Auto-resonant Control of the H-Bridge Resonant Converter for Inductive Power Transfer Applications

Delft University of Technology

Auto-resonant Control of the H-Bridge Resonant Converter for Inductive Power Transfer

Applications

Grazian, Francesca; van Duijsen, Peter; Roodenburg, Bart; Soeiro, Thiago Batista; Bauer, Pavol DOI

10.1109/ISIE45063.2020.9152592 Publication date

2020

Document Version

Accepted author manuscript Published in

2020 IEEE 29th International Symposium on Industrial Electronics, ISIE 2020 - Proceedings

Citation (APA)

Grazian, F., van Duijsen, P., Roodenburg, B., Soeiro, T. B., & Bauer, P. (2020). Auto-resonant Control of the H-Bridge Resonant Converter for Inductive Power Transfer Applications. In 2020 IEEE 29th International Symposium on Industrial Electronics, ISIE 2020 - Proceedings: Proceedings (pp. 1593-1598). [9152592] (IEEE International Symposium on Industrial Electronics; Vol. 2020-June). IEEE .

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Auto-resonant Control of the H-Bridge Resonant

Converter for Inductive Power Transfer Applications

Francesca Grazian, Peter van Duijsen, Bart Roodenburg, Thiago Batista Soeiro, and Pavol Bauer

Delft University of Technology, Delft 2628 CD, The Netherlands

Email: (F.Grazian, P.J.vanDuijsen, B.Roodenburg, T.BatistaSoeiro, P.Bauer)@tudelft.nl

Abstract—In inductive power transfer applications that use resonant compensation networks, the commonly employed H-bridge inverter should be kept operating in soft-switching to ensure high power efficiency and low irradiated electromagnetic noise. To achieve so, the zero-crossing detection circuit for the resonant current or voltage must be fast and accurate in any operating condition. This paper researches the concept of an auto-resonant control for the typical H-bridge resonant converter used in wireless charging systems. In the method proposed here, the reference levels for the zero-crossing detection of the inverter’s current are automatically adapted depending on the slope of the current itself at the zero-crossing. In this way, it is possible to compensate for the circuit delay even in the presence of parameters’ variation and to ensure that the soft-switching is always maintained. The functionality of this control method is proven first mathematically, and then with circuit simulations. The core steps for the implementation are described with the support of functional blocks. Finally, the system start-up strategy is explained, which uses an auxiliary timed oscillator to modulate the inverter with a fixed 50% duty cycle at a higher frequency than the nominal. This guarantees that the start-up is in the inductive region and, thus, the zero-voltage switching turn-on. Once the detection circuits sense the current flow, the oscillator is automatically disabled, and the nominal power transfer starts. Index Terms—Control, inductive power transfer, inverter, soft-switching, wireless charging, zero voltage switching.

I. INTRODUCTION

Over the last decade, wireless charging of electric vehicles (EVs) has been gaining popularity because, in some specific applications, it has clear advantages with respect to the tradi-tional charging through cable. For example, wireless charging is convenient in dynamic charging and to charge autonomous-driven EVs. To make this technology competitive, the power transfer efficiency from the source to the EV battery needs to be maximized, and that can be achieved by minimizing the power losses of each power conversion stage.

The most used method in EV wireless charging is inductive power transfer (IPT) that uses magnetic resonant coupling. Thereby, the power is transferred from the transmitter to the receiver coil through an 85 kHz magnetic field [1]. This magnetic field is generated by a time-varying current that flows in the transmitter coil which, in turn, is produced by an inverter. In EV wireless charging applications, the most used inverter topology is the full- or H-bridge converter that is composed of four switching units, as shown in Fig. 1. In the H-bridge inverter, there are two main sources of power losses: the conduction and the switching losses. For a certain input

Vref+, Vref-M A B a b Control f, D Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver

freq, D IAB Vref+, Vref-V+, V-Q, Q Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver

freq, D IAB Vref+, Vref-V+, V-Q, Q IAB measurement and signal conditioning IAB Vi,B differentiation at the zero-crossing Computation of Vref=f( Vdiff ) Vi,B Vdiff+ V diff-Vref+ V ref-Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver freq, D

V+,

V-Creation of the short pulses p1 and p2 Shoot-through protection Correlation between p1 and p2 V+, V-Q, Q Q, Q p1, p2 Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver freq, D Vref+, Vref-V+, V-Q, Q Vref+, Vref-IAB Comparator for the positive slope of Vi,A Comparator for the negative slope of Vi,A IAB measurement and signal conditioning Vref+ Vref-Vi,A V+, V -V+ V -IAB Reference voltage computation Current’s zero-crossing detection freq, D Vref+, V ref-Q, Q

Creation of the short pulses p1 and p2

Setting of the dead time tdead Correlation between p1 and p2 V+, V -Q, Q p1, p2 Vref+, V ref-Comparator for the positive slope of Vi,A Comparator for the negative slope of Vi,A

IAB measurement and signal conditioning Vref+ V ref-Vi,A V -Vi,B differentiation at the zero-crossing Vi,B Vdiff+ V diff-Vref+ V ref-iAB Vin Cin Vout Iout Cout i1 i2 L1 L2 R1 R2 Control freq, D iAB iAB measurement and signal conditioning Computation of Vref=f( Vdiff ) iAB iAB V+,V -V+ iAB V+, V

-Creation of the control signals to the gate driver

Iin

Fig. 1. General circuit used in EV wireless charging.

voltage and power level, the conduction losses can be limited by choosing unipolar switches, such as MOSFETs, with low on-state resistance. This choice generally leads to higher costs because it may require large areas of the semiconductor chip. On the other hand, the switching losses can be suppressed by operating the inverter in a soft-switching condition. Especially while employing Silicon MOSFETs, the turn-on transition at the zero-voltage switching (ZVS) needs to be ensured to discharge the drain-source capacitance of the switches and to avoid the reverse recovery of the anti-parallel diodes. As explained in [2]–[6], it is possible to achieve the ZVS turn-on in all inverter’s active semicturn-onductors when the resturn-onant current lags the generated square-wave voltage, i.e. when the system operates in the inductive region. This means that the H-bridge inverter has to be modulated at a switching frequency slightly higher than the natural frequency of the passive compensation network. This can be realized by switching the inverter’s legs just before the resonant current’s zero-crossing. Therefore, the zero-crossing detection of the resonant current is a fundamental part of the H-bridge inverter’s control loop.

There are some factors that make the current’s zero-crossing detection challenging. The accuracy and precision of the zero-crossing detection should be relatively high in the inverter’s operating frequency range 79-90 kHz defined by [1], [7]. Moreover, due to manufacturing tolerances, temperature rise, drop of magnetic core permeability, or degradation, the com-ponents of the resonant circuit might assume sightly different values from the theoretical design. In those cases, the actual resonant frequency would change and, to maximize the power transfer efficiency, the operating frequency needs to be re-tuned within the allowed frequency range. Additionally, in EV wireless charging applications, the operating condition is not fixed because there are two important parameters that can vary. The first parameter is the coupling factor between the transmitter and receiver coils, which strongly depends on their

TABLE I

EXAMPLE OF DELAYS INTRODUCED BYICS AND THE GATE DRIVER.

Component Delay (ns)1 Type Name ∆ton ∆tof f Operational amplifier TL071 100 100 Differential comparator LM211 115 165 Gate driver IR 2110 120 94 Total 335 359

1 _{From the datasheet}

alignment. The other parameter is the equivalent output load that changes while the EV battery is being charged. The zero-crossing detection must be fast and adaptable in order to keep the ZVS turn-on at any operating condition.

In the literature, different current’s zero-crossing detection circuits can be found. The need for a fast inverter control loop to detect the zero-crossing accurately is already acknowledged in [8]–[11]. In [8], [9] and [11], the inverter’s control has been implemented by combining analog electronics with an FPGA because of the required fast processing speed. Another alter-native widely used is the phase-locked loop (PLL). However, the detection of the zero-crossing must be precise, and analog-to-digital converters (ADCs) generally do not have enough resolution for these relatively high frequencies. To overcome this problem, analog current measurements are typically used followed by a comparator that performs the detection. Since the detection and the conditioning circuit themselves introduce a fixed time delay, it is important to take it into account in the implementation. In [10], the time delay is compensated by setting a manually adjustable reference to the comparator that depends on the measured current peak. However, in a real application, this reference needs to be adjusted automatically. This paper describes the concept of an auto-resonant control for the H-bridge inverter used in inductive power transfer applications. This concept consists of adapting the reference levels of the current’s zero-crossing detection depending on the slope of the current at the zero-crossing. In this way, the operation is always adjusted to be close to the actual resonant frequency of the circuit while ensuring the ZVS turn-on of the inverter. The general features and the analysis of the current’s zero-crossing detection are explained in Section II. The proposed concept of the auto-resonant current’s zero-crossing detection is described in Section III. Since this control concept has an analog implementation, the start-up of the power transfer is critical. The start-up strategy is addressed and explained in Section IV, and it has been validated in combination with the auto-resonant control through circuit simulations. Finally, conclusions on the concept of the auto-resonant control are described in Section V.

II. CURRENT’S ZERO-CROSSING DETECTION A general circuit schematic of a wireless charging system is shown in Fig. 1. The inverter’s control loop takes as input the measurement of the resonant current iAB= iAB(t), and it

gives as output the operating frequency and duty cycle to the switches’ gate driver. Ideally, the zero-crossing of iAB could

be detected by comparing the measured current waveform to

0 0 0 2 ‐2 5.246 0.1 11.126 ‐0.1 4.887 2.99 10.791 ‐2.93 0.01 ‐0.01 5 0 1.00E‐09 0.001 3.15E‐03 2 ‐2 5.246 0.2 11.126 ‐0.2 4.887 2.99 10.791 ‐2.93 0.02 ‐0.02 5 0 2.00E‐09 0.002 6.30E‐03 2 ‐2 5.246 0.3 11.126 ‐0.3 4.887 2.99 10.791 ‐2.93 0.03 ‐0.03 5 5 3.00E‐09 0.003 9.45E‐03 2 ‐2 5.246 0.4 11.126 ‐0.4 4.887 2.99 10.791 ‐2.93 0.04 ‐0.04 5 5 4.00E‐09 0.004 1.26E‐02 2 ‐2 5.246 0.5 11.126 ‐0.5 4.887 2.99 10.791 ‐2.93 0.05 ‐0.05 5 5 5.00E‐09 0.005 1.58E‐02 2 ‐2 5.246 0.6 11.126 ‐0.6 4.887 2.99 10.791 ‐2.93 0.06 ‐0.06 5 5 6.00E‐09 0.006 1.89E‐02 2 ‐2 5.246 0.7 11.126 ‐0.7 4.887 2.99 10.791 ‐2.93 0.07 ‐0.07 5 5 7.00E‐09 0.007 2.21E‐02 2 ‐2 5.246 0.8 11.126 ‐0.8 4.887 2.99 10.791 ‐2.93 0.08 ‐0.08 5 5 8.00E‐09 0.008 2.52E‐02 2 ‐2 5.246 0.9 11.126 ‐0.9 4.887 2.99 10.791 ‐2.93 0.09 ‐0.09 5 5 9.00E‐09 0.009 2.84E‐02 2 ‐2 5.246 1 11.126 ‐1 4.887 2.99 10.791 ‐2.93 0.1 ‐0.1 5 5 1.00E‐08 0.01 3.15E‐02 2 ‐2 5.246 1.1 11.126 ‐1.1 4.887 2.99 10.791 ‐2.93 0.11 ‐0.11 5 5 1.10E‐08 0.011 3.47E‐02 2 ‐2 5.246 1.2 11.126 ‐1.2 4.887 2.99 10.791 ‐2.93 0.12 ‐0.12 5 5 1.20E‐08 0.012 3.78E‐02 2 ‐2 5.246 1.3 11.126 ‐1.3 4.887 2.99 10.791 ‐2.93 0.13 ‐0.13 5 5 1.30E‐08 0.013 4.10E‐02 2 ‐2 5.246 1.4 11.126 ‐1.4 4.887 2.99 10.791 ‐2.93 0.14 ‐0.14 5 5 1.40E‐08 0.014 4.41E‐02 2 ‐2 5.246 1.5 11.126 ‐1.5 4.887 2.99 10.791 ‐2.93 0.15 ‐0.15 5 5 1.50E‐08 0.015 4.73E‐02 2 ‐2 5.246 1.6 11.126 ‐1.6 4.887 2.99 10.791 ‐2.93 0.16 ‐0.16 5 5 1.60E‐08 0.016 5.04E‐02 2 ‐2 5.246 1.7 11.126 ‐1.7 4.887 2.99 10.791 ‐2.93 0.17 ‐0.17 5 5 1.70E‐08 0.017 5.36E‐02 2 ‐2 5.246 1.8 11.126 ‐1.8 4.887 2.99 10.791 ‐2.93 0.18 ‐0.18 5 5 1.80E‐08 0.018 5.67E‐02 2 ‐2 5.246 1.9 11.126 ‐1.9 4.887 2.99 10.791 ‐2.93 0.19 ‐0.19 5 5 1.90E‐08 0.019 5.99E‐02 2 ‐2 5.246 2 11.126 ‐2 4.887 2.99 10.791 ‐2.93 0.2 ‐0.2 5 5 2.00E‐08 0.02 6.30E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.21 ‐0.21 5 5 2.10E‐08 0.021 6.62E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.22 ‐0.22 5 5 2.20E‐08 0.022 6.93E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.23 ‐0.23 5 5 2.30E‐08 0.023 7.25E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.24 ‐0.24 5 5 2.40E‐08 0.024 7.56E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.25 ‐0.25 5 5 2.50E‐08 0.025 7.88E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.26 ‐0.26 5 5 2.60E‐08 0.026 8.19E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.27 ‐0.27 5 5 2.70E‐08 0.027 8.51E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.28 ‐0.28 5 5 2.80E‐08 0.028 8.82E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.29 ‐0.29 5 5 2.90E‐08 0.029 9.14E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.3 ‐0.3 5 5 3.00E‐08 0.03 9.45E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.31 ‐0.31 5 5 3.10E‐08 0.031 9.77E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.32 ‐0.32 5 5 3.20E‐08 0.032 1.01E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.33 ‐0.33 5 5 3.30E‐08 0.033 1.04E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.34 ‐0.34 5 5 3.40E‐08 0.034 1.07E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.35 ‐0.35 5 5 3.50E‐08 0.035 1.10E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.36 ‐0.36 5 5 3.60E‐08 0.036 1.13E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.37 ‐0.37 5 5 3.70E‐08 0.037 1.17E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.38 ‐0.38 5 5 3.80E‐08 0.038 1.20E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.39 ‐0.39 5 5 3.90E‐08 0.039 1.23E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.4 ‐0.4 5 5 4.00E‐08 0.04 1.26E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.41 ‐0.41 5 5 4.10E‐08 0.041 1.29E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.42 ‐0.42 5 5 4.20E‐08 0.042 1.32E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.43 ‐0.43 5 5 4.30E‐08 0.043 1.35E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.44 ‐0.44 5 5 4.40E‐08 0.044 1.39E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.45 ‐0.45 5 5 4.50E‐08 0.045 1.42E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.46 ‐0.46 5 5 4.60E‐08 0.046 1.45E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.47 ‐0.47 5 5 4.70E‐08 0.047 1.48E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.48 ‐0.48 5 5 4.80E‐08 0.048 1.51E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.49 ‐0.49 5 5 4.90E‐08 0.049 1.54E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.5 ‐0.5 5 5 5.00E‐08 0.05 1.58E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.51 ‐0.51 5 5 5.10E‐08 0.051 1.61E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.52 ‐0.52 5 5 5.20E‐08 0.052 1.64E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.53 ‐0.53 5 5 5.30E‐08 0.053 1.67E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.54 ‐0.54 5 5 5.40E‐08 0.054 1.70E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.55 ‐0.55 5 5 5.50E‐08 0.055 1.73E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.56 ‐0.56 5 5 5.60E‐08 0.056 1.76E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.57 ‐0.57 5 5 5.70E‐08 0.057 1.80E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.58 ‐0.58 5 5 5.80E‐08 0.058 1.83E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.59 ‐0.59 5 5 5.90E‐08 0.059 1.86E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.6 ‐0.6 5 5 6.00E‐08 0.06 1.89E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.61 ‐0.61 5 5 6.10E‐08 0.061 1.92E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.62 ‐0.62 5 5 6.20E‐08 0.062 1.95E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.63 ‐0.63 5 5 6.30E‐08 0.063 1.98E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.64 ‐0.64 5 5 6.40E‐08 0.064 2.02E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.65 ‐0.65 5 5 6.50E‐08 0.065 2.05E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.66 ‐0.66 5 5 6.60E‐08 0.066 2.08E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.67 ‐0.67 5 5 6.70E‐08 0.067 2.11E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.68 ‐0.68 5 5 6.80E‐08 0.068 2.14E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.69 ‐0.69 5 5 6.90E‐08 0.069 2.17E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.7 ‐0.7 5 5 7.00E‐08 0.07 2.21E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.71 ‐0.71 5 5 7.10E‐08 0.071 2.24E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.72 ‐0.72 5 5 7.20E‐08 0.072 2.27E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.73 ‐0.73 5 5 7.30E‐08 0.073 2.30E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.74 ‐0.74 5 5 7.40E‐08 0.074 2.33E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.75 ‐0.75 5 5 -6 -3036 V 0 5.88 11.76 t (μs) iAB vAB (b) 0 5.88 11.76 t (μs) iAB vAB (a)

Fig. 2. Inverter voltage vABand current iABat: (a) ZCS, (b) ZVS operation.

-6 0 6 T-Дtoff 2 -6 0 6 v Vref+ V ref-VIOFF VIOFF -2 0 2 0 5.88 11.76 t (μs) (a) -3 0 3 0 5.88 11.76 t (μs) (b) T-Дton t1 t2 i,A V ( V ) V ( V ) V ( V ) V ( V )

Fig. 3. Voltage signal of the sensed resonant current, vi,A, and the reference

voltages Vref +, Vref −for: (a) compensating the delays times ∆ton, ∆tof f

introduced by the analog circuits (T =_f1), (b) detecting the current IOF F

which guarantees ZVS turn-on.

0 V (ground reference voltage). An example of inverter ZCS operation is shown in Fig. 2(a) for the operating frequency of 85 kHz. However, in a real circuit, all the integrated circuits (ICs), analog logics and the gate drivers introduce a time delay ∆t in the control signal, which would make the commutation happen at a different current. After the current measurement, the zero-crossing detection circuit comprises of at least one operational amplifier (opamp) to perform the signal condition-ing and one differential comparator to realize the detection. For example, Table I shows the typical delay times caused by the commercial opamp TL071, the differential comparator LM211 and the gate driver IR2110. Considering iAB to be a

85 kHz sinusoidal current with an amplitude of 6 A, the actual inverter’s switching current can be calculated by using (1), where I is the current’s root mean square. As a result, the total delay in Table I would make the inverter switching at about 1 A instead of at the zero-crossing. Therefore, if the ground potential is considered as the reference for the comparator (Vref=0 V), the inverter would operate in hard-switching.

iAB = iAB(t) =

√

2I sin(2πf t) =√2I sin(ωt) (1) Given a control circuit, ∆t can be considered to be constant during the operation, and it is possible to compensate for the delay ∆t, by imposing Vref6=0 V. However, both the

amplitude and the frequency of iAB might vary depending on

the operating condition and the circuit parameters’ variation. As a consequence, even though the ∆t is nearly constant, the value of Vref that compensates for ∆t varies in different

operating conditions. To prove this, let’s still consider that iAB is an 85 kHz sinusoidal current. Fig. 3 shows the voltage

signal conditioning stage which result in a 85 kHz sinusoidal voltage with 1:1 amplitude ratio with respect to the original current iAB, and the voltage references Vref + and Vref − for

the zero-crossing detection of iAB at the positive and negative

slope, respectively. In this example, the amplitude of iAB is

assumed to vary in the range of 4-6 A. To compensate for the constant delays ∆ton, ∆tof f in Table I, Fig. 3(a) shows

that the value of Vref might vary considerably when the

amplitude of iAB varies. The variation of Vref would be even

more accentuated if the operating frequency also changes. This means that, to achieve an accurate zero-crossing detection, the reference Vref must be adjusted during the operation. At this

point, it is interesting to compute the mathematical expression of Vref to verify what observed in Fig. 3(a).

The expression of iAB in (1) can be rewritten in terms of

the voltage signal representing the measured current vi,A as

shown in (2). Moreover, since the time instants T₂− ∆tonand

T −∆tof fin Fig. 3(a) are close to the current’s zero-crossings,

the small-angle approximation in (3) can be considered to be valid. By using this approximation, (2) simplifies in (4).

vi,A(t) = √ 2Vi,Asin(ωt) (2) −π 6 < ωt < π 6 → sin(ωt) ≈ ωt (3) vi,A(t) = √ 2Vi,Asin(ωt) ≈ √ 2Vi,Aωt (4)

The value of Vref +, Vref − can be computed as described

in (5), (6), respectively. By observing the small-angle approx-imation of Vref +and Vref −, it is possible to notice that their

first term is equal to the approximated time-derivative of vi,A

in (7). By combining (5) and (6) with (7), it can be concluded that, according to the small-angle approximation in (3), Vref +

and Vref − are equivalent to (8) and (9), respectively. This

means that the references Vref +, Vref − depend on the slope

of iAB(t) during the zero-crossing.

Vref += √ 2Vi,Asin(ω∆ton) ≈ √ 2Vi,Aω∆ton (5) Vref −= √ 2Vi,Asin(ω∆tof f) ≈ √ 2Vi,Aω∆tof f (6) d dtvi,A(t) ≈ √ 2Vi,Aω (7) Vref +≈ d dtvi,A(t)∆ton (8) Vref −≈ d dtvi,A(t)∆tof f (9)

If both the amplitude and frequency of iAB(t) are constant,

the delays ∆tonand ∆tof f could be compensated by choosing

fixed values for Vref + and Vref −. On the other hand, if the

operating frequency is fixed, Vref +and Vref − would depend

only on the amplitude of iAB. In that case, the approach in [10]

could be used which adjusts Vref + and Vref − by measuring

the amplitude of iAB. However, since the allowed frequency

range for the EV wireless charging is 79-90 kHz, the operating frequency can be fine-tuned to match the actual resonant frequency of the circuit that can differ due to components’ variations. This means that, for the same current amplitude, the slope at the zero-crossing could be different depending on

Reference voltage computation Current’s zero-crossing

detection freq, D

Vref+, V

ref-Q, Q

Creation of the short pulses p1 and p2

Setting of the dead time tdead Correlation between p1 and p2 V+, V -Q, Q p1, p2 Vref+, V ref-Comparator for the positive

slope of Vi,A

Comparator for the negative slope

of Vi,A

iAB measurement and signal conditioning Vref+ V ref-vi,A V -vi,B differentiation at the zero-crossing vi,B Vdiff+ V diff-Vref+ V ref-Control f, D iAB iAB measurement and signal conditioning Computation of Vref=f( Vdiff ) iAB V+, V -V+ iAB V+, V

-Creation of the control signals to the gate driver

iAB

Fig. 4. Block diagram of the auto-resonant control’s concept.

the period’s duration. Therefore, it is chosen to adjust Vref +

and Vref − based on the _dtdiAB at the zero-crossing of iAB.

To achieved the soft-switching operation, the ZVS turn-on of the inverter must be ensured. This can be dturn-one by switching the inverter’s legs before the zero-crossing of iABas

shown in Fig. 2(b). In particular, [2]–[6] define the minimum current IOF F that ensures the ZVS turn-on. By considering the

same measured current signal vi,A, and assuming that VIOF F

corresponds to the measurement of IOF F, theoretically the

ZVS turn-on is achieved when the references Vref +, Vref −

are equal to VIOF F as shown in Fig. 3(b), where IOF F =

2 A. According to [2]–[6], the value of IOF F depends on

the MOSFET’s drain-source blocking voltage Vds,of f,

drain-source capacitance Cds and the half-bridge switching dead

time tdead. This means that, for the same MOSFET, DC input

voltage Vin, and tdead, the value of IOF F can be considered

to be constant. According to Fig. 3(b), the IOF F detection

instants t1 and t2 would vary if either the amplitude or the

frequency of iAB changes. However, the comparators would

follow this change and detect IOF F by keeping Vref + and

Vref − constant at VIOF F. Therefore, for the same IOF F, the

references Vref +and Vref −do not have to be adjusted to keep

the ZVS turn-on at different operating conditions. III. CONCEPT OF THE AUTO-RESONANT CONTROL The concept of the auto-resonant control has been derived from the mathematical analysis in Section II, and it is summa-rized by the block diagram in Fig. 4. The auto-resonant control can be used with H-bridge inverters likewise in Fig. 1, and it can work with any resonant compensation network as long as the inverter’s current approximates a sinusoid. The key element in this control is that the current’s zero-crossing detection uses the adjustable references Vref +and Vref − such that the ZVS

turn-on is kept at different operating conditions. The value of

Vref +and Vref −depends on two terms: the delay times ∆ton,

∆tof f introduced by the control circuit, and the detection

voltage VIOF F equivalent to IOF F. The first term is adjusted

depending on the slope of iAB at the zero crossing, according

term is also fixed. The two different reference voltages Vref +

and Vref −are computed separately to detect more accurately

the zero-crossing of iAB for the positive slope (_dtdiAB>0) and

negative slope (_dtdiAB<0) of iAB, respectively. This choice

has been made because it might exist an asymmetry between the on- and off-reaction time of the components, as it can be noticed from the delays in Table I. This approach also takes into account the potential difference between the positive and negative half-wave of iAB, which can be due to the fact that

the resonant oscillations are not perfectly sinusoidal.

Fig. 5(a) shows the measured current signal vi,A with the

updated references Vref +, Vref −that are composed of the two

terms explained above. This means that Vref + and Vref − in

Fig. 5(a) correspond to the updated time instants t1+ ∆tof f

and t2+ ∆tof f which differ to the ones in Fig. 3.

The auto-resonant control is composed of three main blocks as shown in Fig. 4. The first block computes the adjusted reference voltages Vref + and Vref −, and it is explained in

Section III-A. The second block performs the current’s zero-crossing detection, and it is described in Section III-B. Finally, the third block creates the suitable control signals for the switches’ gate driver, and it is discussed in Section III-C. A. Reference voltage computation block

This first block can be divided into three main parts, as shown in Fig. 4. First, the inverter’s current iAB is measured

and, after a signal conditioning stage, the measured current Vi,B is generated. Then, the measured Vi,Bis differentiated at

both the positive and negative zero-crossings. The rectified values of both differentiation produce the voltages Vdif f +

and Vdif f − for the positive and negative slope, respectively.

Finally, the adjusted reference voltages Vref + and Vref − are

computed as function of Vdif f +and Vdif f −, respectively. Fig.

5(a) shows a qualitative representation of Vref + and Vref −.

B. Current’s zero-crossing detection block

This second block can be divided into two main parts, as shown in Fig. 4. First, iABis measured. Then, the conditioned

measurement Vi,A is used as main input for two different

comparators. Additionally, the comparator for the positive slope’s zero-crossing takes as inverting input the reference voltage Vref +, while the comparator for the negative slope’s

zero-crossing takes as non-inverting input the reference voltage

Vref −. In this way, the positive and negative slopes are

treated separately. Each comparator gives a high output (+5 V) when Vi,Ais greater than the reference, and low output (0 V)

otherwise. As a result, the outputs are the two square waves V+and V−produced by the comparators, which might overlap

because they are completely independent of each other. The outputs V+ and V− are qualitatively shown in Fig. 5(b).

C. Creation of the control signals to the gate driver block This third block can be divided into three main parts, as shown in Fig. 4. Since the signals V+ and V− coming from

the comparators might overlap, they need to be related to each other such that the control signal to the gate driver is

0 0 0 2 ‐2 5.246 0.1 11.126 ‐0.1 4.887 2.99 10.791 ‐2.93 0.01 ‐0.01 5 5 0 5 0 5 1.00E‐09 0.001 3.15E‐03 2 ‐2 5.246 0.2 11.126 ‐0.2 4.887 2.99 10.791 ‐2.93 0.02 ‐0.02 5 5 0 5 0 5 2.00E‐09 0.002 6.30E‐03 2 ‐2 5.246 0.3 11.126 ‐0.3 4.887 2.99 10.791 ‐2.93 0.03 ‐0.03 5 5 0 5 0 5 3.00E‐09 0.003 9.45E‐03 2 ‐2 5.246 0.4 11.126 ‐0.4 4.887 2.99 10.791 ‐2.93 0.04 ‐0.04 5 5 0 5 0 5 4.00E‐09 0.004 1.26E‐02 2 ‐2 5.246 0.5 11.126 ‐0.5 4.887 2.99 10.791 ‐2.93 0.05 ‐0.05 5 5 0 5 0 5 5.00E‐09 0.005 1.58E‐02 2 ‐2 5.246 0.6 11.126 ‐0.6 4.887 2.99 10.791 ‐2.93 0.06 ‐0.06 5 5 0 5 0 5 6.00E‐09 0.006 1.89E‐02 2 ‐2 5.246 0.7 11.126 ‐0.7 4.887 2.99 10.791 ‐2.93 0.07 ‐0.07 7.00E‐09 0.007 2.21E‐02 2 ‐2 5.246 0.8 11.126 ‐0.8 4.887 2.99 10.791 ‐2.93 0.08 ‐0.08 8.00E‐09 0.008 2.52E‐02 2 ‐2 5.246 0.9 11.126 ‐0.9 4.887 2.99 10.791 ‐2.93 0.09 ‐0.09 9.00E‐09 0.009 2.84E‐02 2 ‐2 5.246 1 11.126 ‐1 4.887 2.99 10.791 ‐2.93 0.1 ‐0.1 5 5 0 5 0 5 1.00E‐08 0.01 3.15E‐02 2 ‐2 5.246 1.1 11.126 ‐1.1 4.887 2.99 10.791 ‐2.93 0.11 ‐0.11 5 5 0 5 0 5 1.10E‐08 0.011 3.47E‐02 2 ‐2 5.246 1.2 11.126 ‐1.2 4.887 2.99 10.791 ‐2.93 0.12 ‐0.12 5 5 0 5 0 5 1.20E‐08 0.012 3.78E‐02 2 ‐2 5.246 1.3 11.126 ‐1.3 4.887 2.99 10.791 ‐2.93 0.13 ‐0.13 5 5 0 5 0 5 1.30E‐08 0.013 4.10E‐02 2 ‐2 5.246 1.4 11.126 ‐1.4 4.887 2.99 10.791 ‐2.93 0.14 ‐0.14 5 5 0 5 0 5 1.40E‐08 0.014 4.41E‐02 2 ‐2 5.246 1.5 11.126 ‐1.5 4.887 2.99 10.791 ‐2.93 0.15 ‐0.15 5 5 0 5 0 5 1.50E‐08 0.015 4.73E‐02 2 ‐2 5.246 1.6 11.126 ‐1.6 4.887 2.99 10.791 ‐2.93 0.16 ‐0.16 5 5 0 5 0 5 1.60E‐08 0.016 5.04E‐02 2 ‐2 5.246 1.7 11.126 ‐1.7 4.887 2.99 10.791 ‐2.93 0.17 ‐0.17 5 5 0 5 0 5 1.70E‐08 0.017 5.36E‐02 2 ‐2 5.246 1.8 11.126 ‐1.8 4.887 2.99 10.791 ‐2.93 0.18 ‐0.18 5 5 0 5 0 5 1.80E‐08 0.018 5.67E‐02 2 ‐2 5.246 1.9 11.126 ‐1.9 4.887 2.99 10.791 ‐2.93 0.19 ‐0.19 5 5 0 5 0 5 1.90E‐08 0.019 5.99E‐02 2 ‐2 5.246 2 11.126 ‐2 4.887 2.99 10.791 ‐2.93 0.2 ‐0.2 5 5 0 5 0 5 2.00E‐08 0.02 6.30E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.21 ‐0.21 5 5 0 5 0 5 2.10E‐08 0.021 6.62E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.22 ‐0.22 5 5 0 5 0 5 2.20E‐08 0.022 6.93E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.23 ‐0.23 5 5 0 5 0 5 2.30E‐08 0.023 7.25E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.24 ‐0.24 5 5 0 5 0 5 2.40E‐08 0.024 7.56E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.25 ‐0.25 5 5 0 5 0 5 2.50E‐08 0.025 7.88E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.26 ‐0.26 5 5 0 5 0 5 2.60E‐08 0.026 8.19E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.27 ‐0.27 5 5 0 5 0 5 2.70E‐08 0.027 8.51E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.28 ‐0.28 5 5 0 5 0 5 2.80E‐08 0.028 8.82E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.29 ‐0.29 5 5 0 5 0 5 2.90E‐08 0.029 9.14E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.3 ‐0.3 5 5 0 5 0 5 3.00E‐08 0.03 9.45E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.31 ‐0.31 5 5 0 5 0 5 3.10E‐08 0.031 9.77E‐02 2 ‐2 4.887 2.99 10.791 ‐2.93 0.32 ‐0.32 5 5 0 5 0 5 3.20E‐08 0.032 1.01E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.33 ‐0.33 5 5 0 5 0 5 3.30E‐08 0.033 1.04E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.34 ‐0.34 3.40E‐08 0.034 1.07E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.35 ‐0.35 3.50E‐08 0.035 1.10E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.36 ‐0.36 3.60E‐08 0.036 1.13E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.37 ‐0.37 5 5 0 5 0 5 3.70E‐08 0.037 1.17E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.38 ‐0.38 5 5 0 5 0 5 3.80E‐08 0.038 1.20E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.39 ‐0.39 5 5 0 5 0 5 3.90E‐08 0.039 1.23E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.4 ‐0.4 5 5 0 5 0 5 4.00E‐08 0.04 1.26E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.41 ‐0.41 5 5 0 5 0 5 4.10E‐08 0.041 1.29E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.42 ‐0.42 5 5 0 5 0 5 4.20E‐08 0.042 1.32E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.43 ‐0.43 5 5 0 5 0 5 4.30E‐08 0.043 1.35E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.44 ‐0.44 5 5 0 5 0 5 4.40E‐08 0.044 1.39E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.45 ‐0.45 5 5 0 5 0 5 4.50E‐08 0.045 1.42E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.46 ‐0.46 5 5 0 5 0 5 4.60E‐08 0.046 1.45E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.47 ‐0.47 5 5 0 5 0 5 4.70E‐08 0.047 1.48E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.48 ‐0.48 5 5 0 5 0 5 4.80E‐08 0.048 1.51E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.49 ‐0.49 4.90E‐08 0.049 1.54E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.5 ‐0.5 5.00E‐08 0.05 1.58E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.51 ‐0.51 5.10E‐08 0.051 1.61E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.52 ‐0.52 5 5 0 5 0 5 5.20E‐08 0.052 1.64E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.53 ‐0.53 5 5 0 5 0 5 5.30E‐08 0.053 1.67E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.54 ‐0.54 5 5 0 5 0 5 5.40E‐08 0.054 1.70E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.55 ‐0.55 5 5 0 5 0 5 5.50E‐08 0.055 1.73E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.56 ‐0.56 5 5 0 5 0 5 5.60E‐08 0.056 1.76E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.57 ‐0.57 5 5 0 5 0 5 5.70E‐08 0.057 1.80E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.58 ‐0.58 5 5 0 5 0 5 5.80E‐08 0.058 1.83E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.59 ‐0.59 5 5 0 5 0 5 5.90E‐08 0.059 1.86E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.6 ‐0.6 5 5 0 5 0 5 6.00E‐08 0.06 1.89E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.61 ‐0.61 5 5 0 5 0 5 6.10E‐08 0.061 1.92E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.62 ‐0.62 5 5 0 5 0 5 6.20E‐08 0.062 1.95E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.63 ‐0.63 5 5 0 5 0 5 6.30E‐08 0.063 1.98E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.64 ‐0.64 5 5 0 5 0 5 6.40E‐08 0.064 2.02E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.65 ‐0.65 5 5 0 5 0 5 6.50E‐08 0.065 2.05E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.66 ‐0.66 5 5 0 5 0 5 6.60E‐08 0.066 2.08E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.67 ‐0.67 5 5 0 5 0 5 6.70E‐08 0.067 2.11E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.68 ‐0.68 5 5 0 5 0 5 6.80E‐08 0.068 2.14E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.69 ‐0.69 5 5 0 5 0 5 6.90E‐08 0.069 2.17E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.7 ‐0.7 5 5 0 5 0 5 7.00E‐08 0.07 2.21E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.71 ‐0.71 5 5 0 5 0 5 7.10E‐08 0.071 2.24E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.72 ‐0.72 5 5 0 5 0 5 7.20E‐08 0.072 2.27E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.73 ‐0.73 5 5 0 5 0 5 7.30E‐08 0.073 2.30E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.74 ‐0.74 5 5 0 5 0 5 7.40E‐08 0.074 2.33E‐01 2 ‐2 4.887 2.99 10.791 ‐2.93 0.75 ‐0.75 5 5 0 5 0 5 -6 -3036_V 0 2.94 5.88 8.82 11.76 p 2 (V) p 1 (V) t (μs) p1 p2 (c) 0 2.94 5.88 8.82 11.76 V-(V ) V + (V) t (μs) V+ V-(b) 0 2.94 5.88 8.82 11.76 Q (V) Q ( V) t (μs) Q Q (d) -6 -3 0 3 6 0 2.94 5.88 8.82 11.76 V ( V ) t (μs) v Vref+ V ref-t1+Дtoff t2+Дton (a) -6 -3 0 3 6 0 2.94 5.88 8.82 11.76 Vo lta ge (V ) t (μs)

Vi,A Vref+

Vref-t1 t2 IOFF IOFF 0 0 5 5 0 5 0 5 0 5 0 5 i,A vi,A vi,A vi,A vi,A vi,A vi,A

Fig. 5. Qualitative waveforms of the auto-resonant control that refer to the block diagram in Fig. 4. (a) Voltage signal of the measured current vi,Aand

the reference voltages Vref +, Vref −for detecting the current IOF F which

guarantees ZVS operation (as in Fig. 2(b)) and that also takes into account the circuit’s delay times ∆ton, ∆tof f. (b) Comparators’ output voltages V+,

V−. (c) Short pulses p1, p2. (d) Output voltages Q, Q to the gate driver.

unique. According to Section III-B, V+ is the control signal

related to the positive slope zero-crossing of iAB, while V− is

related to the negative slope zero-crossing of iAB. This means

that the part of interest of V+ and V− is just the one related

to their rising edge. Therefore, to guarantee that there is no overlap, V+ and V− are shorten into the short pulses p1 and

p2 which only focus on their parts of interest. The qualitative

representation of p1 and p2 is shown in Fig. 5(c). After this,

the intervals of time in which p1and p2assume the high value

are again extended, such that each signal switches to low as soon as the other one becomes high. In this way, it is possible to make sure that the control signals coming from the positive

Timed Oscillator OUT GND RESET p1,start p2,start V+ V-p1 p1,start V+ V- p2 p2,start Vi,A V ref-Vref+ Vdiff+ V diff-Vi,B Q f(Vdiff-) f(Vdiff+) Q (a) (b) Toggle switch + + + + +

Fig. 6. Summarized circuit schematic used to simulate: (a) the auto-resonant control concept with (b) the start-up strategy.

Vin Cin Cout

L1 L2 M I1 I2 R2 R1 A B a b Iout Iin Q1 Q2 Q3 Q4 D1 D2 D3 D4 Vout Control IAB freq, D Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver

freq, D IAB Vref+, Vref-V+, V-Q, Q Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver

freq, D IAB Vref+, Vref-V+, V-Q, Q IAB measurement and signal conditioning IAB Vi,B differentiation at the zero-crossing Computation of Vref=f( Vdiff ) Vi,B Vdiff+ V diff-Vref+ V ref-Reference voltage computation Current’s zero-crossing detection Creation of the control signals

to the gate driver freq, D Vref+, Vref-V+, V-Q, Q Vref+, Vref-IAB Comparator for the positive slope of Vi,A Comparator for the negative slope of Vi,A IAB measurement and signal conditioning Vref+ Vref-Vi,A V+, V -V+ V -IAB

The timed oscillator is manually enabled

The short-start pulses p1,start and

p2,start are created (f > f0)

The current iAB starts flowing

The timer is automatically disabled Are the comparators creating a

switching output? Yes

No

Fig. 7. Flow chart of the start-up strategy used in the auto-resonant control.

and the negative slope of iAB are not overlapping. Finally,

the last step consists in setting the optimal dead time tdead

because, as explained in [6], it is important that tdeadis long

enough to ensure the actual ZVS turn-on. At this point, the gating control signals Q and Q are created, and they are sent to the gate driver. The qualitative representation of Q and Q is shown in Fig. 5(d). The control signal Q becomes high in correspondence of the positive-slope zero-crossing of iAB.

This means that, among the switches in Fig. 1, Q1 and Q4 are conducting when Q is high, while Q2 and Q3 are open. The complementary operation is valid for the control signal Q.

IV. START-UP STRATEGY

The concept of the auto-resonant control explained in Sec-tion III can be implemented by using only analog components, as shown in Fig. 6(a). The analog implementation complicates the start-up of the power transfer because, when iAB is zero,

the comparators will not work. Therefore, to start the power transfer, the circuit needs to excite the flow of IAB.

The designed start-up strategy uses a timed oscillator, as shown in Fig. 6(b). To start the wireless charging process, the timed oscillator is manually enabled by connecting the oscillator’s ground pin (GND), that is initially floating, to the ground reference voltage. After this, the generation of the starting short pulses p1,start and p2,start begins which

substitute the nominal short pulses p1and p1during the

start-up. The frequency of this initial pulses is higher than the nominal to ensure that the operation is in the inductive region. As a result, during the start-up, the control signals to the gate driver Q and Q have higher frequency than the nominal. The current iAB starts building up in the inverter and, once its

amplitude is high enough, the comparators starts generating the output voltages V+ and V−. As soon as the comparators

start working, the oscillator is automatically disabled through its RESET pin, which voltage is changed from high to low (from 5 V to 0 V). At this point, p1,start and p2,start stop,

and p1 and p2start driving the outputs Q and Q. This start-up

strategy is summarized in the flow chart of Fig. 7.

The start-up strategy has been validated together with the auto-resonant control explained in Section III by simulating the circuit in Fig. 6. The resulting waveforms are shown in Fig. 8. The simulation has been executed with the main coils used in [6] connected to a S-S compensation network tuned at 85 kHz. Additionally, referring to Fig. 1, the other parameters selected are: Vin= 48 V, tdead= 140 ns, and RL= 10 Ω.

Fig. 8(a) shows that initially the RESET pin of the oscillator is set to high which enables the starting short pulses p1,start

and p2,startin Fig. 8(b). This leads to the start of the control

signals to the gate driver Q, Q in Fig. 8(c). Fig. 8(d) shows that the first current oscillation is detected by that the positive slope comparator because the signal V+ changes from high

to low. Therefore, the oscillator is automatically disabled by setting its RESET pin to low. However, the comparators cannot detect the second current oscillation because its amplitude is

0 5 0 20 40 60 80 100 120 140 V (V) t (μs) RESET (a) 0 20 40 60 80 100 120 140 V-(V ) V + (V) t (μs) V+ V-(d) 0 5 0 5 0 20 40 60 80 100 120 140 Q (V) Q (V) t (μs) Q Q (c) 0 5 0 5 0 20 40 60 80 100 120 140 p (V) p ,start (V) t (μs) p1,start p2,start p1 p2 (b) 0 5 0 5 -30 -15 0 15 30 -70 -35 0 35 70 0 20 40 60 80 100 120 140 I (A) V (V) t (μs) vAB iAB 0 20 40 60 80 100 120 140 V (V) t (μs) Vref+ V ref-(f) 0 2.5 -2.5 (e)

Fig. 8. Simulated waveforms of the start-up strategy described in Section IV. (a) Voltage to the RESET pin of the timed oscillator. (b) Starting short pulses p1,start, p2,start, and nominal short pulses p1, p2. (c) Control signals to

the gate driver Q, Q. (d) Output voltages V+, V−from the comparators. (e)

Output voltage vAB and current iAB of the H-bridge inverter. (f) Voltage

references Vref +, Vref −to perform the detection of iAB.

not high enough. As a consequence, the oscillator is enabled again and two starting short pulses p1,start and p2,start are

sent. After this, the nominal operation of the system starts. In Fig. 8(e), it is possible to notice that the ZVS turn-on is kept for the whole start-up transient even in the presence of current peaks higher than the 9 A steady-state peak current.

This is realized by the auto-resonant control that adjusts the detection references Vref +, Vref −, as shown in Fig. 8(f).

V. CONCLUSION

This paper explains the concept of an auto-resonant control for the H-bridge inverter typically used in inductive power transfer applications. This concept consists of adapting the reference levels of the current’s zero-crossing detection de-pending on the slope of the current at the zero-crossing. In this way, the operation is always adjusted to be close to the actual resonant frequency of the circuit, and the ZVS turn-on of the inverter is ensured at different operating cturn-ondititurn-ons. By using the small-angle approximation, it has been proven mathematically that the reference levels for the zero-crossing detection depend on the derivative of the current around the zero-crossing. This justifies why the slope of the current is considered instead of its amplitude for the reference levels’ computation. In particular, it has been decided to treat the pos-itive and the negative-slope zero-crossing separately because of possible asymmetries in the current waveform and in delay times. The concept of auto-resonant control is explained with functional blocks and with qualitative operating waveforms. Finally, the strategy of the power transfer’s start-up is also addressed. It has been proven with simulations that it is possible to use a timed oscillator to generate high-frequency pulses to excite a current flow in the inverter that makes the comparators work. As soon as the switching output from the comparators is sensed, the oscillator is disabled such that the power transfer can continue with its nominal operation.

REFERENCES

[1] J2954 RP: Wireless Power Transfer for Light-Duty Plug-In/ Electric Vehicles and Alignment Methodology, SAE International Std., Apr. 2019. [2] R. Steigerwald, “A comparison of half-bridge resonant converter

topolo-gies,” IEEE Transactions on Power Electronics, 1988.

[3] B. Lu, W. Liu, Y. Liang, F. Lee, and J. van Wyk, “Optimal design methodology for llc resonant converter,” in Twenty-First Annual IEEE Applied Power Electronics Conference and Exposition (APEC), 2006. [4] S. Li, W. Li, J. Deng, T. D. Nguyen, and C. C. Mi, “A double-sided

lcc compensation network and its tuning method for wireless power transfer,” IEEE Transactions on Vehicular Technology, 2015.

[5] T. Kan, T.-D. Nguyen, J. C. White, R. K. Malhan, and C. C. Mi, “A new integration method for an electric vehicle wireless charging system using lcc compensation topology: Analysis and design,” IEEE Transactions on Power Electronics, 2017.

[6] F. Grazian, P. van Duijsen, T. B. Soeiro, and P. Bauer, “Advantages and tuning of zero voltage switching in a wireless power transfer system,” in IEEE PELS Workshop on Emerging Technologies: Wireless Power (WoW), 2019.

[7] F. Grazian, W. Shi, J. Dong, P. van Duijsen, T. B. Soeiro, and P. Bauer, “Survey on standards and regulations for wireless charging of electric vehicles,” in AEIT International Conference of Electrical and Electronic Technologies for Automotive (AEIT AUTOMOTIVE), 2019.

[8] H. L. Li, A. P. Hu, and G. A. Covic, “Fpga controlled high frequency resonant converter for contactless power transfer,” in IEEE Power Electronics Specialists Conference, 2008.

[9] H. L. Li, A. P. Hu, G. A. Covic, and C. Tang, “A new primary power regulation method for contactless power transfer,” in IEEE International Conference on Industrial Technology, 2009.

[10] W. Fu, B. Zhang, and D. Qiu, “Study on frequency-tracking wireless power transfer system by resonant coupling,” in IEEE 6th International Power Electronics and Motion Control Conference, 2009.

[11] A. J. Moradewicz and M. P. Kazmierkowski, “Contactless energy trans-fer system with fpga-controlled resonant converter,” IEEE Transactions on Industrial Electronics, 2010.