Calculation method of electric power lines magnetic field strength based on cylindrical spatial harmonics.
Introduction. One of the problems solved by the designers of overhead transmission lines (TL) in assessing their environmental safety is determination of dimensions [+ or -]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 [+ or -] [X.sub.s] limits [1, 2] of the value of the module of the magnetic field (MF) strength vector [B.sub.l] produced by TL at the height [h.sub.0] of the earth's surface. Under these restrictions, the value of the module [B.sub.l] away -[X.sub.s] [greater than or equal to] x [greater than or equal to] [X.sub.s] from the TL should be less than the specified value [B.sub.s] of the magnetic field strength. Borders (-[X.sub.s]; + [X.sub.s]) of the strip of alienation by the parameter [B.sub.s] are determined by the calculated dependence (magnetograms) of the TL magnetic field strength module [B.sub.l] (Fig. 1).
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 , 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.
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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] form a symmetrical system:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where a = [e.sup.j4[pi]/3].
Under what assumptions spatial harmonic analysis of the magnetic field of the TL can be made in a cylindrical coordinate system (r, [phi], Y) which Y-axis passes through the center of a circle of minimum radius [r.sub.min] where all current filaments fit (Fig. 2).
Relation (1) allows to represent module of the magnetic field strength B(x) of three-phase line at an arbitrary point in space P as the modulus of the sum of the magnetic field strengths [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively of three independent closed broaching circuits A - 0, B - 0 [??] C - 0 (see Fig. 2).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)
When the selected track (along the Y-axis) of passing of inverse wires with currents, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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).
Vector potential [A.sub.(A-0)Y] of the magnetic field of such a circuit in the arbitrary point of the space P(r, [phi], Y) is determined as a sum of the corresponding vector potentials [A.sub.ay], [A.sub.oy] of the current of the phase A and the opposite current and taking into account  it can be determined by the relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
Relation (3) can be represented as Fourier series after that for the external region (r [greater than or equal to] [r.sub.min]) it will have the known form 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where [a.sub.an], [b.sub.an] are the amplitudes of the n-th order of the magnetic field's vector potential of the current circuit A-0
[a.sub.an] = [([r.sub.a]).sub.n] cos n[[phi].sub.a], [b.sub.an] = [([r.sub.a]).sup.n] sin n[[phi].sub.a]. (5)
Magnetic vector potential's harmonics (4) determine also the corresponding harmonics of its magnetic field strength [B.sub.ar] and [B.sub.a[phi]].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
Magnetic field strength module [B.sub.an] of the n harmonic in the point P(r, [phi], Y) will be dependent on the rcoordinate
[B.sub.an] = [[mu].sub.0] [I/2[pi] x [r.sup.n+1]] [square root of ([([a.sub.an]).sup.2] + [([b.sub.an]).sup.2])]. (8)
Table 1 represents values of amplitudes [a.sub.an], [b.sub.an] of two first harmonics for the circuit A-0 in the coordinate system X, Y, Z (Fig. 2).
This format of the amplitudes [a.sub.an], [b.sub.an] 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 [a.sub.bn], [b.sub.bn] and [a.sub.cn], [b.sub.cn] of circuits B - 0 and C - 0 are respectively determined.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
The structure of series (6), (7) is such that as r increases the contribution of high-order harmonic components in the magnetic field strength [B.sub.r] and [B.sub.[phi]] reduces.
So, the magnetic field strength at a distance of two-wire line x [greater than or equal to] [r.sub.mm] 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 .
Comparison of the calculation results (Fig. 3) shows that the distance from the transmission line axis at a distance of more than [r.sub.min] the error of the proposed method in comparison with the exact method  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 [a.sub.ln] and [b.sub.ln] 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:
[a.sub.ln] = [a.sub.an] + [[alpha].sup.2] [a.sub.bn] + [alpha] [a.sub.cn], [b.sub.ln] = [b.sub.an] + [[alpha].sup.2] [b.sub.bn] + [alpha] [b.sub.cn]. (10)
The first significant harmonic of single-circuit TL is the harmonic of the order (n = 1). Its amplitudes [a.sub.ln] and [b.sub.ln] taking into account (5), (10) equal:
[a.sub.l1] = [x.sub.a] + [[alpha].sup.2] x [x.sub.b] + [alpha] x [x.sub.c], [b.sub.l1] = [z.sub.a] + [[alpha].sup.2] x [Z.sub.b] + a x [z.sub.c]. (11)
It should be note that values of the amplitude [a.sub.l1] and [b.sub.l1] of the first harmonic (n =1) do not depend on the beginning of the selected coordinate system Z, Y, Z.
Knowledge of amplitudes of the first harmonic [a.sub.l1] and [b.sub.l1] of the magnetic field of the TL allows by using the relation (7) to build its magnetogram
[B.sub.l](x) [approximately equal to] [B.sub.l1](x) = [[mu].sub.0]I [absolute value of ([square root of [([a.sub.l1]).sup.2] + [([B.sub.l1]).sup.2])]/2[pi] x ([(h - [h.sub.0]).sup.2] + [x.sup.2]). (12)
where h is the distance from the ground level (Fig. 1) to the center of the circle [r.sub.min] which fit all current lines of the TL.
For ease of calculation the distance h can be set equal to the average height [h.sub.a], [h.sub.b], [h.sub.c] of the respectively suspension of phase conductors A, B and C
h [approximately equal to] 1/3 ([h.sub.a] + [h.sub.b] + [h.sub.c]). (13)
After simple but cumbersome transformations the relation (12) can be reduced to the form:
[B.sub.l](x) [approximately equal to] [[mu].sub.0]I (14)
where [d.sub.rms] is the mean square distance between the wires of the TL
[d.sub.rms] = [square root of ([([d.sub.AB]).sup.2] + [([d.sub.BC]).sup.2] + [([d.sub.CA]).sup.2],
where [d.sub.AB], [d.sub.BC], [d.sub.CA] 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 [+ or -] [X.sub.s] of the exclusion for a given parameter [B.sub.l]
[+ or -] [X.sub.s] = [square root of ([[mu].sub.0] x I x [d.sub.rms]/2[square root of 2] x [pi] x [B.sub.l])] - [(h - [h.sub.0]).sup.2]. (15)
This relationship establishes a mutual relationship between the size [+ or -][X.sub.s] of the strip of alienation and TL characteristics--its current (I) loading and designs (profile) of its pillars, namely the average height h of wires suspension points and mean square distance [d.sub.rms] 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.
For the cable <<flat laying>>
[B.sub.l](x) [approximately equal to] [[mu].sub.0]I [[square root of 3] x d/ 2 [pi] x ([(h + [h.sub.o]).sup.2] + [x.sup.2]). (16)
For the cable laying by <<triangle>>
[B.sub.l](x) [approximately equal to] [[mu].sub.0]I [[square root of 3] x d/ 2 [square root of 2] [pi] x ([(h + [h.sub.o]).sup.2] + [x.sup.2]). (17)
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.
[1.] Pravyla ulashtuvannja elektroustanovok 5-te vyd., pererobl. j dopovn. (stanom na 22.08.2014) [Electrical Installation Regulations. 5 edition, Revised and enlarged (as of 22/08/2014)]. Kharkiv, Fort Publ., 2014. 800 p. (Ukr).
[2.] Stepanov I.M. Constructive modification reducing the intensity of the magnetic field on the tracks of overhead and cable power lines. ELEKTRO. Elektrotekhnika, elektroenergetika, elektrotekhnicheskaia promyshlennost'--ELEKTRO. Electrical engineering, power industry, electrical industry, 2009, no. 3. pp. 36-41. (Rus).
[3.] Rozov V.Yu., Reutskyi S.Yu., Pelevin D.Ye., Pyliugina O.Yu. The magnetic field of power transmission lines and the methods of its mitigation to a safe level. Tekhnichna elektrodynamika--Technical Electrodynamics, 2013, no. 2, pp. 3-9. (Rus).
[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).
A.V. Yerisov (1), K.D. Pielievina (1), D.Ye. Pelevin (1), Candidate of Technical Science,
(1) State Institution <<Institute of Technical Problems of Magnetism of the NAS of Ukraine>>, 19, Industrialna Str., Kharkiv, 61106, Ukraine. phone +380 572 992162, e-mail: email@example.com, firstname.lastname@example.org
Caption: Fig. 1. Magnetograms of the TL
Caption: Fig. 2. Representation of a three-phase line as three independent circuits
Caption: Fig. 3. Magnetograms of [B.sub.l] of a two-wire line at unit current I
Caption: Fig. 4. <<Triangle>> (a) and <<flat>> (b) cable line laying
Table 1 Amplitudes of the magnetic field strength harmonics for the current circuit A--0 Amplitude of Relations for the circuit with coordinates harmonics [x.sub.a], [z.sub.a] [a.sub.a1] [x.sub.a] [b.sub.a1] [z.sub.a] [a.sub.a2] [([x.sub.a]).sup.2] - [(z.sub.a]).sup.2] [b.sub.a2] 2[x.sub.a][z.sub.a]
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|Title Annotation:||Theoretical Electrical Engineering and Electrophysics|
|Author:||Yerisov, A.V.; Pielievina, K.D.; Pelevin, D.Ye.|
|Publication:||Electrical Engineering & Electromechanics|
|Date:||Mar 1, 2016|
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