Calculation method of electric power lines magnetic field strength based on cylindrical spatial harmonics

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UDC 621.3. 013
doi: 10. 20 998/2074−272X. 2016.2. 04
A.V. Yerisov, K.D. Pielievina, D. Ye. Pelevin
Purpose. Simplification of accounting ratio to determine the magnetic field strength of electric power lines, and assessment of their environmental safety. Methodology. Description of the transmission lines of the magnetic field by using techniques of spatial harmonic analysis in the cylindrical coordinate system is carried out. Results. For engineering calculations of electric power lines magnetic field with sufficient accuracy describes their first spatial harmonic magnetic field. Originality. Substantial simplification of the definition of the impact of the construction of transmission line poles on the value of its magnetic field and the bands of land alienation. sizes. Practical value. The environmentally friendly projection electric power lines on the level of the magnetic field. References 6, tables 1, figures 4.
Key words: electric power line, magnetic field, environmental safety, cylindrical spatial harmonics.
На основе пространственного гармонического анализа магнитного поля в цилиндрической системе координат предложен метод расчета индукции магнитного поля линий электропередачи. Показано, что магнитное поле линий электропередачи с достаточной для инженерных расчетов точностью описывается первой цилиндрической пространственной гармоникой. Использование предложенного метода позволяет существенно упростить определение влияния конструкции опор линий электропередачи на величину их магнитного поля и на ширину полос отчуждения земельных участков. Библ. 6, табл. 1, рис. 4.
Ключевые слова: линия электропередачи, магнитное поле, цилиндрические пространственные гармоники.
Introduction. One of the problems solved by the designers of overhead transmission lines (TL) in assessing their environmental safety is determination of dimensions ±Xs of the trackside width, as shown in Fig. 1. Among the factors which determine the width of the strips are installed on their border ±Xs limits [1, 2] of the value of the module of the magnetic field (MF) strength vector Bl produced by TL at the height h0 of the earth'-s surface. Under these restrictions, the value of the module Bl away -Xs & gt- x & gt- Xs from the TL should be less than the specified value Bs of the magnetic field strength. Borders (-Xs- +Xs) of the strip of alienation by the parameter Bs are determined by the calculated dependence (magnetograms) of the TL magnetic field strength module Bl (Fig. 1).
The goal of the work is to simplify the settlement of relations to determine the MF strength of the TL and evaluate their environmental safety.
The goal of the work proposed to be carried through the use of cylindrical space harmonics to calculate the magnetic field strength of the TL.
Presentation of research materials. At the description of the TL magnetic field we assume that:
• Phase conductor lines are parallel current filaments of infinite length and infinitely small diameter.
• Line currents 1a, 1B Ic form a symmetrical system:
IB = a I, IC = a I.
1a = I
where a = ej4n!3.
Under what assumptions spatial harmonic analysis of the magnetic field of the TL can be made in a cylindrical coordinate system (r, p, Y) which Y-axis passes through the center of a circle of minimum radius rmin where all current filaments fit (Fig. 2).
Relation (1) allows to represent module of the magnetic field strength Bi (x) of three-phase line at an arbitrary point in space P as the modulus of the sum of the magnetic field strengths BA0(P), BB0(P), BC0(P)
respectively of three independent closed broaching circuits A — 0, B — 0 h C — 0 (see Fig. 2).
Fig. 1. Magnetograms of the TL
Problem definition. To simplify the calculation of the MF of the TL in the far field (at the border of the exclusion zone of the TL) multidipole transmission line models [3], based on the use of spherical spatial harmonics are utilized. At the same calculation relations are quite complex, and final calculation results are, as a rule, in numerical format, which complicates the practical need to establish cause — effect relationships between design parameters of transmission lines and distribution of their MF strength.
Bs (P) =
BA-0(P) + BB-0(P) + BC-0(P)
When the selected track (along the Y-axis) of passing of inverse wires with currents, _ 1a, _ 1B and _ 1c the position of each of three circuits define respectively filaments coordinates of phases A, B, C.
Spatial harmonic analysis of the MF of a closed current circuit. There is a closed current circuit, for example, A — 0 (Fig. 2).
© A.V. Yerisov, K.D. Pielievina, D. Ye. Pelevin
& lt-p
— / _L_

Fig. 2. Representation of a three-phase line as three independent circuits
Vector potential A (A-0)Y of the magnetic field of such a circuit in the arbitrary point of the space P (r, q& gt-, Y) is determined as a sum of the corresponding vector potentials Aay, Aoy of the current of the phase A and the opposite current and taking into account [4] it can be determined by the relation
(A-0)y — oy
= Aoy + Aay = L0~T~ln r —
¦ /& quot-0 (r 2 + (ra)2 — 2rra cos (^ -Va))•
Relation (3) can be represented as Fourier series after that for the external region (r & gt- rmin) it will have the known form [5]
= f 7
(aan cosnc + ban sinnq& gt-)
where aan, ban are the amplitudes of the n-th order of the magnetic field'-s vector potential of the current circuit A-0
aan = (ra)n cosn^, ban = (ra)& quot- sinn^a • (5) Magnetic vector potential'-s harmonics (4) determine also the corresponding harmonics of its magnetic field strength Bar and BaV.
Mo dA (A-0)Y = Mo I V sin + ban cos nq)
B —
nar ~ '-

B =Mo_ dA (A-0)y a& lt-p 2k dr
L1 ^ (aan cosnc + ban sinnq& gt-)
. — y
2k ^
J (aan)2 +(ban)2 • (8)
n=1 r
Magnetic field strength module Ban of the n harmonic in the point P (r, g& gt-, Y) will be dependent on the r-coordinate
Ban =
Table 1 represents values of amplitudes aan, ban of two first harmonics for the circuit A-0 in the coordinate systemX, Y, Z (Fig. 2).
Table 1
Amplitudes of the magnetic field strength harmonics for the current circuit A — 0
Amplitude of harmonics Relations for the circuit with coordinates xa, za
aa1 xa
bai Za
aa2 (x) — (Za)2
ba2 2xVZa
This format of the amplitudes aan, ban representation harmonizes well with the design document for TL pylons which regulates coordinates of points of suspension of its wires with respect to earth surface.
By analogy with (5) amplitudes of harmonics abn, bbn and acn, bcn of circuits B — 0 and C — 0 are respectively determined.
abn = a ¦ (rb)n cos npb, bbn = a ¦ (rb)n sin npb, acn = a2 ¦ (rc)n sinnq) c, bcn = a1 ¦ (rc)n sinnq) c•
cn c c cn c
The structure of series (6), (7) is such that as r increases the contribution of high-order harmonic components in the magnetic field strength Br and Bv reduces.
So, the magnetic field strength at a distance of two-wire line x & gt- rmm is described mainly by its first (n = 1) harmonic constructed as illustrated by equation (8) magnetogram in Fig. 3. It also presents the results of calculations by the Biot-Savart-Laplace low in accordance with [6].
2 r
h = 3'-min
Calculation by the Biot-Savart-Laplace low
0,8 1,6 2,4 3,2 4 x
in parts of Trnin
Bi (x) «Bn (x) = Mo1-
_ h0y + x& quot-
where h is the distance from the ground level (Fig. 1) to the center of the circle rmin which fit all current lines of the TL.
For ease of calculation the distance h can be set equal to the average height ha, hb, hc of the respectively suspension of phase conductors A, B and C
h * 1/3 {ha + hb + hc). (13)
After simple but cumbersome transformations the relation (12) can be reduced to the form:

Bi (x) «Mo1
242x ¦ ((
(h — ho)2 + x2)'-
where drms is the mean square distance between the wires of the TL
drms =^(dAB)2 + {dBC)2 + {dCA)2, where dAB, dBC, dCA is the distance between the suspension points on a support phase wires A and B, B and C, C and A, respectively.
Analytical representation of magnetograms (14) permits to determine the size of the band ±Xs of the exclusion for a given parameter Bi
, X Mo ¦1 ¦ drms (h h)2
±Xs =iHS^B-(h-ho) •
Fig. 3. Magnetograms of Bl of a two-wire line at unit current 1
Comparison of the calculation results (Fig. 3) shows that the distance from the transmission line axis at a distance of more than rmin the error of the proposed method in comparison with the exact method [6] does not exceed 10%, which confirms the possibility of using the first cylindrical space harmonics to calculate the MF of the TL at the boundary of their protected areas.
The magnetic field of single-circuit TL. Single circuit lines have one set of phase conductors. Their relative positions to each other and the Earth'-s surface determines the design of the (profile) of a TL pylon.
According to that shown in Fig. 2 «magnetic» interpretation of the transmission line, amplitudes aln and bln of harmonics of its magnetic field taking into account (1) and (2) are presented in the form of a sum corresponding to the amplitude of its independent circuits A-0, B-0, C-0:
2 2
aln = aan + ^ abn +acn, bin = ban +a bbn +abcn. (10)
The first significant harmonic of single-circuit TL is the harmonic of the order (n = 1). Its amplitudes aln and bln taking into account (5), (10) equal:
ai1 = xa +a ¦ xb +a-xc, b^ = za +a ¦ zb +a-zc. (11)
It should be note that values of the amplitude al1 and bn of the first harmonic (n = 1) do not depend on the beginning of the selected coordinate system X, Y, Z.
Knowledge of amplitudes of the first harmonic au and bl1 of the magnetic field of the TL allows by using the relation (7) to build its magnetogram
This relationship establishes a mutual relationship between the size ±Xs of the strip of alienation and TL characteristics — its current (1) loading and designs (profile) of its pillars, namely the average height h of wires suspension points and mean square distance drms between them.
Underground cable TL. Magnetograms of underground cable lines, similar to the single-circuit air TL are determined by the first (n = 1) harmonic of their magnetic field strength.
Below relations for magnetograms for two most commonly used cable laying (Fig. 4) obtained by taking into account (8) and (14) are presented.
d--«& gt- n
h «(ha 1 hb hc)
A f_f C
A m fi
C iV
d d
a b
Fig. 4. «Triangle» (a) and «flat» (b) cable line laying
For the cable «flat laying»
Bi (x) «Mo1
2 nh + ho)2 + x2) • For the cable laying by «triangle»
Bl (x) «Mo1-p-(d 2−2
2V2 ¦ ^¦((h + ho)2 + x2

1. It is shown that for the calculation of the magnetic field strength of transmission lines on the border of protected zones with limited accuracy (less than 10%), the first cylindrical space harmonic of its magnetic field can be used.
2. The simplified calculation relations of the magnetic field strength of the TL based on cylindrical spatial harmonics, allowing to simplify the calculation of the TL magnetic field distribution and assess the impact of the TL design peculiarities on the width of the land rights of way to ensure environmental safety are proposed.
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4. Shtafl M. Elektrodinamicheskie zadachi v elektricheskikh mashinakh i transformatorakh [Electrodynamic problems in electrical machines and transformers]. Moscow, Leningrad, Energiia Publ., 1966. 200 p. (Rus).
5. Jablonski P. Cylindrical conductor in an arbitrary time-harmonic transverse magnetic field. Przeglqd Elektrotechniczny — Electrotechnical Review, 2011, no. 5, pp. 49−53.
6. Rozov V. Yu., Reutskyi S. Yu., Pyliugina O. Yu. Method of calculating the magnetic field of three-phase power lines. Tekhnichna elektrodynamika — Technical Electrodynamics, 2014, no. 5, pp. 11−13. (Rus).
Received 04. 12. 2015
A.V. Yerisov1, K.D. Pielievina1,
D. Ye. Pelevin1, Candidate of Technical Science, 1 State Institution «Institute of Technical Problems of Magnetism of the NAS of Ukraine», 19, Industrialna Str., Kharkiv, 61 106, Ukraine. phone +380 572 992 162,
e-mail: erisov@yandex. ua, pelevindmitro@ukr. net
How to cite this article:
Yerisov A.V., Pielievina K.D., Pelevin D. Ye. Calculation method of electric power lines magnetic field strength based on cylindrical spatial harmonics. Electrical engineering & amp- electromechanics, 2016, no. 2, pp. 24−27. doi: 10. 20 998/2074−272X. 2016.2. 04.

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