Dual-Side Phase-Shift Control for Strongly Coupled Series–Series Compensated Electric Vehicle Wireless Charging Systems

: Wireless power transfer (WPT) for electric vehicles is an emerging technology and a future trend. To increase power density, the coupling coefﬁcient of coils can be designed to be large, forming a strongly coupled WPT system, different from the conventional loosely coupled WPT system. In this way, the power density and efﬁciency of the WPT system can be improved. This paper investigates the dual-side phase-shift control of the strongly coupled series–series compensated WPT systems. The mathematical models based on the conventional ﬁrst harmonic approximation and differential equations for the dual-side phase-shift control are built and compared. The dual-side phase-shift angle and its impact on the power transfer direction and soft switching are investigated. It is found that synchronous rectiﬁcation at strong couplings can lead to hard switching because the dual-side phase shift in this case is over 90 ◦ . In comparison, a relatively high efﬁciency and soft switching can be realized when the dual-side phase shift is below 90 ◦ . The experimental results have validated the analysis.


Introduction
Compared with the conventional conductive power transfer, wireless power transfer (WPT) [1][2][3][4][5][6] has many advantages, such as safety, automation, convenience, and feasibility to various working environments, such as mining and underwater situations. Thus, WPT has broad application scenarios, such as wireless sensor networks, consumer electronics, implantable medical devices, domestic appliance, electric vehicles (EVs), electric vessels, and even space solar station. WPT via magnetic induction is the most popular WPT technology and has received tremendous attention both from academia and industry. Wireless charging for EVs based on magnetic induction is currently one of the research hot topics [7].
Power density is one of the key indicators for WPT systems. To increase power density, the coupling coefficient of the two coupled coils can be designed to be large, forming a strongly coupled WPT system, where the coupling coefficient is normally larger than 0.5. This system has a totally different feature from the conventional loosely coupled WPT systems, whose coupling coefficient is around 0.2. The current waveforms of the strongly coupled WPT systems are distorted and can be discontinuous [8], whereas in loosely coupled WPT systems the current waveforms are sinusoidal.
(1) Synchronous rectification is normally conducted to the secondary-side rectifier of the loosely coupled WPT systems to improve efficiency [9,10]. However, it is found in strongly coupled WPT systems that this will result in the primary-side inverter working in hard switching, leading to decreasing efficiency and potential circuit failures. Additionally, in loosely coupled WPT systems, dual-side phase-shift control is seldomly used to regulate the secondary-side charging current due to the fact that too much reactive power will be introduced if the phase difference between the primary-side and secondary-side voltages is not 90 • . However, in strongly coupled WPT systems, since the coupling coefficient is large, the efficiency can still be high even though the phase difference is not 90 • . Thus, the secondary-side charging current and even bidirectional power flow can be easily regulated. The contributions of this paper include building the mathematical model of dual-side phase-shift control for strongly coupled WPT systems; (2) Investigating the performance of dual-side phase-shift control for strongly coupled WPT systems; (3) Revealing that synchronous rectification is not suitable for strongly coupled WPT system because soft switching can be lost; (4) Experimentally validating the model and analysis.
This paper studies the dual-side phase-shift control of the strongly coupled seriesseries compensated WPT systems. The mathematical models based on first harmonic approximation (FHA) and differential equations are built in Section 2. Experimental results are offered in Section 3 to validate the analysis. Section 4 concludes this paper.

Mathematical Modelling
The topology of a series-series compensated EV wireless charging system with fullbridge converters is shown in Figure 1. S 1 -S 8 are the active switches. V INV (V REC ), u 1 (u 2 ), i 1 (i 2 ), L 1 (L 2 ), C 1 (C 2 ), u C1 (u C2 ) are the respective primary-side (secondary-side) dc voltage, ac voltage, ac current, self-inductance, capacitance, and capacitor voltage. M is the mutual inductance. seldomly used to regulate the secondary-side charging current due to too much reactive power will be introduced if the phase difference bet mary-side and secondary-side voltages is not 90°. However, in strongly systems, since the coupling coefficient is large, the efficiency can still though the phase difference is not 90°. Thus, the secondary-side chargin even bidirectional power flow can be easily regulated. The contribution include building the mathematical model of dual-side phase-shift contr coupled WPT systems; (2) Investigating the performance of dual-side phase-shift control for stro WPT systems; (3) Revealing that synchronous rectification is not suitable for strongly system because soft switching can be lost; (4) Experimentally validating the model and analysis.
This paper studies the dual-side phase-shift control of the strongly co series compensated WPT systems. The mathematical models based on first proximation (FHA) and differential equations are built in Section II. Experim are offered in Section III to validate the analysis. Section IV concludes this p

Mathematical Modelling
The topology of a series-series compensated EV wireless charging sys bridge converters is shown in Figure 1. S1-S8 are the active switches. VINV (V (i2), L1 (L2), C1 (C2), uC1 (uC2) are the respective primary-side (secondary-side) voltage, ac current, self-inductance, capacitance, and capacitor voltage. M inductance. The load is modelled as a voltage-source load. The voltage gain of the GV is defined as

First Harmonic Approximation
FHA is normally utilized to model the WPT system. Based on FHA, t circuit is shown in Figure 2. U1 (U2) and I1 (I2) are the fundamental comp inverter (rectifier) ac voltage and current, respectively. R1 and R2 are the e sistance of the transmitter and receiver, respectively. With 180° phase shift w of the inverter and the rectifier, we have The load is modelled as a voltage-source load. The voltage gain of the WPT system G V is defined as

First Harmonic Approximation
FHA is normally utilized to model the WPT system. Based on FHA, the equivalent circuit is shown in Figure 2. U 1 (U 2 ) and I 1 (I 2 ) are the fundamental components of the inverter (rectifier) ac voltage and current, respectively. R 1 and R 2 are the equivalent resistance of the transmitter and receiver, respectively. With 180 • phase shift within the legs of the inverter and the rectifier, we have (2) The model in Figure 2 can be established as The model in Figure 2 can be established as where ω is the working angular frequency and the parameters in bold phasor quantities. At resonance where .
According to FHA, the transmitter and receiver currents are constant. With the dual-side phase-shift angle of β, by which U2 leads U1, the ph is plotted in Figure 3 [11]. The rectifier dc current and the output power can as and Pout change sinusoidally with the dual-side phase-shift angle β At resonance where and ignoring R 1 and R 2 when their voltage drops are significantly smaller than U 1 and U 2 , we have According to FHA, the transmitter and receiver currents are constant. With the dual-side phase-shift angle of β, by which U 2 leads U 1 , the phasor diagram is plotted in Figure 3 [11]. The rectifier dc current and the output power can be calculated as where ω is the working angular frequency and the parameters in b phasor quantities. At resonance where .
According to FHA, the transmitter and receiver currents are const With the dual-side phase-shift angle of β, by which U2 leads U1, th is plotted in Figure 3 [11]. The rectifier dc current and the output powe as and Pout change sinusoidally with the dual-side phase-shift an

Time-Domain Modelling
The waveforms of the series-series compensated WPT system are be discontinuous [10,12]. Therefore, FHA is no longer valid. Simila I REC and P out change sinusoidally with the dual-side phase-shift angle β.

Time-Domain Modelling
The waveforms of the series-series compensated WPT system are distorted and can be discontinuous [10,12]. Therefore, FHA is no longer valid. Similar to [12], the time domain modelling based on differential equations is conducted. The typical waveforms are shown in Figure 4. Assume that the cycle is T. At t = 0, u 1 is rising, and at t 0 , u 2 is falling. In the first half cycle, S 1 and S 4 are on and S 2 and S 3 are off. In the second half cycle, S 1 and S 4 are off and S 2 and S 3 are on. Due to symmetry, only the first half cycle is considered.
When 0 < t < t 0 , S 1 , S 4 , S 5 , and S 8 are all on, namely u 1 and u 2 are both positive, the differential equations can be given as where the subscript "−1" denotes that the symbols are in the interval of 0 < t < t 0 .
Electr. Veh. J. 2021, 12, x FOR PEER REVIEW domain modelling based on differential equations is conducted. are shown in Figure 4. Assume that the cycle is T. At t = 0, u1 is risin In the first half cycle, S1 and S4 are on and S2 and S3 are off. In the s S4 are off and S2 and S3 are on. Due to symmetry, only the first hal When 0 < t < t0, S1, S4, S5, and S8 are all on, namely u1 and u differential equations can be given as where the subscript "−1" denotes that the symbols are in the inter When t0 < t < T/2, S2, S3, S5, and S8 are all on, namely u1 is pos the differential equations can be given as When t 0 < t < T/2, S 2 , S 3 , S 5 , and S 8 are all on, namely u 1 is positive and u 2 is negative, the differential equations can be given as where the subscript "−2" denotes that the symbols are in the interval of t 0 < t < T/2. With considerations of the boundary conditions, namely the differential equations can be solved. The rectifier dc current and the output power can be calculated as where k is the coupling coefficient, defined as t 0 indicates the phase-shift instant when the secondary-side rectifier is switching. The relationship between t 0 and β is Thus, (11) and (12) are transformed into For synchronous rectification, there should be one more boundary condition as i 2-1 (t 0 ) = i 2-2 (t 0 ) = 0, which is the natural zero crossing point of the receiver current. By solving this, the phase-shift angle between u 2 and the u 1 is larger than 90 • . In this condition, i 1-1 (0) is larger than 0, indicating that soft switching is lost.

Comparison of FHA and Model Based on Differential Equations
Dividing (15) by (6) yields the ratio of the rectifier dc current from the differential equations over that from FHA, namely The ratio of the rectifier dc currents in (17), varying with the phase difference between u 2 and u 1 , namely β, is plotted in Figure 5. At strong couplings, the rectifier dc currents are larger than the predictions from FHA between the phase-shift angles of 60 • and 120 • , and smaller at other phase-shift angles. This shows the inaccuracy of FHA at strong couplings.
In Figures 2 and 3, the corresponding voltage and current (U 1 and I 1 , U 2 and I 2 ) are defined in an associate reference direction. This means that when the active power of the voltage source is positive, the voltage source acts as an actual source and the active power is transferred from the voltage source to the outside circuit; when the active power of the voltage source is negative, the voltage source acts as a sink and the active power is transferred from the outside circuit to the voltage source. Thus, in Figure 3, when U 2 leads U 1 or β ∈ (0, 180 • ), active power is transferred from the primary side to the secondary side; World Electr. Veh. J. 2022, 13, 6 6 of 10 when U 2 lags behind U 1 or β ∈ (−180 • , 0), active power is transferred from the secondary side to the primary side. This is also true for strongly coupled cases.

Experimental Validation
An experimental prototype is implemented to validate the analysis, as shown ure 7. The parameters of the experimental prototype are tabulated in Table 1. Th type with the coupling coefficient of 0.64 is a typical strongly coupled WPT syste is set to 200 V. Three voltage gains are selected: 0.5, 1.0, and 2.0, with correspond of 100 V, 200 V, and 400 V.
The measured dc-dc efficiency, the rectifier current, and the output power un chronous rectification and phase shift are illustrated in Figure 8. The data denoted chronous rectification" indicate that synchronous rectification is achieved at these shift angles, which are larger than 90° and locate at the range of ZCS, as shown in 6. Thus, with synchronous rectification, soft switching is lost. By changing the ph angle between 0 and 90°, the output current, namely the rectifier current, can b Attention should be paid to the range of [0, 180 • ] where active power is transferred from the primary side to the secondary side. As can also be seen from Figure 3, when 0 < β < 90 • , I 1 lags behind U 1 , which means the input impedance is inductive. Thus, zero voltage switching (ZVS) can be achieved for both the primary-side inverter and secondaryside rectifier. When 90 • < β < 180 • , I 1 leads U 1 , and zero current switching (ZCS) can be achieved for both the primary-side inverter and secondary-side rectifier. When MOSFETs are used, ZVS is preferred, indicating that the dual-side phase-shift angle should be in the range of [0, 90 • ]. The variations of the rectifier currents or output power with the dual-side phase-shift angle β are plotted in Figure 6.

Experimental Validation
An experimental prototype is implemented to validate the analysis, as shown ure 7. The parameters of the experimental prototype are tabulated in Table 1. Th type with the coupling coefficient of 0.64 is a typical strongly coupled WPT syste

Experimental Validation
An experimental prototype is implemented to validate the analysis, as shown in Figure 7. The parameters of the experimental prototype are tabulated in Table 1    The measured dc-dc efficiency, the rectifier current, and the output power under synchronous rectification and phase shift are illustrated in Figure 8. The data denoted as "synchronous rectification" indicate that synchronous rectification is achieved at these phase-shift angles, which are larger than 90 • and locate at the range of ZCS, as shown in Figure 6. Thus, with synchronous rectification, soft switching is lost. By changing the phase shift angle between 0 and 90 • , the output current, namely the rectifier current, can be easily regulated, as shown in Figure 8b. In the wide regulation range, the dc-dc efficiency can still be high, different from the loosely coupled wireless power transfer systems.  The experimental waveforms when the voltage gain is 2.0 are shown in Figure 9. With synchronous rectification in strongly coupled WPT systems, the phase-shift angles are always larger than 90 • , and the dc-dc efficiency is smaller than that with a 90 • phase shift.

Conclusions
This paper presented the modelling and dual-side phase-shift control coupled series-series compensated WPT systems. There are issues with sy

Conclusions
This paper presented the modelling and dual-side phase-shift control of the strongly coupled series-series compensated WPT systems. There are issues with synchronous rectification in strongly coupled series-series compensated WPT systems: the efficiency is not the highest and the inverter may work in hard switching. In comparison, with a 90 • phase shift, the inverter will always work in soft switching and the efficiency is the highest. Moreover, with dual-side phase-shift regulation in strongly coupled series-series compensated WPT systems, the output current can be easily regulated by changing the phase difference between the primary-side and the secondary-side converters with relatively high dc-dc efficiency over a wide regulation range, which is a different characteristic from the loosely coupled WPT systems.

Conflicts of Interest:
The authors declare no conflict of interest.