# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 3 Calculation Example--Electric Conductor with Double Insulation  
An example of the application of the method described above is the analysis of the stationary temperature field in an elliptical electric wire. The conductor is composed of a conducting core carrying alternating current and of two different layers of insulation (Figure 1 for \(N=3\)). The conductor core was modeled as a porous body [35; 36; 37] consisting of helically stranded wires with air between them. The efficiency of the heat source in the core is defined as below  
\[g_{1}=\frac{P_{1}}{V_{1}}=\frac{k_{s}k_{n}R_{DC}|I|^{2}}{\pi a_{1}b_{1}l}=k_{s }k_{n}\rho(T_{max})\frac{|I|^{2}}{S_{1}\cdot\pi a_{1}b_{1}}, \tag{7}\]  
where \(P_{1}\) is the power of heat losses released in the core due to the alternating current flowing through it, with root-mean-square value \(|I|\); \(V_{1}\) denotes the volume of conductor core length \(l\); \(R_{DC}\) is the direct current resistance of conductor core length \(l\); \(\rho(T_{max})\) denotes the electrical resistivity of the core at sustained temperature \(T_{max}\); \(S_{1}\) denotes the sum of the cross-sections of core forming bundles (\(S_{1}<\pi a_{1}b_{1}\)); \(k_{n}\) is the skin effect coefficient; and \(k_{s}\) is the coefficient that takes into account the elongation of the core caused by twisting the bundles [38]. No heat sources (\(g_{2}=g_{3}=0\)) in double insulation were assumed (displacement currents were omitted at low frequency). An important factor affecting the reliability of electrical wires is the thermal field. Too high a temperature (greater than \(T_{max}\)) can cause a number of disadvantageous phenomena. This includes, for example, internal mechanical stresses that can cause the wires to shift more or less. In the case of short circuits, the temperature can be so high that the delaminated wires form a kind of cage in the conductor. High temperature is also highly damaging to electrical insulating materials. It accelerates the ageing of these materials and, in extreme cases, destroys them. For the reasons mentioned above, thermal field analysis is an important technical task. The geometry of the system (Figure 1, \(N=3\)) and the assumed constant value of coefficient \(\alpha\) on the outer perimeter \(\eta=\eta_{3}\) imply that relationships (6) must be both periodical and even with respect to the \(\psi\) coordinate of the elliptical-cylindrical system. This results in zeroing of the constants and coefficients \(D_{i}=D_{ni}=0\) for \(i=1,\ 2,\ 3\). Inaddition, the heat flux cannot take on an infinite value at the focus (for \(\eta=0\), \(\psi=0\)). In order to verify this condition, the heat flux \(\stackrel{{\rightarrow}}{{q}}_{1}\) in the core was calculated based on the Fourier law \(\stackrel{{\rightarrow}}{{q}}_{1}=-\lambda_{1}\cdot grad[T_{a}+v_{1 }(\eta,\psi)]\) and relationship (6) for \(i=1\). It turned out that, at the singular point (focus), the limit \(\stackrel{{\rightarrow}}{{q}}_{1}\) existed in infinity. Therefore, the \(\eta\) and \(\sinh(n\eta)\) functions were rejected in solution (6) (it was assumed \(B_{1}=B_{n1}=0\) in (6)). Upon taking into account the above remarks and introducing new constants and coefficients, the following reduced solutions were obtained in all conductor zones (in addition, \(g_{2}=g_{3}=0\) occurs in the insulations)  
\[v_{1}(\eta,\psi)=-\frac{g_{1}c^{2}}{8\lambda_{1}}[\cosh(2\eta)+\cos(2\psi)]+A_ {0}+\sum\limits_{n=1}^{\infty}A_{n}\cosh(n\eta)\cos(n\psi)\  
where  
\[f(n)=\cosh^{2}(n\eta_{1})-\frac{\lambda_{1}}{\lambda_{2}}\sinh^{2}(n\eta_{1}), \tag{17}\]  
\[g(n)=\frac{\lambda_{1}-\lambda_{2}}{2\lambda_{2}}\sinh(2n\eta_{1}), \tag{18}\]  
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