__________________________________________
* Czestochowa University of Technology.
Dariusz KUSIAK*
Zygmunt PIĄTEK*
Tomasz SZCZEGIELNIAK*
ASPECT OF THE ELLIPTICAL FIELD IN THE SCREENED FLAT THREE-PHASE HIGH CURRENT BUSDUCT.
PART I – IN THE INTERNAL AREA OF THE SCREEN
The paper shown the elliptical magnetic field in the high current screened busduct in the case of the both the external and internal proximity effect. For the characterisation of complex vector values for such a field it is proposed that the length of the longer ellipse semi axis as indicated by the end of the vector within one period be used. Part I describes of elliptic field in the internal area of the screen of the flat three-phase high current busduct.
1.INTRODUCTION
Eddy currents within the shield (internal proximity effect) and within the neighbouring parallel conductor (shield) changes the distribution of the magnetic field within the shield (conductor) and its vicinity. The size of these changes depends on factors λ and α, and so the conductivity of the conductor or shield and its diameter, current frequency in the phase conductor and the respective geometric configuration between this conductor and the shield or a second neighbouring conductor. Current density has only one constant for the axis Oz and generates the so-called reverse reaction magnetic field, which is added as a vector to the magnetic field generated by phase currents [1].
One of the structural solutions for the construction of high current busducts is provided in the shape of the so-called flat three-pole high current busduct (figure 1) [2].
The variable intensity values of magnetic fields emitted by such busducts are large, even in rated current conditions [2, 3].
Component amplitudes of magnetic field intensity are not generally identical, and furthermore these components have different starting phases, i.e.
) , ( ) ,
(r Θ H r Θ
Hr Θ and r(r,Θ)Θ(r,Θ) . In his work [4] M. Krakowski introduces a nonnegative value in the form of complex vector norms
) , ( ) , ( )
,
(r Θ H r Θ H* r Θ
H (1)
R1 R2
R3 R4
d d
L3 L2 L1
μ0 e
Fig. 1. Three-pole screened flat high current transmission line
A such defined norm does not consider differing starting phases (r(r,Θ)Θ(r,Θ) ) of magnetic field intensity components, thus does not provide for the elliptical nature of this field.
From work [5-10] it is seen that magnetic fields at point P( Θr, ) may be presented as the superposition of two fields flowing in opposite directions. The sum of these fields is presented in the ellipse – figure 2.
Im
Re
2
Ha
Hb t=0 H2
H1
H H2
1
2 1
Fig. 2. Elliptic magnetic field [11]
The value of the longer semi axis then has the form
) , ( ) , ( ) , , ( max ) ,
( 1 2
) , 0
( H r Θ t H r Θ H r Θ
Θ r H
T
a t
(2)
whereas the shorter semi axis is expressed as
) , ( ) , ( ) , , ( min ) ,
( 1 2
) , 0
( H r Θ t H r Θ H r Θ
Θ r H
T
b t
(2a)
The ellipse angle to the actual axis
( , ) ( , )
2 ) 1 ,
(r Θ 1 r Θ 2 r Θ
Φ (3)
2. ELLIPTICAL MAGNETIC FIELDS IN THE INTERNAL AREA OF THE SCREEN
The total of magnetic fields in the internal area of the screen (r R3) is defined with the following formula [5]
) , ( )
, ( )
, ( )
,
( 1int int2 int3
int r Θ H r Θ H r Θ H r Θ
H (4)
Components vector H1int(r,Θ) are expressed as
nΘd p R
d R r r d r Θ I
r H
n n
n n n
n n n
r 1 sin
2 1 ) , (
1 3 3
1 int 1
1
(5)
and
nΘd p R
d R r r d r I
r Θ I r H
n n
n n n
n n n
Θ 1 cos
2 1 ) 2
, (
1 3 3
1 1
int 1
1
(5a)
The vector Hint2 (r,Θ) has only one tangent component in the following form
r Θ I
r HΘ
) 2 ,
( 2
int
2 (6)
Components vector Hint3 (r,Θ) are expressed with the formulas d nΘ p R
d R r r d r Θ I
r H
n n
n n n
n n
r 1 sin
) 2 , (
1 3 3
1 int 3
3
(7)
and
d nΘ p R
d R r r d r I r Θ I r H
n n
n n n
n n
Θ 1 cos
2 ) 2
, (
1 3 3
1 3
int 3
3
(7a)
The phase currents are symmetric
1 3
1
2 ]
3 exp[j2
and 3 ]
j2
exp[ I I I
I (8)
The magnetic field components (4) can be compared with the following field
2 1
0 2 R
H I
, and obtain their relative forms. Then we derive formulas for the relative components of the total magnetic field in the internal area of the screen (0r R3 or 0 ) of the 3-phase, 3-pole, flat, high-current busduct. These components have the form [5]
1
1 1
int sin
3 j2 exp 1
) , (
n
n- n n n n
n n
r ξ nΘ
d p Θ
h
(9)
and
1
1 1
int cos
3 j2 exp 1
) , (
n
n- n n n n
n n
Θ ξ nΘ
d p Θ
h
(9a)
while 0 and 0 Θ2. In the above formulas
) j 2 ( ) j 2 ( ) j 2 ( ) j 2
( 1 1 1
1
n n n n
n I K I K
p (9b)
and
) j 2 ( ) j 2 ( ) j 2 ( ) j 2
( 1 1 1
1
n n n n
n I K I K
d (9c)
where kR4 for
1
2
k , (0 1)
4
3
R
R ,
R3
d
(01) and
R4
r
.
The functions in the above formulas In1( 2j), Kn1( 2j), In1( 2j) and )
j 2
1(
Kn are the modified Bessel functions of the firsts and second order, respectively i.e.: n-1 and n+1 [5].
a) b)
Fig. 3. The distribution of relative radial component values for the total magnetic field in the internal area of the screen of the flat three-phase high current busduct: a) the modulus, b) the argument
a) b)
Fig. 4. The distribution of relative tangent component values for the total magnetic field in the internal area of the screen of the flat three-phase high current busduct:
a) the modulus, b) the argument
The distribution of these components for total magnetic field in the internal area of the screen of the flat three-phase high current busduct is depicted in figures 3 and 4.
The set of arguments for the radial and tangential field components are different and therefore at each point measured in a magnetic field, the field is elliptic, which is characterised by lengths ha( Θr, ) and hb( Θr, ) of the ellipse semi axes [1] – figure 5.
The highest value of the magnetic field intensity, and precisely the value of its module should be determined from the formula (2) – figure 6.
The comparison of relative value h indicated in formula (2) with value a h h indicated in formula (1) in the internal area of the screen with the determined value of angle Θ for variable parameter is presented in figure 7 [12].
Fig. 5. The distribution of relative values of lengths ellipsis semi axes of the magnetic field in the internal area of the screen of the flat three-phase high current busduct
Fig. 6. The distribution of relative the total magnetic field modulus in the internal area of the screen of the flat three-phase high current busduct
Fig. 7. The distribution of relative values ha and h h within the function of parameter in the internal area of the screen
3. CONCLUSIONS
The resultant magnetic field has two constants of differing amplitudes and commencing phases. As a result the field is an elliptic field. For its characterisation we should use values (measurements) for length Ha( Θr, ) for the longer and
) , ( Θr
Hb for the shorter ellipsis semi axis and angle Φ( Θr, ) between the longer semi axis and axis of abscissa. These values may be indicated after defining complex constant values for the magnetic fields – formulae (2), (2a) and (3) respectively.
In transmission lines values Ha( Θr, ), Hb( Θr, ) and Φ( Θr, ) change depending on the position of point P( Θr, ). From the calculations performed above (figure 5) it results that the form of the magnetic field is a narrow ellipse, where
) , ( )
,
(r Θ H r Θ
Hb a .
In technological processes, a significant value is the highest value for magnetic field intensity. In the case of elliptic fields, this value is accepted as length
) , ( Θr
Ha of the longer ellipsis semi axis. In busducts seen in practise for power frequencies, the value of parameter is between 5 and 20. This means that, of the flat three-phase high current busduct, value Ha( Θr, ) is lower than the norm value
) , ( Θr
H indicated in formula (1) – figure 7. Within the scope of 212 the difference between Ha( Θr, ) and H( Θr, ) can reach a dozen or so percent .
REFERENCES
[1] Piątek Z., Kusiak D., Szczegielniak T.: Elliptical magnetic field in the high current busducts (in Polish), Przegląd Elektrotechniczny, ISSN 0033-2097, R. 86 NR 4/2010, p.101-106.
[2] Piątek Z.: Modeling of lines, cables and high-current busducts (in Polish), Wyd. Pol.
Częst., Czestochowa 2007.
[3] Nawrowski R.: High-current air or SF6 insulated busducts (in Polish), Wyd. Pol.
Poznańskiej, Poznań 1998.
[4] Krakowski M.: Theoretical electrotechnics. Electromagnetic fields ( in Polish), WN PWN, Warsaw 1995.
[5] Kusiak D.: Magnetic field of two- and three-pole high current busducts, Dissertation doctor (in Polish), Pol. Częst., Wydz. Elektryczny, Częstochowa 2008.
[6] Baron B., Piątek Z.: Impedance and magnetic field of a tubular conductor of finite length (in Polish), XXIII IC SPETO, Gliwice-Ustroń 2000, ss. 477-484.
[7] Piątek Z, Kusiak D., Szczegielniak T.: Magnetic field of the shielded conductor (in Polish), Przegląd Elektrotechniczny, ISSN 0033-2097, R. 85, Nr 5, 2009, ss. 92-95.
[8] Piątek Z.: Impedances of Tubular High Current Busducts. Series Progress in High- Voltage technique, Vol. 28, Polish Academy of Sciences, Committee of Electrical Engineering, Wyd. Pol. Częst., Częstochowa 2008.
[9] Piątek Z., Kusiak D., Szczegielniak T.: Magnetic field of the three phase flat high current busduct (in Polish), Zesz. Nauk. Pol. Śl. 2009, Elektryka, z.1(209), pp. 51-65 [10] Piątek Z., Kusiak D., Szczegielniak T.: Elliptic field problems in a screened
symmetrical busduct, XXXIII IC SPETO, Gliwice-Ustroń 2010.
[11] Kusiak D.: Description of elliptic field in unshielded bifilar transmission line (in Polish), XV Conference Computer Applications in Electrical Engineering, ISBN 978-83-89333-34-6, Poznań 2010.
[12] Kusiak D.: Use elliptical magnetic fields in unscreened flat three phase busduct, XXXIII IC SPETO, Gliwice-Ustroń 2010.
This work is supported by Polish Ministry of Science and Higher Education under the project N N 511 312540.
ASPEKT POLA ELIPTYCZNEGO W EKRANOWANYM PŁASKIM TRÓJFAZOWYM TORZE WIELKOPRĄDOWYM.
CZĘŚĆ I – W OBSZARZE WEWNĘTRZNYM EKRANU
W artykule przedstawiono eliptyczne pole magnetyczne występujące w ekranowanym torze wielkoprądowym w przypadku zewnętrznego i wewnętrznego zjawiska zbliżenia. Do charakteryzowania wartości zespolonego wektora takiego pola zaproponowano długość dłuższej półosi elipsy jaką zakreśla koniec tego wektora w ciągu jednego okresu. Część I opisuje pole eliptyczne w obszarze wewnętrznym ekranu trójfazowego płaskiego toru wielkoprądowego.
