Cuk PFC Converter Based on Variable Inductor

Abstract

When the input inductor operates in discontinuous current mode (DCM), the Cuk converter can automatically achieve power factor correction (PFC) function with only a simple voltage mode control loop. However, the conventional Cuk PFC converter suffers from high intermediate capacitor voltage because of the lack of feedback of the intermediate capacitor voltage and relatively low power factor (PF). In this paper, a Cuk PFC converter using variable inductor which varies with the transient rectified input voltage is proposed to enhance the PF and reduce the intermediate capacitor voltage by injecting a controlled DC bias current into the auxiliary winding of the variable input inductor. The operating principles of the proposed Cuk PFC converter based on variable inductor are analyzed in detail, and the analysis of PF, the voltage of intermediate capacitor, and design considerations are provided. To verify the feasibility of the proposed scheme and compare the characteristics of both the traditional and proposed Cuk PFC converter, a 108W experimental prototype of the proposed converter is built and tested. The experimental results show that the proposed Cuk PFC converter can significantly enhance the PF, decrease the intermediate capacitor voltage, and increase efficiency compared with the traditional Cuk PFC converter.

1. Introduction

The increasing popularity and widespread use of various power electronic devices in power grids has resulted in more and more serious harmonic pollution, so power factor correction (PFC) converters have become particularly essential in the AC–DC conversion [1,2,3,4,5]. Among the commonly used PFC converters, the Boost PFC converter has the advantages of high power factor (PF) and small total harmonic distortion (THD) [2,3,4,5]. However, the Boost converter can only operate when the output voltage is higher than the input voltage, so it is hard to be used in the applications requiring low output voltage. The Buck PFC converter can meet the low output voltage requirement, but the significant harmonic distortion occurs in the input current because of the dead time when the transient rectified input voltage is smaller than output voltage [5,6,7]. Buck-Boost PFC can achieve the wider output voltage range, yet its input current ripple is large which increases the difficulty in designing the input LC filter, and the twice line frequency output voltage ripple is big [8,9].

Slobodan Cuk of California Institute of Technology proposed and studied the Cuk converter which can achieve the regulation of the output voltage which is lower or higher than the input voltage [10]. As a fourth-order converter, the input stage of the Cuk circuit is comparable to the Boost circuit; the output stage of the Cuk circuit is similar to the Buck circuit, and the current ripples of input current and output current are small. Therefore, the Cuk converter is suitable for a PFC converter to achieve low input current ripple and output voltage ripple. The Cuk PFC converters are widely used in different applications such as LED drivers, motor drives, chargers for plug-in electric vehicles, etc. [11,12,13,14,15,16].

A Cuk PFC converter with the switched inductor technology is proposed in [12], which has the advantages of high voltage gain, low current stress, and high efficiency, but it has the large twice line frequency ripple of output current from the experimental results. An LED driving circuit based on the DCM Cuk converter is proposed in [13], and it can achieve PFC and regulate LED current at the same time, but the twice line frequency output current ripple is too large. In [16], a peak and valley current control method for Cuk PFC converters is proposed to realize the electrolytic capacitors elimination, which uses a combination of digital and analog control to reduce the output filter capacitor volume by making the ripple power buffered on the intermediate capacitor through the larger voltage fluctuations on the intermediate capacitor and avoiding the ripple power from entering the output side. However, the voltage of the intermediate capacitor is high at 300 V with 110 Vac input voltage, and an intermediate capacitor voltage that is too high affects the method in [16] to use for the applications of 220 Vac input voltage. The Cuk PFC converter with decoupling diode is proposed in [10], which will facilitate the reduction of the twice line frequency output current ripple, but the distortion of the input current can be found from the input current expression equation in [10]. In recent years, bridgeless Cuk PFC converters have attracted a lot of attention because of low power losses [17,18,19]. However, the cost and size are enhanced because the number of input inductors is increased, and the converter only operates at low input voltage according to the verification results in [17,18,19], so the problem of the intermediate capacitor’s high voltage still exists. According to the above-mentioned literature on Cuk PFC converters, it is hard to obtain high power factor, low intermediate capacitance voltage, and small output voltage ripple simultaneously, so it is important to study a simpler and more reliable solution to improve the performance of Cuk PFC converters.

Variable inductor is a new technology to control the saturation level of the inductor core for the purpose of changing the inductor value. In terms of variable inductor core structure, the most commonly used magnetic cores are the toroid core, the quad-U core, and the double-E core [20,21,22,23]. In order to make more effective use of variable inductance technology, the operation principle of the variable inductor with double-E core was analyzed in detail [23]. The model and simulation of variable inductor based on the double-E core were deduced by SPICE to optimize the design approach, which combined magnetic and electrical behavior [24,25,26]. To improve the calculation accuracy of variable inductor reluctance paths, more factors have been taken into account using the finite element analysis [27].

The technology of variable inductor has been widely used in many applications, such as chargers for electric vehicles, wireless power transfer equipment, LED drivers, PFC converters, etc. [28,29,30,31,32,33,34,35,36,37,38,39]. The bidirectional DC–DC converter based on variable inductor for electric vehicles was proposed in [28,29], which enhances the current processing ability of the inductor and reduces the twice line frequency output current ripple. In [30], a variable inductor-based wireless power transfer system was proposed to reduce the current through the switches, improve the efficiency, and reduce EMI, also eliminating circulating current to ensure zero-voltage switching. In [31,32,33,34], a variable inductor is introduced to replace the resonant inductor of the resonant converter, which ensures that the resonant converter can operate at the resonant frequency to improve conversion efficiency. In recent years, the variable inductor has also been extensively used in Boost PFC which operates in critical conduction mode (CRM) [35,36,37,38,39]. In [35], to solve the problem of the variable range of switching frequency of CRM Boost PFC, the variable inductor is used to achieve the constant switching frequency over a wide range of input voltage. This method improves the converter efficiency but reduces the PF. In order to balance the power factor and the variable range of switching frequency of CRM Boost PFC, segment variable inductance control is put forward in [37], which reduces the switching frequency variable range to ensure unit PF.

A Cuk PFC converter based on variable inductor is proposed to enhance the PF and reduce the intermediate capacitor voltage in this paper, which varies with the transient rectified input voltage by injecting a controlled DC bias current into the auxiliary winding of the variable input inductor. The operation principles of the Cuk PFC converter based on variable inductor are analyzed in detail, and the analyses of PF, the voltage of intermediate capacitor, and operating principle of variable inductor are presented. To verify the feasibility of the proposed scheme, a 108W experimental prototype is built and tested. The experimental results indicate that the proposed Cuk PFC converter based on variable inductor can significantly enhance the PF, decrease the intermediate capacitor voltage, and increase efficiency compared with the traditional Cuk PFC converter. The Cuk PFC converters in [12,13] have the significant twice line frequency ripple of output current. However, the proposed Cuk PFC converter based on variable inductor has no obvious twice line frequency output ripple. Suffering from the high intermediate capacitance voltage problem, the Cuk PFC converters in [16,17,18,19] can only operate 90~135 Vac low input voltage; the proposed Cuk PFC converter can operate in the 90~240 Vac wide input voltage range. Compared with the converter which can operate at 220 Vac input voltage in [15], the intermediate capacitance voltage drops from 500 V in [15] to 410 V for the proposed Cuk PFC converter at 220 Vac input voltage.

This paper consists of 5 sections as follows. In Section 2, the basic operating theory of the traditional Cuk PFC converter is derived, and its shortcomings are analyzed. In Section 3, the Cuk PFC converter with variable inductor is proposed, and the operating theory and key characteristics of the proposed converter are analyzed. In Section 4, the comparative experimental results are given and analyzed, and the conclusions are summarized in Section 5.

2. Operating Principle of the Conventional Cuk PFC

The main circuit diagram and key operation waveforms of the traditional Cuk PFC converter are presented in Figure 1 and Figure 2. From Figure 1, the main circuit of the conventional Cuk PFC converter consists of a rectifier bridge, an input LC filter L_f and C_f, two diodes D₁~D₂, a power switch S₁, an input inductor L₁, an output inductor L₂, an intermediate capacitor C₁, and an output filter capacitor C_o.

Cuk PFC can automatically realize PFC and simplifies the control circuit if both inductors operate in DCM with only a simple voltage mode control loop, so both output inductor L₂ and input inductor L₁ of the Cuk PFC converter operate in DCM in this paper. The waveforms of both inductor currents during two switching cycles are shown in Figure 2.

As can be seen from Figure 2, it is supposed that the input inductor current i_L1 drops to 0 firstly. There are four operating modes of the Cuk PFC converter during one switching cycle.

Mode I: During this time period, the driving signal v_g turns the switch S₁ on; the input inductor L₁ starts to be charged with energy by the rectified voltage v_Rec through diode D₁ and switch S₁. Meanwhile, the output inductor L₂ is charged with energy by the voltage difference of the intermediate capacitor C₁ and output capacitor C_o. The rising slope of both inductor currents can be presented, respectively, as

$$\frac{d{i}_{L1}}{dt}=\frac{v_{\mathrm{Re}c}(t)}{L_1}$$

(1)

$$\frac{d{i}_{L2}}{dt}=\frac{V_{C1}-V_o}{L_2}$$

(2)

where v_Rec(t) represents the rectified voltage; V_C1 denotes the voltage of intermediate capacitor C₁; V_o denotes output voltage.

Mode II: In this mode, the driving signal v_g turns the switch S₁ off, and the rectified voltage v_Rec and input inductor L₁ start to release energy to the intermediate capacitor C₁ through the diode D₂. The output capacitor C_o stores energy from the output inductor L₂ through D₂. The drop slope of both inductors can be given, respectively, as

$$\frac{d{i}_{L1}}{dt}=\frac{V_{C1}-v_{\mathrm{Re}c}(t)}{L_1}$$

(3)

$$\frac{d{i}_{L2}}{dt}=\frac{V_o}{L_2}$$

(4)

Mode III: During this operating mode, the input inductor current i_L1 drops to 0. The output capacitor C_o still stores energy from the output inductor L₂ through diode D₂. The output inductor current i_L2 continues to drop.

Mode IV: The switch S₁ is still off, and diode D₂ is in reverse bias during this operating mode. The output inductor current i_L2 drops to 0, and the output capacitor C_o provides power to the load.

According to Equation (1), the peak current i_{L1_pk}(t) flowing through the input inductor L₁ of the conventional Cuk PFC converter during one switching cycle can be expressed as

$$i_{L1\_\mathrm{pk}}(t)=\frac{V_M|\sin (\omega t)|}{L_1}{t}_{\mathrm{on}}$$

(5)

where t_on represents the conduction time of switch S₁ during a switching period.

When the converter enters in steady-state operation, the discharge time t_off1 of the input inductor can be obtained from the volt-second balance of the input inductor L₁ as

$$t_{\mathrm{off}1}=\frac{V_M|\sin (\omega t)|}{(V_{C1}-V_M|\sin (\omega t)|)}{t}_{\mathrm{on}}$$

(6)

Therefore, according to Equations (5) and (6), the rectified input current is equal to the average current i_L1(t) through the input inductor during one switching cycle, and the rectified input current can be expressed as

$$\left|i_{\mathrm{in}}\left(t\right)\right|=i_{L1}(t)=\frac{i_{L1\_\mathrm{pk}}(t_{\mathrm{on}}+t_{\mathrm{off}1})}{2T_{\mathrm{s}}}=\frac{V_M{t}_{on}{}^2|\sin (\omega t)|}{2T_{\mathrm{s}}{L}_1[1-\frac{V_M}{V_{C1}}|\sin (\omega t)|]}$$

(7)

where T_s is a switching cycle.

It is known from Equation (7) that there is a time component in the denominator of the converter input current i_in(t), so the input current waveform of the conventional Cuk PFC converter will be distorted. For the sake of the convenience in analyzing the distortion of the input current waveform, the input current shown as Equation (7) is normalized with the base of V_Mt_on²/2T_SL₁(1 − V_M/V_C1), and the normalized input current i* waveform is illustrated in Figure 3. It can be found that the input current is related to the ratio of V_M/V_C1 for the conventional Cuk PFC converter. The larger the ratio of V_M/V_C1 is, the more serious the distortion of the input current is, and as the ratio of V_M/V_C1 gradually decreases, the input current distortion is reduced and gradually tends to be sinusoidal. In other words, the higher the voltage of the intermediate capacitor is, the closer the input current is to a sinewave.

According to Figure 2 and Equation (2), the peak current i_{L2_pk}(t) of the output inductor L₂ can be given as

$$i_{L2\_\mathrm{pk}}(t)=\frac{(V_{C1}-V_o)}{L_2}{t}_{\mathrm{on}}$$

(8)

The output current and the average output inductor current are equal, so the output current I_o is expressed as

$$I_{\mathrm{o}}=i_{L2}\left(\mathrm{t}\right)=\frac{i_{L2\_\mathrm{pk}}(t_{\mathrm{on}}+t_{\mathrm{off}2})}{2T_{\mathrm{s}}}$$

(9)

where t_off2 is the discharge time of output inductor L₂.

Because of the volt-second balance of inductor L₂, t_off2 is derived as

$$t_{\mathrm{off}2}=\frac{(V_{C1}-V_o)}{V_o}{t}_{\mathrm{on}}$$

(10)

Substituting Equation (10) into (9), the output current I_o can be rewritten as

$$I_{\mathrm{o}}=\frac{(V_{C1}-V_{\mathrm{o}})V_{C1}{t}_{\mathrm{on}}{}^2}{2T_{\mathrm{s}}{L}_2{V}_o}$$

(11)

From Equation (11), t_on can be derived as

$$t_{\mathrm{on}}=\sqrt{\frac{2T_{\mathrm{s}}{L}_2{V}_{\mathrm{o}}{I}_{\mathrm{o}}}{(V_{C1}-V_{\mathrm{o}})V_{C1}}}$$

(12)

By neglecting power loss, the input power should be equal to output power, so the input power can be given as

$$P_{\mathrm{in}}=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{i}_{\mathrm{in}}(t)}{V}_M\left|\sin (\omega t)\right|d(\omega t)=U_{\mathrm{o}}{I}_{\mathrm{o}}$$

(13)

According to (7) and (12), Equation (13) can be rewritten as

$$\frac{2}{(V_{C1}-V_{\mathrm{o}})}\cdot \frac{M{V}_M^2}{2\pi V_{C1}}{\displaystyle {\int }_0^{\pi }\frac{{\sin }^2(\omega t)}{1-\frac{V_M}{V_{C1}}\left|\sin (\omega t)\right|}}d(\omega t)=1$$

(14)

where M = L₂/L₁. According to the circuit parameters listed in

Table 1. Key Circuit Parameters.

Symbol	Design Parameter	Value
Vin_RMS	RMS input voltage	90~240 V
fL	Grid frequency	50 Hz
Vo	Rated output voltage	72 V
Io	Rated output current	1.5 A
fs	Switch frequency	67 kHz
n1: n2: n3	n1: n2: n3 (EI40 core)	24:95:95
LV	Variable inductor (proposed Cuk PFC)	75 μH~410 μH
L1	Input inductor (conventional Cuk PFC)	75 μH
L2	Output inductor	180 μH
C1	Intermediate capacitor	200 μF
Co	Output capacitors	200 μF
D1, D2	Diodes	STTH12R06FP
S1	Main switch	FCP190N60
S2	Time-multiplexing output control switches	FDD86367

, the relationship curves of the intermediate capacitor voltage V_C1 and RMS input voltage V_{in_RMS} with different ratios M are shown in Figure 4. From Figure 4, it can be observed that the intermediate capacitor voltage V_C1 increases as the V_{in_RMS} increases when M remains unchanged, and the intermediate capacitor voltage V_C1 increases as the ratio M increases at a certain input voltage V_{in_RMS}.

According to (7) and (13), the PF of the conventional Cuk PFC converter can be derived as

$$PF=\frac{\sqrt{2}{P}_{\mathrm{in}}}{V_M{I}_{\mathrm{in}\_\mathrm{RMS}}}=\frac{\sqrt{\frac{2}{\pi }}{\displaystyle {\int }_0^{\pi }\frac{{\sin }^2(\omega t)}{1-\frac{V_M}{V_{C1}}\left|\sin (\omega t)\right|}}d(\omega t)}{\sqrt{{\displaystyle {\int }_0^{\pi }\frac{{\sin }^2(\omega t)}{{[1-\frac{V_M}{V_{C1}}\left|\sin (\omega t)\right|]}^2}}d(\omega t)}}$$

(15)

According to (14) and (15), the relation curves of the PF of the conventional Cuk PFC and RMS input voltage V_{in_RMS} with different ratios M are shown in Figure 5. Observing Figure 5, the PF decreases slightly as the V_{in_RMS} increases in the range from 90 V to 240 V when M is a constant value, and the PF decreases as the ratio M decreases when the converter input voltage V_{in_RMS} is a fixed value.

From the above analysis, for the conventional Cuk PFC converter, the intermediate capacitance voltage V_C1 and PF are both related to the ratio M. The larger the ratio M is, the higher the PF is. Meanwhile, the intermediate capacitor voltage V_C1 also increases, which will result in the need for higher breakdown voltage MOSFET and diode, so it is challenging to achieve high PF and low intermediate capacitor voltage at the same time.

3. Operation Principle and Performance Analysis of Cuk PFC Converter Based on Variable Inductor

3.1. Operation Principle

To reduce the intermediate capacitor voltage V_C1 and increase the PF, a Cuk PFC converter with a variable inductor is proposed as shown in Figure 6. The proposed converter replaces the constant input inductor with the variable inductor and adds the calculation unit and control unit of variable inductor to realize the change of inductance. The main circuit of the proposed converter contains a rectifier bridge, filter inductor L_f, filter capacitor C_f, variable inductor L_V, power switch S₁, decoupling diode D₁, intermediate capacitor C₁, freewheeling diode D₂, output capacitor C_o, and output inductor L₂.

The proposed Cuk PFC converter control circuit based on variable inductor mainly consists of output control circuit, variable inductor calculation unit, and variable inductor control circuit. The output control circuit uses voltage mode control; the variable inductor calculation unit is mainly composed of STM32; and the variable inductor control circuit is mainly composed of an opamp, bias resistor R_bias, and Mosfet S₂. The operation principle of the control circuit is described as follows: the output error signal is obtained by amplifying the difference between the reference voltage V_Ref and feedback signal from the output voltage sampling circuit; the negative input of the comparator is connected to the error signal, and the positive input of the comparator is connected to the sawtooth wave signal v_saw; S₁ is turned on at the start of the switching cycle, and S₁ is turned off when v_saw reaches the error signal. Through sending the rectified input voltage, the sampling voltage, and the calculated intermediate capacitor voltage to the variable inductor calculation unit, the inductance of the variable inductor can be obtained. The bias voltage of the variable inductor control circuit is obtained by calculating the inductance of the variable inductor through bias voltage calculation unit. Through a voltage-controlled current source which is composed of an opamp, bias resistor R_bias, and Mosfet S₂ which is operated at saturation region, the bias current i_bias is obtained from the bias voltage v_bias, and the required inductor is obtained by introducing the bias current into the auxiliary winding of the variable inductor through the variable inductor control circuit.

The proposed converter also operates in DCM. Key waveforms including the output inductor current i_L2 and input inductor current i_LV during half line cycle are shown in Figure 7.

With the same derivation method of input current in Section 2, the input current of the proposed converter can be obtained as

$$i_{\mathrm{in}}{}_{\_VI}(t)=\frac{V_M{t}_{\mathrm{on}}{}^2\sin (\omega t)}{2T_{\mathrm{s}}{L}_V[1-\frac{V_M}{V_{C1}}|\sin (\omega t)|]}$$

(16)

According to Equation (16), if the variable inductor L_V located in the denominator is variable with ωt during the half line cycle to make the denominator of Equation (16) become a constant value, then the purpose of correcting the input current can be achieved. L_V is supposed to meet the formula as

$$L_V[1-\frac{V_M}{V_{C1}}|\sin (\omega t)|]=L_{\mathrm{initial}}$$

(17)

where L_initial is a constant value. According to (17), the input current of (16) can be rewritten as

$$i_{\mathrm{in}}{}_{\_VI}(t)=\frac{V_M{t}_{\mathrm{on}}{}^2\sin (\omega t)}{2T_{\mathrm{s}}{L}_{\mathrm{initial}}}$$

(18)

Observing (18), the switch conduction time t_on is unchanged when the Cuk PFC converter is at the stable operation state, so the input current is standard sinewave.

According to (15) and (18), the PF of the proposed converter can be obtained as

$$PF=\frac{\sqrt{2}{P}_{\mathrm{in}}}{V_M{I}_{\mathrm{in}\_\mathrm{RMS}}}=\frac{\sqrt{2}\cdot \frac{t_{\mathrm{on}}{}^2}{L_{\mathrm{initial}}}\cdot \frac{V_M^2}{2\pi T_{\mathrm{s}}}{\displaystyle {\int }_0^{\pi }{\sin }^2(\omega t})d(\omega t)}{\frac{t_{\mathrm{on}}{}^2}{L_{\mathrm{initial}}}\cdot \frac{V_M^2}{2T_{\mathrm{s}}}\sqrt{\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }{\sin }^2(\omega t})d(\omega t)}}=1$$

(19)

According to (19), unity PF can be achieved by the proposed converter.

3.2. Range of Inductor Variation

According to (18), the expression of the variable inductor L_V can be rewritten as

$$L_V=\frac{L_{\mathrm{initial}}}{[1-\frac{V_M}{V_{C1}}|\sin (\omega t)|]}$$

(20)

According to the circuit parameters listed in Table 1, the initial value of the variable inductor L_initial is 75 µH, and from Equation (20) and combined with the relation of the intermediate capacitor voltage V_C1 and the input voltage V_{in_RMS}, the range of inductor variation can be plotted in a half line cycle as shown in Figure 8.

From Figure 8, it can be obtained that the variable inductor L_V varies about 75 µH~410 µH when the converter input voltage V_{in_RMS} varies from 90 V to 240 V, and the variable inductor L_V varies with the rectified input voltage during the half line cycle.

3.3. Operating Principle of Variable Inductor

In this paper, the variable inductor L_V uses an EI core as a magnetic core, and the basic model schematic of the variable inductor is shown in Figure 9. The middle leg is wound with n₁ turns as the main winding of the input inductor of the proposed Cuk converter; the air gap l₀ is opened to prevent the core from fast saturation; the auxiliary winding which is used to control the inductance is symmetrically wound on the side legs of the EI core; and the number of auxiliary winding turns on both side legs is the same as n₂ and n₃.

Because the air gap is opened in the middle leg, the permeability of the magnetic material is not the same, so the relative permeability of the middle winding is recorded as µ₁, and the air permeability of the air gap is recorded as µ₀. In order to reduce the induction potential on the auxiliary winding due to the change of current in the main winding, it is required that the number of auxiliary winding turns of the left and right legs is the same, and the relative permeability of the left leg is recorded as µ₂ and is recorded as µ₃ for the right leg.

From Figure 9, it can be observed that the flux generated by the main winding flows from the middle leg through the left and right legs. When the main winding current i_LV changes, the induced electromotive forces generated on the auxiliary windings in series with the left and right legs eliminate each other due to the opposite polarity, reducing the effect of the change in the main current on the inductor; when the bias current i_bias changes, the flux generated by the auxiliary windings on both sides only flows in the side legs. Based on the above analysis, the equivalent reluctance model of the variable inductor using the EI core is illustrated in Figure 10.

As shown in Figure 10, F_ac is the AC magnetomotive force; F_dc is the DC magnetomotive force generated by the bias current; Φ₁, Φ₂, and Φ₃ are the fluxes of the middle, left, and right windings, respectively; R₁, R₂, and R₃ are the reluctances of the middle, left, and right magnetic circuits; and R₀ is the middle leg air gap reluctance, according to Ohm’s law of the magnetic circuit which can be expressed as

$$\{\begin{array}{l}{R}_1=\frac{l_1}{{\mu }_0{\mu }_1{A}_1}\\ R_2=\frac{l_2}{{\mu }_0{\mu }_2{A}_2}\\ R_3=\frac{l_3}{{\mu }_0{\mu }_3{A}_3}\\ R_0=\frac{l_0}{{\mu }_0{A}_0}\end{array}$$

(21)

where A₁, A₂, and A₃ denote the section of the middle, left, and right legs, respectively; A₀ is the section of the air gap; l₁, l₂, and l₃ are the length of the middle leg, left leg, and right leg, respectively; l₀ is the length of air gap.

Because of the characteristics of core structure and winding symmetry, it is derived that

$$\{\begin{array}{c}{A}_1=A_0\\ A_2=A_3\\ l_2=l_3\\ n_2=n_3\end{array}$$

(22)

From the definition of the magnetomotive force and its relationship with the reluctance, the relation of magnetic field intensity and reluctance is deduced as

$$F=Hl=\Phi R$$

(23)

According to Figure 10 and Equation (23), combined with the Kirchhoff’s law of the magnetic circuit, it is obtained that

$$\{\begin{array}{l}{H}_1{l}_1+H_0{l}_0-H_3{l}_3=n_1{i}_1-n_3{i}_3\\ H_1{l}_1+H_0{l}_0+H_2{l}_2=n_1{i}_1+n_2{i}_2\\ {\Phi }_2={\Phi }_3+{\Phi }_1\end{array}$$

(24)

where H₁, H₂, H₃, and H₀ are the magnetic field strengths of the middle, left, right, and airgap legs. Substituting (23) into (24), it is derived as

$$\{\begin{array}{l}{\Phi }_1(R_1+R_0)-{\Phi }_3{R}_3=n_1{i}_1-n_3{i}_3\\ {\Phi }_1(R_1+R_0)+{\Phi }_2{R}_2=n_1{i}_1+n_2{i}_2\\ {\Phi }_2={\Phi }_3+{\Phi }_1\end{array}$$

(25)

By simplifying (25), the expression for the main flux of the intermediate winding or variable inductor can be obtained as

$${\Phi }_1=\frac{(R_3-R_2)n_2{i}_2+(R_3+R_2)n_1{i}_1}{R_2{R}_3+R_2(R_1+R_0)+R_3(R_1+R_0)}$$

(26)

Substituting (21) into (26), the relationship between the main flux and the permeability of each segment and length can be deduced as

$$\begin{array}{l}{\Phi }_1=\frac{1}{\left[\frac{l_2}{{\mu }_0({\mu }_2+{\mu }_3)A_2{n}_1{i}_1}+\frac{l_1}{{\mu }_0{\mu }_1{A}_1{n}_1{i}_1}+\frac{l_0}{{\mu }_0{A}_1{n}_1{i}_1}\right]}\\ +\frac{1}{\left[\frac{l_2}{{\mu }_0({\mu }_2-{\mu }_3)A_2{n}_2{i}_2}+\frac{l_1}{{\mu }_0{\mu }_1{A}_1(\frac{{\mu }_2-{\mu }_3}{{\mu }_2+{\mu }_3})n_2{i}_2}+\frac{l_0}{{\mu }_0{A}_1(\frac{{\mu }_2-{\mu }_3}{{\mu }_2+{\mu }_3})n_2{i}_2}\right]}\end{array}$$

(27)

Combining the relationship between inductor and magnetic flux, the expression for variable inductor is derived as

$$L_V=\frac{n_1{\Phi }_1}{i_1}=\frac{1}{\left(\frac{l_2}{2{\mu }_0{\mu }_{\mathrm{var}}{A}_2{n}_1{}^2}+\frac{l_1}{{\mu }_0{\mu }_1{A}_1{n}_1{}^2}+\frac{l_0}{{\mu }_0{A}_1{n}_1{}^2}\right)}$$

(28)

where µ_var is equal to the relative permeability of left and right legs, and it can be changed according to the value of the injected DC bias current i_bias, which presents a negative correlation trend of the variable inductance and the bias current i_bias. As the bias current i_bias increases, µ_var gradually decreases, resulting in a decrease in the inductance of the variable inductor until the bias current reaches i_{bias_max}, which completely saturates the left and right legs, and the variable inductor decreases to minimum value.

3.4. The Voltage of Intermediate Capacitor

Because of power conservation, substitution of (16) into (13), the relationship between the input voltage and the voltage of the intermediate capacitor of the proposed converter is given as

$$\frac{L_2}{L_{\mathrm{initial}}}\cdot \frac{2}{(V_{C1}-V_o)}\cdot \frac{V_M^2}{2\pi V_{C1}}{\displaystyle {\int }_0^{\pi }{\sin }^2(\omega t})d(\omega t)=1$$

(29)

The variation of the intermediate capacitor voltage V_C1 with input voltage V_{in_RMS} can be calculated and plotted, as shown in the blue curve in Figure 11. To facilitate comparison with the intermediate capacitance voltage of the traditional Cuk PFC converter, the input inductor and output inductor are chosen in Table 1, and the red curve shown in Figure 11 is the conventional Cuk PFC when M = L₂ /L₁ = 2.4.

From Figure 11, it is obvious that the proposed converter has a significant reduction in the intermediate capacitor voltage V_C1 during the variation of input voltage V_{in_RMS} from 90 V to 240 V compared with the traditional Cuk PFC converter.

3.5. Peak Current of Input Inductor

Substituting Equation (20) into (5), the envelope curves of the peak current of the input inductor comparison of both the traditional and proposed converters can be plotted for a half line cycle, as depicted in Figure 12.

From Figure 12, the peak current of the input inductor of the proposed Cuk PFC converter is significantly reduced, which will help to diminish the current stress of the switch.

3.6. Conditions of Both Input and Output Inductors Operating in DCM

In order to make the output inductor current i_L2 operate in DCM, it should be ensured that the sum of the on-time t_on and the off-time t_off2 of the output inductor is less than one switching cycle T_s.

$$t_{\mathrm{on}}+t_{\mathrm{off}2}<T_{\mathrm{s}}$$

(30)

Substituting Equations (10) and (12) into the above equation, the output inductor current i_L2 operating in DCM can be obtained as

$$L_2\le \frac{(V_{C1}-V_{\mathrm{o}})\cdot U_{\mathrm{o}}\cdot T_{\mathrm{s}}}{2\cdot V_{C1}\cdot I_{\mathrm{o}}}$$

(31)

According to (31) and combined with the analysis of the intermediate capacitor voltage V_C1, the critical inductor value of the output inductor L₂ operating in DCM when the input voltage V_{in_RMS} is 90 V~240 V can be obtained as shown in Figure 13.

According to (13) and (18), the relationship of t_on and output power with variable inductor can be obtained as

$$\frac{t_{\mathrm{on}}{}^2}{L_{\mathrm{initial}}}\cdot \frac{V_M^2}{2\pi T_s}{\displaystyle {\int }_0^{\pi }{\sin }^2(\omega t})d(\omega t)=P_{\mathrm{o}}$$

(32)

The relationship between the initial inductance of variable inductor and the on-time t_on is derived as

$$t_{\mathrm{on}}=\sqrt{\frac{4T_{\mathrm{s}}{L}_{\mathrm{initial}}{P}_{\mathrm{o}}}{V_M^2}}$$

(33)

Within the half line period, the higher the transient rectified input voltage is, the larger the input inductor current peak is, and the longer the inductor current i_LV discharge time within the half line cycle is. When |sin(ωt)| = 1 at the input voltage peak V_M, the i_LV discharge time reaches the maximum, and the maximum discharge time can be derived as

$$t_{\mathrm{off}1\_\max }=\frac{V_M}{(V_{C1}-V_M)}{t}_{\mathrm{on}}$$

(34)

The limit of the input inductor current i_LV in DCM can be obtained as

$$t_{\mathrm{on}}+t_{\mathrm{off}1\_\max }<T_{\mathrm{s}}$$

(35)

Substituting Equations (33) and (34) into (35) and simplifying the above equation, the limit of the initial inductance of variable inductor can be deduced as

$$L_{\mathrm{initial}}\le \frac{{(V_{C1}-V_M)}^2\cdot V_M^2\cdot T_{\mathrm{s}}}{4P_{\mathrm{o}}{V}_{C1}^2}$$

(36)

From (36) and combined with the circuit parameters, it is obtained that the initial inductance of variable inductor variation range is about 84 µH~152 µH at the input voltage V_{in_RMS} of 90 V~240 V, and its variation curve is shown in Figure 14.

To ensure that both input and output inductors operate in DCM over the input voltage variation range, 75 µH is selected as the initial inductance of the variable inductor to meet the critical value at 90 Vac, and 180 µH is chosen as the output inductor to meet the DCM of the output inductor of the proposed Cuk PFC converter.

4. Experimental Results

For the purpose of checking the correctness of the theoretical analysis, a 108 W experimental prototype has been built in the laboratory to compare the experimental results of both the traditional and proposed Cuk PFC converters. The specifications and the circuit parameters are listed in Table 1. The pictures of the experimental prototype and the experimental platform are presented in Figure 15 and Figure 16, respectively.

The experimental results of input current i_in, input voltage v_in, output voltage V_o, and intermediate capacitor voltage V_C1 for the traditional Cuk PFC converter and the proposed Cuk PFC converter with variable inductor when the input voltage is 110 Vac and 220 Vac are illustrated, respectively, in Figure 17 and Figure 18. According to Figure 17a and Figure 18a, it can be observed that the conventional Cuk PFC converter can obtain a stable output voltage at 72 V. Harmonic distortion occurs in the input current, and the voltage of the intermediate capacitor reaches 280 V and 520 V when the input voltage is 110 Vac and 220 Vac, respectively. According to Figure 17b and Figure 18b, it can be noticed that the proposed Cuk PFC converter can obtain a stable output voltage at 72 V without obvious twice line frequency output voltage ripple; the waveform of input current is corrected to sine-wave with variable inductor; and the intermediate capacitor voltage is reduced to 240 V and 410 V when the input voltage is 110 Vac and 220 Vac, respectively. Therefore, compared with the conventional Cuk PFC converter, the input current of the proposed Cuk PFC converter based on variable inductor can obtain near-ideal sine-wave input current, and the intermediate capacitor voltage of the proposed converter is significantly reduced.

The inductor current waveforms experimental results of both converters at 110 Vac are illustrated in Figure 19, which shows that the peak input inductor current decreases from 8 A for the conventional Cuk PFC converter to 6 A for the proposed Cuk PFC converter. It also can be obtained that the peak input inductor current decreases from 7.5 A to 5.3 A by variable inductor control at 220 Vac in Figure 20.

The variable inductor current i_LV is slightly distorted as illustrated in Figure 19d and Figure 20d, which is due to the small bias current applied to the auxiliary winding, and the inductance of the variable inductor is more easily affected by the main winding current. The operating point of the variable inductor is shifted towards the saturation region [35,36]. It also causes the actual inductance of the variable inductor to not reach the theoretical calculation value, which leads to a higher actual inductor current than the theoretically derived value.

The comparison data of the input current harmonic content test results of both the traditional and proposed converters are presented in Figure 21. The experimental data of PF, efficiency, and THD are illustrated in Figure 22. From Figure 21 and Figure 22c, compared with the traditional Cuk PFC converter, the proposed converter is easier to meet the requirement of IEC61000-3-2 class D. The THD and third harmonic current are significantly reduced by using the variable inductor. From Figure 22a, the PF of the proposed converter with variable inductor is enhanced over the input voltage variation range. Due to the application of the variable inductor, the PF is increased from 0.982 to 0.995 at 110 Vac, and the PF is enhanced from 0.975 to 0.981 at 220 Vac. Because of the influence of the variable inductor main current on the inductor saturation level, the actual inductor value does not reach the theoretical calculated value, which makes the PF of experimental test results with variable inductor lower than the theoretical analysis result. From Figure 22b, the efficiency is improved obviously by using the variable inductor, and the efficiency can be higher than 90% at 90 Vac input voltage.

5. Conclusions

A Cuk PFC converter based on variable inductor is proposed in this paper, which uses real-time variations of inductor to solve the problem of the traditional Cuk PFC converter including low power factor and high intermediate capacitor voltage. The operation principles of the proposed converter are analyzed in detail, and the analyses of PF, the voltage of intermediate capacitor, and design considerations are provided. For the purpose of verifying the feasibility of the proposed scheme, an experimental prototype of 108W was built and tested. The experimental test results indicated that the proposed Cuk PFC converter based on variable inductor can significantly enhance the PF, decrease the intermediate capacitor voltage, and increase efficiency.