__________________________________________
* Częstochowa University of Technology.
Paweł JABŁOŃSKI*
APPROXIMATE BEM ANALYSIS OF TIME-HARMONIC MAGNETIC FIELD DUE TO THIN-SHIELDED WIRES
Analysis of electromagnetic field of filamentary parallel wires with time-harmonic currents enclosed with a thin conductive shield is considered in this paper. The model uses the boundary element method (BEM), but the thin shell is treated specially by using an approximate quasi-analytical solution. Such an approach allows avoiding some numerical troubles connected with the small thickness of the layer, and leads to a system of equations with fewer unknowns when compared to the conventional BEM. Numerical tests confirm its usability in the considered class of problems.
1.INTRODUCTION
Wires with currents are often enclosed in conductive shields to separate them physically from the neighborhood. Analytical analysis of magnetic field and determination of the Joule power losses in such a configuration is possible only in specific cases [1]. Real configurations often require numerical analysis, e.g. by means of the Finite Element Method (FEM), the Collocation Method for integral formulation, or the Boundary Element Method (BEM). The latter allows taking into account the infinite neighborhood, and gives the field values only on boundaries, which is sufficient to evaluate the power transmitted across a boundary. No matter what method is used, one problem arises: because the shield has often very thin walls in relation to the other dimensions, a very fine discretization of the shield and its neighborhood is required [2]. In addition, nearly singular integrals appear in BEM in such a case. This paper presents an approximate BEM-based approach to this problem. It can be regarded as a generalization of model considered in [3, 4].
2.PROBLEMDESCRIPTION 2.1. Problem description
Let us consider K long parallel filamentary wires enclosed in a tubular shielding (Fig. 1). The wires carry time-harmonic currents I1, I2, …, IK of angular
frequency ω. The shielding has conductivity γ = const, magnetic permeability μr = const, and its thickness d is relatively small. The cross section of the whole configuration is assumed constant. The goal is to determine the magnetic field and possibly other quantities (like Joule power losses in the shield). For simplicity, the wires are assumed filamentary, although it is possible to consider a more realistic case with wires of finite cross-sections. The shield region is referred to as Ω1, whereas the external and internal insulating regions are Ω0 and Ω2, respectively.
Ω1
Ω0
Ω2 I1
x y z
γ, μr
S1 S2
I2
I3 d
Fig. 1. Filamentary wires placed in a thin shield
2.2. Governing equations
In the coordinate system the z axis of which is oriented along the wires, the magnetic vector potential A has only a z component. With fringing neglected, it depends on x and y, only. Maxwell’s equations lead to the following equation for the phasor of the z component of A:
k
k k
kδ x x δ y y I
μ V μγ A ωμγ
A j ( ) ( )
2 , (1)
where V is the scalar electric potential (its phasor), and δ is the Dirac’s delta. In non-conducting regions (γ = 0), the equation simplifies to the Poisson equation:
2 , 0 ,
) ( )
) (
2 (
m y
y δ x x δ I μ A
m k Ω I
k k
k
m . (2)
Solution of this equation can be written as A(m) = A'(m) + As
(m),where A'(m) satisfies the Laplace equation, and
m
k Ω
I k k
m k
y y x π x
I y μ
x
A 2 2
) 0 (
s ( ) ( )
ln 1 ) 2
,
( , (3)
where (xk, yk) are coordinates of kth filament.
In the conducting shield (γ ≠ 0) Eq. (1) remains unchanged. In BEM, however, it is better to make it homogeneous by using Lorentz gauge, what leads to
) 0
1 2 ( ) 1
2 (
A κ A , (4)
where
γ μ ωμ
κ2 j r 0 . (5)
Such an approach can lead to a discontinuity in A on the boundaries of the shell.
The discontinuity can be eliminated on the inner surface so that the continuity relationships for A on the shell boundaries take the following form:
0 ) (
, ) (
2 1
) 1 ( ) 2 ( )
1 ( ) 0
(
S
S C A A
A
A , (6)
where C is a constant to be determined. The continuity of the tangent components of the magnetic field intensity give the relationships as follows:
1 0 ,
1 0
2
1 1
) 1 (
r 2
) 2 (
1 ) 1 (
r 0
) 0 (
S
S n
A μ n A n
A μ n
A . (7)
Determination of constant C requires one more equation, which can be delivered by the Ampère’s law written for contour Γ1:
1
1 1
) 1 (
r 0
1 d
Γ
Θ n Γ
A μ
μ , (8)
where Θ1 is the total current enclosed by contour Γ1 (cross-section of boundary S1).
3.BEMBASEDMODELS 3.1. Standard BEM model
The problem can be solved with use of BEM. The conventional approach with the constant approximation for field and its normal derivative in each boundary element leads to the following equations:
,
, ,
2 s 2 2 2 2 2 2 2 2
1 2 1 2 1 1 1 1 1 2 1 2 1 1 1 1
0 s 0 1 0 1 0 1 0 1
A Q G A H
Q G Q G A H A H
A Q G A H
(9)
with the following continuity relationships:
, ,
, ,
1 1 2 2 2 0
1 r 1 1
2 2 1
2 1
1 0 1
rQ
Q Q
Q
A A A
A
μ μ
C
(10) and the Ampère’s law:
1 0 r 1 1
1Q μ μ Θ
L , (11)
where Glm and Hlm – BEM matrices corresponding to Ωm and Sl, L1 – row vector of lengths of boundary elements on boundary S1, Alm
– column vector of nodal values of A on boundary Sl in domain Ωm, Qlm − column vector of nodal values of ∂nA on boundary Sl in domain Ωm, Asm − nodal values of the source potential As from filamentary currents placed in domain Ωm, 1 – column vector of ones, 0 – matrix of zeroes of appropriate dimensions. More details on forming the BEM equations as well as on evaluating the elements of BEM matrices can be found in [5-6].
The above equations can be formed into a single system as follows:
1 2 s 0 s
2 2 1 2 1 1 0 1
1
2 2 2 2 1 0 1 0 1
1 1 1 2 1 1 1 1 r
r
C Θ μ
μ A
A 0
A Q A Q
0 0 0 0 L
0 H G 0 0
1 0 0 H G
0 H G H G
, (12)
The number of equations in the resulting equation system equals twice the number of boundary nodes plus one. The main disadvantage of Eq. (12) is that it is not efficient for very thin shells. Indeed, in such a case, so called nearly-singular integrals appear, and their numerical evaluation can be time-consuming and inaccurate. To avoid this, another approach is proposed below in such a case.
3.2. Approximate BEM model
The proposed approach bases on the assumption that if the shell is sufficiently thin it can be regarded locally as a fragment of infinite plate of thickness d. For such a plate one obtains [4]
d κ
κξ A
ξ d κ ξ A
A sinh
sinh )
( ) sinh
( 1 2
, (13)
where A1 and A2 are values of A on boundary S1 and S2, respectively, at corresponding points, and ξ varies from 0 to d. By differentiating with respect to ξ one can express ∂nA on boundaries S1 and S2 in terms of A1 and A2. This process leads to the following approximate relationships:
1 1 1 2 1 2 1 2 1 1 1
1 A A , Q A A
Q σ τ σ τ , (14)
where
d κ κ τ d κ κ
σ coth , csch . (15)
Omitting the second of Eqs. (9), and incorporating Eqs. (14) together with (10) and (11), one can form the following system of equations:
1 2 s 0 s 2
2 1 1
1 1
2 2 2 2 2
2
0 1 0
1 0 1
r 0
r
r r
r r
Θ
μ C
τ μ
σ
μ σ μ
τ
μ τ μ
σ
A A A
A
L L
0 H G G
1 G H
G
. (16)
The number of unknowns equals the number of boundary nodes plus one. The dimension of the main matrix is almost twice less than in Eq. (12), what significantly shortens the computation time. Moreover, it does not contain the nearly singular integrals. The disadvantage is that it uses an approximate quasi- analytical solution for the shell. Numerical tests, however, confirm its usability.
This model is referred to as ABEM in the subsequent paragraphs.
4.NUMERICALEXAMPLES
Both models were implemented in Mathematica and tested in various geometrical, excitation and material configurations. In all cases the boundary elements with constant approximation of field and its normal derivative were used.
To map the geometry sufficiently accurately, however, quadratic approximation was used. The BEM coefficients were calculated analytically where possible and numerically in all other cases (the Gaussian quadrature with special treatment for singular and nearly singular cases).
Two configurations were considered. The first one is a symmetrical double wire line in a tubular shield of internal radius R2 and thickness d (Fig. 2a), for which two dimensionless parameters were defined: δ = d/R2 and k = R2/Δ, with Δ as the skin depth for the shield. The second configuration is a flat three-phase line in a rectangular shield 2w×w (Fig. 2b), for which δ = d/w, k = w/Δ. Figures 3-8 show values of |A| and |∂nA| on boundaries S1 and S2 for exemplary values of parameters. Values of A are given in units of μ0I, and values of ∂nA – in μ0I/R2 for tubular shield and μ0I/w for rectangular shield. BEM1 concerns very accurate integration, whereas BEM2 and ABEM – a 10-degree Gaussian quadrature.
a) b)
I R2
d
a R1
−I a a
d
2w
w L3 L2 L1
Fig. 2. Benchmark problems: a) 2-wire line in tubular shield, b) 3-phase line in rectangular shield
The first problem has a theoretical solutions which can be used to estimate the accuracy of numerical computations and the conditions in which the ABEM model gives acceptable results. Detailed analysis shows that the ABEM model should work sufficiently well at least if the total current is 0 and d/R2 << 1.
Fig. 3. Values of |A| on boundaries S1 and S2 of the tubular shield for δ = 0.1
Fig. 4. Values of |∂nA| on boundaries S1 and S2 of the tubular shield for δ = 0.1
Fig. 5. Values of |A| on boundaries S1 and S2 of the tubular shield for δ = 0.01
Fig. 6. Values of |∂nA| on boundaries S1 and S2 of the tubular shield for δ = 0.01
Fig. 7. Values of |A| on boundaries S1 and S2 of the rectangular shield for δ = 0.01
Fig. 8. Values of |∂nA| on boundaries S1 and S2 of the rectangular shield for δ = 0.01
If the shield is thin enough, the ABEM model gives practically the same values as the conventional BEM. The BEM results, however, are occupied with much larger computational effort (larger equation system, nearly-singular integrals). If the accuracy of evaluating the BEM integrals is lowered, errors of BEM can be
unacceptably large, whereas they remain almost unchanged in ABEM. This is because no nearly-singular integrals occur in ABEM (for sufficiently regular boundary).
5.CONLUDINGREMARKS
The presented ABEM model bases on BEM, but it uses a quasi-analytical solution in the thin shell. In comparison with the conventional BEM, it allows reducing considerably the number of equations and the computation time.
Numerical tests as well as analysis of known theoretical solutions indicate that it should work if the thickness of the shield is sufficiently small and the total current in the configuration is 0. The ABEM model can be an efficient method for evaluating the magnetic field of thin wires enclosed in shields of thin walls.
REFERENCES
[1] Piątek Z., Impedances of tubular high current busducts, Series Progress in high- voltage technique, Vol. 28, Wyd. Pol. Częst., Częstochowa 2008.
[2] Krähenbühl L., Muller D.: Thin layers in electrical engineering. Example of shell models in analyzing eddy-currents by boundary and finite element methods, IEEE Transactions on Magnetics, 29 (1993), 2, 1450-1455.
[3] Jabłoński P.: Approximate BEM analysis of thin electromagnetic shield, Proceedings of XXXIV IC-SPETO 2011, Gliwice-Ustroń, 18-21.05.2011, 17-18.
[4] Jabłoński P.: Approximate BEM analysis of thin magnetic shield of variable thickness, Proceedings of XXI Sympozjum PTZE, Zamek Lubliniec 5-8.06.2011, 81-83.
[5] Kurgan E.: Analiza pola magnetostatycznego w środowisku niejednorodnym metodą elementów brzegowych. Rozprawy Monografie 81, Uczelniane Wyd.
Nauk.-Dyd., Kraków 1999.
[6] Jabłoński P.: Metoda elementów brzegowych w analizie pola elektromagnetyczne- go. Wyd. Pol. Cz., Częstochowa 2003.
PRZYBLIŻONA ANALIZA HARMONICZNEGO POLA MAGNETYCZNEGO OD PRZEWODÓW UMIESZCZONYCH W CIENKOŚCIENNEJ OSŁONIE ZA POMOCĄ ZHYBRYDYZOWANEJ METODY ELEMENTÓW BRZEGOWYCH
Praca dotyczy analizy harmonicznego pola magnetycznego od długich, równoległych, cienkich przewodów umieszczonych w cienkościennej przewodzącej osłonie.
Zaproponowany model wykorzystuje metodę elementów brzegowych (MEB), ale pole w cienkiej osłonie modelowane jest w sposób przybliżony za pomocą zależności półanalitycznej. Takie podejście pozwala uniknąć pewnych kłopotów numerycznych związanych z małą grubością warstwy, a ponadto prowadzi to układu równań z mniejszą liczbą niewiadomych w porównaniu z tradycyjną MEB. Numeryczne testy potwierdzają jego użyteczność w rozpatrywanej klasie zagadnień.
