# Synchronous Rectification Method for CLLC Resonant Converters Based on State-Trajectory Models
## 1 Introduction
An accurate time-domain modeling method for an asymmetric CLLC resonant converter has been proposed by [20], which achieves accurate SR and a wide voltage regulation range parameter design, but the modeling method is very complicated and is only for the asymmetric CLLC resonant converter, and the computational process is complicated, although sensorless synchronous rectification can be achieved. Some scholars [21] proposed a hybrid modulation digital synchronous rectification algorithm for the CLLC resonant converter based on time-domain modeling, which has a great advantage in the hybrid modulation of the CLLC resonant converter but is not applicable to single frequency control. Although the aforementioned techniques can achieve sensorless SR, their calculation processes remain overly complex. To solve this problem, some scholars have proposed a simplified [22; 23] sensorless SR method by reasonably simplifying the circuit and the calculation process. While these methods reduce the computational complexity, they suffer from poor accuracy. The state-trajectory model can transform abstract state formulas into concrete state trajectories [24], simplify the modeling process using geometric principles, and obtain more accurate results. Therefore, introducing the state-trajectory model into the sensorless SR of the CLLC resonant converter can give full play to the advantages of low computation, low complexity, and high accuracy. The state-trajectory method effectively addresses key challenges in CLLC converter modeling by transforming complex coupled differential equations into intuitive geometric relationships on a voltage-current phase plane, where resonant states appear as circular arcs and their intersections represent mode transitions. This approach provides three significant advantages: (1) substantially reducing the model complexity through the geometric visualization of transient behaviors, (2) enabling the direct extraction of switching timings from the angular and radial parameters of the trajectories, and (3) improving the computational efficiency by eliminating the need for iterative numerical solutions while maintaining modeling accuracy. The simplified CLLC resonant converter model proposed in this paper is developed based on the state-trajectory model. By representing the resonant capacitor voltage and resonant inductor current of the resonant tank on the coordinate axes [22], the operating state of a one-cycle CLLC resonant converter is plotted making the abstract operating state of the circuit visualized and concrete. The delayed on-time and duty cycle of the dynamic synchronous rectifier signals are calculated using essential DC variables such as input voltage, output voltage, and output current resulting in more accurate expressions for the synchronous rectifier method. The sections of this paper are organized as follows:  
In Section 2, the process of synchronous rectification in the CLLC resonant converter is analyzed; the principles and driving signals of the converter for synchronous rectification under both under-resonance (where the resonance point is a special case of under-resonance) and over-resonance conditions are discussed. Section 3 describes the principle of the state-trajectory-based SR scheme proposed in this paper, derives the specific mathematical expressions for the driving signals, and presents the simulation results. Experimental results are provided in Section 4 to validate the synchronous rectification method proposed in this paper. Finally, Section 5 summarizes the paper.