IEC 62428:2008
(Main)Electric power engineering - Modal components in three-phase a.c. systems - Quantities and transformations
Electric power engineering - Modal components in three-phase a.c. systems - Quantities and transformations
IEC 62428:2008 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.
Energie électrique - Composantes modales dans les systèmes a.c. triphasés - Grandeurs et transformations
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.
General Information
Standards Content (Sample)
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
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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
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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 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 ⎟
⎝ ⎠
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
– 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∗
⎜⎟
T = 1a a T = 1a a
⎜⎟ ⎜⎟
T =aa 1
⎜⎟
⎜⎟ ⎜⎟
⎜⎟ 11 1 11 1
⎜⎟ ⎜⎟
aa 1
⎝⎠ ⎝⎠ ⎝⎠
, , .
−1 T∗ T ∗
T from Table 1 is equal to T , so that TT =⋅3 E .
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.
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)
⎛⎞
symmetrical 11 1 1a a
⎛⎞
⎜⎟
components negative-sequence (2)
⎜⎟
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
⎜ 3 ⎟ ⎜ 3 3⎟
non-rotating frame
0 −
− 1
⎜ 2 2 ⎟ ⎜ 2 2⎟
zero-sequence 0
(Clarke components) 1 1 1
⎜ 3 ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
− − 1
2 2 2
2 2
⎝ ⎠ ⎝ ⎠
direct-axis d
⎛ ⎞
c − s 1 c c c
⎛ ⎞
1 1 1 2 3
⎜ ⎟
⎜ ⎟
quadrature-axis q
dq0 components,
c − s 1 ⎜− s − s − s ⎟
⎜ ⎟
2 2 1 2 3
rotating frame
⎜ ⎟
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
components,
aa 1 ⎜⎟1a a
space phasor s*
⎜⎟
⎜⎟
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
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 .
In case of synchronous machines ϑ is given by ϑ = Ω(t)dt , where Ω is the instantaneous angle velocity of the
∫
rotor.
– 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)
⎛⎞
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 0
⎛ ⎞
⎜ ⎟ 1 − −
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⎟
⎜ ⎟
(Clarke components) 1 1 1
⎜ ⎟
3 ⎜ ⎟
⎜ 1 1⎟
− −
⎜ ⎟ 2 2 2
⎝ ⎠
2 2
⎝ ⎠
direct-axis d
⎛ ⎞
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
⎜ ⎟
⎜ ⎟ 1 1 1
1 ⎜ ⎟
c − s
(Park components)
⎜ ⎟
3 3 2 2 2
⎝ ⎠
⎝ ⎠
c
c
space phasor s 2
⎛⎞
11 1 1a a
⎛⎞
space phasor
⎜⎟
conjugated complex
1⎜⎟ 1
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,
⎜⎟
⎜⎟
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 .
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 contain the relations of the αβ0 components and the dq0 components with the
space phasor components in the power-variant and the power-invariant form.
62428 © IEC:2008 – 13 –
Table 3 – Clark, Park and space phasor components – modal transformations in the power-variant form
T T
T T
T ∗ ∗
•()g g g •()g g g •()g g g • (g g g ) • (g g g )
α β 0 d q 0
1 2 3
s s 0 r r 0
jjϑϑ−
⎛ ⎞
⎛⎞ee 2
11 2
⎛⎞
⎛ g ⎞ ⎛1 0 0⎞ 1 0 1 ⎛ c − s 1⎞
⎜ ⎟
1 1 1
⎜⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟ 1
1 2
⎜ 3 ⎟ 2 jjϑϑ−
g = 0 1 0 − 1 c − s 1 aa 2 ae ae 2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎟
2 2 2
⎜⎟
⎜ 2 2 ⎟
2 2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟2 ⎜⎟
⎜ ⎟ jjϑϑ−
g 0 0 1 1 3 c − s 1
⎜⎟
3 3 3 aa 2
⎝ ⎠ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠
− − 1 ae a e 2
⎝⎠
⎝⎠
⎝ 2 2 ⎠
1 1
⎛ ⎞
1 − − jjϑϑ−
⎛⎞
⎛ g ⎞ ⎜ 2 2⎟ ⎛1 0 0⎞ ⎛c − s 0⎞ ⎛ 1 1 0⎞ ee 0
α 1 1
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
2 1 1⎜⎟
⎜ 3 3⎟ jjϑϑ−
g = 0 − 0 1 0 s c 0 − j j 0 −je je 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
β 1 1
⎜⎟
⎜ 2 2⎟
3 2 2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1 1 1 ⎜⎟
⎜ ⎟
g 0 0 1 0 0 1 0 0 2 00 2
0 ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎝⎠
2 2 2
⎝ ⎠
−jjϑϑ
⎛ ⎞
g c c c c s 0 1 0 0 ⎛⎞ee 0 1 1 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
d 1 2 3 1 1
⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
2 1⎜⎟ 1
−jjϑϑ
g = ⎜− s − s − s ⎟ − s c 0 0 1 0 −je je 0 − j j 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
q 1 2 3 1 1
⎜⎟
3 2 2
⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1 1 1
⎜⎟
⎜ ⎟
g 0 0 1 0 0 1 00 2 0 0 2
⎝ 0⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎝⎠
2 2 2
⎝ ⎠
jjϑϑ jϑ
⎛ g ⎞ ⎛⎞
1a a
⎛1 j 0⎞ ⎛⎞eje 0 ⎛1 0 0⎞ ⎛⎞e0 0
s
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟
2 ⎜⎟ ⎜⎟
∗ 2
−−jjϑϑ − jϑ
⎜ ⎟
g = 1a a 1 − j 0 ej−e 0 0 1 0 0e 0
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟
⎜⎟
s
⎜ ⎟ 3
⎜⎟ ⎜ ⎟ ⎜ ⎟
11 1
⎜⎟ ⎜⎟
⎜ g ⎟ ⎜⎟ 0 0 1 00 1 0 0 1 00 1
22 2 ⎝ ⎠ ⎝ ⎠
⎝⎠ ⎝⎠
⎝ ⎠
⎝⎠
−−jjϑϑ −ϑ
⎛⎞
−−jjϑϑ − jϑ
eae ae
⎛ g ⎞
⎛⎞ ⎛⎞
eje 0 ⎛1 j 0⎞ e00 ⎛1 0 0⎞
r
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟
2 ⎜⎟ ⎜⎟
∗ jjϑϑ jϑ jjϑϑ jϑ
⎜ ⎟
g = ⎜⎟eae ae ej−e 0 1 − j 0 0e 0 0 1 0
⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜⎟
r
⎜ ⎟
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜⎟
11 1
⎜ ⎟
g 00 1 0 0 1 00 1 0 0 1
⎜⎟
⎝ ⎠ ⎝ ⎠
22 2 ⎝⎠ ⎝⎠
⎝ ⎠
⎝⎠
2π 2π 2π 2π
c = cosϑ , c = cos(ϑ − ) , c = cos(ϑ + ) , s = sinϑ , s = sin(ϑ − ) , s = sin(ϑ + ) .
1 2 3 1 2 3
3 3 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.
∫
– 14 – 62428 IEC:2008
©
Table 4 – Clark, Park and space phasor components – Modal transformations in the power-invariant form
T T
T T
T ∗ ∗
() ()
•()g g g • g g g • g g g • (g g g ) • (g g g )
α β 0 d q 0
1 2 3
s s 0 r r 0
⎛ 1⎞
⎛ ⎞ jjϑϑ−
1 0
⎜ ⎟
c − s ⎛⎞ee 1
⎛⎞11 1
⎜ 1 1 ⎟
⎛1 0 0⎞ 2
⎛ g ⎞
1 2
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎟
1 1
2 2
3 jjϑϑ−
2 1 1 2 1
⎜ ⎟
g = 0 1 0 − c − s aa 1 ae ae 1
⎜ ⎟ ⎜ ⎟ ⎜⎟
2 ⎜ 2 2 ⎟
⎜⎟
3 2 2 3
2 2
⎜ ⎟
⎜ ⎟ 3
⎜ ⎟ ⎜ ⎟
⎜⎟
2 2
g ⎜⎟ jjϑϑ−
0 0 1 1
3 ⎜⎟
⎝ ⎠ 3
⎝ ⎠ ⎜ 1 1⎟
c − s aa 1 ae a e 1
⎜ ⎟
3 3
− − ⎝⎠
⎜ ⎟ ⎝⎠
2 2 ⎝ ⎠
⎝ ⎠
1 1
⎛ ⎞
1 − − jjϑϑ−
⎛⎞
g ⎜ 2 2⎟ ⎛1 0 0⎞ ⎛c − s 0⎞ ⎛ 1 1 0⎞ ee 0
⎛ ⎞
1 1
α
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟
1 1⎜⎟
⎜ 3 3⎟ jjϑϑ−
g =
0 − 0 1 0 s c 0 − j j 0 −je je 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
β 1 1 ⎜⎟
3⎜ 2 2⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1 1 1 ⎜⎟
g
⎜ ⎟
0 0 1 0 0 1 0 0 2
⎝ 0⎠
⎜ ⎟ 00 2
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎝⎠
2 2 2
⎝ ⎠
⎛ ⎞ −jjϑϑ
⎛⎞
c c c c s 0 1 0 0 ee 0 1 1 0
⎛ g ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜ ⎟
d 1 2 3 1 1
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1⎜⎟ 1
−jjϑϑ
2⎜ ⎟
g = − s − s − s − s c 0 0 1 0 −je je 0 − j j 0
⎜ ⎟
q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1 2 3 1 1 ⎜⎟
⎜ ⎟
⎜ ⎟ 2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
1 1 1
⎜⎟
g
0 0 1 0 0 1 0 0 2
⎜ ⎟
⎝ 0⎠
⎝ ⎠ ⎝ ⎠ 00 2 ⎝ ⎠
⎝⎠
2 2 2
⎝ ⎠
jjϑϑ jϑ
⎛⎞
g 1a a
⎛ ⎞ ⎛⎞
⎛1 j 0⎞ eje 0 ⎛1 0 0⎞ ⎛⎞e0 0
s
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟
⎜⎟
1 1 1 ⎜⎟
∗ 2
−−jjϑϑ − jϑ
⎜ ⎟
g = 1a a 1 − j 0 ej−e 0 0 1 0 0e 0
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟
⎜⎟
s
⎜ ⎟
3 2
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜⎟ ⎜⎟
g
⎜⎟11 1 0 0 2 0 0 1 00 1
00 2
⎝ ⎠ ⎝ ⎠
⎝ 0⎠ ⎝⎠
⎝⎠
⎝⎠
−−jjϑϑ −jϑ
⎛⎞
−−jjϑϑ − jϑ
eaeae
⎛ g ⎞
⎛⎞
eje 0 ⎛1 j 0⎞ ⎛⎞e00 ⎛1 0 0⎞
r
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟
1 ⎜⎟ 1
2 1 ⎜⎟
∗ jjϑϑ jϑ jjϑϑ jϑ
⎜ ⎟
g =⎜⎟eae ae ej−e 0 1 − j 0 0e 0 0 1 0
⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜⎟
r
⎜ ⎟
3 2
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜⎟
⎜ g ⎟
0 0 2 00 1 0 0 1
11 1
⎜⎟ 00 2
0 ⎝ ⎠ ⎝ ⎠
⎝ ⎠ ⎝⎠ ⎝⎠
⎝⎠
2π 2π 2π 2π
c = cosϑ , c = cos(ϑ − ) , c = cos(ϑ + ) , s = sinϑ , s = sin(ϑ − ) , s = sin(ϑ + ) .
1 2 3 1 2 3
3 3 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.
∫
62428 © IEC:2008 – 15 –
Tables 5 and 6 contain the relations between modal components, if the original quantities are
sinusoidal ones of the same frequency and can be written as r.m.s. phasors.
Table 5 – Transformation matrices in the power-variant form for phasor quantities
T T
T
•()G G G
•()G G G •()G G G
(1) (2) (0) α β 0
1 2 3
⎛ ⎞
11 1
⎛⎞
⎛ G ⎞ ⎛1 0 0⎞ 1 0 1
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜⎟
2 ⎜ 3 ⎟
G = 0 1 0 aa 1 − 1
⎜ ⎟ ⎜ ⎟
2 ⎜⎟
⎜ 2 2 ⎟
⎜ ⎟
⎜ ⎟
⎜⎟2
⎜ ⎟
G 0 0 1 1 3
aa 1
3 ⎝ ⎠ ⎜ ⎟
⎝ ⎠ − − 1
⎝⎠
⎝ 2 2 ⎠
⎛ ⎞ 2
G
⎛⎞
(1) 1a a
⎜ ⎟
⎛1 0 0⎞ ⎛1 j 0⎞
⎜⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟ 1
G = 2
(2)
1a a 0 1 0 1 − j 0
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟ 3 2
G ⎜⎟
⎜ ⎟ ⎜ ⎟
(0)
⎝ ⎠ ⎜⎟11 1 0 0 1 0 0 2
⎝ ⎠ ⎝ ⎠
⎝⎠
1 1
⎛ ⎞
1 − −
⎛G ⎞ ⎜ ⎟ ⎛ 1 1 0⎞ ⎛1 0 0⎞
2 2
α
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ 3 3⎟
G = 0 − − j j 0 0 1 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
β
⎜ ⎟
2 2
⎜ ⎟
⎜ ⎟ ⎜ ⎟
1 1 1
⎜ ⎟
G 0 0 1 0 0 1
0 ⎜ ⎟ ⎝ ⎠ ⎝ ⎠
⎝ ⎠
2 2 2
⎝ ⎠
Table 6 – Transformation matrices in the power-invariant form for phasor quantities
T
T T
() •()G G G
•()G G G • G G G
α β 0
1 2 3 (1) (2) (0)
⎛ ⎞
1 0
⎜ ⎟
11 1
⎛ ⎞ ⎛⎞ 2
G ⎛1 0 0⎞
⎜ ⎟
⎜ ⎟ ⎜ ⎟
1⎜⎟
2 1 1
⎜ ⎟
⎜G ⎟ = 0 1 0 aa 1 −
⎜ ⎟
⎜⎟
3 2 2
3 ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜⎟
G 0 0 1
aa 1 3
⎝ 3⎠ ⎝ ⎠ ⎜ 1 1⎟
⎝⎠
− −
⎜ ⎟
2 2
⎝ ⎠
⎛⎞
1a a
⎛ ⎞
1 0 0 1 j 0
G ⎛ ⎞ ⎛ ⎞
(1)
⎜ ⎟
⎜⎟
⎜ ⎟ ⎜ ⎟
1 1
⎜ ⎟
G = 1a a 0 1 0 1 − j 0
⎜⎟
⎜ ⎟ ⎜ ⎟
(2)
⎜ ⎟
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟
G 11 1 0 0 1 0 0 2
⎜⎟
(0) ⎝ ⎠ ⎝ ⎠
⎝ ⎠
⎝⎠
1 1
⎛ ⎞
1 − −
⎛G ⎞ ⎛ 1 1 0⎞ ⎛1 0 0⎞
⎜ 2 2⎟
α
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ 3 3⎟
G = 0 − − j j 0 0 1 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
β
3⎜ 2 2⎟
⎜ ⎟ 2
⎜ ⎟ ⎜ ⎟
1 1 1
⎜ ⎟
G 0 0 2 0 0 1
⎜ ⎟ ⎝ ⎠ ⎝ ⎠
⎝ 0⎠
2 2 2
⎝ ⎠
– 16 – 62428 © IEC:2008
5 Decoupling in three-phase a.c. systems
5.1 Decoupling in case of steady-state operation with sinusoidal quantities
Figure 1 gives the example of a three-phase a.c. system with inductive coupling between
three line conductors L1, L2, L3 or three phase elements of a three-phase motor or generator
with the neutral point earthed via an impedance Z .
N
NOTE Subscripts L1, L2, L3 are introduced for the line-to-neutral voltages at the locations Q or N and for the line
currents (see 3.2).
R jX
Q I 1 11 N
L1 L1
jX = jX
12 21
R jX
I
2 22
L2 jX = jX
L2
13 31
jX = jX
23 32
jX
R 33
I
L3
L3
Z
N
U
L1Q
U
L2Q
U U
L3Q
N
E
IEC 1237/08
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
N N N
From Figure 1 follows:
⎛⎞⎛⎞
UZ Z Z ⎛⎞I ⎛U⎞
L1Q L1L1 L1L2 L1L3 L1 N
⎜⎟⎜⎟
⎜⎟ ⎜ ⎟
, (16)
UZ=+Z Z I U
L2 Q L2 L1 L2 L2 L2 L3 L2 N
⎜⎟⎜⎟
⎜⎟ ⎜ ⎟
⎜⎟ ⎜ ⎟
⎜⎟⎜⎟
UZ Z Z I U
L3 Q L3 L1 L3 L2 L3 L3 L3 N
⎝⎠ ⎝ ⎠
⎝⎠⎝⎠
UZ=+I U (17)
LQ LL N
If the original quantities of Equation (17) are substituted by the modal components using
Equation (2), the following equation is found:
TU = Z T I + T U (18)
MQ L M MN
−1
−1
Equation (18) multiplied with T leads to Equation (19), because TT =E :
−1
U = T Z T I + U = Z I + U (19)
MQ L M MN M M MN
62428 © IEC:2008 – 17 –
The following modal components are therefore defined:
Modal impedance matrix
−1
Z = T Z T (20)
M L
Modal voltage vector in Q
−1
U = T U (21)
MQ LQ
Modal current vector
−1
I = T I (22)
M L
Modal voltage vector in N
−1
UT= U (23)
MN N
If Z is finite, the following equation is obtained from Figure 1:
N
U ZZZ I
⎛⎞ ⎛ ⎞⎛ ⎞
N NNN L1
⎜⎟ ⎜ ⎟⎜ ⎟
U =ZZZ I (24)
N NNN L2
⎜⎟ ⎜ ⎟⎜ ⎟
⎜⎟ ⎜ ⎟⎜ ⎟
U ZZZ I
⎝⎠N ⎝NNN⎠⎝L3⎠
UZ= I (25)
N NL
Then Equation (23) is changed to:
−1
U = T Z T I = Z I (26)
MN N M MN M
Table 7 gives the modal voltages and the modal impedances for the symmetrical components
and the αβ0 components in the power-variant (reference-conductor-invariant) and the power-
invariant form in case of phasor quantities under the following conditions:
• Symmetrical system with:
UU= a
UU= a , , from which results: UU++U = 0
L2 Q L1Q L3 Q L1Q
L1Q L2 Q L3 Q
• Impedance matrix cyclic and symmetric:
Z ==ZZ =Z ,
L1L1 L2 L2 L3 L3 A
Z =Z =Z =Z ===ZZZ
L1L2 L2 L1 L1L3 L3 L1 L2 L3 L3 L2 B
• Impedance matrix only cyclic:
Z ==ZZ =Z ,
L1L1 L2 L2 L3 L3 A
Z ==ZZ =Z ,
L1L2 L2 L3 L3 L1 B
Z ==ZZ =Z
L1L3 L2 L1 L3 L2 C
From Table 7 it can be seen, that decoupling is possible if the impedance matrix Z has
L
diagonal cyclic symmetry or, if the symmetrical components are used, also in that case if the
– 18 – 62428 © IEC:2008
impedance matrix Z has only cyclic symmetry. Figure 2 demonstrates this result. A voltage
L
source does not exist in the zero-sequence system.
Table 7 – Modal voltages and impedances in case of phasor quantities
Power-variant form Power-invariant form
Modal components
(1) (2) (0) αβ0 (1) (2) (0) αβ0
symmetrical system of
source voltages
U U
⎛⎞ ⎛⎞ U
⎛⎞UU⎛ ⎞ U ⎛⎞
L1Q ⎛⎞ L1Q
L1Q
L1Q M1Q L1Q
− ⎜⎟ ⎜⎟
1⎜⎟⎜ ⎟ ⎜⎟ ⎜⎟
0 −j 30 −jU
TUU= = U
L1Q L1Q
L2 Q M2 Q 2
⎜⎟⎜ ⎟ ⎜⎟ ⎜⎟ ⎜⎟
⎜⎟
⎜⎟
⎜⎟ ⎜⎟ ⎜⎟
⎜⎟⎜ ⎟
UU 0
0 0
L3 Q M3 Q ⎝⎠ ⎝⎠
⎝⎠⎝ ⎠ ⎝⎠ ⎝⎠
a
UU⎛⎞
⎛⎞
NM1N
⎜⎟
−1⎜⎟
⎛ 0 ⎞ ⎛0 0 0 ⎞⎛ I ⎞ ⎛ 0 ⎞ ⎛0 0 0 ⎞⎛ I ⎞
M1 M1
TUU==
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
NM2N
⎜⎟
⎜⎟
0 = 0 0 0 I 3 0 = 0 0 0 I
⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎜⎟
⎜⎟
M2 M2
UU
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
NM3N ⎜ ⎟ ⎜ ⎟
⎝⎠
⎝⎠
U 0 0 3 Z U 0 0 3 Z
I I
N N
⎝ N⎠ ⎝ ⎠⎝ M3⎠ ⎝ N⎠ ⎝ ⎠⎝ M3⎠
b
−1
T Z T = Z with
Independent of the given modal components
L M
⎛ ⎞
⎛ Z Z Z ⎞ Z 0 0 ⎛ Z − Z 0 0 ⎞
A B B M1 A B
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
Z = Z Z Z Z =⎜ 0 Z 0 ⎟ =⎜ 0 Z − Z 0 ⎟
⎜ ⎟
L B A B M M2 A B
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
Z Z Z 0 0 Z 0 0 Z + 2 Z
⎝ B B A⎠ ⎝ M3⎠ ⎝ A B⎠
c
only for symmetrical components (1) (2) (0)
−1 Z 00
⎛⎞
M1
T Z T = Z with
L M
⎜⎟
Z =00Z
MM2
⎜⎟
⎛ Z Z Z ⎞ ⎜⎟
A B C
00 Z
⎜ ⎟ ⎝⎠M3
Z = Z Z Z
⎜ ⎟
L C A B
⎛⎞
ZZ++aaZ 0 0
⎜ ⎟ AB C
Z Z Z
⎜⎟
⎝ B C A⎠
=+0aZZ+aZ 0
⎜⎟
d
AB C
⎜⎟
00 Z++ZZ
⎜⎟
ABC
⎝⎠
a 2
UU= a , UU= a , UU++U = 0
L2 Q L1Q L3 Q L1Q L1Q L2 Q L3 Q
b
U ===UUU , see Figure 1.
L1N L2 N L3 N N
c
A: L1L1 = L2L2 = L3L3, B: L1L2 = L2L3 = L3L1
d
A: L1L1 = L2L2 = L3L3, B: L1L2 = L2L3 = L3L1, C: L1L3 = L2L1 = L3L2
62428 © IEC:2008 – 19 –
Z
I M1
Q M1 N
U
M1Q
U
M1N
0M1
Z
I M2
Q M2
N
U
M2Q
U
M2N
0M2
Z
Q I M3
M3
N
Z = 3Z
M3N N
U
M3Q
U
M3N
0M3
IEC 1238/08
Figure 2 – Three decoupled systems which replace the coupled three-phase a.c. system
of Figure 1 under the described conditions (see text)
Decoupling is also possible if an admittance matrix, for instance for the line-to-line and the
line-to-earth capacitances, is given instead of the impedance matrix.
5.2 Decoupling under transient conditions
For the coupled three-phase a.c. system in Figure 1 the following differential equation can be
found:
⎡⎤
uL⎛⎞LLu
⎛⎞ ⎛⎞
Ri00 i
L1Q ⎛ ⎞⎛⎞ L1L1 L1L2 L1L3 ⎛⎞ L1N
LL1 L1
⎢⎥
⎜⎟
⎜⎟ ⎜⎟
d
⎜ ⎟⎜⎟ ⎜⎟
uR=+00i LLL i+u (27)
⎢⎥⎜⎟
⎜⎟L2 Q L L2 L2 L1 L2 L2 L2L3 L2 ⎜⎟L 2N
⎜ ⎟⎜⎟ ⎜⎟
dt
⎢⎥⎜⎟
⎜ ⎟⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
00 Ri i
uLLLu
LL3 L3
⎝ ⎠⎝⎠ ⎝⎠
⎝⎠L3Q ⎢ L3 L1 L3 L2 L3 L3 ⎥⎝⎠L3N
⎝⎠
⎣⎦
d
u = R i + ()L i + u (28)
LQ L L L L LN
dt
If the original quantities of Equation (28) are substituted by the modal components using
Equation (2), the following equation is obtained:
d
T u = R T i +()L T i + T u (29)
L L
MQ M M MN
dt
−1
Equation (29) multiplied from the left hand side with T leads to Equation (30), because
−1
TT =E .
d d
−1 −1 −1 −1
u = T R T i + (T L T i ) + (T T)T L T i + u (30)
L L L
MQ M M M MN
dt dt
– 20 – 62428 © IEC:2008
Introducing the modal resistance matrix:
−1
T R T = R (31)
L M
and the modal inductance matrix:
−1
T L T = L, (32)
L
M
Equation (30) leads to:
d d
−1
u = R i +()L i + (T T) L i + u (33)
M M M
MQ M M M MN
dt dt
The additional matrix term of Equation (33) in comparison to Equation (28) for the original
quantities can be found in connection to Table 2 as follows:
• αβ0 components and space-phasor transformation in a non-rotating frame:
d
−1
T T = 0 (34)
dt
• dq0 quantities:
0 − 1 0
⎛ ⎞
⎜ ⎟
d dϑ
−1
T T = 1 0 0 (35)
⎜ ⎟
dt dt
⎜ ⎟
0 0 0
⎝ ⎠
• space phasor transformation in a rotating (angle-dependent) frame:
⎛ j 0 0⎞
⎜ ⎟
d dϑ
−1
T T = 0 − j 0 (36)
⎜ ⎟
t t
d d
⎜ ⎟
0 0 0
⎝ ⎠
If the neutral point in Figure 1 is earthed through a resistance R and an inductance L in
N N
series, the voltage vector then can be expressed by:
⎛⎞
u
RRR i ⎡L L L i ⎤
L1N ⎛ ⎞⎛⎞ ⎛ ⎞⎛⎞
N N N L1 N N N L1
⎜⎟
d
⎢ ⎥
⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟
u=+R R R i LLL i (37)
⎜⎟
L2 N N N N L2 N N N L2
⎜ ⎟⎜⎟ ⎢⎜ ⎟⎜⎟⎥
dt
⎜⎟
⎜ ⎟⎜⎟ ⎜ ⎟⎜⎟
⎢ ⎥
⎜⎟
RRR i L L L i
u
N N N L3 N N N L3
⎝ ⎠⎝⎠ ⎣⎝ ⎠⎝⎠⎦
L3 N
⎝⎠
d
u = R i + (L i ) (38)
LN N L L L
dt
If the original quantities of Equation (38) are substituted by the modal components using
Equation (2), the following equation is obtained:
d
T u = R T i +()L T i (39)
N N
MN M M
dt
−1 −1
Equation (39) multiplied from the left side with T leads to Equation (40), because TT =E :
d d
−1 −1 −1 −1
u = T R T i + (T L T i ) + (T T )T L T i (40)
N N N
MN M M M
dt dt
62428 © IEC:2008 – 21 –
Introducing the modal resistance matrix:
−1
T R T = R (41)
N MN
and the modal inductance matrix
−1
T L T = L (42)
N
MN
Equation (40) leads to:
d d
−1
u = R i +()L i + (T T ) L i (43)
MN MN M MN M MN M
dt dt
The additional matrix term of Equation (43) in comparison to Equation (38) for the original
quantities can be found in connection to Table 2 as follows:
u = 0 , u = 0 , u = 0 , u = 0 ,
αN dN sN rN
∗ ∗
u = 0 , u = 0 , u = 0 , u = 0 ,
βN qN
sN rN
d
u = 3R i + 3 L i with R = 3 R and L = 3 L
0N N 0 N 0 MN N MN N
dt
d
In the special case of steady-state operation where u → U , i → I , i → jω I ,
MQ MQ M M M M
dt
u → U and if T is time invariant, Equation (30) leads to Equation (19) and
MN MN
Equation (43) leads to Equation (26).
Table 8 gives the expressions for modal voltages and inductances for αβ0 components,
dq0 components, and space phasor components under transient conditions.
– 22 – 62428 IEC:2008
©
Table 8 – Modal voltages and inductances under transient conditions
Power-variant form: F = 1 and R = 1, Power-invariant form: F = and R =
2 2
Modal components
αβ0 ss*0 dq0 rr*0
j(ω t+ϕ ) j((ω −Ω )t+ϕ −ϑ )
⎛⎞uu⎛ ⎞
⎛ Q ⎞ ⎛ Q 0 ⎞
L1Q M1Q
ˆ ˆ
ˆ u e ˆ u e
⎛u cos(ω t + ϕ )⎞ ⎛u cos((ω − Ω )t + ϕ − ϑ )⎞
Q Q
Q Q ⎜ ⎟ Q Q 0 ⎜ ⎟
−1⎜⎟⎜ ⎟
⎜ ⎟ ⎜ ⎟
−j(ω t+ϕ ) −j((ω−Ω )t+ϕ −ϑ )
Tuu== ⎜ Q ⎟ ⎜ Q 0 ⎟
L2 Q M2 Q ˆ ˆ ˆ ˆ
F u sin(ω t + ϕ ) R u e F u sin((ω − Ω )t + ϕ − ϑ ) R u e
⎜⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟
Q Q Q Q Q 0 Q
⎜ ⎟ ⎜ ⎟
⎜⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟
uu
0 ⎜ 0 ⎟ 0 ⎜ 0 ⎟
L3 Q M3 Q
⎝⎠⎝ ⎠
⎝ ⎠ ⎝ ⎠
⎝ ⎠ ⎝ ⎠
a
power-variant form: D = 1, power-invariant form: D = 3
uu⎛⎞
⎛⎞
NM1N
⎜⎟
−1⎜⎟
0 0 0 0 ⎛ i ⎞ 0 0 0 ⎛ i ⎞
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
T uu==
M1 M1
NM2N ⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟
d
D 0 = 0 0 0 i + 0 0 0 i
⎜ ⎟ ⎜ ⎟
⎜⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜⎟
M2 M2
uu
dt
⎝⎠NM3N
⎝⎠ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
u 0 0 3 R i 0 0 3 L i
⎝ N⎠ ⎝ N⎠ ⎝ N⎠
b ⎝ M3⎠ ⎝ M3⎠
−1
T L T = L with
L M independent of the modal components:
⎛ L 0 0 ⎞ ⎛ L − L 0 0 ⎞
⎛ L L L ⎞ M1 A B
A B B
⎜ ⎟ ⎜ ⎟
⎜ ⎟
L = 0 L 0 = 0 L − L 0
⎜ ⎟ ⎜ ⎟
L = L L L M M2 A B
⎜ ⎟
L B A B
⎜ ⎟ ⎜ ⎟
⎜ ⎟
0 0 L 0 0 L + 2 L
L L L M3 A B
⎝ ⎠ ⎝ ⎠
⎝ B B A⎠
c
−1
T L T = L with only for ss*0 and rr*0 components :
L M
⎛⎞
LL++aaL 0 0
L 00 A BC
⎛ L L L ⎞ ⎛⎞
M1
A B C
⎜⎟
⎜ ⎟
⎜⎟
L==00LL⎜⎟0 +aL+aL 0
L = L L L
⎜ ⎟ MM2 AB C
⎜⎟
L C A B
⎜⎟
⎜ ⎜⎟
⎟
00 L⎜⎟0 0 L
...
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