# Trajectory Control Approach for Single-Stage Soft-Switching Grid-Tied Inverters
## 3 State Trajectory Control  
### Steady State Operation  
Resonant operations in the filter were analyzed using the state equations in the \(m_{C}\)-\(j_{L}\) state plane. The trajectory control method directly controls the energy state of the resonant filter circuit in the time domain to reach a steady state with minimal switching. In switching using the trajectory control method, a steady state is reached with minimal switching, and the energy amount of the resonance circuit is directly controlled so that it is possible to respond immediately to the situation in the resonance circuit. The state differential equation obtained from the equivalent is as follows:  
\[L\frac{di_{L}}{dt}=k_{dc}V_{dc}-k_{ac}\frac{V_{ac}}{2N}-v_{C},\ \ \ C\frac{dv_{C}}{dt}=i_{L} \tag{4}\]  
The state-transition matrix is derived as follows. \[\frac{dX}{dt}= \begin{bmatrix}0&-1/L\\ 1/C&0\end{bmatrix} X+ \begin{bmatrix}1/L&-1/2 NL\\ 0&0\end{bmatrix} U \tag{5}\]  
where \(X=[i_{L}v_{C}]^{\mathrm{T}}\) is the state vector and \(U=[k_{dc}V_{dc}k_{ac}V_{ac}]^{\mathrm{T}}\) is the source vector. The initial conditions of the current of the filter inductor and the voltage of the filter capacitor are defined as follows:  
\[i_{L}(t_{0})=I_{L0},\ \ v_{c}(t_{0})=V_{C0} \tag{6}\]  
The AC-side bridge switches when the inductor current is zero, so \(I_{L0}\) is negligible in the equation. Using the defined initial conditions, the normalized filter inductor current \(j_{L}\) and capacitor voltage \(m_{C}\) expressions are derived as Equations (7) and (8). \[m_{C}(t)=M_{F}+\big{(}M_{C_{0}}-M_{F}\big{)}cos(\omega_{0}(t-t_{0}))+J_{L0}sin (\omega_{0}(t-t_{0})) \tag{7}\]  
\[j_{L}(t)=J_{L0}cos(\omega_{0}(t-t_{0}))+(M_{C0}-M_{F})sin(\omega_{0}(t-t_{0})) \tag{8}\]  
where \(M_{F}\)= \(k_{dc}-k_{ac}\frac{M_{ac}}{2N}\). The state plane trajectory of the \(j_{L}\) and \(m_{c}\) in the resonant filter forms circular trajectory centered at the pivots \(P=(M_{F},\ 0)\) at each switching mode. The pivots of the circular motion move four times during one switching cycle. The radius of the circle \(R_{d}\) is determined by the initial conditions and remain constant during the switching mode. \(R_{d}\) on the state plane can be derived as follows:  
\[R_{d}^{2}=\{M_{C0}-M_{F}\}^{2}+J_{L0}^{2} \tag{9}\]  
Figure 6: Filter voltage and current \(m_{C}\), \(j_{L}\) waveforms in first quadrant; the \(m_{C}\)–\(j_{L}\) plane. Depending on the switching mode, \((k_{dc},\ k_{ac})\) and the location of the pivot in the \(m_{C}\)-\(j_{L}\) state plane is determined as shown in Table 2. In Figure 7, the \(m_{C}\)-\(j_{L}\) state plane trajectories are seen during the steady-state operation in the first and third quadrants. In the first and third quadrants of the \(v_{ac}\)-\(i_{ac}\) state plane, energy is transferred from the DC side to the AC side, with the pivot circulating in the sequence \(P_{1}\)-\(P_{2}\)-\(P_{3}\)-\(P_{4}\). In the second and fourth quadrants of the \(v_{ac}\)-\(i_{ac}\) state plane, energy is transferred from the AC side to the DC side, and the pivot circulates in the reverse sequence, \(P_{4}\)-\(P_{3}\)-\(P_{2}\)-\(P_{1}\). The amount of change in the capacitor voltage \(v_{C}\) represents the charge stored in the capacitor, and this is directly related to the load current which is the average value of the inductor current in a series resonant configuration. The exact closed-form solution of the angle \(\gamma\) between the state planes when operating in the first quadrant follows. \[\gamma=\pi-cos^{-1}\left(\frac{4-\left(R_{ref}-M_{ac}\right)^{2}-R_{ref}^{2}}{ -2\left(R_{ref}-M_{ac}\right)R_{ref}}\right) \tag{10}\]  
The grid side voltage and the inductor current are in phase, and the AC side current \(i_{ac}\) coincides with the half-cycle average of the inductor filter current during steady state operation.