EN 62428:2008

SLOVENSKI STANDARD
01-januar-2009
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Electric power engineering - Modal components in three-phase a.c. systems - Quantities
and transformations (IEC 62428:2008)
Elektrische Energietechnik - Modale Komponenten in Drehstromsystemen - Größen und
Transformationen (IEC 62428:2008)
Energie électrique - Composantes modales dans les systèmes a.c. triphasés -
Grandeurs et transformations (CEI 62428:2008)
Ta slovenski standard je istoveten z: EN 62428:2008
ICS:
01.060 9HOLþLQHLQHQRWH Quantities and units
29.020 Elektrotehnika na splošno Electrical engineering in
general
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN 62428
NORME EUROPÉENNE
September 2008
EUROPÄISCHE NORM
ICS 01.060; 29.020
English version
Electric power engineering -
Modal components in three-phase a.c. systems -
Quantities and transformations
(IEC 62428:2008)
Energie électrique -  Elektrische Energietechnik -
Composantes modales Modale Komponenten
dans les systèmes a.c. triphasés - in Drehstromsystemen -
Grandeurs et transformations Größen und Transformationen
(CEI 62428:2008) (IEC 62428:2008)

This European Standard was approved by CENELEC on 2008-08-01. 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 62428:2008 E
Foreword
The text of document 25/382/FDIS, future edition 1 of IEC 62428, prepared by IEC TC 25, Quantities and
units, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as EN 62428 on
2008-08-01.
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) 2009-05-01
– latest date by which the national standards conflicting
with the EN have to be withdrawn (dow) 2011-08-01
Annex ZA has been added by CENELEC.
__________
Endorsement notice
The text of the International Standard IEC 62428:2008 was approved by CENELEC as a European
Standard without any modification.
In the official version, for Bibliography, the following notes have to be added for the standards indicated:
IEC 60909-0 NOTE  Harmonized as EN 60909-0:2001 (not modified).
IEC 61660 NOTE  Harmonized in EN 61660 series (not modified).
__________
- 3 - EN 62428:2008
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

1)
IEC 60050-141 - International Electrotechnical Vocabulary - -
(IEV) -
Part 141: Polyphase systems and circuits

1)
Undated reference.
IEC 62428
Edition 1.0 2008-07
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Electric power engineering – Modal components in three-phase a.c. systems –
Quantities and transformations

Energie électrique – Composantes modales dans les systèmes a.c. triphasés –
Grandeurs et transformations
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
S
CODE PRIX
ICS 01.060; 29.020 ISBN 2-8318-9921-4
– 2 – 62428 © IEC:2008
CONTENTS
FOREWORD.3
1 Scope.5
2 Normative references .5
3 Terms, definitions, quantities and concepts .5
3.1 General .5
3.2 Terms and definitions .5
4 Modal transformation.7
4.1 General .7
4.2 Power in modal components.8
4.3 Established transformations .10
5 Decoupling in three-phase a.c. systems .16
5.1 Decoupling in case of steady-state operation with sinusoidal quantities.16
5.2 Decoupling under transient conditions .19
Bibliography.23

Figure 1 – Circuit, fed by a three-phase voltage source with U , U , U at the
L1Q L2Q L3Q
connection point Q and earthed at the neutral point N via the impedance
Z = R + j X .16
N N
N
Figure 2 – Three decoupled systems which replace the coupled three-phase a.c.
system of Figure 1 under the described conditions (see text) .19

Table 1 – Power-variant form of modal components and transformation matrices.11
Table 2 – Power-invariant form of modal components and transformation matrices.12
Table 3 – Clark, Park and space phasor components – modal transformations in the
power-variant form.13
Table 4 – Clark, Park and space phasor components – Modal transformations in the
power-invariant form .14
Table 5 – Transformation matrices in the power-variant form for phasor quantities .15
Table 6 – Transformation matrices in the power-invariant form for phasor quantities.15
Table 7 – Modal voltages and impedances in case of phasor quantities .18
Table 8 – Modal voltages and inductances under transient conditions.22

62428 © IEC:2008 – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
ELECTRIC POWER ENGINEERING –
MODAL COMPONENTS IN THREE-PHASE AC SYSTEMS –
QUANTITIES AND TRANSFORMATIONS

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 62428 has been prepared by IEC technical committee 25:
Quantities and units.
The text of this standard is based on the following documents:
FDIS Report on voting
25/382/FDIS 25/390/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.

– 4 – 62428 © 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.
62428 © IEC:2008 – 5 –
ELECTRIC POWER ENGINEERING –
MODAL COMPONENTS IN THREE-PHASE AC SYSTEMS –
QUANTITIES AND TRANSFORMATIONS

1 Scope
This International Standard deals with transformations from original quantities into modal
quantities for the widely used three-phase a.c. systems in the field of electric power
engineering.
The examination of operating conditions and transient phenomena in three-phase a.c.
systems becomes more difficult by the resistive, inductive or capacitive coupling between the
phase elements and line conductors. Calculation and description of these phenomena in
three-phase a.c. systems are easier if the quantities of the coupled phase elements and line
conductors are transformed into modal quantities. The calculation becomes very easy if the
transformation leads to decoupled modal systems. The original impedance and admittance
matrices are transformed to modal impedance and admittance matrices. In the case of
decoupling of the modal quantities, the modal impedance and admittance matrices become
diagonal matrices.
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 60050-141, International Electrotechnical Vocabulary (IEV) – Part 141: Polyphase
systems and circuits
3 Terms, definitions, quantities and concepts
3.1 General
Quantities in this standard are usually time-dependent. These quantities are for instance
electric currents, voltages, linked fluxes, current linkages, electric and magnetic fluxes.
For quantities the general letter symbol g in case of real instantaneous values, g in case of
complex instantaneous values and G in case of phasors (complex r.m.s. values) are used.
NOTE Complex quantities in this standard are underlined. Conjugated complex quantities are indicated by an
additional asterisk (*). Matrices and column vectors are printed in bold face type, italic.
3.2 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-141 and the
following apply.
3.2.1
original quantities
quantities g or G of a three-phase a.c. system
NOTE Subscripts 1, 2, 3 are generally used in this standard; additional letters may be put, for instance L1, L2, L3
as established in IEC 60909, IEC 60865 and IEC 61660.

– 6 – 62428 © IEC:2008
3.2.2
modal components
quantities g , g or G found by a transformation from the original quantities according to
M
M
M
Clause 3
NOTE Additional subscripts 1, 2, 3 are used.
3.2.3
column vector of quantities
column matrix containing the three original quantities or modal components of a three-phase
a.c. system
NOTE Column vectors are described by g or g and G or G , respectively.
M
M
3.2.4
modal transformation
matrix equation T g = g for a column vector g containing the three unknown modal
M M
quantities, where g is a column vector containing the three given original quantities and T is a
3 × 3 transformation matrix
NOTE The transformation can be power-variant or power-invariant, see Tables 1 and 2.
3.2.5
inverse modal transformation
−1
solution g = T g of the modal transformation that expresses a column vector g containing
M M
−1
the three modal quantities as a matrix product of the inverse transformation matrix T by a
column vector g containing the three original quantities
3.2.6
transformation into symmetrical components
Fortescue transformation
linear modal transformation with constant complex coefficients, the solution of which converts
the three original phasors of a three-phase a.c. system into the reference phasors of three
symmetric three-phase a.c. systems — the so-called symmetrical components — , the first
system being a positive-sequence system, the second system being a negative-sequence
system and the third system being a zero-sequence system
NOTE 1 The transformation into symmetrical components is used for example for the description of asymmetric
steady-state conditions in three-phase a.c. systems.
NOTE 2 See Tables 1 and 2.
3.2.7
transformation into space phasor components
linear modal transformation with constant or angle-dependent coefficients, the solution of
which replaces the instantaneous original quantities of a three-phase a.c. system by the
complex space phasor in a rotating or a non-rotating frame of reference, its conjugate
complex value and the real zero-sequence component
NOTE 1 The term “space vector” is also used for “space phasor”.
NOTE 2 The space phasor transformation is used for example for the description of transients in three-phase a.c.
systems and machines.
NOTE 3 See Tables 1 and 2.
3.2.8
transformation into αβ0 components
Clarke transformation
linear modal transformation with constant real coefficients, the solution of which replaces the
instantaneous original quantities of a three-phase a.c. system by the real part and the

62428 © IEC:2008 – 7 –
imaginary part of a complex space phasor in a non-rotating frame of reference and a real
zero-sequence component or replaces the three original phasors of the three-phase a.c.
system by two phasors (α and β phasor) and a zero-sequence phasor
NOTE 1 The power-variant form of the space phasor is given by g = g + j g and the power-invariant form is
α β
s
given by g = (g + j g ) .
α β
s
NOTE 2 The αβ0 transformation is used for example for the description of asymmetric transients in three-phase
a.c. systems.
NOTE 3 See tables 1 and 2.
3.2.9
transformation into dq0 components
Park transformation
linear modal transformation with coefficients sinusoidally depending on the angle of rotation,
the solution of which replaces the instantaneous original quantities of a three-phase a.c.
system by the real part and the imaginary part of a complex space phasor in a rotating frame
of reference and a real zero-sequence component
NOTE 1 The power-variant form of the space phasor is given by g = g + j g and the power-invariant form is
d q
r
given by g = (g + j g ) .
d q
r
NOTE 2 The dq0 transformation is normally used for the description of transients in synchronous machines.
NOTE 3 See Tables 1 and 2.
4 Modal transformation
4.1 General
The original quantities g , g , g and the modal components g , g , g are related to each
1 2 3
M1 M2 M3
other by the following transformation equations:
⎛ ⎞
g ⎛t t t ⎞ g
⎛ ⎞
11 12 13
⎜ ⎟
M1
⎜ ⎟ ⎜ ⎟
⎜ ⎟
g = t t t g (1)
⎜ ⎟
⎜ ⎟
21 22 23
M2
⎟
⎜
⎜ ⎟ ⎜ ⎟
g t t t ⎜ g ⎟
⎝ 3⎠
⎝ 31 32 33⎠
⎝ M3⎠
or in a shortened form:
g = T g (2)
M
The coefficients t of the transformation matrix T can all be real or some of them can be
ik
complex. It is necessary that the transformation matrix T is non-singular, so that the inverse
relationship of Equation (2) is valid.
−1
g = T g (3)
M
If the original quantities are sinusoidal quantities of the same frequency, it is possible to
express them as phasors and to write the transformation Equations (2) and (3) in an analogue
form with constant coefficients:

– 8 – 62428 © IEC:2008
⎛ ⎞ ⎛ ⎞⎛ ⎞
G t t t G
1 11 12 13 M1
⎜ ⎟ ⎜ ⎟⎜ ⎟
⎜G ⎟ =⎜t t t ⎟⎜G ⎟ (4)
2 21 22 23 M2
⎜ ⎟ ⎜ ⎟⎜ ⎟
G t t t G
⎝ 3⎠ 31 32 33 M3
⎝ ⎠⎝ ⎠
G = T G (5)
M
−1
G = T G (6)
M
4.2 Power in modal components
Transformation relations are used either in the power-variant form as given in Table 1 or in
the power-invariant form as given in Table 2.
For the power-invariant form of transformation, the power calculated with the three modal
components is equal to the power calculated from the original quantities of a three-phase a.c.
system with three line conductors and a neutral conductor, where u , u and u are the line-
1 2 3
to-neutral voltages and i , i and i are the currents of the line conductors at a given
1 2 3
location of the network. In a three-phase a.c. system with only three line conductors, u , u
1 2
and u are the voltages between the line conductors and a virtual star point at a given
location of the network.
The instantaneous power p expressed in terms of the original quantities is defined by:
∗
⎛ ⎞
i
⎜ ⎟
∗ ∗ ∗ ∗ T ∗
⎜ ⎟
p = u i + u i + u i =()u u u i = u i (7)
1 1 2 2 3 3 1 2 3 2
⎜ ⎟
∗
⎜i ⎟
⎝ ⎠
∗ ∗
NOTE The asterisks denote formally the complex conjugate of the currents i , i , i . If these are real, i , i ,
1 2 3 1 2
∗
i are identical to i , i , i .
3 1 2 3
If the relationship between the original quantities and the modal components given in
Equation (2) is introduced for the voltages as well as for the currents:
u = T u and i = T i (8)
M M
taking into account
T T T T
()
u = T u = u T, (9)
M M
the power p expressed in terms of modal components is found as:
T T ∗ ∗
p = u T T i. (10)
M M
T ∗
For the power-variant case where T T is not equal to the unity matrix an example is given
at the end of this section. In case of
T ∗
TT =E (11)
62428 © IEC:2008 – 9 –
with the matrix E being the unity matrix of third order, Equation (10) changes to
∗
⎛ ⎞
i
M1
⎜ ⎟
T ∗ ∗ ∗ ∗ ∗
⎜ ⎟
p = u i =()u u u i = u i + u i + u i. (12)
M M M1 M2 M3 M2 M1 M1 M2 M2 M3 M3
⎜ ⎟
∗
⎜i ⎟
M3
⎝ ⎠
T ∗ −1 T∗
The condition TT =E or T = T means that the transformation matrix T is a unitary
matrix.
Because the Equations (7) and (12) have identical structure, the transformation relationship
with a unitary matrix is called the power invariant form of
...