# Calculation of Main Circuit Steady-State Parameters for Capacitor Commutated Converter System
## 3 Calculation Model and Method of the Main Circuit Parameters
### Determination of the Capacitance Value
#### 3.1.2 Steady-State Mathematical Model of the CCC  
Taking the trigger of valve 1 as an example, the instantaneous line-to-neutral AC source voltages are:  
\[\left\{ \begin{array}{l}u_{\rm a}=\frac{\sqrt{2}}{\sqrt{3}}E\sin(\omega t+ \alpha+\frac{\pi}{6})\\ u_{\rm b}=\frac{\sqrt{2}}{\sqrt{3}}E\sin(\omega t+\alpha-\frac{\pi}{2})\\ u_{\rm c}=\frac{\sqrt{2}}{\sqrt{3}}E\sin(\omega t+\alpha+\frac{5\pi}{6})\\ \end{array} \right. \tag{4}\]  
where \(E\) is the root mean square (rms) value of the line-to-line voltage at the valve side. In the interval \([0,\mu]\), valve 5 is commutated to valve 1 and valve 6 is on. The equivalent circuit of the converter is shown in Figure 5.  
According to [PERSON]'s current law, the currents satisfies:  
\[i_{\rm a}+i_{\rm c}=I_{\rm d} \tag{5}\]  
Integrate both sides of (5) to obtain:  
\[\frac{1}{\omega C}\int_{0}^{\mu}i_{\rm a}d\theta+\frac{1}{\omega C}\int_{0}^{ \mu}i_{\rm c}d\theta=\frac{1}{\omega C}\int_{0}^{\mu}I_{\rm d}d\theta \tag{6}\]  
According to (1) and (2), the relationship between \(\Delta U_{1}\) and \(\Delta U_{2}\) is:  
\[\Delta U_{1}+\Delta U_{2}=\frac{\mu I_{\rm d}}{\omega C} \tag{7}\]  
Figure 5: Equivalent circuit of the commutation process in the interval \([0,\mu]\). Substituting (7) into (3), the peak capacitor voltage \(U_{\rm c}\) is:  
\[U_{\rm c}=\frac{\pi I_{\rm d}}{3\omega C} \tag{8}\]  
According to [PERSON]'s voltage law, the differential equation of the commutation process can be derived as:  
\[u_{\rm a}(t)-u_{\rm c}(t)=L_{\rm a}\frac{di_{\rm a}}{dt}+u_{\rm ca}(t)-L_{\rm c }\frac{di_{\rm c}}{dt}-u_{\rm cc}(t) \tag{9}\]  
Differentiate both sides of (9) to obtain:  
\[\frac{d^{2}i_{\rm a}}{dt^{2}}+\frac{1}{LC}i_{\rm a}=\frac{1}{2 LC}I_{\rm d}+ \frac{1}{2L}\big{(}u_{\rm a}{}^{\prime}(t)-u_{\rm c}{}^{\prime}(t)\big{)} \tag{10}\]  
The solution of (10) is:  
\[i_{\rm a}(t)=A\cos(\omega_{0}t)+B\sin(\omega_{0}t)+\frac{1}{2}I_{\rm d}+\frac{ \sqrt{2}\omega}{2L(\omega_{0}^{2}-\omega^{2})}E\cos(\omega t+\alpha) \tag{11}\]  
where \(\omega_{0}\) is the angular frequency of oscillation during the commutation process, satisfying:  
\[\omega_{0}=\sqrt{1/LC} \tag{12}\]  
The boundary conditions are:  
\[\left\{ \begin{array}{l}i_{\rm a}(\omega t=0)=0\\ i_{\rm a}(\omega t=\mu)=I_{\rm d}\end{array} \right. \tag{13}\]  
Substitute (13) into (11) to obtain:  
\[A+\frac{1}{2}I_{\rm d}+\frac{\sqrt{2}\omega}{2L(\omega_{0}^{2}-\omega^{2})}E \cos\alpha=0 \tag{14}\]  
\[A\cos(\frac{\omega_{0}}{\omega}\mu)+B\sin(\frac{\omega_{0}}{\omega}\mu)-\frac {1}{2}I_{\rm d}+\frac{\sqrt{2}\omega}{2L(\omega_{0}^{2}-\omega^{2})}E\cos(\mu +\alpha)=0 \tag{15}\]  
When \(\omega t=0\), substitute (11) into (9) to obtain:  
\[\frac{1}{2L}\big{(}\frac{2\pi I_{\rm d}}{3\omega C}-\Delta U_{2}\big{)}-B \omega_{0}+\frac{\sqrt{2}\omega_{0}^{2}}{2L(\omega_{0}^{2}-\omega^{2})}E\sin \alpha=0 \tag{16}\]  
When \(\omega t=\mu\), substitute (11) into (9) to obtain:  
\[\frac{1}{2L}\big{(}\frac{2\pi I_{\rm d}}{3\omega C}-\Delta U_{1}\big{)}+A \omega_{0}\sin(\frac{\omega_{0}}{\omega}\mu)-B\omega_{0}\cos(\frac{\omega_{0} }{\omega}\mu)+\frac{\sqrt{2}\omega_{0}^{2}}{2L(\omega_{0}^{2}-\omega^{2})} \frac{\sqrt{2}}{\sqrt{3}}E\sin(\mu+\alpha)=0 \tag{17}\]  
The DC voltage of the six-pulse CCC is:  
\[u_{\rm d}=\left\{ \begin{array}{ll}-\frac{1}{2}u_{\rm a}-\frac{1}{2}u_{\rm c }+u_{\rm b}-\frac{3\Delta U_{2}}{2}+\frac{3\theta I_{\rm d}}{2\omega C}&\theta \in(0,\mu)\\ -u_{\rm a}+u_{\rm b}-\frac{\pi I_{\rm d}}{3\omega C}-2\Delta U_{2}+\frac{2 \theta I_{\rm d}}{\omega C}&\theta\in(\mu,\frac{\pi}{3})\end{array} \right.