CROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Patent Application No. 61/764,019, filed February 13, 2013 and titled SYSTEM AND METHOD FOR IMPROVED WIRELESS POWER TRANSFER, the content of which is hereby incorporated herein by reference in its entirety.

TECHNICAL FIELD

[0002] The present disclosure is directed towards a system and method for wireless power transfer.

BACKGROUND

[0003] Wireless power transfer (WPT) via magnetic coupling is seen as an effective way to transfer power over relatively large air gaps. Numerous applications of the technology have been considered, including charging batteries of portable electronics and electric vehicles. This inductive transfer method usually includes a source and a receiver. However, electromagnetic field emissions in uncoupled portions of the system should be minimized to meet emission and efficiency standards, thus preventing unnecessary power losses. The emissions losses are further exaggerated when the frequency of the power source is not constant.

[0004] Accordingly, for at least the aforementioned reasons, there is a need for improved systems and methods for wireless power transfer.

SUMMARY

[0005] This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description of Illustrative Embodiments. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.

[0006] Disclosed herein are systems and methods for wireless power transfer. The power transfer system includes a receiver configured to wirelessly receive power for powering an electronic device, a power source, and at least one transmitter operably coupled to the power source for wirelessly transferring power generated by the power source. Wherein when the at least one transmitter is operably coupled to the receiver, the power source and the at least one transmitter operate together in a first mode such that the power source generates power at a first level and the at least one transmitter transfers the generated power to the receiver. Wherein when the at least one transmitter is not operably coupled to the receiver, the power source and the at least one transmitter operate together in a second mode such that the power source generates power at a second level and the transmitter does not wirelessly transfer power.

BRIEF DESCRIPTION OF THE DRAWINGS

[0007] The foregoing summary, as well as the following detailed description of preferred embodiments, is better understood when read in conjunction with the appended drawings. For the purposes of illustration, there is shown in the drawings exemplary embodiments; however, the presently disclosed invention is not limited to the specific methods and instrumentalities disclosed. In the drawings disclosed throughout the detailed description, one or more systems having a power transfer device that efficiently utilizes power transfer from an energy source with minimal waste are provided.

[0008] FIG. 1 is a block diagram of a wireless power transfer system according to embodiments of the present subject matter;

[0009] FIG. 2 is a circuit diagram showing the source track and the series-parallel compensated LCC receiver;

[0010] FIG. 3A is a graph showing that current gain of parallel LC receiver is negative;

[0011] FIG. 3B is a graph showing that current gain of using the LCC receiver is a positive value;

[0012] FIG. 4A is a graph showing efficiency of an example system as a function of the load resistance, which controls the voltage gain;

[0013] FIG. 4B are graphs that show the increase in Q _to tai as the current gain reduces;

[0014] FIG. 5 is a circuit diagram showing the reduction of the switching loss of inverter switches generated by the reactive current of the uncoupled source tracks when multi source tracks are connected to the inverter in parallel;

[0015] FIG. 6A is a graph showing the series-compensated source track in a coupled condition with the receiver; [0016] FIG. 6B is a graph showing the series-compensated source track in an uncoupled condition with the receiver;

[0017] FIG. 7A-D are graphs showing input current of an inverter and source track coil in both the coupled and uncoupled condition;

[0018] FIG. 8A is a circuit diagram of a multi-resonant receiver with an LC filter at the output;

[0019] FIG. 8B is a circuit diagram of a multi-resonant receiver with the LC filter and the load replaced with an ideal current source;

[0020] FIG. 9 is a circuit diagram of a receiver equivalent circuit, the rectifier, the filter and the load are replaced with equivalent resistances;

[0021] FIG. 10A is a graph showing a timing diagram of the voltages for Zone 1 operation with emphasis on zero-voltage crossings of V _ac (t);

[0022] FIG. 10B is a graph showing a timing diagram of the voltages for Zone 3 operation with emphasis on zero-voltage crossings of V _ac (t);

[0023] FIG. 11 A-C are graphs showing a timing diagram of a Fourier expansion of the current I _ac for 11 A mI, _ac =l/3, 1 IB mI, _ac >l/3 and 11C mI, _ac <l/3;

[0024] FIG. 12 is a graph showing a zero-crossing angle θ _ζ , and voltage harmonic ratio mv,ac vs current harmonic ratio

[0025] FIG. 13 is a graph showing a timing diagram of I _ac ,i and I _ac ,3 for current ratio

[0026] FIG. 14 is a graph showing a timing diagram of V _oc ,i , V _oc ,3, V _ac ,i , and V _ac ,3 for current ratio (Zone 2);

[0027] FIG. 15 is a graph showing a timing diagram of I _ac ,i and I _ac ,3 for current ratio

[0028] FIG. 16 is a graph showing a timing diagram of V _oc ,i , V _oc ,3, V _ac ,i , and V _ac ,3 for current ratio (Zone 1);

[0029] FIG. 17 is a graph showing a timing diagram of I _ac ,i and I _ac ,3 for current ratio

[0030] FIG. 18 is a graph showing a timing diagram of V _oc ,i , V _oc ,3, V _ac ,i , and V _ac ,3 for current ratio (Zone 3);

[0031] FIG. 19 is a graph of simulation and analytical results of normalized equivalent resistances Ra _C ,i and Ra _C ,3;

[0032] FIG. 20 is a graph of simulation and analytical results of normalized power

[0033] FIG. 21 is a graph of voltage harmonics versus duty ratio D of a phase controlled inverter;

[0034] FIG. 22 is a circuit diagram of an exemplary model of the primary (transmitter) of an IPT system;

[0035] FIG. 23 is a circuit diagram of an exemplary detailed model of a primary

(transmitter) with L-C-L-C compensation circuit in the form of a Cauer 1 ladder network; and

[0036] FIG. 24 is a graph of a timing diagram of the transmitter's signals.

DETAILED DESCRIPTION

[0037] While the disclosure of the technology herein is presented with sufficient details to enable one skilled in this art to practice the invention, it is not intended to limit the scope of the disclosed technology. The inventors contemplate that future technologies may facilitate additional embodiments of the presently disclosed subject matter as claimed herein. Moreover, although the term "step" may be used herein to connote different aspects of methods employed, the term should not be interpreted as implying any particular order among or between various steps herein disclosed unless and except when the order of individual steps is explicitly described.

[0038] WPT via magnetic coupling is seen as an effective way to transfer power over relatively large air gaps. Numerous applications of the technology for providing power to electronic devices may be implemented via the systems and methods disclosed herein. For example, the systems and methods disclosed herein may be used for, but not limited to, charging batteries of portable electronics and electric vehicles.

[0039] In accordance with embodiments, FIG. 1 illustrates a block diagram of a wireless power transfer system. The system provides sectionalized (referred to as "lumped") source coils that are commensurable with the receiver to maintain the high coupling coefficient. This approach may be used to solve both the efficiency and the emission challenges in dynamic charging applications. Referring to FIG. 1, a source track 102 includes multiple source track coils (also referred to herein as a "transmitter") 104 that cover the entirety or a substantial amount of the area where the receiver coil 106 is expected to be placed with each unloaded coil 108 commeasurable with the receiver coil 106. The source track coils 104 can be compensated so that the coil resonance occurs at a frequency offset from the operating frequency of the system 100 in unloaded or uncoupled coils of the source-receiver. The result may be a relatively weak field in the uncoupled coil sections. When the receiver coil 106 is strongly coupled with the source track 102, the magnetic field 110 of the source track may be automatically increased to transfer the power required from the receiver coil 106 as the resonant frequency of source track 102 is brought to the operating frequency through the reflected reactive load from the receiver coil 106. Additional details are provided in the present disclosure.

[0040] In wireless power transfer systems, the parallel compensated receiver may be used since it boosts the voltage to the load and it can be easy to decouple with source track by controlling the switch of the receiver driver. The load impedance reflected back onto the source track by the parallel compensated receiver can be described as:

[0041] Typically, the reflected reactance results in higher power supply VA rating, and therefore, increased inverter current, without contributing the real power to the load. This is considered as the disadvantage of the parallel compensated receiver. In order to overcome this disadvantage, the LCL compensated receiver, which reflects purely a real load onto the source track and a unity input power factor may be employed. The system can be configured to have a substantial reactive power reflected onto the source, and to use this property as a way to tune the source circuit.

[0042] FIG. 2 shows the source track and the series-parallel compensated LCC receiver. The resonant coil of the LCC receiver is series-parallel compensated with capacitors C 1 and C2 to form a resonant tank.

[0043] The values of two capacitors are chosen for the resonance of receiver to occur at the operating frequency of system as described in (2)-(4):

Ci \\ C ₂ = C (4)

[0044] The resonant frequency of the receiver is derived as: [0045] The quality factor of the system can be calculated by finding the current and voltage boost factor (Qi & Qv). I _sc is the short circuit current of the receiver coil and the current boost factor of receiver (Qi) can be defined as:

^2

n (6) lSC L ₂ - ) ² Cl

[0046] Here It is the input current of rectifier. The voltage boost factor of receiver (Qv) can be defined as:

Q - ^v L ₌ ^ . _{n =} Q . _{n (7)} v Voc ωΐ ₂ ^ ^w

Vix,ad is the input voltage of the rectifier, and Q is the quality factor of the parallel compensated receiver. The quality factor (Qtotai) of the LCC receiver is obtained by multiplying Qi and Qv:

Qtotai = Qi - Qv = ^r - n ² = Q - n ² (8)

[0047] From (8), the quality factor of the series-parallel compensated LCC receiver depends on the ratio (n) of capacitors used for forming resonant tank, and is different with that of traditional parallel compensated receiver. In addition, the current of the LCC receiver can be defined as:

I in = e ^■ n + e ^■ Qtotai ^■ j (9) It = Isc - n (10) Ic ₂ ⁼ he " Qtotai ' j (1 1)

[0048] Here Ii _n is the coil current, I _t is the input current of rectifier, Ic2 is the current of parallel capacitor. Rioad can be replaced by the effective load (R _eq ) seen by resonant tank before the rectifier. The value of Re _q is determined by:

Req ⁼ ~ ^Load (12)

[0049] The impedance (Zm) of the LCC receiver seen by the open circuit voltage is calculated as:

Z _in = ja>L ₂ +— + (R _eq II— ₌ ^l2∑ll _{( 13)}

[0050] The load impedance reflected back onto the source track by the LCC receiver can be defined by: [0051] It is noticed that if n is chosen to be greater than one, the amount of reflected reactive load from the LCC receiver can be larger than that of the parallel compensated receiver obtained in (1). It means that the LCC receiver can shift the resonant frequency of the source track more than the parallel compensated receiver can. In FIG. 2, the source track coil 104 is

compensated with C _s as considering the reactive load reflected from the receiver.

η · ) = ΔΧ (v ]ω1 ₅ + - _Γ - ΔΧ = 0) (15)

[0052] Here, ΔΧ is the amount of reactive load which may be reflected from the LCC receiver in the coupled condition. When the source track is coupled with the LCC receiver, the impedance of the s

(16)

[0053] Here R _r is the real load reflected from the receiver. This is because all reactive components are cancelled out by the reflected reactive load at the operating frequency. The reflected load from both the LCC receiver and parallel compensated receiver is summarized in Table 1. THE SOURCE

[0054] The current gain of the source track in case of using the parallel LC receiver is calculated as:

Gain[dB] = 20 log ₁₀ . ^eceive . = 20 log ₁₀ ± (17)

\ ^L no receiver \ ¾

[0055] The current gain is always a negative value since a quality factor in wireless power transfer systems is chosen to be greater than one. In contrast, the current gain of the LCC receiver can be a positive value if the ratio (n) of series and parallel capacitors is chosen to be larger than the quality factor. The current gain of the source track in case of using the LCC receiver is described as:

Gain[dB] = 20 1og ₁₀ —— (18)

Qtotal [0056] As shown in FIG. 3 A, the current gain of parallel LC receiver is a negative value. In contrast, FIG. 3B shows that the current gain in case of using the LCC receiver is a positive value. As a result, the electromagnetic field emission can be limited since the current flowing in the source track is reduced in the uncoupled condition. However, the ratio (n) cannot be increased up to very high value because the efficiency of the receiver decreases due to the increase of a circulating current by the ratio (n). Therefore, all parameters of circuit should be chosen to maximize the current gain of source track while the system satisfies the expected overall efficiency. FIG. 4A shows the efficiency of the proposed system as a function of the load resistance (which controls the voltage gain). It may be concluded that a larger voltage gain would be beneficial. On the other hand, as shown in FIG. 4B it is apprarent that with the increase in Qtotai the current gain reduces, making the difference in current between the coupled and uncoupled sections ever smaller for higher quality factors. Therefore, a tradeoff between efficiency and current gain may be needed.

[0057] The source track 500 can reduce the switching loss of inverter switches generated by the reactive current of the uncoupled source tracks when multi source tracks are connected to the inverter in parallel. The configuration of the source track 500 is shown in FIG. 5.

[0058] The basic principle is to make the input impedance (Zi _n ) 502 to be very high in the uncoupled condition with the receiver 106. A LC series filter 504 and parallel capacitor 506 are added to the series compensated source track 500. This resulting impedance is:

[0059] The value of Csi is chosen to make the denominator of impedance (Zi) to be zero at the operating frequency. Then, the impedance seen by the inverter 508 becomes very high. An additional bandpass LC filter (formed by LF and CF) is added to eliminate the high order harmonics. Therefore, the inverter current may be almost zero. When the source track 500 is coupled with the receiver, the impedance (Z ₂ ) may be changed as a pure real load by the reflected load from the receiver. At this condition, the impedance (Zi) is defined as:

Zx = , ^Rr ,2 ^ Rr - ⁾ C _Si R _r j (20) l+(a)C _Sl R _r ) l+(o)C _Sl R _r )

[0060] Here, R _r is the reflected real load from the receiver. In (20), the impedance has a little capacitance component. This can be removed as making the LC series filter to have a sufficiently small inductance component at the operating frequency. As a result, the coil current as well as the inverter current can be automatically controlled by using the source track 500 because the impedance (Zi, Z ₂ ) is changed as the coupling condition.

[0061] As an example, the parameters are selected as listed in Table II. The operating frequency of the source power supply is 100 kHz. The experiment was implemented at the output power of 300W. To validate the effect of using the series-parallel LCC receiver, the coil current of the source track was measured under both the uncoupled and coupled condition with the receiver.

TABLE II THE SYSTEM PARA ETERS

[0062] As an example, FIG. 6 shows the current and voltage of the series-compensated source track at the load resistor (4.2Ω). FIG. 6A shows the series-compensated source track in a coupled condition with the receiver. FIG. 6B shows the series-compensated source track in an uncoupled condition with the receiver. The current in the coupled condition is 9.25 A, and the current in the uncoupled condition is 2.82A. The current gain of the source track coil is 10.32dB at the coupling factor (k=0.224). The quality factor of the receiver is 1.50. The overall efficiency of system is 82.03%. When the load is changed to 6.24Ω, the current in the coupled condition is 6.62 A, and the current in the uncoupled condition is 2.72 A. The current gain of the source track coil is 7.73dB at the same coupling factor. The quality factor is 2.23. The efficiency of system is 84.77%. This means that the efficiency can be improved as increasing the quality factor, even though the current gain is decreased, in this example.

[0063] FIGs. 7A-D shows the exemplary input current of inverter and source track coil in case of using the new proposed source track. In the uncoupled condition, the inverter current is reduced up to 511mA. The current gain of source track coil is 10.46dB. The efficiency of system is 80.45%). It is decreased a little compared to the serise compensated source track because of additional capacitors and inductor. [0064] The WPT system has been designed by using a series compensated source track and a series-parallel compensated LCC receiver. The advantage of using the LCC receiver has been described and compared to parallel compensated receiver in terms of reflected load to the source track. In addition, the source track is developed to reduce the switching loss of the inverter switches generated by the reactive current of the uncoupled source tracks when multi source tracks are connected to the inverter.

[0065] Selection of an optimal signal frequency is one of the most complex problems in the design of a system for wireless power transfer. Indeed, regarding the same previous derivation and expressions, a higher frequency leads to a better design with a higher power transfer. Although it is true in terms of transferred power, the situation changes when the second order effects are taken into consideration, since the overall efficiency reduces. A part of the problem lays in the fact that the power converters that would process high power at very high frequency are not readily available. If they exist, they typically results in either low efficiency or complicated and unreliable design. The direct impact of the higher switching frequency is the increase of switching losses at the primary (transmitter) side, and even so to some extent at the receiver side, if the power is rectified and delivered to the load in a controllable manner. It may be beneficial to somehow detach the coupling that exist between the switching frequency of the power converter and the frequency of the magnetic field, with the former kept as low as possible for selected signal frequency.

[0066] Another problem may be the lack of signal multiplex when the power is transferred wirelessly. While this approach is almost necessity when the communication signals are distributed both by wire or wirelessly, there was no design that have applied that for wireless power transfer. Indeed, all available designs tried to filter out any input signal of "unwanted" frequency and augment the targeted one for the power transfer.

[0067] As described herein, this WPT system solves both the problems described above. The system is able to preserve a low switching frequency by engaging the switching harmonics in the power transfer as well, the same ones that have been filtered out as described herein. The wireless power multiplex is allowed by redesigning the resonant circuits at both the primary and secondary to resonate at more than one frequency. The multi-resonant topologies reported applications only included the design of multi -resonant analog filter inductors integrated on printed boards or the systems for energy transfer between two reactive elements. According to our best knowledge, the multi -resonant topologies have not been applied for the wireless power transfer so far. The part of the explanation for the deficit is that the rectifier system at the receiver side makes the whole system nonlinear which introduces some difficulties in the design procedure. This particularly issue is described herein with a solution for the system that exploits the first and the third harmonic is demonstrated.

[0068] A thorough analysis of the multi-resonant system is provided. Besides the analytical description, Simulink-based numerical simulations are included to prove the proposed design procedure. As described herein, a design of a multi resonance receiver is explained. The reader should note the derivation of the expressions of equivalent resistances that corresponds to two harmonics applied for the power transfer, and which can be used to replace the nonlinear rectifier at the output. That way the circuit is linearized and the well-established Laplace's concept can be used for the resonant circuit design. Additionally, a ladder LC Cauer 1 topology of the compensation circuit is used and formulas for inductance and capacitance values are derived, as functions of the load characteristics and open-circuit voltages. As an example, this particular topology has been selected due to very low sensitivity to non-exact value of its components, which is a feature for the IPT designs.

[0069] A similar ladder LC Cauer 1 topology is used at the primary side, and its design is described herein. In this description, the goal was to design a load independent track current, but this time that resonates at two or more resonant frequencies (signal harmonics).

[0070] As described herein, a framework of a multi-resonance receiver design may be given. The analysis that follows primarily treats the receivers resonating at two signal harmonics, but the analysis can also be generalized for more than two. Although the main objective is to select a topology and derive formulas for resonant circuit design, the central part of this section is derivation of equivalent resistances R _ac ,i and R _ac ,3 that allow apparent linearization of the receiver circuit and application of the Laplace transform.

[0071] To focus our attention to other important aspects of the resonant tank designing procedure, the output power conditioner may be replaced with a simple LC filter. Inclusion of a boost or some other converter between the rectifier and the load would only simplify the design since it would allow an arbitrary equivalent resistance Rdc for specified output power and voltage conditions, which would not be the case for an LC filter applied. A parallel compensation tank Ci- L2-C2 800 in a π configuration has been chosen, as it is shown in FIG. 8A. Combined with the self- inductance of the receiving coil L \ 802, the resonant circuit creates an L-C-L-C ladder

configuration. [0072] Assuming that the cut-off frequency of the LC filter is sufficiently small enough compared to the signal frequency, and accepting large value of the filter inductance Lf 804, the harmonics in the he 806 can be neglected and the average value of Vdc 808 voltage would be approximately equal to the output voltage:

^ dc ^~ ^dc,avg ^ out ' "

[0073] Consequently, the filter and the load can be replaced with an ideal current source 810, as it is exemplified in 8B:

[0074] The circuit may be further simplified by replacing the rectifier and current source such that the circuit would properly model the power transfer from the ac to the dc side of the system. Although it was an easy task for single-frequency systems (resulting in ² l%*Rdc equivalent resistance), it is much more difficult if more than one frequency are present at the input of the rectifier. One of the difficulties may be the calculation of the dc value of the rectified voltage as a function of two ac signal at the input of the bridge. Indeed, the rectified voltage may depend on both the amplitude ratio and phases of the two harmonics present at the input of the rectifier: ( ²⁴ )

Θ kΛ /

where the phase of harmonic n is selected to be zero (θ _η = ) without losing the generality of the approach, and n and k are two odd numbers (n<k). As may be explained later, cases when these two signals are in phase or anti phase ((%=0 or (%=π) may be particularly interesting. By introducing the parameter mv.ac as the ratio of the ac voltage harmonics at the input of the rectifier, the expression for V _ac can be written as:

« , _ac = ±¾ (25)

ac,n c (ή = c,n, _m ^H ^{n0 + m} v, _ac sin(*fl¾i)) . (26)

[0075] The positive sign in (25) corresponds to in-phase harmonics V _ac ,n and V _ac ,k, where the negative sign describes the anti-phase case. These two cases are analyzed below for the specific situation where the first and the third harmonics are applied, and the values for the equivalent ac resistances R _ac ,i and R _ac ,3 has been derived there as the function of parameter mv,ac and the load resistances RL. Assuming that value for R _ac ,p is known, the circuit can be simplified even more, as it is shown in 9. As one can see, R _AC 900 might be different for different harmonics, which is denoted in by using p in the subscript.

[0076] The voltage signal at the input of the diode rectifier consists of the first and the third harmonics:

[0077] The voltage ratios of interest belong to the Zone 1 where mv, _a ≥\ (mi, _a c≥l/ ), and Zone 3, where mv,ac≤- .395 (mi,ac≤ ). In both cases, voltage V _ac (t) may have total of 6 zero- crossings in one period, as it was demonstrated in 10A for Zone 1 , and FIG. 10B for Zone 3. In simulated examples that have generated the described figures, 1 V-diode voltage drop is introduced to allow simultaneous isualization of both V _ac (t) and rectified V _ac (t). For the rectified voltage:

it is not difficult to derive the avera e value over a period by exploiting the symmetry of the V _ac {t):

^ _,» ≤ -0-395) = -}|Γ _« (<?) <?

[0078] The final expressions for both cases can be written as:

where "+" sign belongs to the Zone 1 solution, while the solution for the Zone 3 includes the sign "- " in front of the expression.

[0079] At the dc side, the output power can be expressed as a function of an equivalent dc resistance Rdc which is equal to the load resistance RL if an ideal LC filter is applied:

[0080] At the ac side, R _ac ,\ and R _AC ,3 may be used to model the individual contribution of each harmonic to power transferred to the load:

[0081] Since the R _ac ,i and R _AC ,3 are not independent:

the total ac power P _ac can be expressed in terms of power delivered at the first harmonic P _a voltage and current ratios:

[0082] After neglecting the losses in the bridge, the ac and dc powers may be the same. It allows a relation to be established between the e uivalent ac and equivalent dc resistances:

[0083] Combination of (31), (34), and (35) may result in wanted expressions for equivalent ac resistances:

[0084] Since Vdc,av _g appears in (36) as a squared variable, the effect of its sign is canceled so the expressions for both Zone 1 and Zone 3 parameters may be identical.

[0085] Although this replacement does not make the circuit linear (do not forget that the equivalent resistance R _a c, _P is now frequency dependent), it allows Laplace operator s to be used to formalize the circuit description at particular frequency. Since the resonant tank behaves as a current source and supplies current to the bridge, transconductance G(s) may be selected as the most appropriate function to describe the characteristics of the tank. With continued reference to FIGs. 9 and 10A-B, it can be written as a function of operator s and circuit parameters:

1

G(s) = ^l ₌ ^„ ) (39)

[0086] For an unloaded system the voltage V _ac = Rac.pLc may resonate at harmonics ηωι and kcoi if the four imaginary poles of the voltage gain:

1

correspond to ±/ ^' (ηωι) and It may be true if the following conditions are satisfied:

(41)

1

(42)

[0087] If (41) and (42) conditions are fulfilled, the transconductance G(s) becomes load-independent,

^G (*) = Γ-7 ( ⁴³ ) which makes possible to command the current of harmonics n and k. After replacing the Laplace operator s with j<x>:

G(j<o) = jG j = 3 ¹ (44) one can see that transconductance gain G( co) contains only the imaginary component G, shifting that way the phase of the input voltage harmonic backward or forward for 90°. The gain changes its sign from a negative to a positive for an increasing ω at the anti-parallel resonant frequency: co _r = ■ ■ (45)

[0088] Depending on the position of the particular harmonic frequencies, the gain might be positive or negative. Let us follow a general approach and assume that algebraic values of the gains at resonant frequencies ηω\ and have two arbitrary values G _n and Gk'.

1

(46)

L _l L ₂ C _l n ³ a^ - ηω _ι (Z^ + Z, ₂ )

L^C^co - kco _l {L _X + L ₂ )

[0089] From the previous expressions, the expressions of the parameters of the ladder resonant circuit ma be extracted:

[0090] From (41), (42), (48), and (49) the resonant tank elements can be determined as the functions of selected harmonics and transconductances:

(k ² - n ² )G _k G _n

(53)

a nk(nG _n - kG _k ) ^' [0091] After studying the previous four equations, it is easy to see that they can result in positive values for circuit elements only if G _n and Gk have different signs. Taking into account the shape of the G(co) curve, it would be possible if the resonant frequencies (ηωι) are at the different sides of the anti-parallel resonance <x> _r which results in G _n < 0 < Gk. It is not difficult now to see that the previous relation is also a sufficient condition to have all positive values of the designed inductors and capacitors.

[0092] The equations (50) - (53) represent a sufficient set of equation for the receiver design and do not depend on R _a c, _P , which makes the proposed design to be load-independent.

However, they assume that coil inductance L\ is a designing parameter, which may not be true if the objective is to develop a compensation circuit for an existing receiving coil. In that case, the frequency ω\ might be varied to satisfy (50). If even the ω\ is predetermined and cannot be adjusted, the system does not have enough parameters to control both resonant frequencies and transconductances. Then, only the ratio between the two transconductances could be specified, and the equation (50) could be then used to calculate actual transconductances.

[0093] To complete this general analysis, let us derive the expression for the input impedance of the resonant tank and the approximate values of the resistances and reactances transferred back into primary circuit at resonant frequencies ηωι and The input impedance, expressed in terms of the circuit parameters is:

z ₍ x ^L 2 ^C i ^C 2 ^R a + ^L 2 ^C + AQ + L ₂ C ₂ ) s ² + ( + L ₂ )s + R _ac

i ^{n [S )} L ₂ C _X C ₂ R + L ₂ C + (R _ac C _{2 +} C _l R _ac ) _{S +} l

[0094] At resonant frequencies (54) becomes simpler since even order terms in the nominator disappear and odd terms constitute the transconductance G(s): i ^{n [S "} G (s) L ₂ C _x C ₂ R _ac + L ₂ C _lS ² + (R _ac C ₂ + ^ Js f 1 ^'

[0095] The next step is to derive the expressions for the impedances reflected to the primary side of the system, by using an approximate formula:

where M is the mutual inductance of primary and secondary coils. It may result in the following expressions:

R _ref A ) = ( ) ² M ² G _n ² R _ac ^ (57)

[0096] Although these general expressions for a two-frequency systems are useful, the analysis that follow may be focus on one specific system: the system that exploits the first and the third harmonic of the input signal to transfer the power to the load (n=l, k=3). Since this is, according to our best knowledge, the first system that uses more than one frequency to transfer power wirelessly, the selected frequencies represent a natural extension from the traditional system that uses only one, base frequency.

[0097] For a specified values of Rac, _P , the harmonics current I _ac ,i and I _ac ,3 are the main quantities that determine delivered power:

^P out = ^P ac _C J ^ - ( ⁶¹ )

[0098] However, they require the transconductances G\ and G ₃ to be selected a priori. If the open- circuit voltages V _oc ,i and V _oc ,3 are known, or at least their ratio mv.oc, by choosing the transconductances, the sharing of the delivered power among the two frequencies may be controlled:

¾ = c Kc t ^V h %3 ^G ¾ ,l ^R a _C ,G v,o - (62)

[0099] For the further discussion, it is important to define the ratio of the currents l _ac\ and Lc,3 and denote it by

- - ^Iac ' ³ (63)

[00100] For the constant open-circuit voltages, a properly designed system may comply designers requirements to generate the specified current ratio mi, _a c At this point, it is important for the reader to note that a diode bridge operation is determined by the voltage at its input side, not by current. Therefore, the resonant tank has to profile its output voltage V _ac to indirectly achieve the required ratio of the current harmonics. [00101] The design procedure, as it is explained above, is able to adjust the magnitude of the transconductance G, while the phase is out of our control and has 90° value for Gk and -90° for Gn. Consequently, the phases of current harmonics l _ac \ and I _ac ,3 are not adjustable parameters, and they are completely determined by the phases of the input voltages V _oc , i and V _oc ,y.

[00102] It is important at this point to mention the following side note: although the calculation of each individual phase angle for the harmonics obeys the well-known rules, one should be particularly careful when comparing the phases of different harmonics. Indeed, some conventions that are valid for the same frequency in terms of the phase shift and mutual signals position may not be in charge when different harmonics are analyzed. For example, phase shift of 90° degrees of both the first and the third harmonics do not maintain the mutual position of the signal, as it would be if they had the same frequency. Luckily, in the discussion below, only two distinctive situations may be interesting. Using the voltage signal V _ac as the example, the first one can be defined as the position of the harmonics that results in positive parameter mv,ac in (26), while the second one is determined by a negative value of the mv.ac The similar definition can be used for other voltage and current multi-harmonic signals. Keeping this addendum in mind, "in-phase" term may be used to describe mv,ac> case, and "anti-phase" to correspond to mv,ac< .

[00103] As described herein, the design of the primary (transmitter) is discussed, it may be clear that V _oc , i and V _oc ,3 can only appear in two distinctive mutual position: they can be in-phase or anti-phase, referring to the definition of these two terms given above. The anti-phase of the voltage harmonics and the phase shift defined by (64) may result in the in-phase current l _ac , \ and Lc,3, while the in-phase input voltage signals may force the same currents to accept anti-phase mutual position.

[00104] The ratio between the third and the first harmonic of a square wave is as shown in 11 A. Consequently, for any ratio different than 1/3, the voltage at the input of the rectifier has to modify the I _ac by inserting an additional voltage zero-crossings, and changing the sign of lac for interval (θ _ζ , π-θ _ζ ) for m/ _ac >l/3, and interval (π-θ _ζ , π+θ _ζ ) for m/ _ac <l/3, as it is exemplified in . In the same figure for each of the periodic current waveforms the expansions into Fourier series are given. [00105] With the understanding of the diode bridge operation, nowexpressions of the equivalent resistances at the first and the third harmonics may be derived. Let us assume that the ratio of the current harmonics mi, _a c and the equivalent dc resistance Rdc=VdclIdc,av _g are known.

[00106] According to the expressions given in 1 IB and 11C, the ratio mi, _a c can be written as:

/ _aC;3 1 l-2cos(30 _z ) _ 1 l-2[4cos ³ 0 _z -3cos0 _z ]

m Ί .ac (65)

I _ac , 3 l-2cos(0 _z ) 3 l-2cos(0 _z )

[00107] By substituting the cosine function with a variable t, a cubic equation may be obtained:

8t ³ + 6t(l + m _Uac ) + 1 - 3m _z (66)

[00108] The previous cubic equation has an analytical solution, and it is given by (67). Exactly one of the zeros is real and in the range t<≡ [0, 1], which corresponds to the angular range ( _z e[0, π/2]. One should be careful while applying the explicate solution (67), since it is valid only while the expression under the square root is positive. When it becomes negative, MATLAB function roots ) or some other numerical method of solving may be an option.

d = 1 2>m,

Had Jon ¹ a d )8a c Had Jon ¹ a d 108a c (67)

3a

. ³ 0.5 V 10.5 27 'a ² d - V27 ² aV ² +108aV

6a 6a

[00109] In order to have the specified ratio between the third and first harmonics mj,ac, the ac current I _ac should have the transition from -hc.avg, to Idc.avg and vice versa at the specified positions inside a period of the signal. However, the current is not the one that forces the diode bridge to switch its state at these particular time instants - it is done by voltage V _ac . The voltage V _a , consists domin the first and the third harmonics in-phase or in anti-phase:

c (0 = (sin(fl¾i) + m _v>ac sin(3fi¾i)) , (68) where mv.ac is positive when the harmonics are in-phase and negative when they are in anti-phase. Voltage zero-crossings are uniquely determined by the ratio of the harmonics amplitudes (or rms values) mv.ac Therefore, mv.ac may be set to an appropriate value to achieve the zero-crossings at positions 0, θ _ζ , π-θ _ζ and periodically further on:

sin(0 _z ) + m _F n (3i? _z ) = O , (69) which results in:

" , _ac = ■ _{2 a} · (70) -3 + 4 sin θ _ζ

[00110] It may be interesting to show the zero-crossing angle θ _ζ and voltage ratio mv,ac as the functions of the designing parameter mi, _ac , and it is shown in FIG. 12. As exemplified in FIG. 12, for mi,ac<l/3 (corresponds to Zone 2 and Zone 3 in FIG. 12), angle θ _ζ is smaller than 600 and consequently the voltage ratio mv,ac is negative. Besides it negative sign, the mv,ac is smaller than 100% which indicate that the third harmonic is smaller than the first one. For mi, _a c>l/3

(corresponds to Zone 1 in FIG. 12), the angle θ _ζ is greater than 600 (implying the positive sign of mv,ac) and third voltage harmonic is greater than the first one.

[00111] The operation of the system in these three zones is demonstrated in the exemplary FIGs. 13 to 18 where timing diagrams of I _ac ,i, Lc,3 , V _oc ,i, V _oc ,3 , V _ac ,i, and V _ac ,3, are shown for (rms), and (rms) during a selected one -period-long time interval. The graphs shown are taken from the exemplary Simulink model of a 500 W, 10 kHz system that may be disclosed in more detail below. The previous observations lead to three very important conclusions:

• Operation in Zone 2 is demonstrated in FIG. 13 and for an exemplary mj, _a c = 0.2 designed value of the current ratio. The diagrams confirm the disclosed above, related to mutual phase position of the signals. 180° phase shift of the signals I _ac ,3 and V _oc ,3 coincides to a negative equivalent resistance (R _a c,3< ) which means that the third harmonic transfers the real power in the opposite direction, from the load to the primary of the IPT system. Certainly, it is not a desirable operational condition, since then the proposed system that exploits two harmonics, transfers less power and produces more losses than a single-frequency-based system.

• When miac is greater than 1/3 (Zone 1), the all four ac signals l _ac ,\, hc,i, V _ac ,i, and V _ac ,3 are in- phase. It results in positive values of R _ac ,i and R _ac ,3 and consequently in proper direction of the power flow at both resonant frequencies. Again, the results of a 500 W 10 kHz Simulink model are used to support this discussion in the similar way it was done for 0<mi, _a c<l/ range earlier. mi,ac>l/3 operation regime is demonstrated in and for .

• For the parameter mi, _a c in Zone 3, each voltage-current pair {l _ac ,\- Vac,\ and Iac,3-V _a c,3) is in phase, although the voltages or current are not by themselves. Consequently, it results again in positive R _ac ,i and R _ac ,3, and a wanted direction of the power flow. This operation mode is illustrated in and . 18 for mi, _a c= -0.5.

• Taking into account the previous observations, portions of the rest of this disclosure may be focused on different cases when mi,ac>l/3 or m/, _ac <0. Since the resonant, design in the identical way for both these two operational regimes, introduces +90° phase shift for the third harmonic and -90° for the first one, one can conclude that only two particular mutual phase positions of the open-circuit voltage result in operation in Zone 1 or Zone 3. Indeed, to make the system operate in-phase after the resonant tank (Zone 1), the open-circuit voltages V _oc ,i and V _oc ,3 have to have phases +90° and -90°, respectively. The operation of V _ac ,i and V _ac ,3 in an anti-phase (Zone 3) requires the phases of the open-circuit voltages V _oc ,i and V _oc ,3 to be 0° and 180°, respectively. As disclosed further herein, the design of the transmitter circuit results in exactly these two phase constellations, which gave us the right to proceed further by analyzing both of these two operational modes.

[00112] The next important step is to derive the expressions for R _ac ,i and R _ac ,3 and determine how the two harmonics share the power transferred to the load as a function of the voltage and current ratios mi, _a c and mv.ac Since the derivation process is quite long, to keep smooth flow of this discussion, it is disclosed below, while in this section only the final expression are rewritten due to completeness:

(73) [00113] Although it seems that R _ac ,i and R _ac ,3 depends on two parameters mi, _a c and mv.ac, it should not be forgotten that there is a direct, one-to-one relation between mi, _a c and mv.ac given by (67) and (70) above. It further means that for a designed system with specified Gi and G3, only the input, open-circuit voltages determine power sharing and resonant resistances.

[00114] In 19 and 20, the analytical and simulation results are presented for R _ac ,i, Rac,3 and Pout. However, instead to plot the absolute values, each of them is normalized: the resistances are normalized by (8/π ² /¾), while the ratio is shown instead of Pout. Exemplary simulation results are extracted from V _ac (t) and I _ac {t) by using Simulink block for Fourier analysis, followed by a block for division of magnitudes of the appropriate voltage and current harmonics. The figures show an excellent matching between the analytical expectations and simulation results.

[00115] Before continuing with a case study of a multi-resonant receiver, let us examine the reflected impedances Z _re f,\ and Z _re /,3 when the first and third harmonics are applied. For n=\ and k=3, the equations (57) - (60) becomes:

i? _re/;1 ( _i¾ ) = « ₁ ^{2 2} G¾ ₁ , (74)

M ⁼ 64G t ^{1 8 Gl Gs + 3 + Gl} ] ' ^{( }}

R _ref , (3 ₁ ) = (3 ₁ ) ² M ² G^R _ac „ (76)

^X ^ ^k ^ ) = ^{{ 3} ^ _G ^{M 2} [ ²⁶ ^3 27 (G* G ₃ ² ] . (77)

[00116] Based on the previous expressions, some interesting observations can be made:

• Th ratio of the resistances reflected back into the primary circuit can be expressed as:

If, for example, the absolute values of the transconductances are the same and nifac is close to 0.9 (where both harmonics deliver approximately the same power), the equivalent transferred resistance at third harmonic may be approximately 11 times greater than the one at the first harmonic: R _ac ,3 ~ 1 lRac,i.

• Reflected reactance at ωι, Xre/ , is capacitive, regardless of the values of transconductances, while Xrefi is capacitive, but for an limited range of transconductances: G3<(9-4D5)*|Gi|. Although the range for X _re f,\ is restricted, it typically includes all designs of a practical importance.

Practical designs very often utilize the identical primary and secondary coils L _P =L\. In that case, the coupling between the coils can be effectively described by using a coupling coefficient kc . M=k _c Lp. For this particular case, it would be interesting to compare the inductance of the primary coil L _p and the reflected "negative inductance" since it would represent an detuning for the rimary circuit. From (75) and (77) the ratio can be written as:

0) for n=\ and k=3, the inductance L _p may be:

[00118] It finally leads to the expression for detuning ratio of the primary inductance of:

[00119] To develop the sense of relative detuning, let us calculate the previous expressions for G ₃ =-Gi and coupling coefficients of _c =0.1 : ,- ₃ =- f, _c 0.1) 0.5% , (84)

— =- Gf, k _c 0.1) 2.5% . (85)

^L P

[00120] Let us apply the theoretical analysis presented in the previous section to design an exemplary receiver for multi-resonant IPT.

[00121] The goal is to design an exemplary receiver that exploits the first and third harmonics to supply to a resistive load of nominal value The self-inductance of the receiving coils is measured at Zi=34.76 μΗ. Although the open-circuit voltages V _oc ,i and Voc,3 depends on the designed based frequency that has been determined yet, let assume that they can be adjusted to V (rms) values by changing the primary current for any base frequency in the range f\ e(9 kHz, 1 1 kHz). It would be desirable to design the system whose quality factors at both resonant frequencies and nominal load stay less than 10 (a value from experience), since it would result in a resonant tank less sensitive to detuning.

Step 1: Determining the equivalent resistances R _ac ,i and R _ac 3:

[00122] Since the induced voltages are equal, it is reasonable to preserve this balance it terms of power delivered to the load at both frequencies as well. Therefore,

will be chosen as the way of sharing power. By reading the value at the abscissa axis for ordinate value of 2 in , it is easy to obtain the value of the parameter mi, _ac that provides equal power sharing (Zone 1 operation):

[00123] The ac current ratio can be now substituted into the set of equations (67) to calculate the zero-crossing position θ _ζ of the total ac voltage:

<¾ [ ^m i,ac = 0.9066) 81.21° . (88)

[00124] Angle θ _ζ can be used as the input for (70) to get the ac voltage ratio mv.ac. m _{V ac} (0 _z = 81.21° ) 1.103 . (89)

[00125] A shortcut to the same outcome of mv.ac might be the application of (73) which would give mv,ac directly, if power and current ratio are known. Finally, the equivalent resistances may be obtained from 71) and (72): 03, R _L 20= ) 23.5$ , (90) R _ac , ( ^m i, _c = 0.907, m _ac fe l03, * _£ 20= ) 301135 . (91)

Step 2: Calculation of transconductances Gi and Gy.

[00126] Since the transconductances represents the ratio of the ac current and open- circuit voltages, the currents from the power delivered may be calculated: 3.126=A , (92)

^R ac l

[00127] The transconductances may be then:

G _{i =} _ i__L QJ _{13 S} _ _{= = GI} j _Gi o.313 S , (94)

G ₃ = -≡ = 0.283 G= jG ₃ yO.283 S . (95)

[00128] At this point it may be advantageous to check the values of the quality factors and Qi at resonant frequencies.

Q ₃ = G ₃ R _ac , 8.82 . (97)

Step 3: Design of the resonant circuit L ₂ , Ci and C2:

[00129] Since in our problem the base frequency is unknown, the equation needs to be rearranged to allow calculation of/i instead of L\

[00130] Now (51) - (53) can be used further to determine the other elements:

L ₂ = 24.87 μΗ

C, = 2.551 μΡ . (99) C ₂ = 3.233 μΡ

[00131] As one way to validate the analytical derivations disclosed herein, a Simulink model has been built and this particular operational condition is simulated. Since there was no an analytical method that may easily include the losses into calculation, they are removed from the Simulink model as well. The result of the analytically obtained values and corresponding values extracted from the simulation results are comparatively shown in Table 3. As, one can see, there is an excellent matching between them, which indirectly proves the presented methodical approach.

[00132] In a similar manner a system can be designed that operates in Zone 3. Table 3: Receiver - a comparative presentation of the analytical and simulation results

[00133] Since the structure and behavior of a multi-resonant receiver is recognized, the next step is to develop a corresponding multi -resonant transmitter structure that may be able to excite receiver coil at certain frequencies and deliver the power. A general topology of the transmitter usually contains a high-frequency power converter, compensation circuit and the primary coil or track. The input power converter generates the excitation voltage that supplies the compensation circuit and the transmitter coil. The output voltage of the converter can be regulated by the phase shift control. The advantage of this control method is that it allows regulation of the voltage and indirectly the current of the primary coil. This may complicate the system and reduces the utilization of the converter rated power which may lead to a reduced efficiency. One or more applications of the phase shift control may be provided in an open loop control configuration to reduce certain harmonics (typically the third one) by selecting a suitable value of the phase angle. To the contrary, the design described herein may exploit the existence of the voltage harmonics.

[00134] With the dead-time interval neglected, the square wave inverter output voltage can be re resented by the followin Fourier series expression:

[00135] From (100) one can see that the voltage harmonics are in-phase or anti -phase and decreases with the harmonic order k. Let us analyze the influence of the duty ratio D to the fundamental and the next two harmonics. The normalized algebraic magnitude of the fundamental, the third and the fifth harmonics are shown in FIG. 21, where the normalization factor has been used to provide an application-independent examination. As one can see, the duty ratio does not affect only the magnitude but also the phase of the harmonics. For example, the fundamental is always positive while the third harmonic becomes negative (changes its phase for 180°) when D gets lower than 2/3. For the future analysis it might be important to draw attention to two particular operational points: D=\ and D=l/3. When D=\, all harmonics are in phase and reach they maximum amplitudes. At D=l/3, the third harmonic is at its maximum, but shifted for 180° degrees, while the first harmonic is reduced by 50% referring to its maximum amplitude.

[00136] The exemplary circuit of a transmitter with that kind of the inverter's model is drawn in due to completeness of the presented material. The primary coil inductance is modeled by inductance L _p 2200, while the Z _re /2202 represents the reflected secondary impedance at particular frequency. Resistive and reactive components of Z _re /2202 are disclosed herein and quantified by (57) - (60) for n ^& and k ^th harmonics of the fundamental frequency i.

[00137] The resonant circuit in FIG. 22 resonates at two or more selected voltage harmonics, allowing multi-frequency current to supply the reflected load. Since the load may vary, a desirable feature of the system would be the load-independence of the primary coil current. It is similar feature to the one looked for with LCC compensation circuit, with the difference that now this independence should be extended to more than one resonant frequency. Therefore, next important step is to select the topology of the compensation circuit, which is suitable for the goals set. The number of used additional reactive components may be kept low (since it results in a compact and easy tunable design), but at the same time to have enough degrees of freedom for tuning. Let us narrow our design to bi-resonant system that exploits only two frequencies to transfer the power. Following the idea of the ladder Cauer's topology as disclosed herein, a similar L-C-L-C ladder is shown in. Compensation elements are represented by Lx-Cx-Li-Ci 2300, the primary coil inductance is denoted by L _p 2302, while i? _re /2304 and X _re f 2306 symbolize the resistance and the reactance reflected from the receiver side (receiver impedance referred to the primary).

[00138] Let us now derive the transconductance G _p that relates the input voltage Vmv 2308 and current in the primary coil I _p 2310: (101)

J

where L _p , _m represents the modified inductance due to reflected reactance X _re f,.

X r,ef ,m

p,m m n.k (102) jmco^

[00139] After rearranging the expression in the denominator it becomes:

1

. + 3) n ) + s ² (Q + Q + i ₂ Q ) + 1) + ¾/, + _S ( + (10

1 ₂ ) ^■

[00140] Evidently from (103), there is a way to make transconductance G _p load and primary-impedance independent, at resonant frequencies ηω\ and Indeed, if the following conditions are satisfied:

[00141] The term s ⁴ Li∑2CiC2+ s ⁴ {L\C\+ L1C2+ Z ₂ C ₂ )+1 becomes zero at both resonant frequencies ηω\ and Conse uently, the transconductance reduces to:

[00142] One can easily see that conditions (104) - (105) and expression for G _p (s) are identical to the expressions disclosed herein for receiver design. Since the design objectives for G _p are the same, (50) - (53) may be reused to calculate the parameters of compensation circuit. The expressions are rewritten here due to completeness:

2 (1 10) mi ^nk { ^{n G} _P ,« - ^kG _P ,k )

where G _p ,„ and G _p ,k represent the designed transconductance at ηω\ and k<x>\. It should be noted by the reader that this approach makes the primary current load independent. Under the term "load" it is assumed not just the reflected resistance i? _re /2304, but also the reflected reactance _re /2306 as long as it is sufficiently small enough not to change the total inductive character of the coil impedance.

[00143] Another important discussion is related to the phases of transconductances. As it was already disclosed herein, selection of G _{p n} < and G _p ,k>0 would result in a feasible set of circuit values Lx-Cx-Li-Ci. Taking into account the phase requirements of the receiver's coil open-circuit voltages (which essentially determines the operation of the receiver in Zone 1 or Zone 3) one can conclude that duty cycle ratio D of the input inverter cannot be picked-up arbitrarily.

[00144] Let us now elaborate the condition that the phase difference of the two engaged signals Vinv.n and Vinv.k (n<k) should satisfy to result in operation in the Zone 1 or Zone 3. On its path from the output of the inverter to the input of the rectifier the signal Vinv.n is shifted in phase three times: by -90 ⁹ through the primary resonant circuit, +90° due to Faraday's voltage induced in the receiving coil and -90° while it has been processed through the receiver's resonant tank. At the same time, the voltage harmonic Vinv.k is shifted three times by the same amount: +90°. Without losing the generality, it can be assumed that phase of Vinv.n at the output of the inverter is zero

Additionally, the whole phase shift of the signal Vinv.n can be taken from the input to the output of the system, scale by factor kin and assign to Vinv.k. In that case the signal Vinv.n is unmoved and corresponding signal V _ac ,n at the input of the rectifier contains zero phase while the signal Vinv.k at the same place has the phase: Vac,k "V,k,0 o "V,k,0

2 2 2 n 2 2 2 ) 2 \ n [00145] Obviously, if the system described herein is used to operate in Zone 1, the phase difference of the voltage signals at the input of the rectifier should be an integer number of the 2π [rad]:

while for the operation in Zone 3 the phase difference needs to be an odd number of π [rad] : θ _{ν 0} +^3 + ^ = (2ί ±) π ί =8,1,2,... (113)

[00146] From (112) and (113), the required initial phase θν, ο that result in a system operating in the particular zone can be derived:

π („ k

Zonel: 6 _vk ^i 2π -^-S -J 4 0,1,2,... (114)

Zone 3: 0 _vkfi ={2i l) π ^3 +- i 3,1,2,... (115)

[00147] As an example, this can be calculated for the harmonics n=3 and k=5, the phase of the Vinv.k for operation in Zone 1 and Zone 3 should be -π/3 and 2π/3, respectively.

[00148] Particularly interesting for practical applications is the case when the fundamental signal (n=l) is used. In that case (114) and (115) become simple:

Zone 1 (n=l): 0 _VkO =2i -π - (3 +k) i =0,1,2,... (116)

Zone3(n=l): 0 _VkO =(2i ±-) π ^(3 *) i $,1,2,... (117)

[00149] After an in-depth analysis, one can see that these expressions becomes even more simpler if the order of the second harmonic is divided into two categories: for k=3, 7, 11,... the required initial phases are:

Zonel(n=l, £=3,7,11,···): d _vk<A =n (118) Zone 3 («=1, £=3,7,11,...): 0 _rkO =O (119) while for the other disjunctive category k=5, 9, 13,... the initial phases have exchanged values:

Zone 1 (n=l, £=5,9,13,...): 6> _i0 =0 (120)

Zone 3 («=1, £=5,9,13,...): θ _ν , ₀ =π (121) [00150] Now it easy to see why the design with n=\ is particularly important: in that case the phase angles (118) - (121) are exactly the angles that can be obtained from a voltage pulse wave at the output of a full bridge phase-shift-controlled inverter. For instance, if the first and the third harmonics are engaged, they satisfy the condition for Zone 1 operation if 0<D<2/3 and conditions for Zone 3 operation if l>D>2/3. Inside these ranges the duty ratio D can be further used to adjust the harmonics magnitudes and regulate the power sharing among them. To determine the magnitudes of transconductances, the desired amounts of transferred power have to be considered:

^P m = { ^V in _v ,m ^G _P ,m f ^R ref,m = (122)

[00151 From (122), the magnitudes of the trans conductance can be calculated as:

where Vmv,m depends on duty ratio D and is defined by:

. ( ιηπΌ . ( ιηπ

sin sin

AV 2 2

^, _m =— ^ ^J ^ m = n, k . (124) π m

[00152] The exemplary design of the 500W, 10 kHz IPT multi -resonant system as disclosed herein, now shows the design of the power transmitter side. As disclosed herein, the exemplary goal is to transfer 500W of power by using the first and the third harmonics (n=l, k=3) of a i=10 kHz system. For this example, it is assumed that the primary and secondary inductances are identical . The high frequency full bridge inverter is supplied from a dc link that generates a stable voltage: ^=169.7 V. All inverter switches and elements of the resonant circuit are assumed ideal (lossless).

Step 1: Calculation of the reflected resistances and reactances R _re f. R _re f3 Xref.i and

[00153] By applying (57) - (60), values for reflected resistances and reactances at resonant frequencies 10 kHz and 30 kHz can be obtained:

R _ref , = 0.1192 , (125)

X _ref = «01 12 , (126) R _ref , = 1.0732 (127)

X _refj = JB1525 (128)

Step 2:Calculation of the transconductances (7 _P ,i<0 and G„,3>0:

[00154] Considering the fact that the first and third harmonic are used and that the receiver has been already designed to operate in Zone 1 , a suitable duty ratio value should be selected. As described herein, this operational conditions requires D from the range (0, 2/3). Let us assume that value D=l/3 is chosen, since it generates the maximum amplitude of the third harmonic. Conse uently, the rms values of the voltage harmonics may be: sm = 50.93 V (rms) (130)

3π

[00155] In order to deliver specified amount of power, the primary coil currents have to be:

l P,3 15.26 A (132)

R e/, 3

[00156] Consequently, the values of the transconductances are: =- 0.6 S (133)

V i,nvA

0.3 3= (134)

V i,nv,3

Step 3: Design of the resonant circuit L L2, Ci and Cr.

[00157] After the transconductances are substituted in (107) - (110), the compensation elements are obtained as: L, = 20.23 μΗ

L _? = 11.85 μΗ

² (135) C _! = 5.84 nF

C ₂ = 5.09 μΡ

[00158] As the simplest way to validate the analytical derivations presented, an exemplary Simulink model has been built and this particular operational condition is simulated. The result of the expected analytical values and corresponding values extracted from the simulation results are comparatively shown in the Table 4. Again, excellent matching between them can be used as an indirect proof of the presented methodical approach. Finally, inverter's output voltage Vinv, primary coil current I _p and its two harmonic components l _p ,\ and I _p are extracted from the simulation results and shown in . As one can see, the error between the simulated and analytically derived values are slightly greater when the whole system is simulated, mainly due to the approximate formula used to calculate the impedances reflected from the secondary to the primary circuit.

Table 4: Transmitter - a comparative presentation of the analytical and simulation results

[00159] While the embodiments have been described in connection with the preferred embodiments of the various figures, it is to be understood that other similar embodiments may be used or modifications and additions may be made to the described embodiment for performing the same function without deviating therefrom. Therefore, the disclosed embodiments should not be limited to any single embodiment, but rather should be construed in breadth and scope in accordance with the appended claims.