A Systematic Approach for Load Monitoring and Power ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 30, NO. 3, MARCH 2015

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A Systematic Approach for Load Monitoring and Power Control in Wireless Power Transfer Systems Without Any Direct Output Measurement Jian Yin, Deyan Lin, Member, IEEE, Chi-Kwan Lee, Senior Member, IEEE, and S. Y. Ron Hui, Fellow, IEEE

Abstract—A systematic method is presented in this paper to show that, based only on the measurements of the input voltage and current, the load impedance of a wireless power transfer system can be instantaneously monitored and load power controlled without using any direct measurement from the load. A new mathematic procedure for deriving the output load information based on input voltage and current is explained. This systematic method, which can be applied to wireless power systems with two or more coils, eliminates the need for sensors and communication devices on the load side, thereby greatly simplifying the power control circuitry. The principle of the load estimation method, the power loss optimization and control scheme are described and favorably verified with measurements obtained from an eight-coil wireless power transfer system. Index Terms—Load estimation, magnetic resonance, power control, wireless power transfer.

I. INTRODUCTION IRELESS power transfer based on the magnetic resonance and near-field coupling of two loop resonators was reported by Nicola Tesla a century ago [1]. Research investigations about transcutaneous energy systems for biomedical implants [2]–[8], the inductive power transfer systems [9], and wireless charging systems for portable equipment such as mobile phones [10] have been reported. Recently, a lot of research efforts have been devoted to the improvement of the performance of wireless power transfer systems such as to increase the transfer distance and to improve the overall system efficiency. Most of the projects have been extended from the traditional two-coil systems to systems with more than two coils. For examples, three-coil systems [11]–[13], fourcoil systems [14]–[16], systems with relay resonators [17]–[19] and domino-resonator systems [20]–[22] have emerged as vari-

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Manuscript received December 11, 2013; revised March 5, 2014; accepted April 9, 2014. Date of publication April 14, 2014; date of current version October 15, 2014. This work was supported by the Hong Kong Research Grant Council under GRF Project HKU 712913. Recommended for publication by Associate Editor C. Fernandez. J. Yin, D. Y. Lin, and C. -K. Lee are with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]). S. Y. R. Hui is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong and also with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2014.2317183

ants of the wireless power transfer systems with potentials of simultaneously extending the transmission distance and overall system efficiency. Wireless power transfer is preferred in applications in which the load should be electrically isolated from the power driving stage. For feedback control, the load conditions must be monitored. Feedback signals provided by a wireless communication system installed on the load side has been reported in [7], [8], and [23]–[25]. In [26], ASK modulation technique is adopted and data from the receiver side is obtained by monitoring the current in the primary (transmitter) circuit. For a two-coil system, Madawala and Thrimawithana [27]–[29] derive the load power from the lumped circuit model of a two-coil system with fixed resonant frequency. In [30] and [31], an energy equilibrium function of a zero voltage switching (ZVS) operated two-coil system is introduced to estimate the load variation. The report in [32] proposes a transient model of a series–series compensated two-coil system to detect the initial load condition by injecting a series of high-frequency signal before startup. So far, these pioneering projects focus primarily on the two-coil wireless power transfer systems. This paper presents a systematic approach to demonstrate that the load condition can be computed with the information of the input voltage and input current only, and the output power can be controlled without using any wired or wireless feedback information obtained directly from the output load in a multiple-coil wireless power transfer system. The power transfer efficiency can be optimized by tuning the input frequency to an optimal value under the detected load condition. A sensitivity analysis is provided to access how accurate the load estimation results will be. The concept is verified by computer simulations based on the proven mathematical model previously reported [20] and confirmed with practical measurements. This paper is an extended version of a conference paper [33]. II. LOAD MONITORING A. General Mathematical Model for an n-Coil System Consider a general wireless power transfer system consisting of n coils as shown in Fig. 1, where the first coil is the transmitter and the nth coil is the receiver. Let Li be the self-inductance, Ri be the coil resistance and Ci be the resonant capacitance of the ith coil, respectively; Mij be the mutual-inductance between the ith coil and the jth coil (obviously Mij = Mj i ), and ZL

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I1 are measureable variables. If all of the system parameters, such as winding self-inductance values, mutual-inductance values, coil resistance values, and resonant capacitance values are known, the remaining unknowns include the loop currents (i.e., I2 to In ) and the load impedance ZL . For closed-loop control purpose, it is necessary to find In (which is the current in the receiver coil) and ZL connected to the output terminal. B. Load Estimation By rearranging I1 to the left-hand side of the equation, the system can be expressed as

Schematic of an n-coil wireless power transfer system.

Fig. 1.

be the load impedance, then the system could be described in a general matrix ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

VS

⎤

⎡

Z1

jωM1 2

⎢ ⎥ ⎢ jωM1 2 Z2 0 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ⎥ .. .. ⎢ . ⎥ . . ⎥ =⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎥ ⎢ jωM1 (n −1 ) jωM2 (n −1 ) ⎣ ⎦ 0 jωM1 n jωM2 n ⎡

I1

⎢ ⎢ I2 ⎢ ⎢ ⎢ .. × ⎢ ⎢ . ⎢ ⎢I ⎢ n −1 ⎣ In

· · · jωM1 (n −1 )

jωM1 n

· · · jωM2 (n −1 )

jωM2 n

..

.

···

⎤

.. .

.. .

Zn −1

jωM(n −1 )n

· · · jωM(n −1 )n

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Zn + ZL

⎡ V −Z I ⎤ S 1 1 ⎥ ⎢ −jωM I 12 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎢ −jωM1(n −1) I1 ⎥ ⎦ ⎣ −jωM1n I1 ⎡ jωM jωM13 12 ⎢ Z2 jωM23 ⎢ ⎢ ⎢ .. .. =⎢ . . ⎢ ⎢ ⎢ jωM2(n −1) jωM3(n −1) ⎣ jωM3n jωM2n

···

jωM1n

···

jωM2n .. .

..

.

· · · jωM(n −1)n ···

Zn

0 ⎤ ⎡ I2 ⎤ ⎢ ⎥ 0⎥ ⎥ ⎢ I3 ⎥ ⎥⎢ ⎥ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥. .⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ 0 ⎦ ⎣ In ⎥ ⎦ ZL In 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2) Since VS and I1 are measureable variables and the parameter values such as Zi , Mij are known, (2) is a set of n linear (1) equations with n independent unknowns (i.e., I2 , I3 , . . ., In , and ZL In , where ZL In represents the output voltage supplied to the load Vo ). Equation (2) can now be expressed in a general form as

where Zi = Ri + j(ωLi − ω 1C i ) is the total impedance of the ith coil, VS is the input voltage vector in the first coil, Ii is the current vector of the ith coil, and ω is the angular frequency of VS . The system model defined above has three major characteristics. 1) In this system matrix equation, the dimension, or order of the matrix is equal to the number of the coils used in the whole system. 2) The coils are magnetically coupled to each other and such coupling does not interact with the load, which means that the load, including both the resistance and the reactance, is independent of the coils in the system. For example, if there is a transformer connected in the load circuit, the transformer windings are not coupled with the resonant coils. 3) The load ZL is not restricted to linear one. It could be nonlinear and time varying, i.e., ZL (ω, t). The focus of this project is to develop a new control method based on the input voltage and current only, because VS and

⎡ I ⎤ ⎡ jωM jωM13 12 2 ⎢ I ⎥ ⎢ jωM23 Z2 ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ .. ⎥ ⎢ .. .. ⎢ . ⎥=⎢ . . ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎢ In ⎥ ⎢ jωM2(n −1) jωM3(n −1) ⎦ ⎣ ⎣ ZL In jωM2n jωM3n ⎡ V −Z I ⎤ S 1 1 ⎢ −jωM I ⎥ 12 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎥. ×⎢ . ⎢ ⎥ ⎢ ⎥ ⎢ −jωM1(n −1) I1 ⎥ ⎣ ⎦ −jωM1n I1

···

jωM1n

···

jωM2n .. .

..

.

· · · jωM(n −1)n ···

Zn

0 ⎤−1 0⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ 0⎥ ⎦ 1

(3)

It is important to note that the vector column on the righthand side of (3) consists of the input voltage, input current, and system parameters, which are known information. By applying Cramer’s rule, the unique solutions for the n unknowns of (3)

YIN et al.: SYSTEMATIC APPROACH FOR LOAD MONITORING AND POWER CONTROL IN WIRELESS POWER TRANSFER SYSTEMS

can be obtained

⎧ Di−1 ⎪ ⎪ ⎨ Ii = D , Dn ⎪ ⎪ . ⎩ ZL In = D

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2≤i≤n (4)

Therefore, the load impedance can be expressed explicitly as ZL =

ZL In Dn = In Dn −1

(5)

where D is the determinant of the n × n coefficient matrix on the right-hand side of (2), Di is the determinant of the n×n coefficient matrix on the right-hand side of (2) with the ith column vector replaced by the column vector on the left-hand side of (2). In this problem jωM13 ··· jωM1n 0 jωM12 Z2 jωM23 ··· jωM2n 0 .. .. .. .. .. . D= . . . . (6) jωM2(n −1) jωM3(n −1) · · · jωM(n −1)n 0 jωM2n jωM3n ··· Zn 1

Dn −1

jωM13 jωM12 Z2 jωM23 .. .. = . . jωM2(n −1) jωM3(n −1) jωM2n jωM3n

... ... ..

.

... ...

0 −jωM12 I1 0 .. .. . . −jωM1(n −1) I1 0 −jωM1n I1 1 VS − Z1 I1

(7) Dn = jωM13 jωM12 Z2 jωM23 .. .. . . jωM2(n −1) jωM3(n −1) jωM2n jωM3n

...

jωM1n

...

jωM2n .. .

..

.

. . . jωM(n −1)n ...

Zn

−jωM12 I1 .. . . −jωM1(n −1) I1 −jωM1n I1 VS − Z1 I1

Fig. 2.

Parallel load in the receiver coil.

The load power can then be determined from Pout = In2 Re (ZL ) or Pout = In2 Re (ZL )

(10)

where In is the rms value of In . The energy efficiency of the system is Pout . Pin

η=

(11)

C. Sensitivity Analysis In the proposed load estimation method, the two variables VS and I1 are measured in real time. It is necessary to evaluate how sensitive the accuracy of the calculated load impedance is with respect to VS and I1 . Based on (5), (7), and (8), the partial derivatives are 1 ∂Dn Dn ∂Dn −1 ∂ZL = − 2 (12) ∂VS Dn −1 ∂VS Dn −1 ∂VS ∂ZL 1 ∂Dn Dn ∂Dn −1 = − 2 ∂I1 Dn −1 ∂I1 Dn −1 ∂I1

(13)

where ∂D n −1 = (−1)n ∂V S

Z2 jωM 2 3 jωM 2 3 Z3 .. .. × . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 )

...

jωM 2 (n −2 )

...

jωM 3 (n −2 )

..

.. .

.

...

Z n −2

. . . jωM (n −2 )(n −1 )

jωM 3 (n −1 ) .. . jωM (n −2 )(n −1 ) Z n −1 jωM 2 (n −1 )

(14)

(8) For a physical system, (3) should have unique solutions and the determinant is nonzero, i.e., D = 0. All earlier discussion assumes the condition that the load ZL is connected in series with the capacitor Cn in the receiver coil. If the load is connected in parallel with the capacitor in the receiver coil as shown in Fig. 2, the element Zn in (3) becomes Zn = Rn + jωLn , and the estimated load ZL = 1+j ωZCL n Z L . It is the equivalent impedance of Cn in parallel with ZL . So the Z practical load impedance ZL = 1−j ω CL n Z . L The voltage across the load is Vo = In ZL

or

Vo = In ZL .

(9)

∂Dn −1 = ∂I1 jωM12 jωM13 Z2 jωM23 .. .. − . . jωM2(n −2) jωM3(n −2) jωM 2(n −1) jωM3(n −1)

...

jωM1(n −1)

...

jωM2(n −1)

..

.. .

.

. . . jωM(n −2)(n −1) ...

Zn −1

jωM12 .. . jωM1(n −2) jωM1(n −1) Z1

(15)

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∂Dn = (−1)n +1 ∂VS Z2 jωM23 jωM23 Z3 .. .. × . . jωM2(n −1) jωM3(n −1) jωM jωM3n 2n

∂Dn = ∂I1 jωM13 jωM12 Z2 jωM23 .. .. − . . jωM2(n −1) jωM3(n −1) jωM2n jωM3n

Several conclusions can be drawn from the observation of (21) and (22). 1) The values of input voltage and current have the same weight in affecting the load estimation error. However, for high efficiency, it is better to have high voltage and . . . jωM2(n −1) jωM2n low current in the coil resonator in order to reduce the . . . jωM3(n −1) jωM3n conduction losses. The voltage rating is, thus, higher than .. .. .. the current rating numerically in most cases and there. . . fore the load estimation error is more sensitive to current ZL ∂ ZL ... Zn −1 jωM(n −1)n < measurement error, i.e., VS > I1 , ∂∂ V . ∂ I S 1 2) The accuracy of load estimation is not only related to VS . . . jωM(n −1)n Zn and I1 , but also the circuit parameters such as Ri , Li , Ci ,, (16) and Mij , the load impedance ZL , and the input angular frequency ω. While most of them can be predetermined fairly accurately, any error arising from component tolerance or temperature effects, for example, will influence the accuracy of this method. 3) The load estimation expression (5) can be further ... jωM1n Z1 simplified into the fraction of two linear combinations of VS and I1 ... jωM2n jωM12 .. .. .. . ∂ Dn ∂ Dn . . . ∂ V S VS + ∂ I 1 I 1 Z = . (23) L ∂ D n −1 ∂ D n −1 . . . jωM(n −1)n jωM1(n −1) ∂ V S VS + ∂ I 1 I 1 ... Zn jωM1n It reveals the inherent structure of the final solution ZL . (17)

The elements in (14)–(17) can be regarded as coefficients which are not related to VS or I1 measurements. The derivations of (14)–(17) are included in Appendix I, from which we can find the following simplified Dn −1 and Dn expressions instead of (7) and (8) Dn −1 = Dn =

∂Dn −1 ∂Dn −1 VS + I1 ∂VS ∂I1

(18)

∂Dn ∂Dn VS + I1 . ∂VS ∂I1

(19)

It indicates that Dn −1 and Dn are linear combinations of VS and I1 . Equations (18) and (19) can be generalized for Di in the whole system as Di =

∂Di ∂Di VS + I1 . ∂VS ∂I1

(20)

Combining (12) and (13) with (18) and (19) we can rewrite the errors of load estimation with respect to VS and I1 errors as ( ∂ D n ∂ D n −1 − ∂∂DVn S−1 ∂∂DI1n )I1 ∂ZL = ∂ V∂SD n ∂−1I1 ∂VS ( ∂ V S VS + ∂ D∂ In1−1 I1 )2

(21)

D n ∂ D n −1 ( ∂ D n −1 ∂∂DI1n − ∂∂ V ∂ZL ∂ I 1 )VS S = ∂ V∂SD n −1 . ∂ D n −1 ∂I1 ( ∂ V S VS + ∂ I1 I1 )2

(22)

The vigorous mathematical deduction steps are shown in Appendix II.

D. Application to a Two-Coil System The proposed methodology can be applied to a wireless power transfer system with two or more coils. It is now illustrated with a two-coil system, where the first coil is the transmitter coil fed with an input voltage VS , and the second coil is the receiver coil, series compensated, terminated with the load ZL . For a two-coil system, (6)–(8) can be expressed as jωM12 0 = jωM12 D = Z2 1 VS − Z1 I1 0 = VS − Z1 I1 D1 = −jωM12 I1 1 jωM12 VS − Z1 I1 D2 = Z −jωM I 2 12 1 = (ωM12 )2 I1 + Z1 Z2 I1 − Z2 VS . Substituting these terms into (4) and the load estimation formula (5), the output current I2 in the second coil, the output voltage Vo , and the load ZL can be expressed as I2 =

VS − Z1 I1 jωM12

Vo =

(ωM12 )2 I1 + Z1 Z2 I1 − Z2 VS jωM12


Fig. 3.

Diagram of power contents in a general wireless power transfer system.

ZL = = where Zin =

(ωM12 )2 I1 + Z1 Z2 I1 − Z2 VS VS − Z1 I1 (ωM12 )2 + Z1 Z2 − Z2 Zin Zin − Z1

VS I1

is the equivalent input impedance

1 Z1 = R1 + j ωL1 − , ωC1

1 . Z2 = R2 + j ωL2 − ωC2

So far every electrical quantity in the whole system and the load impedance are available. Similarly, the error evaluation equations are (ωM12 )2 I1 ∂ZL =− ∂VS (VS − Z1 I1 )2 ∂ZL (ωM12 )2 VS = . ∂I1 (VS − Z1 I1 )2 III. OUTPUT POWER CONTROL Fig. 3 depicts the structure of a general wireless power transfer system driven by a power inverter with negligible source impedance. Because the coil resonators do not have magnetic cores and their associate core losses, the power loss in the transmission channel is total conduction loss in all the coils, that is n Ii2 Ri (24) Ploss = i=1

where Ii is the rms value of Ii . Fig. 4 describes a general control scheme for wireless power system. The input voltage VS and input current I1 are sensed and fed to an estimator, which uses the algorithms previously explained to determine the load impedance ZL from (5), the

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output voltage Vo from (9), the output current Io = In from (4), the output power Pout from (10) and the energy efficiency η from (11). These variables calculated from the estimator can then be fed into an appropriate controller for control purpose. It has been pointed out that the peak energy efficiency operating frequency varies slightly with the load impedance [20]. Fig. 5 describes the relationship between the energy efficiency and the operating frequency under different load conditions. For each load condition there is an optimum efficiency point corresponding to an optimal frequency. Thereby the transmission channel can be optimized by only adjusting the operating frequency so as to achieve optimum efficiency. The relationship of the optimal frequency can be obtained either by direct calculation for simple systems or from a predetermined look-up table for large systems as long as the range of the load impedance is known. If the load varies from 10 to 100 Ω, the input frequency should follow the trajectory shown in Fig. 5 to make the wireless power transfer system always operate under optimum efficiency condition. Since the current of each coil, the input power and the output power can be calculated simultaneously one can further control the input voltage to make the load power achieve the desired power lavel under the optimal frequency, even though the load is varying. In summary, the proposed method enables the load monitoring and load power control without any direct measurement from the load. The analysis described previously assumes a sinusoidal driving function. This assumption can be justified because a good wireless power transfer system should use resonators with highquality factors, which are essentially well-tuned filters. This point has been illustrated in a simulation study that compares the use of a sinusoidal voltage and a rectangular voltage as the driving function. The results are recorded in Table III in Appendix III. The system parameters are given in Tables I and II in the following section. It can be seen that the third harmonic power loss is only 1.6% while the fifth harmonic loss is 0.53%. These small harmonic power losses do not alter the conclusion of this study. The load is assumed resistive because high-frequency resonant converter with unity power factor [34] is preferred in the receiver circuit in order to ensure continuous current (and thus continuous power flow) in the receiver circuit. IV. SIMULATION RESULTS AND EXPERIMENTAL VERIFICATION In order to verify the proposed systematic method for identifying the load impedance for wireless power transfer system, an eight-coil wireless power domino-resonator system (see Fig. 1) is adopted in this study. Fig. 6 shows the setup of the hardware system. An oscilloscope and its probes are used to sample input voltage and current. The sampled real-time data are transmitted to the microprocessor. After a series of programmed computations, the objective input voltage amplitude and operating frequency are derived, and are transmitted to a function generator to generate a sinusoidal reference signal. This signal is further amplified by a power amplifier as the input voltage to drive

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Fig. 4.

Control diagram of WPTS (with known system parameters).

Fig. 5.

Relationship of the energy efficiency and the operating frequency for the eight-coil system. TABLE I MUTUAL INDUCTANCE OF EACH PAIR OF COILS OF THE EIGHT-COIL DOMINO-RESONATOR SYSTEM

TABLE II CAPACITANCE VALUE OF EACH SERIES CAPACITOR OF THE EIGHT-COIL DOMINO-RESONATOR SYSTEM

the transmitter coil of the wireless power system. A practical resistor bank is used as the variable load. The system parameters and setup details are provided in Tables I and II, where each coil has the same parameters, i.e., R{1−8} = 0.9998 Ω, L{1−8} = 82.03 μH. In order to reduce the influence of parameter errors, all the parameters except the

load impedance in this system above are calculated with the help of a genetic algorithm [34]. In this project, a simple control scheme is adopted. A look-up table of the operating frequency with load resistance has been set up for the frequency control. The load resistance calculated by the proposed method is used to determine the frequency. The output power is regulated to be


Fig. 6.

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Schematic diagram of the hardware setup in the control stage. Fig. 8. Input power, practical and estimated output power variations under loss optimization and constant output power control.

Fig. 7. Comparison of (practical) set resistance and estimated resistance without using averaging technique.

Fig. 9.

about 10 W. Although this experimental setup is not an optimal implementation method for fast response because the data processing through the digital storage oscilloscope is relatively slow, the setup nevertheless provides a platform for testing the algorithms of the proposed systematic method. The load resistance is changed by step changes of 10 Ω with time. Fig. 7 shows the step changes of the actual load resistance with time and the calculated load resistance. The calculated values match the actual ones well. The good agreement confirms that the proposed method for load monitoring is feasible. The power loss optimization and constant output power control are also implemented simultaneously in this load change experiment. Using the same load profile in Fig. 7, other results are shown in Figs. 8–11. Using the input power feeding the transmitter coil as the input power, the output power and the efficiency can be calculated online using (10) and (11). Fig. 8 displays the measured input power and output power of the eight-coil system, together with the estimated output power.

It can be observed that the output power can be maintained at about 10 W even when there are step changes in the load resistance. At the same time the system efficiency is kept optimum by adjusting the input operating frequency. Fig. 9 shows the measured and theoretical frequency of the eight-coil system. The variation of the frequency follows the lookup table for the load-dependent optimal system energy efficiency as reported in [20] and [22]. The measured and calculated energy efficiency results of the system are plotted in Fig. 10. The energy efficiency can be maintained well above 70%, which is a result that can be achieved with the use of the maximum energy efficiency method (and not with the maximum power transfer method). The root-mean-square magnitudes of the input ac voltage and ac current in the transmitter coil are recorded in Fig. 11. Their variations are responses determined by the proposed method to the step changes of the load resistance and the constant output power control requirement.

Comparison between measured and calculated operating frequency.

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APPENDIX I From (7) and (8), the expressions of Dn −1 andDn can be further developed

D n −1

Fig. 10.

Comparison between theoretical and measured efficiency.

jωM 1 2 Z 2 . = .. jωM 2 (n −1 ) jωM 2 n

. . . V S − Z 1 I1

jωM 2 3

...

.. .

..

Root-mean-square amplitude of input ac voltage and input ac current.

V. CONCLUSION A systematic approach to load monitoring and output power control of a wireless power system without using any direct measurements from the load is presented. Based only on the input voltage and current measurements, the proposed method can derive a range of variables which can be used for output power control. Such variables include the load impedance, output voltage, output current, output power and the loop currents. The proposed method has been verified with a wireless power domino-resonator system consisting eight coil resonators. The practical measurements agree well with the calculated ones. The contribution of the paper includes a new mathematical procedure for deriving the output load information based on the input voltage and current. The proposed load monitoring and load power control method can be generalized to wireless power systems with two or more magnetically coupled coils. The experimental setup reported here is not the optimal implementation for fast response. Further work is being conducted to implement the proposed idea with the use of digital controller.

jωM 3 n

...

jωM 1 2 jωM 1 3 Z jωM 2 3 2 . . . . = . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 )

jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . + . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 ) = (−1)n V S Z2

.

jωM 3 (n −1 ) . . .

jωM 2 3 jωM Z3 23 . . . . × . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 )

0

...

jωM 1 (n −1 )

...

jωM 2 (n −1 )

..

. . .

.

−jωM 1 2 I1 . . . −jωM 1 (n −2 ) I1 −jωM 1 (n −1 ) I1 V S − Z 1 I1

. . . jωM (n −2 )(n −1 )

jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . = . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 )

Fig. 11.

−jωM 1 2 I1 0 .. .. . . −jωM 1 (n −1 ) I1 0 −jωM 1 n I1 1

jωM 1 3

...

Z n −1

...

jωM 1 (n −1 )

VS

...

jωM 2 (n −1 )

0

..

. . .

. . .

.

. . . jωM (n −2 )(n −1 )

0

...

0

Z n −1

...

jωM 1 (n −1 )

...

jωM 2 (n −1 )

..

. . .

.

Z n −1

...

jωM 2 (n −2 )

...

jωM 3 (n −2 )

..

. . .

.

...

jωM 3 (n −1 ) . . . jωM (n −2 )(n −1 ) Z n −1 jωM 2 (n −1 )

Z n −2

. . . jωM (n −2 )(n −1 )

jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . − I1 . . jωM 2 (n −2 ) jωM 3 (n −2 ) jωM 2 (n −1 ) jωM 3 (n −1 )

−jωM 1 2 I1 . . . −jωM 1 (n −2 ) I1 −jωM 1 (n −1 ) I1 −Z 1 I1

. . . jωM (n −2 )(n −1 ) ...

...

jωM 1 (n −1 )

...

jωM 2 (n −1 )

..

. . .

.

. . . jωM (n −2 )(n −1 ) ...

Z n −1

jωM 1 2 . . . jωM 1 (n −2 ) jωM 1 (n −1 ) Z1


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TABLE III NUMERICAL HARMONIC ANALYSIS

jωM 1 2 jωM 1 3 Z2 jωM 2 3 .. .. D n = . . jωM 2 (n −1 ) jωM 3 (n −1 ) jωM 2 n jωM 3 n jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . = . . jωM 2 (n −1 ) jωM 3 (n −1 ) jωM 2 n jωM 3 n

...

jωM 1 n

...

jωM 2 n

..

.. .

. . . jωM (n −1 )n ...

jωM 2 3

jωM Z3 23 . . . . × . . jωM 2 (n −1 ) jωM 3 (n −1 ) jωM 2 n jωM 3 n

Zn

...

jωM 1 n

VS

...

jωM 2 n

0

..

. . .

. . .

.

. . . jωM (n −1 )n

0

...

0

jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . + . . jωM 2 (n −1 ) jωM 3 (n −1 ) jωM 2 n jωM 3 n = (−1)n + 1 V S Z2

.

−jωM 1 2 I1 .. . −jωM 1 (n −1 ) I1 −jωM 1 n I1 V S − Z 1 I1

Zn

...

jωM 1 n

...

jωM 2 n

..

. . .

.

Fig. 12. Schematics for wireless power transfer system with different inputload combinations.

APPENDIX II Substitute (18) and (19) into (12) ∂ZL ∂VS

−jωM 1 2 I1 . . . −jωM 1 (n −1 ) I1 −jωM 1 n I1 −Z 1 I1

. . . jωM (n −1 )n ...

Zn

. . . jωM 2 (n −1 )

jωM 2 n

. . . jωM 3 (n −1 )

jωM 3 n

..

. . .

. . .

Z n −1

jωM (n −1 )n

.

...

. . . jωM (n −1 )n

jωM 1 2 jωM 1 3 Z2 jωM 2 3 . . . . − I1 . . jωM 2 (n −1 ) jωM 3 (n −1 ) jωM 2 n jωM 3 n

Zn

...

jωM 1 n

...

jωM 2 n

..

. . .

.

. . . jωM (n −1 )n ...

Zn

jωM 1 2 . . . . jωM 1 (n −1 ) jωM 1 n Z1

So the corresponding partial derivatives can be derived as written in (14) to (17).

=

=

=

( ∂∂DVn S−1 VS +

Dn − ( ∂∂ V VS + S

∂ D n −1 ∂ Dn ∂ I 1 I1 ) ∂ V S

∂ Dn ∂ I1

I1 ) ∂∂DVn S−1

Dn2 −1 Dn ( ∂∂ V S

∂ D n −1 ∂ I1

−

∂ D n −1 ∂ D n ∂ V S ∂ I1

)I1

Dn2 −1 ∂ D n ∂ D n −1 ∂ V S ∂ I1

−

∂ D n −1 ∂ D n ∂ V S ∂ I1

D2

I1 . I2n

Substitute (18) and (19) into (13), ∂ ZL ∂ I1

=

(

∂ D n −1 ∂VS

VS +

∂ D n −1 ∂ I1

∂Dn n I 1 ) ∂∂DI n −( ∂∂ D I1 ) V VS + ∂ I 1

= =

(

∂ D n −1 ∂ D n ∂VS ∂ I1

∂ D n −1 ∂ D n ∂VS ∂ I1

1

S

D n2 −1 n − ∂∂ D V

S

∂ D n −1 ∂ I1

∂ D n −1 ∂ I1

)V S

D n2 −1

n − ∂∂ D V

D2

S

∂ D n −1 ∂ I1

VS I 2n

.

Here the general solution in (4) is used, i.e., Dn −1 = DIn . APPENDIX III A simulation study has been conducted on an eight-coil wireless power transfer system with the same circuit parameters as in Section IV. The operating frequency is chosen to 530 kHz, which is the maximum efficiency frequency when the system is driving by sinusoidal voltage source and the load resistance Rload = 11.7 Ω. Fig. 13 shows the simulated input and output waveforms of the system when the driving function is a sinusoidal voltage [as shown in Fig. 12(a)]. The corresponding

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[8] [9] [10] [11] [12] [13]

Fig. 13.

Simulation results for Fig. 12(a).

[14] [15]

[16]

[17] [18] [19]

Fig. 14.

Simulation results for Fig. 12(b).

results based on a driving function of a rectangular voltage [as shown in Fig. 12(b)] are shown in Fig. 14. The results of the analysis are listed in Table III. It is noted that the input currents in Fig. 12(a) and 12(b) are essentially sinusoidal. The power losses due to the third and fifth harmonics are 1.6% and 0.53%, respectively. These harmonic power losses are negligible because the resonators are well-tuned filters.

[20] [21] [22]

[23] [24]

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[31] Z. Wang, X. Lv, Y. Sun, X. Dai, and Y. Li, “A simple approach for load identification in current-fed inductive power transfer system,” in Proc. IEEE Int. Conf. Power Syst. Technol., 2012, pp. 1–5. [32] Z. Wang, Y. Li, Y. Sun, C. Tang, and X. Lv, “Load detection model of voltage-fed inductive power transfer system,” IEEE Trans. Power Electron., vol. 28, no. 11, pp. 5233–5243, Nov. 2013. [33] J. Yin, D. Lin, C. K. Lee, and S. Y. R. Hui, “Load monitoring and output power control of a wireless power transfer system without any wireless communication feedback,” in Proc. IEEE Energy Convers. Congr. Expo., Denver, USA, Sep. 15–19, 2013, pp. 4934–4939. [34] K. Kusaka and J. Itoh, “Experimental verifications and design procedure of an AC–DC converter with input impedance matching for wireless power transfer systems,” in Proc. IEEE Energy Convers. Congr. Expo., Denver, USA, Sep. 15–19, 2013, pp. 2574–2581. [35] S. Y. R. Hui, D. Lin, J. Yin, and C. K. Lee, “Method for parameter identification, load monitoring and output power control of wireless power transfer systems,” U.S. Patent application, US 61/862,627, Aug. 6, 2013.

Jian Yin was born in Jinan, China, in 1984. He received the B.E. degree from Shandong University, China, in 2007. He is currently working toward the Ph.D. degree with the Department of Electrical and Electronic Engineering, The University of Hong Kong. During 2008 to 2009, he worked as an Engineer in SDEPCI, China, to design power substations. His research interests include sensorless motor drives and wireless power transfer technologies.

Deyan Lin (M’09) was born in China, in 1972. He received the B.Sc. and M.A.Sc. degrees from Huazhong University of Science and Technology, Wuhan, China, in 1995 and 2004, respectively, and the Ph.D. degree from the City University of Hong Kong, Kowloon, in 2012. He is currently a Research Associate with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam. From 1995 to 1999, he was a Teaching Assistant in the Electrical Engineering Department at Jianghan University, Wuhan, where he became a Lecturer later. From 2008 to 2009, he was a Senior Research Assistant with the City University of Hong Kong. His current research interests include memristors, and modeling, control, simulation of gas-discharge lamps, and wireless power transfer.

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Chi-Kwan Lee (M’08–SM’14) received the B.Eng. and Ph.D. degrees in electronic engineering from the City University of Hong Kong, Kowloon, in 1999 and 2004, respectively. He was a Postdoctoral Research Fellow in the Power and Energy Research Centre at the National University of Ireland, Galway, from 2004 to 2005. In 2006, he joined the Centre of Power Electronics in City University of Hong Kong as a Research Fellow. From 2008 to 2011, he was a Lecturer of electrical engineering at the Hong Kong Polytechnic University. He was a Visiting Academic at Imperial College London from 2010 to 2013. Since January 2012, he has been an Assistant Professor at the Department of Electrical and Electronic Engineering, The University of Hong Kong. His current research interests include applications of power electronics to power systems, advanced inverters for renewable energy and smart grid applications, reactive power control for load management in renewable energy systems, wireless power transfer, energy harvesting, and planar electromagnetics for high-frequency power converters.

S. Y. Ron Hui (F’03) received the Ph.D. degree at Imperial College London, U.K., in 1987. He is presently Chair Professor of power electronics at The University of Hong Kong (HKU) and Imperial College London. At HKU, he holds the Philip Wong Wilson Wong Endowed Professorship in Electrical Engineering. He has published more than 200 technical papers, including more than 170 refereed journal publications and book chapters. More than 50 of his patents have been adopted by industry. He has been appointed twice as an IEEE Distinguished Lecturer by the IEEE Power Electronics Society in 2004 and 2006. He served as one of the 18 Administrative Committee members of the IEEE Power Electronics Society and was the Chairman of its Constitution and Bylaws Committee from 2002 to 2010. Dr. Hui received the Excellent Teaching Award in 1998. He won an IEEE Best Paper Award from the IEEE IAS Committee on Production and Applications of Light in 2002, and two IEEE Power Electronics Transactions Prize Paper Awards for his publications on Wireless Battery Charging Platform Technology in 2009 and on LED system theory in 2010. His inventions on wireless charging platform technology underpin key dimensions of Qi, the world’s first wireless power standard, with freedom of positioning and localized charging features for wireless charging of consumer electronics. In November 2010, he received the IEEE Rudolf Chope R&D Award from the IEEE Industrial Electronics Society, the IET Achievement Medal (The Crompton Medal) and was elected to the Fellowship of the Australian Academy of Technological Sciences and Engineering. He is an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. Since 2013, he has been an Editor of the IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS.