# Determination of the Operating Temperature of the Gas-Insulated Transmission Line +
## 2 Magneto-Thermal Modeling
 \tag{11}\]  
In the above formulas, \(I_{0}\!\left(\Gamma r\right)\), \(K_{0}\!\left(\Gamma r\right)\), \(I_{1}\!\left(\Gamma R_{1}\right)\), \(K_{1}\!\left(\Gamma R_{1}\right)\) and \(I_{1}\!\left(\Gamma R_{2}\right)\), \(K_{1}\!\left(\Gamma R_{2}\right)\) are the modified Bessel functions of the first and second kind and order 0 or 1 [32] and the \(\underline{\Gamma}\) is the complex propagation constant in the phase conductor:  
\[\Gamma=\sqrt{\mathrm{j}\omega\mathrm{\mu_{0}}\gamma}=k+\mathrm{j}k \tag{12}\]  
in which \(k\) is the attenuation constant  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]  
where \(\delta\) is the electrical skin depth of the electromagnetic wave penetration into the conducting environment, \(\omega\) is the angular frequency, \(\gamma\) stands for the electrical conductivity of the phase conductor and \(\mathrm{\mu_{0}}=4\pi\ 10^{-7}\ \mathrm{H}\cdot\mathrm{m}^{-1}\) is the magnetic permeability of the vacuum. In turn, power losses in the outer screens (E1 and E3) were almost the same and can be written down as follows [29]:  
\[P_{s13}=P_{e0}+P_{e13} \tag{14}\]  
where  
\[P_{e0}=\frac{\Gamma_{e}^{*}\,{I_{1}}\,{I_{1}}^{2}}{4\,\pi\,\gamma_{e}\,\beta _{e}^{2}\,R_{4}}\frac{a}{d_{0}} \tag{15}\]  
in which  
\[\begin{array}{ll}a_{0}&=b\,\,b^{*}\!\  
\[\beta_{e}=\frac{R_{3}}{R_{4}}\ (\ 0\leq\beta_{e}\leq 1) \tag{23}\]  
Power \(P_{e13}\) has a following form  
\[P_{e13}=\frac{\Gamma\ ^{*}\ l\ l_{1}^{2}}{2\ \pi\ \gamma\ R_{4}}\sum_{n=1}^{ \infty}A_{n}^{2}\left(\frac{R_{4}}{d}\right)^{2n}\frac{a_{ne}}{b\ \ \ \ b^{*}_{ne}} \tag{24}\]  
where  
\[A_{n}=-\frac{1}{2}\Big{[}(1+2^{-n})+{\rm j}\ \sqrt{3}\ (1-2^{-n})\Big{]}=A_{n} \exp[{\rm j}\ \varphi_{n}] \tag{25}\]  
\[b_{ne}=I_{n-1}(\Gamma\ R_{4})\ K_{n+1}(\Gamma\ R_{3})-I_{n+1}(\Gamma\ R_{3}) \ K_{n-1}(\Gamma\ R_{4}) \tag{26}\]  
\[b^{*}_{ne}=I^{*}_{n-1}(\Gamma\ R_{4})\ K^{*}_{n+1}(\Gamma\ R_{3})-I^{*}_{n+1}( \Gamma\ R_{3})\ K^{*}_{n-1}(\Gamma\ R_{4}) \tag{27}\]  
\[a_{ne}=I_{n+1}(\Gamma\ R_{4})\ K_{n+1}(\Gamma\ R_{3})\left[I^{*}_{n+1}(\Gamma \ R_{3})\ K^{*}_{n}(\Gamma\ R_{4})+I^{*}_{n}(\Gamma\ R_{4})\ K^{*}_{n+1}( \Gamma\ R_{3})\ \right]\]  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]0  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]1  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]2  
Power losses in the middle screen (E2) are [29]  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]3  
where  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]4  
and  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]5  
In the thermal analysis, the material properties, such as electrical conductivity, are temperature-dependent; therefore, nonlinear thermal analysis was performed. The electrical conductivity can be expressed by the following equation [33]:  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]6  
where: \(\gamma_{20}\) is the electrical conductivity at 20 \({}^{\circ}\)C and \(\alpha_{20}\) is the temperature coefficient. The heat generated in the high-current bustuct is transferred to the ambient air by natural convection and radiation on the internal surface of the enclosure. Therefore, the total heat transfer coefficient \(\alpha\) is a sum of the radiation \(\alpha_{\rm r}\) and convection \(\alpha_{\rm c}\) heat transfer coefficient. If radiation heat flow occurs between two surfaces \(F1\) and \(F2\), then the radiation heat transfer coefficient can be calculated by [33]:  
\[k=\sqrt{\frac{\omega\mathrm{\mu_{0}}\gamma}{2}}=\frac{1}{\delta} \tag{13}\]7  
where \(\varphi_{FF1}\) is a configuration coefficient of surfaces \(F1\) and \(F2\). The configuration coefficient depends on the shape and the mutual location of the surfaces. In the case of the tubular high-current bustuct, surface \(F1\) is placed concentric to surface \(F2\) (Figure 2).