## 4 Case of Electromagnetic and Energy Conversion Systems
### [PERSON]'s Equations  
[PERSON] was a Scottish physicist and mathematician. He is mainly known to have unified, into a single set of equations, the [PERSON]'s equations, electricity, and magnetism, containing a key revision of [PERSON]'s theorem. It is the most unified model of electromagnetism. The equations established by [PERSON] involve an association of three experimental laws discovered by his predecessors:  
* Maxwell-Gauss law relates the electric displacement \(d\) to the free electric charge density \(q\).
* Maxwell-Ampere law links the \(h\)-magnetic field to the time-rate of the electric displacement \(d\) and the free total current density \(j\).
* Maxwell-Faraday law relates the electric field e to the time-rate of the magnetic induction \(b\).  
Around 1865, [PERSON] produced a harmonious synthesis of these experimental laws. As mentioned in the introduction, he introduced into an equation a term based on the called displacement current, whose presence guarantees the coherence of the unified edifice. As mentioned before, after the publication of the initial [PERSON]'s equations in 1865 and in 1873, [PERSON] rewrote these equations as four vector partial differential equations.  
This equation system can be formulated mathematically under different functional forms of the considered problem. One of the most common is the basic full-wave electromagnetic formulation given by:  
\[\mathbf{\
abla}\times\mathbf{H}=\mathbf{J} \tag{4}\]  
\[\mathbf{J}=\sigma\,\mathbf{E}+\mathbf{j}\,\omega\,\mathbf{D}+\mathbf{J}\mathbf{e} \tag{5}\]  
\[\mathbf{E}=-\mathbf{\
abla}\,\mathbf{V}-\mathbf{j}\,\omega\,\mathbf{A} \tag{6}\]  
\[\mathbf{B}=\mathbf{\
abla}\times\mathbf{A} \tag{7}\]  
where \(\mathbf{H}\) and \(\mathbf{E}\) are the magnetic and electric fields, \(\mathbf{B}\) and \(\mathbf{D}\) are the magnetic and electric inductions, and \(\mathbf{A}\) and \(\mathbf{V}\) are the magnetic vector and electric scalar potentials. \(\mathbf{J}\) and \(\mathbf{J}_{\mathbf{e}}\) are the total and source current densities, \(\sigma\) is the electric conductivity, and \(\omega\) is the frequency pulsation. The symbol \(\mathbf{\
abla}\) is a vector of partial derivative operators, and its three possible implications are gradient (product with a scalar field), divergence and curl (dot and cross products, respectively, with a vector field).  
The magnetic and electric behavior laws respectively between **B/H** and **D/E** are characterized by the permeability \(\mu\) and the permittivity \(\varepsilon\).  
It may be noted that the above harmonic representation that seems simpler cannot be applied in the case of free transient variations, as for example, a capacitor discharge.  
The solution of Equations (4)-(7) determines in a system the concerns of electromagnetic fields for a frequency pulsation, accounting for the magnetic materials behaviors through the permeability, for eddy currents in electric conductors through the electric conductivity and for displacement currents in dielectrics through the permittivity.  
Often, EMSs involve fields other than EM, for example, mechanic, thermic, material, etc. In some cases, the influence of these other fields could be negligible, and it will then be possible to solve the problem correctly with only the [PERSON]'s equations, see, e.g., [3,4]. Nevertheless, in such a case, there is yet a conflict with real situations. In realistic applications, the electric current is delivered by a non-perfect current source as supposed by the elegant form in (4)-(7) that considers the current delivered by a current source. The committed assumption is that we consider in (4)-(7) the value of \(\mathbf{J_{e}}\), which is known and could be imposed. In general, the current is delivered by a voltage source through an external electric circuit. We will consider this question later in the paper.  
In general, to model an EMS, we need to account for other fields in addition to the EM field through coupling of the corresponding governing equations.