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1102

IEEE TRANSACTIONS

ON

INDUSTRY APPLICATIONS, VOL. 26, NO.

6.

NOVEMBERIDECEMBER 1990

Derating of Induction Motors Due to

Waveform Distortion

Abstract-Electrical motors are designed on the basis ofbalanced

three-phase sinusoidal input voltage. Nonsinusoidal voltage hasdetri-

mental effects on induction motor performance, and derating ofthe

machine is required. IEEE Standard 519 suggests that no deratingof the

motor would be necessary for a harmonic content of up to

5070.

The

derating of induction motors due to harmonic distortion isdiscussed in

detail.

INTRODUCTION

HE OUTPUT of an induction motor depends mainly on

T eating, and the motor’s life is shortened by overheating.

The temperature rise resulting from losses is, therefore, a

major factor in determining the machine output rating. The

presence of harmonics in the applied voltage can cause

excessive heating.

The amount of voltage distortion, measured by a “distor-

tion factor” (DF) and defined by IEE E Standard 519

[l]

as

1 I 2

sum of squares of am plitudes

of all harmonic voltages

square of amplitude of

fundamental voltage

] 1)

F =

I

is used to establish harmonic limits. On industrial power

systems, the voltage distortion is limited to

5 .

However, no

limit is specified in regard to the individual harmoniccontent.

Derating of ‘‘N EM A design B” induction motors of different

output ratings and for two types of enclosures (drip-proofand

totally enclosed) due to different cases of harmonic distor-

tions are discussed in this paper.

ANALYTICALA CK G RO U N D

General Assump tions

grounded.

ing conditions.

1) The motors are nonskewed, Y -connected, and un-

2)

The analysis is limited to full-load steady-state operat-

3) The principle of superposition applies.

Equivdent Circuit Parameters

The simplified equivalent circuit of a three-phase induction

motor is shown

in

Fig. 1. It is to be remembered that this

Paper PID 90-29, approved by the Petroleum and ChemicalIndustry

Committee of the IEE E Industry Applications Society forpresentation at the

1989 Petroleum and Chemical Industry Technical Conference, SanDiego,

CA, September 11-13. Manuscript released for publicationMarch

8,

1990.

The authors are with the Department

of

Electrical Engineering and

Computer Science, University of Colorado at Denver, 1200 LarimerStreet,

Campus

Box 110

Denver, CO

80204.

IEEE

Log

Number 9038525.

Stator Quantities Rotor Quantities

s x ‘R/s ‘R

2

7

>

I

I 1 4

Magnetic Core

Fig. 1.

Equivalent circuit

of

induction m otor.

equivalent circuit does not take into account any time or

space harmonics. The various resistances and reactances are

referred to the stator winding and are expressed in per unit

(pu) at the machine base. These parameters ( R , ,

R,,

X,

=

W L , , X , , X R , )

are assumed to be constants. This is true

only for a given operating condition. They vary with changes

in motor current, speed, voltage, and temperature

[2].

Motor Losses

The total motor losses (P, , , ) consist of iron, winding,

mechanical, and stray losses.

Iron Losses (Pf,):

For an alternating magnetic field, losses

that occur in the iron consist of hysteresis loss and eddy

current loss. In general, for sinusoidal flux, the losses in

Watt/kg of iron (for a given lamination thickness) can be

expressed as

p f e = k,fBi

+

k, fBm)’

(2 )

where the first term accounts for the hysteresis loss andthe

second term for the eddy current loss. The constants

kh

nd

k ,

depend on the properties of the material.

B,

is the

maximum flux density and is proportional to the air-gap

voltage E. If the flux density

B,,,

s uniform over the

cross-sectional area

A

of the core,

where

E

is the rms value of the air-gap voltage,

C ,

is

a

machine constant, and

f is

the frequency. Substituting

3)

in

(2)

gives

‘fe = [ kh l/f) -k k e ] E Cm ) 2 . ( 4 )

Winding Losses Pcu,and Pcu2):

These losses are stator

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990 IEEE

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SEN

AND LANDA:

DERATING

OF INDUCTION MOTORS DUE TO WAVEFORM DISTORTION

1103

and rotor Z2R losses caused by the current flowing through

the respective winding.

Mechanical Losses (Pmech):The mechanical losses com-

prise of friction and windage losses. They are approximately

proportional to the square of the speed and to the contact

surface area. These losses will be assumed to be unaffected

by voltage harmonic distortion.

Stray-Load Losses (Psr roy):

hese are additional iron and

eddy current losses caused by the increase in air-gapleakage

flux with load, and by high-frequency pulsation fluxes.These

losses can be divided into six components as follows:

1

the eddy current loss in the stator copper

W ,

due to slot

leakage flux (normally neglected);

2) the losses in the motor end structure

We

due to end

leakage flux;

3)

the high-frequency rotor and stator surface losses

W

due to zig-zag leakage flux;

4) the high-frequency tooth pulsation and rotor

Z2R

losses

W , , also due to the zig-zag leakage flux;

5) the six-times-frequency (for three-phase machines) ro-

tor Z2R losses

w,

due to circulating currents induced

by the stator belt leakage flux;

6) The extra iron losses in motors with skewed slots

W ,

due to skew leakage flux (neglected here because of the

nonskewed assumption).

The equations for these components given

in [3]

are (with

some changes in notation)

We = CeZ:fl 5 )

w,

=

W l

zo)2Bg2

(6)

(7)

w

= Cbk,.,,,R,I;

(8)

W , = C , k , , R , ( C , Z ~ C I Z t )

where

C,, C, , C, , C, , CO,

nd

C ,

are constants that depend

on the m achine and other empirical factors,

k , ,

and

k , ,

are

the skin effect coefficients for the rotor ba rs at the statorslot

harmonic frequency and at the phase belt frequency, respec-

tively, B , is the average flux density over the effective

air-gap area,

Io

is the no-load current, Z is the stator

current, and

f,

s the line frequency. For typical standard

NEMA design

B

machines operating at full load, the losses

can be distributed as [4]

Pmech

0.09

Pcul =

0 . 37

Pcu2 =

0 . 1 8

Pfe =

0.20

Pstray

0 . 1 6

Pdiss=

1 oo

Thermal Stress

[2]

[SI

The allowable hot-spot temperature in the stator winding

determined by the insulation class governs the output of a

machine. Assuming a lumped parameter approach, Figs. 2

and

3

show the simplified thermal equivalent circuit of a

drip-proof radially cooled and a totally enclosed fan-cooled

machine.

Air Temp

b)

Thermal network.

Fig.

2.

Drip-proof radially cooled squirrel-cage motor. (a) Air flow.(b)

R j i .

Air Temp

( b )

Thermal network.

Fig. 3 .

Totally enclosed fan-cooled squirrel-cage motor. (a) Air

f low.

(b)

The temperature rise of the stator winding T is then (for

a drip-proof radially cooled machine)

Twl = PculRewl 9)

where

R e w ,

s the thermal resistance of the stator end-wind-

ing. F or a totally enclosed fan-cooled machine, thetempera-

ture rise is

~ w l

pcu1Ri

+

(PcuI

+

p c u 2

+

Pfe

+

stray) ,

+ (Pcul + Pcu2 + Pfe + Pstray + Pmech) ,

10)

where R i , R j , and R , are the thermal resistances of theslot

insulation, stator backiron (including air-gap between the

stator core and outside frame of the motor), and outside

frame and moving air, respectively.

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1104

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS. VOL. 26, N O. 6.NOVEMBERIDECEMBER 1990

DERATINGF INDUCTIONMOTORS

UE

O

HARMONICS

Voltage Waveshape

general form as

The nonsinusoidal supply voltage can be expressed

in

a

n

( t )

=

Jz V ,

s i n w t

+ v,sin kwt + 0,) 11)

where

V I

is the fundamental voltage, v k represents har-

monic voltages of order k and 6 , is the harmonic phase

angle. The fundamental and the 4,7,

10,

13;

* , [ 3 n +

11,

n

=

1 , 2 ,

* , order voltage harmonics contribute to a

rotating magnetomotive force (MMF)

in

the direction of

motion and hence results in the development of a .positive

torque.

T h e 2 , 5 , 8 , 1 1 ; . . , [ 3 n + 2 ] , n = 0 , 1 , 2 ; * .,

or-

der of harmonics results in a rotating MMF in a direction

opposite to the direction of motion of the rotor and hence

contributes to a negative torque. The

3 , 6 , 9 ,

12; e [3n

31,

n

= 0, 1 , 2 , order of harmonics produces no rotating

MMF and, therefore, no torque.

[

k = 2

Equivalent Circuit Parameters

Based on assumption 3) , the temperature rise of the motor

can be computed by superposition as if a series of indepen-

dent generators were supplying the motor. Each generator

would represent one

of

the voltage terms in (1 1). Each of

these voltages would produced stator and rotor currents.

Evaluation of the reactances and resistances in these series

of equivalent circuits ought to be done at the corresponding

harmonic frequencies.

The actual frequency of the current

in

the stator is

[k

f

]

and in the rotor

[

k * f,

sk],

where

f

is the fundamental

frequency and sk is the slip for the kth harmonic. At these

frequencies, skin and proxim ity effects should beconsidered

in

inductances and resistances. The synchronous speed corre-

sponding to the applied frequency [

k * f,

is

[ k N , ] , N ,

being the synchronous speed in revolutions per m inutecorre-

sponding to the fundamental (i.e.,

N ,

=

120

.fl

p , p

is the

number of poles). Therefore, the slip sk at any speed N of.

the rotor is given by

kN,

+

N ,

Sk =

( 1 2 )

kNS

The plus sign has been used to account for the fact thatsome

harmonics result

in

rotating MMF’s in the same direction as

the motion of the rotor, while others result in rotating

MMF’s

in

the opposite direction of the motion of the rotor.

In terms of the slip s corresponding to the fundamental),

N

= 1 s ) N , .

(13)

Therefore,

kN, + 1 s ) N , k + (1

S

. (14)

k

=

w s

k

To arrive at appropriate values for the inductances

in

the

circuit with the motor operating with nonsinusoidal voltage,

the effects of harmonic voltage and currents on the degreeof

RSk

‘Sk

Rpk ‘Rk

“k

~

(b )

tion.

(a)

Fundamental. b)

k,

harmonic.

Fig. 4.

Equivalent circuit

for

induction motor with nonsinusoidal excita-

saturation must also be considered. The effect of saturationis

to limit the flux

in

the iron of the flux paths, causing a

reduction

in

the inductances.

Eflect of Saturation

To determine the effect of saturation in the no-load current

due to harmonic distortion, a new modified magnetizing

reactance is calculated as

Xm,

= Xrn

lMf

(15)

where the factor

M ,

is defined as

M =

/=//-

n = O

( 1 6 )

where i , and i , , are the instantaneous magnetizingcurrent

corresponding to the flux density wave for normal and abnor-

mal conditions, respectively.

Results of the increase

in

no-load current due to saturation

were verified experimentally by lab tests on a 2-hpinduction

motor

[ 6 ] .

Determination of Harmonic Currents

When the motor is running near the fundamental syn-

chronous speed, the harmonic equivalent circuit is quite

similar to the locked rotor equivalent circuit for theparticular

harmonic being considered. The magnetizing branch may be

neglected since the magnetizing reactance for the kth har-

monic

( k X , )

is much greater than the rotor leakage

impedance for the kth harmonic ( Z , , ) . For a similarreason,

the resistors

R

and R , , representing core (and m echanical)

losses for the fundamental and the different harmonics are

also neglected. Fig. 4(a) shows the sim plified equivalent

circuit for the fundamental component and Fig. 4(b) for the

kth harmonic. The kth harmonic current is then given as

‘k

I , =

[ (

R S k

+ R R k

/ ’ I 2

+ ( x . S k +

X R k ) 2 ]

where Vk is the voltage due to the kth harmonic,

R

and

R

the stator and rotor resistances and

x s k

and X , , the

stator and rotor reactances for the kth harmonic. The total

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SE N A N D L A N D A : D E R A T I N G

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I N D U C T I O N M O T O R S D U E T O W A V E FO R M D I ST OR T I O N

1105

harmonic current is

(18)

Motor Losses

Iron Losses:

The presence of time harmonics results in

higher saturation of the magnetic paths. Consequently, iron

losses will increase. This increase can be estimated bysubsti-

tuting the corresponding harmonic voltages and frequencies

in (4).

Winding Losses: Assuming the skin effect in the stator

winding to be negligible, the stator copper loss on anonsinu-

soidal supply is proportional to the square of the total rms

current. If R , is the stator resistance, the loss per phaseis

P C U I = C I S R S (19)

I,,, =

JI 1; .

p c u , =

w

1;)

(20)

(21)

Then

where

I f R ,

represents the loss due to the fundamental and

I i R ,

accounts for the losses due to harmonics.

When the rotor conductor depth is appreciable (as in large

machines) skin effect should be taken into account. Loss due

to each harmonic must be considered separately and then

added. In the case of a deep bar machine rotor, the totalrotor

copper loss per phase is

n

pcu2

=

i k R R k -k i R R . (22)

k = 2

Here, the first term represents the loss due to harmonics,and

the second term will give the rotor copper loss due to the

fundamental. In orde r to maintain constant output torque,the

last term of (22) will vary an am ount given by (23), whichis

proportional to the resultant torque produced by theharmonic

currents

:

n

T k = z i k R R k / s k .

(23)

k = 2

Stray-Load Losses:

Voltage harmonics significantly affect

these losses. They can be estimated by adapting

(5)-(8)

to the

motor with harmonics as follows:

r

n 1

where

ZA

is the no-load current corrected for saturation and

I ; is the total stator current including harmonics.

RESULTS ND DI SCUSSI ONS

Using the equations derived in the preceding sections, a

computer program is developed. The temperature rise of

induction motors of different ratings (Table I) and twotypes

TABLE I

THREE-PHASENDUCTION M O TO R H A R A C TER I STI C S

7]

Full- Resistances and Reactan ces Rotor

Load in Per Unit Based on Full- Bar

Rating Slip Load kVA and Rated Voltage Height

(hP) ( ) R.s R R X ,

X.7

X

( m m )

Up to5 4.5

0.055 0.055

1.9 0.048 0.072 15

5-25 3 .0

0.040 0.04 2 .6

0.064 0.096 25

25-200

2.5 0 .030

0.03 3.2 0.068 0.102 35

200-1000 1.75 0.025

0.02

3.6 0 .068 0 .102

45

TABLE I1

H A R M O N I CONTENT

F

W A V EF O R M SSED

Waveform Voltage in

of

Harmonic Fundamental

k a b

C

2

3

4

5

6

7

8

9

10

11

12

13

17

19

5.00 0. I7

0.21

0.26

5.00 3.41

0.87

0.25

0.26

0.61

0.26

0.56

0.43

1.21

0.87

0.78

DF =

5 % 5 % 5 %

of enclosures are determined for three different cases of 5%

voltage distortion (Table 11). Triplen harmonics areneglected

in the calculations.

The thermal resistances R ; ,

R j ,

and R , are determined

for a 5-hp and for a 100-hp machine. The values found for

the 5-hp machine are used as representatives of machines

ranging from 0 to 25 hp, and those found for the 100-hp

machine f or machines ranging from 25 to

1000

hp. Insulation

class B is assumed for all cases.

Slot leakage inductance reduction d ue to harmonic distor-

tion is considered equal to 10 for all waveforms and rotor

slot leakage inductance is taken to be 30% of total rotor

leakage for all machines. In all the computations, an ANSI

M-22 steel and 1 O-T maximum air-gap magnetic flux density

is used.

The results of the increases in temperature with respect to

full-load norm al operating conditions defined by

where TH and TN are the hot-spot temperature of the ma-

chine when supplied with nonsinusoidal (or harmonic) and

sinusoidal voltages, respectively, are shown in Figs.

5

and 6.

Figs.

5

and

6

reveal that the second harmonic (case a) in a

nonsinusoidal voltage has the most pronouncecl effect on the

temperature rise. The fifth harmonic only or the assorted

higher order harmonics of small magnitudes (cases b and c)

do not have any appreciable effect on the temperature rise

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL.

26,

NO. 6 , NOVEMBERIDECEMBER 1990

17.5

15.0

12.5

H 10.0

7 - 5

References

a

- harm.

wavefozm a

b - harm. waveform b

c

- harm. waveform C

1

lo oi

.5

0 5 25 200 1000

OTOR RATINGS

HP)

Temperature increase on drip-proof radially cooled inductionmotor.

ig. 5 .

Reference9

b -

harm.

waveform b

-

harm.

vaveform

C

a

- ham waveform a

a

c

5 25 200 1000

PaTOR RATINGS HP)

Fig. 6 . Temperature increase on totally enclosed fan-cooledinduction

motor.

(less than 5 ). It is also clear that, for drip-proofradially

cooled machines, the percentage variation of temperaturerise

reduces as the size increases (negative slopes in Fig. 5 ) ,

whereas it increases in totally enclosed fan-cooled machines

(positive slopes in Fig.

6 ) .

Based on the initial results, it is decided to calculate the

derating due to second harmonics (case a) only. T odetermine

the derating, a computer program reduces the output power

of the machine until the temperature due to the abnormal

condition is less than or equal to the normal operating

temperature corresponding to class B insulation. Thederating

due to harmonic distortion is then calculated as

29)

pout H

derating,

= 1

ou t

where Pout

and Pout re the output power

of

the machine

1.00 t

I 1

0.95

D:

0.90

Lr

0.85

z

* 0.80

W

CI

0 70

4

I

I

I

c

0 5

25

200 1000

MOTOR RATINGS HP)

Derating factors due to harmonic voltage distortion. Dottedlines:

drip-proof radially cooled motor. Solid line: totally enclosedfan-cooled

motor.

Fig. 7.

when supplied with nonsinusoidal and with sinusoidal volt-

age, respectively. The results are shown in Fig.

7.

When supplied with waveform a, machines of both types

of enclosure and for all ratings would require derating at

5

harmonic distortion. Fig. 7 also shows the derating due to

8 % second harmonic.

CONCLUSION

While, in the cases of harmonic waveforms b and c,

5

limitations on harmonic content is acceptable, results forcase

a are significant. We find that a restriction for the second

harmonic should be included on the harmonic distortion

limits estab lished by IEEE Standard 519 [l], and derating

in

some cases should be considered for less than 5 % harmonic

distortion.

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SEN AND LANDA: DERATING O F INDUCTION M OTORS DUE TO WAVEFORMDISTORTION

1107

Drip-proof machines are found to be less affected by

harmonic distortion than the totally enclosed machines. Ef-

ficiency plays a very important role in the degree ofderating.

Less

efficient machines would require

a

higher derating. It is

also clear that smaller machines (less than 5 hp) areaffected

more by the harmonics than are larger machines.

REFERENCES

IEEE Guide

f o r

Harmonic Control and Reactive Compensation

of Static Power Converters, ANSIIIE EE Standard 519, Apr.1981.

E. Levi , Polyphase

Motors:

A Direct Approach to Their Design.

New York: Wiley, 1984.

P. L. Alger, G. Angst, and E. J . Davies, “Stray-load lossesin

polyphase induction machines,” AIEE Trans. Power App . Syst .,

vol. 78, pt. 111-A, pp. 349-357, June 1959.

J . C. Andreas,

Energy Ejicient Electric

Motors. New York:

Marcel Dekker, 1982.

Optimization

of

Induction

Motor

Ejic iency , vol. 1 EPRI, Univ.

of Colorado, Boulder, July 1885.

H. A. Landa, “Derating of induction motors due to harmonicsand

voltage unbalance,” M.S. thesis, Univ. of Colorado, Denver,Nov.

1987.

E. Clarke, Circuit Analysis of AC Power Systems. London: Wi-

ley, 1950.

B. N. Gafford, W. C. Duesterhoeft , Jr . , and C. C. Mosher,111,

“Heat ing of induction m otors on unbalanced voltages,” AIEETrans.

P o w er A p p . Sy s t . , pp. 282-288, June 1959.

E.

A. Klingshirm and H. E. Jordan, “Polyphase induction motor

performance and losses on non-sinusoidal voltage sources,”IEEE

T rans . P ow e r A p p . S y s t . , vol. PAS-87, pp. 624-631,M ar. 1968.

P. G . Cummings, “Estimating effect of system harmonics onlosses

and temperature rise of squirrel-cage motors,” IEEE paperno.

PCIC-85-7, 1985.

Pankaj K.

Sen

(S’71-M ’74-SM ’90) received the

B

S E E degree (with honors) from Jadavpur Uni-

versity, Calcutta, India, and the M.E and Ph D

degrees in electrical engineering from the Techni-

cal University of Nova Scotia, Halifax, NS.

Canada

He is currently an Associate Professor of Elec-

trical Engineering at the University of Colorado,

Denver His research interests include application

problems in machines and power systems and

power engineering education

Dr Sen is a Registered Professional Engine er in the State ofColorado

Hector

A . Landa (S’87-M’87) was born in

Paysandu, Uruguay, in 1956 He received the B S

degree in electromechanical engineering from the

University of Concepcion del Uruguay, Argentina,

and the M

S

degree in electrical engineering from

the University of Colorado, Denver

Since 1988 he has been employed by the U S

Government Bureau of Reclamation

in

the Electric

Power Branch as a Research Electrical Engineer

His areas of special interest are power systems

harmonics and computer simulation

Mr. Landa is a member of Eta Kappa Nu