Introduction.

The present thesis deals with the determination and analysis of the current density and magnetic vector potential distribution in an infinite cylindrical conductor carrying a time varying current. The material of the conductor is assumed to be isotropic, homogeneous, nonmagnetic having non-linear properties. The problem examined is two dimensional. The determination of the time varying electromagnetic field in the conductor amounts to the solution of a parabolic differential equation with appropriate initial and boundary conditions. Diffusion of the electromagnetic field into an infinite metallic object having rectangular or circular cross-section has been thoroughly studied and analysed in the case of sinusoidal current flowing in the above conductors, Hallen [1.1], Gosselin and Gauthier [1.2]. The recent trend toward the design of electromagnetic accelerators which carry an applied ramp current has resulted in more stringent specifications and a demand for economy and reliability of various parts used in such apparatus, Garg and Vechio [1.3], and Perry [1.4]. In turn, the more exact specifications demand the development of more advanced and accurate mathematical methods for predicting the current density and magnetic field density, especially for a circular or rectangular conductor, carrying a predetermined net current.

Several methods can be found in Angot [1.5] for solving the diffusion equation governing the magnetic field intensity or current density vector, subject to various boundary conditions. Mocanu [1.6] used analytical techniques when the partial differential equation describing the distribution of the above field quantaties is linear. This paper presents by far the most complete analytical account of the subject emphasizing the physical formulation of the electromagnetic diffusion but does not give quantitative numerical results. In the literature, only very simple static problems have been formulated and solved analytically for the magnetic vector potential. However, once the coefficients (the electrical conductivity and the magnetic reluctivity of the material) become functions of other quantities, for example, the temperature, this technique no longer holds. Moreover, in many applications, the geometry is no longer simple. One way around these complications is to use finite difference and finite element techniques in order to solve the problem.

An extensive treatment of finite differences is found in Smith [1.7] Kreiss [1.8], and Lapidus and Pinder [1.9]. The books by Collatz [1.10] and Durand [1.11] are extremely useful for an overall impression of current thinking, but the numerically solved diffusion equation with the aid of finite difference method are not given in terms of the magnetic vector potential. Collatz and others present these equations in terms of the magnetic field intensity. Moreover, in the particular problems considered, the voltage is suddenly applied. This indeed is the way in which the problem was formulated before the 1970's. Only with the advent of computers has it been favorable to work with the magnetic vector potential. The finite difference and finite element scheme are formulated in terms of the magnetic vector potential. The current density vector existing at any point in the conductor can be derived. This modern approach is prefered to the direct approach of setting up the equation in terms of the current density and solving for the current density because the direct approach poses difficulties in finite difference or finite element techniques. Moreover, once the magnetic vector potential is calculated, the other important field quantaties, namely the magnetic flux density and the electric field intensity, are obtained easily.

Zienkiewicz [1.12] and Ciarlet [1.13] are well cited authors about finite elements. For a discussion of existing finite element techiques for transient electromagnetic diffusion problems, the reader should consult the most recent literature: Lowther and Silvester [1.14]. Silvester [1.15] and, later on, Chari [1.16] published extensively on finite element formulations in two dimensions using model decomposition to study sinusoidal non-linear problems with important applications to electrical engineering. Tandon and Chari [1.17] have worked on and developed an algorithm which uses finite elements in space and finite differences in time. They discussed both Crank-Nicolson and Euler backwards formulations in time and also accounted for varying reluctivity through the use of a predictor-corrector scheme. However, they restricted their experimental verification to a periodic problem. Soon afterwords, Kamar [1.18] developed a forward difference one-dimensional finite element program but also restricted his study to time periodic problems. He formulated his problem using weighted residuals in time and showed that for certain approximation functions, the time finite element formulation reduces to the time finite difference formulation. Armstrong and Biddlecombe [1.19] at Rutherford Labs in England formulated and implemented finite elements in time. Infolytica Corporation and Vector Fields Unlimited expanded a version of this program. Konrad, Chari and Csendes [1.20] developed a method for imposing a current constraint for multiple conductors which is extendable to finite difference formulations of these problems. Pillsbury [1.21] formulated and discussed finite elements in space and finite differences in time, but restricted his studies to that of the first harmonic. Bedrosian [1.22] wrote up a loose description of an algorithm using a functional variation approach. He also encode a program TDTEMP which was very similar to that used by Tandon and Chari [1.17]. Weiss and Garg [1.23] worked on WEMAP which uses Crank-Nicolson time finite differences and second order finite elements in space. Moreover, their description is extended to multiconductor problems. Lately, Perry [1.4] and Garg and Weiss [1.3] presented results on transient current calculations. However, they restricted their study to a specialized problem with complicated geometry.

The computer program TDTEMP models the diffusion of current density into a conductor as time varies. The finite element method is used to model the two dimensional cross-section of a conductor. Moreover, current pulses (rather than the usual sinosoidal steady-state currents) passing through the conductor are studied. This complication leads to the developement of a time domain finite difference method without resorting to frequency decomposition. Also, because of the high currents passing through the conductor, the thermal heating and subsequent change in electrical conductivity had to be considered. The thermal diffusion effects are disregarded because they occured on a much slower time scale than the current diffusion times of the presented data. Ferromagnetic materials are not consider, and thus a strongly non-linear equation, namely one in which the reluctivity, n, is a function of the magnetic field intensity and displays hysterisis effects, does not have to be solved. It should be pointed out that this additional problem is not much more difficult to formulate and solve than ours is.

In this thesis, the geometry is simplified to that of a cylindrical conductor. The reason for the geometry is twofold, the most important of which is that many conductors are cylindrical and the second is that the analytical solution to this problem is obtained. Thus, in the comparison, the accuracy of the computed solutions is verified easily.

The problem is formulated from Maxwell's equations and the relavent electromagnetic theory is briefly discussed. The analytical solution for this problem is presented in terms of the current density and the relationship between the current density and the magnetic vector potential is given. The current density is the prefered quantity with which many scientists work because it has a physical representation, namely, the current per unit area. This gives the heating losses resulting from currents flowing through the conductor. These losses are called the (J(r,t))2/s losses. The finite difference and finite element methods are formulated in terms of the magnetic vector potential because of the simplicity of the algorithm. The current density is then derived.

The finite difference method is described for angle independent polar coordinates and implemented as a computer program FDT. The finite element method is described from the approach of weighted residuals in the two-dimensional space domain. The time derivatives are approximated with finite differences. The presented results were calculated using a packaged program TDTEMP. The results are presented for a conductor having a circular cross-section, since both programs could work on this geometry and the analytic solution is available.

References:

[1.1] Hallen, E., *Electomagnetic Theory*, John Wiley and Sons Inc., New York, (1962).

[1.2] Gosselin, P. Rochon, and Gauthier, N., " Study of eddy currents in a cylindrical wire: an undergraduate laboratory experiment," *American Journal of Physics*, 50(5), May 1982, pp. 440-443.

[1.3] Garg, V. K., Weiss, J., and Vechio, R. M., "A Parametric Study of Electromagnetic Launcher Rails Using the WEMAP System." *Workshop on Electromagnetic Field Computation*, Schnectady Section of IEEE, October 20 and 21, 1986, Schenectady, NY 12345.

[1.4] Perry, M. P., "A Coupled-Circuit Transient Electromagnetic Formulation with External Constraints," *Workshop on Electromagnetic Field Computation*, Schnectady Section of IEEE, October 20 and 21, 1986, Schenectady, NY 12345.

[1.5] Angot, A., *Compléments de mathématiques*, Masson et Cie, Paris, (1972)

[1.6] Mocanu, C. I., "The Equivalent Schemes of Cylindrical Conductors at Transient Skin Effect," Transaction Paper, IEEE, August 6, 1971.

[1.7] Smith, G. D., *Numerical Solution of Partial Differential Equations, Finite Difference Methods*, Oxford University Press, 1978.

[1.8] Kreiss, H. O. ,* Numerical Methods for Solving Time Dependent Problems for Partial Differential Equations*. Les Presses de L'Universite de Montreal, C. P. 6128, Succ. "A", Montreal, (1978)

[1.9] Lapidus, L. and Pinder, G. F., * Numerical Solution of Partial Differential Equations in Science and Engineering*, John Wiley and Sons, New York, (1982)

[1.10] Collatz, L. *The Numerical Treatment of Differential Equations*, Springer-Verlag, Berlin-Gottingen-Heidelberg, (1960).

[1.11] Durand, E. *Électrostatique, Tome II.-Problèmes généraux, Conducteurs*, Masson et Cie, Paris, (1966)

[1.12] Zienkiewicz, D. C., *The Finite Element Method*, McGraw Hill, London, 3rd ed., (1977).

[1.13] Ciarlet, Ph. G., *Numerical Analysis of the Finite Element Method*, Les Presses de l'university de Montreal, C.P. 6128, Succ. "A", Que., Canada H3C-3J7, (1976).

[1.14] Lowther, D. A. and Silvester, P. P., *Computer-Aided Design in Magnetics*, Springer-Verlag, New York, (1986).

[1.15] Silvester, P. P. and Ferrari, R. L., Finite Elements for Electrical Engineers, Cambridge University Press, (1983).

[1.16] Chari, M.V.K., Lecture Notes from: "A First Course in Finite Element Methods," General Electric Company, Corporate Research and Development, Schnectady, NY (1979).

[1.17] Tandon, S.C. and Chari, M.V.K. "Transient Solution of the Diffusion Equation by the Finite Element Method," *Journal of Applied Physics* (53), 3, March 1981, pp. 2431-2432.

[1.18] Kamar, A.M. , "Solution fo Nonlinear Eddy Current Problems Using Residual Finite Element Methods for Space and Time Discretization," *IEEE Transactions on Magnetics*, Vol. MAG-19, No. 5, September 1983.

[1.19] Armstrong, A.G.A.M., and Biddlecombe, C., "The PE2D Package for Transient Eddy Current Analysis". *IEEE Trans MAG-18*. p 411-415, (1982).

[1.20] Konrad A., Chari, M.V.K., and Csendes, Z.J. "New Finite Elements," *IEEE MAG*.-18, No. 2, March 1982, pp. 450-453 (refer to eqs 6-11 on p. 451).

[1.21] Pillsbury, RD., Jr. NMLMAP-A Two-dimensional Finite Element Program for Transient or Static, Linear or Nonlinear Magnetic Field Problems," *IEEE Trans. on Magnetics* , pp. 406-410 (March 1984)

[1.22] Bedrosian, G., Private communications on TDTEMP, November 1984 (see Perry [1.4]).

[1.23] Weiss, J. and Garg, V. K. "Finite Element Solution of Transient Eddy Current Problems in Multiply-Excited Magnetic Systems." *Westinghouse Report* 1986.