## 2 Generation of Converter Transformer Inrush Current and Its Time-Frequency Characteristics  
### Generation Mechanism of No-Load Inrush Current  
Due to the deep saturation of the core during the closing of the converter transformer, the excitation impedance is reduced, resulting in an internal inrush current. The no-load closing equivalent circuit of the converter transformer is shown in Figure 1. The mechanism of the magnetizing inrush current is as follows:\(r_{\rm s}\) and \(L_{\rm s}\) are the resistance and inductance of the AC power supply; \(r_{\sigma}\) and \(L_{\sigma}\) are the resistance and inductance of the primary side of the converter transformer and \(r_{\rm m}\) and \(L_{\rm m}\) are the excitation resistance and excitation inductance of the converter transformer. Suppose the power supply voltage is \(u_{\rm s}(t)=U_{\rm m}\sin(\omega t+\theta)\), the total resistance of the circuit is \(r=r_{\rm s}+r_{\sigma}+r_{\rm m}\), the total inductance of the circuit is \(L=L_{\rm s}+L_{\sigma}+L_{\rm m}\), and the circuit satisfies Formula (1).  
\[L\frac{\mathrm{d}i}{\mathrm{d}t}+ri=U_{\rm m}\sin(\omega t+\theta) \tag{1}\]  
In the formula, \(i\) is the excitation current; \(t\) is the closing time of converter transformer; \(u_{\rm m}\) is the amplitude of the power supply voltage and \(\omega\) is the angular velocity of the converter transformer under power frequency. When the saturation degree of the core deepens, both \(x_{\rm m}\) and \(r_{\rm m}\) decrease with the saturation of the magnetic circuit. Since \(r_{\rm m}=2\pi fL_{\rm m}\), \(L\) and \(r\) decrease. Since \(u_{\rm s}\) is constant, the value of the excitation current \(i\) will rise sharply.  
When the core of the converter transformer is unsaturated, the relationship between the flux linkage \(\psi\) and the excitation current \(i\) is \(\psi=Li\). By using [PERSON]'s voltage law, we can obtain  
\[\Psi(t)=\Psi_{\rm m}\sin(\omega\,t+\theta-\varphi)+[\,\Psi_{\rm r}-\Psi_{\rm m }\sin(\theta-\varphi)]e^{-\frac{t}{L}t} \tag{2}\]  
In the formula, \(\Psi_{\rm m}\) is the excitation flux linkage, and its expression is \(\Psi_{\rm m}=\frac{LI_{\rm m}}{\sqrt{t^{2}+(\omega L)^{2}}}\); \(\varphi=\arctan\frac{\omega L}{r}\) is the closing angle of the converter transformer; \(\Psi_{\rm r}\) is the residual magnetism inside the core of the converter transformer; \(\omega\) is the power frequency angular velocity; \(\theta\) is the circuit; \(L\) is the total inductance of the circuit; and \(r\) is the total resistance of the circuit. Because the reactance inside the circuit is much larger than the resistance, and when the closing angle \(\varphi=90^{\circ}\), Equation (2) is  
\[\Psi=-\Psi_{\rm m}\cos(\omega t+\theta)+(\Psi_{\rm m}\cos\theta+\Psi_{\rm r})e ^{-\frac{t}{L}t} \tag{3}\]  
The magnetization curve is approximately regarded as two straight lines connected, and the formula of the amplitude of the excitation current is  
\[i=\frac{\Psi-\Psi_{\rm s}}{L_{\rm m}}=\frac{-\phi_{\rm m}\cos(\omega t+\theta )+(\phi_{\rm m}\cos\theta+\phi_{\rm r})e^{-\frac{t}{L}t}-\phi_{\rm s}}{L_{\rm m }/N} \tag{4}\]  
In the formula, \(\Psi_{\rm s}\) is the flux linkage corresponding to the knee point; \(\Phi_{\rm m}\) is the excitation flux; \(\Phi_{\rm r}\) is the remanence of iron core; \(\Phi_{\rm s}\) is the knee-point flux; and \(N\) is the number of coil turns.  
Figure 1: Single-phase converter transformer no-load closing circuit diagram.