Energy and Power Engineering, 2013, 5, 1139-1146
doi:10.4236/epe.2013.54B217 Published Online July 2013 (http://www.scirp.org/journal/epe)
Model of Three-Phas e Transmission Line with the Theory
of Modal Decomposition Implied
Rodrigo Cleber da Silva, Sérgio Kurokawa
Department of Electrical Engineering, Unesp, Ilha Solteira, Brasil.
Received February, 2013
ABSTRACT
This paper shows the development of transmission line model, based on lumped element circuit that provides answers
directly in the time and phase domain. This model is valid to represent the ideally transposed line, the phases of each of
the small line segments are separated in their modes of propagation and the voltage and current are calculated at the
modal domain. However, the conversion phase-mode-phase is inserted in the state equations which describe the currents
and voltages along the line of which there is no need to know the user of the model representation of the theory in the
line modal domain.
Keywords: π-circuits; Electromagnetic Transients; Transmission Lines; Transmission Line Models
1. Introduction
It is known that a transmission line power has a basic
characteristic that is the fact of their parameters of
longitudinal and transverse are distributed along the
length of the line. This characteristic with the fact that
their parameters of longitudinal line vary as a function of
frequency, makes the line of electric power transmission
an element with a certain particularities that should be
taken into consideration when their representation [1].
The distributed nature of the longitudinal parameters
of a transmission line must be taken into account,
especially in studies and simulations of transient
electromagnetic resulting from operations of maneuvers
and atmospheric discharges occurring in the line or
around to it. In these situations, the transient current and
voltage of the line assume characteristics such that a full
understanding of the same is only possible if these
quantities are treated as waves that propagate along the
transmission line. This particularity makes many
researchers, since 1960, working on developing of the
models of transmission lines focused on analysis of
electromagnetic transients that occur in the electrical
system due to disturbances verified on the line.
The longitudinal and transverse parameters of a power
transmission line are defined in terms of geometric
characteristics of the line, the environment surrounding
their conductor (in case the air, in the case of airlines
(overhead line) and the electrical and magnetic
characteristics of the conductors and the ground on
which the line was constructed [2]. All these factors
make with that the longitudinal parameters of the airlines
are variable in frequency. For analysis of transient
electromagnetic resulting from operations of maneuvers
and switching that occur in single-phase transmission
line may be represented such a line by means cascade of
the π -circuit [3].
The model line, in which it is represented by a cascade
π-circuit can be used to represent polyphase lines. In this
case, the line of n phases coupled due to the mutual
impedances to be separated into its n modes of
propagation which behave in the modal domain, as n
single-phase fully decoupled. Then, each propagation
mode can be represented by a cascade π-circuit [3]. Once
the calculated current and voltage in each of the modes
of propagation, it is possible to obtain the currents and
voltages in the phases of the line. The conversion phase-
mode-phase occurs through the use of a modal
transformation matrix appropriate [4].
The polyphase representation of lines using the model
described above has the advantage of the fact that a
model developed directly in the time domain, which can
be easily implemented in classical software used in
simulations of electromagnetic transients in power
systems [5] (as an example of such programs, it is
possible to cite the Electromagnetic Transients Program -
EMTP [6]). This model can also be described as the
equations of state and be used independently without the
need to have a program type EMTP [7-10].
2. Development of the Model

R. C. da SILVA, S. KUROKAWA
1140
The Figure 1 shows a three phase transmission line ide-
ally transposed.
The transmission line shown in Figure 1 may be rep-
resented by a cascade of line segments as shown in Fig-
ure 2.
In Figure 2, V1J(t) and V1(j-1)(t) are the voltages at
terminals of the j-th segment of the first phase of the line
while V2j(t) and V2 (j 1)(t) the voltages at terminals of the
segment of the j-th row and second stage, V3j(t) and
V3(j-1)(t) are the voltages at terminals of the j-th third
stage segment of the line. The currents i1j(t), i2j(t) and i3j(t)
are the references to the phases 1, 2 and 3, respectively,
of the line segment.
In Figure 2, R11 is resistance proper in the phase 1, 2
and 3 of the line segment, while L11 is the proper
inductance of the in the phases 1, 2 and 3. Note that the
terms of mutual inductance and resistance are represented
by R12 and L12. This line segment has partial capacitances
C01 and C12.
The longitudinal and transversal parameters of the line
segment shown in Figure 2, can be written in matrix
form as (1)-(3):

11 12 12
1211 12
12 12 11
RRR
[R]RRR
RRR








(1)
1
23
45
(9.27; 24.4)
(7.51; 36)
3.6 m
Figure 1. Transmission line three phase.
Figure 2. Representation of a line segment by discrete cir-
cuit elements.

11 12 12
1211 12
12 12 11
LLL
[L]LLL H
LLL









(2)

11 12 12
1211 12
12 12 11
CCC
[C]CCC F
CCC





(3)
In equations (1) - (3) we have:
11 11
d
RR
n

 (4)
12 12
d
RR
n

 (5)
11 11
d
LL
n

 (6)
12 12
d
LL
n

 (7)
11 11
d
CC
n

 (8)
12 12
d
CC
n

 (9)
In equations (4) - (9), d is the length of the transmission
line and n is the number of segments in which the line is
divided, of the type shown in Figure 2.
In the frequency domain can be written matrices of
longitudinal impedance and transverse admittance of the
line, as being:
[Z][R]j [L]

 (10)
[Y]j [C]

 (11)
The matrices modal admittance and impedance of the
line segment shown in Figure 2 are written as:
1
mV
[Z][T][Z][T ]

I
I
V
m
(12)
1
mI V
[Y][T ][Y][T]

 (13)
Substituting (10) and (11) in (12) and (13), re-
spectively, and manipulating these equations, it is
possible to obtain:
11
mVI V
[Z ][T] [R][T]j[T] [L][T]

 (14)
1
mI
[Y]j[T][C][T]

 (15)
Equations (14) and (15) can be written as:
mm
[Z ][R]j[L ]

 (16)
m
[Y ]j[C]
m

 (17)
which:






1
mV
RTRT


 I
(18)

R. C. da SILVA, S. KUROKAWA 1141






1
mV
LTLT


 I
(19)






1
mI V
CTCT



(20)
The matrices [Rm], [Lm] and [Cm], shown in (19)-(20),
are formed by the longitudinal parameters of the line
segment shown in Figure 2 represented it in the modal
domain.
Once the line is considered ideally transposed, it is
possible to consider [TI] as the matrix of Clarke [5] as
shows (21).
Ck
21
0
63
111
T623
111
623













(21)
When using the matrix the Clarke with the
transformation matrix, it is known that the matrix [TI] is
equal to the transposed [TV].
Substituting (1)-(3) and (21) in (18)-(20) and making
the appropriate mathematical operations are obtained:

MA
mMB
MC
R00
[R]0R0
00R






(22)

MA
mMB
MC
L00
[L]0L0 H
00L









(23)

MA
mMB
MC
C00
[C]0C0 F
00C





(24)
which:
MA11 12
RRR (25)
MB1112
RRR (26)
MC 1112
RR2R (27)
MA11 12
LLL (28)
MB1112
LLL (29)
MC 1112
LL2L (30)
MA11 12
CCC (31)
MB11 12
CCC (32)
MC 1112
CC2C (33)
In the equations (22)-(24) that the matrices [Rm], [Lm]
and [Cm] that the elements are not on the main diagonal
are all zero. This means that the line segment shown in
Figure 2 is represented in modal domain for three
propagation modes uncoupled. Each of these propagation
modes can be represented by a π-circuit as shown in
Figure 3.
Applying the Kirchhoff's laws in circuit mode
propagating A, B and C shown in Figure 3, we have:
MAj MA
MAj MAj
MA MA
MA(j1)
MA
dI (t)R1
I(t) E(t)
dt LL
1
E(t)
L
 

(34)
MBj MB
MBj MBj
MB MB
MB( j1)
MB
dI (t)R1
I(t) E(t)
dt LL
1
E(t)
L
 

(35)
MCj MC
MCj MCj
MC MC
MC( j1)
MC
dI (t)R1
I (t)E (t)
dt LL
1
E(t)
L
 

(36)
MAj
MA
MA
dE (t)2I(t)
dt C
 (37)
MBj
MB
MB
dE (t)2I(t)
dt C
 (38)
MCj
MC
MC
dE (t)2I(t)
dt C
 (39)
Writing the currents and voltages of the modes A, B
and C of the line segment in the form of vectors is
obtained:
MAj
Mj MBj
MCj
I(t)
[I ]I(t)
I(t)











(40)
MAj
Mj MBj
MCj
E(t)
[E ]E(t)
E(t)











(41)
i
m j
'
2
m
C '
2
m
C
L’
m
R’
m
v
m j
v
m (j-1 )
Figure 3. Representation of a line segment in modal do-
main.

R. C. da SILVA, S. KUROKAWA
1142
MA( j1)
M(j1)MB(j1)
MC( j1)
E(t
[E] E(t)
E(t








)
)




(42)
The relationship between the magnitudes and phase
and mode are written as:
1
MjI Fj
[I][T ][i]

 (43)
1
MjV Fj
[E][T ][V ]

 (44)
which:
1j
Fj2 j
3j
i(t)
[i ]i(t)
i(t)





(45)
1j
Fj2 j
3j
V(t)
[V ]V(t)
V(t)





(46)
Substituting (45) and (46) and (21) in (43) and (44), we
obtain:
MA1j2 j3j
21 1
E(t)V(t) V(t) V(t)
66 6
 (47)
MB2 j3 j
11
E(t)V (t)V (t)
22
 (48)
MC1j2 j3 j
111
E(t)V(t) V(t) V(t)
333
 (49)
MA1j2 j3j
21 1
I(t)i(t)i(t)i(t)
666
 (50)
MB2 j3 j
11
I(t)i (t)i (t)
22
 (51)
MC1j2 j3 j
111
I(t)i(t) i(t) i(t)
333
 (52)
Substituting (25)-(33) and (47)-(52) in (34)-(39) and
making the appropriate mathematical manipulation, we
obtain:
1j
11j21j32j42j33j
43j 21(j1) 42(j1)
43(j1)
di (t)ai(t)aV(t) ai(t) aV(t) ai(t)
dt
aV(t)aV(t)aV(t)
aV(t)


 
 

(53)
2j
31j4 1j12j2 2j
3 3j43j41(j 1)22(j 1)
43(j1)
di (t)ai(t) aV(t) ai(t)aV(t)
dt
ai(t)aV(t)aV(t)aV(t)
aV(t)


 
 

3j
31j41j3 2j
42j 13j23j
41(j 1)42(j 1)
23(j1)
di (t)ai(t)aV(t)ai(t)
dt
aV(t)+ai(t)aV(t)
aV(t)aV(t)
aV(t)


 



(55)
1j
51j6 2j
63j
dV (t)ai(t)ai(t)
dt
ai(t)


(56)
2j
61j52j
53j
dV (t)ai(t)ai(t)
dt
ai(t)


(57)
3j
61j 62j
53j
dV (t)ai (t)ai(t)
dt
ai(t)


(58)
which:
11 121112
1
11 121112
2(RR )R2R 1
a3LL L2L


 



(59)
2
11 121112
12 1
a3L LL 2L

 



(60)
11 121112
3
11 121112
RR R2R 1
a3L LL2L






(61)
4
11 121112
111
a3L LL 2L





(62)
5
11 121112
22 1
a3C CC2C





(63)
6
11 121112
21 1
a3C CC2C

 



(64)
Writing the equations (53)-(58) as state equation is
obtained:
[x][A][x] [B][u(t)]


 (65)
where:

t1j1j 2j
2j 3j3j
di(t)dV (t)di(t)
x
dt dtdt
dV(t)di(t)dV (t)
dtdt dt







(66)

(54)
Being i1j, i2j i3j and currents at the terminal of the line
segment, V1j, V2j, V3j tensions in the terminal segment of
the line, all in the time domain.
The vector is the vector containing the derivatives of
the vector [x].

R. C. da SILVA, S. KUROKAWA 1143
123434
566
341234
656
343412
665
aaaaaa
a0a0a0
aaaaaa
[A] a0a0a0
aa aaaa
a0a0a0






















(67)

244
424
442
aaa
000
aaa
B000
aaa
000













(68)

1( j1)
2( j1)
3( j1)
V(t)
V(t)
u(t)
V(t)








(69)
The equation (65) describes the currents and voltages
in the line segment shown in Figure 2 directly in the
phases. The solution of equation (65) can be obtained by
methods of numerical integration [11].
In equation (65) was obtained for a one line segment
as shown in Figure 2. However, the same reason may be
used to obtain the equations of current and voltage when
the line segments is represented by n circuit of the type
shown in Figure 2, connected in cascade. Thus, it is
shown that the currents and voltages phase along the line,
when it is represented by a cascade n circuit with type
shown in Figure 2, rewriting the equation of state (65)
we have:

t
111n 11 1n212n
212n313n313n
[x]i(t)i(t) v(t)v(t) i(t) i(t)
v(t)v(t) i(t)i(t) v(t) v(t)]
 
 
(70)
The vector [x], in (70) has 6n elements and consists of
the current longitudinal and the voltages transversal in
the phases of the line represented by a cascade of n
circuits of the type shown in Figure 2. Thus, the
quantities i1j(t), i2j(t) and i3j(t) correspond to the currents
in the phases 1, 2 and 3, respectively, the j-th circuit
represented by a segment equal to the circuit shown in
Figure 2. Similarly, the magnitudes V1j(t), V2j(t) and
V3j(t) corresponding to the voltages in phases 1, 2 and 3
in the j-th segment.
123434
576767
341234
675767
343412
676757
[A][A][A] [A][A] [A]
[A][A ][A ][A ][A ][A ]
[A ][A][A][A][A ][A]
[A] [A][A ][A][A ][A ][A ]
[A] [A][A] [A] [A] [A]
[A ][A ][A ][A ][A ][A ]










(71)
The matrix [A] in (71), is a square matrix of dimension
6n. This array consists of 36 square submatrices of
dimension n, which obeys the following formation rules:
- Submatrices [A1] and [A3]: these sub matrices have
non zero elements only on the main diagonal and is
written as:
p
p
p
p
p
a
a
[A ]
a
a

















 (72)
Equation (72) is valid for p = 1 and p = 3, p being the
element that is calculated from the equations (59)-(64).
- Submatrices [A2] and [A4]: these submatrices have
nonzero elements only on the main diagonal and at low
subdiagonal and are written as:
p
pp
p
p
p
p
p
a
aa
a
[A ]
a
aa






















(73)
The equation (73) is valid for p = 2 and p = 4, p being
the element that is calculated from the equations
(59)-(64).
- Submatrices [A5] and [A6]: these submatrices have
nonzero elements only on the main diagonal and top
subdiagonal and are written as:
pp
pp
p
p
p
p
aa
aa
1
[A ]2aa
2a




















 (74)
The equation (74) is valid for p = 5 and p = 6, p being
the element that is calculated from (59)-(64).
The submatrices [A7] rest are zero matrices.
The matrix [B] is of order 6n x 3 which are written as:

11 1213
(2n 1)1(2n 1)2(2n 1)3
(4n 1)1(4n 1)2(4n 1)3
bbb
000
bb b
B
000
bb b
000
 
 
































(75)

R. C. da SILVA, S. KUROKAWA
1144
which:
11 2
b
a (76)
12 4
b
a (77)
13 4
b
a (78)

4
2n11
b
a


 (79)

2
2n1 2
b
a


 (80)

4
2n13
b
a


 (81)

4
4n11
b
a


 (82)

4
4n1 2
b
a


 (83)

6
4n13
b
a


 (84)
The elements a2 and a4 are calculated from equations
(59)-(64).
The vector [u (t)] is written as:
1
2
3
V(t)
[u(t)]V(t)
V(t)






(85)
And V1(t), V2(t) and V3(t) are sources connected in the
phases 1, 2 and 3 in the sending end of the line.
It is noted that in (65) consists of only quantities and
parameters which are in the field of phase and allows
obtaining directly in the time domain, the currents and
voltages along phase the length thereof.
3. Validation of the Proposed Model
For to validation of the proposed model, the results
obtained were compared with those obtained with the
model EMTP. This is one three-phase line model in
software ATPD raw. It is also a model for discrete
parameters.
The two models used to represent a hypothetical three-
phase line with the lines ideally transposed, as shown in
Figure 4.
1
23
45
(9.27; 24.4)
(7.51; 36)
3.6 m
Figure 4. Representation of three phase transmission line.
Assumed that each phase of line consists of a 1 cm
radius of the conductor which shown in Figure 4. The
parameters that were calculated in line frequency of 60
Hz, taking into account of the soil and Skin effect. Once
obtained the following values for the parameters of
longitudinal and transversal line.
 
0.6667 0.46670.4667
R'0.46670.66670.4667 /km
0.4667 0.46670.6667









(86)


1.50.5167 0.5167
L'0.51671.50.5167 mH/km
0.5167 0.51671.5





(87)
 
0.0075-0.0018-0.0018
C'-0.00180.0075-0.0018 F/km
-0.0018 -0.00180.0075






(88)
It is considered that the transmission line shown in
Figure 4, 100 km in length, had one of its phases
energized by a source of constant voltage of 440 kV,
while the receiving end of the line were open. The
configuration described above is shown in Figure 5.
The model proposed and the EMTP model used to
simulate the voltages in the phases at the receiving end of
the line shown in Figure 5.
In the simulations with the model proposed was
represented by 100 circuits of the line (shown in Figure
2) connected in cascade. In the ATPD raw the line was
also represented by 100 discrete circuits (each repre-
senting a small segment of the line) connected in
cascade.
Figures 6, 7 and 8 shows, respectively, the voltages at
the receiving end in the phases 1, 2 and 3 of the line. The
curve shows of the results obtained with the model and
the curve b and the software EMTP.
Figures 6, 7 and 8 show that there is no difference
between the results obtained with the two models. Thus
we can say that the considerations made during the
development of the model are correct. Importantly, the
proposed model allows obtaining the results directly in
the field of phases.
Figure 5. Validation test of the model for three phase
transmission line.

R. C. da SILVA, S. KUROKAWA 1145
00.5 11.5 22.5 33.5
0
200
400
600
800
1000
Tempo [ms]
Tensão [kV]
(a)
(b)
Figure 6.Voltage in the receiving end of the line in phase 1:
proposed model (curve a) and EMTP model (curve b).
00.5 11.522.533.5
-400
-200
0
200
400
Tempo [ms]
Tensão [kV]
(a)
(b)
Figure 7. Voltage in the receiving end on the phase 2: pro-
posed model (curve a) and EMTP mode l (c urve b).
00.5 11.5 22.5 33.5
-400
-200
0
200
400
Tempo [ms]
Tensão [kV]
(a)
(b)
Figure 8. Voltage in the receiving end on the phase 3: pro-
posed model (curve a) and EMTP mode l (c urve b).
4. Conclusions
This article was shown to develop a new model for
three-phase transmission line ideally transposed. The
model allows calculating the currents and voltages from
phase along the line, directly in the time domain and
without the use of modal transformation matrix explicitly.
The currents and voltages from phase of the line are
written in the form of equations of state, and the state
matrices [A] and [B] have only line parameters in the
domain of phases. The results obtained with the model
shown is identical to the results obtained with the model
EMTP, confirming that the model are designed correctly.
The main advantage of this model is that any user can
use this program without prior knowledge about the
theory of modal decomposition.
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5. Acknowledgements
Thanks to CNPq and FAPESP for funding the research.
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