44 Three-Dimensional Analy SIS 44.1 Introduction 44.2 The Field equations Low Frequency Fields. Statics Limit 44.3 Numerical Methods C. W. Trowbridge Finite Elements. Edge Elements.Integral Methods Vector Fields. Inc 44.4 Modern Design Environment 44.1 Introduction The three-dimensional analysis of electromagnetic fields the use of numerical techniques exploiting he best available computer systems. The well-found laboratory will have at its disposal a range of machines that allow interactive data processing with access to software that provides geometric modeling tools and has sufficient central processing unit( CPU) power to solve the large(>10,000)set of algebraic equations involved, see Section 44.4 44.2 The Field equations The classical equations governing the physical behavior of electromagnetic fields over the frequency range dc to light are Maxwells equations. These equations relate the magnetic flux density(B), the electric field intensity (E), the magnetic field intensity(H), and the electric field displacement(D)with the electric charge density (p)and electric current density (). The field vectors are not independent since they are further related by the material constitutive properties: B=HH,D=EE, and j= oE where u, E, and o are the material permeability, permittivity, and conductivity, respectively. In practice these quantities may often be field dependent, and furthermore, some materials will exhibit both anisotropic and hysteretic effects. It should be strongly stated that accurate knowledge of the material properties is one of the most important factors in obtaining reliable Because the flux density vector satisfies a zero divergence condition(div B=0), it can be expressed in terms f a magnetic vector potential A, ie,B=curl A, and it follows from Faraday s law that E=-(aA/at+ vv), where V is the electric scalar potential. Neither a nor V is completely defined since the gradient of an arbitrary scalar function can be added to a and the time derivative of the same function can be subtracted from v without affecting the physical quantities E and B. These changes to a and v are the gauge transformations, and uniqueness is usually ensured by specifying the divergence of a and sufficient boundary conditions. If V. A=-(uoV+ HEaV/at)(Lorentz gauge)is selected, then the field equations in terms of A and V are c 2000 by CRC Press LLC© 2000 by CRC Press LLC 44 Three-Dimensional Analysis 44.1 Introduction 44.2 The Field Equations Low Frequency Fields • Statics Limit 44.3 Numerical Methods Finite Elements • Edge Elements • Integral Methods 44.4 Modern Design Environment 44.1 Introduction The three-dimensional analysis of electromagnetic fields requires the use of numerical techniques exploiting the best available computer systems. The well-found laboratory will have at its disposal a range of machines that allow interactive data processing with access to software that provides geometric modeling tools and has sufficient central processing unit (CPU) power to solve the large (>10,000) set of algebraic equations involved, see Section 44.4. 44.2 The Field Equations The classical equations governing the physical behavior of electromagnetic fields over the frequency range dc to light are Maxwell’s equations. These equations relate the magnetic flux density (B), the electric field intensity (E), the magnetic field intensity (H), and the electric field displacement (D) with the electric charge density (r) and electric current density (J). The field vectors are not independent since they are further related by the material constitutive properties: B = mH, D = eE, and J = sE where m, e, and s are the material permeability, permittivity, and conductivity, respectively. In practice these quantities may often be field dependent, and furthermore, some materials will exhibit both anisotropic and hysteretic effects. It should be strongly stated that accurate knowledge of the material properties is one of the most important factors in obtaining reliable simulations. Because the flux density vector satisfies a zero divergence condition (div B = 0), it can be expressed in terms of a magnetic vector potential A, i.e., B = curl A, and it follows from Faraday’s law that E = –(]A/]t + ¹V), where V is the electric scalar potential. Neither A nor V is completely defined since the gradient of an arbitrary scalar function can be added to A and the time derivative of the same function can be subtracted from V without affecting the physical quantities E and B. These changes to A and V are the gauge transformations, and uniqueness is usually ensured by specifying the divergence of A and sufficient boundary conditions. If ¹ • A = –(msV + me]V/]t) (Lorentz gauge) is selected, then the field equations in terms of A and V are: C. W. Trowbridge Vector Fields, Inc