## 4 Construction of Electromagnetic-Force-Flow Coupling Platform for Converter Transformer
### The Establishment of Electromagnetic-Force-Flow Coupling Relationship  
According to the actual closing conditions of the converter transformer, the current amplitude of the inrush current was calculated, and the current was applied to the winding coil of the converter transformer as the excitation source. The current density \(\mathbf{J}_{\mathrm{e}}\) in the winding can be obtained by using Formula (18).  
\[\mathbf{J}_{\mathrm{e}}=\frac{\mathbf{i}_{\mathrm{t}}}{S} \tag{18}\]  
In the formula, \(\mathbf{J}_{\mathrm{e}}\) is the current vector density of the winding; \(\mathbf{i}_{\mathrm{t}}\) is the instantaneous value of the magnetizing inrush current flowing through the winding; and \(S\) is the sectional area of the winding wire. The equation of the unit micro-element on the winding subjected to the electric force \(\mathbf{F}_{\mathrm{v}}\) under the excitation current is shown in Formula (19).  
\[\mathbf{F}_{\mathrm{v}}=\mathbf{B}\times\mathbf{J}_{\mathrm{e}} \tag{19}\]  
In the formula, \(\mathbf{F}_{\mathrm{v}}\) is the electric force of the winding micro-element under the inrush current and \(\mathbf{B}\) is the magnetic induction intensity. When the winding is in a balanced state, the sum of the gradient of the internal stress and the external applied force is zero. Only considering the influence of the electric force on the winding, the internal force of the winding should meet the balance equation as shown in Equation (20).  
\[0=\
abla\cdot S+\mathbf{F}_{\mathrm{v}} \tag{20}\]  
The relationship between the local velocity \(\mathbf{u}_{2}\) and the average velocity \(\mathbf{u}_{\mathrm{tr}}\) transmitted by the vibration signal generated by the winding in the oil reflects the distribution of the velocity in the fluid, which is considered to satisfy Equation (21).  
\[\mathbf{u}_{2}=\mathbf{u}_{\mathrm{tr}} \tag{21}\]  
The motion of insulating oil satisfies the momentum conservation equation. The left term represents the acceleration of insulating oil, the right term represents the contribution of oil flow pressure gradient and viscous force and the external force \(\mathbf{F}_{\mathrm{o}}\) also affects the acceleration of insulating oil. The specific equations are shown in Equations (22) and (23).  
\[\rho(\mathbf{u}\cdot\
abla)\mathbf{u}=\
abla\cdot(-p\mathbf{I}+\mathbf{K})+\mathbf{F}_{\mathrm{o}} \tag{22}\]  
\[\mathbf{K}=\mu[\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}] \tag{23}\]  
In the formula, \(\rho\) is the density of insulating oil; \(\mathbf{u}\) is the speed of insulating oil in the box and tubing; \(p\) is the pressure of insulating oil; \(\mathbf{I}\) is the unit matrix, \(\mathbf{K}\) is the stiffness matrix and \(\mu\) is the dynamic viscosity coefficient of insulating oil.  
At the same time, the internal insulating oil of the converter transformer also satisfies the mass conservation equation, that is, the density of the insulating oil multiplied by the divergence of its flow velocity is zero, which means that when the insulating oil flows, the mass will not be created or disappeared, and will only be redistributed inside the box and tubing. The specific equation is shown in Equation (24).  
\[\rho\
abla\cdot\mathbf{u}=0 \tag{24}\]  
Considering the influence of pressure and fluid flow on the stiffness of the equipment, the expression of the force \(\mathbf{F}_{\mathrm{A}}\) applied by the fluid to the converter-transformer component is shown in Equation (25).  
\[\mathbf{F}_{\mathrm{A}}=(-p\mathbf{I}+\mathbf{K})\cdot\mathbf{n} \tag{25}\]  
In the formula, \(\mathbf{F}_{\mathrm{A}}\) represents the force of fluid on solid; \(n\) is a normal vector.