Difference between revisions of "A Review of Maxwell's Equations & Computational Electromagnetics (CEM)"

[[Image:Maxwell1.png|right|600px]]

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:<math> \nabla . \mathbf{D} = \rho </math>

:<math> \nabla \times \mathbf{E} = - \dfrac{\partial \mathbf{B}}{\partial t} </math>

where &epsilon; is the permittivity in F/m, &mu; is the permeability in H/m, and &sigma; is the electric conductivity of the medium in S/m.

:<math> \nabla . \mathbf{D} = \rho </math>

 

:<math> \nabla . \mathbf{B} = \rho_m  </math>

 

 

 

The following constitutive relationships now hold:

 

 

 

where &sigma;_m is the magnetic conductivity of the medium in &Omega;/m.

 

 

 

 

 

Combining Maxwell's equations, we can arrive at the electric and magnetic wave equations:

 

 

 

where '''&nabla;'''² is the Laplacian operator. The wave equations are hyperbolic partial differential equation in space coordinates and time, which must be solved subject to the proper initial and boundary conditions.

 

 

The electric field boundary conditions at the interface between two material media are:

 

 

 

 

 

 

 

 

 

 

In certain applications, it is advantageous to cast Maxwell's equations in an integral form. These can be done using the theorems of vector calculus. In an isotropic, time-invariant and homogeneous medium, the integral forms of Maxwell's equations are given by:

 

 

 

 

 

where V is a closed region of the space, S the surface boundary and C is a path.

 

 

In a time-harmonic system operating at a given frequency f, the time dependence of the fields takes the form of <math> e^{j\omega t} </math>, where <math> j = \sqrt{-1} </math>, and &omega; = 2&pi;f is the angular frequency. In that case, the time derivative is <math> {\partial}/{\partial t} = j\omega </math>, and Maxwell's curl equations reduce to:

 

 

 

and the continuity equations reduce to:

 

 

 

 

 

 

 

Under the time-harmonic assumption, the electric and magnetic fields can further be expressed in terms of an electric scalar potential &Phi;, a magnetic scalar potential &Psi;, a vector electric potential '''F''' and a vector magnetic potential '''A''' in the following form:

 

 

 

 

 

 

 

 

 

 

 

Sometimes it is useful to define the Hertz vector potentials as:

 

 

 

In that case, the electric and magnetic fields can be fully expressed in terms of these two vector potentials:

 

 

 

{{Note|For historical reasons, it is customary in electrostatic problems to directly set the magnetic flux density '''B'''  equal to &nabla; &times; '''A'''. [[EM.Cube]]'s Static Module ([[EM.Ferma]]) uses that convention for definition of the vector magnetic potential '''A''', which is different by a factor of &mu; from the definition of the '''A''' vector in the electrodynamic discussion presented in this section. That would also change the source term of the Helmholtz equation by the same factor.}} 

 

 

The Green’s functions are the analytical solutions of boundary value problems when they are excited by an elementary source. This is usually an infinitesimally small vectorial point source.  The total electric ('''E''') field and total magnetic ('''H''') field can be expressed in terms of the volume electric current source '''J''' and volume magnetic current source '''M''' in the following way:

 

 

 

 

 

where '''E^inc''' and '''H^inc''' are the incident electric and magnetic fields, respectively, and V_J and V_M are the volumes containing the electric and magnetic current sources, respectively. The above equations involve four dyadic Green's functions types: dyadic electric-field Green’s functions due to electric current sources '''G_EJ''', dyadic magnetic-field Green’s functions due to electric current sources '''G_HJ''', dyadic electric-field Green’s functions due to magnetic current sources '''G_EM''', and dyadic magnetic-field Green’s functions due to magnetic current sources '''G_HM''' . The incident or impressed fields represent the source terms and provide the excitation of the structure.

 

In order for the Green’s functions to be computationally useful, they must have analytical closed forms. This can be a mathematical expression or a more complex recursive process. It is no surprise that only very few electromagnetic boundary value problems have closed-form Green’s functions. Among [[EM.Cube]]'s computational modules, [[EM.Libera]] is based on the free-space Green's functions, whereas [[EM.Picasso]] is based on the dyadic Green's functions of an arbitrary multilayer planar structure.

 

 

 

:<math> \mathbf{ \overline{\overline{G}}_{EJ}(r|r') = (\overline{\overline{I}} + \nabla\nabla) } G_{\Lambda} (\mathbf{r|r'}), \quad G_{\Lambda} (\mathbf{r|r'}) = \frac{ e^{-jk_0 \mathbf{|r-r'|}} }{ 4\pi \mathbf{|r-r'|} } </math>

 

 

 

 

 

 

 

 

 

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