Methods of Electromagnetic Field Analysis*

This paper presents a discussion of ideas involved in various mathematical methods of electromagnetic field analysis and of the inter-relations between these ideas. It stresses the points of contact between circuit and field theories and their mutually complementary character. While the field theory focuses our attention on the electromagnetic state as a function of position in space, the generalized circuit theory is preoccupied with the electromagnetic state as a function of time. The points of contact between the field and circuit theories are many. Thus, Maxwell's equations are identical with Kirchhoff's equations (really Lagrange-Maxwell equations) of certain three-dimensional networks in which only the adjacent meshes are coupled. The integral equations for the electrical current in conductors embedded in dielectric media are also Kirchhoff equations of certain networks containing infinitely many meshes with a coupling between every two meshes. From the point of view of electrical performance the difference between a physical network of lumped elements and a continuous network, such as a resonator, is due to a certain difference in the distribution of the zeros and poles of associated impedance functions in the complex impedance plane. Similarly, the difference between ordinary transmission lines and wave guides is due to a difference in the distribution o(natural propagation constants. The paper ends with a general discussion of the discontinuities in wave guides, idealized boundary conditions for simplification of electromagnetic problems, and the analytical character of field vectors regarded as functions of the complex oscillation constant.