Three-dimensional vector potential analysis for machine field problems

436 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAC-18, NO. 2, MARCH 1982

THREE-DIMENSIONAL VECTOR POTENTIAL ANALYSIS FOR MACHINE FIELD PROBLEMS (Invited)

M.V.K. Chari, A . Konrad, M.A. Palmo, and J. D’Angelo

ABSTRACT

Modern electrical plant and machinery have to be designed to operate at high power densities at minimum cost and optimal efficiency with a high degree of reliability during operation. These stringent requirements necessitate accurate performance prediction at t h e design stage. As a first and important step in this process, the magnetic field distribution must be evaluated taking full account of the geometrical complexity of the field region, magnetic saturation of the iron parts, and circulating currents in conducting media. The advent of digital computers has spurred the development of sophisticated numerical techniques to accomplish this task with a high degree of precision. In this paper, three-dimensional vector potential solution methods for linear and nonlinear diffusion and magnetostatic field problems are presented. The methods are illustrated by numerical examples where feasible.

This research has been partially funded by the Electric Power Research Institute, Palo Alto, California under EPRI contract RP 1288-1.

INTRODUCTION

With increasing emphasis on energy conservation in modern power systems and the need for plant availability and reliability of operation, it has become important to design electrical apparatus to be operated at high power densities efficiently. economically, and with a high degree of reliability. In order to achieve optimization and predict machine performance accurately at the design stage, i t is essential to obtain a clear understanding of the electromagnetic field distribution in the presence of magnetic saturation in iron parts as well as induced currents in conducting media and their impact on power losses, heating. and forces on different parts of the machine geometry.

Field computation has been traditionally carried out by analytical and analog methods [1,2], using simplified machine representation and boundary conditions. With the advent of digital computers, i t has becomes possible to apply numerical methods of magnetic field analysis in engineering design in a practical manner. Electromagnetic field analysis had concentrated during the last decade on two-dimensional machine representation including iron saturation and eddy current effects [3-101, using scalar or vector potential representation. Such representations were not only easy and economical to apply, but were also adequate for most practical field problems. With the ever increasing need for improved efficiency, high reliability, and economical designs of electrical apparatus, the necessity for accurate modeling of the three-dimensional geometry, material characteristics and disposition of source and induced currents has been recognized. Progress in computer technology, software development, and graphics are adding a new perspective and dimension to machine modeling. Most likely, the decade of the eighties will evidence considerable activity in the area of three-dimensional electromagnetic field analysis.

Pioneering work in three-dimensional magnetic field computation was carried out with finite difference schemes, the most notable amongst these being the work of Wolff and Muller [ I l l , who employed a scalar potential representation for the magnetic field. Quasi three-dimensional solutions by axisymmetric modeling for scalar and vector fields [12,131; and by hybrid methods [14,15] have also been reported in the literature. An integral equation formulation and its successful application to the determination of flux-distribution in an accelerator magnet was first presented by Trowbridge and his associates [141.

Scalar potential functions, while they are convenient. utilize only a single unknown variable i n their formulation and are not completely adequate for three-dimensional representation of current regions, especially in eddy current diffusion problems. Since an electrical machine embodies a large volume of current-carrying windings of complex geometry and eddy-current regions, it has become essential to explore three- dimensional vector potential solutions. Until recently, vector potential methods did not receive much attention, owing to their analytical complexity and the resulting large system of equations. Various formulations and solution methods have been proposed [16-191, nevertheless, a good deal of research work is required in this relatively new area.

The authors are with Corporate Research and Development, General Electric Company, Schenectady, NY. 12301.

In this paper, the generalized formulation of the three-dimensional eddy current diffusion problem based on a vector potential solution is presented, and an attempt is made to present a comprehensive analysis by the finite element method. The associated variational formulation and functional minimization are described. First order element representation, prescription of the potential-function in each element, and the derivation of the coefficient matrices and forcing functions are discussed. The method of specializing the diffusion equation solution to the magnetostatic problem is also described. The paper also discusses the development of a new equation solution, namely the pre-conditioned conjugate gradient method, in order to handle the large system of simultaneous equations resulting from the discrete field representation; as well as a vector graphics algorithm to illustrate the three-dimensional field solution in a realistic manner. The analysis developed in the paper is applied to several illustrative examples and consistency checks with solutions by other methods and tests, where feasible, are included.

THE THREE-DIMENSIONAL ELECTROMAGNETIC FIELD PROBLEM AND THE NONLINEAR DIFFUSION EQUATION

The electromagnetic field problem can be analyzed in terms of a vector potential function taking into consideration magnetic saturation and eddy current effects. The associated partial differential equation can be derived from Maxwell’s equations subject to the following assumptions:

1. The magnetic field is quasi-stationary at power frequencies such that displacement currents can be neglected.

2 . Material characteristics are isotropic and the 5 - H characteristics are single valued.

3. Temperature effects of electrical conductivity are neglected.

4. Eddy currents and material conductivity in current sources and stator laminations are neglected.

With the aforesaid assumptions, the following Maxwell’s equations hold

vxTI=5 (1)

V x E = - - - a B ar

The constitutive relations applicable to the field problem are given by

T7= ” B (3 )

7 = U E (4) v.B = 0 (5)

Where Ei is the magnetizing force

B is the magnetic induction or flux density

7 is the current density

Y is the reciprocal permeability which is both position and field dependent

E is the electric field

u is the conductivity

lntroducing a vector potential function A such that

E = v x A ( 6 )

and substituting in (3) and solving (1) and (3) simultaneously, one obtains

v x v v x A = 7 (7)

In (2) if we substitute for 0 from (6) and E from (4) and introducing a scalar potential function &, we have

-

VX-=--VXAfVX(V~) J a U a 1

(8)

Since the operator a/a I is not a function of the spatial coordinates, we may remove the curl operator form both sides of (8) with the result

0018-9464/S2/0300-0436$00.75 0 1982 IEEE

- a2 J = - u - + + V +

a t (9)

Substituting for ?from (9) in (7), one obtains

V X wV X A =-IT - - - + u V + - a2

a t (loa!

Equation (loa) is the nonlinear diffusion equation or the curl-curl equation of the three-dimensional electromagnetic field problem.

Substituting the source current density ?, for ]in (71, one obtains the nonlinear Poisson equation for eddy current free source region, as

v x uv x A = 7, ([Ob)

Combining (loa) and (lob) into a single equation representing the respective cases in the different regions, one obtains

v X v V X 2 = - - - + u v 4 + 3 s ua 2 a t

(10c)

Divergence Condition on Vector Potential x It is evident from (1) that the divergence of the current density is

zero since taking the divergence on both sides of (1) yields

V.(V x H ) = 0 . 3 (11)

From vector identities, it is known that div curl of a vector is always zero, so that

V.(V x 71) = v . J=o (12)

We can, therefore, assert that the divergence of the current density is always zero.

Taking the divergence of both sides of (9) and setting i t to zero, there is

Since a / a t is independent of spatial coordinates, (13) becomes. for constant U ,

a a t

-u - (V.A) + aV2+ = 0 (14)

Equation (14) imposes a generalized divergence condition on A in terms of a scalar potential +.

Integrating (14) within the limits of --co to t a s shown in Appendix C, we have

u v . 2 - @ J ' V*l#JdT = 0 (15) - m

Equations ( 1 0 ~ ) and (15 ) can be written in matrix form as

Equation (16) may be formally written as

9*= . / (17)

where 9 is the matrix operator in (16), + is the vector of unknowns, and ./is the forcing function.

The operator 9 in (17) is symmetric. From a variational viewpoint, symmetric operators possess some useful properties, but not dimensional compatibility of the terms in the associated functional. Therefore, the time integration converting (14) into (15) is performed for making the operator %1 in (17) symmetric and at the same time for maintaining dimensional compatibility.

Alternative Formulations for the Diffusion Equation

I f we choose a vector potential function 2 and a scalar potential function +, such that

B = V x ( A - V q J ) ( 1 8)

and substitute for B in (3), and manipulate (1) and (2), one obtains the following set of coupled equations

+ u v * + T S a t

+ n Lv2+ a t

437

(19)

(20)

I t may be observed that the scalar pcential function l#J in (16) and (20) and the divergence condition on A_ are intended to ensure zero divergence of the current density vectors J , and ?T in the implementation of the numerical solution. However, if the zerodivergence of 7, can be strictly imposed, (methods of imposition of .J, = 0 explicitly are discussed in a later section), then the diffusion equation formulation may be simplified by eliminating the scalar potential 4, as

v x v V X A + u - = ? 5 , a 2 a t

In (21). the divergence of 2 follows that of the source j,$, ab i n magnetostatic problems. (See reference 1221.)

Variational Formulation and the Energy Related Functional

It is well known that the solution to a boundary value problem can be best obtained by first formulating the associated partial differential equation in variational terms by means of an energy related expression called a functional. The field region is then subdivided into finite elements by means of rectangular and triangular prisms or tetrahedra. In each of these elements, an approximate solution of the field problem is prescribed in terms of nodal values of the potential function. The functional is then extremized by setting its first derivative with respect to each of the nodal potentials to zero. This process yields a system of linear algebraic equations, which when solved, provides the required solution to the field problem. The finite element procedure is best illustrated by the flow chart of Fig. 1.

I I FORMULATE THE FIELD PROBLEM BY THE CURL CURL EQUATION

L OBTAIN A VARIATIONAL EXPRES- SION IN TERMS OF AN ENERGY RELATED FUNCTIONAL

I Ir

SUBDWIDE THE FIELD REGION INTO DISCRETE FINITE ELEMENTS

t DEFINE AN APPROXIMATE FIELD

t SOLUTION IN EACH ELEMENT

MINIMIZE THE FUNCTIONAL WITH RESPECT TO THE APPROXIMATE SOLUTION *

I SOLVE THE RESULTING SET OF LINEAR ALGEBRAIC EQUATIONS AND OBTAIN THE VECTOR POTEN- TIAL SOLUTION

Fig. 1 Flow chart illustrating the finite element procedure.

From variational calculus, i t is known that the solution I,IJ which satisfies the differential equation (17) also renders the corresponding energy functional stationary. The functional expression for a symmetric operator as given in (17) is of the form

.7= <*, @*> - 2 <*J> (22)

In (22), the scalar product is defined by the equation

< u, V > = s uT.V dR (23) R

43 8

where Tdenotes the transpose of u

Expanding $, we have

(24) Expanding (24) as shown in Appendix A and applying the vector form

of Green's identity, one obtains

- 2 7 . 3 , ) d R

Substituting for V x 2 in (40), one obtains

Equations (25) and (26) represent the energy functional for the three-dimensional nonlinear diffusion equation in terms of a three- component vector potential A and a scalar potential function $. The reluctivity Y is field dependent for nonlinear problems and constant for linear cases.

The steady state linear diffusion equation can be modeled in variational terms as above, and the functional will be of the form of equation (40) with the following substitutions

Boundary Conditions for the Three-dimensional Electromagnetic Field Problem

The energy functional for the three-dimensional nonlinear diffusion equation (IOc) including the Eneralized divergence condition of (14) on the magnetic vector potential A has been derived in (25).

It may be noted that (25) involves only first derivatives of the field quantities, while second derivatives appear in (IOc). This difference arises from an integration by parts performed in setting u p ( 2 5 ) . The same integration by parts also gives rise to the surface integral terms of ( 2 5 ) .

Various approximate methods may be constructed for solving the electromagnetic field (16). Of interest here are, in particular. finite element methods created by rendering Ihe functional of ( 2 5 ) stationary on a Hil- bert space of preselected approximating functions. In such methods, two forms of interelement and boundary conditions normally occur: those which must be enforced explicitly on the space of approximaling functions (principal boundary conditions) and those which will be satisfied au- tomatically in some weighted-mean sense (natural boundary conditions). The requirement that the surface integral terms vanish imposes certain natural boundary conditions and natural interface conditions in the functional (25). There are two such terms: and since the vector and scalar potentials are independent variables. the two integrals will be required to vanish independenlly.

The object of this section is to state ( 1 ) the natural boundary conditions that are imposed on the electromagnetic fields by requiring the two boundary integrals to vanish, the (2) the natural interface conditions that are implied by this requirement at material interfaces within the solution regions. Since an interface may be purely imaginary (a separation between two regions of like material properties) as well as a physically real boundary, the natural interface conditions will in turn imply the natural interelement continuity conditions for any finite elemenl method, as shown in Appendix B.

The Magnetostatic Formulation

The magnetostatic field formulation and energy functional expression are obtained from (16) and (25) respectively by substituting ,$ = 0 and v = 0. However, the Coulomb gauge must be implemented in the differential equation, as shown in Reference [221. The functional formulation may be obtained from (25) as described above or alternatively as shown in Reference [22], which is more general and applicable to other nonfinite element operators.

Finite Element Representation of the Three-Dimensional Field Region

A segment of the three-dimensional field region and its discretization by finite elements is illustrated in Fig. 2. Regular eight-noded rectangular brick elements and six-noded triangular prisms are used, which represents first-order finite elements. I t is also possible to use tetrahedral elements, which possess certain useful properties, but are cumbersome in their application in respect to subdividing the field region and graphical visualization. In the following section, the respective elements wi l l be described in greater detail.

Fig. 2 Segment of a three-dimensional field region

Triangular Prism. A first-order six-noded triangular prism element is illustrated in Fig. 3, in which the vector potential function is defined as a linear combination of the vertex values such that

6

A,, = b A A t , h I,, = X.Y Or Z (28) h = I

where in are the interpolation polynomials which are functions of the geometry. These are defined as

bi, = ( I f p h ) 2 I k = I , 2 ...6 p = 1 for k = 1,2 ,3 / I = - 1 for k = 4,5 ,6 (29)

5,. are the area coordinates given by

(a i + bAx + thy) 2A t h = (30)

where a,, b,, c , are geometric coefficients, and A is the area of the triangular face.

Rectangular Brick Element. An eight-noded brick element is illustrated in Fig. 4. The solution function is represented in terms of the nodal values and interpolation functions as

8

A,, = 2, 5 1 182 x,Y O r z (3 1)

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-= a .F a Auk

0 lu-x,y or 2

k-1.2 . . .a

(34) Functional minimization is achieved by substituting for the functional

in (34), the expression of (25) and carrying out the indicated differentiation.

In order to .ensure potential continuity at the nodes, the terms associated with a given node in each element are added together and this process is carried over all the nodes of all the elements subdividing the field region. The resulting global matrix equation is solved by an efficient equation solution algorithm to obtain the vector potential solution.

Methods of Implementing the Zero-divergence of the Source Current- density

The zero-divergence of the current-density vector must be implemented in the source region, by assigning the appropriate current density components to each element. The simplest way, where feasible, is illustrated in Figs. (a) and (b). Figure (a) shows a quadrant of a rectangular coil. Figure (b) shows the respective current density components in each of the elements for uniform current density such as in a wire wound coil.

p = 1 for k = 1,4,5,8

p = -1 for k = 2,3,6,7

< k ( l+pf)( l+q?l)( l+rh) = 1 for k = 1,2,5,6

8

q = -1 for k = 3,4,7,8

r = 1 for k = 1,2,3,4

r = -1 for k = 5,6,7,8 (32)

Fig. 3 Six-noded triangular prism.

2 f f : '2 - - 3 4

Fig. 4 Eight-noded brick element

The Tetrahedral Element. The tetrahedral element is a three- dimensional simplex and is, therefore, rotationally invariant. Although, i t is somewhat cumbersome to generate and poses visualization problems in grid assembly, i t is perhaps the simplest element which permits prescription of a complete interpolation polynomial for the shape functions. Con- sidering the first-order tetrahedral element shown in Fig. 5, the potential function may be written as

4

= f, < k Auk Iu - XJ, O F T

where c k = (ak+bkxfcky+dkz)/6 v V = volume of the element

(33)

in progressive modulo 4.

FiL. 5 First-order tetrahedral element.

Functional Minimization and Matrix Assembly

The solution to the electromagnetic field problem in the three- dimensional geometry is obtained by minimizing the energy functional of (25) with respect to each of the nodal values of potentials. The necessary condition for minimization is that the first variation of the functional must equal zero. Thus

J Fig. (a) Fig. (b)

If the conductors are made of flat conducting material instead of small cross-section wires, then one may perform a 2-D analysis with imposed magnetic field a, obtgn the current density components by differentiating the solution for the H, and impose the same in the 3-D model appropri- ately in each element. Such a two step solution was carried out as illustrated in Figs. 6 and 7. -

H? = 0

..". Fig. 6 2-D solution of the conducting region to determine current-

density components.

Fig. 7 Determination of 7 i z from 3D vector potential solution using current densities obtained from the 2D solution of Fig. 6.

440

A third, but perhaps a more general method, consists of obtaining a magnetic scalar potential solution in the conductor. By differentiating the solution, one obtains the current densities which are then implemented in the vector 3-D solution.

Solution of Nonlinear Equations

At present the only practical methods that are being explored are the chord and Newton-Raphson algorithms which have been extensively reported in the literature. These are being currently applied to the solution of the nonlinear 3-D magnetostatic problem. Their extension to the diffusion equation solution will require the development of a time dependent solution for this class of problems.

Solution of Simultaneous Equations

The functional minimization of (34) involves solving a large number of simultaneous algebraic equations. Their number is large because of the three-dimensional structure of the problem, and to a lesser extent, because a multicomponent vector field is sought. For problems of practical value, 5000 to 50,000 equations may easily be involved. The method adopted here for dealing with large numbers of equations is a preconditioned conjugate gradient method, in which the preconditioning matrix is computed by an incomplete Choleski decomposition. Such techniques have recently come into prominence.

The conjugate gradient method was first proposed by Hestenes and Stiefel 1191 as a method for solving the matrix equation of order N

sx = y (36)

where the coefficient matrix S is symmetric and positive definite. The method relies on the fact that there exists a set of column vectors p orthogonal with respect to S , so that

< P , , SPJ” = 6 , j (37)

N such vectors must exist; they span the N-space in which the solution to (37) must lie. The solution is thus expressible in the form

A’

,=I x = X c,pi

Unlike iterative methods, the Hestenes and Stiefel method is guaranteed to converge in at most N steps as a result of (38). Unfortunately, its use has not been widespread because generating all N vectors requires of the order of N 3 operations and, thus, can be very time-consuming for large N.

The preconditioned conjugate gradient technique is similar to the Hestenes and Stiefel algorithm in principle, with the difference that (36) is first rewritten in the form

(B-‘SB-I)BTx = BY’ (39)

Clearly B could be any invertible matrix. In practice, it is necessary to choose B diagonal, triangular, or otherwise easy to invert. Further, B is so chosen that the new coefficient matrix is as nearly similar to the unit matrix as can conveniently be achieved. If the new coefficient matrix were exactly similar to the unit matrix, the conjugate gradient method would converge in one step. Such similarity is not easy to achieve. How- ever, a very great acceleration in the convergence rate i s achieved by computing B as the incomplete Choleski decompose of S: the triangular decomposition is undertaken in the normal fashion, but the structure of B is forced to be the same as that of S . In other words, no fill-in is allowed during decomposition; nonzero elements are allowed in B only wherever S has nonzero elements. The new, or “preconditioned,” coefficient matrix typically yields satisfactory convergence in a number of steps proportional to the square root of the number of variables, with a proportionality constant of the order of 1 or 2. [201

The work per conjugate gradient step required by this method is approximately (SM+4) N multiply-and-add operations, where M is the aver- age number of nonzero entries per row in the coefficient matrix. The amount of computer storage required is approximately (2M+4)N floating-point words. Since Mdepends on the type of finite element employed, but not on the tot.al number of elements, the memory requirement is seen to rise linearly with N , exactly like the classical iterative methods. The total computing time varies, however, as the number of variables N to the three-halves power. In summary, this technique comes close to matching the memory requirements of the most memory-efficient solution methods known, while it provides, for large problems, a shorter computing time than any other technique.

Graphics

Three-Dimensional Field Plotting. An important aspect of electromagnetic field computations is computer plotting of the results. For two-.dimensional problems, magnetic field plotting is performed by draw- ing equipotential contour lines which (conveniently) also represent magnetic field lines. In three dimensions, magnetic field plotting is not so easy: three-dimensional magnetic fields must be drawn in perspective on a two-dimensional page and contour lines of vector potential do not, in general, represent magnetic field lines. In the following sections, both contour plotting and vector plotting methods will be described.

Contour Plotling. The procedure traditionally employed to plot three-dimensional magnetic field solutions is based on a two-dimensional contour plotting algorithm. In this procedure, one takes slices of the three-dimensional field at symmetry planes in the solution and plots contours of the normal vector potential component on the slices. Figure 8 is an example of such a plot.

Fig. 8 The magnetic field lines inside a circular conductor plotted in terms of vector potential contours.

Vector Plotting. Since contour plotting is strictly valid only at symmetry planes, there is a need to develop a procedure for plotting vectors to represent the magnetic field distribution in three-dimensional space.

One method is to draw arrows at a large number of points in the field region, the size and direction of each arrow indicating the magnitude and direction of the magnetic field at that location. However, a simple arrow, such as that used in drafting, is a two-dimensional figure which cannot in- dicate a field component going into or out of the plane of the plot. To in- dicate direction in three-space, a three-dimensional arrow is required. The shape used for the arrow head may be cone shaped. Paint the cone’s base black, to distinguish the inward and outward orientation of the field. Figures 9, 14, and 19 are examples of a vector field plot using three- dimensional cones with their bases shaded.

Fig. 9 The magnetic field inside a circular conductor plotted using cone-shaped vector elements.

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Applications

The three-dimensional magnetostatic finite-element analysis was applied to illustrative examples, such as square and circular cross-section bars and solenoids, a three-limbed inductor, square cross-section coil, a two-winding transformer, and a segment of a turbine generator on no load.

The first results obtained by the three-dimensional diffusion equation solution on a square cross-section solenoid are also illustrated.

The flux plot obtained for the circular solenoid is shown in Fig. 10. Finite element discretization of a square cross-section bar is shown in Fig. 11, and the equipotential contours (flux lines) are illustrated in Fig. 12.

Table 1 compares the flux densities obtained by the three-dimensional magnetostatic solution with those obtained by a well-proven two-

Fig. 10 Flux distribution in a slice of a circular solenoid.

Fig. 11 Finite element discretization of a square cross section conductor.

Fig. 12 Flux distribution in a square cross section conductor.

Table 1

COMPARISON OF 3-D MAGNETOSTATIC SOLUTION WITH A WELL-ESTABLISHED 2-D SOLUTION

Conductor

Airspace

Element Number (Fig. 11)

47 48 55 56 79 80 87 88

7 8

16 23 40 72 63 95

112 119 127 128 31 32

103 104

lux Density (Wb/m2

2-D

0.0176 0.0176 0.0176 0.0176 0.0176 0.0176 0.0176 0.0176

0.0144 0.0144 0.0206 0.0206 0.0157 0.0157 0.0157 0.0157 0.0206 0.0206 0.0144 0.0144 0.0144 0.0144 0.0144 0.0144

3-D

0.0176 0.0176 0.0176 0.0176 0.0176 0.0176 0.0176 0.0176

0.0144 0.0144 0.0206 0.0206 0.0157 0.0157 0.0157 0.0157 0.0206 0.0206 0.0144 0.0144 0.0144 0.0144 0.0144 0.0144 -~

dimensional analysis technique of the bar cross-section. The agreement between the results of the respective methods is exceedingly good.

Iron Core Inductor. The field distribution in the iron core inductor is illustrated in Fig. 13(a).

Vector plot of the flux-densities in the inductor and a comparison of measured flux-densities with 3-D finite element results are illustrated in Figs. 13(b) and 13(c) respectively.

Table 2 shows a comparison of flux densities obtained from 2-D and 3-D solutions for the iron core inductor.

-

Fig. 13 (a) Field contours in a half model in a section of an iron core inductor.

i / : . . . / . . . ' . . ./

Fig. 13 (b) Vector plot of flux densities in the iron core inductor.

442

1600 - 1400 -

v) 1200 - v)

1000-

- 30 FINITE ELEMENT SOLUTION coo- TET RESULTS

2 400

0 ' I h ' r I ' l I '

0 10 20 30 40 50 60 70 80 90 100

CURRENT mA

Fig. 13 (c) Variation of flux-density with primary current in the iron core inductor.

Table 2

FLUX DENSITY COMPARISON (B in Gauss)

3-D Percent E r z

,330 49.400 49.100 .610 85.620 85.300 ,380

100.73 100.40 ,320 122.52 122.00 ,430 150.04 149.50 ,360 172.93 172.30 ,370 188.17 187.40 .410 445.96 443.60 .530 750.56 746.32 .570

1035.1 1030.0 ,500

Square Cross-section Coil. A further example of 3-D magnetostatic solution for an air core coil is illustrated by the vector plot of flux- densities shown in Fig. 14.

1 IRON CORE 2 SECONDARY WINDING 3 PRIMARY WINDING

Fig. 15 Outline of a two-winding transformer (front view).

Fig. 16 Flux distribution in a two-winding transformer (section through mid-plane in front view).

Fig. 17 Flux distribution in a two-winding transformer (section through mid-plane in end view).

1 IRON CORE 2 WINDINGS 3 AIR SPACE

Fig. 14 Vector plot of flux densities in an air-core coil

Two-Winding Transformer. Figure 15 shows the outline of a two- winding transformer in front view. The primary and secondary windings are on the center limb of the three-limb core. The flux plots obtained in the mid-plane sections for the front and side views are shown in Figs. 16 and 17. It may be noted that these are scalar equipotential plots obtained for the respective components of the vector potential orthogonal to the cross section.

A three-dimensional vector plot of the flux densities obtained for the two-winding transformer model of Fig. 18 is illustrated in Fig. 19.

Fig. 18 Sectional view of an octant of a two-winding transformer.

Fig. 19 Vector plot of flux densities in the octant of a two-winding transformer.

Turbine Generator. A three-dimensional section of a turbine generator, including the end iron region, rotor windings, and rotor forgings, is illustrated in Fig. 20. The magnetic field distribution obtained under no- load conditions with the field windings excited is shown in Fig. 21.

t ROTOR IRON

2 AIR GAP

3 ROTOR WlNDlNQS

4 STATOR IRON

5 END REGION AIR SPACE

Fig. 20 A three-dimensional section of a turbine generator.

Fig. 21 Magnetic flux distribution in a section of a turbine generator. (Magnitude and direction of the cones illustrate the’flux density vector.)

3-D Diffusion Solurion. Figure 22 shows the eddy current profile in a square cross-section solenoid as obtained from the three-dimensional diffusion equation solution.

Conclusions

In this paper, three-dimensional diffusion equation and magnetostatic field analyses based on the finite element method have been presented. The necessary condition for the choice of the vector potential function is shown. Implementation of this condition in the variational formulations

443

‘ALUMINUM BAR

Fig. 22 (a) 3-D diffusion equation solution in a quadrant of a solenoid; (contours of -A) Freq = 0.

FREO = 1 5 Hz CT=32r10-an-m

-WINDINGS RelAl

Fig. 22 (b) 3-D diffusion equation solution in a quadrant of a solenoid; (contours of -A) Freq = 15 HZ.

yields energy-related functiotials, which implicitly satisfy material interface and boundary conditions.

The field region was discretized by first order finite elements, such as rectangular bricks, triangular prisms, and tetrahedra. The solution to the field problem is obtained by minimizing the functional with respect to the vector potential values at the nodes of the elements, and solving the resulting set of linear algebraic equations. An iterative solution technique, known as the conjugate gradient method, is employed with suit- able preconditioning to accelerate convergence.

The three-dimensional field analysis was applied to illustrative problems such as circular and square section conductors and solenoids. A comparison is made, with excellent agreement, between the results of flux densities obtained by the three-dimensional field analysis and well- established two-dimensional solutions and tests values where available.

The vector potential analysis is also applied to a three-limb iron core inductor, a two-winding transformer, and a segment of a turbine generator on no-load. The results are presented graphically by means of contour and vector plots for particular cases. Finally, the 3-D diffusion equation solution is applied to a square cross-section solenoid.

ACKNOWLEDGEMENT

The authors wish to acknowledge the contributions made by Dr. P. Silvester and Dr. Z.J. Csendes of McGill University.

This research was partially funded by the Electric Power Research Institute under RP 1288-1, “Improvement in Accuracy of Prediction of Electrical Machine Constants.”

APPENDIX A

ENERGY RELATED FUNCTIONAL FOR THE’3-D NONLINEAR DIFFUSION EQUATION

Expanding Equation (24) of Section 2.2

444

We shall now consider the first term in the above volume integral given by

. ~ I = ~ ~ . V X V V X / ~ O ' ~ ~ (A-2)

Using the vector identity

V ' ( A x B ) = B . ( V x 2 ) - 2 ' ( ( V x B ) (A-3)

and substituting UV x 2 for B, one obtains

v . (2 x vv x 2) = v(V x 2) ' (V x 2) - 2 '(V x vv x 2) (A-4)

Integrating both sides of (A-4) above over the volume of the field region and rewriting, we have

9, =s, u(v x 2) . (V x 2)dn

- p . (2 x uv x (A-5)

The second volume integral on the right-hand side of (A-5) may be transformed as a surface integral by using the divergence theorem, so that

(A-6)

Equation (A-7) represents the vector form of Green's identity as reported by Konrad in Ref. [211.

Let us next consider the expression

Since Vz, the Laplacian operator, is time invariant, i t can be brought outside the time integral, so that (A-8) reduces to the form

If we define a variable @ such that

@ =sf +dr -- expression (A-9) can be rewritten as

,gz = JL f V2@d0

(A-9)

From the vector identity

v ' (u2) = (vu) ' A + utv .A) (A-12)

v ' (+m) - (V$) ' (m) + $(V . m) (A-13)

By substituting for 2 and 6 for u, there is

Integrating both sides of (A-13) over the volume of the field region

,g3 = & + V ~ C D ~ ~ = s (m . m ) d n - J V . ( $ m ) d n (A-14)

Applying the divergence theorem to the second volume integral on

and rearranging terms, one obtains

n CL

the right-hand side of (A-14), there is

6 , = s, (-1 . (m)dn - f i r ( d m ) ' ds (A- 15)

Finally by substituting (A-7) and (A-15) in (A-11, the expression for the nonlinear energy functional is obtained as

APPENDIX B

BOUNDARY CONDITIONS

The Electric Potential Boundary Integral

The natural interface conditions and natural boundary conditions arising from the electric scalar potential boundary integral are somewhat easier to visualize than those associated with its magnetic counterpart, since the quantities involves are scalars. It will be required that the integral

must vanish. This requirement implies that certain natural boundary conditions, as well as certain interface conditions at material interfaces, must be met.

Consider the interface conditions first. Suppose the region in question is simply connected. (In practical cases, this supposition usually means only that the problem model must include all of the physical problem region, without exclusion of any parts.) Suppose further that it is divided into two regions, as in Fig. 1, along a boundary surface such that the two simply connected subregions created by the division contain two distinct materials, say 1 and 2. The integral of (B-1) may then be rewritten as two closed surface integrals, taken over the two subregions,

The bounding surfaces of the two subregions may be divided each into an exterior portion (S1 and S2, respectively) and an interface portion SO.

Fig. 1 The entire region of integration, bounded by the closed surface S, may be subdivided into two regions by the interface surface SO. SO may be selected to follow some natural surface, such as a material interface. Region 1 is bounded by S I , a portion of S , and SO; similarly for S2.

Since an artificial subdivision of the region of integration cannot change the value of the functional of (25), region by region integration in (B-2) must yield the same result as integration over the whole region, (B-1). This can only happen if the contributions arising from the interface SO to both integrals in (B-4) exactly cancel,

Here the numeric subscripts identify the two subregions; the normal direction n, however, is taken as positive from region 1 into region 2.

The electric scalar potential must be continuous throughout any problem region. It is, therefore, not obligatory in (B-3) to identify by subscripts on which side of the interface the potential is to be evaluated, and (B-3) may be written as

(B-4)

It follows from variational principles that the natural interface condition prevailing at the material interface is

- -

But, by definition,

@ ( t ) = 1 ' 4 ( r ) d 7 (B-6)

Differentiating with respect to the upper limit t, and substituting in -m

Equation (B-5), there results

The notation has been chosen to emphasize that derivatives of the same potential, but evaluated on the two sides of the interface, appear on

445

For physical clarity, it is useful to recall the definition of the vector potential,

B = v x A (8-14)

and thus to rewrite (B-12) in perhaps the more lucid form

the two sides of (8-7). Equation (8-7) is thus the natural electric potential interface ,condition associated with the stationary functional (25). Physically, it may be interpreted as saying that the normal component of its gradient (i.e., the normal electric potentiaPfield component) has at the interface a jump equal to the resistivity contrast across the interface,

a@ _ _ = - n ‘ 1 P I

a n ‘ 2 a@ ~2

(B-8)

This natural interface condition immediately implies the natural boundary condition to be met at an exterior surface. Since it provides no contribution to the functional, the space exterior to S may be regarded as having zero conductivity. The relevant condition at the exterior boundary of S is thus obtained from (25) as the special case which arises when the adjoining region has zero conductivity,

Physically, this equation may be interpreted to say that at the region boundary, the electric potential field must be parallel to the bounding surface. This requirement is hardly surprising, since it echoes the well- known natural boundary condition encountered in electrostatics.

The Magnetic Vector Potential Boundary Integral

The magnetic vector potential boundary integral term may in principle be treated in a similar fashion to the electric scalar potential term. The details are different, however, because the magnetic potential is a vector quantity. The integral in question is

I , =$,(A x vv x A) ‘ ds 03-91

Again, let the region of integration S be subdivided into two parts. The integral (B-9) then becomes

1, (AI x VlV x AI) ‘ ds + $s2”s0 (22 x qv x 22) ’ ds (B-10)

This form may in turn be rewritten, grouping together the exterior and interior surface integrals, in exactly the same way as for the scalar electric potential case above:

(B-11)

The subscripts 1 and 2 refer to the two regions which the interface SO separates; the norma1,direction on the interface is assumed to point from region 1 into region 2. No subscript is shown with the vector potential A itself, for although its derivatives need no’ all be continuous across the interface, all values of the components of A must be continuous. By vir- tue of (B-91, (B-11) becomes

(B- 12)

Since the area element dS has the direction of the normal, the in- tegrand in fact contains only the normal component of the vector quantity in square brackets. Let a right-handed local coordinate system (n,p,q) be chosen at some point on the surface SO, in the manner indicated in Fig. 2. One may choose the coordinate directions so that n is the normal direction pointing from region 1 into 2, and p is parallel to the direction of 2 at that point. Expanding in the usual manner,

Fig. 2 At any point of a surface element ds on the separating surface SO, a right-handed local coordinate system (n,p,q) may be defined, with the coordinate n normal to the surface and p,q two mutually orthogonal directions in the surface.

(B-15)

The natural interface condition thus imposed by stationarity of the overall functional is thus

V l ( f 2 x B), = v * ( % x B)2 (B- 16)

Physically, this equation may be interpreted as saying that the tangential magnetic flux density has at the interface a discontinuity equal to the permeability contrast between the two media:

where the subscript “tang” indicates that part of the vector B which lies in the p-q plane, and which is thus itself a two-component vector. Note that since the permeability is assumed to be a scalar quantity, the direction of the tangential vector is the same on both sides of the interface.

The natural boundary condition at an exterior boundary of the overall region S may again be deduced by noting that the space external to S does not contribute to the overall functional; hence it is indistinguishable from an infinitely-extending, zero-reluctivity (infinitely permeable) space. Setting the reluctivity to zero, (B-16) yields the natural boundary condition

Evan& = 0 (B-18) This condition will be satisfied in a weighted mean sense by approxi-

mate solutions arrived at by finding stationary points of the diffusive-field functional (25) for the three-dimensional nonlinear electromagnetic field problem. As above, note that the tangential flux density is a vector quantity, since it possesses components in both coordinate directions p,q lying in the surface.

The boundary conditions described by (B-9) and (B-18) are natural to the functional (25) and, therefore, they need not be explicitly imposed on any approximate solution.

APPENDIX C

LIMITS OF THE INTEGRAL St ( Id,

Consider a general term f(t,x,y), whose integral with respect to time

[ ’ I . ( T , x , y ) d 7 = F( f , x , y ) - F(/ , ,x ,y) (C-1)

The first term on the right-hand side of (C-1) is a function of time, but the second is only a space function. However, the second integral may prove cumbersome in any minimization. It is, therefore, advisable to choose to to be the starting time when F( t , , x , y ) = 0. In general, this choice is not easy. Therefore, one may have to go back in time a long way before F(r,,x,y) = 0, for all x,y in the region. There is only one starting time f ( , when F = 0; and that is infinitely far back. Hence, we assert F(-m,x , y ) = 0.

-m

may be written as

I t must be noted that the aforesaid argument does not permit us to use forms such as F = E ’ ~ ! Rather, one must set

F = u(f--1,)E”‘ (C-2)

U ( T ) = 1, T > 0 (C-3)

= o , T < 0 (C-4)

and choose to go very far back so as to guarantee that all transients have died down. Otherwise there remaihs the difficult problem of finding E-J -

i.e., of evaluating the limit

I--‘% Lim e -Jwr

Since E J w ‘ denotes “sinusoidal variation since a very long time ago,” one might say --oo (which implies a limit and not a value) really means “a long time ago also.” Here, however, it will be taken to mean “even long ago,” in keeping with (C-2) through (C-4). Thus, the limits of integration of (C-2) are --m and /.

446

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