Three-Phase Single-Stage AC-DC Converter Using Series–Series Compensation Circuit in Inductive-Power-Transfer-Based Small Wind Power Generation System

Abstract

In this paper, a three-phase single-stage AC-DC converter for an IPT-based small wind power generation system (WPGS) with an S-S compensation circuit is proposed. It applies a three-phase single-stage AC-DC converter to improve the input power factor (PF), efficiency, and reliability in small WPGSs. Also, inductive power transfer (IPT) was applied to compensate for brush wear in the nacelle of small and medium-sized wind turbines while ensuring electrical safety. In conditions of the three-phase Permanent Magnet Synchronous Generator (PMSG) voltage (80~260 V_rms) for the wind turbine and the load (150~1000 W), it was verified that the desired output voltage below 3% can be controlled through the fixed link voltage (V_Link) control without wireless communication. A 1 kW prototype was built and tested to demonstrate its applicability to the rotation of small and medium-sized wind turbine nacelles instead of brushes and slip rings.

1. Introduction

Small-scale wind power systems ranging from 1 kW to 3 kW are being increasingly applied to grid-connected wind power systems for self-generation in urban and rural communities. Given their installation near residential areas, efficiency, reliability, and electrical stability are particularly important [1,2,3].

Existing small-scale wind power generation systems (WPGSs) are being upgraded to improve efficiency by boosting the generation voltage of Permanent Magnet Synchronous Generators (PMSGs) from 80 V_rms to 250 V_rms. These systems incorporate a three-phase power factor correction (PFC) AC-DC converter and a non-isolated DC-DC converter to enhance the input power factor and overall performance [4,5,6].

By applying a PFC AC-DC converter at the output of the PMSG, the three-phase AC generation voltage and current can be made in phase, thereby improving the power factor (PF) and achieving a nearly sinusoidal AC current waveform. However, the use of a non-isolated DC-DC converter necessitates the use of a low-frequency transformer (50 Hz or 60 Hz) at the output of the grid-connected inverter, which limits the compactness of small-scale wind power systems.

To achieve compactness, high-frequency transformers (HFTR) are used in the DC-DC converter stage, providing isolation and a wide output voltage range. Nevertheless, the power conversion devices of small-scale WPGSs are complex due to the two-stage configuration of PFC and DC-DC converters, leading to increased circuit complexity and control challenges [7,8,9].

To address these issues, recent research has focused on developing small-scale WPGSs that utilize a single-stage AC-DC converter, which can improve the power factor (PF) and provide isolated output within a single power conversion unit. This approach allows for a more compact and lightweight design, reducing the weight burden on the nacelle, and can also be expanded for grid-connected applications [10,11].

However, in practical applications, the three-phase wind turbine generators can rotate with the nacelle due to wind speed or direction changes. This makes it challenging to ensure electrical stability when connecting the generator voltage to the power conversion device located in a fixed tower as it can lead to issues such as cable twisting.

Slip rings and brushes, as shown in Figure 1a, are used to address this issue. However, when power is transmitted through the brushes during nacelle rotation, issues such as sparks and contact wear arise, leading to reduced reliability of the wind power system [12]. Therefore, to reduce disadvantages such as brush wear and decreased reliability caused by nacelle rotation in three-phase medium and small-scale wind turbines, recent research has focused on wireless power transfer (WPT) systems for Inductive Power Transfer (IPT)-based small-scale wind power generation systems (WPGSs) [13].

The basic structure of the IPT-based small-scale WPGS is illustrated in Figure 1b. The primary contactless coupler is fixed to the shaft below the main bearing unit of the nacelle tower, allowing it to rotate with the nacelle according to wind direction. The secondary coupler, which maintains a fixed air gap with the primary side, is secured inside the tower. The fixed secondary coupler is connected to a high-frequency diode rectifier and a high-voltage DC link, which can be connected to a grid-tied inverter. Additionally, for IPT-based wireless power transfer, compensation circuits such as S-S (series–series) and S-P (series–parallel) circuits are commonly used [14,15]. These main circuit topologies for wireless power transfer are being studied for application in small-scale wind power systems.

Although single-stage converter modules for IPT-based three-phase small wind power systems have been reviewed, they are complex because they implement three-phase circuits based on single-phase designs, making the main circuit configuration complicated. Additionally, the need for duty control to manage gain variations due to load changes or wind speed variations can lead to hard switching at minimum duty during light or no-load conditions [8].

Furthermore, in IPT-based small WPGSs, it is essential to maintain a constant output voltage (V_o) at the secondary-side rectifier output, which serves as the input supply voltage for the grid-tied inverter. This typically requires controlling the output voltage (V_o) via wireless communication such as Bluetooth or Wi-Fi. However, failure in wireless communication control can result in unstable output voltage (V_o), potentially causing issues in the overall wind power system.

To address this issue, a DC-DC converter for the stabilized power supply must be applied at the secondary rectifier output. However, this approach can lead to increased power conversion losses, reducing efficiency.

This paper proposes an IPT-based three-phase single-stage AC-DC converter that integrates a PFC stage for power factor correction and a wireless power transfer DC-DC stage into a single power conversion unit. This design aims to improve the power factor and ensure electrical safety by addressing brush wear in the nacelle of small wind power systems.

In the proposed IPT-based three-phase single-stage AC-DC converter, the gain characteristics of the link voltage (V_Link) according to the value of the boost inductor (L_B) in the PFC stage for three-phase power factor correction were analyzed and presented. The DC-DC stage for wireless power transfer applies an S-S (series–series) compensation circuit, which shows no gain variation at the resonant frequency due to the air gap of less than 10 mm between the primary and secondary sides of the contactless coupler.

Through a 1 kW prototype experiment, it was demonstrated that, in the small wind turbine generation voltage range (112 V_rms to 250 V_rms), according to wind speed, the secondary output voltage (V_o) could operate within a 3% variation range solely by controlling the link voltage (V_Link) in the PFC stage, without the need for wireless communication control. Furthermore, comparative experiments were conducted to control the secondary output voltage (V_o) in real time via wireless communication, demonstrating the feasibility of applying this system to small wind power systems.

2. Three-Phase Single-Stage AC-DC Converter in Inductive-Power-Transfer-Based Wind Power Systems

2.1. Three-Phase Single-Stage AC-DC Converter

This section discusses an IPT-based three-phase single-stage AC-DC converter that uses the generated line voltage (V_LL: 110 V_rms to 250 V_rms) from a three-phase small wind turbine as the input power source. The converter aims to improve the power factor (PF) and total harmonic distortion (THD_i) of the three-phase AC power source and control the isolated output voltage (V_O: 400 V_DC) for grid connection. The IPT-based three-phase single-stage AC-DC converter, as shown in Figure 2a, consists of input filter inductors (L_F = L_FA, L_FB, L_FC) and input filter capacitors (C_F = C_FA, C_FB, C_FC), boost inductors (L_B = L_B1, L_B2, L_B3), input rectifier diodes, voltage divider link capacitors (C₁, C₂), a flying capacitor (C_B), voltage divider capacitors (C_B1, C_B2), circulating diodes (D₁, D₂), and four main switching devices (Q₁, Q₂, Q₃, Q₄). Additionally, in the IPT-based small wind power system, the air gap between the primary and secondary sides of the contactless coupler is fixed to be less than 1 cm, and, due to minimal coupling variation, the leakage inductances (L_l1, L_l2) and magnetizing inductance (L_m) parameters of the coupler do not vary significantly. Therefore, an S-S (series–series) compensation circuit with resonant capacitors (C_r1, C_r2) is applied.

Therefore, near the resonant frequency of the S-S compensation circuit, where the gain does not vary, the switching devices (Q₁, Q₂, Q₃, Q₄) operate alternately at a fixed switching frequency with a 50% duty cycle controlled by the phase control (D₁T). A square wave voltage (V_P) with a 50% duty cycle at one-fourth of the link voltage (V_Link) is always applied to the terminal voltage (V_P) across the primary S-S compensation circuit (a-b). This ensures that the primary switching devices achieve zero-voltage switching (ZVS) under all load and input voltage conditions, and the voltage corresponding to the resonant circuit gain characteristic is transferred to the secondary side of the coupler. Additionally, by applying a doubler rectification method composed of secondary rectifier diodes and doubler capacitors (C_o1, C_o2), the voltage applied to the load is approximately doubled compared to the voltage applied to the secondary side of the contactless coupler. According to the phase control (D₁T) of the switching devices (Q₁/Q₄, Q₂/Q₃), when the switching devices Q₁ and Q₂ (or Q₃ and Q₄) are turned on simultaneously, the three-phase phase voltages (V_AN, V_BN, V_CN) are applied to the boost inductors (L_B1, L_B2, L_B3), storing energy. When the switching device Q₁ (or Q₃) is turned off, the energy stored in the boost inductors (L_B1, L_B2, L_B3) is reset, operating in discontinuous mode. The boost inductor currents (I_B1, I_B2, I_B3), which operate in discontinuous mode, regulate the PFC stage link voltage (V_Link) to control the output voltage (V_o) of the rectifier stage at the secondary side of the contactless coupler. The detailed operation of this process is described in Section 2.2.

2.2. Operation Modes of Inductive-Power-Transfer-Based Three-Phase Single-Stage AC-DC Converter

The operation modes of the IPT-based three-phase single-stage AC-DC converter are illustrated in Figure 3, corresponding to the time periods (t₀~t₉) shown in Figure 2b. The operation modes for each period are as follows.

MODE 1 (t₀–t₁)/MODE 2 (t₁–t₂): The primary resonant current (I_P) flows due to the previously negative resonant current through the path of the anti-parallel diode of switching device Q₂ → voltage divider capacitor (C_B1) → primary winding of the contactless coupler (N_P) → resonant capacitor (C_r1) → anti-parallel diode of switching device Q₂, and simultaneously through the path of the anti-parallel diode of switching device Q₂ → flying capacitor (C_B) → voltage divider capacitor (C_B2) → primary winding of the contactless coupler (N_P) → resonant capacitor (C_r1) → anti-parallel diode of switching device Q₂. At this time, at t₀, Q₂ turns on with zero-voltage switching (ZVS). In the t₀–t₁ period of MODE 1, the filter capacitor voltages of the a-phase and c-phase (V_CFA, V_CFC) are applied to the boost inductors L_B1 and L_B3, respectively. The boost inductor currents (I_B1, I_B3) flow through the path of filter capacitors (C_FA, C_FC) → boost inductors (L_B1, L_B3) → input rectifier diode → switching devices (Q₁, Q₂) → filter capacitors (C_FA, C_FC) → boost inductors (L_B1, L_B3), and begin to store energy in the boost inductors (L_B1, L_B3). Simultaneously, the energy stored in the boost inductor (L_B2) is reduced through the path of boost inductor (L_B2) → filter capacitor (C_FB) → anti-parallel diode of switching device Q₂ → flying capacitor (C_B) → circulating diode (D₂) → voltage divider link capacitor (C₂) → input rectifier diode → boost inductor (L_B2), and is completely reset at t₁. Since switching devices Q₁ and Q₂ remain turned on, a quarter of the link voltage (V_Link) is applied to the primary resonant circuit, allowing the primary and secondary resonant currents (I_P, I_S) to flow. In the t₁–t₂ period of MODE 2, the primary resonant current (I_P) flows through the path of voltage divider capacitor (C_B1) → switching device (Q₂) → resonant capacitor (C_r1) → primary winding of the contactless coupler (N_P) → voltage divider capacitor (C_B1), and simultaneously through the path of link capacitor (C₁) → switching devices (Q₁, Q₂) → resonant capacitor (C_r1) → primary winding of the contactless coupler (N_P) → voltage divider capacitor (C_B2) → circulating diode (D₂) → link capacitor (C₁). Additionally, in the secondary voltage doubler rectifier, the secondary resonant load current (I_S) flows through the path of the secondary winding of the contactless coupler (N_S) → secondary resonant capacitor (C_r2) → output rectifier diode → output capacitor (C_o)/load → doubler capacitor (C_o2) → secondary winding of the contactless coupler (N_S), delivering power to the load.

MODE 3 (t₂–t₃): The period t₂ to t₃ is the dead time for the switching devices (Q₁, Q₄). At t₂, switching device Q₁ turns off, and the boost inductor currents (I_B1, I_B3) and the primary resonant current (I_P) begin to charge and discharge the parasitic capacitors (C_p) of switching devices Q₁ and Q₄, as well as the parasitic capacitors (C_jp) of circulating diodes D₁ and D₂. The voltages of the parasitic capacitor (C_p) of switching device Q₁ and the parasitic capacitor (C_jp) of circulating diode D₂ are charged to V_Link/2. In contrast, the voltages of the parasitic capacitor (C_p) of switching device Q₄ and the parasitic capacitor (C_jp) of circulating diode D₁ are fully discharged to 0. Afterward, when the negative current flows through the anti-parallel diode of switching device Q₄, if Q₄ turns on at t₃, it operates with zero-voltage switching (ZVS).

MODE 4 (t₃–t₄): At t₃, switching device Q₄ turns on, and the energy stored in the boost inductors L_B1 and L_B3 is reset due to the differential voltage between the filter capacitor voltages (V_CFA, V_CFC) and the link capacitor (C₁) voltage, which is half of the link voltage (V_Link). Consequently, the reset slope of the boost inductor currents (I_B1, I_B3) gradually decreases, flowing through the path of filter capacitors (C_FA, C_FC) → boost inductors (L_B1, L_B3) → input rectifier diode → link capacitor (C₁) → circulating diode (D₁) → switching device (Q₂) → filter capacitors (C_FA, C_FC) → boost inductors (L_B1, L_B3). At this time, a quarter of the link voltage (V_Link) is applied to the primary resonant circuit terminals (a-b), allowing the primary and secondary resonant currents (I_P, I_S) to flow. The primary resonant current (I_P) follows two distinct paths. The first path flows through the voltage divider capacitor (C_B1), the switching device (Q₂), the resonant capacitor (C_r1), the primary winding of the contactless coupler (N_P), and returns to the voltage divider capacitor (C_B1). Concurrently, the second path involves the flying capacitor (C_B) or the link capacitor (C₂), passing through the switching device (Q₂), the resonant capacitor (C_r1), the primary winding of the contactless coupler (N_P), the voltage divider capacitor (C_B2), and subsequently either the flying capacitor (C_B) or, alternatively, the switching device (Q₄) leading to the link capacitor (C₂). The secondary voltage doubler rectifier operates with the secondary resonant load current (I_S) flowing through the path of the secondary winding of the contactless coupler (N_S) → secondary resonant capacitor (C_r2) → output rectifier diode → output capacitor (C_o) → output voltage divider capacitor (C_o2) → secondary winding of the contactless coupler (N_S).

MODE 5 (t₄–t₅): The period t₄ to t₅ is the dead time for the switching devices Q₂ and Q₃. At t₄, switching device Q₂ turns off, and the boost inductor current I_B3 and the primary resonant current (I_P) cause the parasitic capacitors (C_p) of switching devices Q₂ and Q₃ to charge and discharge to half the link voltage (V_Link) and 0 volts, respectively. At this time, the polarity of the voltage (V_P) at the primary resonant circuit terminals (a–b) reverses, causing the voltage polarity of the secondary winding (N_S) of the contactless coupler also to reverse. At t₅, the voltage of the parasitic capacitor (C_p) of switching device Q₃ discharges from V_Link/2 to 0 volts, and when the negative current flows through the anti-parallel diode of switching device Q₃, Q₃ turns on with zero-voltage switching (ZVS) at t₅.

MODE 6 (t₅–t₆): At t₅, switching device Q₃ turns on with zero-voltage switching (ZVS), and, during the period t₅ to t₆, both switching devices Q₃ and Q₄ remain turned on. During this period, the boost inductor current (I_B3) flows through the path of boost inductor (L_B3) → input rectifier diode → link capacitor (C₁) → circulating diode (D₁) → flying capacitor (C_B) → anti-parallel diode of switching device Q₃ → filter capacitor (C_FC) → boost inductor (L_B3), and is quickly reset due to the reverse bias provided by the link voltage (V_Link). A voltage of −V_Link/4 is applied to the primary resonant circuit terminals (a-b), allowing the primary and secondary resonant currents (I_P, I_S) to flow. However, the primary resonant current (I_P) previously flowing affects the ZVS operation of switching device Q₃ by flowing simultaneously through the paths of anti-parallel diode of switching device Q₃ → resonant capacitor (C_r1) → primary winding of the contactless transformer (N_P) → voltage divider capacitor (C_B2) → anti-parallel diode of switching device Q₃ and anti-parallel diode of switching device Q₃ → resonant capacitor (C_r1) → primary winding of the contactless transformer (N_P) → voltage divider capacitor (C_B1) → flying capacitor (C_B) → anti-parallel diode of switching device Q₃.

MODE 7 (t₆–t₇): Since both switching devices Q₃ and Q₄ remain turned on, the filter capacitor voltage (V_CFB) of phase b is applied to the boost inductor L_B2, storing energy. During this time, the boost inductor current (I_B2) flows through the path of the filter capacitor (C_FB) → switching device (Q₃) → switching device (Q₄) → input rectifier diode. Simultaneously, a voltage of −V_Link/4 is applied to the primary resonant circuit terminals (a-b), causing the primary resonant current (I_P) to flow through the path of voltage divider capacitor (C_B2) → primary winding of the contactless coupler (N_P) → resonant capacitor (C_r1) → switching device (Q₃) → voltage divider capacitor (C_B2). Concurrently, the primary resonant current also flows through the path of the link capacitor (C₂) → circulating diode (D₁) → voltage divider capacitor (C_B1) → primary winding of the contactless coupler (N_P) → resonant capacitor (C_r1) → switching device (Q₃) → switching device (Q₄), delivering power to the secondary voltage doubler rectifier. The secondary voltage doubler rectifier operates with the secondary resonant current (I_S) flowing through the path of the secondary winding of the contactless coupler (N_S) → output voltage divider capacitor (C_o1) → output capacitor/load → output rectifier diode → secondary resonant capacitor (C_r2) → secondary winding of the contactless coupler (N_S), delivering power.

MODE 8 (t₇–t₈): The interval t₇ to t₈ is the dead time for the switching devices Q₁ and Q₄. At t₇, when switching device Q₄ turns off, the boost inductor current I_B2 and the primary resonant current I_P cause the parasitic capacitors (C_p) of switching devices Q₁ and Q₄, as well as the parasitic capacitors (C_jp) of circulating diodes D₁ and D₂, to begin discharging and charging simultaneously. The voltages of the parasitic capacitor (C_p) of switching device Q₄ and the parasitic capacitor (C_jp) of circulating diode D₁ are charged to V_Link/2. In contrast, the voltages of the parasitic capacitor (C_p) of switching device Q₁ and the parasitic capacitor (C_jp) of circulating diode D₂ are discharged to 0. Subsequently, the negative current begins to flow through the anti-parallel diode of switching device Q₁, and, at t₈, Q₁ turns on with zero-voltage switching (ZVS). The subsequent operation repeats from MODE 1.

3. Analysis of Power Factor Correction Operation Characteristics in a Three-Phase Single-Stage AC-DC Converter

The three-phase single-stage AC-DC converter applied to the IPT-based small wind power system controls both the PFC stage power factor compensation and the DC-DC stage, which employs an S-S compensation circuit on the primary and secondary sides of the contactless coupler according to the phase control (D₁T) of the main switching devices (Q₁, Q₂, Q₃, Q₄) at a fixed switching frequency. Therefore, because the output voltage (V_O) of the DC-DC stage and the power factor (PF) must be controlled solely through phase control (D₁T), the PFC stage boost inductors (L_B1, L_B2, L_B3) must be designed to operate in Discontinuous Conduction Mode (DCM) rather than in Continuous Conduction Mode (CCM) or Critical Conduction Mode (CRM). Figure 4 shows the AC phase voltages (V_AN, V_BN, V_CN) of a three-phase PMSG small wind turbine operating in the range of 10 to 47 Hz depending on wind speed and the currents (IB1, IB2, IB3) flowing through the boost inductors (LB1, LB2, LB3) of the three-phase single-stage AC-DC converter. It also enlarges the AC phase voltages (VAN, VBN, VCN) and the boost inductor currents (I_B1, I_B2, I_B3) in the interval 0 < ωt ≤ π/3. In the enlarged waveform, the negative voltages of the three-phase AC phase voltages (V_AN, V_BN, V_CN) are represented in absolute values. The rising and reset slopes of the boost inductor currents (I_B1, I_B2, I_B3) are divided into two or three segments depending on the magnitudes of the three-phase AC phase voltages (V_AN, V_BN, V_CN) in the interval 0 < ωt ≤ π/3, the link voltage (V_Link), and the load [R_Link = (4N)²R_DC]. The critical angle that differentiates the two or three slopes of the boost inductor currents (I_B1, I_B2, I_B3) is defined as $\mathsf{\Phi }$_{cr_x} (x = a, b, c), and this is represented in Equation (1) by applying the flux balance condition.

Here, taking the a-phase of the PMSG small wind turbine three-phase AC phase voltages (V_AN, V_BN, V_CN) as the reference, $\mathsf{\Phi }$_{cr_a, t}, the critical angle, is symmetric about π/2 during the half-cycle (0 < ωt ≤ π). Using this, the condition for $\mathsf{\Phi }$_{cr_a} during the interval 0 < ωt ≤ π/2 for each phase is represented as shown in Equation (1), where $\mathsf{\Phi }$_cr is defined as shown in Equation (2).

$${\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{a}}=\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\pi }}{2}}{, \mathrm{if} \mathsf{\Phi }}_{\mathrm{cra}}>{\displaystyle \frac{\mathsf{\pi }}{2}} \\ {\mathsf{\Phi }}_{\mathrm{cr}}{, \mathrm{if} \mathsf{\Phi }}_{\mathrm{cra}}\le {\displaystyle \frac{\mathsf{\pi }}{2}} \end{array}\right.$$

(1)

$${\mathsf{\Phi }}_{\mathrm{cra}}{=\sin }^{-1}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{Link}}}{{2\mathrm{V}}_{\mathrm{pk}}}}\left(1-{2\mathrm{D}}_1\right)\right)$$

(2)

As depicted in Figure 4, the boost inductor current (I_B1) of the a-phase operates in Discontinuous Conduction Mode (DCM), resulting in two slopes (rise and reset) within the interval 0 < ωt ≤ Φ_{cr_a}. The period during which the main switching devices (Q₁/Q₂ or Q₃/Q₄) are simultaneously turned on is defined as D₁T, where D_i, i = 1,2,3, is the duty ratio and T is the switching period. During D₁T, the a-phase voltage across the filter capacitor C_FA (V_AN = V_pksin(ωt)) is applied to the boost inductor L_B. This causes the boost inductor current I_B1 to increase, storing energy in the inductor. The D₂T interval is defined as the period when switching device Q₁ turns off, allowing the reset current to flow into the voltage divider capacitor C₁ and transferring energy.

During the D₁T period, when the a-phase voltage (V_AN = V_pksin(ωt)) is applied to the boost inductor (L_B1), causing the boost inductor current (I_B1) to rise and store energy, the highest peak point is denoted as I_{pk_a}, as shown in Equation (3). The reset period D₂T can be expressed using the flux balance condition in Equation (4). During the D₂T period, the energy stored in the boost inductor is quickly reset by the differential voltage (V_AN − V_Link/2) between the lower a-phase voltage (V_AN = V_pksin(ωt)) and the voltage of the link capacitors (C₁, C₂) (V_Link/2), causing the boost inductor current (I_B1) to reset completely to zero.

$${\mathrm{I}}_{\mathrm{pk}\_\mathrm{a}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)}{{\mathrm{L}}_{\mathrm{B}1}}}{\mathrm{D}}_1\mathrm{T}$$

(3)

$${\mathrm{D}}_2\mathrm{T}={\displaystyle \frac{{2\mathrm{D}}_1{\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)}{{\mathrm{V}}_{\mathrm{Link}}-{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)}} \mathrm{T}$$

(4)

Therefore, during the period 0 < ωt ≤ D₂T of the switching period (T), the instantaneous average current (<I_LBa1>) of the boost inductor current (I_LBa1) with two slopes is expressed using I_{pk_a}, D₁T, and D₂T, as shown in Equation (5).

$$\langle {\mathrm{I}}_{{\mathrm{L}}_{\mathrm{B}}\mathrm{a}1}\rangle ={\displaystyle \frac{1}{\mathrm{T}}}{{\displaystyle \int }}_0^{\left({\mathrm{D}}_1{\mathrm{T}+\mathrm{D}}_2\mathrm{T}\right)}{\mathrm{I}}_{{\mathrm{L}}_{\mathrm{B}}\mathrm{a}1}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)={\displaystyle \frac{{\mathrm{D}}_1^2{\mathrm{V}}_{\mathrm{Link}}\sin \left(\mathsf{\omega }\mathrm{t}\right)}{{2\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}}\left(\frac{{\mathrm{V}}_{\mathrm{Link}}}{{\mathrm{V}}_{\mathrm{pk}}} - 2\sin \left(\mathsf{\omega }\mathrm{t}\right)\right)}},$$

(5)

where fs denotes the switching frequency (=1/T).

In the interval Φ_{cr_a} < ωt ≤ π/2, where the boost inductor current (I_LBa2) operates with three rising and reset slopes, the increased a-phase voltage (V_AN = V_pksin(ωt)) results in the boost inductor current (I_LBa2) having a gentler slope during the reset period D₂T. This is due to the reduced differential voltage (V_AN − V_Link/2) between the a-phase voltage (V_AN = V_pksin(ωt)) and the link capacitor (C₁) voltage (V_Link/2). Consequently, the boost inductor current (I_LBa2) is not fully reset within the half-switching period (T/2), resulting in three slopes. The D₃T interval occurs when switching devices Q₁ and Q₂ are turned off, and the boost inductor current (I_LBa2) resets to zero. During this interval, the large differential voltage between the a-phase voltage (V_AN = V_pksin(ωt)) and the combined voltage of link capacitors C₁ and C₂ (V_Link) causes the boost inductor current (I_LBa2) to reset quickly, allowing it to operate in Discontinuous Conduction Mode (DCM).

In the D₂T period, where three slopes appear, the relationship between the boost inductor voltage (V_LB1 = V_pksin(ωt) − V_Link/2) and the boost inductor current (I_LBa2) is expressed in Equation (6). As shown in Figure 4, using Equations (3), (4), and (6), the current I_{mid_a}, which is the point where the switching devices Q₁ and Q₂ turn off and the a-phase boost inductor current (I_LBa2) begins to reset quickly, can be obtained as shown in Equation (7).

$${\mathrm{V}}_{\mathrm{LB}1}{=\mathrm{L}}_{\mathrm{B}1}{\displaystyle \frac{\mathrm{di}}{\mathrm{dt}}}{=\mathrm{L}}_{\mathrm{B}1}{\displaystyle \frac{{\mathrm{I}}_{\mathrm{mid}\_\mathrm{a}}{-\mathrm{I}}_{\mathrm{pk}\_\mathrm{a}}}{{\mathrm{D}}_2\mathrm{T}}}={\displaystyle \frac{{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right){-\mathrm{V}}_{\mathrm{Link}}}{2}}$$

(6)

$${\mathrm{I}}_{{\mathrm{mid}}_{-}\mathrm{a}}={\displaystyle \frac{{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right){-\mathrm{V}}_{\mathrm{Link}}}{{2\mathrm{L}}_{\mathrm{B}1}}}\left({\displaystyle \frac{1}{2}}{-\mathrm{D}}_1\right){\mathrm{T}+\mathrm{I}}_{{\mathrm{pk}}_{-}\mathrm{a}}={\displaystyle \frac{1}{{4\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}}\left[{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)+\left({2\mathrm{D}}_1-1\right){\mathrm{V}}_{\mathrm{Link}}\right]$$

(7)

Therefore, the reset duty period D₃T can be obtained using the flux balance condition, as shown in Equation (8).

$${\mathrm{D}}_3\mathrm{T}={\displaystyle \frac{{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)+\left({2\mathrm{D}}_1-1\right){\mathrm{V}}_{\mathrm{Link}}}{4\left({\mathrm{V}}_{\mathrm{Link}}{-\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)\right)}}\mathrm{T}$$

(8)

Here, the area under the current in the period 0 < ωt ≤ D₁T, where the boost inductor current (I_LBa2) rises, is defined as X_a. The area of the period D₁T < ωt ≤ D₂T, where the boost inductor current (I_LBa2) is not fully reset, is defined as Y_a. The area of the interval D₂T < ωt ≤ D₃T, where the boost inductor current (I_LBa2) is fully reset to zero, is defined as Z_a. These are represented in Equations (9)–(11), respectively.

$${\mathrm{X}}_{\mathrm{a}}={\displaystyle \frac{1}{2}}{\mathrm{D}}_1{\mathrm{TI}}_{{\mathrm{pk}}_{\_\mathrm{a}}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)}{{2\mathrm{L}}_{\mathrm{B}1}}}{\left({\mathrm{D}}_1\mathrm{T}\right)}^2$$

(9)

$${\mathrm{Y}}_{\mathrm{a}}={\displaystyle \frac{1}{2}}\left({\mathrm{I}}_{{\mathrm{pk}\_}_{\mathrm{a}}}{+\mathrm{I}}_{{\mathrm{mid}\_}_{\mathrm{a}}}\right)\left({\displaystyle \frac{1}{2}}{-\mathrm{D}}_1\right)\mathrm{T}={\displaystyle \frac{{\mathrm{T}}^2\left(\frac{1}{2}{-\mathrm{D}}_1\right)}{{8\mathrm{L}}_{\mathrm{B}1}}}\left[\left({4\mathrm{D}}_1+2\right){\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)+\left({2\mathrm{D}}_1-1\right){\mathrm{V}}_{\mathrm{Link}}\right]$$

(10)

$${\mathrm{Z}}_{\mathrm{a}}={\displaystyle \frac{1}{2}}{\mathrm{D}}_3{\mathrm{TI}}_{{\mathrm{mid}\_}_{\mathrm{a}}}={\displaystyle \frac{{\mathrm{T}}^2}{{8\mathrm{L}}_{\mathrm{B}1}}}{\displaystyle \frac{{\left[{2\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)\left({2\mathrm{D}}_1-1\right){\mathrm{V}}_{\mathrm{Link}}\right]}^2}{4\left({\mathrm{V}}_{\mathrm{Link}}{-\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)\right)}}$$

(11)

When operating with three slopes within one cycle, the instantaneous average current (<I_LBa2>) of the boost inductor current I_LBa2 in the interval Φ_{cr_a} < ωt ≤ π/2 − Φ_{cr_a} can be obtained by summing the three areas defined by X_a, Y_a, and Z_a, as shown in Equation (12).

$$\langle {\mathrm{I}}_{{\mathrm{L}}_{\mathrm{B}}\mathrm{a}2}\rangle ={{\displaystyle \int }}_0^{\left({\mathrm{D}}_1{\mathrm{T}+\mathrm{D}}_2{\mathrm{T}+\mathrm{D}}_3\mathrm{T}\right)}{\mathrm{I}}_{{\mathrm{L}}_{\mathrm{B}}\mathrm{a}2}\mathrm{dt}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{Link}}\left({8\mathrm{D}}_1^2+2\right){\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)-{\left({2\mathrm{D}}_1-1\right)}^2{\mathrm{V}}_{\mathrm{Link}}}{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}\left({\mathrm{V}}_{\mathrm{Link}}{-\mathrm{V}}_{\mathrm{pk}}\sin \left(\mathsf{\omega }\mathrm{t}\right)\right)}}$$

(12)

To determine the critical angle Φ_{cr_x} for each phase as shown in Figure 4, it can be simplified to the interval 0 < ωt ≤ π/3, which is 1/6 of one cycle (2π), rather than the interval 0 < ωt ≤ π/2 by expressing the three-phase AC phase voltages (V_AN, V_BN, V_CN) in absolute values. Using this, and applying Equation (1), the critical angle Φ_{cr_χ} for each phase can be determined using the logical expressions for the interval 0 < ωt ≤ π/3, as shown in Equations (13)–(16).

$${\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{a}}=\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\pi }}{3}}{ \mathrm{if} \mathsf{\Phi }}_{\mathrm{cra}}> {\displaystyle \frac{\mathsf{\pi }}{3}}\\ {\mathsf{\Phi }}_{\mathrm{cr}}{ \mathrm{if} \mathsf{\Phi }}_{\mathrm{cra}}\le {\displaystyle \frac{\mathsf{\pi }}{3}}\end{array}\right.$$

(13)

$${\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}1}=\left\{\begin{array}{c}0 \mathrm{if} 0\ge -{\displaystyle \frac{\mathsf{\pi }}{3}}{+\mathsf{\Phi }}_{\mathrm{cra}}\\ -{\displaystyle \frac{\mathsf{\pi }}{3}}{+\mathsf{\Phi }}_{\mathrm{cra}} \mathrm{if} 0<-{\displaystyle \frac{\mathsf{\pi }}{3}}{+\mathsf{\Phi }}_{\mathrm{cra}}\le {\displaystyle \frac{\mathsf{\pi }}{6}}\\ {\displaystyle \frac{\mathsf{\pi }}{6}} \mathrm{if} {\displaystyle \frac{\mathsf{\pi }}{6}}<-{\displaystyle \frac{\mathsf{\pi }}{3}}{+\mathsf{\Phi }}_{\mathrm{cra}}\end{array}\right.$$

(14)

$${\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}2}=\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\pi }}{3}} \mathrm{if} {\displaystyle \frac{2\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}}\ge {\displaystyle \frac{\mathsf{\pi }}{3}}\\ {\displaystyle \frac{2\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}} \mathrm{if} {\displaystyle \frac{\mathsf{\pi }}{6}}\le {\displaystyle \frac{2\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}}\le {\displaystyle \frac{\mathsf{\pi }}{3}}\\ {\displaystyle \frac{\mathsf{\pi }}{6}} \mathrm{if} {\displaystyle \frac{\mathsf{\pi }}{6}}>{\displaystyle \frac{2\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}}\end{array}\right.$$

(15)

$${\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{c}}=\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}} \mathrm{if} 0< {\displaystyle \frac{\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}}\le {\displaystyle \frac{\mathsf{\pi }}{3}}\\ 0 \mathrm{if} 0\ge {\displaystyle \frac{\mathsf{\pi }}{3}}{-\mathsf{\Phi }}_{\mathrm{cra}}\end{array}\right.$$

(16)

As shown in Figure 4, the instantaneous reset currents (I_{Link_a}, I_{Link_b}, I_{Link_c}) of the boost inductors (L_B1, L_B2, L_B3) are determined according to the critical angle Φ_{cr_x} for each phase. During the phase control (D₁T) switching operation, where the switching devices Q₁/Q₂ or Q₃/Q₄ are turned on, the energy stored in the boost inductors (L_B1, L_B2, L_B3) is transferred to the voltage divider link capacitors (C₁, C₂) at the output of the input rectifier diode over the D₂T or D₂T + D₃T time period. This transfer of energy establishes each link voltage (V_{Link_χ1}, V_{Link_χ2}) based on the areas (Y_a, Y_a + Z_a) corresponding to the slopes of the two and three reset current segments of the boost inductor currents (I_B1, I_B2, I_B3). At this time, the instantaneous average reset currents (<I_{Link_χ1}>, <I_{Link_χ2}>) are defined as the average values of the reset currents of the boost inductors for each phase according to the critical angle Φ_{cr_x} over one switching period (T), as specified in Equations (17) and (18). Here, θ_x is 0 for the a-phase, −2π/3 for the b-phase, and +2π/3 for the c-phase.

$$\langle {\mathrm{I}}_{{\mathrm{Link}\_}_{\mathrm{x}}1}\rangle ={\displaystyle \frac{1}{\mathrm{T}}}{{\displaystyle \int }}_{{\mathrm{D}}_1\mathrm{T}}^{{\mathrm{D}}_2\mathrm{T}}{\mathrm{i}}_{{\mathrm{Link}\_}_{\mathrm{x}}1}\mathrm{dt}={\displaystyle \frac{{\mathrm{D}}_1^2{\left({\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\right)}^2}{{\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}\left({\mathrm{V}}_{\mathrm{Link}}{-2\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\right)}}={\displaystyle \frac{{\mathrm{V}}_{{\mathrm{Link}\_}_{\mathrm{x}}1}}{{\mathrm{R}}_{\mathrm{Link}}}}$$

(17)

$$\langle {\mathrm{I}}_{{\mathrm{Link}\_}_{\mathrm{x}}2}\rangle ={\displaystyle \frac{1}{\mathrm{T}}}{{\displaystyle \int }}_{{\mathrm{D}}_1\mathrm{T}}^{{\mathrm{D}}_2{\mathrm{T}+\mathrm{D}}_3\mathrm{T}}{\mathrm{I}}_{{\mathrm{Link}\_}_{\mathrm{x}}2}\mathrm{dt}={\displaystyle \frac{{16\mathrm{D}}_1^2{\left({\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\right)}^2+\left({2-8\mathrm{D}}_1^2\right){\mathrm{V}}_{\mathrm{Link}}{\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)-{\left[\left({2\mathrm{D}}_1-1\right){\mathrm{V}}_{{\mathrm{Link}\_}_{\mathrm{x}}2}\right]}^2}{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}\left({\mathrm{V}}_{\mathrm{Link}}{-\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\right)}}={\displaystyle \frac{{\mathrm{V}}_{{\mathrm{Link}\_}_{\mathrm{x}}2}}{{\mathrm{R}}_{\mathrm{Link}}}}$$

(18)

To determine the instantaneous link voltages V_{Link_χ1} and V_{Link_χ2} using the instantaneous average reset currents <I_{Link_χ1}> and <I_{Link_χ2}>, the secondary side load is reflected to the primary side using the link equivalent load resistance R_Link of the PFC AC-DC stage. This is expressed in Equation (19). Here, N represents the turn ratio of the contactless coupler between the primary and secondary coils (N₁/N₂).

$${\mathrm{R}}_{\mathrm{Link}}={\left(4\mathrm{N}\right)}^2{\mathrm{R}}_{\mathrm{DC}}={\displaystyle \frac{{\left({4\mathrm{NV}}_{\mathrm{o}}\right)}^2}{{\mathrm{P}}_{\mathrm{o}}/\mathsf{\eta }}},$$

(19)

where η is the efficiency of the single-stage AC-DC converter.

By examining the relationships among Equations (17)–(19), the instantaneous link voltages V_{Link_χ1} and V_{Link_χ2} can be expressed using Equations (20) and (21).

$${\mathrm{V}}_{{\mathrm{Link}\_}_{\mathrm{x}}1}{=\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\left(1+{\displaystyle \frac{{\mathrm{R}}_{\mathrm{Link}}\sqrt{\frac{{\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}\left(\frac{{\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}{+\mathrm{D}}_1^2\right)}}{{\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}}\right)$$

(20)

$${\mathrm{V}}_{{\mathrm{Link}\_}_{\mathrm{x}}2}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{pk}}\sin \left({\mathsf{\omega }\mathrm{t}+\mathsf{\theta }}_{\mathrm{x}}\right)\left\{\left({-8\mathrm{D}}_1^2+2+\frac{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}\right)\pm \sqrt{{\left({-8\mathrm{D}}_1^2+2+\frac{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}\right)}^2{+64\mathrm{D}}_1^2\left(\frac{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}+{\left({2\mathrm{D}}_1-1\right)}^2\right)}\right\}}{2\left\{\frac{{32\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{B}1}}{{\mathrm{R}}_{\mathrm{Link}}}+{\left({2\mathrm{D}}_1-1\right)}^2\right\}}}$$

(21)

As a result, the average link voltages <V_{Link_x}> can be determined according to the slopes of the boost inductor currents (I_B1, I_B2, I_B3) using Equations (20) and (21), and are given by Equations (22)–(24). Therefore, within the interval 0 < ωt ≤ π/3, the final three-phase link voltage V_Link is represented as the sum of the average link voltages for each phase, <V_{Link_a}>, <V_{Link_b}>, and <V_{Link_c}>, as shown in Equation (25).

$$\langle {\mathrm{V}}_{\mathrm{Link}\_\mathrm{a}}\rangle ={\displaystyle \frac{3}{\mathsf{\pi }}}\left\{{{\displaystyle \int }}_0^{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{a}}}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{a}1}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)+{{\displaystyle \int }}_{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{a}}}^{\mathsf{\pi }/3}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{a}2}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)\right\}$$

(22)

$$\langle {\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}}\rangle ={\displaystyle \frac{3}{\mathsf{\pi }}}\left\{\begin{array}{c}{{\displaystyle \int }}_0^{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}1}}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}1}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)+{{\displaystyle \int }}_{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}1}}^{\mathsf{\pi }/6}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}2}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)+\\ {{\displaystyle \int }}_{\mathsf{\pi }/6}^{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}2}}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}2}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)+{{\displaystyle \int }}_{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{b}2}}^{\mathsf{\pi }/3}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}1}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)\end{array}\right\}$$

(23)

$$\langle {\mathrm{V}}_{\mathrm{Link}\_\mathrm{c}}\rangle ={\displaystyle \frac{3}{\mathsf{\pi }}}\left\{{{\displaystyle \int }}_0^{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{c}}}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{c}2}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)+{{\displaystyle \int }}_{{\mathsf{\Phi }}_{\mathrm{cr}\_\mathrm{c}}}^{\mathsf{\pi }/3}{\mathrm{V}}_{\mathrm{Link}\_\mathrm{c}1}\mathrm{d}\left(\mathsf{\omega }\mathrm{t}\right)\right\}$$

(24)

$${\mathrm{V}}_{\mathrm{Link}}={\displaystyle \frac{1}{\sqrt{3}}}\left({\mathrm{V}}_{\mathrm{Link}\_\mathrm{a}}{+\mathrm{V}}_{\mathrm{Link}\_\mathrm{b}}{+\mathrm{V}}_{\mathrm{Link}\_\mathrm{c}}\right)$$

(25)

Figure 5 shows the simulation results of the boost inductor values (L_B = L_B1 = L_B2 = L_B3) and the PFC stage link voltage (V_Link) variations according to phase control (D₁T) obtained using Equation (25) at a 1 kW link equivalent load resistance (R_Link). As can be seen from the simulation results, a smaller boost inductance (L_B1, L_B2, L_B3) results in a wider boost control range, but a higher V_Link increases the boost inductor currents (I_B1, I_B2, I_B3), which reduces efficiency. Considering that one-quarter of the link voltage (V_Link) is applied to the primary side of the contactless coupler and the remainder is applied to the secondary side according to the turn ratio, the maximum range of the link voltage is set to 600 V to control an output voltage of 400 V at the rated load (1 kW). Consequently, the boost inductor is selected as 60 µH.

4. Design of a Contactless Coupler for Inductive-Power-Transfer-Based Wind Power Systems

The contactless coupler utilized in the small wind power system based on inductive power transfer (IPT) was engineered with a PM11 material circular core (R/r/H: 100 mm/5 mm/4 mm) and a spiral winding technique, applied to both the primary and secondary pads of equivalent area. To address core saturation, the core thickness was chosen as 4 mm, and a minimal air gap of 10 mm was maintained between the primary and secondary sides of the coupler. To protect against stray magnetic fields in the actual nacelle environment, aluminum shielding plates (R’/H’: 140 mm/4 mm) were installed above and below the circular core to mitigate the stray flux on the primary and secondary sides of the coupler [16]. While the ferrite core enhances the effective inductance by reducing magnetic reluctance within the flux path, the aluminum shielding plates contribute to increased magnetic reluctance and decreased effective inductance due to eddy currents generated when exposed to the magnetic field. To minimize eddy current losses in the aluminum shielding plates, a 0.3 mm gap was introduced between the ferrite core and the aluminum shielding. The design of the ferrite core and aluminum shielding plates is illustrated in Figure 6.

To minimize reactive power and maximize power transfer, the number of turns on the primary and secondary sides needed to determine the primary-side magnetizing inductance (L_m) of the contactless coupler can be calculated when the resonant frequency (f_r) and the switching frequency (f_s) are identical, with the input and output voltage gains being constant in the series–series compensation circuit, regardless of the AC equivalent resistance (R_ac). As shown in Figure 7 and Figure 8, when the contactless coupler has a single turn on both the primary and secondary sides, the magnetizing inductance from the perspective of the secondary side (L_m0) and the mutual inductance (M₀) for a single turn are equivalent. Therefore, the mutual inductance (M₀) for a single turn, which is used to determine the number of turns for the contactless coupler, was extracted using Maxwell 3D (Ansys Electronics Desktop 2021 R2) electromagnetic field analysis based on the determined coupler geometry (R/r/H: 100 mm/5 mm/4 mm) and the air gap (10 mm), resulting in a mutual inductance (M₀) of 375.75 nH, as shown in Figure 7. Based on this, with a fixed output voltage (V_o: 400 V_DC) and a rated output load of 1 kW, the number of turns on the primary side (N₁) and secondary side (N₂) of the contactless coupler was determined to be N₁ = 13 turns and N₂ = 19 turns, respectively (refer to [17] for more detail). Accordingly, the contactless coupler was fabricated as shown in Figure 9. Additionally, for wireless power transfer, international standards specify the frequency ranges for high-power devices (200 W to 2.4 kW). The WPC Qi standard designates an inductive power transfer frequency band of 87 kHz to 205 kHz. According to the wireless power transfer standard SAE-J2954 for electric vehicles, the standard power transfer system operates within a frequency range of 79 to 90 kHz, with a nominal power transfer frequency of 85 kHz [16]. In this study, as discussed in Section 5 (refer to Figure 11b), the resonant frequency (f_r) was designed to be 80 kHz, maintaining consistent voltage gain characteristics regardless of the AC equivalent resistance (R_ac) in the series–series compensation circuit. The switching frequency (f_s) was chosen to be 85 kHz, which is higher than the resonant frequency (f_r), to accommodate parameter variations in the resonant gain characteristics and ensure zero-voltage switching operation under all operating conditions.

5. Analysis of Gain Characteristics in DC-DC Stage with Series–Series (S-S) Compensation Circuit

Figure 10 shows the AC equivalent model of the series–series (S-S) compensation circuit applied to the proposed three-phase single-stage AC-DC converter. The model consists of the leakage inductances (L_l1, L_l2) on the primary and secondary sides, the magnetizing inductance (L_m), the resonant capacitors (C_r1, C_r2) on the primary and secondary sides, and the AC equivalent resistance (R_ac). The internal resistance components of the primary and secondary windings have been neglected. C_r1 and C_r2 are the resonant capacitors on the primary and secondary sides, respectively, and R_ac was derived by converting the DC load resistance (R_DC) into the AC equivalent resistance (R_ac = 8R_DC/π²). The voltage gain characteristics of the series–series (S-S) compensation circuit in the DC-DC stage of the IPT-based three-phase single-stage AC-DC converter were analyzed using the Fundamental Harmonic Approximation (FHA). In this study, only the fundamental frequency was considered, ignoring the third, fifth, and seventh harmonics. Equation (26) represents the input–output gain G_V of the S-S compensation circuit in terms of the coupling coefficient k, the magnetizing inductances (L₁, L₂), and the leakage inductances (L_l1, L_l2) of the contactless coupler. The voltage gain curve as a function of frequency, derived from the AC equivalent model, is shown in Figure 11b.

The resonant frequencies of the series–series (S-S) compensation circuit include three frequencies: f_m, f_r, and f_p. The first resonant frequency, f_m, is the resonant frequency associated with the primary and secondary resonant capacitors (C_r1, C_r2), the secondary-side magnetic inductance (L₂), and the magnetizing inductance (L_m), as shown in Equation (27). At the resonant frequency (f_m), the switching operation can always occur with a leading current, making it suitable for hard switching. The second resonant frequency, f_r, is the resonant frequency associated with the leakage inductances (L_l1, L_l2) of the primary and secondary sides and the resonant capacitors (C_r1, C_r2), as depicted in Equation (28). At the frequency f_r, the AC equivalent resistance R_ac does not affect the input–output voltage gain characteristics, maintaining a consistent gain characteristic despite changes in R_ac. The third resonant frequency, f_p, is the resonant frequency between the primary-side magnetic inductance (L₁) and the resonant capacitor (C_r1), as shown in Equation (29). This frequency (f_p) is where the gain characteristics are most significantly affected by changes in the equivalent AC resistance (R_ac). Particularly, as the load increases, a bifurcation phenomenon occurs where the slope of the gain characteristics decreases. Thus, if a fixed alternating switching operation is performed at the third resonant frequency (f_p) under rated-load conditions, the primary-side resonant current (I_P) may lead the phase of the primary-side square wave voltage (V_P), leading to hard switching.

In the contactless coupler used for a three-phase single-stage AC-DC converter in an IPT-based small wind power generation system, the primary and secondary sides of the coupler have a fixed air gap (10 mm) and rotate during coupling. Therefore, the parameters of the primary and secondary sides of the coupler remain almost unchanged and constant. Therefore, in this paper, based on the contactless coupler parameters shown in

Table 1. Parameters of the IPT-based loosely coupled transformer.

Core size (R/r/H)	100 mm/5 mm/4 mm
Aluminum shield size (R’/H’)	140 mm/4 mm
Transformer ratio (N1/N2)	N(N1/N2) = 0.684 (13/19)
Conditions of air gap	10 mm
Pri./Sec. self-inductance (L1/L2)	87.44 µH/166.3 µH
Magnetizing inductance (Lm)	66.76 µH
Pri./Sec. leakage inductance (Ll1/Ll2)	20.68 µH/23.7 µH
Coupling coefficient (k)	0.809
Resonant frequency (fr)	80 kHz
Pri./Sec. resonant capacitor (Cr1/Cr2)	203 nF/173 nF

and resonant capacitors C_r1: 203 nF and C_r2: 167 nF, the switching frequency (f_s) was set around the second resonant frequency f_r (80 kHz) to ensure that the gain remains constant (G_V = 1) despite load changes. At a fixed switching frequency (fs: 85 kHz), the link voltage (V_Link) can be controlled through phase control (D₁T). As shown in Figure 5, the boost inductor value (60 µH) was set to allow the link voltage (V_Link) to be controlled up to a maximum range of 600 V to regulate the output voltage of 400 V at rated load (1 kW). The gain characteristics with load changes (500 W~1.5 kW) and phase control (D₁T) are shown in Figure 11a. Therefore, as shown in Figure 11b, at a fixed switching frequency (f_s) near the resonant frequency (f_r), the input–output voltage gain remains constant and is unaffected by changes in the equivalent load resistance (R_ac). Under these conditions, without feedback through wireless communication, the secondary-side output voltage (V_o) can be controlled within a certain range solely by sensing the primary-side link voltage (V_Link) and using phase control (D₁T).

$${\mathrm{G}}_{\mathrm{v}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{S}}}{{\mathrm{V}}_{\mathrm{P}}}}=\left|{\displaystyle \frac{{\mathrm{s}}^3{\mathrm{R}}_{\mathrm{ac}}{\mathrm{C}}_{\mathrm{r}1}{\mathrm{C}}_{\mathrm{r}2}\mathrm{Nk}\sqrt{{\mathrm{L}}_1{\mathrm{L}}_2}}{\left({1+\mathrm{s}}^2\mathrm{A}\right)\left({1+\mathrm{s}}^2\mathrm{B}\right){+\mathrm{sR}}_{\mathrm{ac}}{\mathrm{C}}_{\mathrm{r}2}{(1+\mathrm{s}}^2\mathrm{C})}}\right|, \mathrm{k}={\displaystyle \frac{{\mathrm{L}}_{\mathrm{m}}}{\mathrm{N}\sqrt{{\mathrm{L}}_1{\mathrm{L}}_2}}}$$

(26)

where

$${\mathrm{A}=\mathrm{C}}_{\mathrm{r}2}{\mathrm{L}}_2(\mathrm{Nk}\sqrt{{\mathrm{L}}_1{\mathrm{L}}_2}){\displaystyle \frac{{1+\mathrm{s}}^2{\mathrm{C}}_{\mathrm{r}2}{\mathrm{L}}_{\mathrm{l}2}}{{1+\mathrm{s}}^2{\mathrm{C}}_{\mathrm{r}1}{\mathrm{L}}_{\mathrm{l}1}}}$$

(27)

$${\mathrm{B}=\mathrm{C}}_{\mathrm{r}1}{\mathrm{L}}_{\mathrm{l}1}$$

(28)

$${\mathrm{C}=\mathrm{C}}_{\mathrm{r}1}{\mathrm{L}}_1$$

(29)

6. Experimental Results

The key specifications of a 1 kW three-phase single-stage AC-DC converter, equipped with a power factor correction (PFC) stage and a contactless coupler integrated with a series–series (S-S) compensation circuit, are outlined in

Table 2. Main input–output specifications and the used devices.

Specification/Parameters	Values
Wind power generator voltage (VLL)/frequency (fG)	112–250 Vrms/20~47 Hz
Output voltage (Vo)/Power (Po)	400 VDC/1 kW
Switching/resonance Frequency (fs/fr)	85 kHz/80 kHz
Dead time (tdead)	350 ns
SIC switching devices (Q1–Q4)	UJ3C0120040KS [1200 V/65 A/35 mΩ]
Circulating diodes (D1, D2)	UJ3D6560KSD [650 V/60 A/1.5 VF]
SIC input rectifier diodes	GP2D050A120B [1200 V/50 A/1.4 VF]
Schottky output diodes	UJ3D6560KSD [650 V/60 A/1.5 VF]
C1, C2/CB, CB1, CB2	1000 µF/3.3 µF
PFC Inductor and Capacitor Parameters
LFa, LFb, LFc	10.8 mH [DKIH-3352-1011-NK]
LB1, LB2, LB3	60 µH
CFa, CFb, CFc	3.3 µF

. To emulate the wind-speed-dependent generator voltage and frequency of a small wind turbine, a 1 kW wind turbine test setup was utilized. This setup included a drive inverter, a drive motor, a 1/4 reduction gearbox, and a Permanent Magnet Synchronous Generator (PMSG).

Table 3. Power generation voltage and frequency specifications according to the RPM of the wind power generator.

Motor RPM	Gear Ratio	Generator RPM	VLL (V)	Pac (W)	Freq(fG) (Hz)
650	10.3	63.11	112.6	150	20.99
750	10.3	72.82	127.8	268	24.85
950	10.3	92.23	167.9	467	31.62
1100	10.3	106.80	190.3	650	37.15
1250	10.3	121.36	218.3	800	41.13
1350	10.3	131.07	232.5	950	43.86
1450	10.3	140.78	249.3	1000	47.12

shows the line-to-line voltage (V_LL) of the wind turbine as a function of the RPM. Based on these parameters, a prototype of the 1 kW system was fabricated and tested.

In the experimental results, as shown in Figure 12, the operation characteristics of the proposed circuit were compared and analyzed using a dsPIC33FJ16GS502 16-bit DSP control board. This experiment was conducted at a fixed switching frequency (f_s: 85 kHz), implementing phase control (D₁T) for the IPT-based three-phase single-stage AC-DC converter. The comparison included controlling the secondary-side output voltage (V_o) via wireless communication versus controlling only the primary-side link voltage (V_Link) without wireless communication. As shown in Figure 13, a three-phase 1 kW wind turbine simulation setup, used as a three-phase alternating current source, an electronic load (Chroma 63206 A, 6 kW, CR Mode (1066 Ω–160 Ω)), and a power meter (HIOKI, PW3390) were prepared for experimental study. This setup was used to verify the operational characteristics of the proposed three-phase single-stage AC-DC converter for small wind turbine systems.

Firstly, this paper describes the experimental setup for controlling the secondary-side output voltage (V_o) of the three-phase single-stage AC-DC converter designed for a small wind power generation system based on IPT technology, aiming to maintain a constant output voltage of 400 V_DC via wireless sensing of V_o.

Figure 14, Figure 15 and Figure 16 illustrate the performance characteristics under various load conditions, ranging from a light load of 150 W to a rated load of 998 W. Specifically, these figures present the waveforms of the boost inductor current (I_B1), AC phase voltage (V_AN), phase current (I_A), link voltage (V_Link), and the terminal voltages and currents of the contactless coupler in the DC-DC stage: primary-side terminal voltage/current (V_P/I_P) and secondary-side terminal voltage/current (V_S/I_S). The generator line-to-line voltage (V_LL) was varied from 110 Vac to 250.2 Vac during these experiments.

The experimental waveforms show that the boost inductor current (I_B1) in the PFC stage operates in discontinuous mode for the controlled output voltage (V_o: 400 V_DC). This operation improves the power factor and makes the phase current waveform (I_A) align with the AC phase voltage (V_AN) as a sinusoidal wave through filtering. Additionally, examining the waveforms of the primary-side square wave voltage (V_P) and current (I_P) applied to the S-S compensation circuit for wireless power transfer shows that the output voltage (V_o) is controlled in response to variations in the link voltage (V_Link) according to phase control (D₁T). Due to the resonance current (I_P) being delayed relative to the square wave voltage (V_P) at the a-b terminals, all primary-side switching devices are confirmed to operate with zero-voltage switching (ZVS). In particular, as illustrated in Figure 17, during the control of the secondary-side output voltage (V_o: 400 V_DC) via wireless communication, the primary-side link voltage (V_Link) was observed to vary from 557 V_DC to 573.2 V_DC, resulting in a variation rate of 2.87%. The experimental results demonstrate that the primary-side link voltage (V_Link) exhibits minimal variation across the entire range of generator line-to-line voltages (110 V_ac to 250.2 V_ac) and varying load conditions. Based on these findings, the feasibility of controlling the output voltage (V_o: 400 V_DC) without sensing the secondary-side output voltage and without relying on wireless communication solely through the control of the primary-side link voltage (V_Link) was also assessed.

Therefore, the average link voltage (V_Link: 565 V_DC) was selected through the experimental results obtained from Figure 17. Figure 18, Figure 19 and Figure 20 show the operation of the link voltage (V_Link: 565 V_DC) control by sensing only the primary-side link voltage (V_Link) without wireless communication. Also, the figures provide waveforms of the boost inductor current (I_B1), phase voltage (V_AN), phase current (I_A), link voltage (V_Link), and the primary-side terminal voltage/current (V_P/I_P) and secondary-side terminal voltage/current (V_S/I_S) of the DC-DC stage of the three-phase single-stage AC-DC converter under varying load conditions from a light load of 150 W to the rated load of 1 kW. The experimental waveforms reveal that, for the S-S compensation circuit, the primary-side square wave voltage (V_P) and current waveforms (I_P) show that the resonance current (I_P), which is delayed relative to the square wave voltage (V_P) at the a-b terminals, causes all primary-side switching devices to operate with zero-voltage switching (ZVS). Furthermore, Figure 21 presents an enlarged view of the boost inductor current (I_B1) waveform under a 478 W load condition. Figure 21a shows the boost inductor current (I_B1) waveform with two slopes in the interval 0 < ωt ≤ Φ_{cr_a}, while Figure 21b depicts the waveform with three slopes in the interval Φ_{cr_a} < ωt ≤ π/2. Therefore, as detailed in Section 2.2, the boost inductor current (I_B1) waveform demonstrates that the waveform exhibits two and three slopes in the intervals 0 < ωt ≤ π/2, corresponding to the difference between the phase voltage (V_AN) and the link voltage (V_Link).

Figure 22 presents the total harmonic distortion (THD_i), power factor (PF), and overall efficiency characteristics of the experimental results obtained by selecting the average link voltage (V_Link: 565 V_DC) from the results shown in Figure 17 while controlling only the primary-side link voltage (V_Link: 565 V_DC) without wireless communication. As shown in Figure 22a, the power factor (PF) and THD_i characteristics indicate that, under load conditions of 154 W or higher, the power factor (PF) remains above 0.99, and the THD_i is below 10%. Specifically, under the rated-load condition (1 kW), a low THD_i of 3.79% is observed. Furthermore, based on the efficiency characteristics graph shown in Figure 22b, it was observed that, from a load condition of 273 W onwards, the system achieves an efficiency of 90.54% within the link voltage control range (V_Link: 565 V_DC). For higher load conditions, the system maintains an efficiency above 90%, reaching a maximum efficiency of 93.61% under the rated load (1 kW).

As illustrated in Figure 22c, without wireless communication and by controlling only the link voltage (V_Link), the output voltage (V_o) under the light load (154 W) is 406.3 V, while under the rated load (1 kW), it is 398.2 V, showing a variation rate of 2.025%.

Therefore, the experimental results demonstrate that stable power control can be achieved by controlling only the link voltage (V_Link) rather than the output voltage via wireless communication for the IPT-based small wind power generation system.

7. Conclusions

This paper analyzes the design and operational characteristics of a three-phase single-stage AC-DC converter for IPT-based small wind power systems. Based on this analysis, the characteristics of the resonant gain and resonant frequencies of the series–series (S-S) compensation circuit, as well as the gain characteristics concerning the boost inductor and load, were examined to select the resonant frequency (f_r) and switching frequency (f_s). Experimental results confirm that the proposed three-phase single-stage AC-DC converter achieves zero-voltage switching (ZVS) for the switching devices (Q₁, Q₂, Q₃, Q₄) across the PMSG line-to-line voltages and load conditions range. Additionally, the system demonstrates an improved power factor, a low total harmonic distortion (THD_i) of 3.79% under rated load (1 kW), and a maximum efficiency of 93.61%. Through the prototyping and experimentation of a 1 kW three-phase single-stage AC-DC converter in an IPT-based small wind power system, it was confirmed that the output voltage variation (2.025%) could be kept minimal using only primary-side link voltage control without real-time wireless communication. This demonstrates that the system can connect to a grid-tied inverter.