Iványi , A.

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Periodica Polytechnica (Budapest University of Technology and Economics)

ELECTRICAL FIELD IN CYLINDRICAL SYMMETRICAL LAYOUTS BY V ARIA TIONAL CALCULUS By A. IVANYI Department of Theoretical Electricity, Technical University Budapest Received February n, 1982 Presented by Prof. Dr. l. V .. \UO To determine the static electrical field in a charge-free (p = 0) region with a homogeneous medium the potential function satisfying the Laplace equation (1) is to be derived [lJ, [2]. It is known [3J, [4J that one way of solving Laplace's equation by variational calculus is to reduce it to the determination of the extremum of a functionaL Accordingly, in a volume v bounded by lines offorce and equipotential surfaces the potential function (f> extremizing the functional 1 J~ I((f» = 2 grad2 (f> d~' (2) among the functions satisfying the prescribed boundary conditions will also solve Laplace equation (1). Thus, the electrical field problem is equivalent to the determination of the function extremizing the functional (2) and satisfying the prescribed boundary conditions. One way to satisfy the boundary conditions is the application of logical R-functions. In the following the variational method applied in [7J to the solution of planar problems will be extended to the determination of the electrical field in cylidrical symmetrical layouts. The medium is presumed to be homogeneous. In a plane qJ = const. of the cylindrical arrangement the bounding curve of the planar region bounded by lines of force and equipotential curves is assumed to be given or approximated analytically. The functions necessary for satisfying of the Dirichlet boundary conditions are constructed in the form of R-functions known from the literature [5J, [6]. The method is illustrated on an example. The examination of cylindrical symmetrical layouts is advantageously carried out in a cylindrical coordinate system r, qJ, z with the axis z coinciding with the axis of the arrangement. Now, since the geometry is independent of the 222 /V,-isY/, A. z Fig, I angle <p, the electrical field has no azimuthal component. This means that the potential function depends 0nly on radius I' and coordinate :::, i.e. c = O. o<p (3) Accordingly, in order to determine the electrical field in cylindrical symmetrical layouts it suffices to examine a plane <p = const. of the arrangement, a planar region Q bounded by lines of force and equipotential curves (Fig. 1). The equipotential curves bounding the planar region Q are T Dj, i = 1, 2, ... , mD. The value of the potential is given on these curves: i= 1, 2, .. . ,mD' (4) This is a Dirichlet-type boundary condition. Curves T Ni, j = 1, 2, ... , mN are the lines of forces bounding the planar region. On these sections the prescribed boundary condition is of Neumann-type: j = 1,2, ... , mN , (5) where v is the normal unit vector pointing away from the region examined. Taking the cylindrical symmetry into account the potential function satisfying two-dimensional Laplace equation ~ 0 ( 6<1>(1', Z)) 02<1>(1', z) = 0 ~ I' ~ + ~? r or or oz- (6) ELECTRICAL FIELD IN LA rOUTS 223 is to be determined. Since the potential function is independent of the angle cp of the coordinate system, the integration by cp between the bounds 0 and 2n: can be carried out, the elementary volume being dv=rdcpdrdz. (7) Hence, the potential function (r, z) extremizes the functional J(<I»=n f[C~~'Z)y + Ci:,Z)Yl'dQ, (8) D where dQ=dr dz. (9) The potential function zeroing the first variation of functional (8) satisfies the homogeneous Neumann-type boundary condition (5) as natural boundary condition. The Dirichlet-type boundary condition (4) can be satisfied by separating the potential function into two terms [4J, [1]: (r, z) = b(r, z) + 7(r, z) . (10) k, z) satisfies the pre-scribed Dirichlet-type boundary condition (4) on the bounding curves r Di, i= 1,2, ... , mD : i = 1, 2, ... , m D . (11 ) The unknown term 7(r, z) of the potential function satisfies a homoge-neous Dirichlet-type boundary condition on the bounding curves r Di, i= 1, 2, ... , mD : i= 1, 2, ... , I1l D . (12) According to Ritz's process, <l>7(r, z) can be approximated by a linear combination of the first n terms of a function set: n <l>7(r, z) ~ L akfk(r, z)wD(r, z), (13) k = 1 where j~(r, z)wD(r, z) is the k-th element of the approximating function set. wD(r, z) is a positive, continuous and at least twice differential able function in 224 nA.vY!. A. the studied region Q. Function wD(r, z) has to be selected so as to satisfy the boundary condition (12) prescribed for $,,(1', z): i=1,2 ... . ,mD' (14) and on the sections ls j • j = 1. 2, ... , Ins (if any) to be of arbitrary nonzero value. In Eg. (13), fk(r, z) is the k-th element of an entire function set defined in the studied planar region, while elk is the k-th coefficient in the approximating function. In order to determine the potential function, functions $0(1', z) and vvD(r, z) satisfying condition (11) and (14), resp., are to be derived. These functions are constructed as shown in [7]. with the aid of the R-functions developed by R vachev [5]. [6J for the description of planar regions. Taking Eq. (13) into account, the function approximating the potential function (l0) satisfying the conditions (4) and (5) is the approximate solution of Laplace equation (6) at parameters elk' k = 1, 2 .... , n zeroing the first derivatives of integral (8). This condition yields a set of linear algebraic equations for coefficients ak- k = 1. 2 ..... 11, permitting their computation [4J. [1]. Application of the method Let us now determine the electrical field of a cylindrical symmetrical layout by means of the above method. A planar section cp = const. of the arrangement is shown in Fig. 2. The potential function $(1'. z) satisfying Laplace equation (6) in the plane 1', z has to be determined. The planar region bounded by lines of force and equipotential curves is once connected. Sections 1 D l' 1 D2 and 1 D3 of the bounding curve are equipotentiaL section r" 1 is a line of force. The boundary conditions for the potential are the following: (15) and V' grad $(1'. z)!r .... , =0. (16) The boundary conditions are of Dirichlet-type on curves r Dl' 1 D2 and 1 D3 and of Neumann-type on curve l S1 ' ELECTRICAL nELD /,\ LA WLTS 225 Fill. :: Using variational principles, the potential function satisfying conditions (15) and (16) and approximately solving the Laplace equation (6) is 11 <P1I (r, z) = <Pb(r, t) + I Qkfk(r, z)wD(r, z) (17) k=1 with (l0) and (13) taken into account. For the construction offunctions <Po(l', z) and wD(r, z) the planar region shown in Fig. 2 is obtained from regions QD1 and QD2 associated with curves 1 D1 ul D3 and 1 D2 (Fig. 3). With Q 11 (ri - r2 2: 0) , as elementary regions, region QD1 (Fig. 3a) is the following: (18) 226 1V.-iNl'/ .. 4. 0) Fiy.3 The R -functions describing the elementary regions of Q D 1 are the following [5]. [6]: In accordance with (18), Q D1 is described by the R-function ELECTRICAL FIELD 1.'1 LA YOUTS 227 With as elementary regions, the region QDZ is the following: (20) The elementary regions of QDZ are described by the R-functions The R-function describing the region QDZ is (21) For the layout shown in Fig. 2, function wD(r, z) satisfying condition (14) is (22) while function ([J ,,(r, z) satisfying condition (15) is (23) The elements of the approximating function have been chosen from the function set consisting of the products of the Chebishev polynomials 1Xr), i = = 0, 1, 2, ... , n 1 and ~{z),j = 0, 1, 2, ... , nz depending on rand z, respectively. Since due to the cylindrical symmetry, the electrical field intensity has no radial component at r = ° in the arrangement seen in Fig. 2, the elements of the approximative function set have been selected to satisfy this condition. In order to make the examination complete, a layout has been at first examined with an electrical field having a singularity along the intersection of the cylinder of radius rz and the sphere of radius p(rz # p). In this case, the 228 II"ASYI .. 4. 'tJ/'JJ? Di::;2 ~3 :4 05 C6 - ,r-r-r-r-r-I / I i / / / I ! I / .I! I .;/ i ___________ ~z Fig. 4 Fig . .' equipotential lines of the potential function approximated by (24) (11 = 6, 11 1 = 1, 112 = 2) have been plotted in Figs 4 and 5 for electrode distances Z2!Z1 =0,5 and Z2/Z1 =0.3, respectively, with the geometrical ratios rdz1 =0.5, r2/z 1 =0.20, plz! =0.25. ELECTRICAL FIELD IS LA r01TS 229 G1.. J 5 :: 6 r- r- r-! / / , I i fi l i ! /' f ' , , / j I i Fiy.6 V 7 C 8 Q S iC Fi!/. 7 In the case r 2 = P where the electrical field intensity is not of extreme value along the intersection of the cylinder and the sphere. the approximation (25) has been used with n=20 (nl =2, n2 =4), and the equipotentiallines of the potential function (17) have been plotted in Figs 6 and 7 with unchanged geometrical dimensions and two different electrode ,distances. 2 Periodica Poiytechnica El. 26/3-4 230 The numerical experience made in the solution of the problem indicates that if the R-functions applied to the construction of the potential function have their derivatives in a finite number of points on the bounding curve of the region defined, irrespective of whether the electrical field intensity is of extreme value in these points, these points should be excluded from the investigation. The potential function is distorted in the neighbourhood of these singular points. By increasing the number of terms in the approximation this distortion can be limited to a narrow vicinity ofthe singular points, but it can't be avoided. Summary The paper deals with the variational solution of static and stationary fields in cylindrical symmetrical layouts of a homogeneous medium. An approximate solution of the two-dimensional Laplace equation exactly satisfying the prescribed Dirichlet-type boundary conditions is constructed in analytical form. The functions needed for satisfying the boundary conditions on curves described at least piecewise analytically are constructed with the aid of R-functions. The application of the method is illustrated on an example. References I. V.i..GO. I.: Theoretical electricity. Tankonyvkiad6, Budapest 1972. (In Hungarian). 2. SIMONYI, K.: Theoretical electricity. Tankonyvkiad6, Budapest 1967. (In Hungarian) 3. MICHLl:--1, S. G.: Variational methods in mathematical physics. Nauka. Moscow 1970. (In Russian) 4. KANTOROVICH, L. V.-KRILOV, V. Y.: Approximate methods of higher analysis. Akademiai Kiad6. Budapest 1953. (In Hungarian) 5. RVACHEV, V. L.: Methods of logical algebra in mathematical physics. Naukova Dumka, Kiev 1974. (In Russian) 6. RVACHEV, V. L.: Theoretical application of logical algebra. Technica. Kiev 1967. (In Russian) 7. I V ,\:--1 Y I, A.: Determination of static and stationary electromagnetic fields by using variation calculus. Per. Poly tech. El. Eng., Vo!. 23. No. 3-4. Budapest 1979. (201-208). 8. BARD!, I.-BIRO, O.-GYIMESI, M.: Calculation of static and stationary fields in inhomogeneous. anisotropic media by the variational method. Per. Poly tech. El. Eng .. Vo!. 20. No. 2. Budapest 1976. (103-118). dr. Ama.lia IVANYI H-1521 Budapest

ELECTRICAL FIELD IN CYLINDRICAL SYMMETRICAL LAYOUTS BY VARIATIONAL CALCULUS

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ELECTRICAL FIELD IN CYLINDRICAL SYMMETRICAL LAYOUTS BY VARIATIONAL CALCULUS

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Periodica Polytechnica (Budapest University of Technology and Economics)