# Trajectory Control Approach for Single-Stage Soft-Switching Grid-Tied Inverters
## 2 Single-Stage Soft-Switching Module-Integrated Grid-Tied Inverters
### Switch Operation Sequences  
The switching sequence depends on the ratio between the switching frequency and the resonant frequency of the filter. The control strategy in this paper operates under continuous conduction mode (DDM) above the resonant frequency of the series resonant filter, using zero-voltage switching on the DC side and zero-current switching on the AC side, functioning as a two-quadrant rectifier. The circuit operation is determined by the state of specific eight MOSFETs, S1, S2, S3, and S4 of the two-quadrant switches on the DC side and St1, St2, Sb1, and Sb2 on the AC side. The upper switches (St1, Sb1) operate during positive grid voltage, while the lower switches are conducting through MOSFET gating or the anti-parallel diode based on operating conditions. During negative grid  
Figure 4: Grid voltage \(v_{ac}\) and current \(i_{ac}\) waveforms; the four quadrants on \(v_{ac}\)-\(i_{ac}\) plane. voltage, the roles are reversed. This switching sequence on the AC side is similar to a cycloconverter, converting high-frequency AC to low-frequency AC. When operating first and fourth quadrants on the \(v_{ac}\)-\(i_{ac}\) plane, MOSFETs St2 and Sb2 on the AC side bridge can remain in a conducting state to minimize switching and conduction losses. Similarly, when operating in the second and third quadrants, the switching operation is applied in the reverse manner. The grid frequency, typically ranging from 50 Hz to 60 Hz, is considerably lower than the switching frequency. Therefore, the circuit operation and the control schemes can be analyzed under quasi-static conditions. This assumption simplifies the modeling process by treating the DC and AC-side voltages as nearly constant relative to the fast-switching dynamics. The switching modes and the trajectories on the \(v_{C}\)-\(i_{L}\) state plane where energy is transferred from the DC side to the AC side in the first quadrant align with the structure of a series resonant converter at the specific operating point [18; 19; 20; 21; 22]. The steady-state switching characteristics and operational modes are initially analyzed based on the behavior of the series resonant converter. The six parameters \(v_{C}\), \(i_{L}\), \(v_{dc}\), \(i_{dc}\), \(v_{ac}\), and \(i_{dc}\) are used to determine the switching modes. In order to simplify the analysis, the parameters are normalized and expressed according to Table 1. The \(v_{dc}\) and \(v_{ac}\) are considered constant in this analysis under quasi-static conditions. The \(v_{dc}\) is represented by a voltage source, \(k_{dc}V_{dc}\), and \(v_{ac}\) is represented by a current-controlled voltage source, \(k_{ac}\frac{V_{dc}}{2N}\). The \(k_{dc}\) varies by controlling the two-quadrant switches on the DC side and the \(k_{ac}\) is determined by the polarity of the inductor current. Figure 5 shows the simplified equivalent circuit of the microinverter with the normalized parameters. The control strategy employs a time-domain based state-plane representation of state variables \(m_{C}\) and \(j_{L}\). Variables \(m_{dc}\) and \(m_{ac}\) change with \(k_{dc}\) and \(k_{ac}\) depending on the operating conditions in a piece-wise linear manner in Equation (3). \[m_{dc}=k_{dc}M_{dc},\ \ m_{ac}=k_{ac}\frac{M_{ac}}{2N}\ \ \ k_{dc},\ k_{ac}=1,-1 \tag{3}\]  
In Equation (3), \(k_{dc}\) and \(k_{ac}\) are 1 or \(-1\), which shift the operational pivots on the \(m_{C}\)-\(j_{L}\) plane and divide the steady-state operation into four distinct switching modes. Figure 6 shows the inductor current and capacitor voltage waveforms for one cycle in the first quadrant of \(m_{ac}\)-\(j_{ac}\). When switching from operation mode 4 to mode 1 for the \(F<1\) CCM operation, the current \(i_{L}\) lags the voltage, so ZVS is achieved by soft-turning on after the anti-parallel diode conducts. As the resonance current increases, energy is accumulated in the resonant filter inductor and the capacitor. When the energy in the filter network reaches a set value, it is hard-turned off and switched to mode 2.