Wireless power system with mobius transform filter

Abstract

A wireless power transmitter includes an amplifier or other rectifier comprising a switching device and a shunt capacitor, a DC power source connected to the amplifier, a low-pass filter network connected to an output of the amplifier, and a load connected to an output of the low-pass filter network. The low-pass filter network is configured to perform a Möbius transformation on an output impedance of the amplifier, wherein the filter function and Möbius transformation takes an output impedance of the amplifier and transforms the output impedance to a straight line in an output impedance plane of the low-pass filter network. The low-pass filter network is configured to maintain a substantially constant output current or voltage to the load across a range of load conditions (e.g., independent of a number of receivers and so on).

Description

Attorney Docket: 100502-2100WIRELESS POWER SYSTEM WITH MOBIUS TRANSFORM FILTERCROSS-REFERENCE TO RELATED APPLICATION

[0001] The present application claims the benefit of and priority to U. S. Provisional Patent Application No. 63 / 749,305 entitled “WIRELESS POWER SYSTEM WITH MOBIUS TRANSFORM FILTER,” filed January 24, 2025, the contents of which being incorporated by reference in their entirety herein.TECHNICAL FIELD

[0002] The present disclosure relates generally to wireless power transfer systems and associated methods. More specifically, the disclosure pertains to power amplifiers, transmitters, rectifiers, receivers, and the like incorporating Mobius transform filters for efficient and robust wireless power transmission and reception.BACKGROUND

[0003] Wireless power transfer systems have gained significant attention in recent years as a convenient method for charging or directly powering electronic devices without the need for physical connectors. These systems typically utilize resonant inductive coupling between a transmitter coil and a receiver coil to transfer energy wirelessly. However, maintaining efficient power transfer across varying load conditions and coupling distances remains a challenge. Traditional wireless power systems often suffer from reduced efficiency and power output as the load or coupling conditions deviate from their optimal design parameters.BRIEF SUMMARY

[0004] According to an aspect of the present disclosure, a wireless power transmitter is provided. The wireless power transmitter includes an amplifier comprising a switching device. The wireless power transmitter includes a DC power source connected to the amplifier. The wireless power transmitter includes a low-pass filter network connected to an output of the amplifier, the low-pass filter network configured to perform a Mobius transformation on an output impedance of the amplifier, wherein the filter function and Mobius transformation takes an output impedance of the amplifier and transforms the output impedance to a straight line in an output impedance plane of the low-pass filter network. The wireless power transmitter includes a loadAttorney Docket: 100502-2100connected to an output of the low-pass filter network, wherein the low-pass filter network is configured to maintain a substantially constant output current or voltage to the load across a range of load conditions.

[0005] According to other aspects of the present disclosure, the wireless power transmitter may include one or more of the following features. The low-pass filter network may comprise a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation, and the load may comprise a transmitter coil. The low-pass filter network may comprise at least two inductors and at least two capacitors. The switching device may be a metal-oxide-semiconductor field-effect transistor (MOSFET) transistor. The wireless power transmitter may further comprise a gate drive circuit configured to provide a square wave signal to control the switching device. The wireless power transmitter may further comprise a feedback loop circuit configured to maintain zero voltage switching (ZVS) of the switching device. The amplifier may be a Class-E amplifier, and the wireless power transmitter may further comprise an auto-tuning circuit configured to adjust a variable reactance to maintain zero voltage switching across the range of load conditions.

[0006] According to another aspect of the present disclosure, a wireless power receiver is provided. The wireless power receiver includes a receiver resonator configured to receive wireless power from a transmitter. The wireless power receiver includes a transmitter coupled to the receiver resonator which induces either a radiofrequency (RF) voltage or an RF current in the receiver resonator. The wireless power receiver includes a rectifier connected to the receiver resonator, the rectifier being active and comprising a switching device. The wireless power receiver includes a low-pass filter network connected between the receiver resonator and the rectifier, the low-pass filter network configured to perform a Mobius transformation on an output impedance of the receiver resonator. The wireless power receiver includes a gate drive control circuit configured to generate a gate drive signal for the switching device of the rectifier, wherein the low-pass filter netw ork is configured to maintain a DC output voltage from the rectifier which remains substantially constant across a range of DC load conditions with the transmitter.

[0007] According to other aspects of the present disclosure, the wireless power receiver may include one or more of the following features. The low-pass filter network may comprise a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation. The low-pass filter network may comprise at least two inductors and at least two capacitors. The switching device may be a metal-oxide-semiconductor field-effect transistorAttorney Docket: 100502-2100(MOSFET) transistor. The gate drive control circuit may be configured to maintain zero voltage switching (ZVS) of the switching device. The rectifier may be a Class-E active rectifier, and the wireless power receiver may further comprise a DC-to-DC converter connected to an output of the Class-E active rectifier. The DC-to-DC converter may be configured to regulate an output voltage or current to a load.

[0008] According to another aspect of the present disclosure, a method of wireless power transmission is provided. The method includes generating, by an amplifier comprising a switching device, a radiofrequency (RF) signal from a DC power source. The method includes filtering the RF signal through a low-pass filter network configured to perform a Mobius transformation on an output impedance of the amplifier, wherein the filter function and Mobius transformation takes an output impedance of the amplifier and transforms the output impedance to a straight line in an output impedance plane of the low-pass filter network. The method includes delivering, to a load connected to an output of the low-pass filter network, a substantially constant output current or voltage across a range of load conditions.

[0009] According to other aspects of the present disclosure, the method may include one or more of the following features. The low-pass filter network may comprise a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation, and the load may comprise a transmitter coil. The method may further comprise transmitting, by the transmitter coil, a magnetic field based on a filtered RF signal output from the low-pass filter network. The low-pass filter network may comprise at least two inductors and at least two capacitors. The switching device may be a metal-oxide-semiconductor field-effect transistor (MOSFET) transistor. The method may further comprise providing, by a gate drive circuit, a square wave signal to control the switching device. The method may further comprise maintaining, by a feedback loop circuit, zero voltage switching (ZVS) of the switching device. The amplifier may be a Class-E amplifier. The method may further comprise adjusting, by an auto-tuning circuit, a variable reactance to maintain zero voltage switching across the range of load conditions. Maintaining the substantially constant output current or voltage may comprise mapping an input impedance point of the low-pass filter network to infinity at an output of the low-pass filter network. The method may further comprise maintaining a stable magnetic field strength of the transmitter independent of changes in loading conditions.

[0010] According to another aspect of the present disclosure, a low-pass filter network is provided. The low-pass filter network includes a first port and a second port. The low-pass filterAttorney Docket: 100502-2100network includes a plurality of inductors and capacitors arranged in a predetermined topology that performs a Mobius transformation on an input impedance, wherein the Mobius transformation transforms a circular impedance locus at the first port of the low-pass filter network to a substantially straight line in an impedance plane of the second port of the low-pass filter network. The low-pass filter network includes a switching device connected to the first port of the low-pass filter network. The low-pass filter network includes a DC power source or a DC load connected to the switching device, wherein the switching device converts RF power to DC power or DC power to radiofrequency (RF) power. The circular impedance locus at the first port of the low-pass filter network is conjugate-matched to a circular locus of RF impedance points of the switching device as a function of power.

[0011] According to other aspects of the present disclosure, the low-pass filter network may include one or more of the following features. The plurality of inductors may comprise at least two inductors connected in series along a signal path, and the plurality of capacitors may comprise at least two capacitors connected between nodes along the signal path and a ground reference. The low-pass filter network may be a four-pole low-pass filter. The low-pass filter network may further comprise a series LC resonator connected at the second port of the low-pass filter network, the series LC resonator comprising a series inductor and a series capacitor. The low-pass filter network may be configured to attenuate harmonic frequencies of a fundamental operating frequency of the switching device. The substantially straight line in the output impedance plane may have a substantially constant imaginary part across a range of real impedance values. The low-pass filter network may be connected between a Class-E amplifier and a transmitter coil in a wireless power transmitter circuit, and the low-pass filter network may be configured to maintain a substantially constant output DC voltage or DC current to a DC load across a range of loading conditions. The low-pass filter network may be connected between a receiver coil and a Class-E active rectifier in a wireless power receiver circuit, and the low-pass filter network may be configured to maintain a substantially constant input voltage or current to the Class-E active rectifier across a range of coupling conditions. The low-pass filter network may further comprise a feedback loop circuit configured to detect a zero voltage switching condition of a switching device and adjust a variable reactance, an operating frequency, or a phase based on the detected zero voltage switching condition. The low-pass filter network may further comprise a phase-locked loop circuit configured to lock a gate drive signal of a switching device in a receiver to a phase of a transmitter,Attorney Docket: 100502-2100wherein the phase-locked loop circuit comprises a voltage-controlled oscillator and a zero-voltage switching (ZVS) detector.

[0012] This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description, and is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. The foregoing general description of the illustrative embodiments and the following detailed description thereof are merely exemplary aspects of the teachings of this disclosure and are not restrictive.BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, with emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

[0014] FIG. 1 illustrates a Class-E switch circuit and associated waveforms according to various embodiments of the present disclosure.

[0015] FIG. 2 shows normalized current and drain voltage waveforms for a Class-E switch according to various embodiments of the present disclosure.

[0016] FIG. 3 depicts a complex plane plot of impedance values for a Class-E circuit according to various embodiments of the present disclosure.

[0017] FIG. 4 illustrates a two-port network diagram representing an electrical circuit configuration according to various embodiments of the present disclosure.

[0018] FIG. 5 shows a circuit schematic of a Class-E switch configuration according to various embodiments of the present disclosure.

[0019] FIG. 6 depicts a circuit schematic of a Class-E amplifier configuration according to various embodiments of the present disclosure.

[0020] FIG. 7 illustrates another circuit diagram of a Class-E amplifier configuration according to various embodiments of the present disclosure.

[0021] FIG. 8 shows a circuit diagram of a Class-E amplifier with load modeling according to various embodiments of the present disclosure.

[0022] FIG. 9 depicts a wireless power transfer system with transmitter and receiver circuits according to various embodiments of the present disclosure.Attorney Docket: 100502-2100

[0023] FIG. 10 illustrates a schematic diagram of a wireless power transfer system according to various embodiments of the present disclosure.

[0024] FIG. 11 shows a circuit schematic of a complementary Class-E wireless power transfer system according to various embodiments of the present disclosure.

[0025] FIG. 12 depicts a two-port network diagram with transfer matrix representation according to various embodiments of the present disclosure.

[0026] FIG. 13 illustrates a two-port low-pass filter network according to various embodiments of the present disclosure.

[0027] FIG. 14 shows a four-pole low-pass filter circuit with a series resonator according to various embodiments of the present disclosure.

[0028] FIG. 15 depicts a four-pole low-pass filter circuit diagram according to various embodiments of the present disclosure.

[0029] FIG. 16 illustrates another four-pole low-pass filter circuit with a series LC resonator according to various embodiments of the present disclosure.

[0030] FIG. 17 shows a Class-E amplifier with a low-pass filter network according to various embodiments of the present disclosure.

[0031] FIG. 18 depicts an experimental test setup for the Class-E amplifier system according to various embodiments of the present disclosure.

[0032] FIG. 19 illustrates a graph of load impedance characteristics for Zero Voltage Switching conditions according to various embodiments of the present disclosure.

[0033] FIG. 20 shows a graph of RF current amplitude versus input power for the Class-E amplifier according to various embodiments of the present disclosure.

[0034] FIG. 21 depicts a graph of output RF power versus input DC power for the Class-E amplifier according to various embodiments of the present disclosure.

[0035] FIG. 22 illustrates a Class-E amplifier circuit with a series LRC resonator according to various embodiments of the present disclosure.

[0036] FIG. 23 depicts a Class-E amplifier circuit with a four-pole Mobius filter and resonant tank circuit according to various embodiments of the present disclosure.

[0037] FIG. 24 shows theoretical current and voltage waveforms for a Class-E switch operating with a 50% duty cycle across different DC current values according to various embodiments of the present disclosure.Attorney Docket: 100502-2100

[0038] FIG. 25A depicts a photograph of an experimental test setup for a Class-E inverter with a Mobius filter and variable RF load according to various embodiments of the present disclosure.

[0039] FIG. 25B shows a photograph of an experimental test setup comprising a Class-E inverter with a Möbius filter on the transmitter side and a Class-E active rectifier with a Möbius filter on the receiver side according to various embodiments of the present disclosure.

[0040] FIG. 26 illustrates an equivalent circuit schematic of the experimental setup of FIG.25A according to various embodiments of the present disclosure.

[0041] FIG. 27A shows measured effective impedance of a Class-E inverter according to various embodiments of the present disclosure.

[0042] FIG. 27B depicts measured output impedance of a Mobius filter according to various embodiments of the present disclosure.

[0043] FIG. 27C illustrates measured drain voltage waveforms of a Class-E inverter according to various embodiments of the present disclosure.

[0044] FIG. 28 shows measured RF current output of a Class-E inverter with a Mobius filter as a function of input DC power according to various embodiments of the present disclosure.

[0045] FIG. 29 depicts measured harmonic content of voltage and current waveforms at the input and output of a Mobius filter according to various embodiments of the present disclosure.

[0046] FIG. 30 shows voltage attenuation of each harmonic at the output of a Mobius filter relative to the input according to various embodiments of the present disclosure.

[0047] FIG. 31 illustrates measured RF load power as a function of input DC power for a Class-E inverter with a Mobius filter according to various embodiments of the present disclosure.

[0048] FIG. 32 depicts an equivalent circuit schematic of the experimental setup of FIG. 25B according to various embodiments of the present disclosure.

[0049] FIG. 33A shows measured effective impedance of a Class-E active rectifier according to various embodiments of the present disclosure.

[0050] FIG. 33B depicts measured input impedance of a Mobius filter in a receiver according to various embodiments of the present disclosure.

[0051] FIG. 33C illustrates measured drain voltage waveforms of a Class-E active rectifier according to various embodiments of the present disclosure.

[0052] FIG. 34 shows measured output DC power from a Class-E active rectifier as a function of input DC power according to various embodiments of the present disclosure.Attorney Docket: 100502-2100DETAILED DESCRIPTION

[0053] The present application relates to wireless power transfer systems and methods including, but not limited to, those utilizing Class-E power amplifiers and rectifiers. Wireless power transfer systems have been developed to enable the transmission of electrical energy without the need for physical connections between power sources and devices. One approach, described by Kurs et al. in “Wireless Power Transfer via Strongly Coupled Magnetic Resonances” (Science, 2007), utilizes strongly coupled magnetic resonances to transfer power over mid-range distances. This method employs self-resonant coils to achieve efficient non-radiative power transfer. However, the efficiency of this system may decrease rapidly as the distance between the transmitter and receiver increases, limiting its practical applications in scenarios where positioning flexibility is required.

[0054] Another technique for wireless power transfer, proposed by Sample et al. in “Analysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power Transfer” (IEEE Transactions on Industrial Electronics, 2011), focuses on adaptive frequency tuning to maintain high efficiency over varying distances and orientations. While this approach may improve the system's robustness to environmental changes, it may require complex control mechanisms and may be sensitive to interference from nearby metallic objects. These limitations may restrict the use of such systems in certain industrial or consumer environments where metal structures are prevalent.

[0055] Another approach, presented by Hui et al. in “A Critical Review of Recent Progress in Mid-Range Wireless Power Transfer” (IEEE Transactions on Power Electronics, 2014), explores the use of domino-effect energy transfer for extending the range of wireless power transmission. This method involves a series of intermediate resonators to relay power from the source to the load. Although this technique may increase the overall transfer distance, it may also introduce additional complexity in system design and control. Furthermore, the efficiency of the system may be affected by the number of intermediate resonators, limiting its scalability for large-scale applications.

[0056] According to various embodiments of the present disclosure, various embodiments for low-pass filters are disclosed which may have an arbitrary number of poles and which may enable a Class-E switch, amplifier, rectifier, etc. to maintain ZVS when the filter is placed between the Class-E switch and a load or source with variable real impedance. Load-independent ZVS mayAttorney Docket: 100502-2100be a consequence of a Möbius transformation performed by the filter on the effective impedance of the Class-E switch, which may be found to lie on a circular arc in the complex plane. A Mobius transformation may be capable of transforming this circular arc into a relatively straight line with a constant imaginary part. This transformation may be suitable for connecting the Class-E inverter or active rectifier to a resonant loop antenna for transmitting or receiving varying RF power. The constant imaginary part of the impedance may allow the tuning of the system to remain unaffected as the load power is varied.

[0057] Accordingly, various embodiments are disclosed herein for Mobius transform filters in wireless power systems, including those incorporating Class-E power amplifiers and rectifiers. In some embodiments, a wireless power system includes a Class-E power amplifier with a low-pass filter network configured to apply a Mobius transformation, which allows the amplifier to maintain a constant output current over a range of load impedances, also referred to as variable load impedances. The constant current characteristic can be advantageous for driving magnetic loop antennas in wireless power transfer applications, as it may result in a stable magnetic field strength regardless of variations in the coupling between transmitter and receiver.

[0058] Various embodiments are also described for a wireless power system having a complementary Class-E rectifier design incorporating similar Mobius transform filters. This rectifier configuration may enable efficient RF-to-DC power conversion while maintaining zerovoltage switching (ZVS) conditions across varying load conditions.

[0059] In some embodiments, methods for analyzing and designing impedance characteristics of Class-E switching networks using complex plane analysis are disclosed. The impedance of a Class-E switch may be represented as a circle in the complex impedance plane as operating conditions vary. This representation may facilitate the design of impedance matching networks that maintain desired operating conditions across a range of loads.

[0060] Moreover, techniques for implementing automatic tuning in both transmitter and receiver circuits are disclosed using Mobius transform filters. These techniques can utilize phase-locked loops (PLLs) and zero-voltage switching (ZVS) detection to maintain optimal operating conditions despite variations in component values, coupling, or environmental factors.

[0061] The Mobius transform filter designs as disclosed herein can offer several advantages over conventional approaches. First, constant output current characteristics in transmitters is provided, leading to more stable magnetic fields for wireless power transfer. Second, improved maintenance of zero-voltage switching conditions in both transmitters and receivers is provided,Attorney Docket: 100502-2100resulting in higher efficiency. Third, reduced sensitivity to variations in load impedance is provided, improving system robustness. Fourth, simplified automatic tuning implementations are provided, which can lead to more reliable operation in dynamic environments. These improvements enable the creation of wireless power transfer systems with enhanced efficiency, reliability, cost, and flexibility compared to existing solutions. These improvements enable the creation of wireless power transfer systems with enhanced efficiency, reliabi lity, and flexibility compared to existing solutions.

[0062] Turning now to the drawings, FIG. 1 illustrates a Class-E switch configuration and associated waveforms. The Class-E switch includes a switching device, such as a metal-oxide-semiconductor field-effect transistor (MOSFET), connected in parallel with a capacitor Cs. The switch Qi is connected to a constant DC voltage source through an inductive choke, which carries a DC current. The switch Qi is also connected to an RF output terminal, which carries an approximately sinusoidal RF current.

[0063] The operation of the Class-E switch may be described using the following equations. Let 5 denote the duty cycle of the gate-drive signal, and let the time t = 0 be defined as the center of the low period of the gate-drive signal. The phase angle of the signal, 0, may be expressed as: 0 = a>twhere co is the angular frequency of the gate drive and RF waveforms.

[0064] The drain voltage waveform can contain many harmonics of the drive frequency. However, if the RF output is connected to a low-pass filter with high impedance at harmonics of the drive frequency, then the RF current waveform can be approximately sinusoidal. In this case, only the fundamental component of the drain voltage waveform may have significant effect on the external circuitry after the low-pass filter.

[0065] The drain voltage waveform may contain many harmonics of the drive frequency. However, if the RF output is connected to a low-pass filter with high impedance at harmonics of the drive frequency, then the RF current waveform may be approximately sinusoidal. In this case, only the fundamental component of the drain voltage waveform may have significant effect on the external circuitry after the low-pass filter.

[0066] The current flowing into the capacitor when the MOSFET is off may be expressed as: IS(θ) = IDC− Ircosθ + Iisinθwhere Irand Iiare the real and imaginary parts of the phasor of the fundamental component of the current, respectively.Attorney Docket: 100502-2100

[0067] The phase angles of the off and on times of the MOSFET may be defined as:0off = — 7r(l — 5)0On = 71(1 - 5) = -0offDuring the time when the MOSFET is off. the drain voltage may be expressed as:VD(0) = [(^ - 0off)IDc ~ (sin0 - sin0oll) / ,. - (cos0 - cos0off) / / swhere1Xs =^Cs~

[0068] In some embodiments, the Class-E switch may be operated in Zero Voltage Switching (ZVS) conditions, where the MOSFET drain voltage is zero at the turn-on time. This condition may be expressed as:^(0on) = O2 [0ODC - sin0onZr] / s= 0

[0069] The Class-E switch may function as an RF generator when DC current flows in one direction, converting DC input power to RF output power. Conversely, when the DC current direction is reversed, the switch may act as an active rectifier, converting RF power into DC power. This bidirectional capability makes Class-E switches versatile components in wireless power transfer systems.

[0070] In some embodiments, the Class-E switch may operate with specific current and voltage waveforms. FIG. 2 shows normalized current and drain voltage waveforms for a Class-E switch. The upper graph in FIG. 2 displays the normalized current (IS / VDC / ^S)) plotted against normalized time (t / T), where T is the period of the switching cycle. The lower graph in FIG. 2 shows the normalized drain voltage (VD / VDC) plotted against the same normalized time axis. The current waveform / s(0) may be expressed as:4(0)=IDC ~ Ircos0 + ZjSin#where IDCis the DC current, Irandare the real and imaginary parts of the phasor of the fundamental component of the current, respectively, and 0 is the phase angle defined as 0 = cot. The drain voltage waveform CD(0) may be described by:VD(0) = [(0 - OoffVoc ~ (sin0 - smdoff)Ir- (cos0 - cos0o / / )ZJ / swhere 6oJfJfis the phase angle at which the switch turns off, and Xs = (liC c ■Attorney Docket: 100502-2100In some embodiments, the Class-E switch may operate in Zero Voltage Switching (ZVS) mode. The ZVS condition may be expressed as:^o(^on)=0where 6onis the phase angle at which the switch turns on.

[0071] The waveforms in FIG. 2 illustrate multiple operating conditions of the Class-E switch. The current waveforms in the upper graph show various sinusoidal-like patterns that intersect near the center points at t / T = 0 and ±0.25. The drain voltage waveforms in the lower graph maintain zero voltage at their endpoints (around ±0.25 t / T), demonstrating the ZVS condition. The dimensionless current parameter ιDCmay be defined as:ιDC≡DC =VDC / XSThe solutions for the DC and RF current components may be expressed as:VDCIr = ^jr^DCXsVDCXs. > ^DC‘DC ~LDCXswhere jr(8) and j (8) are functions of the duty cycle 8.

[0072] The waveforms in FIG. 2 demonstrate how the Class-E switch maintains ZVS operation across various operating conditions, as indicated by the drain voltage returning to zero at the switch turn-on times for all plotted waveforms. The complex impedance characteristics of a Class-E switch may be represented in a complex plane plot, as illustrated in FIG. 3. This plot shows the dimensionless RF impedance, <’(iDC, 5), as a function of the dimensionless DC current parameter, iDC, for a fixed duty cycle 8.

[0073] In FIG. 3, the real part of the dimensionless impedance is plotted on the horizontal axis, denotedas 5)], while the imaginary part is plotted on the vertical axis, denoted as 3[ G‘DC, )]• As iDCvaries from negative infinity to positive infinity, the dimensionless impedance traces out a circle in the complex plane. The dimensionless RF impedance may be expressed as:rZ(iDC, 6) _ _ ur(6) + iui(6)iDCaiDC’ Xs JA5)iDC+ ihWwhere %sis defined as:Attorney Docket: 100502-2100Xs= 7 a>7Cs~The functions ur(5). u,-(5). / ,.(5). and / ,(5) are defined as:> cos(n:5)sin(7r5) + 7r(l — 5)ur(<5)+_ 5)COS(7T5) 2sin2(rr5) + TT(1 — 5)cos(7r5)sin(7r5) — TT2(1 — 5)2uf(<5) = - 7rsin(7r5)Jr^ sin(7r< S)— itsin(7r5) — 7r(l — 5)cos(?r5)

[0074] The circle in FIG. 3 intersects the imaginary axis at two points, labeled <0and oo These points represent the dimensionless impedance values when iDCis zero and infinity, respectively. The expressions for these points are:ur(8) sin(7r5)cos(7r5) + 7r(l — 8) <0(5) = <(0,5) = i^- = -i1 / - - - - Jt(°)nui (£) <«(5) = limi ^iDC, 8) = -i—^jr(8) 2sin2(7r5) + rr(l — 5)cos(7r5)sin(7r5) — rr2(l — 5)27r2(l — 5)

[0075] In FIG. 3, the circle is rendered with a dashed line in its upper portion and a solid line in its lower portion. The dashed region indicates where the drain voltage VD(0) becomes negative for part of the cycle, which may be undesirable in some applications.

[0076] This complex impedance representation provides insight into the behavior of the Class-E switch under various operating conditions and may be used to design impedance matching networks and optimize circuit performance.

[0077] In some embodiments, two-port network analysis can be used to characterize and optimize the behavior of the impedance transformation networks used in conjunction with Class-E amplifiers and rectifiers. FIG. 4 illustrates a basic two-port network configuration, where an input impedance Zin is connected to a first port on the left side, and a two-port network labeled Z is shown in the center. The voltages and currents at each port are denoted as Vi, Ii for the input port and Vo, Io for the output port.Attorney Docket: 100502-2100

[0078] The relationship between the voltages and currents at the two ports may be described using the following equations:)=00I0+ ZnI.=Wo +znhwhere Zoo, Zoi. Zio, and Zu are the elements of the impedance matrix Z.

[0079] In some embodiments, the two-port network may be represented using a T-matrix. The T-matrix representation relates the voltages and currents at the two ports as follows:)=Too^i + TotIQ=Tio^i + T'n^iwhere Too, Toi, Tio, and Tn are the elements of the T-matrix.

[0080] The output impedance of the two-port network may be expressed as a Mobius transformation of the input impedance:„ _ _ T00Zin— T01out j T 7 Tl0 ~yll

[0081] This Mobius transformation is a key aspect of the disclosure, as it allows for the design of impedance networks that maintain Zero Voltage Switching (ZVS) conditions over a wide range of load impedances. By carefully selecting the elements of the T-matrix, the impedance transformation may be tailored to map the desired input impedance of the Class-E switch to the required output impedance for maintaining ZVS.

[0082] In some embodiments, the T-matrix elements may be chosen such that Toi = 0 and Tio = 0. Under these conditions, the Mobius transformation simplifies to a scaling operation:TooZo(JuILtL =—rp ZiInII,' when T0U11 = 0, ' and T1J0U = 0111Alternatively, when Too = 0 and Tn = 0, the Mobius transformation becomes an inversion operation composed with a scaling operation:zToi 1out = ~T 7, when Too= 0, and T1±= 0110 in

[0083] These special cases of the Mobius transformation may be utilized to design impedance networks that provide the desired impedance transformation while maintaining ZVS conditions for the Class-E amplifier or rectifier. By implementing such two-port impedance networks, the disclosure enables the maintenance of Zero Voltage Switching across a wide range of real load impedances, thereby improving the efficiency and reliability of wireless power transfer systems.Attorney Docket: 100502-2100

[0084] In some embodiments, a Class-E amplifier may be configured with various impedance networks to achieve specific output characteristics. FIG. 5 shows a basic Class-E switch configuration. The configuration may include a MOSFET driven by an RF square wave generator at its gate. A DC voltage source may be connected to the drain of the MOSFET through an RF choke inductor. A shunt capacitor may be connected in parallel with the drain-source terminals of the MOSFET. The drain node may connect to a two-port network represented by an impedance block.

[0085] FIG. 6 illustrates a Class-E amplifier configuration that builds upon the basic switch arrangement. In this configuration, the two-port impedance network may connect to a load resistance. The impedance network may be designed to transform the load impedance to achieve desired operating conditions for the Class-E amplifier.

[0086] In some embodiments, the Class-E amplifier may be configured to provide a constant current output. FIG. 8 shows a configuration where the output section includes a load conductance in parallel with an induced voltage source. The impedance network, represented by the block labeled “Z,” may be designed using a Mobius transform filter to map specific impedance points. This mapping may allow the amplifier to maintain a constant output current across varying load conditions. The Mobius transform filter may be characterized by the following equation:7_ Toplin ~ ^01out T 7 T110zin— 711where Zoutis the output impedance, Zinis the input impedance, and Too. T01, T10, and T1Xare elements of the transfer matrix for the filter network.

[0087] In some embodiments, the Class-E amplifier may be configured to operate with an induced voltage at the output. FIG. 7 shows a configuration where the output section includes a load resistance in series with a voltage source. This arrangement may be useful in applications such as wireless power transfer, where an induced voltage may be present in the receiver coil. The effective impedance of the Class-E switch may be described by the following equation:_ u,.(5) + iUi(8)iDCDC> °) ~ Xs • z ox,. ■ •Jr(S)lDC+ IJiwhere Xs is the reactance of the shunt capacitor, ιDCis a dimensionless current parameter, 8 is the duty cycle, and ur, ui, jr, and j are functions of the duty cycle.Attorney Docket: 100502-2100

[0088] By carefully designing the impedance network using Mobius transform filters, the Class-E amplifier may be configured to maintain constant current or voltage output under varying load conditions, making it suitable for applications such as wireless power transfer systems.

[0089] In some embodiments, a wireless power transfer system may include a transmitter circuit and a receiver circuit. FIG. 9 shows an example configuration of such a system. The transmitter circuit may include a Class-E amplifier with a MOSFET switch, inductors, and capacitors arranged in a predetermined topology. The receiver circuit may include a complementary arrangement with its own MOSFET switch, inductors, and capacitors, forming a Class-E rectifier.

[0090] FIG. 11 illustrates another implementation of a wireless power transfer system. In this configuration, the transmitter may use a P-channel MOSFET while the receiver may use an N-channel MOSFET. The transmitter and receiver circuits may be coupled through their respective LC networks, enabling wireless power transfer between them.

[0091] In some embodiments, phase correction may be implemented using Zero Voltage Switching (ZVS) detection. FIG. 10 shows a wireless power transfer system that may incorporate ZVS detection for phase correction. The gate drive control circuitry in the receiver may monitor the drain voltage of the MOSFET to detect zero-voltage conditions.

[0092] The ZVS detection may be described mathematically. Let Vo(0) represent the drain voltage as a function of phase angle 9. The ZVS condition may be expressed as:^on) = 0where 0onis the phase angle at which the MOSFET turns on.

[0093] In some embodiments, gate drive signals for active rectifiers may be generated using various methods. One approach may involve capacitive pickup. The voltage across a capacitor in the receiver circuit may be sensed and used to generate a timing signal for the gate drive. Another method may implement inductive pickup. The voltage induced in a secondary winding coupled to the receiver's main inductor may provide timing information for the gate drive signal. A phase-locked loop (PLL) may also be used to generate gate drive signals. The PLL may compare the phase of a locally generated oscillator signal with the received RF signal to maintain synchronization. The mathematical representation of a basic PLL may be given by:deb— = KdK0F(s)sm(<prei- (p~)Attorney Docket: 100502-2100where <p is the phase of the voltage-controlled oscillator, Kdis the phase detector gain, Kois the VCO gain, F(s) is the loop filter transfer function, andrefis the reference phase.

[0094] In some implementations, the gate drive signal may be generated using a Variable Frequency Oscillator (VFO) controlled by the ZVS detection circuit. The frequency of the VFO may be adjusted based on the error signal from the ZVS detector to maintain optimal switching conditions.

[0095] In some embodiments, low-pass filter networks may be used in conjunction with Class-E amplifiers to achieve desired impedance transformation and output characteristics. Various configurations of low-pass filters may be employed, including two-pole and four-pole designs, as well as filters incorporating series LC resonators.

[0096] FIG. 13 shows a basic two-pole low-pass filter configuration. The filter includes an inductor LI connected in series between the input and output ports, and a capacitor Cl connected between the output port and ground. The impedance matrix for this two-pole low-pass filter may be expressed as:1 1sCj Z01-,Z = SC-I n _.[-ZooL1 1JZio ZnJsLsCr1sCrwhere:1 / o° “ "1Zoi = Zio = -^1Zn =a)C1The corresponding T-Matrix for this filter may be expressed as:1 -sL.T= [r,r2 JsCr—I^C^s — 1

[0097] FIG. 14 illustrates a two-pole low-pass filter with an additional series LC resonator. This configuration adds an inductor Lo and capacitor Co in series with the input port. The impedance matrix for this filter may be expressed as:Attorney Docket: 100502-2100 1 1 1—— + sL0+ ——2 _ |-s^isCo s(u _ Zoo Zoi1J“l[Xw Znstl +7c?where:Zoo=1-1—CI)LQ 6C)C£1Zoi=Zio= —(,i) Cri1Zn = " C -a) TC1The T-Matrix for this filter configuration may be expressed as:CQC^LQS2+ Cr+ CoCQC-^LQLIS4+ (C^Li + + CQLQ^S2+ 1T = [ So ]sC1— C1L1s2— 1

[0098] FIG. 15 shows a four-pole low-pass filter configuration. This filter includes two inductors (Li and L2) connected in series along the top path, with two capacitors (Ci and C2) connected from the midpoints to ground. The impedance matrix for this four-pole low-pass filter may be expressed as:C1L2S2+ 1_ S(<C1C2L2S2+ Q + C2) s(C1C2L2s2+ C2+ CJ, “L1 C1C2L1L2s4+ (C2L2+ C2L! + QLJs2+ 1JS{C1C2L2S2+ C2+ CJ S(C1C2L2S2+ C2+ Q) The corresponding T-Matrix for this filter may be expressed as:j. _ r L2s2+ 1—(Li + L2)S — C1L1L2S3^I^2^2S 3+ (Q + C?)s~CIC2L1L2S4— (C2L2+ C2Lt+ C1L1)s2— 1

[0099] FIG. 16 illustrates a four-pole low-pass filter with an additional series LC resonator. This configuration adds an inductor Lo and capacitor Co in series with the input port of the four-pole filter shown in FIG. 15. The impedance matrix for this filter may be expressed as:(XL2s2+ 1 1 1 _ S(^C1C2L2S2+ Q + C2) sC0s(C1C2L2s2+ C2+ C-^) ~L1 CIC2LIL2S4+ (C2L2+ C2Lr+ C-^L-^s2+ 1S^C1C2L2S2+ C2+ Cj) S(CIC2L2S2+ C2+ C-J _.. Zoo Zoi-,1Xw XuAttorney Docket: 100502-2100where:1Zoo — ^foa)C0(j)C^(j)C2(joLi2coC2toCj^01a)C1CjoC2o)L2— (JOC2— a)C1— a> C1a> C2(i)Lra)L2T O)C2coL2T <z)C2toLi + — 1— O)C^CC> C2U)L2+ coC2+ coC^

[0100] In some embodiments, these low-pass filter configurations may be used in conjunction with Class-E amplifiers to achieve desired impedance transformation and output characteristics. The choice of filter topology and component values may depend on the specific requirements of the application, such as output impedance, harmonic suppression, and efficiency considerations.

[0101] In some embodiments, a Class-E amplifier may be combined with a low-pass filter network to achieve specific performance characteristics. FIG. 17 shows a circuit diagram of such a configuration. The circuit includes a MOSFET switch with a parallel capacitor Cs, connected to a DC voltage source Vdc through an RF choke. A square wave input signal drives the MOSFET. Following the switch is a low-pass filter network composed of two inductors (L 1 and L2) and three capacitors (Ci, C2, and C3) arranged in a predetermined topology. The output of the filter network may be terminated with an impedance Zout.

[0102] The inductor and capacitor values in the low-pass filter may be chosen to map the input impedance point Zeff(?r / 2, 5) to infinity at the output. This configuration may cause the output current. Io, to be constant and load-independent. The output current amplitude may be proportional to the DC voltage, Vdc.

[0103] In some embodiments, this amplifier configuration may be suitable for driving a resonant magnetic loop antenna with a constant RF current to produce a constant load-independent magnetic field for wireless power transfer. The impedance matrix for the four-pole low-pass filter shown in FIG. 17 may be expressed as:C1L2s2+ 1_ S(C1C2L2S2+ + C2) s(C1C2L2S2T ^2 + ^1)“L1 ^1^2^1^254+ (^2^2 + ^2^1 + C1L1)S2+ 1s(C1C2L2s2d" C2+ Q) + ^2 + Q) The corresponding T-Matrix may be given by:_ C1L2s2+ 1 (fl + ^2)SC-\ L-LL2S2+ (6\ + C2)s CIC2L1L2S4(C2f2d" ^2^1 d" flfl)521Attorney Docket: 100502-2100In some embodiments, setting T01= 0 and 7j0= 0 may yield the following solution:Ci + C2Ci + C2C1& = -^r^k2&CiC2s2This solution may result in the following T-Matrix:When an impedance. Zin. is connected to port 1. this T-Matrix may represent a Mobius transform equal to a scaling operation. The impedance, Zout, at port 0 may be expressed as:7 = -Z°2.7. = (~C1 / C2)y =£^out rp ^in f p [p \ ^in \p ) in* 11This configuration may allow the Class-E amplifier to output an approximately constant RF current amplitude over a wide range of loading conditions with relatively small changes in the imaginary impedance.

[0104] In some embodiments, an experimental setup may be used to test and validate the performance of a Class-E amplifier system with a low-pass filter for wireless power transfer applications. FIG. 18 shows a photograph of such an experimental setup. The setup may include a magnetic loop antenna mounted on a surface. In some embodiments, a resonant RF load may be used in the experimental setup. This load is shown in FIG. 18 on the left side of the photograph. The resonant RF load may be inductively coupled to the magnetic loop antenna. The coupling between the load and the antenna may be varied by moving the load to change the loading conditions on the loop antenna.

[0105] For each loading condition, measurements may be taken of the RF voltage and current at the output of the low-pass filter. Additionally, the DC input power and the RF power received by the resonant RF load may be measured. The results of these measurements may be analyzed and presented in graphical form, shown in FIG. 19, FIG. 20, and FIG. 21.

[0106] FIG. 19 shows a graph of the load impedance measured at the output of the low-pass filter. The x-axis represents the real part of the impedance QR[Z]) in ohms (Q), while the y-axis shows the imaginary part of the impedance (3[Z]j also in ohms. The real part of the impedance may be controlled by adjusting the coupling to the resonant RF load. The imaginary part may be automatically controlled to maintain the Class-E amplifier in a state of Zero Voltage Switching (ZVS).Attorney Docket: 100502-2100

[0107] As seen in FIG. 19, the imaginary part of the impedance may vary by only about 2 as the real part is varied from 0 Q to 14.4 Q. This small variation in the imaginary' impedance may be close to the design goal of zero change in imaginary impedance. The slight deviation from the ideal case may be attributed to component tolerances and approximations made in the mathematical model of the Class-E amplifier.

[0108] FIG. 20 presents a graph showing the amplitude of the RF current in the magnetic loop antenna as a function of input power. The x-axis represents the input DC power in watts, ranging from 0 to 100 W, while the y-axis shows the output RF current amplitude in amperes, ranging from 0 to 5 A. The graph demonstrates that the RF current amplitude remains approximately constant over the entire range of input power. The current amplitude may only drop by about 7.4% when maximally loaded, indicating the effectiveness of the low-pass filter design in maintaining a stable RF current output across different loading conditions.

[0109] FIG. 21 displays a graph of the output RF power measured from the resonant RF load versus the input DC power. The x-axis represents the input DC power in watts, ranging from 0 to 100 W, while the y-axis shows the output RF power also in watts, ranging from 0 to 80 W. The relationship between output and input power may be approximated by a linear relation with a constant offset. The x-intercept of this linear relationship may represent the power required to maintain the RF current in the loop antenna in the absence of an additional external load.

[0110] The experimental results shown in FIG. 19, FIG. 20, and FIG. 21 may demonstrate that the model of the impedance of a Class-E amplifier / rectifier and Mobius transform low-pass filter presented in this work is capable of achieving practical results. The low-pass filter may allow the Class-E amplifier to output an approximately constant RF current amplitude over a wide range of loading conditions with relatively small changes in the imaginary impedance. These small changes in imaginary impedance may be sufficiently small to be mitigated by an auto-tuning circuit.

[0111] In some embodiments, the impedance of the Class-E switch may be represented by the following equation:ur(δ) + iui(δ)ιDC−χSjr(δ)ιDC+ iji(δ)where χSis the reactance of the shunt capacitor, ιDCis the normalized DC current, δ is the duty cycle, and ur(δ), ui(δ), jr(δ), and ji(δ) are functions of the duty cycle.Attorney Docket: 100502-2100

[0112] The experimental results may validate the effectiveness of using this impedance model in conjunction with a Möbius transform low-pass filter to achieve stable RF current output in a Class-E amplifier system for wireless power transfer applications.

[0113] The experimental measurements presented demonstrate that the model of the impedance of a Class-E amplifier / rectifier and Möbius transform low-pass filter described as disclosed herein may achieve practical results. The low-pass filter allows the Class-E amplifier to output an approximately constant RF current amplitude over a wide range of loading conditions with relatively small changes in the imaginary impedance. These small changes in imaginary impedance may be sufficiently small to be mitigated by an auto-tuning circuit.

[0114] The implementation of Möbius transform filters in Class-E amplifiers and rectifiers for wireless power transfer systems provides several advantages. In some embodiments, the system may maintain a stable field strength which may be independent of changes in loading conditions or environmental perturbations. This may reduce the influence of receivers on each other in a multi-receiver system.

[0115] In some embodiments, each receiver may maintain itself in a state of resonance with the frequency of the externally applied field. This state of resonance may maximize the power transfer efficiency while also ensuring that the transmitter system sees a mostly real reflected impedance.

[0116] The described system may accommodate varying loading conditions without requiring re-tuning of the transmitter LC resonator. This may aid in the maintenance of a stable field strength without relying upon corrections from an auto-tuning feedback loop, which may have limitations in tuning range and speed.

[0117] The practical implementation of these concepts in a wireless power transfer system demonstrates the effectiveness of the theoretical model. The experimental results show that the RF current amplitude in the magnetic loop antenna may remain approximately constant over a wide range of loading conditions. This characteristic may be particularly useful for applications requiring consistent field strength, such as wireless charging systems for multiple devices.

[0118] The combination of Class-E amplifiers and rectifiers with Möbius transform low-pass filters may provide a robust and efficient solution for wireless power transfer systems. The ability to maintain constant current output and accommodate varying load conditions may offer significant advantages in practical applications, improving the reliability and efficiency of wireless power transfer technologies.Attorney Docket: 100502-2100

[0119] In some embodiments, a series LRC circuit may be used to convert the non-sinusoidal drain voltage of a Class-E switch into a sinusoidal voltage across a resistive load and to shape the drain waveform to achieve ZVS. FIG. 22 illustrates such a configuration, where a first switching device QI is connected to a DC voltage source through an RF choke inductor. A first capacitor Cl is connected in parallel with the first switching device QI, forming the shunt capacitance of the Class-E topology. The output of the first switching device QI connects to a series resonant circuit comprising a first inductor LI, a second inductor L2, and a second capacitor C2. A third capacitor C3 is connected in parallel with a transmitter coil L3 at the output.

[0120] However, this approach may have several drawbacks. First, the tuning may only work over a narrow bandwidth of drive frequencies. Second, in order to achieve ZVS, the tuning of the LRC circuit may need to be adjusted based on the resistive load. Third, a series LRC circuit is only a two-pole filter. In some situations, a two-pole filter may not be sufficient to prevent harmful interference caused by harmonics of the drive frequency.

[0121] In some embodiments, a four-pole Möbius filter may be inserted between the Class-E switch and a resonant tank circuit to address the limitations of the series LRC approach. FIG. 23 illustrates such a configuration. The circuit includes a first switching device QI connected to a DC voltage source through an RF choke inductor. A first capacitor C 1 is connected in parallel with the first switching device QI, forming the shunt capacitance of the Class-E topolog)’. The output of the first switching device QI connects to a four-pole low-pass filter network comprising a first inductor LI, a second inductor L2, a second capacitor C2, and a third capacitor C3. The first inductor LI and second inductor L2 are connected in series along the signal path, with the second capacitor C2 connected between the junction of the first inductor LI and second inductor L2 and ground, and the third capacitor C3 connected at the output of the second inductor L2. A transmitter coil L3 is connected to the output of the filter network in parallel with a load resistance Rload.

[0122] With a suitable choice of filter components, the first switching device QI may maintain ZVS as the load resistance Rioad is varied. This configuration may have an arbitrary number of poles and may enable the Class-E switch to maintain ZVS when the filter is placed between the Class-E switch and a load or source with variable real impedance.

[0123] The load-independent ZVS characteristic may be a consequence of a Möbius transformation performed by the filter on the effective impedance of the Class-E switch. As described previously, the effective impedance of the Class-E switch lies on a circular arc in theAttorney Docket: 100502-2100complex plane. A Möbius transformation may transform this circular arc into a straight line with a constant imaginary part. This transformation may be suitable for connecting the Class-E inverter or active rectifier to a resonant loop antenna for transmitting or receiving varying RF power. The constant imaginary part of the impedance may allow the tuning of the system to remain unaffected as the load power is varied.

[0124] FIG. 24 illustrates theoretical current and voltage waveforms for the Class-E switch operating with a 50% duty cycle across different values of DC current. The waveforms are plotted for negative, zero, and positive DC current values, represented by the dimensionless quantity iDC. The figure demonstrates that ZVS may be maintained at all values of DC current, with the drain voltage returning to zero at the turn-on instant regardless of whether the DC current is negative, zero, or positive. Because the current flowing into the circuit is assumed to flow into a tuned circuit that blocks harmonic frequencies, the interaction between the Class-E switch and the rest of the circuitry may be approximated by neglecting the harmonics and considering only the fundamental components of the drain voltage waveform. This approximation may simplify the analysis while still capturing the behavior of the Class-E switch under varying load conditions.

[0125] FIG. 25A shows an experimental setup used to validate the performance of a Class-E inverter with a Möbius filter and variable RF load. The setup includes a transmitter loop L3 constructed from 14 AWG stranded copper wire arranged along the boundary of a rectangular acrylic frame measuring approximately 56 cm by 118 cm. The receiver coil L4 appears as a square structure with approximate dimensions of 52 cm, visible as a black square with a white border positioned adjacent to the transmitter loop. The loading conditions on the transmitter may be varied by adjusting the position of the receiver resonator relative to the transmitter loop, which changes the magnetic coupling between the transmitter coil L3 and the receiver coil L4. Electronic circuitry associated with the Class-E inverter and Möbius filter network is mounted on a platform near the transmitter loop, with test leads and connectors extending to laboratory measurement equipment visible in the background. FIG. 26 is an equivalent schematic of FIG. 25 A.

[0126] FIG. 25B shows a photograph of an experimental setup comprising a Class-E inverter with a Möbius filter on the transmitter side and a Class-E active rectifier with a Möbius filter on the receiver side, and FIG. 32 shows an equivalent circuit thereof. The Möbius filter of the receiver may comprise the seventh capacitor C7 and the third inductor L5. Unlike the Möbius filter described in the transmitter section, the impedance matrix may be chosen such that Z0 is transformed to infinity. This may cause the input of the Möbius filter to be high impedance whenAttorney Docket: 100502-2100the load draws zero power, which may result in the receiver current IRX being close to zero when the receiver is unloaded.

[0127] As the load power is increased, the input impedance of the Möbius filter may fall, and the amplitude of IRX may increase, causing power to be drawn from the transmitter through the mutual inductance between the transmitter coil L3 and the receiver coil L4. The receiver coil L4 is visible as a white circular wire loop with a diameter of approximately 35 cm, constructed from 14 AWG stranded copper wire. The second switching device Q2 may be switched using a square wave generated by a voltage-controlled crystal oscillator (VCXO).

[0128] The VCXO frequency may be controlled by a ZVS detector in a feedback loop that keeps the second switching device Q2 always in a state of ZVS. The feedback loop may ensure that the phase of the gate drive of the second switching device Q2 is locked to the phase of the transmitter, enabling synchronized operation between the transmitter and receiver circuits. The receiver DC output may be connected to a variable DC load to enable testing across different loading conditions. Electronic circuitry for both the transmitter and receiver is visible on the test platform, with measurement equipment and test leads arranged for characterizing the wireless power transfer performance under varying load and coupling conditions.

[0129] FIG. 27A shows the measured effective impedance of the Class-E inverter, w here the data points lie along a circle in the complex impedance plane, as predicted by the theoretical model describing the impedance characteristics of the Class-E switch. FIG. 27B shows the measured output impedance of the Möbius filter, demonstrating that the imaginary part of the impedance remains nearly constant over all loading conditions, which confirms the impedance transformation properties of the filter network. FIG. 27C shows the measured drain voltage waveforms of the Class-E inverter, where the duty cycle was estimated to be approximately δ ≈ 0.38 based on zero-crossing times, and ZVS was maintained over the full loading range.

[0130] FIG. 28 shows the measured RF current output of the Class-E inverter with a Mobius filter as a function of input DC power. The figure displays the amplitude of the RF current in the magnetic loop antenna of the transmitter as a function of loading conditions. The RF current amplitude remains approximately constant over the entire range of input DC power, and only drops by about 6.8% when maximally loaded. This behavior demonstrates that the Möbius filter configuration may enable the Class-E inverter to maintain a substantially stable output current despite variations in the power delivered to the load.Attorney Docket: 100502-2100

[0131] FIG. 29 shows the measured harmonic content of the voltage and current waveforms at the input and output of the Möbius filter of the Class-E inverter, with amplitudes expressed relative to the fundamental frequency component. FIG. 29 displays the harmonic content of the voltage waveform at the input and the voltage and current waveforms at the output of the Möbius filter. The Möbius filter may be effective at attenuating the harmonic content of the drain waveform of the first switching device QI. The resonant loop antenna formed by the fourth capacitor C4 and the transmitter coil L3 may provide additional filtering which further reduces the harmonic content of the transmitter current. In some cases, only the second harmonic may be large enough to be detected, with all higher harmonics falling below the noise floor of the test equipment.

[0132] FIG. 30 shows the voltage attenuation of each harmonic at the output of the Möbius filter relative to the same harmonic amplitude at the input of the filter. The Möbius filter provides an attenuation close to the asymptotic 80 dB / decade expected for a four-pole filter. FIG. 31 shows the measured RF load power as a function of input DC power for the Class-E inverter with a Möbius filter. The relationship between RF load power and input DC power is approximately linear with a small quadratic term. A quadratic fit is shown as a dotted curve overlaid on the measurement data points. The maximum system efficiency achieved was 81%, which occurred at an input DC power of 94.4 W and an RF load power of 76.2 W. The approximately linear relationship with a small quadratic correction may indicate that losses in the system scale with power in a predictable manner, with the quadratic term potentially representing second-order effects such as resistive losses that increase with current squared.

[0133] FIG. 32 is an equivalent schematic of FIG. 25B. The circuit diagram illustrates a wireless power transfer system with a Class-E inverter on the transmitter side and a Class-E active rectifier on the receiver side. The transmitter section includes a first switching device QI connected to a DC input voltage VIN through an input choke inductor L6. A first capacitor Cl is connected in parallel with the first switching device QI, forming the shunt capacitance of the Class-E topology. The output of the first switching device QI connects to a four-pole low-pass filter network comprising a first inductor LI, a second inductor L2, a second capacitor C2, and a third capacitor C3. A fourth capacitor C4 is connected in series with a transmitter coil L3, forming a resonant transmitter antenna circuit. The transmitter section includes a ZVS detector connected to the first switching device QI, a low-pass filter, and an electrically controlled variable reactance element for autotuning.Attorney Docket: 100502-2100

[0134] The transmiter coil L3 is magnetically coupled to a receiver coil L4 through mutual inductance M. The receiver includes the receiver coil L4 connected in parallel with a fifth capacitor C5. A sixth capacitor C6 is connected in series with a third inductor L5 along the signal path. A seventh capacitor C7 and an eighth capacitor C8 are connected to ground at respective nodes in the receiver filter network. A second switching device Q2 is configured as part of a Class-E active rectifier. An output choke inductor L7 connects the second switching device Q2 to an output terminal providing an output voltage VOUT and output current IOUT to a load resistance RLOAD. A ninth capacitor C9 is connected across the output terminals. The ZVS detector and voltage-controlled crystal oscillator (VCXO) in the receiver form a phase-locked loop that locks the gate drive signal of the second switching device Q2 to the phase of the transmitter, enabling synchronized operation between the transmitter and receiver circuits.

[0135] FIG. 33A shows the measured effective impedance of the Class-E active rectifier in the receiver. The data points lie along a circle in the complex impedance plane, as predicted by the theoretical model describing the impedance characteristics of the Class-E switch when operating as an active rectifier. This circular impedance locus confirms that the Class-E active rectifier exhibits the same fundamental impedance behavior as the Class-E inverter, with the impedance tracing a circular arc as the DC current parameter varies with changing load conditions.

[0136] FIG. 33B shows the measured input impedance of the Mobius filter in the receiver. The imaginary part of the impedance remains nearly constant over all loading conditions, demonstrating that the Möbius transformation performed by the receiver filter network successfully converts the circular impedance locus of the Class-E active rectifier into a substantially straight line with constant reactive component. This impedance transformation property may allow the receiver to maintain proper tuning with the transmitter frequency regardless of variations in the power being delivered to the DC load.

[0137] FIG. 33C shows the measured drain voltage waveforms of the Class-E active rectifier across different loading conditions. The duty cycle was estimated to be approximately δ ≈ 0.44 based on zero-crossing times of the drain voltage waveform. ZVS was maintained over the full loading range, with the drain voltage returning to zero at the tum-on instant for all measured operating points. The maintenance of ZVS conditions in the receiver across varying load conditions may contribute to high conversion efficiency and reduced switching losses in the Class-E active rectifier.Attorney Docket: 100502-2100

[0138] FIG. 34 shows the measured output DC power from the Class-E active rectifier as a function of input DC power to the Class-E inverter. The figure displays the DC output power of the receiver plotted against the DC input power of the transmitter. The relationship between output DC power and input DC power follows a quadratic function, with a quadratic fit shown as a dotted curve overlaid on the measurement data points. The maximum DC-to-DC efficiency of the wireless power transfer system was 69%, which occurred at an input power of 64.0 W and a load power of 44.3 W. This efficiency measurement represents the overall power conversion from the DC input of the transmitter through the wireless link to the DC output delivered to the load resistance. The quadratic relationship may indicate that system losses include both fixed overhead components and load-dependent components that scale with the square of the current or power level.

[0139] The features, structures, or characteristics described above may be combined in one or more embodiments in any suitable manner, and the features discussed in the various embodiments may be interchangeable, if possible. In the following description, numerous specific details are provided in order to fully understand the embodiments of the present disclosure. However, a person skilled in the art will appreciate that the technical solution of the present disclosure may be practiced without one or more of the specific details, or other methods, components, materials, and the like may be employed. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the present disclosure.

[0140] Although the relative terms such as “on,” “below,” “upper,” and “lower” are used in the specification to describe the relative relationship of one component to another component, these terms are used in this specification for convenience only, for example, as a direction in an example shown in the drawings. It should be understood that if the device is turned upside down, the “upper” component described above will become a “lower” component. When a structure is “on” another structure, it is possible that the structure is integrally formed on another structure, or that the structure is “directly” disposed on another structure, or that the structure is “indirectly” disposed on the other structure through other structures.

[0141] In this specification, the terms such as “a,” “an,” “the,” and “said” are used to indicate the presence of one or more elements and components. The terms “comprise,” “include,” “have,” “contain,” and their variants are used to be open ended, and are meant to include additional elements, components, etc., in addition to the listed elements, components, etc. unless otherwise specified in the appended claims.Attorney Docket: 100502-2100

[0142] The terms “first” “second,” etc. are used only as labels, rather than a limitation for a number of the objects. It is understood that if multiple components are shown, the components may be referred to as a “first” component, a “second” component, and so forth, to the extent applicable.

[0143] The terms “about” and “substantially,” unless otherwise defined herein to be associated with a particular range, percentage, or related metric of deviation, account for at least some manufacturing tolerances between a theoretical design and manufactured product or assembly, such as the geometric dimensioning and tolerancing criteria described in the American Society of Mechanical Engineers (ASME®) Y14.5 and the related International Organization for Standardization (ISO®) standards. Such manufacturing tolerances are still contemplated, as one of ordinary skill in the art would appreciate, although “about,” “substantially,” or related terms are not expressly referenced, even in connection with the use of theoretical terms, such as the geometric “perpendicular,” “orthogonal,” “vertex,” “collinear,” “coplanar,” and other terms.

[0144] The above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.

Claims

Attorney Docket: 100502-2100CLAIMSTherefore, the following is claimed:

1. A wireless power transmitter, comprising:an amplifier comprising a switching device;a DC power source connected to the amplifier;a low-pass filter network connected to an output of the amplifier, the low-pass filter netw ork configured to perform a Mobius transformation on an output impedance of the amplifier, wherein the filter function and Mobius transformation takes an output impedance of the amplifier and transforms the output impedance to a straight line in an output impedance plane of the low-pass filter network; anda load connected to an output of the low-pass filter network, wherein the low -pass filter network is configured to maintain a substantially constant output current or voltage to the load across a range of load conditions.

2. The wireless power transmitter of claim 1, wherein the low-pass filter network comprises a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation, and the load comprises a transmitter coil.

3. The wireless power transmitter of claim 2, wherein the low-pass filter network comprises at least two inductors and at least two capacitors.

4. The wireless power transmitter of claim 1, wherein the switching device is a metal -oxide-semiconductor field-effect transistor (MOSFET) transistor.

5. The wireless power transmitter of claim 1, further comprising a gate drive circuit configured to provide a square wave signal to control the switching device.

6. The wireless power transmitter of claim 5, further comprising a feedback loop circuit configured to maintain zero voltage switching (ZVS) of the switching device.Attorney Docket: 100502-21007. The wireless power transmiter of any one of claims 1-6. wherein the amplifier is a Class-E amplifier, and the wireless power transmiter further comprising an auto-tuning circuit configured to adjust a variable reactance to maintain zero voltage switching across the range of load conditions.

8. A wireless power receiver, comprising:a receiver resonator configured to receive wireless power from a transmiter; a transmiter coupled to the receiver resonator which induces wither a radiofrequency (RF) voltage or an RF current in the receiver resonator;a rectifier connected to the receiver resonator, the rectifier being active and comprising a switching devic;a low-pass filter network connected between the receiver resonator and the rectifier, the low-pass filter network configured to perform a Mobius transformation on an output impedance of the receiver resonator; anda gate drive control circuit configured to generate a gate drive signal for the switching device of the rectifier, wherein the low-pass filter network is configured to maintain a DC output voltage from the rectifier which remains substantially constant across a range of DC load conditions with the transmiter.

9. The wireless power receiver of claim 8, wherein the low-pass filter network comprises a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation.

10. The wireless power receiver of claim 9, wherein the low-pass filter network comprises at least two inductors and at least two capacitors.

11. The wireless power receiver of claim 8, wherein the switching device is a metal-oxide-semiconductor field-effect transistor (MOSFET) transistor.

12. The wireless power receiver of claim 8, wherein the gate drive control circuit is configured to maintain zero voltage switching (ZVS) of the switching device.Attorney Docket: 100502-210013. The wireless power receiver of any one of claims 8-12. wherein the rectifier is a Class-E active rectifier, and the wireless power receiver further comprises a DC-to-DC converter connected to an output of the Class-E active rectifier.

14. The wireless power receiver of claim 13, wherein the DC-to-DC converter is configured to regulate an output voltage or current to a load.

15. A method of wireless power transmission, comprising:generating, by an amplifier comprising a switching device, a radiofrequency (RF) signal from a DC power source;filtering the RF signal through a low-pass filter network configured to perform a Mobius transformation on an output impedance of the amplifier, wherein the filter function and Mobius transformation takes an output impedance of the amplifier and transforms the output impedance to a straight line in an output impedance plane of the low-pass filter network; and delivering, to a load connected to an output of the low-pass filter network, a substantially constant output current or voltage across a range of load conditions.

16. The method of claim 15, wherein the low-pass filter network comprises a plurality of inductors and capacitors arranged in a predetermined topology to perform the Mobius transformation, and the load comprises a transmitter coil.

17. The method of claim 16, further comprising transmitting, by the transmitter coil, a magnetic field based on a filtered RF signal output from the low-pass filter network.

18. The method of claim 16, wherein the low-pass filter network comprises at least two inductors and at least two capacitors.

19. The method of claim 15, wherein the switching device is a metal-oxide-semiconductor field-effect transistor (MOSFET) transistor.

20. The method of claim 15. further comprising providing, by a gate drive circuit, a square wave signal to control the switching device.Attorney Docket: 100502-210021. The method of claim 20, further comprising maintaining, by a feedback loop circuit, zero voltage switching (ZVS) of the switching device.

22. The method of claim 15, wherein the amplifier is a Class-E amplifier.

23. The method of claim 22, further comprising adjusting, by an auto-tuning circuit, a variable reactance to maintain zero voltage switching across the range of load conditions.

24. The method of claim 15, wherein maintaining the substantially constant output current or voltage comprises mapping an input impedance point of the low-pass filter network to infinity at an output of the low-pass filter network.

25. The method of any one of claims 15-24, further comprising maintaining a stable magnetic field strength of the transmitter independent of changes in loading conditions.

26. A low-pass filter network, comprising:a first port and a second port;a plurality of inductors and capacitors arranged in a predetermined topology that performs a Mobius transformation on an input impedance, wherein the Mobius transformation transforms a circular impedance locus at the first port of the low-pass filter network to a substantially straight line in an impedance plane of the second port of the low-pass filter network;a switching device connected to the first port of the low-pass filter network; and a DC power source or a DC load connected to the switching device, wherein the switching device converts RF power to DC power or DC power to radiofrequency (RF) power;wherein the circular impedance locus at the first port of the low-pass filter network is conjugate-matched to a circular locus of RF impedance points of the switching device as a function of power.

27. The low-pass filter network of claim 26, wherein the plurality of inductors comprises at least two inductors connected in series along a signal path, and the plurality ofAttorney Docket: 100502-2100capacitors comprises at least two capacitors connected between nodes along the signal path and a ground reference.

28. The low-pass filter network of claim 26, wherein the low-pass filter network is a four-pole low-pass filter.

29. The low-pass filter network of claim 26, further comprising a series LC resonator connected at the second port of the low-pass filter network, the series LC resonator comprising a series inductor and a series capacitor.

30. The low-pass filter network of claim 26, wherein the low-pass filter network is configured to attenuate harmonic frequencies of a fundamental operating frequency of the switching device.

31. The low-pass filter network of claim 26, wherein the substantially straight line in the output impedance plane has a substantially constant imaginary part across a range of real impedance values.

32. The low-pass filter network of any one of claims 26-31, wherein the low-pass filter network is connected between a Class-E amplifier and a transmitter coil in a wireless power transmitter circuit, and wherein the low-pass filter network is configured to maintain a substantially constant output DC voltage or DC current to a DC load across a range of loading conditions.

33. The low-pass filter network of any one of claims 26-32, wherein the low-pass filter network is connected between a receiver coil and a Class-E active rectifier in a wireless power receiver circuit, and wherein the low-pass filter network is configured to maintain a substantially constant input voltage or current to the Class-E active rectifier across a range of coupling conditions.

34. The low-pass filter network of any one of claims 26-33, further comprising a feedback loop circuit configured to detect a zero voltage switching condition of a switching deviceAttorney Docket: 100502-2100and adjust a variable reactance, an operating frequency, or a phase based on the detected zero voltage switching condition.

35. The low-pass filter network of any one of claims 26-34. further comprising a phase-locked loop circuit configured to lock a gate drive signal of a switching device in a receiver to a phase of a transmitter, wherein the phase-locked loop circuit comprises a voltage-controlled oscillator and a zero-voltage switching (ZVS) detector.

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