The starting point for seeking the distribution of the turns inside the stranded coil from Figure 1 is the equations of steady state current flow [6]. These equations, taking into account relation between vector potential *T*_{0} and current density vector *J*_{0} (*i*.*e*. *J*_{0} = *curl* *T*_{0}), can be expressed as:
$$\begin{array}{}{\displaystyle {\left.curl\left({\gamma}^{-1}curl{\mathit{T}}_{0}\right)=0\right|}_{{\mathit{\Omega}}_{c}}}\end{array}$$(1a)

and
$$\begin{array}{}{\displaystyle {\left.curl\left(curl{\mathit{T}}_{0}\right)=0\right|}_{{\mathit{\Omega}}_{n}}}\end{array}$$(1b)

and are solved with respect to following boundary conditions:

$$\begin{array}{}{\displaystyle {\left.{\mathit{T}}_{0}=0\right|}_{{\mathit{\Gamma}}_{0}+{\mathit{\Gamma}}_{\mathit{\Omega}}},}\end{array}$$(2a)

$$\begin{array}{}{\displaystyle {\left.{\mathit{T}}_{0}={\mathit{\tau}}_{0}\right|}_{{\mathit{\Gamma}}_{To}}}\end{array}$$(2b)

and
$$\begin{array}{}{\displaystyle {\left.curl{\mathit{\tau}}_{0}=0\right|}_{{\mathit{\Gamma}}_{To}}}\end{array}$$(2c)

where: *γ* is the material conductivity, and *τ*_{0} is the function that describes distribution of the potential *T*_{0} on the boundary surface *Γ*_{T}_{0} (see Figure 1).

Taking into account the boundary conditions (2) in (1), the equations describing distribution of current vector potential *T*_{0} inside region of considered coil will have following form:

$$\begin{array}{}{\displaystyle {\left.curl\left({\gamma}^{-1}curl{\mathit{T}}_{0}\right)\right|}_{{\mathit{\Omega}}_{c}}+{\left.curl\left({\gamma}^{-1}curl{\mathit{\tau}}_{0}\right)\right|}_{{\mathit{\Gamma}}_{{T}_{0}}\to {\mathit{\Omega}}_{c}}=0}\end{array}$$(3a)

$$\begin{array}{}{\displaystyle {\left.curl\left(curl{\mathit{T}}_{0}\right)\right|}_{{\mathit{\Omega}}_{n}}+{\left.curl\left(curl{\mathit{\tau}}_{0}\right)\right|}_{{\mathit{\Gamma}}_{{T}_{0}}\to {\mathit{\Omega}}_{n}}=0}\end{array}$$(3b)

In general, the equations (3), obtained by means of *T*_{0} method, are solved employing numerical methods based on the space discretization. To solve (3) the 3D Edge Element Method has been applied [9, 10], for which equations (3) can be expressed in the following form:

$$\begin{array}{}{\displaystyle {\mathit{k}}_{s}^{T}{\mathit{R}}_{{\mathit{\Omega}}_{c}}{\mathit{k}}_{s}^{}{\mathit{i}}_{0c}={\mathit{R}}_{{\mathit{\Omega}}_{c}}{\mathit{i}}_{0\tau}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Omega}}_{c}}\end{array}$$(4a)

$$\begin{array}{}{\displaystyle {\mathit{k}}_{s}^{T}{\mathit{R}}_{{\mathit{\Omega}}_{n}}{\mathit{k}}_{s}^{}{\mathit{i}}_{0n}={\mathit{R}}_{{\mathit{\Omega}}_{n}}{\mathit{i}}_{0\tau}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Omega}}_{n}}\end{array}$$(4b)

where: *i*_{0c} and *i*_{0n} are the vectors describing the edge values of the potential *T*_{0} inside region (*Ω*_{c}) of the coil and beyond this region (*Ω*_{n}) respectively [6]; **k**_{s} is the transposed loop matrix of EFN [10], and the vector *i*_{0τ} describes given source of edge values of *T*_{0} on surface *Γ*_{T0}. The matrices *R*_{Ωc} and *R*_{Ωn} are the coefficient matrices of the EE equations for considered regions *Ω*_{c} and *Ω*_{n} in the coil calculated from the following expressions:

$$\begin{array}{}{\displaystyle {\mathit{R}}_{{\mathit{\Omega}}_{c}}=\underset{{V}_{e}}{\int}{\mathit{w}}_{fq}^{}{\gamma}^{-1}{\mathit{w}}_{fp}^{}dv}\end{array}$$(5)

$$\begin{array}{}{\displaystyle {\mathit{R}}_{{\mathit{\Omega}}_{n}}=\underset{{V}_{e}}{\int}{\mathit{w}}_{fq}^{}{\mathit{w}}_{fp}^{}dv}\end{array}$$(6)

where: *w*_{fq}, *w*_{fq} are interpolation functions of facet element [10].

For determining the edge values (*i*_{0τ}) of potential *T*_{0} on the boundary surface *Γ*_{T0}, the relation between edge quantities *i*_{0τ} and facet currents of EFN network (facet quantities *i*_{f0} of current density vector *J*_{0}, where *J*_{0}curl*T*_{0}) has been employed [10]. To determine the *i*_{0τ}, it is necessary to include also condition that current density in normal direction to surface *Γ*_{T}_{0} is equal to zero [6]:

$$\begin{array}{}{\displaystyle \left.{\mathit{k}}_{s}{\mathit{i}}_{0\tau}^{}=0\right|{}_{{\mathit{\Gamma}}_{{T}_{0}}}}\end{array}$$(7)

In discussed approach the equations (7) are solved first, next obtained distribution of *i*_{0τ} vector is introduced to (4) and solved. Nevertheless, to solve equations (7) (at the stage of meshing of the domain) it is necessary to choose one or higher number of cross-sections of the coil on which the current density is specified. Our experience shows that using single cross-section per coil is sufficient for most of the cases, however when modeling the coils of complex geometry of current path to achieve faster convergence of the solving equations (7), it is convenient to increase number of cross-sections. Next, basing on the selected cross-sections the right hand sides of the equations (7) are determined. In the paper, equations (7) are solved for given values of the edge quantities *i*_{0τPuPv} of the potential *T*_{0} of edges *P*_{u}*P*_{v}. These edges of the mesh elements correspond to *P*_{i}*P*_{j} edges of selected cross-section of the coil (Figure 2) and lay on the surface *Γ*_{T0} (Figure 1). For example, for the cross-section illustrated in Figure 2 values *i*_{0τ}*P*_{u}*P*_{v} are determined by the following formula [6]:

$$\begin{array}{}{\displaystyle {i}_{0\tau ,{P}_{u}{P}_{v}}=\underset{{P}_{u}}{\overset{{P}_{v}}{\int}}{\mathbf{T}}_{0}\mathrm{d}\mathbf{l}=\underset{{P}_{u}}{\overset{{P}_{v}}{\int}}{\tau}_{0}\mathrm{d}\mathbf{l}=\frac{{z}_{c}{i}_{c}}{{l}_{i,j}}\cdot {l}_{u,v}}\end{array}$$(8)

Figure 2 Definition of bonduary function *τ*_{0} for the edge *P*_{i}*P*_{j} of the cross-section of coil shown in Figure 1

where: *i*_{c} is the value of the current in coil, *z*_{c} is the number of turns, *l*_{u},_{v} and *l*_{i},_{j} are the lengths of the edges *P*_{u}*P*_{v} and *P*_{i}*P*_{j}, respectively (see Figure 2).

The presented above formulas are the same as for the CF and MF of *T*_{0} method. The difference between formulations lays only in the different way of calculation of conductivity *γ* of the medium. In the classical approach of *T*_{0} method, as noted in [8], “it is assumed that *γ* is constant and uniform over whole coil domain *Ω*_{c}, (*i*.*e*. *γ* = *γ*_{c}), whereas in the modified method the value of the conductivity of each finite element *γ*_{i} of the domain (*i*.*e*. *i*^{th} EFN for presented EEM formulation) is determined iteratively to achieve the homogeneity of the current density distribution inside the coil. In k^{th} iteration, the conductivity of *i*^{th} element is calculated according to the following formula:

$$\begin{array}{}{\displaystyle {\gamma}_{i}=\left\{\begin{array}{}{\gamma}_{c}& \text{for}\phantom{\rule{thinmathspace}{0ex}}k=1\\ {\gamma}_{c}\cdot \prod _{k=2}^{M}\left(\frac{{J}_{av}^{(k-1)}}{{J}_{e,i}^{(k-1)}}\right)& \text{for}\phantom{\rule{thinmathspace}{0ex}}k\ge 2\end{array}\right.}\end{array}$$(9)

where: $\begin{array}{}{\displaystyle {J}_{e,i}^{(k-1)},{J}_{\phantom{\rule{thinmathspace}{0ex}}av}^{(k-1)}}\end{array}$ are the current density in *i*^{th} facet element and global current density in the coil for (*k*-1)^{th} iteration, and symbol *M* is assumed number of refinements (iterations) of current density distribution homogenization”. It is worth to notice that the modification of the *T*_{0} method, proposed by authors, is applicable for the coils of constant cross-section of the current path along studied domain.

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