## 1 Introduction
However, it is challenging for the converter to achieve both high PF and low intermediate capacitor voltage simultaneously during normal operation and the converter can only operate under low input voltage [16; 17; 18]. In [19], a Cuk PFC converter utilizing a variable inductor is proposed, which makes both output and input inductors operate in DCM. The implementation of variable inductor (VI) control effectively enhances the PF of the converter. However, at high input voltages, the peak current in the output inductor becomes significantly large, which increases the current stress on the switch, ultimately reducing the efficiency of the converter. Variable inductor technology has changed the characteristics of inductor components, transforming the inductor from a fixed inductance component to a variable inductance component. VI technology can add an extra control variable to the switching power supply circuit, allowing the switching power supply to change from only adjusting the duty cycle to simultaneously adjusting both the duty cycle and power inductance. Therefore, VI technology has attracted considerable interest from many researchers. Recent advancements in science and technology have spurred rapid development in magnetic materials, leading researchers to propose various types of variable inductors. Commonly used variable inductors include DC auxiliary winding types such as the toroid core VI, double-E core VI, and quad-U core VI [20; 21; 22; 23; 24]. These designs utilize DC magnetic flux to drive core saturation, thereby altering the core equivalent magnetic permeability to adjust the inductance value. A variable inductor offers inherent flexibility of the inductance variation, which allows converters to handle more complex operating conditions. Variable inductors have been extensively applied in many cases such as PFC converters, electric vehicles, electronic ballasts, and LED drivers [24; 25; 26; 27; 28; 29; 30]. In order to improve the current handling ability and reduce the size of the magnetic component of the power inductor, a toroid core variable inductor, which enhances the inductance value by de-saturating the core with DC magnetic flux generated by a direct current, is proposed to replace the conventional power inductor in the bidirectional DC-DC converter in [24]. In [25], a double-E core type variable inductor is employed in a critical conduction mode (CRM) Boost PFC circuit proposed to change the inductance value, thereby increasing the switching period, minimizing switch turn-off losses, and enhancing system efficiency. In [26], a new analog control method is utilized to improve the PF of Boost PFC operated at CRM. It has been pointed out that the poor PF is due to the input current phase lead caused by the presence of an input capacitor. It can increase PF by producing a lagging current utilizing a variable inductor to compensate for the input current phase lead. In [28], the power of the fluorescent lamp is adjusted by controlling the inductance value to achieve the purpose of dimming, thereby enhancing the linear dimming range and achieving high efficiency and high PF. A unidirectional resonant switched-capacitor step-up converter for OLED driving is proposed in [29]. This converter combines a variable inductor with a capacitor in series to control the OLED current and adjust the brightness. Variable inductor technology is a technique that can change the magnitude of inductance by controlling the saturation level of the inductance core to operate the inductance in a nonlinear region, thereby changing the magnetic permeability of the magnetic circuit and achieving changes in inductance value. In recent years, some scholars have also conducted research on inductors working in nonlinear regions in switch-mode power supplies and proposed corresponding analysis methods to better understand their working characteristics and their effect on converter systems.