Verification of Fortescue’s
Symmetrical Components Theorem for
Four-Phase Unbalanced System with MATLAB
Dr. Iranna Korachagaon
Professor – Department of Electrical Engineering
Annasaheb Dange College of Engineering and Technology, Ashta – 416301, Sangli, India
Email: irannamk[at]gmail.com, Cell No.: +91 986 0088 339
Abstract – Symmetrical components concepts are developed for a typical four-phase unbalanced system. A systematic
approach for building the symmetrical components transformation matrix has been discussed with the numerical example to
make the concept clear. Further the MATLAB code is also provided for the verification of the results. This serves the purpose
of additional experiment in power system course for undergraduate students in Electrical Engineering. This also invokes the
interest in the reader, to think logically to extend the applications of various electrical engineering theorems to the near near
vicinity of their application area.
Key Words – Fortescue, Symmetrical Components, four-phase,
I.
INTRODUCTION
Charles Le Geyt Fortescue (1876 – 1936) was an
engineer with the Westinghouse Electric and
Manufacturing Company in Pittsburgh, PA, where he
spent his entire professional career. His classic paper
‘Method of Symmetrical Coordinates Applied to the
Solution of Poly-phase Networks’ appeared in the
Transactions of the American Institute of Electrical
Engineers (AIEE) in 1918. This paper occupied 88
pages of the Transaction, with an additional 24 pages
being devoted to discussions. This paper contained
some 303 numbered equations and greatly facilitated
the analysis of unbalanced poly-phase systems by
converting problems into equivalent symmetrical
systems. V. Karapetoff, suggested that the term
‘Symmetrical Components’ was a more correct and
descriptive expression than ‘Symmetrical Coordinates’.
Later in 1933, with C.F. Wagner and R.D. Evans,
Fortescue,
authored
the
book
‘Symmetrical
Components’ [1].
From the literature, it was observed that, plenty of
discussions and the numericals have been solved on the
analysis of the concept of three-phase symmetrical
components. Hence an effort to make a systematic
analysis of the symmetrical components for four-phase
systems was thought of. This paper is aimed at
addressing the under-graduate students of Electrical
Engineering, to present the parallelism and contrasts
between the four-phase and the three-phase formulas
may throw further light on the general theory and
applications of symmetrical components.
II.
CONCEPT OF SYMMETRICAL COMPONENTS
Symmetrical components allow unbalanced phase
quantities such as currents and voltages to be replaced
by three separate balanced symmetrical components.
Based on the C.L. Fortescue’s theory, three phase
unbalanced phasors a–b–c, of a three-phase system can
be resolved into three balanced systems of phasors as
follows.
1. Positive (+ve) sequence components consisting
of a set of three-phase components with a
phase sequence a–b–c.
2. Negative
(–ve)
sequence
components
consisting of a set of three-phase components
with a phase sequence a–c–b.
3. Zero (0) sequence components consisting of
three single-phase components, all equal in
magnitude but with the same phase angles.
Considering the three-phase unbalanced currents
[Iabc], with the ‘symmetrical components transformation
matrix (SCTM), denoted as [A], which transforms [Iabc]
into component currents [Ia012]. This can be represented
in matrix notations as,
[Iabc] = [A] × [Ia012]
(1)
where,
𝟏 𝟏 1
A = [ 𝟏 𝒂𝟐 𝑎 ]
(2)
1 𝑎 𝑎2
And the operator ‘𝑎’ is defined as,
𝑎 = 1∠1200 = −0.5 + 𝑗0.866
𝑎2 = 1∠2400 = −0.5 − 𝑗0.866
(3)
1 + 𝑎 + 𝑎2 = 0
Therefore, for the symmetrical components of the
currents, we have,
[Ia012] = [A]-1 × [Iabc]
(4)
Similar expressions exist for voltages. Thus the
unbalanced phase voltages in terms of the symmetrical
components could be represented [2].
Those familiar with the three-phase technique will
be interested in observing the resemblances and the
differences between the three-phase and the four-phase
techniques.
1
III.
FOUR-PHASE SYSTEM REPRESENTATION
The conventional concept of symmetrical
components of three-phase systems – positive, negative
and zero-phase sequence components, is peculiar to
three-phase system and not very adaptable to a system
with greater number of phases, because a four-phase
system must have four components (n-phase system n
components), and the concept of positive, negative and
zero-phase sequence components offers no clue as to
how to find an appropriate fourth component for the
four-phase system [3].
To understand the similarities and the contrasts
between a three-phase and four-phase system, a typical
numerical is presented.
Let,
𝐼𝑎
3∠310°
[𝐼𝑏 ] = [5∠105°]
(5)
𝐼𝑐
4∠012°
Be a three-phase (n=3) unbalanced system and the
360°
operator, 𝑎 =
= 1200 = 1∠1200
𝑛
On solving for the symmetrical components, using
equation (4), with predefined symmetrical components
transformation matrix as given in equation (2), we
obtain,
𝐼𝑎012
1.8852∠36.4890° 3.3495∠ − 106.4364° 1.3761∠ − 8.6327°
[𝐼𝑏012 ] = [1.8852∠36.4890° 3.3495∠133.5636°
1.3761∠111.3673° ]
1.8852∠36.4890°
3.3495∠13.5636° 1.3761∠ − 128.6327°
𝐼𝑐012
On similar lines,
Let,
𝐼𝑎
3∠310°
𝐼𝑏
5∠105°
[ ]=[
]
𝐼𝑐
4∠012°
𝐼𝑑
7∠230°
Superscript ‘0’ indicates the zero-sequence. As
there are only two sequences possible, ‘1’ indicates
positive sequence, ‘2’ indicates negative sequence, ‘3’
indicates positive sequence and so on.
The preparation of the symmetrical components
transformation matrix is automatic. The element A[2,2]
of the matrix shall have (n-1) as superscript to the
operator element. This is seen as 2 for three-phase and 3
for four-phase system. The next element A[2,3] shall be
the operator with superscript as 1. And the next element
A[2,4] shall have the superscript as 2. Likewise the
matrix A is constructed.
IV.
Implementation of MATLAB program for
determining the symmetrical components for the threephase and four-phase unbalanced systems along with
the results are given in Appendix –I, and II respectively.
V.
[1]
[2]
Be a three-phase (n=4) unbalanced system and the
operator,
360°
𝑎 =
= 900 = 1∠900 = 𝑎1
(8)
𝑛
On solving for the symmetrical components, using
equation (4), we obtain,
𝐼𝑎0123
0.4999∠ − 88.6434° 6.7879∠ − 119.0536° 8.2125∠ − 12.9194°
6.5919∠115.0558°
𝐼𝑏0123
0.4999∠ − 88.6434° 6.7879∠150.9464°
8.2125∠77.0806°
6.5919∠ − 64.9442° ]
=[
0.4999∠ − 88.6434° 6.7879∠ − 29.0536°
8.2125∠167.0806°
6.5919∠25.0558°
𝐼𝑐0123
0.4999∠ − 88.6434°
6.7879∠60.0536°
8.2125∠ − 102.9194° 6.5919∠ − 154.9442°
[𝐼𝑑0123 ]
(9)
The symmetrical components transformation
matrix for four-phase system is defined as,
𝟏 𝟏 1 1
𝟏 𝒂𝟑 𝑎1 𝑎2
A=[
]
(10)
1 𝑎1 𝑎2 𝑎3
1 𝑎2 𝑎3 𝑎1
CONCLUSION
In this paper, the basic concepts of symmetrical
components are discussed in brief to support the
analysis of four-phase unbalanced system. A systematic
logical approach for the construction of the symmetrical
components transformation matrix is discussed.
(6)
(7)
MATLAB PROGRAM AND RESULTS
[3]
REFERENCES
James E. Brittain, ‘Scanning the Past: Charles L. G. Fortescue
and Method of Symmetrical Components’, Proceedings of
IEEE, Vol. 86(5), pp. 1020-1021, May 1998.
Hadi Sadat, Power Systems Analysis, Tata McGraw Hill
Edition, India, pp. 400-406, 2002.
Boyajian A., ‘Symmetrical-Componenets Analysis of the FourPhase System, Transactions – Electrical Engineering, pp. 48-51,
1943.
Authors Biography:
Iranna Korachagaon, (B.Engg., 1991, M.Tech., 2001, Ph.D., 2012)
has more than 20 years of experience in industry and academia. His
research interests include – Power Systems, Renewable Energy
Applications, Energy Management & Audit and
Teaching-Learning Techniques. He has
published more than 18 research papers in
reputed International/ National Journals/
Conferences and authored one text book. He is
life member of professional bodies such as
MIE, MISTE, MSESI, MGMM. Presently he is
serving as Professor, Department of Electrical
Engineering, Annasaheb Dange College of
Engineering & Technology, Ashta-416301, Sangli, India. Email
Address: irannamk@gmail.com , Cell. +91 98600 88339.
www.irannamk.com
2
Appendix - I
% To find Symmetrical Components for 3-phase Unbalanced System
Q=[3 310
5 105
4 12]
i=sqrt(-1);
n=3; % Number of Phases
a=1.0*(cos((360/n)*pi/180)+i*sin((360/n)*pi/180));
a2=a^2;
% Conversion from Polar to Cartesian Form of Matrix I
Ia=Q(1)*(cos(Q(4)*pi/180)+i*sin(Q(4)*pi/180));
Ib=Q(2)*(cos(Q(5)*pi/180)+i*sin(Q(5)*pi/180));
Ic=Q(3)*(cos(Q(6)*pi/180)+i*sin(Q(6)*pi/180));
% Define symmetrical components transformation matrix
A= [1
1
1
1
a2
a
1
a
a2];
A012=inv(A)*[Ia;Ib;Ic];
% To check the results
Add0=A012(1)+A012(2)+A012(3); % This is Ia
Add1=A012(1)+a2*A012(2)+a*A012(3); % This is Ib
Add2=A012(1)+a*A012(2)+a2*A012(3); % This is Ic
% Symmetrical Components in Cartesian Form
Symm3= [A012(1) A012(2) A012(3)
A012(1) a2*A012(2) a*A012(3)
A012(1) a*A012(2) a2*A012(3)];
% Symmetrical Components in Polar Form
Sym3Abs=abs(Symm3)
Sym3Angle=angle(Symm3)*180/pi
% End of the Program
Results
Q =
3
5
4
310
105
12
Sym3Abs =
1.8852
1.8852
1.8852
3.3495
3.3495
3.3495
1.3761
1.3761
1.3761
Sym3Angle =
36.4890 -106.4364
-8.6327
36.4890 133.5636 111.3673
36.4890
13.5636 -128.6327
3
Appendix - II
% To find Symmetrical Components for 4-phase Unbalanced System
Q=[3 310
5 105
4 12
7 230]
i=sqrt(-1);
n=4; % Number of Phases
a=1.0*(cos((360/n)*pi/180)+i*sin((360/n)*pi/180));
a2=a^2;
a3=a^3;
% Conversion from Polar to Cartesian Form of Matrix Q
Ia=Q(1)*(cos(Q(5)*pi/180)+i*sin(Q(5)*pi/180));
Ib=Q(2)*(cos(Q(6)*pi/180)+i*sin(Q(6)*pi/180));
Ic=Q(3)*(cos(Q(7)*pi/180)+i*sin(Q(7)*pi/180));
Id=Q(4)*(cos(Q(8)*pi/180)+i*sin(Q(8)*pi/180));
% Define symmetrical components transformation matrix
A=[1
1
1
1
1
a3 a
a2
1
a
a2 a3
1
a2 a3 a];
A0123=inv(A)*[Ia;Ib;Ic;Id];
A0123rho=abs(A0123);
A0123theta=angle(A0123)*180/pi;
% To check the results
Add0=A0123(1)+A0123(2)+A0123(3)+A0123(4); % This is Ia
Add1=A0123(1)+a3*A0123(2)+a*A0123(3)+a2*A0123(4); % This is Ib
Add2=A0123(1)+a*A0123(2)+a2*A0123(3)+a3*A0123(4); % This is Ic
Add3=A0123(1)+a2*A0123(2)+a3*A0123(3)+a*A0123(4); % This is Id
% Symmetrical Components in Cartesian Form
Symm4=[A0123(1) A0123(2) A0123(3) A0123(4)
A0123(1) a3*A0123(2) a*A0123(3) a2*A0123(4)
A0123(1) a*A0123(2) a2*A0123(3) a3*A0123(4)
A0123(1) a2*A0123(2) a3*A0123(3) a*A0123(4)];
% Symmetrical Components in Polar Form
Symm4Abs=abs(Symm4)
Symm4Angle=angle(Symm4)*180/pi
% End of the Program
Results
Q =
3
5
4
7
310
105
12
230
Symm4Abs =
0.4999
0.4999
0.4999
0.4999
6.7879
6.7879
6.7879
6.7879
8.2125
8.2125
8.2125
8.2125
6.5919
6.5919
6.5919
6.5919
Symm4Angle =
-88.6434 -119.0536 -12.9194 115.0558
-88.6434 150.9464
77.0806 -64.9442
-88.6434 -29.0536 167.0806
25.0558
-88.6434
60.9464 -102.9194 -154.9442
4