## 3 Operating Principle and Performance of DCM-CRM Cuk PFC Converter Utilizing VI Control  
### Operating Principle of VI  
The principle of the variable inductor is based on the core structure characteristics, wherein a bias winding is integrated with the main winding of the power inductor. By injecting DC current through the bias winding, the saturation level of certain sections of the magnetic circuit can be controlled. This, in turn, alters the magnetic permeability of those sections, thereby allowing the overall inductance value to be adjusted. The variable inductor employed in this study utilizes an EI core, with its basic schematic diagram presented in Figure 4. The main winding of the input inductor is wound with \(n_{3}\) turns on the center leg of the EI core, while an air gap of \(l_{0}\) is introduced. To control the inductance, auxiliary windings are symmetrically wound on both outer legs of the EI core. Each auxiliary winding consists of an equal number of turns, denoted as \(n_{1}\) and \(n_{2}\), respectively. An air gap is introduced in the center leg of the EI core, altering its permeability. The relative permeability of the center leg is denoted as \(\mu_{3}\), while the permeability of the air gap is represented by \(\mu_{0}\). To minimize the induced electromotive force in the auxiliary windings caused by changes in the main winding current, the auxiliary windings are wound with an equal number of turns on the left and right outer legs of the core. These windings are connected in series with opposite polarities. The relative permeabilities of the left and right outer legs are denoted as \(\mu_{1}\) and \(\mu_{2}\), respectively. In Figure 4, \(l_{3}\), \(l_{1}\), \(l_{2}\), and \(l_{0}\) represent the lengths of the center leg, the left outer leg, the right outer leg, and the air gap, respectively, where \(l_{1}=l_{2}\). The areas of the center leg, left outer leg, and right outer leg are denoted as \(A_{3}\), \(A_{1}\), and \(A_{2}\), respectively. The cross-sectional area of the air gap is represented as \(A_{0}\), with \(A_{1}=A_{2}\), \(A_{3}=A_{0}\). The magnetic fluxes in the center leg, left outer leg, and right outer leg are denoted as \(\Phi_{3}\), \(\Phi_{1}\), and \(\Phi_{2}\), respectively. According to [PERSON]'s second law, the relationship between the magnetomotive force \(F\) and magnetoresistance \(R\) can be obtained. \[F=iN=Hl=\Phi R \tag{21}\]  
According to the formula derived from the variable inductance flux theory in [22], the magnetic flux of the main winding of the variable inductance of the EI magnetic core can be obtained as follows  
\[\Phi_{3}=\frac{(R_{2}-R_{1})n_{1}i_{1}+(R_{2}+R_{1})n_{3}i_{3}}{R_{1}R_{2}+(R_ {1}+R_{2})(R_{3}+R_{0})} \tag{22}\]  
According to Equation (22), the fundamental expression for the variable inductor model depicted in Figure 4 can be derived as follows  
\[L_{V}=\frac{n_{3}\Phi_{3}}{i_{3}}=\left[(\frac{l_{1}}{2\mu_{0}\mu_{\rm var}A_ {1}n_{3}^{2}})+(\frac{l_{3}}{\mu_{0}\mu_{3}A_{3}n_{3}^{2}})+(\frac{l_{0}}{\mu _{0}A_{3}n_{3}^{2}})\right]^{-1} \tag{23}\]  
When selecting the magnetic core, the number of winding turns, the effective length of the magnetic circuit, and the effective cross-sectional area for magnetic flux are predetermined. Once the air gap \(l_{0}\) is determined, only the core permeability, \(\mu_{\rm var}=\mu_{1}=\mu_{2}\), is variable, and it changes in response to the bias current \(i_{\rm bias}\). When \(i_{\rm bias}\) is applied to \(n_{1}\) and \(n_{2}\), a bias magnetic flux \(\Phi_{\rm bias}\) is established along the outer legs of the EI core. As \(\Phi_{\rm bias}\) increases, the permeability of the core along this path decreases, and the magnetic flux \(\Phi_{\rm bias}\) shifts the operating point on the B-H curve toward the nonlinear saturation region. Concurrently, the passage of \(i_{\rm LV}\) through \(n_{3}\) induces a main magnetic flux \(\Phi_{\rm LV}\).