Design and Implementation of Three-Winding Coupled Inductor Applied in High Step-Up DC/DC Converter Combined with Voltage Multipliers

Abstract

By combining a coupled inductor with voltage multipliers, the voltage gain of a boost converter can be improved significantly. This method has good application prospects in renewable energy generation and in DC microgrids. A coupled inductor is the core component of the high step-up DC/DC converter and has serious impact on its performance. However, shortage in the methods used to design the coupled inductor have limited the applications of such converters. By analyzing the operating modes of the high step-up DC/DC converter with a three-winding coupled inductor combined with two voltage multipliers, accurate and simplified models of currents in the three windings are established. Furthermore, a design methodology for a multi-winding coupled inductor is put forward, in which a method of calculating the boost inductance and product areas (AP) and a method for selecting the magnetic core are established. The influence of winding arrangements and loss evaluations of the coupled inductor are also investigated. Finally, a 200 W prototype converter with an input of 20 V and output of 200 V is prepared and tested. The correctness of the current models of and design methods used on coupled inductor are verified. More important, the method proposed to design multi-winding coupled inductors can be applied to design high step-up DC/DC converters with different topologies.

1. Introduction

Traditional fossil energies are facing the crises of exhaustion and causing serious environmental problems. Renewable energies have been developed rapidly and used more and more widely in recent years. However, the voltage generated by renewable sources, such as photovoltaic (PV) panels and fuel cells (FCs), is relatively low, with a value less than 60.0 V. This is a big challenge to their applications. High step-up DC/DC converters are usually used as the interface between renewable energy sources and the alternating current (AC) grid or direct current (DC) microgrids. Due to these applications, high step-up DC/DC converters have attracted tremendous research interest [1,2,3].

Different methods have been developed to improve the voltage gains of traditional DC/DC converters, such as multi-stage, voltage multiplier (VM) and coupled inductors. A quadratic boost converter is also formed with two cascaded boost converters [4,5]. In cascaded and quadratic converters, power semiconductor devices suffer from high voltage stress equal to the output voltage. VM techniques can be divided into two categories, switched-capacitor and switched-inductor [6,7,8,9,10]. The large volume of the switched-inductor limits its application in high step-up converters. Current spikes in the charging and discharging processes of switched-capacitors may threaten power switches and result in more difficulties in designing against electromagnetic interference (EMI). Current spikes can be depressed by inserting a resistor or inductor into the charge and discharge circuits for the switched-capacitor [11]. The efficiency and voltage gain of the converter may be decreased by the inserted resistors and inductors.

Including the advantages of simplicity in structure and the low cost of the high step-up DC/DC converters based on coupled inductors, the voltage conversion ratio of the converter can be adjusted by changing the turn ratios of the coupled inductor. A large number of high step-up DC/DC converters have been developed based on double-winding coupled inductors [12,13,14,15,16,17,18] and three-winding coupled inductors [19,20,21,22,23,24,25,26,27,28]. In these converters, the winding connected to the voltage source can be considered the boost inductor, and other windings integrated with voltage multipliers have the function of booster transformers to increase the voltage gain. Using combinations of coupled inductors with voltage multipliers is the most promising technology to realize high voltage gains.

In a traditional DC/DC converter, the magnetic element has serious impacts on the efficiencies and volume [29,30,31,32,33]. More seriously, an inappropriately designed magnetic element may result in the core entering a saturation state and an abnormal temperature rise, which seriously threaten the stability of the converter. In high step-up DC/DC converters based on coupled inductors, the influence of the magnetic element on the performance of the converter becomes more significant. However, complexity in the operating modes of the high step-up DC/DC converter results in more difficulties in the design of the coupled inductor. Up to the present, no report on methods of designing coupled inductors has been found. This has become one of the most serious restrictions in the applications of high step-up DC/DC converters.

In this paper, a design method for the three-winding coupled inductor applied in a high step-up DC/DC boost converter is put forward and verified with a 200 W prototype converter. The following contributions are achieved.

• With analysis of the operating modes of the converter, accurate models and simplified models for currents in the three windings are established. (Section 2)
• A design methodology for the coupled inductor is established. The boost inductance of the converter is determined based on the ripple in the input current of the converter. Furthermore, a product areas (AP) method of selecting the magnetic core is obtained. The impact of the winding arrangement on the coupled inductor and a method of estimating losses in the three windings are achieved also. (Section 3)
• A prototype of a 200 W DC/DC converter with an input of 20 V and output of 200 V is prepared and tested. The correctness of the current models and design method are verified. (Section 4)

The method proposed in this paper can be also used to design multi-winding coupled inductors applied in high step-up DC/DC converters with different topologies. Application studies on these converters can be promoted.

2. Current Models of the Three-Winding Coupled Inductor

Accurate models of currents in the windings of the coupled inductor are the foundation of the design of magnetic elements and selecting power semiconductor devices. Accurate modeling can be achieved by analyzing the operating modes of the high step-up DC/DC converter. A high step-up boost converter with a three-winding coupled inductor and two voltage multipliers is presented in Figure 1. Winding W₃, diode D₂ and clamping capacitor C₃ form the basic voltage multiplier (labeled with orange dashed line). Winding W₂, capacitors C₁ and C₂, switch S and the basic voltage multiplier compose the extended voltage multiplier, which is labeled with a green dashed line in Figure 1. n₁, n₂ and n₃ are the turn numbers of windings W₁, W₂ and W₃, respectively.

Equivalent circuits and waveforms for the high step-up DC/DC converter operating in continuous conduction mode (CCM) in a complete switching circle are presented in Figure 2 and Figure 3, respectively.

2.1. Operating Modes of the High Step-Up DC/DC Boost Converter

Mode 1 (t₀-t₁): Switch S and diode D₃ are on, while diodes D₁, D₂ and D_o are off. The energy of the load resistor R_L originates from the output capacitor C_o and the variation of the voltage of the capacitor C_o can be neglected (right blue loop in Figure 2a). Energy from the voltage source V_in is stored in the coupled inductor (left green loop). Energy stored in capacitors C₂ and C₃ is transferred to the clamping capacitor C₁ (center orange loop).

The conducting of diode D₃ results in the windings W₂ and W₃ being in series, and the equivalent inductance can be calculated as L₂₃ = L₂ + L₃ + M₂₃. L₂ and L₃ are inductances for the windings W₂ and W₃. M₂₃ is the mutual inductance between the two windings. By applying Kirchhoff’s voltage law to the left green loop and center orange loop in an equivalent circuit in Figure 2a, the following equation set is obtained

$$\left\{\begin{array}{l} V_{in} = L_1\times {\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M1} -M_{1,23}\times {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M1}\\ V_{L23M1}=M_{1,23}\times {\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M1}- L_{23}\times {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M1}\end{array}\right.$$

(1)

where L₁ is the inductance for the winding W₁ and M_1,23 is the mutual inductance between winding 1 and the equivalent inductance for windings W₂ and W₃ in series. V_L23M1 is the voltage across the equivalent inductance L₂₃ and can be calculated as the difference between the voltage across the clamping capacitor C₁ and the summation of voltages across the capacitors C₂ and C₃. By solving Equation (2), slopes for the currents in the three windings can be achieved

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M1}={\displaystyle \frac{V_{in}\times L_{23}-V_{L23M1}\times M_{1,23}}{L_1\times L_{23}-M_{1,23}^2}}\\ {\left. {\displaystyle \frac{d{i}_2}{dt}}\right|}_{M1}= {\displaystyle \frac{V_{in}\times M_{1,23}-V_{L23M1}\times L_1}{L_1\times L_{23}-M_{1,23}^2}}\end{array}\right.$$

(2)

Mode 2 (t₁-t₂): Switch S is turning off and diode D₁ is turning on. The energy of the load resistor R_L is provided by capacitor C_o. Energies stored in the leakage inductances for the three windings of the coupled inductor are transferred to the three clamping capacitors. High frequency oscillations and ripples are formed across these capacitors. The lasting time of this mode is very short and the currents for the three windings can be regarded as constant. This mode is usually neglected in most studies.

Mode 3 (t₂-t₃): The origin of the energy of the load R_L is the output capacitor C_o, which is the same as those in the two previous modes. The polarities of the three windings are reversed and two novel loops are formed (green loop and orange loop in Figure 2c). The voltage across the winding W₁ in this mode V_dis can be calculated using amp-second balance.

$$V_{dis}={\displaystyle \frac{V_{in}\times D}{1-D}}=V_{in}\times {\displaystyle \frac{D}{D^{\prime }}}$$

(3)

where D is the duty ratio and equal to the ratio of the conduction time to the period. Similar to the analysis of Mode 1, Kirchhoff’s voltage law is applied to the left green loop and center orange loop in an equivalent circuit. Decreasing slopes for the currents in the three windings of the coupled inductor can be achieved

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M3}=-{\displaystyle \frac{V_{dis}\times L_{23}-V_{L23M3}\times M_{1,23}}{L_1\times L_{23}-M_{1,23}^2}}\\ {\left. {\displaystyle \frac{d{i}_2}{dt}}\right|}_{M3}={\displaystyle \frac{-V_{dis}\times M_{1,23}+V_{L23M3}\times L_1}{L_1\times L_{23}-M_{1,23}^2}}\end{array}\right.$$

(4)

where V_L23M3 is the difference between the voltages across the capacitors C₁ and C₃, V_L23M3 = V_C1 − V_C3. In the ending time of Mode 3, the currents in windings W₂ and W₃ of the coupled inductor are decreased to zero.

Mode 4 (t₃-t₄): Switch S and diode D₃ are off. Diode D₁ is on. From time t₃, diodes D₂ and D_o enter a conducting state, and energy dissipated by the load R_L mainly originates from the coupled inductor. In the end of this mode t₄, the current in winding W₁ is equal to the summation of the currents in windings W₂ and W₃. The conduction of the diodes D₂ and D_o can be considered resonant processes determined by the junction capacitances of diodes and the leakage inductances of the windings W₂ and W₃. These processes will be studied intensively.

In the loop formed by winding W₃, diode D₂ and a clamping capacitor C₃ (labeled with a blue dashed line in Figure 2d), the junction capacitance for the diode D₂, C_JD2, is significantly smaller than the capacitance of the clamping capacitor C₃, and the leakage inductance of the winding W₃, L_3Lk, is also smaller than the conductance of the winding. The current in the loop (the same current following through winding W₃) increases with the resonance process formed by the junction capacitance and the leakage inductance. The lasting time for this mode is one quarter of the period of the resonance, which can be calculated as $T_{Res1M3}=2\pi \times \sqrt{L_{3Lk}\times C_{JD2}}$.

Similarly, in the increasing process for the current following through the winding W₃ determined by the loop formed by diodes D_o and D₁, clamping capacitors C₁ and C₂ and winding W₂ are in a resonance process. The period for the resonance is $T_{Res2M3}=2\pi \times \sqrt{L_{2Lk}\times C_{JDo}}$. C_JD0 and L_2Lk are the junction capacitor of diode D_o and the leakage inductance of the winding W₂. This means that the currents in windings W₂ and W₃ reach their positive maximum values at the same time as t₄.

Neglecting the difference in the impedances of the above two loops, the currents in the windings of W₂ and W₃ in the end of this mode t₄ are

$$i_2(t_4)=i_3(t_4)={\displaystyle \frac{1}{2}}{i}_1(t_4)$$

(5)

Mode 5 (t₄-t₅): The equivalent circuit for this mode is the same as that of the Mode 4 (Figure 2d) and no change of the switching states for the power devices has occurred. Energy stored in the coupled inductor and clamping inductor C₁ are transferred to the output capacitor C_o and load R_L in this mode, and the current in winding W₁ is the same as the summation of the currents of windings W₂ and W₃. Windings W₂ and W₃ are in parallel, and the equivalent inductance can be calculated as

$$L_{23}^{\prime }={\displaystyle \frac{L_2\times L_3-M_{23}^2}{L_2+L_3-2\times M_{23}}}$$

(6)

where M₂₃ is the mutual inductance between windings W₂ and W₃. In most prepared coupled inductors with the same turn number for the two windings W₂ and W₃, $L_{23}^{\prime }$ is equal to the inductance of winding W₂ or winding W₃. The inductance of winding W₁ is in series with the equivalent inductance $L_{23}^{\prime }$. The equivalent inductance of the three windings of the coupled inductance is $L_{eqM5}=L_1+L_{23}^{\prime }+M_{1,23}^{\prime }$. $M_{1,23}^{\prime }$ is the mutual inductance between the inductance of winding W₁ and equivalent inductance $L_{23}^{\prime }$. The slope for the current in winding W₁ is

$${\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M5}={\displaystyle \frac{V_o-V_{in}-V_{C1}}{L_{eqM5}}}$$

(7)

The summation of the slopes of the currents in windings W₂ and W₃ is the same as that of winding W₁. The ratio of the slopes of the windings W₂ and W₃ is determined by the impedances of two loops formed by the winding W₃, diode D₂ and clamping capacitor C₃ (labeled with a dashed blue line), and winding W₂, clamping capacitors C₁ and C₂, diodes D_o and D₁, output capacitor C_o and load R_L (in the orange dashed line) (in Figure 2d), respectively. In the following analysis, this difference is neglected.

Mode 6 (t₅-t₆): Switch S is in the transient process of turning on and diode D₁ is in the transient process of turning off. The equivalent circuit for the converter is presented in Figure 2e. High frequency ripples originating from the oscillations between the leakage inductances of the three windings in the coupled inductor and the junction capacitors of the power devices are formed. Due to the short time of this mode, it can be neglected, which is similar to Mode 2.

Mode 7 (t₆-t₇): Switch S and diodes D_o and D₂ are on. Diodes D₁ and D₃ are off. Three loops are formed in this mode (Figure 2f). Comparing with Mode 5, the polarities of voltages across the three windings of the coupled inductor are reversed. From the left loop in Figure 2f (in green), it can be seen that the voltage across the winding W₁ is V_in. The voltages across the windings W₂ and W₃ are V_o − V_C1 and V_C3, respectively. Slopes for the currents in the three windings of the coupled inductor can be calculated with the following equation set

$$\left\{\begin{array}{l} V_{in}=L_1\times {\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M7}-M_{12}\times {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M7}-M_{13}\times {\left.{\displaystyle \frac{d{i}_3}{dt}}\right|}_{M7}\\ V_o-V_{C1}=M_{21}\times {\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M7}-L_2\times {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M7}+M_{23}\times {\left.{\displaystyle \frac{d{i}_3}{dt}}\right|}_{M7}\\ V_{C3}=M_{31}\times {\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M7}+M_{32}\times {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M7}-L_3\times {\left.{\displaystyle \frac{d{i}_3}{dt}}\right|}_{M7}\end{array}\right.$$

(8)

where M_ij (1 ≤ i,j ≤ 3 and i ≠ j) is the mutual inductance between winding i and winding j and M_ij = M_ji. The achieved slopes for the currents in the three windings of the coupled inductor are listed in Equation (9).

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{d{i}_1}{dt}}\right|}_{M7}={\displaystyle \frac{V_{in}(L_2{L}_3-M_{23}^2)-(V_o-V_{C1})(L_3{M}_{12}+M_{12}{M}_{23})-V_{C3}(L_2{M}_{13}+M_{12}{M}_{23})}{L_1{L}_2{L}_3-L_1{M}_{23}^2-M_{13}^2{L}_2-M_{12}^2{L}_3-2M_{12}{M}_{13}{M}_{23}}}\\ {\left.{\displaystyle \frac{d{i}_2}{dt}}\right|}_{M7}={\displaystyle \frac{V_{in}(L_3{M}_{12}+M_{13}{M}_{23})-(V_o-V_{C1})(L_1{L}_3-M_{13}^2)-V_{C3}(M_{12}{M}_{13}+L_1{M}_{23})}{L_1{L}_2{L}_3-L_1{M}_{23}^2-M_{13}^2{L}_2-M_{12}^2{L}_3-2M_{12}{M}_{13}{M}_{23}}}\\ {\left.{\displaystyle \frac{d{i}_3}{dt}}\right|}_{M7}={\displaystyle \frac{V_{in}(L_2{M}_{13}+M_{12}{M}_{23})-(V_o-V_{C1})(L_1{M}_{23}+M_{12}{M}_{13})-V_{C3}(-L_1{L}_2+M_{12}^2)}{L_1{L}_2{L}_3-L_1{M}_{23}^2-M_{13}^2{L}_2-M_{12}^2{L}_3-2M_{12}{M}_{13}{M}_{23}}}\end{array}\right.$$

(9)

Mode 8 (t₇-t₈): Switch S is on, and diodes D₁, D₂ and D_o are off. The currents in windings W₂ and W₃ are decreased to negative maximal values in resonance transient processes, which are similar to the resonances in Mode 4. The lasting time for this mode can be calculated as one quarter of the period determined by the junction capacitance of diode D₂. The equivalent inductance L₂₃ formed by windings W₂ and W₃ in series. The period of the resonance can be calculated with Equation (10).

$$T_{Res3M8}=2\pi \times \sqrt{L_{23}\times C_{JD3}}$$

(10)

where C_JD3 is the junction capacitor for diode D₃. L₂₃ is the equivalent inductance with a value of L₂₃ = L₂ + L₃ + M₂₃, which is the same as that of Mode 1. In the ending time of this mode t₈, the operating of the converter enters Mode 1 and the complete cycle is realized.

2.2. Simplified Models for Currents in the Three Windings of the Coupled Inductor

Current in the three windings of the coupled inductor can be divided into two parts, the alternating current (AC) component and direct current (DC) component. As energies in the coupled inductor are stored in Mode 1 and released in Mode 5, other transient modes are neglected. The simplified waveforms for these currents are presented in Figure 4. The AC components of these currents can be calculated with the slopes obtained in the previous analysis of the operating modes. However, the DC components for these currents are still unclear, and the design requirements of the inductor can not be satisfied.

The DC components for currents in the three windings are calculated first. The charge balance of a capacitor in DC/DC converters means that the quantity of charge injected into the capacitor is equal to that released in a switching period, which is another form of amp-second balance [35]. By applying a charge balance to the capacitor C_o, the capacitor can be neglected in calculations of the quantity of charges following through the load R_L. The charges through the load R_L in a switching period (presented with a green dashed line in the bottom of Figure 4) are

$$Q_o=I_o\times T_s={\displaystyle \frac{V_o}{R_L}}\times T_s$$

(11)

where V_o is the output of the high step-up converter, I_o is the output current of the converter and T_s is the switching period. The charges through the R_L are the same as those following through diode D_o (orange dashed line in Figure 4), Q_Do = Q_o. The current of the diode D_o in conducting state I_Do is

$$I_{Do}={\displaystyle \frac{Q_{Do}}{(1-D)\times T_s}}={\displaystyle \frac{I_o}{1-D}}$$

(12)

The charges following through winding W₂ in Mode 5 are the same as those following through diode D_o. Moreover, the quantities of the charges through winding W₂ in Mode 1 and Mode 5 are the same. These relationships in the charges are also suitable for analysis on winding W₃. Models for estimating the DC currents in the windings W₂ and W₃ are listed as follows

$$\left\{\begin{array}{l}{I}_{L23on}=-{\displaystyle \frac{I_o}{D}}\\ I_{L23off}= I_{Do}\end{array}\right.$$

(13)

where I_L23on and I_L23off are the DC components for currents in windings W₂ and W₃ in Modes 1 and 5, respectively.

As charges following through winding W₁ in Mode 5 Q_L1off are the summation of those in windings W₂ and W₃, this current can be calculated with Equation (18). The power balance means that the input power of a DC/DC converter is equal to the output power in a switching cycle. The DC current in winding W₁ in a switching cycle is M × I_o. M is the voltage gain of the converter. The charges through winding W₁ in Mode 1 are (M − 2) × I_o. The DC component of the current in winding W₁ in Mode 1 can be calculated.

$$\left\{\begin{array}{l}{I}_{L1on}=I_{L1}-I_{L1off}={\displaystyle \frac{M-2}{D}}\times I_o\\ I_{L1off}=2\times I_{Do}={\displaystyle \frac{2}{1-D}}\times I_o\end{array}\right.$$

(14)

Ripples in the currents of the three windings of the coupled inductor will be investigated in the following section. Ripples in the three currents (in Figure 4) are formed by the differences in the DC components and AC components in Mode 1 and 5. Differences in the DC components of the three windings can achieved

$$\left\{\begin{array}{l}{i}_{L1ripDC}= I_{L1on}-I_{L1off} =({\displaystyle \frac{M-2}{D}}-{\displaystyle \frac{2}{1-D}})\times i_o\\ i_{L23ripDC}=I_{L23on}-I_{L23off}=({\displaystyle \frac{1}{D}}+{\displaystyle \frac{1}{1-D}})\times i_o\end{array}\right.$$

(15)

Although the AC components of the currents in the three windings of the coupled inductor can be calculated with their slopes in Modes 1 and 5 (Equations (2) and (7)), these calculations are complicated, and the characteristics of the coupled inductor are not included. Most coupled inductors are prepared with magnetic cores, and the coupled coefficients between different windings in the inductor are close to 1. A simplified model of the coupled inductor in Mode 1 is presented in Figure 5.

The three-winding coupled inductor operating in Mode 1 is modeled with a two-terminal network (from the green arrow in the left of Figure 5). L_B is the boost inductance and L₂₃ is the inductance of the windings W₂ and W₃ in series. Inductance L₂₃ is coupled to the boost inductance with an ideal transformer with a turn ratio of n₁:(n₂ + n₃). Inductances for the isolated winding W₁ and the two windings W₂ and W₃ in series are L₁ and L₂₃, respectively. The relationship between the two inductances is

$$L_{23}=L_1\times {({\displaystyle \frac{n_2+n_3}{n_1}})}^2$$

(16)

$L_{23}^{\prime }$ is the inductance of L₂₃ transformed to the primary side of the ideal transformer with a value of L₁, which is the same as that of winding W₁. This is attributed to the tight coupling between the three windings and the same magnetic flux in these windings.

From analysis of the operating modes of the coupled inductor in an entire switching period, it can be confirmed that the variations of the currents following through windings W₁ and W₂ are zero, and the energy is mainly stored in winding W₁. The slope for the current following in the two-terminal network is di_Cp/dt = V_in/L₁. This current slope is divided into two equal parts, the slope for the current in the boost inductance and the slope for the current in the inductance $L_{23}^{\prime }$. The same inductance leads to the equal current slope with a value of V_in/(2 × L₁).

The slope for the current in the boost inductance means that its equivalent value is two times the inductance of winding W₁. This is beneficial to depress the AC component in the ripple in the current of winding W₁. Similarly, the current slope for the equivalent inductance L₂₃ can be achieved and presented in Equation (17).

$$\left\{\begin{array}{l}{\left. {\displaystyle \frac{d{i}_1}{dt}}\right|}_{M1}={\displaystyle \frac{V_{in}}{2\times L_1}}\\ {\left.{\displaystyle \frac{d{i}_{23}}{dt}}\right|}_{M1}={\displaystyle \frac{V_{in}}{2\times L_1}}\times {\displaystyle \frac{n_1}{n_2+n_3}}\end{array}\right.$$

(17)

The slopes of the three windings in Mode 1 can be estimated with the inductance of winding W₁ and turn ratios of the three windings, which can reduce the difficulty of designing the coupled inductor significantly.

Energies stored in the coupled inductor are transferred to the load of the converter in Mode 5. Windings W₂ and W₃ are in parallel, and their equivalent inductor is in series with winding W₁. To balance the currents in windings W₂ and W₃, the two windings have the same turn numbers. The equivalent inductance of the three windings of the coupled inductor can be estimated with the following equation

$$L_{cpM5}={({\displaystyle \frac{n_1+n_2}{n_1}})}^2\times L_1$$

(18)

Voltage across the equivalent inductance can be calculated as

$$V_{LcpM5}=({\displaystyle \frac{n_1+n_2}{n_1}})\times V_{dis}$$

(19)

The slope for current in winding W₁ in Mode 5 is

$$\left\{\begin{array}{l}{\left. {\displaystyle \frac{d{i}_1}{dt}}\right|}_{M5}={\displaystyle \frac{V_{dis}}{L_1}}\times {\displaystyle \frac{n_1}{n_1+n_2}}\\ {\left.{\displaystyle \frac{d{i}_{23}}{dt}}\right|}_{M5}={\displaystyle \frac{V_{dis}}{2\times L_1}}\times {\displaystyle \frac{n_1}{n_1+n_2}}\end{array}\right.$$

(20)

The slopes for currents in windings W₂ and W₃ are half of the slope of current in winding W₁.

Finally, a model of the current in winding W₁ is established

$$i_1(t)=\left\{\begin{array}{l} {\displaystyle \frac{M-2}{D}}\times i_o+{\displaystyle \frac{V_{in}}{2\times L_1}}\times (t-{\displaystyle \frac{D\times T_s}{2}}) 0<t\le D\times T_{\mathrm{s}}\\ {\displaystyle \frac{2}{D^{\prime }}}\times i_o-{\displaystyle \frac{V_{dis}}{L_1}}\times {\displaystyle \frac{n_1}{n_1+n_2}}\times [t-{\displaystyle \frac{(2D+D^{\prime })\times T_s}{2}}] D\times T_s<t\le T_s\end{array}\right.$$

(21)

Similarly, models of the currents in windings W₂ and W₃ are obtained

$$i_{23}(t)=\left\{\begin{array}{l} {\displaystyle \frac{-1}{D}}\times i_o+{\displaystyle \frac{V_{in}}{2\times L_1}}\times {\displaystyle \frac{n_1}{n_2+n_3}}\times (t-{\displaystyle \frac{D\times T_s}{2}}) 0<t\le D\times T_s\\ {\displaystyle \frac{1}{D^{\prime }}}\times i_o-{\displaystyle \frac{V_{dis}}{2\times L_1}}\times {\displaystyle \frac{n_1}{n_1+n_2}}\times [t-{\displaystyle \frac{(2D+D^{\prime })\times T_s}{2}}] D\times T_s<t\le T_s\end{array}\right.$$

(22)

With the established models, currents in the three windings of the coupled inductor can be estimated, with the boost inductance determined by winding W₁ and the turn numbers of the three windings, which can significantly reduce the difficulty of designing the coupled inductor. High-frequency parasitic effects, such as the leakage inductances and parasitic capacitors in the three windings of the coupled inductor, are neglected in the current models. The errors of the models cannot be neglected in the analysis of the converter operating at high frequencies.

3. Design on the Three-Winding Coupled Inductor

In the previous analyses, accurate and simplified models of the currents in the three windings of the coupled inductor were established. These are foundation of design of the coupled inductor. A novel method of designing the inductor will be proposed in the following section. The method is divided into four parts: determining the required conductance for winding W₁ in the coupled inductor, selecting a magnetic core with the product areas (AP) method, determining the windings’ arrangement and evaluating the loss of the coupled inductor.

3.1. Determining the Required Inductance of Winding W₁ in the Coupled Inductor

Ripples in the current following through winding W₁ (in Equation (23)) can be divided into two parts: the difference in the DC components for currents in Modes 1 and 5 and the AC component determined by the boost inductance.

$$\begin{array}{l}{i}_{L1rip}=({\displaystyle \frac{M-2}{D}}-{\displaystyle \frac{2}{1-D}})\times I_o+{\displaystyle \frac{V_{in}}{L_1}}\times {\displaystyle \frac{D\times T_s}{2}}+{\displaystyle \frac{V_{dis}}{L_1}}\times {\displaystyle \frac{n_1}{n_1+n_2}}\times {\displaystyle \frac{D^{\prime }\times T_s}{2}}\\ =\underset{Difference in DC Currents}{\underbrace{({\displaystyle \frac{M-2}{D}}-{\displaystyle \frac{2}{1-D}})\times I_o}}+\underset{AC compnents}{\underbrace{{\displaystyle \frac{V_{in}\times D\times T_s}{L_B}}\times (1+{\displaystyle \frac{n_1}{n_1+n_2}})}}\end{array}$$

(23)

From Equation (23), it can be found that the DC components of the current in winding W₁ play an important role in the formation of its ripple, and this ripple can be depressed by properly increasing the duty of the converter. More important, a high duty is also required by the high voltage gain. The voltage gain of the high step-up converter in Figure 1 can be calculated with the following equation

$$M={\displaystyle \frac{N_1+N_2+2}{1-D}}$$

(24)

where N₁ and N₂ are the turn ratios for windings W₂ and W₃ to winding W₁. More details of calculating the voltage gain of the high step-up converter can be found in the Appendix A. To decrease the ripple in the current of winding W₁ and improve the voltage gain of the converter, the duty is chosen as 0.60. With a parallel coiling structure for windings W₂ and W₃, the leakage flux can be depressed significantly. This means that the two windings have the same turn numbers and turn ratios N₁ and N₂ have the same value. Using Equation (24), the turn ratios N₁ and N₂ can be estimated at a value of 1.00. Finally, the achieved turn ratio for the three windings of the coupled inductor is 1:1:1, which is also found in refs. [19,20,24,27,28].

Furthermore, the required inductance for winding W₁ of the coupled inductor can be achieved, which is half of the boost inductance

$$L_1={\displaystyle \frac{V_{in}\times D\times T}{2\times (i_{L1rip}-i_{L1ripDC})}}\times (1+{\displaystyle \frac{n_1}{n_1+n_2}})$$

(25)

where i_L1rip is the total ripple of the current in winding W₁ and i_L1ripDC is the difference between the DC components of the currents in Modes 1 and 5 (right item in Equation (22)).

The parameters of the high step-up DC/DC converter are listed in

Table 1. High step-up DC/DC converter specifications.

Specifications	Values
Input voltage Vin	20 V
Ripple in current of winding W1 iL1rip	10 A
Output voltage Vo	200 V
Switching frequency fs	40 kHz
Output maximum power Po	200 W

. They are the foundation of the design of the three-winding coupled inductor. The difference in the DC components of the ripple of the current in winding W₁ is 7.88 A, and the AC component in the current is 2.12 A. Furthermore, the required inductance for winding W₁ can be obtained with Equation (25). The achieved result is 109.67 µH.

3.2. Product Areas (AP) Method for Magnetic Core Selecting

Selecting an appropriate core is an important work in the design of the magnetic elements. The product areas (AP) method has been widely used in designs of traditional transformers and inductors. However, no report on the AP method being used to design a multi-winding coupled inductor has been found. In Mode 1 (Figure 2a), the voltage is applied to winding W₁ and the following relation is formed

$$V_{in}=L_1\times d{i}_1/dt =n_1\times d\varphi /dt ={\displaystyle \frac{n_1\times \Delta B\times A_e}{D\times T_s}}$$

(26)

where n₁ is the turn number for winding W₁. ΔB is the variation of the magnetic induction intensity. A_e is the effective area of the magnetic core.

Assuming the converter operates in the linear region of the hysteresis loop, the following relationship can be established

$${\displaystyle \frac{\Delta B}{B_m}}={\displaystyle \frac{\Delta I_{L1}}{I_{L1}}}$$

(27)

where B_m is the peak magnetic induction intensity. ΔI_L1 is the ripple of the current in winding W₁ and I_L1 is the peak value for the current. By defining k_IL1 = ΔI_L1/I_L1, the variation of the magnetic induction intensity can be calculated as ΔB = B_m × k_IL1. The desired effective area of the magnetic core A_e is

$$A_e={\displaystyle \frac{V_{in}\times D\times T}{n_1\times k_{IL1}\times B_m}}$$

(28)

The winding area of the magnetic core is determined by the root-mean-square (RMS) values of the currents in the three windings

$$A_w={\displaystyle \frac{n_1\times I_{L1rms}+n_2\times I_{L2rms}+n_3\times I_{L3rms}}{j\times k_w}}$$

(29)

where I_Lirms is the RMS value of the current in the winding i. j is the current density of the three windings. k_w is the utilization factor of the winding window.

The areas product formula for selecting the magnetic core of the coupled inductor can be established

$$AP=A_e\times A_w={\displaystyle \frac{D\times V_{in}\times {\displaystyle \sum _{1\le i\le 3}\frac{n_i}{n_1}\times I_{Lirms}}}{f\times k_{IL1}\times B_m\times j\times k_w}}$$

(30)

Using the simplified models of currents in the three windings of the coupled inductor in Equations (21) and (22), RMS values for these currents are calculated and listed in

Table 2. RMS values for currents in the three windings and Litz wires to coil these windings.

Windings	RMS Value of Current	Litz Wires
W1	10.81 A	200 strands/AWG40
W2 and W3	2.05 A	50 strands/AWG40

. Furthermore, the peak magnetic induction intensity, current density and utilization factor of the winding window are set as 0.36 T, 7.50 A/mm² and 0.60, respectively. The obtained AP value for the three-winding coupled inductor is 28,262.60 mm⁴. The corresponding value for the ETD39 core manufactured by the Ferroxcube company using the material 3C95 is 29,287.50 mm⁴.

3.3. Winding Arrangement for the Coupled Inductor

With the selected magnetic core, the turn number for the winding W₁ can be estimated using Equation (34). Furthermore, the turn numbers for the other two windings are calculated with the turn ratios of N₁ and N₂ calculated using Equation (24).

$$n_1={\displaystyle \frac{V_{in}\times D\times T_s}{k_{IL1}\times B_m\times A_e}}$$

(31)

To avoid the core entering a saturation state, the variation of the magnetic induction intensity is set as 0.25 T and smaller than the peak magnetic induction intensity in Equation (28). The effective area of the ETD 39 core is 125.0 mm². The turn number for winding W₁ in Equation (31) is 24.25. In preparing the coupled inductor, the three windings are coiled with the same turns number of 24.0.

The boost inductance L₁ based on the achieved turn number and magnetic core without an air gap can be estimated as

$$L_1^{\prime }={\displaystyle \frac{n_1^2\times u\times A_e}{l_e}}$$

(32)

where u and l_e are the permeability and effective length of the magnetic circuit of the magnetic core. The achieved inductance is 2.12 mH. An air gap must be implemented in the center leg of the magnetic core. The length of the gap δ can be calculated as

$$L_1={\displaystyle \frac{n_1^2}{1/A_L+\delta /({\mu }_0\times A\times \delta )}}$$

(33)

where u₀ is the permeability of the vacuum and A is the area of center leg of the core. A_L can be calculated as A_L = u × A_e/l_e. The obtained length of the air gap is about 1.00 mm.

Selecting the appropriate copper wires to coil windings in the coupled inductor is another important work. To reduce losses caused by eddy effects, Litz wires are usually used to prepare transformers and inductors. The required effective sectional areas of the Litz wires for coiling the three windings are 1.44 mm² and 0.27 mm², respectively. Considering the actual dimensions of the ETD39 core former, round Litz wire of 200 strands/AWG40 with an effective sectional area of 1.54 mm² and diameter of 1.90 mm is adopted to coil winding W₁. Windings W₂ and W₃ are coiled with Litz wire of 50 strands/AWG40. The diameter of the Litz wire and its effective sectional area are 1.00 mm and 0.38 mm².

The winding arrangement has significant impacts on the performance of the coupled inductor. A concentric winding structure is used. Three windings of the coupled inductor are coiled on the central leg of the core in a sequence of W₁-W₂-W₃ (presented in Figure 6a). The two inner layers are winding W₁ and the two outer layers are windings W₂ and W₃. No interleaving is formed between winding W₁ and the other two windings. Weak coupling may lead to a large leakage inductance. By dividing winding W₁ into two parts, two different winding arrangements with a strong coupling are formed. Corresponding coiling sequences for the four layers of the windings are 1/2W₁-W₂-W₃-1/2W₁ and 1/2W₁-W₂-1/2W₁-W₃ (in Figure 6b,c).

To investigate the impact of winding arrangement on the performance on the coupled inductor, a finite element analysis (FEA) on the coupled inductors was implemented with Maxwell package. In simulations, the eddy current was selected as a solution type to calculate the inductances and magnetic induction intensity vectors. In the meshing processes, the maximum lengths for the elements in the core and the windings were set as 1.0 mm and 0.5 mm, respectively. The convergence criterion was that the difference for the energies in two adjacent calculations had to be smaller than 0.5% of the total energy. The achieved inductances for winding W₁ and the leakage inductances with different winding arrangements are listed in

Table 3. Inductance of winding W₁ and leakage inductances of the coupled inductor with different winding arrangements.

Winding Structure	Inductance of Winding W1	Leakage Inductance
W1-W2-W3	105.94 µH	5.27 µH
1/2W1-W2-W3-1/2W1	107.33 µH	3.03 µH
1/2W1-W2-1/2W1-W3	106.65 µH	2.91 µH

. From Table 3, it can be found that the difference in the inductances of windings W₁ with different winding arrangements can be neglected. The inductance is determined by the turn number of winding W₁ and the length of the air gap in the center leg of the magnetic core. Interleaving between the three windings of the coupled inductor can suppress magnetic flux leakage and decrease the leakage inductance. The minimum leakage inductance for the coupled inductor is 2.91 µH in a winding sequence of 1/2W₁-W₂-1/2W₁-W₃, which is about 55.2% of the leakage inductance with a winding sequence of W₁-W₂-W₃.

Comparing the current density in the three windings of the coupled inductor with different coiling sequences (in Figure 7), it can be found that the interleaving between adjacent layers of different windings can improve the uniformity of current in the coupled inductor. The highest current density in the coupled inductor with a winding sequence of W₁-W₂-W₃ is located at the interface between winding W₁ and winding W₂, which originates from the proximity effect in the two windings (in bottom of Figure 7a). This is consistent with the highest magnetic induction intensity for the magnetic leakage (in the top of Figure 7a). The uniformity of the current densities of the coupled inductor with the coiling sequences 1/2W₁-W₂-W₃-1/2W₁ and 1/2W₁-W₂-1/2W₁-W₃ is improved significantly (at the bottom of Figure 7b,c). The highest magnetic induction intensity of the magnetic leakage in the coupled inductors in the two winding sequences is depressed significantly (at the top of Figure 7b,c). This is consistent with the achieved leakage inductances in Table 3. The coiling sequence of 1/2W₁-W₂-1/2W₁-W₃ is an appropriate choice to prepare the coupled inductor.

3.4. Loss Evaluation for the Three-Winding Coupled Inductor

Losses in the coupled inductor can be divided into two parts, winding loss P_wind and core loss P_core. Winding loss mainly originates from the energy dissipations by AC resistances of the windings and can be calculated as

$$P_{wind}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{1\le i\le 3}{R}_{ACWi}}\times I_{Lirms}^2$$

(34)

where R_ACWi is the AC resistance of the winding i of the coupled inductor and I_Lirms is the RMS value of the current in the winding i. The ratio of the AC resistance to the DC resistance of the winding realized with multilayer foils is defined as F_R and calculated with the Dowell equation [36].

$$\begin{array}{ll}\hfill F_R& ={\displaystyle \frac{R_{skin}+R_{prox}}{R_{dc}}}=F_{Rskin}+F_{Rprox}\hfill \\ & =A{\displaystyle \frac{\sinh (2A)+\sin (2A)}{\cosh (2A)-\cos (2A)}}+A{\displaystyle \frac{2(N_l^2-1)}{3}}{\displaystyle \frac{\sinh A-\sin A}{\cosh A+\cos A}}\hfill \\ & =A{\phi }_1+A{\displaystyle \frac{2(N_l^2-1)}{3}}{\phi }_2\hfill \end{array}$$

(35)

where F_Rskin and F_Rprox are ratios for the resistances caused by the skin effect and the proximity effect. A is the thickness of the foil and N_l is the layer number of the foils. For windings realized with Litz wire, effective layer number N_ll for the winding is modeled by the strand number k and N_l is the real layer number of the winding [37]

$$N_{ll}=\sqrt{k}{N}_l$$

(36)

Considering the skin depth, the effective thickness for the strand layers in Litz wire A_ef is modeled

$$\left\{\begin{array}{l}{A}_{ef}={({\displaystyle \frac{\pi }{4}})}^{0.75}\sqrt{\eta }{\displaystyle \frac{d_{str}}{\delta }}\\ \eta =\sqrt{{\displaystyle \frac{d}{p}}}=\sqrt{{\displaystyle \frac{\sqrt{k}{d}_{str}}{p}}}\\ \delta ={\displaystyle \frac{1}{\sqrt{\pi f{\mu }_0\sigma }}}\end{array}\right.$$

(37)

where η is the equivalent porosity factor for the Litz wire windings, p is the sum of the diameter and the distance between two adjacent wires, δ is the skin depth, μ₀ is the free space permeability, f is the operating frequency, and σ is the conductivity. With the effective layer number N_ll and effective thickness A_ef (in Equations (36) and (37)) for Litz wire, the AC resistances of the three windings in the coupled inductor with winding sequences of 1/2W₁-W₂-1/2W₁-W₃ are calculated and presented in Figure 8 with dashed lines.

The Steinmetz equation is widely used in calculating core loss, which is listed as follows [38].

$$P_{core}=K\times f^{\alpha }\times \Delta B^{\beta }\times C_T$$

(38)

where K, α, and β are parameters related to the core material, f is the switching frequency of the converter (in kHz) and ΔB is the peak-to-peak magnetic induction intensity (in mT) of the converter with the maximum output power. C_T is the temperature coefficient for core loss.

$$C_T=c_0-c_{1}t-c_2{t}^2+c_3{t}^3$$

(39)

In the switching frequency range from 20.0 to 200.0 kHz, K, α and β are 5.46 × 10⁻¹⁴, 1.70 and 2.74, respectively. The parameters for the temperature coefficient C_T, c₀, c₁, c₂ and c₃ are 1.00, 2.27 × 10⁻³, 1.28 × 10⁻⁵ and 3.88 × 10⁻⁷. The unit of the core loss is mW/cm³.

4. Experimental Results

A coupled inductor with coiling sequence of 1/2W₁-W₂-1/2W₁-W₃ and air gap length of 1.05 mm in the center leg was prepared and is presented in Figure 9a. The inductor was measured with LCR meter Chroma 11025 (Chroma, Burlington, VT, USA). Under the frequency of 40.0 kHz, the inductance for winding W₁ was 105.04 µH, which is consistent with the results of theoretical calculations in a value of 106.65 µH (in Table 3). The error for the inductor was about 1.53% of the measured result. The measured leakage inductance at 40.0 kHz was 3.46 µH and slightly larger than the results of theoretical calculations with a value of 0.58 µH. The difference between the leakage inductance as measured and calculated was about 16.76%. The measured AC resistances at the frequency of 40.0 kHz for the three windings of the coupled inductor were 67.2, 118.7 and 135.5 mΩ (in Figure 8). Due to the resistances of the cables and the kelvin clip of the LCR meter, the measured results were slightly larger than the real resistances. These are consistent with the results of theoretical calculations at values of 68.5, 114.8 and 137.6 mΩ made with the improved Dowell equation for Litz wires.

To compare with other coupled inductors applied in high step-up DC/DC converters, the parameters for the converters and coupled inductors are listed in

Table 4. Comparison between designed inductor and other coupled inductors applied in high step-up DC/DC converters.

Converter	Voltage Gain	Rated Power	Switching Frequency	Magnetic Core	Turn Ratio n1:n2:n3	Inductance of Winding W1	Urcore
[39]	${\displaystyle \frac{1+2N_{31}-N_{21}}{(N_{31}-N_{21})(1-D)}}$	160 W	50 kHz	EE42/21/20	29:11:24	220 µH	7.05 W/cm3
[27]	${\displaystyle \frac{2+D+N_{31}-N_{21}}{(1-N_{21})(1-D)}}$	200 W	60 kHz	EE42/21/20	1:0.61:0.63	200 µH	8.81 W/cm3
[40]	${\displaystyle \frac{(1+N_2)(1-D)+2+N_3}{1-D}}$	200 W	50 kHz	ETD44/22/15	20:40:40	130 µH	11.23 W/cm3
[20]	${\displaystyle \frac{2+N_{21}+3N_{21}D+2N_{31}-2N_{31}D}{{(1-D)}^2}}$	250 W	70 kHz	EE42/21/15	1:1:1	16 µH	14.45 W/cm3
[19]	${\displaystyle \frac{3+2N_{21}+N_{31}(2+D)}{1-D}}$	500 W	50 kHz	EE55/28/21	18:7:7	80 µH	9.61 W/cm3
This paper	${\displaystyle \frac{N_1+N_2+2}{1-D}}$	200 W	40 kHz	ETD39/20/13	24:24:24	105 µH	17.39 W/cm3

. The turn ratios for the coupled inductors in refs. [20,27] were normalized. The switching frequency had a significant impact on the volume of the magnetic element. The switching frequencies for these converters (Table 4) range from 40 kHz to 70 kHz, and the difference can be neglected. Parameter U_rcore, which describes the utilization rate of the magnetic core, can be defined as

$$U_{rcore}=P_o/V_e$$

(40)

where P_o is the rated output power of the converter and V_e is the effective volume of the core. A lower U_rcore means a lower utilization rate of the magnetic core. The U_rcore for the coupled inductor designed in this paper is higher than those of other coupled inductors listed in Table 4. This means that the proposed design method for a coupled inductor is correct.

To further verify the designed coupled inductor, a prototype of the high step-up DC/DC converter was prepared (Figure 9b). Its testing platform is presented in Figure 9c. The coupled inductor is presented in the upper part of the PCB board (Figure 9b). An N channel MOSFET IRFB38N20DPBF from the Infineon company (Neubiberg, Germany) was used as the switch S. To reduce the number of types of components, all diodes D₁, D₂, D₃ and D_o were realized with ultra-fast recovery diodes MUR1640CTG manufactured by the Onsemi semiconductor company. The capacitance for the clamping capacitor C₁ was 100.0 μF and the capacitance for C₂ and C₃ were adopted with the same value of 47.0 μF. The capacitances for input filter and output filter capacitors (C_in and C_o) were 94.0 Μf and 200.0 μF, respectively. The main components of the converter are listed in

Table 5. Main components in prepared high step-up converter.

Components	Parameters
Magnetic Core	ETD39/PC95
n1:n2:n3	24:24:24
Inductance of winding 1	105.04 μF
MOSFET S	IRFB38N20DPBF
D1, D2, D3 and Do	MUR1640CTG
Cin	94.0 μF
Co	200.0 μF
C1	100.0 μF
C2 and C3	47.0 μF

.

The measured waveforms of the high step-up converter with an output power of 80.0 W are presented in Figure 10. The duty of the driving signal of the switching MOSFET S was about 0.61 and the output of the converter was 199.74 V (waveforms in green and wine in Figure 10a). These are consistent with the results of the analysis of the operating modes. The correctness of the operating principle of the converter is preliminarily validated.

The average value of the current in winding W₁ (orange waveform in Figure 10a) was about 4.39 A, which is slightly larger than the theoretical value (4.28 A with an efficiency of 93.27%). This difference can be neglected. The RMS value for the measured current in winding W₁ was 4.86 A. The theoretical value obtained with the model proposed in this paper is 4.56 A. The measured ripple in the current of winding W₁ was 5.31 A. The difference in the DC components of this current between Mode 1 and Mode 5 is 3.96 A. The AC component for the ripple was 1.62 A (in Equation (21)). The ripple in the theoretical calculation is 5.58 A. The difference in the ripples between measured and theoretically calculated is 0.27 A, which is about 5.08% of the measured ripple. The correctness of the current model for the current in winding W₁ is verified.

The measured rising slope of the current in winding W₁, SLL_1M1 (labeled with a green dashed line in the waveform in Figure 10a) is 89.36 mA/µs, and the slope form model established in this paper is 94.88 mA/µs. The difference between the measured and theoretically calculated slope is 6.18 mA/µs (about 6.2% of the measured slope). Similarly, the measured descending slope SLL_1M5 (labeled with a dashed red line) and the calculated slope are 149.53 mA/µs and 142.32 mA/µs, respectively. The measured rising slope and falling slopes for the current in winding W₁ are consistent with the results from the model in Equation (21). The correctness of the model of the current in winding W₁ and the design method for the three-winding coupled inductor are verified.

The waveforms for currents in windings W₂ and W₃ are presented in the middle and at the bottom of Figure 10b,c). The average values for the two currents are zero, which is consistent with the results of analysis of the operating modes of the high step-up DC/DC converter in Section II. The measured rising slopes of the two currents in Mode 1 (SLL_2M1 and SLL_3M1, labeled with a black dashed line) are −52.16 mA/µs, which are similar to the established model, which had a value −47.44 mA/µs. The negative sign means that the currents follow out the dotted terminals of the two windings. The measured falling slopes in the two windings are 46.15 mA/µs and 102.13 mA/µs (SLL_2M5 and SLL_3M5, labeled with blue and purple dashed lines). The summation of the two slopes is equal to the corresponding slope of the current in winding W₁ (142.32 mA/µs). This is consistent with the results of operating mode analysis on the Mode 5.

The efficiencies of the prototype of the high step-up converter were tested with loads from 10 W to 200 W. The results are presented in Figure 11. The efficiencies of the high step-up DC/DC converter are higher than 90.0% in the tested range from 20 W to 200 W, in which the appropriately designed coupled inductor plays an important role. The peak efficiency of the converter is about 94.3%. The losses in the coupled inductor with an output of 80.0 W were analyzed in detail. The RMS values for currents in the three windings W₁, W₂ and W₃ of the coupled inductor were 4.86 A, 0.86 A and 0.87 A, respectively. With the measured AC resistances of these windings at 40 kHz, the power loss in the three windings of the coupled inductor in the windings was 1.78 W. With the variation of 100 mT of the magnetic induction intensity in the coupled inductor, the power loss of the magnetic core could be restrained to a value of 1.20 W. The total loss of the converter is about 2.98 W, which is about 55.4% of the total power loss of the converter.

5. Conclusions

Due to the prospects for application in renewable energy generation and DC microgrids, high step-up DC/DC converters have attracted tremendous research interest. By combining a multiple-winding coupled inductor with voltage multipliers, the voltage gain of a boost converter can be improved significantly. However, the lack of a method of designed multi-winding coupled inductors has become an important restriction on the applications of high step-up DC/DC converters.

By analyzing the operating modes of a boost converter combined with a three-winding coupled inductor and double voltage multipliers, accurate models of currents in the three windings of the coupled inductor are established. Furthermore, simplified models of these currents are proposed. The relationship between the ripple in the input current of the coupled inductor (Winding W₁) and the boost inductance is established.

A method of designing a coupled inductor is proposed, in which a method of estimating the required inductance for winding W₁ of the coupled inductor, an improved product areas (AP) method to select a magnetic core, the influence of the windings’ arrangement and a loss evaluation for the coupled inductor are investigated in detail.

A prototype of the 200 W DC/DC converter with an input of 20 V and an output of 200 V was prepared and testified. The results indicate that the measured waveforms are consistent with the established models of currents in a coupled inductor. The efficiency of the converter is higher than 90.0% in the testing range for outputs from 20 W to 200 W. The correctness of the method of designing a multi-winding coupled inductor has been verified. The proposed method can be applied to the design of multi-winding coupled inductors, especially inductors applied in high step-up DC/DC converters operating as interfaces between renewable sources and DC microgrids.