ISM to Accompany Electric Machinery and Power System Fundamentals 1/e

Chapter 7: Induction Motors

7-1. A dc test is performed on a 460-V -connected 100-hp induction motor. If VDC = 21 V and IDC = 72 A, what is the stator resistance R1 ? Why is this so? SOLUTION If this motor's armature is connected in delta, then there will be two phases in parallel with one phase between the lines tested.

VDC

R1

R1

R1

Therefore, the stator resistance R1 will be

VDC I DC

=

R1(R1 + R1 ) R1 + (R1 + R1 )

=

2 3

R1

R1

=

3 VDC 2 I DC

=

3 ?? 21 V ?? = 0.438 2 ? 72 A ?

7-2. A 220-V three-phase six-pole 50-Hz induction motor is running at a slip of 3.5 percent. Find: (a) The speed of the magnetic fields in revolutions per minute

(b) The speed of the rotor in revolutions per minute

(c) The slip speed of the rotor

(d) The rotor frequency in hertz

SOLUTION

(a) The speed of the magnetic fields is

nsync

= 120 fe P

= 120(50 Hz)

6

= 1000 r/min

(b) The speed of the rotor is

nm = (1 - s) nsync = (1 - 0.035) (1000 r/min) = 965 r/min

(c) The slip speed of the rotor is

nslip = snsync = (0.035)(1000 r/min) = 35 r/min

(d) The rotor frequency is

fr

=

nslip P 120

=

(35

r/min) (6)

120

= 1.75

Hz

7-3. Answer the questions in Problem 7-2 for a 480-V three-phase four-pole 60-Hz induction motor running at a slip of 0.025.

SOLUTION

(a) The speed of the magnetic fields is

114

nsync

= 120 fe P

= 120(60

4

Hz)

= 1800

r/min

(b) The speed of the rotor is

nm = (1 - s) nsync = (1 - 0.025) (1800 r/min) = 1755 r/min

(c) The slip speed of the rotor is

nslip = snsync = (0.025) (1800 r/min) = 45 r/min

(d) The rotor frequency is

fr

=

nslip P 120

=

( 45

r/min ) ( 4)

120

= 1.5

Hz

7-4. A three-phase 60-Hz induction motor runs at 715 r/min at no load and at 670 r/min at full load. (a) How many poles does this motor have? (b) What is the slip at rated load? (c) What is the speed at one-quarter of the rated load? (d) What is the rotor's electrical frequency at one-quarter of the rated load?

SOLUTION (a)

This machine has 10 poles, which produces a synchronous speed of

nsync

= 120 fe P

= 120(60

10

Hz)

= 720

r/min

(b) The slip at rated load is

s = nsync - nm ?100% = 720 - 670 ?100% = 6.94%

nsync

720

(c) The motor is operating in the linear region of its torque-speed curve, so the slip at ? load will be s = 0.25(0.0694) = 0.0174

The resulting speed is

nm = (1 - s) nsync = (1 - 0.0174) (720 r/min) = 707 r/min

(d) The electrical frequency at ? load is

fr = sfe = (0.0174) (60 Hz) = 1.04 Hz

7-5. A 50-kW 440-V 50-Hz two-pole induction motor has a slip of 6 percent when operating at full-load conditions. At full-load conditions, the friction and windage losses are 520 W, and the core losses are 500 W. Find the following values for full-load conditions: (a) The shaft speed nm

(b) The output power in watts

(c) The load torque load in newton-meters

(d) The induced torque ind in newton-meters

115

(e) The rotor frequency in hertz

SOLUTION

(a) The synchronous speed of this machine is

nsync

= 120 fe P

= 120(50

2

Hz)

= 3000

r/min

Therefore, the shaft speed is

nm = (1 - s) nsync = (1 - 0.06) (3000 r/min) = 2820 r/min

(b) The output power in watts is 50 kW (stated in the problem).

(c) The load torque is

load

=

POUT m

=

50 kW

(2820 r/min)?? 2 rad ???? 1 min ??

= 169.3 N m

? 1 r ?? 60 s ?

(d) The induced torque can be found as follows:

Pconv = POUT + PF&W + Pcore + Pmisc = 50 kW + 520 W + 500 W = 51.2 kW

ind

=

Pconv m

=

51.2 kW

(2820 r/min)?? 2 rad ????1 min ??

= 173.4 N m

? 1 r ?? 60 s ?

(e) The rotor frequency is

fr = sfe = (0.06) (50 Hz) = 3.00 Hz

7-6. A three-phase 60-Hz two-pole induction motor runs at a no-load speed of 3580 r/min and a full-load speed of 3440 r/min. Calculate the slip and the electrical frequency of the rotor at no-load and full-load conditions. What is the speed regulation of this motor [Equation (4-57)]?

SOLUTION The synchronous speed of this machine is 3600 r/min. The slip and electrical frequency at noload conditions is

s nl

=

nsync - nnl nsync

? 100%

=

3600 - 3580 3600

? 100%

=

0.56%

fr,nl = sfe = (0.0056) (60 Hz) = 0.33 Hz

The slip and electrical frequency at full load conditions is

s fl

=

nsync - nnl nsync

? 100%

=

3600 - 3440 3600

? 100%

=

4.44%

fr,fl = sfe = (0.0444) (60 Hz) = 2.67 Hz

The speed regulation is

SR = nnl - nfl ?100% = 3580 - 3440 ?100% = 4.1%

nfl

3440

116

7-7. A 208-V four-pole 60-Hz Y-connected wound-rotor induction motor is rated at 15 hp. Its equivalent circuit components are

R1 = 0.220

R2 = 0.127

X M = 15.0

X1 = 0.430

X2 = 0.430

Pmech = 300 W

Pmisc 0

Pcore = 200 W

For a slip of 0.05, find (a) The line current IL

(b) The stator copper losses PSCL

(c) The air-gap power PAG

(d) The power converted from electrical to mechanical form Pconv

(e) The induced torque ind

(f) The load torque load

(g) The overall machine efficiency

(h) The motor speed in revolutions per minute and radians per second

SOLUTION The equivalent circuit of this induction motor is shown below:

IA

R1

jX1

jX2

R2

+

0.22 j0.43

j0.43 0.127

V

j15 jXM

-

R2

?? ?

1

- s

s

?? ?

2.413

(a) The easiest way to find the line current (or armature current) is to get the equivalent impedance Z F

of the rotor circuit in parallel with jX M , and then calculate the current as the phase voltage divided by the sum of the series impedances, as shown below.

IA

R1

jX1

jXF

RF

+

0.22 j0.43

V

-

The equivalent impedance of the rotor circuit in parallel with jX M is:

ZF =

1= 1 +1

1

1 +

1

= 2.337 + j0.803 = 2.4719?

jX M Z2 j15 2.54 + j0.43

117

The phase voltage is 208/ 3 = 120 V, so line current I L is

IL

= IA

=

R1 +

V jX1 + RF

+

jX F

=

0.22 +

1200? V j0.43 + 2.337 +

j0.803

I L = I A = 42.3 - 25.7? A

(b) The stator copper losses are

PSCL

=

3I

2 A

R1

=

3(42.3

A)2

(0.22

)

=

1180

W

(c)

The air gap power is

PAG

=

3I

2 2

R2 s

= 3I A2 RF

(Note that

3I A2RF

is equal to

3I 2 2

R2 s

, since the only

resistance in the original

rotor circuit was

R2

/s,

and the resistance in the Thevenin equivalent circuit is RF . The power consumed by the Thevenin equivalent circuit must be the same as the power consumed by the original circuit.)

PAG

=

3I

2 2

R2 s

=

3I

2 A

RF

=

3(42.3

A)2 (2.337

)

= 12.54

kW

(d) The power converted from electrical to mechanical form is

Pconv = (1 - s) PAG = (1 - 0.05) (12.54 kW) = 11.92 kW

(e) The induced torque in the motor is

ind

=

PAG sync

=

12.54 kW

(1800 r/min)?? 2 rad ????1 min ??

= 66.5 N m

? 1 r ?? 60 s ?

(f) The output power of this motor is

POUT = Pconv - Pmech - Pcore - Pmisc = 11.92 kW - 300 W - 200 W - 0 W = 11.42 kW

The output speed is

nm = (1 - )s nsync = (1 - 0.05) (1800 r/min) = 1710 r/min

Therefore the load torque is

load

=

POUT m

=

11.42 kW

(1710 r/min)?? 2 rad ????1 min ??

= 63.8 N m

? 1 r ?? 60 s ?

(g) The overall efficiency is

= POUT ?100% = POUT ?100%

PIN

3V I A cos

=

3(120

11.42 kW

V) (42.3 A) cos 25.7?

? 100%

=

83.2%

(h) The motor speed in revolutions per minute is 1710 r/min. The motor speed in radians per second is

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