketch the cross section of a reluctance stepper motor that has six stator poles and eight rotor poles Your sketch should be similar to Figure $17.33(\mathrm{a})$ on page $845 .$ Label the windings and specify the sequence of activation for clockwise rotation. What is the rotation angle per step?

Nicole Hoffman

Numerade Educator

Problem 57

Use the Web to find two or more sources for stepper motors.

Vishal Gupta

Numerade Educator

Problem 58

List several advantages of brushless dc motors compared to conventional dc motors.

Narayan Hari

Numerade Educator