Rotating electrical machines - Part 29: Equivalent loading and superposition techniques - Indirect testing to determine temperature rise

IEC 60034-29 provides various indirect load tests, the purpose of which is to determine the temperature rise of rotating electrical machines, including a.c. induction machines, a.c. synchronous machines and d.c. machines. The test methods in some cases provide, in addition, means of measuring or estimating other parameters such as losses and vibration. Applies to machines covered by IEC 60034-1 when they cannot be loaded to a specific condition. Is applicable to both motors and generators.

Drehende elektrische Maschinen - Teil 29: Verfahren der äquivalenten Belastung und Überlagerung – Indirekte Prüfung zur Ermittlung der Übertemperatur

Machines électriques tournantes - Partie 29: Techniques par charge équivalente et par superposition - Essais indirects pour déterminer l'échauffement

La CEI 60034-29 fournit différents essais en charge indirects destinés à déterminer l'échauffement des machines électriques tournantes, y compris les machines à induction à courant alternatif, les machines synchrones à courant alternatif et les machines à courant continu. Dans certains cas, les méthodes d'essai permettent en outre de mesurer ou d'estimer d'autres paramètres tels que les pertes et les vibrations. S'applique aux machines couvertes par la CEI 60034-1 lorsqu'il est impossible de les soumettre à un point de charge spécifique. Est applicable tant aux moteurs qu'aux génératrices.

Električni rotacijski stroji - 29. del: Nadomestna obremenitev in tehnika superpozicije - Posredno preskušanje za določanje povišanja temperature (IEC 60034-29:2008)

General Information

Status
Published
Publication Date
10-Jul-2008
Technical Committee
Drafting Committee
Parallel Committee
Current Stage
6060 - Document made available
Due Date
11-Jul-2008
Completion Date
11-Jul-2008

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SIST EN 60034-29:2008
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Rotating electrical machines - Part 29: Equivalent loading and superposition techniques -

Indirect testing to determine temperature rise (IEC 60034-29:2008)

Drehende elektrische Maschinen - Teil 29: Verfahren der äquivalenten Belastung und

Überlagerung – Indirekte Prüfung zur Ermittlung der Übertemperatur (IEC 60034-
29:2008)

Machines électriques tournantes - Partie 29: Techniques par charge équivalente et par

superposition - Essais indirects pour déterminer l'échauffement (CEI 60034-29:2008)

Ta slovenski standard je istoveten z: EN 60034-29:2008
ICS:
29.160.01 Rotacijski stroji na splošno Rotating machinery in
general
SIST EN 60034-29:2008 en,fr

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
SIST EN 60034-29:2008
---------------------- Page: 2 ----------------------
SIST EN 60034-29:2008
EUROPEAN STANDARD
EN 60034-29
NORME EUROPÉENNE
July 2008
EUROPÄISCHE NORM
ICS 29.160 Supersedes EN 61986:2002
English version
Rotating electrical machines -
Part 29: Equivalent loading and superposition techniques -
Indirect testing to determine temperature rise
(IEC 60034-29:2008)
Machines électriques tournantes - Drehende elektrische Maschinen -
Partie 29: Techniques par charge Teil 29: Verfahren der äquivalenten
équivalente et par superposition - Belastung und Überlagerung -
Essais indirects pour déterminer Indirekte Prüfung zur Ermittlung
l'échauffement der Übertemperatur
(CEI 60034-29:2008) (IEC 60034-29:2008)

This European Standard was approved by CENELEC on 2008-06-01. CENELEC members are bound to comply

with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard

the status of a national standard without any alteration.

Up-to-date lists and bibliographical references concerning such national standards may be obtained on

application to the Central Secretariat or to any CENELEC member.

This European Standard exists in three official versions (English, French, German). A version in any other

language made by translation under the responsibility of a CENELEC member into its own language and notified

to the Central Secretariat has the same status as the official versions.

CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Cyprus, the

Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,

Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,

Sweden, Switzerland and the United Kingdom.
CENELEC
European Committee for Electrotechnical Standardization
Comité Européen de Normalisation Electrotechnique
Europäisches Komitee für Elektrotechnische Normung
Central Secretariat: rue de Stassart 35, B - 1050 Brussels

© 2008 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.

Ref. No. EN 60034-29:2008 E
---------------------- Page: 3 ----------------------
SIST EN 60034-29:2008
EN 60034-29:2008 - 2 -
Foreword

The text of document 2/1476/FDIS, future edition 1 of IEC 60034-29, prepared by IEC TC 2, Rotating

machinery, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as

EN 60034-29 on 2008-06-01.
This European Standard supersedes EN 61986:2002.
The following dates were fixed:
– latest date by which the EN has to be implemented
at national level by publication of an identical
national standard or by endorsement (dop) 2009-03-01
– latest date by which the national standards conflicting
(dow) 2011-06-01
with the EN have to be withdrawn
Annex ZA has been added by CENELEC.
__________
Endorsement notice

The text of the International Standard IEC 60034-29:2008 was approved by CENELEC as a European

Standard without any modification.
__________
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SIST EN 60034-29:2008
- 3 - EN 60034-29:2008
Annex ZA
(normative)
Normative references to international publications
with their corresponding European publications

The following referenced documents are indispensable for the application of this document. For dated

references, only the edition cited applies. For undated references, the latest edition of the referenced

document (including any amendments) applies.

NOTE When an international publication has been modified by common modifications, indicated by (mod), the relevant EN/HD

applies.
Publication Year Title EN/HD Year
IEC 60034-1 2004 Rotating electrical machines - EN 60034-1 2004
Part 1: Rating and performance
1) 2)
IEC 60034-2-1 - Rotating electrical machines - EN 60034-2-1 2007
Part 2-1: Standard methods for determining
losses and efficiency from tests (excluding
machines for traction vehicles)
Undated reference.
Valid edition at date of issue.
---------------------- Page: 5 ----------------------
SIST EN 60034-29:2008
---------------------- Page: 6 ----------------------
SIST EN 60034-29:2008
IEC 60034-29
Edition 1.0 2008-03
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Rotating electrical machines –
Part 29: Equivalent loading and superposition techniques – Indirect testing to
determine temperature rise
Machines électriques tournantes –
Partie 29: Techniques par charge équivalente et par superposition – Essais
indirects pour déterminer l’échauffement
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
CODE PRIX
ICS 29.160 ISBN 2-8318-9598-7
---------------------- Page: 7 ----------------------
SIST EN 60034-29:2008
– 2 – 60034-29 © IEC:2008
CONTENTS

FOREWORD...........................................................................................................................0H3

INTRODUCTION.....................................................................................................................1H5

1 Scope...............................................................................................................................2H6

2 Normative references .......................................................................................................3H6

3 Symbols and units ............................................................................................................4H6

4 General test requirements ................................................................................................5H7

5 Superposition method.......................................................................................................6H8

5.1 Basic principles .......................................................................................................7H8

5.1.1 General .......................................................................................................8H8

5.1.2 Temperature rise .........................................................................................9H9

5.1.3 Estimation of temperature rise from reduced load tests ...............................10H9

5.2 Induction motors....................................................................................................11H10

5.2.1 Applicable tests .........................................................................................12H10

5.2.2 Method of reduced voltage and rated current.............................................13H10

5.2.3 Method of rated voltage and reduced current.............................................14H13

5.2.4 Method combining tests at reduced voltage and reduced current ...............15H14

5.3 Synchronous machines .........................................................................................16H14

5.3.1 Method of open circuit, short circuit, zero excitation ..................................17H14

5.3.2 Method of zero power factor and open circuit loading ................................18H16

5.4 DC machines.........................................................................................................19H17

6 Equivalent load method ..................................................................................................20H17

6.1 Principles ..............................................................................................................21H17

6.1.1 General .....................................................................................................22H17

6.1.2 Temperature rise .......................................................................................23H18

6.2 Induction motors....................................................................................................24H18

6.2.1 Forward short-circuit test ...........................................................................25H18

6.2.2 Modulated frequency method.....................................................................26H19

6.2.3 DC injection...............................................................................................27H20

6.2.4 Mixed-frequency or bi-frequency method ...................................................28H21

6.3 Synchronous machines – Zero power factor ..........................................................29H24

7 Preferred methods..........................................................................................................30H26

Annex A (informative) Example calculation ..........................................................................31H28

Figure 1 – Graphical superposition method for induction motors ...........................................32H12

Figure 2 – Derivation of field winding temperature rise at rated load (synchronous

machines) .............................................................................................................................33H16

Figure 3 – Test circuit for d.c.-injection equivalent load test..................................................34H20

Figure 4 – Mixed-frequency test – Generators in series ........................................................35H21

Figure 5 – Mixed-frequency test – Series transformer ...........................................................36H22

Figure 6 – Combination of torque and current in a mixed-frequency test ...............................37H23

Figure 7 – Rotor-feeding mixed-frequency method................................................................38H24

39H27

Table 1 – Preferred methods.................................................................................................

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SIST EN 60034-29:2008
60034-29 © IEC:2008 – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
______________
ROTATING ELECTRICAL MACHINES –
Part 29: Equivalent loading and superposition techniques –
Indirect testing to determine temperature rise
FOREWORD

1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising

all national electrotechnical committees (IEC National Committees). The object of IEC is to promote

international co-operation on all questions concerning standardization in the electrical and electronic fields. To

this end and in addition to other activities, IEC publishes International Standards, Technical Specifications,

Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC

Publication(s)”). Their preparation is entrusted to technical committees; any IEC National Committee interested

in the subject dealt with may participate in this preparatory work. International, governmental and non-

governmental organizations liaising with the IEC also participate in this preparation. IEC collaborates closely

with the International Organization for Standardization (ISO) in accordance with conditions determined by

agreement between the two organizations.

2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international

consensus of opinion on the relevant subjects since each technical committee has representation from all

interested IEC National Committees.

3) IEC Publications have the form of recommendations for international use and are accepted by IEC National

Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC

Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any

misinterpretation by any end user.

4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications

transparently to the maximum extent possible in their national and regional publications. Any divergence

between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in

the latter.

5) IEC provides no marking procedure to indicate its approval and cannot be rendered responsible for any

equipment declared to be in conformity with an IEC Publication.

6) All users should ensure that they have the latest edition of this publication.

7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and

members of its technical committees and IEC National Committees for any personal injury, property damage or

other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and

expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC

Publications.

8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is

indispensable for the correct application of this publication.

9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of

patent rights. IEC shall not be held responsible for identifying any or all such patent rights.

International Standard IEC 60034-29 has been prepared by IEC technical committee 2:

Rotating machinery. It cancels and replaces IEC 61986:2002 which is withdrawn.
The text of this standard is based on the following documents:
FDIS Report on voting
2/1476/FDIS 2/1491A/RVD

Full information on the voting for the approval of this standard can be found in the report on

voting indicated in the above table.

This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.

A list of all parts of IEC 60034 series, under the general title Rotating electrical machines, can

be found on the IEC website.
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SIST EN 60034-29:2008
– 4 – 60034-29 © IEC:2008

The committee has decided that the contents of this publication will remain unchanged until

the maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in

the data related to the specific publication. At this date, the publication will be

• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
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SIST EN 60034-29:2008
60034-29 © IEC:2008 – 5 –
INTRODUCTION

The object of this standard is to provide various indirect load tests, the purpose of which is to

determine the temperature rise of rotating electrical machines, including a.c. induction

machines, a.c. synchronous machines and d.c. machines. The test methods in some cases

provide, in addition, means of measuring or estimating other parameters such as losses and

vibration, but the methods are not designed specifically to provide such data.

The proposed test methods are considered equivalent, the choice relying only on the location,

the testing equipment and the machine type, and the test result accuracy.

This standard should not be interpreted as requiring any or all of the tests on any given

machine. Particular tests are subject to a special agreement between the manufacturer and

the purchaser.

NOTE As the methods reproduce only approximately the thermal conditions of the machines under rated

condition, temperature-rise measurement results achieved from tests with these methods may be taken as the

basis for the evaluation of machine heating in accordance with 8.10 of IEC 60034-1 by agreement between the

manufacturer and the purchaser.
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SIST EN 60034-29:2008
– 6 – 60034-29 © IEC:2008
ROTATING ELECTRICAL MACHINES –
Part 29: Equivalent loading and superposition techniques –
Indirect testing to determine temperature rise
1 Scope

This International Standard applies to machines covered by IEC 60034-1 when they cannot be

loaded to a specific condition (rated or otherwise). It is applicable to both motors and

generators.
2 Normative references

The following referenced documents are indispensable for the application of this document.

For dated references, only the edition cited applies. For undated references, the latest edition

of the referenced document (including any amendments) applies.
IEC 60034-1:2004, Rotating electrical machines – Part 1: Rating and performance

IEC 60034-2-1, Rotating electrical machines – Part 2-1: Standard methods for determining

losses and efficiency from tests (excluding machines for traction vehicles)
3 Symbols and units
For the purposes of this document, the following symbols and units apply.
K slope factor of temperature rise, K/W

NOTE 1 The full name of K is “slope factor of the straight line characterizing variation of temperature rise with

losses”, see IEC 60027-4, item 901.
Δθ temperature rise, K
θ temperature, °C
P power, loss, W
I current, A
R resistance, Ω
X reactance, Ω
U voltage, V
E e.m.f., V
f frequency, Hz
f main/auxiliary frequency, Hz
1,2
Δt time interval, s
T torque, N·m
J moment of inertia, kg·m
cosϕ power factor
γ method uncertainty, %

NOTE 2 The definition implies that γ > 0 means test temperature rise is higher than at actual load condition.

---------------------- Page: 12 ----------------------
SIST EN 60034-29:2008
60034-29 © IEC:2008 – 7 –
δ amplitude of frequency deviation, Hz
λ ratio of auxiliary voltage to main voltage
σ correction factor
ω angular frequency, rad/s
Subscripts
m, n, o, p test conditions

1, 2, 3, etc. machine component (e.g. stator winding, rotor winding, stator core, etc.)

NOTE 3 If not indicated otherwise, numbers 1, 2, 3 will be used as assigned above.

t test
f field
a ambient, referring to reference coolant (see IEC 60034-1, 8.2)
c due to constant losses
L leakage
N rated value
equiv equivalent-load test
super superposition test
4 General test requirements
Measurement of the electrical parameters shall be made as follows.

a) The class of accuracy of measuring instruments shall be not greater than 0,5.

b) The measuring range of analogue instruments shall be chosen with a view to the

measured values being higher than 30 % of the full-scale range. These requirements need

not be complied with in the case of the three-phase power measurement by means of two

wattmeters, but the currents and voltages in the measured circuits shall be at least 20 %

of the rated currents and voltages of the wattmeters being used. The range of the other

measuring instruments shall be chosen in such a way that the measuring errors are not

increased.

c) The waveform and symmetry of the supply voltage at the machine terminals shall be in

accordance with the requirements of Clause 7 of IEC 60034-1.

d) Each line current shall be measured. The arithmetic average value shall be used to

determine the machine operating point.

NOTE When using the two-wattmeter method, it is acceptable to measure only two currents.

e) Power input to a three-phase machine shall be measured by either two single-phase

wattmeters connected as in the two-wattmeter method, or one polyphase wattmeter, or

three single-phase wattmeters. The total power read on a wattmeter shall be reduced by

the amount of the I R loss in the voltage circuits or in the current circuits of the

instruments in accordance with their connection whenever this loss is a measurable

portion of the total power.

Unless otherwise indicated all electrical quantities to be measured are root-mean-square

values.
---------------------- Page: 13 ----------------------
SIST EN 60034-29:2008
– 8 – 60034-29 © IEC:2008
5 Superposition method
5.1 Basic principles
5.1.1 General

Superposition tests may be applied to any d.c. or a.c. machine. The method comprises a

series of tests at operating conditions other than rated load, for example: reduced load, no load,

short circuit, reduced voltage, positive (inductive) or negative (capacitive) reactive load.

The method allows the full-load temperature rise of various component parts of the machine

to be deduced. For each component, the loss shall be known at each particular test condition

and at full load. The machine should be tested with the same cooling conditions as when

operating at rated load. Hence, a locked-rotor test will not be suitable as the air-flow

distribution and magnitudes will be incorrect.

On completion of the individual tests, a series of equations based on equivalent thermal

circuit theory is constructed, each equation being of the form:
Δθ = K P + K P + K P
1m 11 1m 12 2m 13 3m
where
Δθ is the measured temperature rise of component 1 for test condition m;
P , P etc. is the loss in component 1, 2, etc. for test condition m;
1m 2m

K , K , etc. are the slope factors of temperature rise determining the temperature rise

11 12
of component 1 due to losses in component 1, and the temperature rise of
component 1 due to losses in component 2, etc.

Components 1, 2, and 3 may be, for example, the stator winding, the rotor winding, and the

stator iron.

In some test conditions, certain losses may be equal to zero, and hence the related term in

the equation disappears. For example, using the above assigned subscripts, a synchronous

machine has K P = 0 at no load and K P = 0 at short circuit.
11 1 13 3

The method is based on the principle that the coefficients K do not change from test to test,

i.e. that the cooling conditions are invariable between tests, which requires the speed to be

the same in each test. The method is also based on the principle of linear thermal conditions

so that temperature rises in one case can be added to those for another case. It requires the

losses in the relevant component parts to be known with sufficient accuracy for each case,

either by calculation or measurement.

When the tests have been completed and the equations compiled, the coefficients K can be

derived by simple arithmetic. These are then used in a final equation with the losses for the

rated load condition to calculate the temperature rise of component 1. By similar means, the

temperature rises at rated load of components 2, 3, etc. can be derived.

If any component loss is temperature dependent (for example, stator copper loss), then the

calculation procedure has to be repeated using values for the loss corrected for the estimated

temperature rise. It is normally necessary to do this iteration once only. For the calculation of

winding temperature rises corrected to a reference ambient temperature equations in closed

form are also provided.

The method may be used to determine the temperature rise of any component at any load if

the losses at that load are known. The slope factors of temperature rise (K , etc.) may be

useful in other thermal modelling studies, for example, in analysing the response to supply

unbalance, voltage reduction, etc.
---------------------- Page: 14 ----------------------
SIST EN 60034-29:2008
60034-29 © IEC:2008 – 9 –

In all superposition tests, correction is necessary for variation in heat exchanger performance

(if one is fitted to the machine), as the thermal performance of the heat exchanger will partly

depend on the total loss in each test.
5.1.2 Temperature rise

When determining the temperature rise values of machine parts by superposition tests, the

variations from the results that should be obtained at rated-load test are always to be

considered. The uncertainty value γ (%) for rated load is defined:
N,super
γ=−( 1)×100

NOTE 1 The uncertainty values obtained by superposition tests may be negative (test temperature rise is lower

than under normal operation) or positive (test temperature rise is larger than under normal operation).

Consequently, for comparing with the temperature rise values given in IEC 60034-1, test

results have to be multiplied with a correction factor σ:
σ =
100
NOTE 2 For negative uncertainty values the correction factor is > 1.
5.1.3 Estimation of temperature rise from reduced load tests

When estimating the temperature rise from tests at reduced load, the losses should be

separated into variable (load) losses and constant (iron, friction and windage) losses. For the

adjustment of temperature rise values, the machine may be considered as a two-component

system (see 40H5.1.1).

NOTE Depending on the enclosure and pole number of the machine, the temperature rise due to the constant

losses can be significant. Tests on large machines have shown that the separation of loss components results in

better agreement between the reduced load estimation and the full load actual test.

When a load test is performed at currents different from rated current, the I²R losses have to

be converted to full load with the squared ratio of the currents, and the resistance R has to be

corrected for total winding temperature. The following equation describes the adjustment of

temperature rise to full load, neglecting the effect of constant losses and of additional load

losses:
I 235 +θ
Nat
Δ⋅ = Δ⋅⎜⎟
11N ⎜⎟ t
⎜⎟N
⎝⎠1t
235θθ ( ) θ
++Δ − ⋅Δ
at 1t 1t
⎝⎠1t
where
I is the rated current;
I is the measured stator current;
θ is the measured temperature of the reference coolant;
Δθ is the measured stator winding temperature rise.

In cases where the portion of the winding temperature rise due to the constant losses is not

known, and the total temperature rise is assumed due to I²R losses only, the calculated

temperature rise will be too large. Therefore, this method can be used only when the effect of

constant losses is low; in most cases, methods taking constant and load losses separately

into account are preferred. Induction machines are tested according to 41H5.2.2.
---------------------- Page: 15 ----------------------
SIST EN 60034-29:2008
– 10 – 60034-29 © IEC:2008

When the temperature rise components due to load loss at reduced current and due to

constant loss are known, the total temperature rise can be calculated by using an iteration

procedure or, alternatively, by a closed-form equation. 42HAnnex A presents an example.

5.2 Induction motors
5.2.1 Applicable tests

In these methods the motor is tested at the following operating points as indicated:

Test m: reduced voltage, with the motor loaded to give rated current,
giving I , P and Δθ at U
1m 1m 1m m
Test n: the same reduced voltage as in test m, but at no load,
I , P and Δθ at U =U
giving
1n 1n 1n n m,
Test o: rated voltage at no load,
giving I , P and Δθ at U =U
,
1o 1o 1o o N
Test p: rated voltage and frequency, at reduced load,
giving I , P and Δθ at U =U .
1p 1p 1p p N
Preferably I is not less than 70 % of rated stator current,
Test q: reduced voltage, with the motor loaded, giving I , P and Δθ at U .
1q 1q 1q q
Preferably I is not less than 70% of rated stator current.

NOTE 1 Lower values of I in tests p and q may be used but will increase the uncertainty.

NOTE 2 Where applicable use Δθ Δθ Δθ Δθ Δθ for the rotor winding of wound rotor machines.

, , , ,
2m 2n 2o 2p 2q
5.2.2 Method of reduced voltage and rated current
5.2.2.1 General

The method requires a variable-voltage supply at rated frequency and either a loading

generator or braking equipment with a rating much less than the rating of the motor under

test. For each of the tests m, n, o the voltage, current, input power and stator winding

temperature rise are measured:

Δθ is the stator winding temperature rise due to rated stator current, quasi rated rotor

current and reduced-voltage iron loss and full friction and windage losses;

Δθ is the stator winding temperature rise due to reduced-voltage no-load stator current,

reduced-voltage iron loss and full friction and windage losses;

Δθ is the stator winding temperature rise due to rated-voltage no-load stator current,

rated-voltage no-load iron loss and friction and windage losses.

It should be noted that with large induction motors there may be cases when test m is not

practicable with a slip below the pull-out slip; operation above the pull-out slip is then an

alternative. To measure the stator winding temperature rise by the resistance method when

the machine is on no-load, some means to rapidly stop the motor should be employed when

shutting down, or the resistance should be measured directly under load (see 8.6.2 of

IEC 60034-1).

The method assumes that the cooling remains unchanged for each test, which implies that the

speed is also virtually unchanged.
The quantity Δθ can be determined with sufficient accuracy by using th
...

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