## 3 Control Loop  
The circuit block diagram of the control loop for the battery charger is illustrated in Figure 3. It is composed of the current shunt, voltage amplifier, voltage divider, error amplifiers, PSFB controller, and isolated gate driver. In the voltage and current feedback loops, the error amplifiers are used to control the PSFB converter, as illustrated in Figure 4. The circuit is composed of the amplifiers (EA\({}_{\rm v}\) and EA\({}_{\rm c}\)), impedance elements (\(Z_{\rm v1}\), \(Z_{\rm v2}\), \(Z_{\rm c1}\), and \(Z_{\rm c2}\)), and a switch S\({}_{\rm w}\). The voltage signal \(V_{o\_em}\) obtained from the voltage divider is proportional to the charger's output voltage \(V_{o}\), while the output voltage \(V_{io\_sen}\) from the voltage amplifier is proportional to the charger's output current \(I_{o}\), as shown in Figure 5 [16]. Moreover, \(V_{ref\_vo}\) serves as the reference command for the charger to op  
Figure 3: Circuit block diagram of the control loop for the battery charger. charging mode, and \(V_{env}\) represents the output voltage from the voltage-loop error amplifier. Similarly, \(V_{ref\_io}\) is the reference command for constant-current charging mode, and \(V_{exc}\) denotes the output voltage of the current-loop error amplifier. Based on \(V_{env}\) and \(V_{exc}\), along with their small-signal perturbations, the input to the PSFB controller adjusts the phase-shifted angle of \(v_{ga}\)-\(v_{gd}\), thereby enabling the charger to maintain either a constant-voltage or constant-current charging mode. When the charger is operated in constant-voltage charging mode, the switch S\({}_{w}\) remains in the off state; conversely, during constant-current mode operation, S\({}_{w}\) is turned on. In Figure 4, the impedance components \(Z_{c1}\) and \(Z_{c2}\) can be configured as a resistor-capacitor network to realize PI control, which increases the DC gain and eliminates the steady-state error. Additionally, the system crossover frequency is appropriately set to meet the required phase margin [20,21]. At the initial start-up of the charger, the charger's output voltage \(V_{o}\) gradually increases due to the soft-start mechanism, initially reaching the constant-voltage output mode before transitioning into constant-current output mode. The corresponding voltage waveforms of the error amplifiers vary over time, as illustrated in Figure 5, where \(V_{env}\) denotes the voltage at the inverting terminal of the EA\({}_{c}\). Figure 4: Voltage and current feedback loops used error amplifiers. Figure 5: The error amplifier used PI control resulted in current overshoot [16]. Since the input impedance of the error amplifier is ideally infinite, \(V_{em}\cong V_{i_{o},sen}\). As \(V_{em}\) varies in direct proportion to \(V_{i_{o},sen}\) and \(I_{o}\), the operating principle is described as follows:  
* When \(V_{em}<V_{ref,i_{o}}\), the EA\({}_{\text{c}}\) output voltage reaches its positive saturation, resulting in \(V_{eac}=V_{eac(\text{max})}\). * As \(I_{o}\) increases and reaches the expected constant current \(I_{o,cc}\), the \(V_{em}\) reaches \(V_{ref,i_{o}}\), prompting the current-loop error amplifier to enter its feedback operation. Due to the relatively slow response of the PI-controlled error amplifier, \(V_{eac}\) gradually decreases over a delay period \(t_{d}\) until it reaches the expected control voltage \(V_{exp,cc}\) for constant-current charging. This delay results in a current overshoot \(I_{o,ss}\), as illustrated in Figure 5.
* This current overshoot \(I_{o,ss}\) may trigger the charger overcurrent protection; therefore, it must be minimized. Although a proportional-integral-derivative control can be employed to accelerate the system response, the converters presented in [22; 23; 24] did not address the suppression of the output current overshoot.