Wide-range power-regulation method for wireless power receiving units by using hybrid multi-level topologies

Abstract

A power-regulated Power Receiving Unit (PRU) of an RWPT system, comprising an ML post-regulation stage via which a load is connected to the PRU; a controller circuit, being adapted to: determine target/predicted values for voltage and current of the a Power Transmit Unit (PTU) of the RWPT system; determine the wireless medium characteristics and resonant frequency of the RWPT system; generate an overall system model by using First Harmonic Approximation (FHA); determine a desired output power; calculate the voltage V.sub.S1 of the first harmonic; use V.sub.S1 to calculate the equivalent reflected impedance Z.sub.o of the load; and calculate the duty-cycle d using the predicted values of the efficiency n the conversion ratio M(D) and the calculated equivalent reflected impedance Z.sub.o.

Claims

1. A power-regulation method for a Power Receiving Unit (PRU) of an RWPT system, comprising: a) connecting a load to said PRU via an ML post-regulation stage; b) determining target/predicted values for voltage and current of a Power Transmitting Unit (PTU) of said RWPT system; c) determining wireless medium characteristics and a resonant frequency of said RWPT system; d) generating an overall system model by using First Harmonic Approximation (FHA); e) determining a desired output power; f) calculating a voltage V.sub.S1 of a first harmonic; g) using V.sub.S1 to calculate an equivalent reflected impedance Z.sub.0 of said load; and h) calculating a duty-cycle d using predicted values of an efficiency of a conversion ratio M(D) and said calculated equivalent reflected impedance Z.sub.0.

2. A method according to claim 1, wherein the ML post-regulation stage is integrated into the PRU or performs step-up, step-down conversion ratios, or both, or is implemented in a Hybrid Multi-Level (HML) topology with a high conversion ratio.

3. A method according to claim 1, further comprising one or more of the following: using HML post-regulation for performing wide impedance matching for the RWPT system; working at optimal operating conditions.

4. A method according to claim 2, wherein the HML post-regulator is cascaded with an AC-DC rectifier stage.

5. A method according to claim 1, wherein the ML post-regulation stage is designed according to specific target wireless operating conditions.

6. A method according to claim 1, wherein the PTU comprises: a) a DC-DC pre-regulation stage; b) a DC-AC power inverter; c) a transmitter resonator, resonating at the same frequency as a resonator of the PRU; d) a power transmitting element; and e) a controller for compensating misalignments between said PTU and said PRU.

7. A method according to claim 1, wherein the PRU comprises: a) a pick-up element; b) a receiver resonator, resonating at the same frequency as a resonator of the PTU; c) an AC-DC rectifier; d) a DC-DC hybrid post-regulation stage; and e) a controller for compensating misalignments between said PTU and said PRU.

8. A method according to claim 1, wherein the power regulation is made for several PRUs simultaneously, wherein each PRU comprises a corresponding HML post-regulator.

9. A method according to claim 2, wherein the PRU comprises a high-conversion HML Buck post-regulation stage.

10. A method according to claim 9, wherein the Buck post-regulation stage comprises: a) four power switches; b) a flying capacitor; c) an output inductor; d) an output capacitor; e) a sensing circuitry for a controller.

11. A power-regulated Power Receiving Unit (PRU) of an RWPT system, comprising: a) an ML post-regulation stage via which a load is connected to said PRU; b) a controller circuit, being adapted to: c) determine target/predicted values for voltage and current of a Power Transmit Unit (PTU) of the RWPT system; d) determine wireless medium characteristics and a resonant frequency of the RWPT system; e) generate an overall system model by using First Harmonic Approximation (FHA); f) determine a desired output power; g) calculate a voltage V.sub.S1 of a first harmonic; h) use V.sub.S1 to calculate an equivalent reflected impedance Z.sub.0 of the load; and i) calculate a duty-cycle d using predicted values of an efficiency of a conversion ratio M(D) and the calculated equivalent reflected impedance Z.sub.0.

12. A power-regulated PRU according to claim 11, in which the controller is adapted to examine duty-cycle ranges and post-regulator performance for corner operating points of the RWPT system.

13. A power-regulated PRU according to claim 11, in which the ML post-regulation stage is integrated into the PRU or performs step-up, step-down conversion ratios, or both, or is implemented in a Hybrid Multi-Level (HML) topology with a high conversion ratio.

14. A power-regulated PRU according to claim 11, in which HML post-regulation is used for performing wide impedance matching for the RWPT system or cascaded with an AC-DC rectifier stage.

15. A power-regulated PRU according to claim 11, in which the ML post-regulation stage is designed according to specific target wireless operating conditions.

16. A power-regulated PRU according to claim 11, in which the PTU comprises: a) a DC-DC pre-regulation stage; b) a DC-AC power inverter; c) a transmitter resonator, resonating at the same frequency as a resonator of the PRU; d) a power transmitting element; and e) a controller for compensating misalignments between the PTU and the PRU.

17. A power-regulated PRU according to claim 11, in which the PRU comprises: a) a pick-up element; b) a receiver resonator, resonating at the same frequency as a resonator of the PTU; c) an AC-DC rectifier; d) a DC-DC hybrid post-regulation stage; and e) a controller for compensating misalignments between the PTU and the PRU.

18. A power-regulated PRU according to claim 11, in which the power regulation is made for several PRUs simultaneously, wherein each PRU comprises a corresponding HML post-regulator.

19. A power-regulated PRU according to claim 11, in which the PRU comprises a high-conversion HML Buck post-regulation stage.

20. A power-regulated PRU according to claim 19, in which the Buck post-regulation stage comprises: a) four power switches; b) a flying capacitor; c) an output inductor; d) an output capacitor; e) a sensing circuitry for a controller.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The above and other characteristics and advantages of the invention will be better understood through the following illustrative and non-limitative detailed description of preferred embodiments thereof, with reference to the appended drawings, wherein:

(2) FIG. 1 shows a block diagram block diagram of a controlled RWPT system with a hybrid multi-level post-regulator at the PRU, according to an embodiment of the invention;

(3) FIG. 2a (prior art) illustrates a typical capacitively-coupled RWPT system;

(4) FIG. 2b illustrates a capacitive medium. modeled as a x-type network:

(5) FIG. 2c shows typical waveforms of the system of the RWPT system;

(6) FIG. 2d illustrates modelling the overall RWPT system:

(7) FIG. 3a (prior art) shows an inductive-based RWPT system with double-sided LC networks;

(8) FIG. 3b (prior art) shows Inductive wireless medium model;

(9) FIG. 3c (prior art) shows typical currents and voltages waveforms of the inductive RWPT system at the PRU;

(10) FIG. 3d (prior art) shows an overall system model by employing first harmonic approximation;

(11) FIG. 4 shows a simplified schematic diagram of the PRU output stages, according to an embodiment of the invention;

(12) FIG. 5 illustrates a capacitively-coupled double-sided LC RWPT system with a wide-input step-down HML buck post-regulation stage;

(13) FIG. 6a-6c illustrate typical waveforms of the multi-level buck converter at steady-state;

(14) FIG. 7 illustrates a Behavioral model of the PRU while the transmitter is current-controlled;

(15) FIG. 8 illustrates a simulation test-bench for the analyzed capacitive RWPT system of the present invention, which has been constructed in PSIM platform;

(16) FIGS. 9a and 9b show the simulation results of the currents and voltages of the transmitter and receiver, respectively;

(17) FIG. 10a shows simulation test-bench in PSIM platform for the behavioral model of the PRU side;

(18) FIG. 10b shows simulation waveforms of the voltage and current;

(19) FIG. 11a-11c show modifications in the behavioral model test-bench;

(20) FIG. 12a-12b show the results for compensated and uncompensated systems;

(21) Waveforms of the PRU behavioral model for step-up medium variation: (a) with compensation, (b) without compensation;

(22) FIG. 13a-13b show waveforms of the PRU behavioral model for step-up medium variation with and without compensation, respectively;

(23) FIG. 14 shows conversion ratio ranges of different post-regulators for the RWPT system;

(24) FIG. 15a shows efficiency measurements of the HML buck as a function of the load current at input voltage of 88 V;

(25) FIG. 15b shows loss breakdown of the HML buck as a function of the input voltage;

(26) FIGS. 16a-16b show experimental results of the system at 75 mm (CM14 pF) misalignment;

(27) FIGS. 17a-17d show experimental measurements of the capacitive RWPT system for step-up medium variation, 14 pF to 20 pF; and

(28) FIGS. 18a-18d show Experimental measurements of the capacitive RWPT system for step-down medium variation, 14 pF to 8 pF.

DETAILED DESCRIPTION OF THE INVENTION

(29) The present invention provides a power regulation method for Power Receiving Units (PRUs) in Resonant Wireless Power Transfer (RWPT) systems by using hybrid multi-level (HML) post-regulators. A behavioral model for the PRU including the additional regulation stage has been developed. By using HML stages for power-regulation, significant wider impedance matching range has been achieved, which results in better spatial freedom of wireless power systems. The method of the present invention also provides design tradeoffs between several post-regulators in terms of efficiency, coupling range and conversion range.

(30) FIG. 1 shows a block diagram block diagram of a controlled RWPT system with a hybrid multi-level post-regulator at the PRU, according to an embodiment of the invention.

(31) The basic topology is an RWPT system with double-sided LC resonant converter configuration 0, 0. The goal is to ensure that the voltage (Vload) across the load will remain constant even though the distance between the receiver and the transmitter varies due to movements of the user. An ML post-regulation stage or an HML post regulation module that consists of a multistage DC-DC converter is used to regulate the voltage Vload and to provide wide impedance matching capability, in order to maintain high efficiency and low losses for any desired power level, under all conditions. The ML post-regulation stage may be integrated into the PRU. The HML post-regulator may be cascaded with an AC-DC rectifier stage.

(32) FIG. 2a (prior art) illustrates a typical capacitively-coupled RWPT system. The wireless medium consists of by two pairs of conductive coupling plates, such as copper foils or aluminum 0, 0, 0. The capacitive medium can be modeled as -type network 0, 0 as shown FIG. 2b (prior art). Assuming loosely-coupled operation, i.e., the coupling capacitances C.sub.M, C.sub.M1, C.sub.M2, are relatively low compared to C.sub.P and C.sub.S 0, 0, 0, the drive frequency, .sub.INV, is near the resonators natural frequency (i.e., .sub.INV.sub.0=1/(2{square root over (L.sub.PC.sub.P)}), =1/(2{square root over (L.sub.SC.sub.S)}), then the currents as well as voltages of the passive components are virtually sinusoidal 0. This is since high-Q operation is naturally facilitated as the effective output impedance of the resonator in the transmitter is relatively high.

(33) FIG. 2c (prior art) shows typical waveforms of the system. While the transmitter and receiver voltages V.sub.P and V.sub.S are square waves, the currents are sinusoidal due to high-Q operation of the circuit. Since a full-bridge inverter is used at the front-end, the transmitter voltage V.sub.P toggles between V.sub.in, while the receiver voltage, V.sub.S, toggles between V.sub.out. It can be also seen, that for both the transmitter and receiver sides the current is in phase with the voltage, whereas the receiver current I.sub.S lags the transmitter current I.sub.P by 90 (same applies for the voltages V.sub.S and V.sub.P). By employing First Harmonic Approximation (FHA), and by utilizing the capacitive medium model (as shown in FIG. 2b), the overall system can be modeled, as shown in FIG. 2d (prior art). VP1 and VS1 represent the first harmonics of the square waves V.sub.P and V.sub.S, where the magnitudes of the first harmonic are 4V.sub.in/ and 4V.sub.out/, respectively; Z.sub.0 is the reflected impedance with respect to the load such that Z.sub.o=8/.sup.2 R.sub.Load. The above outcomes are obtained under the assumption that the transmitter and receiver are resonating at the same frequency.

(34) FIG. 3a (prior art) shows an inductive-based RWPT system with double-sided LC networks. The wireless inductive medium can be modeled as T-type network 0, as shown in FIG. 3b (prior art). Assuming that the inductive system is also loosely-coupled, and that both the transmitting and receiving sides are resonating at the same frequency, and by employing FHA on the inductive RWPT system, the behavior of the main currents and voltages of the inductive system can be extracted(in a similar manner to the capacitive-based system analysis).

(35) FIG. 3c (prior art) shows the currents and voltages of the receiver before and after the rectifier stage, V.sub.S1 is the voltage of the first harmonic and where I.sub.S1 is the current of the first harmonic. Assuming that I.sub.out is filtered by an output capacitor, the average output current and voltage are

(36) $\begin{array}{cc}\{\begin{array}{c}{I}_{\mathrm{out},\mathrm{AVG}}=\frac{2}{}{I}_{S1,\mathrm{PK}}\\ V_{\mathrm{out},\mathrm{AVG}}=\frac{}{4}{V}_{S1,\mathrm{PK}}\end{array}.& \left(1\right)\end{array}$

(37) Similarly to the capacitive system, by employing FHA the overall inductive RWPT system can be modeled, as shown in FIG. 3d (prior art).

(38) The above analytical relationships and waveforms behavior shown in FIG. 2c and FIG. 3c, are valid for both covered RWPT systems.

(39) Equivalent Impedance Reflection

(40) FIG. 4 shows a simplified schematic diagram of the PRU output stages, according to an embodiment of the invention. Assuming steady-state conditions, the post-regulation can be considered as a three-port stage that comprises input power, output power and a control signal, such that the output voltage can be represented in a generic form as follows 0:

(41) $\begin{array}{cc}{V}_{Load}=M\left(D\right).Math.V_{out},& \left(2\right)\end{array}$
where V.sub.out is the voltage after the rectifier stage, and M(D) is the conversion ratio of the post-regulator at steady-state. M(D) is a function of the duty-cycle d(t). By taking into consideration the efficiency factor in such regulators, it can be assumed that the generic form of the output current is

(42) $\begin{array}{cc}{I}_{Load}\frac{I_{\mathrm{out}}}{M\left(D\right)}& \left(3\right)\end{array}$
where I.sub.out is the current after the rectifier stage and is the efficiency of the regulator. By substituting (2) into (3), the equivalent input resistance of the post-regulator can be expressed as

(43) $\begin{array}{cc}{R}_{eq}=\frac{V_{out}}{I_{out}}=\frac{V_{\mathrm{Load}}}{I_{\mathrm{Load}}{M}^2\left(D\right)}=\frac{R_{\mathrm{Load}}}{M^2\left(D\right)},& \left(4\right)\end{array}$

(44) By using the FHA relationships discussed above, the reflected impendence seen from the input of the rectifier stage, Z.sub.o, can be expressed with respect to the load (R.sub.Load) and the conversion ratio M(D) as

(45) $\begin{array}{cc}{Z}_O=\frac{8}{{}^2}{R}_{eq}=\frac{8}{{}^2}\frac{R_{\mathrm{Load}}}{M^2\left(D\right)}.& \left(5\right)\end{array}$
Z.sub.o can be also expressed in a straightforward manner as

(46) $\begin{array}{cc}{Z}_O=\frac{V_{S1}^2}{2P_O}.Math.V_{S1}^2=2Z_O{P}_O& \left(6\right)\end{array}$
where P.sub.o=P.sub.Load/ is the equivalent power of Z.sub.o.

(47) From (5) and (6) it can be seen that for given target output power and finite efficiency of the post-regulation, a wider range of M(d) enables a wider range of Z.sub.o. A wider range of Z.sub.o entails better compensation for V.sub.S variations, which are primarily due to medium variations and the cross-coupling relationships between the transmitter and receiver, as will be discussed later on.

Example: Capacitive RWPT with Hybrid Multi-Level Buck Post-Regulator

(48) FIG. 5 illustrates a capacitively-coupled double-sided LC RWPT system with a wide-input step-down HML buck post-regulation stage (a step-down converter is a DC-to-DC power converter which steps down voltage from its input to its output). The HML buck converter has several distinguishing advantages over the conventional buck converter 0. First, assuming that the charge balance across the flying capacitor, C.sub.f is maintained, the voltage swing at the switching node is half of the input voltage, which in the context of the WTP system, is half of the rectified voltage of V.sub.S. The effective switching frequency (1/T.sub.SW,eff) at the switching node, V.sub.SW is twice the switching frequency f.sub.SW=1/T.sub.SW. These features entail better power density and less efforts on the overall heat dissipation of the converter. In addition, in a conventional buck converter, due to the blocking voltage Vas, requirement of the switches 0, 0, both of the switches need to be rated for maximum input voltage of V.sub.S. In the multi-level buck converter there are four switches which can be rated for V.sub.S/2. Since silicon area for switch realization is approximately proportional to V.sub.ds.sup.2, the overall semiconductor areas of multi-level and conventional buck converters can be equivalent 0, 0. Typically, the volume of the output filter is significantly larger than that of the semiconductor components. In the case of the HML buck converter, due to the combined reduction of the voltage swing at the switching node, and two times higher effective switching frequency, the output filter is further reduced. Thus, the additional switches virtually do not introduce volumetric penalty 0. The multi-level buck converter does not require any additional peripherals, and the core control is the same as for the conventional buck converter, with potentially improved dynamics.

(49) FIGS. 6a-6c illustrate typical waveforms of the multi-level buck converter at steady-state. The operation of the buck converter is divided into four switching phases, in which, Q.sub.1 and Q.sub.4 are complementary, Q.sub.2 and Q.sub.3 are also complementary, and Q.sub.2 is delayed from Q.sub.1 in T.sub.SW/2. It can be seen that for a case that V.sub.Load<VS/2 (FIG. 6a), the switching node toggles between 0 to V.sub.S/2; for V.sub.Load=V.sub.S/2 (FIG. 6b), and neglecting the flying capacitor charge balance mismatches, the switching node is virtually constant and equals to V.sub.S/2=V.sub.Load, and the inductor current is also virtually constant I.sub.L=I.sub.Load; for V.sub.S/2<V.sub.Load<V.sub.S FIG. 6c) the switching node toggles between V.sub.S/2 to V.sub.S. It can be observed that for the entire operation region, the maximum voltage drop over the inductor (and the switches) is V.sub.S/2.

(50) The conversion ratio function M(D) of the HML buck converter is identical to the conventional buck converter and is equals to d(t). Thus, by using (5), the equivalent reflected impudence of the multi-level buck, Z.sub.O,ML-Buck, is

(51) $\begin{array}{cc}{Z}_{O,\mathrm{ML}-\mathrm{Buck}}=\frac{8}{{}^2}\frac{R_{\mathrm{Load}}}{d^2}.& \left(7\right)\end{array}$

(52) From (7), the duty ratio d can be calculated as

(53) $\begin{array}{cc}d=\sqrt{\frac{8}{{}^2}\frac{R_{\mathrm{Load}}}{Z_{O,\mathrm{ML}-\mathrm{Buck}}}}.& \left(8\right)\end{array}$

(54) One of the methods to control transmitters in RWPT systems, and in particular in multi-receiver RWPT systems, is by constant current regulation 0, 0, 0. By employing such a control approach on the analyzed capacitive system, the voltage V.sub.CP can be treated as constant. Therefore, the circuit in FIG. 5 can be simplified to the circuit shown in FIG. 7. By applying Kirchhoff's voltage law on the circuit of FIG. 7, the voltage V.sub.CS can be expressed as

(55) $\begin{array}{cc}{V}_{\mathrm{CS}}=V_{CP}\frac{Z_S}{Z_S+Z_{CM}}& \left(9\right)\end{array}$$\{\begin{array}{c}{Z}_S=Z_{\mathrm{CS}}.Math.\left(Z_{\mathrm{LS}}+Z_{O,\mathrm{ML}-\mathrm{Buck}}\right)\\ Z_{\mathrm{CM}}=1/j{C}_M\end{array}.$

(56) The voltage V.sub.S1 can be expressed as

(57) 0$\begin{array}{cc}{V}_{S1}=V_{\mathrm{CS}}\frac{Z_{O,\mathrm{ML}-\mathrm{Buck}}}{Z_{O,\mathrm{ML}-\mathrm{Buck}}+j{L}_S}.& \left(10\right)\end{array}$

(58) By substituting (9) into (10), and rearranging the expression, V.sub.S1 is expressed as

(59) $\begin{array}{cc}{V}_{S1}=V_{\mathrm{CP}}=\frac{Z_{O,\mathrm{ML}-\mathrm{Buck}}}{Z_{O,\mathrm{ML}-\mathrm{Buck}}+j{L}_S+\frac{1-{}^2{L}_S{C}_S+j{C}_S{Z}_{O,\mathrm{ML}-\mathrm{Buck}}}{j{C}_M}},& \left(11\right)\end{array}$
and if operation in resonance is assumed, V.sub.S1 can be simplified to

(60) $\begin{array}{cc}{V}_{S1}=V_{\mathrm{CP}}\frac{Z_{O,\mathrm{ML}-\mathrm{Buck}}}{Z_{O,\mathrm{ML}-\mathrm{Buck}}\left(1+C_S/C_M\right)+j{L}_S}.& \left(12\right)\end{array}$

(61) By substituting the expression in (6) into (12), and after some manipulations, V.sub.S1 can be further expressed as

(62) $\begin{array}{cc}{V}_{S1}=\frac{\frac{V_{\mathrm{CP}}{C}_M}{C_M+C_S}+\sqrt{{\left(\frac{V_{\mathrm{CP}}{C}_M}{C_M+C_S}\right)}^2-j4L_S}}{2}.& \left(13\right)\end{array}$

(63) RWPT systems operate in the MHz range. Therefore, the resonators' typical capacitance values are in the range of hundreds of pF, the inductances are tens to hundreds H, and the mutual coupling capacitance, C.sub.M, is in the order of a few pF. As a result, the expression in (13) can be simplified to generic expressions as follows

(64) $\begin{array}{cc}\frac{V_{\mathrm{CP}}{C}_M}{C_M+C_S}4L_S.Math.V_{S1}=\frac{V_{\mathrm{CP}}{C}_M}{C_M+C_S}.& \left(14\right)\end{array}$

(65) Having the relationships given in (6), (8) and (14), the duty ratio can be calculated for any operating point as a function of the operating conditions of the transmitter, medium variations (translates to changes in C.sub.M), and the PRU resonator. The system may be adapted to work at optimal operating conditions, at which maximum efficiency is obtained.

(66) Using the above analysis, generalized design guidelines to examine and design an end-to-end RWPT system with post-regulation stage is performed as follows: Given target/predicted values for the power transmitting unit (PTU), which may be modeled as a current-controlled module. 1) transmitter's voltages and currents, the wireless medium characteristics medium characteristics, resonators values and resonant frequency. 2) Assume operation in resonance and extract the overall system model by employing FHA. 3) Given the target output power, substitute (6) into (12), and calculate V.sub.S1. 4) Use V.sub.S1 to calculate Z.sub.o from (6). 5) Given nominal operating point, based on the chosen post-regulator, insert the predicted efficiency , M(d), and Z.sub.o to (5), and calculate the duty-cycle d. 6) Examine the duty-cycle ranges and post-regulator performance for corner operating points of the RWPT system (which determine the desired voltage-current points at which the RWPT system should operate).

(67) In one embodiment, the PTU comprises a DC-DC pre-regulation stage, a DC-AC power inverter, a transmitter resonator for resonating at the same frequency as the resonator of the PRU a power transmitting element and a controller for compensating misalignments between said PTU and the PRU.

(68) In one embodiment, the PRU comprises a pick-up element; a receiver resonator, resonating at the same frequency as the resonator of the PTU; an AC-DC rectifier; a DC-DC hybrid post-regulation stage; and a controller for compensating misalignments between said PTU and the PRU.

(69) In one embodiment, the power regulation is made for several PRUs simultaneously, wherein each PRU comprises a corresponding HML post-regulator.

(70) The PRU comprises a high-conversion HML buck post-regulation stage.

(71) In one embodiment, the buck post-regulation stage comprises four power switches; a flying capacitor; an output inductor; an output capacitor; a sensing circuitry for a controller.

(72) FIG. 8 illustrates a simulation test-bench for the analyzed capacitive RWPT system of the present invention, which has been constructed in PSIM platform (of PowerSim, Inc.PSIM is an electronic circuit simulation software package, designed specifically for use in power electronics and motor drive simulations). The input voltage was 25 V, the medium mutual capacitance C.sub.M=14 pF, at resonant frequency 0=3 MHz. The resonators for the transmitter and receiver have been chosen to be L.sub.P=20.8 H, C.sub.P=120 pF and L.sub.S=21.65 H, C.sub.S=130 pF. The output stage comprises load resistance R.sub.Load=10, output capacitor C.sub.Load=90 F, output inductor L=10 H, and flying capacitor, C.sub.fly=10 F. The switching frequency of the post-regulator, f.sub.SW, was 250 kHz, and the target output power was 40 W, with a constant load voltage V.sub.Load=20 V. The overall parameters and values of the test-bench are summarized in Table I.

(73) TABLE-US-00001 TABLE I SIMULATION TEST-BENCH VALUES AND PARAMETERS AT NOMINAL OPERATION Parameter Value/Type Input voltage V.sub.in 25 V Transmitter resonator 20.8 H, 120 pF Receiver resonator 21.6 H, 130 pF Resonant frequency f.sub.0 3 MHz Coupling capacitance C.sub.M 14 pF Load resistance R.sub.Load 10 Output capacitor C.sub.Load 90 F Output inductor L 10 H Flying capacitor C.sub.fly 10 F Post-regulator switching 250 kHz (500 kHz frequency f.sub.SW effective) Load voltage V.sub.Load 20 V Load power P.sub.Load 40 W

(74) FIGS. 9a and 9b show the simulation results of the currents and voltages of the transmitter and receiver, respectively. It can be seen from FIG. 9a that the voltage V.sub.P toggles between 25 to 25 V. The sinusoidal voltage on the medium input, V.sub.CP, peaks at 1.2 kV, where the phase difference between V.sub.P and V.sub.CP is 90 as expected.

(75) FIG. 9b shows the receiver waveforms with respect to V.sub.CP. It can be seen that the sinusoidal voltage on the medium output, V.sub.CS, lags V.sub.CP by 90. In addition, the voltage V.sub.S toggles between 88 to 88 V, which implies that the rectified voltage, i.e., the input voltage of the multi-level buck is 88 V. Moreover, given the target load conditions, the duty-cycle of the post-regulator is d=0.225, i.e., V.sub.Load=20 V, which is in excellent agreement with the theoretical predictions in (6), (8) and (14). Assuming 100% efficiency for the simulation test-bench, and by using expressions (1) and (3), the load current is found to be I.sub.Load=2 A, which is also in good agreement with the target load specifications.

(76) FIG. 10a shows a simulation test-bench, where, V.sub.CP is assumed constant (1.2 kV peak), C.sub.M, L.sub.S and C.sub.S have been set according to Table I, and based on the expression in (7), the equivalent reflected impedance is Z.sub.o=160.3.

(77) FIG. 10b shows simulation results for the PRU first harmonics of the voltage and current. It can be seen that the current, I.sub.S1, toggles between 0.75 to 0.75 A, which is in excellent agreement with the theoretical predictions and the end-to-end simulation results FIG. 10b). The first harmonic voltage V.sub.S1 peaks at 112 V as expected from the results in FIG. 10b. Considering the rms values obtained by the behavioral model, it can be observed that the equivalent output power is 40 W as targeted.

(78) FIGS. 11a-11c show modifications in the behavioral model test-bench. The coupling capacitance C.sub.M has been replaced by a continues variable capacitor model FIG. 11b), and the reflected impedance, Z.sub.o, has been replaced continues variable resistor model (FIG. 11c). The methodology to model continuous-time elements has been employed and adopted based on 0. By doing so, the behavioral model of the PRU is further analyzed for on-the-fly variations in C.sub.M, while adjusting Z.sub.o accordingly.

(79) The first medium variation has been carried out by changing the initial coupling capacitance from 14 pF to 20 pF (i.e. better coupling), this implies that Z.sub.o has been adjusted from 160.3 to 308, or in terms of the duty-cycle of the post-regulator, from d=0.225 to d=0.16.

(80) FIG. 12a-12b show the results for compensated and uncompensated systems. It can be seen that after the variation, for the compensated system FIG. 12a), the current drops from 0.75 A to 0.5 A, and the voltage increases to 157 V, while the overall output power maintained at 40 W. On the other hand, for the uncompensated system (FIG. 12b), it can be seen that both the current and voltage increase to 0.93A and 150 V, respectively. More importantly, the output power increases to 70 W, meaning that the overall system specifications are not obtained.

(81) C.sub.M has been varied from 14 pF to 8 pF (i.e. worse coupling), this translates to adjusting Z.sub.o to 49 and d=0.4. 0a shows that after the transition due to medium variation, the tuned system sustains 40 W operation, where the current and voltage settles on 1.2 A and 61 V, respectively. Ob shows that for the given medium variation, the output power of the non-tuned system significantly drops to 15 W.

(82) To further highlight the advantages of the HML buck converter as a post-regulation stage, given the above operating conditions, a more thorough comparison between buck and multi-level buck over wide range of the coupling capacitance has been carried out, as shown in FIG. 14. Loss estimation has been done for both topologies for various duty-cycles. It can be seen that for the given RWPT system, at extremely weak coupling (C.sub.M<6 pF), in terms of efficiency, a conventional buck converter has reasonable efficiency over 91%. However, as the operating points move toward the strongly-coupled region (C.sub.M>14 pF), where high-conversion ratios are required, the efficiency of the buck converter drops below 80%, whereas the HML buck converter remains above 80% efficiency up to C.sub.M40 pF.

(83) It should be noted that the blocking voltage of the chosen switches for both converters is V.sub.ds=100 V. Therefore, in practice, for the given RWPT system regardless the efficiency criteria, the conversion range of the buck converter is limited to d=0.2, while the HML buck can be potentially pushed to 0.1 duty-cycle. It should be further emphasized that by compromising on the operation range such that 0.35d0.9, the switches of the buck converter can be optimized with lower blocking voltage V.sub.ds=60 V, resulting in better efficiencies (similar to the HML buck) as illustrated by the dashed line in FIG. 14.

(84) FIG. 14 also depicts the conversion range of a Series-Capacitor (SC) buck converter 0, to exemplify more design considerations of post-regulators in WPT. For the sake of simplicity, the parameters of the SC buck are same as for the previous converters, and by using the expression in (5), the duty ratio for the SC buck can be expressed as follows

(85) $\begin{array}{cc}{d}_{\mathrm{SC}-\mathrm{Buck}}=\sqrt{\frac{32}{{}^2}\frac{R_{\mathrm{Load}}}{Z_O}}.& \left(15\right)\end{array}$

(86) Although SC buck converter is limited to maximum 50% duty-cycle, it can be seen that if the system is operated at C.sub.M>15 pF while maintaining significantly wider operating range, SC buck converter would be a better candidate to perform such post-regulation task.

(87) The first set of the experimental validation has been carried out by characterizing the operation range and the power conversion efficiency of the post-regulation stage. FIG. 15a shows efficiency measurements of the HML buck at nominal conditions (i.e., input voltage is 88 V and load voltage is 20 V) as a function of the load current. It can be seen that the efficiencies are above 96% over wide range of operating points. FIG. 15b shows losses breakdown of the post-regulator at various input voltages. It can be seen that as the input voltage increases, the switching losses become dominant, which implies that for the case of conventional buck converter, the switching losses will be higher, thereby further limiting the medium misalignment range.

(88) FIG. 16a shows waveforms of the resonant WPT system has been examined for 40 W load output power, at 75 mm misalignment, which translates to 14 pF, such that the transmitter's resonant current, I.sub.P, is regulated to 2.5 A peak, and the voltage V.sub.P toggles between 35 and 35 V. The rectifier voltage at the receiver side, V.sub.S, toggles between 90 and 90 V, while the current, Is, peaks at 1 A. Although there is slight imperfection in the rectifier behavior due to parasitic capacitances, the experimental results are in good agreement with the theoretical predictions in Section V.

(89) FIG. 16b shows the main waveforms of the HML buck at steady-state operation. It can be observed that load voltage is 20 V, the effective switching frequency is 500 kHz, and the effective duty-cycle is 22.4% (half of the duty ratio of the switching node).

(90) FIG. 17a-17d show experimental measurements for step-up in the medium, i.e., from 75 mm misalignment to fully aligned couple, which translates to coupling variation of 14 pF to 20 pF (a step-up converter is a DC-to-DC power converter which steps up voltage from its input to its output). FIG. 17a depicts the PRU's main waveforms without post-regulator. It can be seen that after variation of the coupler both the V.sub.S and Is increase to peak values of 115 V and 1.5 A, respectively. This results in load output power of 85 W, approximately twice than the target load power. On the other hand, FIG. 17b shows the PRU's main waveforms with post-regulations stage. Although V.sub.S has not change much, it can be seen the Is drops by 20%, and the load power is well regulated to 40 W. FIG. 17c shows a zoom-in view on the transmitter and receiver main waveforms at steady-state with the post-regulator, and FIG. 17d shows the HML post-regulator main waveforms, where the effective duty-cycle is 16.66%.

(91) FIGS. 18a-18d show Experimental measurements of the capacitive RWPT system for step-down medium variation, 14 pF to 8 pF. The system has been also examined for step-down variation from 75 mm to 150 mm (14 pF to 8 pF). FIG. 18a shows that without post-regulation the load power drops by a factor of two to the range 20 W, while for the regulated system (FIG. 18b) the target load power is preserved. FIG. 18c depicts a zoom-in view on the transmitter and receiver main waveforms at steady-state with the post-regulator. It can be seen that the transmitter voltage, V.sub.P, slightly increases to 40 V peak, due to low coupling at such misalignment (which adds more losses to the end-to-end system), while the receiver's voltage, V.sub.S, drops to 55 V, i.e., 37% duty-cycle, which is in good agreement with the theoretical predictions. FIG. 18d shows post-regulator main waveforms.

(92) While some embodiments of the invention have been described by way of illustration, it will be apparent that the invention can be carried out with many modifications, variations and adaptations, and with the use of numerous equivalents or alternative solutions that are within the scope of persons skilled in the art, without exceeding the scope of the claims.

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