THE magnetic field has been widely used for the transfer of

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1 148 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 1, FEBRUARY 2004 Power Transfer Capability and Bifurcation Phenomena of Loosely Coupled Inductive Power Transfer Systems Chwei-Sen Wang, Grant A. Covic, Senior Member, IEEE, and Oskar H. Stielau Abstract Loosely coupled inductive power transfer (LCIPT) systems are designed to deliver power efficiently from a stationary primary source to one or more movable secondary loads over relatively large air gaps via magnetic coupling. In this paper, a general approach is presented to identify the power transfer capability and bifurcation phenomena (multiple operating modes) for such systems. This is achieved using a high order mathematical model consisting of both primary and secondary resonant circuits. The primary compensation is deliberately designed to make the primary zero phase angle frequency equal the secondary resonant frequency to achieve maximum power with minimum VA rating of the supply. A contactless electric vehicle battery charger was used to validate the theory by comparing the measured and calculated operational frequency and power transfer. For bifurcation-free operation, the power transfer capability and controllability are assured by following the proposed bifurcation criteria. Where controllable operation within the bifurcation region is achievable, a significant increase in power is possible. Index Terms Bifurcation, compensation, electromagnetic coupling, inductive power transfer. I. INTRODUCTION THE magnetic field has been widely used for the transfer of power or information. Traditionally, good coupling is fundamental for effective transfer of significant amounts of power such as in transformers and induction motors. Recently, developments in modern power electronics have enabled many new loosely coupled applications such as contactless battery charging across large air gaps [1] [3]. Other examples include material handling systems [4] [6] and people movers [7], where the secondary systems are electrically isolated and move along a long track. Electrical isolation is essential for power supplies in harsh environments such as mining and outdoor systems. The advantages of such loosely coupled inductive power transfer (LCIPT) systems are reliability and low maintenance. An LCIPT system may be envisaged as shown in Fig. 1. It consists of two independent mutually coupled electrical systems. The primary (stationary) part produces an essentially constant current in the track or coil inductance using a suit- Manuscript received July 15, 2002; revised June 11, Abstract published on the Internet November 26, This work was supported in part by the Foundation of Research, Science and Technology (FRST), New Zealand. C.-S. Wang and G. A. Covic are with the Department of Electrical and Computer Engineering, The University of Auckland, Auckland, New Zealand ( ga.covic@auckland.ac.nz). O. H. Stielau was with the Department of Electrical and Electronic Engineering, The University of Auckland, Auckland, New Zealand. He is now at 152B Forrest Hill Road, Forrest Hill, Auckland, New Zealand. Digital Object Identifier /TIE Fig. 1. Structure of LCIPT system. able resonant high frequency switching power supply with primary compensation to minimize the VA rating of the supply. The pickup (secondary) can move with respect to the track. Here compensation is also often required, to enhance the power transfer capability. A switched mode controller is normally used to control the power flow from the pickup coil to the load. In more complex systems a number of individual pickups can exist, supplied by a single primary. The power supply and controller normally control both the frequency and the primary current to achieve maximum power transfer capability to the load. Both fixed- and variable-frequency control can be used. Fixed frequency controllers are much simpler, but increase the required VA rating of the power supply. Variable-frequency controllers ideally operate at the zero phase angle point of the load impedance seen by the power supply in order to minimize the VA rating of the supply. However, the ideal control point becomes difficult to determine if more than one zero phase angle condition exists in the frequency spectrum, as can occur as the load is increased. If a variable-frequency controller can not deal with uncertainty in the bifurcation region, the operational frequency of the power supply will either drift away from the desired operating position or move unstably between several undesirable operating conditions. Consequently, the power transfer capability will drop significantly. To date, some analysis has been undertaken to determine the bifurcation criterion for selected topologies and applied to particular applications [2], [4], [8], but no general criterion has been found. Without a thorough theoretical analysis of the interactions between the primary and secondary subsystems, it is usually quite difficult to achieve good designs of the LCIPT system. For example, many designs calculate the primary capacitance by compensating only the primary self-inductance [5], [6], [9]. This is acceptable if the reflected impedance is negligible in comparison to the primary self-inductance. Moreover, bifurcation-free operation is normally assumed. In this paper, a general analysis is proposed for the design of LCIPT systems using the load model as seen by the power /04$ IEEE

2 WANG et al.: POWER TRANSFER CAPABILITY AND BIFURCATION PHENOMENA OF LCIPT SYSTEMS 149 Using a standard mutual inductance coupling transformer model and assuming sinusoidal voltages and currents [4], [8], the induced voltage in the secondary due to the primary current is equal to, while the reflected voltage in the primary due to the secondary current is equal to, with M the mutual inductance between the primary and secondary and the operational frequency. The circuit in Fig. 2(b) represents this coupling model. III. POWER TRANSFER CAPABILITY The power transfer capability of LCIPT system can be determined using an identical approach to that developed in [4]. Here, the load impedance of the secondary is calculated as a lumped impedance whose value depends on the secondary compensation as given by series compensated secondary parallel compensated secondary. (1) The loading effect of the secondary on the primary circuit is shown in Fig. 2(c) as a reflected impedance. This impedance is dependent on the transformer coupling and operating frequency, and is given by Fig. 2. Topology and modeling. (a) Basic topologies. (b) Equivalent coupling circuit. (c) Primary circuits with reflected impedance. supply. This high order mathematical model combines both the primary and secondary subsystems. The theory is developed with the assumptions of sinusoidal voltages and currents under steady-state conditions. Moreover, the primary capacitance is deliberately designed to compensate both the primary self-inductance and the reflected impedance. This forces the zero phase angle frequency of the load model to equal the secondary resonant frequency. At this operating condition, maximum power transfer is achieved with minimum VA rating of the power supply. Normalization of the system parameters is employed to achieve a general description of various LCIPT systems enabling general bifurcation criteria to be determined for variable frequency controllers operating under zero phase angle control. The theory is then applied to the design of a contactless electric vehicle battery charger. Finally, the power transfer capability and bifurcation phenomena of the proposed design are identified and verified practically using a suitable experimental rig. II. TOPOLOGIES Four basic topologies labeled as SS, SP, PP, and PS for LCIPT systems are shown in Fig. 2(a), where the first S or P stands for series or parallel compensation of the primary winding and the second S or P stands for series or parallel compensation of the secondary winding. The subscripts p and s stand for the primary and secondary respectively, while the resistance R represents the load on the secondary. In practice, a rectifier, filter and switched-mode controller is normally used to drive the load that may be inductive or capacitive. However, it is usual to represent this load as an equivalent resistance. Substituting (1) into (2), the reflected resistance and reactance can be derived as and (2) series compensated secondary parallel compensated secondary (3) series compensated secondary parallel compensated secondary where the operators and represent the real and imaginary components of the corresponding variable, respectively. The power transferred from the primary to the secondary is then simply the reflected resistance multiplied by the square of the primary current as given by There is theoretically no limit to the power transfer capability if the system is operated at the secondary resonant frequency given by The reflected resistance at this secondary resonant frequency can be calculated from (3) using (4) (5) (6) (7)

3 150 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 1, FEBRUARY 2004 TABLE I PARAMETERS DEFINED AT! described in Section III, the load impedance power supply is seen by the series compensated primary parallel compensated primary. (8) The real part of the load impedance is the load resistance representing the real power transfer, while the imaginary part is the load reactance indicating the reactive power flow. In order to minimize the VA ratings of the power supply, it is desirable to operate at the zero phase angle frequency of the load impedance. At this frequency, the load reactance seen by the power supply is zero, eliminating reactive power flow. This zero phase angle frequency must be designed to equal the secondary resonant frequency to ensure maximum power transfer meets the required power. To achieve these objectives, a new approach is proposed in this paper. Here the primary capacitance is selected to compensate both the primary self-inductance and the reflected impedance. The design solutions are given in Table I(b) for the four basic topologies in Fig. 1(a). They are calculated using (9) The result of (7) is shown as in Table I(a) for both series and parallel-compensated secondary systems. It can be shown that both the reflected resistance and the power transfer capability assuming constant primary current for series compensation increase to infinity when the load is reduced to zero. A similar result arises for parallel-compensated systems as the load increases to infinity. The reflected reactance at this frequency can be calculated similarly from (4) and is shown as for each topology in Table I(a). As can be seen, a series compensated secondary has zero reflected reactance, whereas a parallel-compensated system reflects a capacitive load. This is one of the major differences between series and parallel-compensated secondary systems. As shown, the required primary compensation capacitance is independent of the load if the primary is series compensated (SS and SP topologies). For parallel-compensated primary (PP and PS topologies), the required primary compensation capacitance is a function of the load. In this case, the primary compensation capacitance must be designed for the required power because it is impractical to allow the primary compensation capacitance to vary with the load. V. NORMALIZATION To facilitate the design of LCIPT systems a general analysis is achieved by normalizing the load model described by (8) using the reflected resistance at the secondary resonant frequency given in Table I(a). In this process, both the frequency and the load impedance are normalized. The operating frequency is normalized using from (6) as (10) IV. LOAD MODEL The load impedance from (7) as is normalized in terms of The primary and secondary resonant circuits present a sensitive load to the power supply. Investigating such load characteristics precisely is crucial to ensure power transfer capability and controllability. This is achievable by modeling the load impedance seen by the power supply. With the loading effect of the secondary modeled by a reflected impedance as (11) This definition is suitable for series compensated primary systems, however it is often much easier to describe the load impedance in terms of its admittance value (conductance and

4 WANG et al.: POWER TRANSFER CAPABILITY AND BIFURCATION PHENOMENA OF LCIPT SYSTEMS 151 susceptance) when dealing with parallel compensated primary systems so that TABLE II NORMALIZED FUNCTIONS IN (13) (16) (12) Substituting (8) into (11), the normalized load resistance and reactance for series compensated primary systems are given by (13) and (14) Substituting (8) into (12), the normalized load conductance and susceptance for parallel-compensated primary systems are derived as (15) and Table II shows the result of this simplification process for each term in (13) (16). A complete derivation is provided in the Appendix for the SS topology. All other topologies follow an identical approach. (16) As shown above, for a series-compensated primary the load resistance is identical to the reflected resistance. The load reactance, however, depends on the primary capacitance and inductance, and also the reflected reactance. For a parallel-compensated primary the load conductance depends on the primary inductance, and also the reflected resistance and reactance, while the load susceptance depends on the primary capacitance and inductance as well as the reflected resistance and reactance. Equations (13) (16) can be simplified if they are written in terms of u, and the quality factors ( and ) associated to the primary and secondary resonant circuits. These quality factors are defined as the ratio of reactive to real power described in [4] and [8], and calculated at the secondary resonant frequency, so that (17) where and are the primary and secondary reactive powers respectively. Table I(c) specifies the resulting values of and. VI. BIFURCATION To illustrate the phenomenon of bifurcation, the PP topology was chosen. This topology is also used for the example discussed in Section VII, so that theoretical and practical results can be compared easily. Similar characteristics can be found for the other three basic topologies. The imaginary component of the normalized load admittance given by (16) is shown in Fig. 3 as a function of and u, with assumed to be 5, which is a typical design choice for LCIPT systems [4], [8]. Similar graphs can be drawn for other values of. It can be seen that the zero phase angle frequency is unique and equal to the secondary resonant frequency only when the primary quality factor is much higher than the secondary quality factor. However, there are three zero phase angle frequencies if the primary quality factor is much lower than the secondary quality factor. When designing LCIPT systems, it is desirable to determine the boundary where bifurcation occurs such that the power transfer capability and the bifurcation phenomena can be identified for both bifurcation-free and bifurcation-allowed operations. This is useful when designing the controller. If bifurcation-free operation is desired, the controller must operate

5 152 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 1, FEBRUARY 2004 TABLE IV BIFURCATION CRITERIA Fig. 3. Example of the normalized load susceptance (PP topology with Q = 5). TABLE III NORMALIZED FUNCTIONS IN (18) within the bifurcation boundary. If the system is allowed to operate in the bifurcation region, the controller must be able to operate at the desired operating mode. The necessary criteria to ensure a unique zero phase angle frequency for each topology can be determined by rearranging (14) and (16) to give (18) where and are functions of,, and as defined in Table III(a) and (b) and (19), below. For clarity, a complete example derivation of these terms for an SS topology is given in the Appendix. As expected, the value of (18) is zero at the secondary resonant frequency corresponding to zero phase angle. As shown in Table III(a), the function associated to each of the four basic topologies is always positive. To ensure the secondary resonant frequency is the only zero phase angle frequency, the function in (18) must be greater than zero. For the four basic topologies, the function is a polynomial of the form (19) and the nonzero coefficients in (19) are given in Table III(b) for the four basic topologies. As shown in this table, is a biquadratic polynomial for SS and SP topologies. For PP and PS topologies, is a bi-quartic polynomial. Normally, both the primary and secondary quality factors are larger than unity. This makes the polynomial coefficient for the SS and SP topologies positive so that if the biquadratic polynomial is to be greater than zero, the discriminant (20) must be less than zero. The necessary criteria for achieving a single zero phase angle frequency for the SS and SP topologies derived from this condition are given in Table IV, and an example derivation is provided for the SS topology in the Appendix. The solution for the PP and PS topologies require the bi-quartic polynomial to be greater than zero and can be solved by the approach proposed by Ludovico Ferrari in the 16th century [10], but is rather complicated and cumbersome. Alternatively, a numerical methodology can be used. Here, an iteration process is used over a practical range of from 1 to 10 and a suitable frequency range of around unity. In practical designs, is normally larger than to ensure bifurcation free operation [4], [8]. The numerical process starts from a significantly large to make positive across the frequency spectrum, and then reduces by a small amount at each iteration step to verify whether intersects with the axis, where bifurcation occurs. The numerical bifurcation boundary of the PP topology is then verified against the bifurcation criterion in Table IV. For the PS topology, is positive across the frequency spectrum if is larger than and, as such, no bifurcation occurs under this condition.

6 WANG et al.: POWER TRANSFER CAPABILITY AND BIFURCATION PHENOMENA OF LCIPT SYSTEMS 153 TABLE V PARAMETERS OF THE CONTACTLESS BATTERY CHARGER Fig. 4. Electromagnetic structure of the contactless charger. VII. VALIDATION A contactless electric vehicle battery charging system having a PP topology was designed using the design methodology proposed in [8] in order to validate the theory. This PP topology was chosen since it is commonly used for high-power industrial applications [4]. The current source characteristic of the parallel-compensated secondary is well suited for battery charging, whereas the parallel-compensated primary is used to generate a large primary current [8]. Using this approach it is expected that the designed system will deliver rated power without exhibiting bifurcation, although (if the design parameters are all maximized which ideally results in the lowest cost system) only a small safety margin will exist between this rated operating condition and the onset of bifurcation. The electromagnetic structure of this system is given in Fig. 4. Here the primary and secondary windings are identical, each having concentrated coils with a magnetic linking path. It is assumed that the secondary winding is attached to the underside of an electric vehicle, while the primary winding is buried in the ground. Once an electric vehicle has stopped over the charging station, electric power is transferred to the vehicle across an air gap via magnetic coupling between the primary coil in the ground and secondary coil on the vehicle. This system was designed to deliver 30 kw across a 45-mm air gap at a nominal frequency of 20 khz with a primary current of 150 A. As described in Section III, there is theoretically no limit to the power transfer capability of a given coupling structure and compensation topology if the system is operated at the secondary resonant frequency. According to the description in Section IV, the primary compensation capacitance can be deliberately designed to make the operational frequency (primary zero phase angle frequency) equal the secondary resonant frequency for any specified power requirement. Theoretically, this capacitance (in the PP topology) is load dependent, but in practice, it must be fixed at some suitable value. Consequently, the operating and secondary resonant frequencies can not be equal for all operating loads and thus the available power is limited. Moreover, bifurcation emerges as the system power demand increases, since while reduces with increasing load, actually increases and this ensures that the criteria of Table IV will eventually not be satisfied. It is therefore necessary to identify the power transfer capability and bifurcation phenomena in every system. The measured coupling and compensation parameters, along with other key system parameters used for the above design, are given in Table V. The primary and secondary quality factors and are calculated at rated load from Table I(c) as 2.51 and 1.96, respectively. The bifurcation boundary given by Table IV as is Since the calculated is very close to, but slightly above this critical boundary, the system should not bifurcate for all normal operating loads, up to and including rated operation. The system is assumed to operate at the primary zero phase angle frequencies calculated by (21) These zero phase angle frequencies are shown as functions of the load in Fig. 5(a). The power transfer capability when operated at each zero phase angle frequency and load can be calculated from (5) and are shown in Fig. 5(b). Here, the curve represents operation at the lowest zero phase angle frequency, while curves and represent operation at higher zero phase angle frequencies, respectively. The power curves were calculated assuming the primary current was controlled at 15 A instead of the rated value of 150 A for practical reasons, as this enables the theoretical curves to be verified in the laboratory. Although the primary current is reduced, all the system characteristics presented are identical except that the power level is scaled down by one-hundredth. As part of the experimental setup, a voltage-fed full-bridge parallel-resonant inverter with a variable-frequency controller was used to drive the designed system. Zero phase angle operation was achieved by controlling the inverter current to follow the voltage across the parallel-compensated primary winding. The measured frequencies and power transfers are shown in the Fig. 5 (and in corresponding figures) as circles at each of the measured loads. In the bifurcation region, control perturbations and transients within the system affect the operational frequency. In order to investigate and measure the system operating at different frequency modes, the turn-on interval of the inverter was adjusted manually to force a shift between these operating modes. Under steady-state conditions the system was found to operate stably at either the lowest or highest zero phase angle

7 154 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 1, FEBRUARY 2004 (a) (a) Fig. 5. Measured and calculated frequency and power (C = 2:21 F). (a) Zero phase angle frequency. (b) Power transfer capability. (b) Fig. 6. Measured and calculated frequency and power (C = 2:38 F). (a) Zero phase angle frequency. (b) Power transfer capability. (b) frequency ( or ), whereas stable operation at the middle frequency is not feasible without forcing the frequency (which was not possible with the controller used). As shown in Fig. 5 the measured results closely follow the theoretical curves despite simplifications in the model that assumes sinusoidal voltages and currents, and ignores losses in the capacitors and inductors. When the system is operated at its design frequency (where is unity) as shown in Fig. 5(a) and (b) the available power is 328 W, which as expected is close to the rated value of 300 W. Because the power transfer capability depends on both the operational frequency and the load resistance, the theoretical maximum power achievable for bifurcation-free operation is 340 W at a slightly higher load where the operational frequency is lower than the desired frequency. As expected from the design approach discussed earlier, the onset of bifurcation occurs at a load of 7.2 very close to (but slightly above) rated load of 6.5 as shown by the dotted lines. Fig. 5(b) also clearly shows why a variable frequency power controller requires a safety margin to ensure bifurcation-free operation over all practical power demands. The safety margin of this design is 10% above the rated load. Once such a controller begins to operate within the bifurcation region, the potential power delivery is significantly less if the controller chooses to operate at compared with.however, assuming a suitable controller could be designed to predict the operating modes and control the frequency, then even greater power delivery is possible once bifurcation has occurred by controlling the frequency to operate at when the predicted power from Fig. 5(b) is determined to be higher than that found at the other possible operating frequencies and. Using the example given, the controller must operate at when the load is less than 9, and be forced to operate at for all higher loads. The maximum available power under such operation can be as high as 365 W. In order to further verify the theory presented in this paper, the system design was deliberately changed to be either well within or far outside the bifurcation boundary (as calculated at the secondary resonant frequency where equals unity). One simple

8 WANG et al.: POWER TRANSFER CAPABILITY AND BIFURCATION PHENOMENA OF LCIPT SYSTEMS 155 VIII. CONCLUSION Fig. 7. Measured and calculated frequency and power (C = 1:87 F). (a) Zero phase angle frequency. (b) Power transfer capability. (a) (b) way to do this is by changing the primary compensation capacitance. The measured and calculated operational frequencies and power transfers with varying load are shown in Figs. 6 and 7. In Fig. 6 the primary compensation capacitance was increased to 2.38 F with the expectation that the system would have a much higher safety margin between its maximum power level and the on-set of bifurcation, but that the power levels would be lower than that achievable in Fig. 5. In Fig. 7 the primary capacitance was decreased to 1.87 F and the system is expected to exhibit bifurcation within the operating load range. Again, measured and calculated values are seen to agree within practical limitations. The resulting system of Fig. 7 is undesirable. While good power transfer capability is possible in the bifurcation region assuming a controller could be designed to utilize this, the disadvantage is the relatively low power transfer capability in the bifurcation-free region, where the controller is forced to operate at. From the three examples, it can be observed for bifurcation-free operation that power transfer is maximized by a design where the operating point is close to but within the bifurcation boundary. This paper has developed an analytical procedure for the design and control of the LCIPT systems operating at or near zero phase angles between the primary input voltage and current. At this operating condition, the voltage and current ratings of the power supply are minimized and so is the cost. It is found that there is theoretically no limit to the power transfer capability of a given electromagnetic coupling structure and compensation topology if the system is operated at the secondary resonant frequency. Therefore, a new approach is proposed to achieve these objectives. Here, the primary compensation capacitance is deliberately designed to make the operational frequency (primary zero phase angle frequency) equal the secondary resonant frequency and as such in theory is able to meet any power transfer requirement without physical changes to the coupling structure and compensation topology. The required primary compensation capacitance is found to be independent of the load if the primary is series compensated. When the primary is parallel compensated, the required primary compensation capacitance is a function of the load. In this case, the primary compensation capacitance must be designed for the required power since it is impractical to allow the primary compensation capacitance to vary with the load. Moreover, multiple operating modes (bifurcation) emerge as the power demand is increased. General bifurcation criteria are developed to identify the power transfer capability and this bifurcation phenomenon, and can be used to facilitate the design of the controller. For bifurcation-free operation, the controller must be designed to operate within the bifurcation boundary. If the system is allowed to operate in the bifurcation region, the controller must be capable of predicting and forcing operation at the desired operating mode. A contactless electric vehicle battery charger with a variable frequency controller was used to validate the theory. The proposed theory was verified by comparing the measured and calculated operational frequency and power transfer using a 300-W test system. APPENDIX The normalized functions and the bifurcation criterion are derived in this appendix for the SS topology. All other topologies follow an identical approach. The reflected resistance at the secondary resonant frequency as shown in Table I(a) is Dividing (3) by (A1) results in Substituting (10) into (A2) results in (A1) (A2) (A3)

9 156 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 1, FEBRUARY 2004 (A16) According to (6), the secondary capacitive reactance equals the secondary inductive reactance and as such the secondary quality factor given in Table I(c) can be represented as Letting (A17) be less than zero results in (A18) Substituting (A4) into (A3) results in Dividing (4) by (A1) results in (A4) (A5) The bifurcation criterion for SS topology derived from (A18) is then (A19) Substituting (10) into (A6) results in Substituting (A4) into (A7) results in The following equation is derived from (A1): Substituting (10) into (A9) results in (A6) (A7) (A8) (A9) (A10) The primary quality factor given in Table I(c) can also be represented as Substituting (A11) into (A10) results in The following equation is also derived from (A1): Substituting (10) into (A13) results in Substituting (A11) into (A14) results in (A11) (A12) (A13) (A14) (A15) Substituting (A8), (A12) and (A15) into (14) results in (A16), as shown at the top of the page. The discriminant given in (20) is then (A17) REFERENCES [1] H. Abe, H. Sakamoto, and K. Harada, A noncontact charger using a resonant converter with parallel capacitor of the secondary coil, IEEE Trans. Ind. Applicat., vol. 36, pp , Mar./Apr [2] R. Laouamer, M. Brunello, J. P. Ferrieux, O. Normand, and N. Buchheit, A multi-resonant converter for noncontact charging with electromagnetic coupling, in Proc. IEEE IECON 97, vol. 2, 1997, pp [3] H. Sakamoto, K. Harada, S. Washimiya, K. Takehara, Y. Matsuo, and F. Nakao, Large air-gap coupler for inductive charger [for electric vehicles], IEEE Trans. Magn., vol. 35, pp , Sept [4] J. T. Boys, G. A. Covic, and A. W. Green, Stability and control of inductively coupled power transfer systems, Proc. IEE Elect. Power Applicat., vol. 147, no. 1, pp , Jan [5] A. Esser and H.-C. Skudelny, A new approach to power supplies for robots, IEEE Trans. Ind. Applicat., vol. 27, pp , Sept./Oct [6] A. W. Green and J. T. Boys, An inductively coupled high frequency power system for material handling applications, in Proc. Int. Power Electronics Conf., vol. 2, Singapore, 1993, pp [7] G. A. Covic, G. Elliott, O. H. Stielau, R. M. Green, and J. T. Boys, The design of a contact-less energy transfer system for a people mover system, in Proc. Int. Conf. Power System Technology, vol. 1, Dec. 2000, pp [8] O. H. Stielau and G. A. Covic, Design of loosely coupled inductive power transfer systems, in Proc Int. Conf. Power System Technology, vol. 1, Dec. 2000, pp [9] T. Bieler, M. Perrottet, V. Nguyen, and Y. Perriard, Contactless power and information transmission, in Conf. Rec. IEEE-IAS Annu. Meeting, vol. 1, 2001, pp [10] J. Gullberg, Mathematics: From the Birth of Numbers. New York: Norton, 1997, pp Chwei-Sen Wang received the B.E. degree in mechanical engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., the M.E. degree in mechanical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., and the M.E. Hons. degree in electrical and electronic engineering from The University of Auckland, Auckland, New Zealand. He is currently a Doctoral Fellow with the Foundation of Research, Science and Technology, New Zealand, while hosted in the Department of Electrical and Electronic Engineering, The University of Auckland. He has been a Research Fellow in the Mechanical Industry Research Laboratory of the Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C., a Lecturer in the Department of Mechanical Engineering at National Chiao Tung University, and a Project Manager with Rechi Precision Company Ltd., Hsinchu, Taiwan, R.O.C. His research area covers automatic production systems, computer graphics, CAD/CAM, refrigerant compressors, room air conditioners, power electronics, and inductively coupled power transfer systems.

10 WANG et al.: POWER TRANSFER CAPABILITY AND BIFURCATION PHENOMENA OF LCIPT SYSTEMS 157 Grant A. Covic (M 88 SM 04) received the B.E. Hons. and Ph.D. degrees from The University of Auckland, Auckland, New Zealand, in 1986 and 1993, respectively, He is a full-time Senior Lecturer in the Department of Electrical and Electronic Engineering, The University of Auckland. His current research interests include power electronics, ac motor control, electric vehicle battery charging, and inductive power transfer. He has consulted widely to industry in these areas. He also has a strong interest in improved delivery methods for electronics and control teaching Dr. Covic received the John Hopkinson Premium Award from the Institution of Electrical Engineers, U.K., in Oskar H. Stielau received the B.Eng., M.Eng., and D.Eng. degrees from Rand Afrikaans University, Johannesburg, South Africa, in 1986, 1988, and 1991, respectively. He currently consults in the high-frequency power electronic field specializing in inductive technologies. Prior to that, he spent two years with the Inductive Power Transfer research group at the University of Auckland, Auckland, New Zealand, and also seven years working in industry, mainly in the induction heating field.

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