SIST EN 62428:2009

Electric power engineering - Modal components in three-phase a.c. systems - Quantities and transformations (IEC 62428:2008)

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.

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)

La CEI 62428:2008 traite des transformations des grandeurs originales en grandeurs modales pour les systèmes a.c. triphasés qui sont largement utilisés dans le domaine de l'énergie électrique. L'étude des conditions de fonctionnement et des régimes transitoires dans les systèmes a.c. triphasés est rendue difficile du fait des couplages résistifs, inductifs ou capacitifs entre les éléments de phase et entre les conducteurs de ligne. Le calcul et la description de ces phénomènes dans les systèmes a.c. triphasés est plus facile si les grandeurs concernant les éléments de phase et les conducteurs de ligne sont transformées en grandeurs modales. Le calcul devient très facile si la transformation conduit à des systèmes modaux découplés. Les matrices d'impédances et d'admittances originales sont transformées en matrices d'impédances et d'admittances modales. Dans le cas où les grandeurs modales sont découplées, les matrices d'impédances et d'admittances deviennent diagonales.

Elektroenergetsko inženirstvo - Modalne komponente v trifaznih izmeničnih sistemih - Veličine in transformacije (IEC 62428:2008)

General Information

• Status: Published
• Publication Date: 07-Dec-2008
• ICS: 01.060 - Quantities and units
• ICS: 29.020 - Electrical engineering in general
• Technical Committee: I11 - Imaginarni 11
• Current Stage: 6060 - National Implementation/Publication (Adopted Project)
• Start Date: 04-Dec-2008
• Due Date: 08-Feb-2009
• Completion Date: 08-Dec-2008
• Ref Project: EN 62428:2008 - Electric power engineering - Modal components in three-phase a.c. systems - Quantities and transformations

Standards Content (Sample)

SLOVENSKI STANDARD
SIST EN 62428:2009
01-januar-2009
(OHNWURHQHUJHWVNRLQåHQLUVWYR0RGDOQHNRPSRQHQWHYWULID]QLKL]PHQLþQLK
VLVWHPLK9HOLþLQHLQWUDQVIRUPDFLMH,(&
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
SIST EN 62428:2009 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN 62428:2009

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SIST EN 62428:2009

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

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SIST EN 62428:2009
EN 62428:2008 - 2 -
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).
__________

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SIST EN 62428:2009
- 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.

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SIST EN 62428:2009

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SIST EN 62428:2009
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
® Registered trademark of the International Electrotechnical Commission
Marque déposée de la Commission Electrotechnique Internationale

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SIST EN 62428:2009
– 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

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SIST EN 62428:2009
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.

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SIST EN 62428:2009
– 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.

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SIST EN 62428:2009
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.

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SIST EN 62428:2009
– 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

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SIST EN 62428:2009
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
1
given by g = (g + j g ) .
α β
s
2
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
1
given by g = (g + j g ) .
d q
r
2
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
⎛ ⎞
1
11 12 13
⎜ ⎟
M1
⎜ ⎟ ⎜ ⎟
⎜ ⎟
g = t t t g (1)
⎜ ⎟
⎜ ⎟
2
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:

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SIST EN 62428:2009
– 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
3
location of the network.
The instantaneous power p expressed in terms of the original quantities is defined by:
∗
⎛ ⎞
i
1
⎜ ⎟
∗ ∗ ∗ ∗ 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 ⎟
3
⎝ ⎠
∗ ∗
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)

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SIST EN 62428:2009
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 transformation.
In connection with Table 2, the following examples can be given:
p = u i + u i + u i
αβ0 αα ββ 0 0
p = u i + u i + u i
dq0 d d q q 0 0
∗ ∗ ∗
p = u i + u i + u i = 2Re{u i }+ u i
∗
s s s s 0 0 s s 0 0
ss 0
∗ ∗ ∗
p = u i + u i + u i = 2Re{u i }+ u i
∗
0 0 0 0
r r r r r r
rr 0
In case of three-phase systems of voltages and currents the complex power is given in
original phasor quantities as follows:
∗
⎛ ⎞
I
⎜ 1⎟
∗ ∗ ∗ ∗ T ∗
⎜ ⎟
S = U I + U I + U I =()U U U I = U I (13)
1 2 3 2
1 2 3 1 2 3
⎜ ⎟
∗
⎜ I ⎟
3
⎝ ⎠
Substituting the modal components by
T T T T ∗ ∗ ∗
()
U = T U = U T and I = T I
M M M
the complex apparent power is found as:
T T ∗ ∗
S = U T T I (14)
M M
T ∗
In case of power invariance, the condition TT =E must also be valid. Then Equation (14)
leads to the following power invariant expression:
∗
⎛ ⎞
I
⎜ M1⎟
T ∗ ∗ ∗ ∗ ∗
⎜ ⎟
()
S = U I = U I + U I + U I = U U U I (15)
M M M1 M1 M2 M2 M3 M3 M1 M2 M3 M2
⎜ ⎟
∗
⎜ I ⎟
M3
⎝ ⎠
The power-variant forms of transformation matrices are given in the Tables 3 and 5. They are
also known as reference-component-invariant transformations, because, under balanced

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SIST EN 62428:2009
– 10 – 62428 © IEC:2008
symmetrical conditions, the reference component (the first component) of the modal
components is equal to the reference component of the original quantities or its complex
phasors, respectively. This is not the case for transformations in a rotating frame.
EXAMPLE According to Table 2 for the power-invariant form of the transformation matrix T it follows:
2 2
⎛⎞ ⎛⎞
11 1 1a a 1a a
⎛⎞
⎜⎟ ⎜⎟
⎜⎟
1 1 1
2 T2 T2∗
T = aa 1 ,    T = 1a a ,    T = 1a a ,
⎜⎟ ⎜⎟
⎜⎟
3 3 3
⎜⎟ ⎜⎟
⎜⎟2
⎜⎟11 1 ⎜⎟11 1
aa 1
⎝⎠
⎝⎠ ⎝⎠
−1 T∗ T ∗
showing that T = T or TT =E , fulfilling the condition for power invariance.
If the transformation matrix T from Table 1 for the power-variant transformation is used, then
the following results are found:
2 2
⎛⎞ ⎛⎞
1a a 1a a
11 1
⎛⎞
⎜⎟ ⎜⎟
T2 T2∗
⎜⎟
2
T = 1a a T = 1a a
⎜⎟ ⎜⎟
T =aa 1
⎜⎟
⎜⎟ ⎜⎟
2
⎜⎟ 11 1 11 1
⎜⎟ ⎜⎟
aa 1
⎝⎠ ⎝⎠ ⎝⎠
, , .
−1 T∗ T ∗
1
T from Table 1 is equal to T , so that TT =⋅3 E .
3
4.3 Established transformations
−1
The most widely used transformation matrices T and their inverse matrices T are given in
the Tables 1 and 2, whereby Table 1 contains the power-variant (reference-component-
invariant) form and Table 2 the power-invariant form of transformation matrices. The
subscripts for the components are chosen to be equal in both cases of Tables 1 and 2, 3 and
4, 5 and 6.
The Tables 3 to 6 give the relations between the different types of modal components.

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SIST EN 62428:2009
62428 © IEC:2008 – 11 –
Table 1 – Power-variant form of modal components and transformation matrices
Component: Subscript:
First M1 T
−1
Modal components
T
Second M2
a
Third M3
positive-sequence (1)
2
⎛⎞
symmetrical 11 1 1a a
⎛⎞
⎜⎟
components negative-sequence (2)
⎜⎟
1
2 2
aa 1 1a a
⎜⎟
⎜⎟
(Fortescue zero-sequence (0) 3
⎜⎟
⎜⎟2
⎜⎟11 1
components) aa 1
⎝⎠
b
⎝⎠
α α
1 1
⎛ ⎞ ⎛ ⎞
1 − −
⎜ 1 0 1⎟ ⎜ 2 2⎟
αβ0 components,
β β
2
⎜ 3 ⎟ ⎜ 3 3⎟
1
non-rotating frame
0 −
− 1
⎜ 2 2 ⎟ ⎜ 2 2⎟
zero-sequence 0
3
(Clarke components) 1 1 1
⎜ 3 ⎟ ⎜ ⎟
1
⎜ ⎟ ⎜ ⎟
− − 1
2 2 2
2 2
⎝ ⎠ ⎝ ⎠
direct-axis d
⎛ ⎞
c − s 1 c c c
⎛ ⎞
1 1 1 2 3
⎜ ⎟
⎜ ⎟
2
quadrature-axis q
dq0 components,
c − s 1 ⎜− s − s − s ⎟
⎜ ⎟
2 2 1 2 3
rotating frame
3
⎜ ⎟
zero-sequence 0
⎜ ⎟
1 1 1
⎜ ⎟
c − s 1
⎝ 3 3 ⎠
(Park components) ⎝ 2 2 2 ⎠
c
c
space phasor s 2
⎛⎞
11 1 1a a
⎛⎞
space phasor
⎜⎟
conjugated complex
1⎜⎟ 2
2
2
components,
aa 1 ⎜⎟1a a
space phasor s*
⎜⎟
3
2
⎜⎟
11 1
non-rotating frame ⎜⎟2
⎜⎟
zero-sequence 0 aa 1 22 2
⎝⎠
⎝⎠
space phasor r jjϑϑ− 2
−−jjϑϑ −jϑ
⎛⎞ee 2⎛⎞
eaeae
⎜⎟
⎜⎟
conjugated complex 1
2
2jjϑϑ− 2
space phasor jjϑϑ jϑ
ae a e 2 eae ae
⎜⎟⎜⎟
space phasor r*
components,
2 3
⎜⎟⎜⎟
2 11 1
jjϑϑ−
⎜⎟
⎜⎟
zero-sequence 0 ae ae 2
22 2
rotating frame
⎝⎠
⎝⎠
c
c
a
All the transformation matrices T given here fulfil the following conditions:
t + t + t = 0 ,  t + t + t = 0 ,  t = t = t .
11 21 31 12 22 32 13 23 33
b
The IEC Standards 60909, 60865 and 61660 have introduced the subscripts (1), (2), (0) for the power-variant form of
the symmetrical components, to avoid confusion, if the subscripts 1, 2, 3 instead of L1, L2, L3 are used.
2π 2π 2π
c
c = cosϑ ,  c = cos(ϑ − ) ,  c = cos(ϑ + ) ,  s = sinϑ ,  s = sin(ϑ − ) ,
1 2 3 1 2
3 3 3
2π
j2π / 3 2 ∗ 2
s = sin(ϑ + ) ,  ae= ,  aa= ,  1a++a=0 .
3
3
In case of synchronous machines ϑ is given by ϑ = Ω(t)dt , where Ω is the instantaneous angle velocity of the
∫
rotor.

---------------------- Page: 17 ----------------------

SIST EN 62428:2009
– 12 – 62428 © IEC:2008
Table 2 – Power-invariant form of modal components and transformation matrices
Component: Subscript:
First M1 T
−1
Modal components
T
Second M2
a
Third M3
positive-sequence (1)
2
⎛⎞
symmetrical 11 1 1a a
⎛⎞
⎜⎟
components negative-sequence (2)
⎜⎟
1 1
2 2
aa 1 1a a
⎜⎟
⎜⎟
(Fortescue zero-sequence (0) 3 3
⎜⎟
⎜⎟2
⎜⎟11 1
components) aa 1
⎝⎠
b
⎝⎠
α α
⎛ ⎞
1
1 1
1 0
⎛ ⎞
⎜ ⎟ 1 − −
2
β β ⎜ 2 2⎟
αβ0 components,
⎜ ⎟
3 ⎜ 3 3⎟
2 1 1 2
non-rotating frame
⎜ ⎟
zero-sequence 0 − 0 −
3 2 2 3⎜ 2 2⎟
2
⎜ ⎟
(Clarke components) 1 1 1
⎜ ⎟
3 ⎜ ⎟
⎜ 1 1⎟
− −
⎜ ⎟ 2 2 2
⎝ ⎠
2 2
2
⎝ ⎠
direct-axis d
⎛ ⎞
1
c − s ⎛ ⎞
⎜ ⎟
1 1
c c c
⎜ ⎟
quadrature-axis q 2 1 2 3
⎜ ⎟
dq0 components,
2 1 2⎜ ⎟
c − s − s − s − s
⎜ ⎟
zero-sequence 0 2 2 1 2 3
rotating frame 3 3
⎜ ⎟
2
⎜ ⎟ 1 1 1
1 ⎜ ⎟
c − s
(Park components)
⎜ ⎟
3 3 2 2 2
⎝ ⎠
2
⎝ ⎠
c
c
space phasor s 2
⎛⎞
11 1 1a a
⎛⎞
space phasor
⎜⎟
conjugated complex
1⎜⎟ 1
2
2
components,
aa 1 ⎜⎟1a a
space phasor s*
⎜⎟
3 3
⎜⎟
non-rotating frame ⎜⎟2
⎜⎟11 1
zero-sequence 0 aa 1
⎝⎠
⎝⎠
space phasor r
jjϑϑ− −−jjϑϑ 2−jϑ
⎛⎞ee 2⎛⎞
eaeae
⎜⎟
conjugated complex ⎜⎟
1 1
2 2
space phasor jjϑϑ− jjϑϑ jϑ
ae ae 2
space phasor r* ⎜⎟eae ae
⎜⎟
components,
3
3
⎜⎟
⎜⎟
2
jjϑϑ−
zero-sequence 0 ⎜⎟⎜⎟11 1
ae a e 2
rotating frame
⎝⎠⎝⎠
c
c
a
All the transformation matrices T given here fulfil the following conditions:
t + t + t = 0 ,  t + t + t = 0 ,  t = t = t .
11 21 31 12 22 32 13 23 33
b
The IEC Standards 60909, 60865 and 61660 have introduced the subscripts (1), (2), (0) for the power-variant form of
the symmetrical components, to avoid confusion, if the subscripts 1, 2, 3 instead of L1, L2, L3 are used.
2π 2π 2π
c
c = cosϑ ,  c = cos(ϑ − ) ,  c = cos(ϑ + ) ,  s = sinϑ ,  s = sin(ϑ − ) ,
1 2 3 1 2
3 3 3
2π 2 ∗ 2
j2π / 3
s = sin(ϑ + ) ,  ae= ,  aa= ,  1a++a=0 .
3
3
In case of synchronous machines ϑ is given by ϑ = Ω(t)dt , where Ω is the instantaneous angle velocity of the
∫
rotor.

Tables 3 and 4 cont
...