A Resonant Tertiary Winding-Based Novel Air-Core Transformer Concept Pooya Bagheri, Wilsun Xu, Fellow, IEEE, and Walmir Freitas, Member, IEEE

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1 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY A Resonant Tertiary Winding-Based Novel Air-Core Transformer Concept Pooya Bagheri, Wilsun Xu, Fellow, IEEE, and Walmir Freitas, Member, IEEE Abstract This paper presents a new concept of air-core transformers, which uses a resonant tertiary winding to create a strong coupling between the primary and secondary transformer windings. A large magnetizing impedance is achieved because of the resonance. Experimental and analytical studies are carried out on a toroidal air-core transformer to assess the effectiveness of the concept. The results show improvements in transformer performance when compared with the results from simple air-core transformers. The power loss caused by the resonant-winding resistance is identified as the main challenging issue for the proposed method. Since the proposed idea is developed and explained based on the three-winding transformer conventional model, the accuracy of the mathematical model is also verified and validated by the experimental tests. Index Terms Air-core transformer, resonant power transmission, three-winding transformers. I. INTRODUCTION I RON CORES, essential parts of power transformers, increase the magnetic coupling between the windings, reduce the magnetizing current required for operation and, consequently, lead to higher efficiency and power factor levels. However, the development of air-core transformers, by eliminating the iron cores from the transformer structure, has been always an attractive goal. Air-core transformers have the following merits when compared with their iron-core counterparts [1]: no power loss associated with the iron core caused by hysteresis and eddy currents; mitigation of harmonics and inrush currents caused by the nonlinear characteristics of the iron core; lower weight, which makes transportation more convenient; no noise from the iron core layers vibration during operation; no need for insulation between the iron core and winding conductors. Manuscript received August 23, 2011; revised January 25, 2012; accepted March 09, Date of publication June 12, 2012; date of current version June 20, This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Paper no. TPWRD P. Bagheri and W. Xu are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 Canada ( pbagheri@ualberta.ca; wxu@ualberta.ca). W. Freitas is with the Department of Electrical Energy Systems, University of Campinas, Campinas, SP, , Brazil ( walmir@ieee.org). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPWRD Despite these advantages, a main drawback has made the use of air-core transformers unpractical in the power industry. A low magnetic coupling exists between the primary and secondary windings, so that the transformer needs high magnetizing currents for operation. These currents require high primary currents which are drawn from the power system and result in high power losses in the transformer primary windings and the conductors of the power transmission/distribution line feeding the transformer. Moreover, the magnetizing current is purely inductive, so that it demands a significant amount of reactive power. In a transformer mathematical model, this requirement can be represented as a low magnetizing inductance. To overcome these fundamental problems, researchers have investigated two main types of solutions. 1) Superconductors: In some studies, the use of superconductors for the windings of an air-core transformer was investigated [1] [4]. The use of superconductors eliminates resistive power losses in the transformer windings. Nevertheless, this method does not address the problems associated with the high magnetizing current. Hence, the application of superconducting air-core transformers is usually suggested only for the places in a power network where reactive power absorption is desired, such as at the end point of an underground transmission line [4], so that the transformer is capable of acting as a reactor as well. In addition, the usage of superconductors increases the costs, the complexity, and the total weight. 2) Resonant power transmission: The idea of resonance has been widely used to increase inductive coupling [5] and has been recently applied to wireless power transmission research [6], [7]. The wireless power transmission scheme has issues similar to those of the air-core transformers because no magnetic channel is present between the sending and receiving coils. The resonance idea addresses these issues by adding series or parallel capacitors to the sending and receiving coils. For the series scheme shown in Fig. 1, the capacitances cancel out the series impedance of the coupling windings at the resonant frequency, creating a low-impedance path between the sending and receiving ends. The aforementioned idea has several variations. For example, the inherent capacitances of coils were used in [7]. The resonance method was also employed in partialcore resonant transformers [8]. In this type of transformer, the problem of low magnetizing inductance is solved by using the capacitive characteristic of the load to prevent high magnetizing current. However, the use of series and shunt capacitors in air-core transformers can lead to issues such as a large voltage drop on the capacitors, high losses /$ IEEE

2 1520 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 Fig. 1. Example of a resonant power transmission scheme. due to the resonant currents, and design limitations since the capacitor is exposed to the operating currents and voltages. In addition, these approaches do not completely solve the problem related to the magnetizing branch. This paper proposes a novel resonant power transfer idea for the development of air-core transformers. Instead of adding parallel or series capacitors to the primary and secondary coils, the idea is to add a resonant tertiary winding. As will be shown later, this winding acts as a high-impedance magnetizing branch for the proposed transformer, thereby creating an efficient air-core transformer scheme. In addition, the use of a tertiary winding may increase the design flexibility since the winding can be optimized for resonance because it is not exposed to the operation currents and voltages. This paper is organized as follows. Section II describes the proposed idea based on the conventional mathematical model of a three-winding transformer. Section III provides the details about the transformer built to test the proposed method. Section IV presents the experimental results from the built transformer to derive the parameters of the analytical model for the theoretical analysis. Section V discusses the main results from the experiments conducted to assess the proposed air-core transformer. In Section VI, the transformer performance with different loading conditions is investigated by using analytical studies, and suggestions to improve the transformer performance are also presented and discussed. Finally, the main conclusions are summarized in Section VII. II. IDEA OF RESONANT TERTIARY WINDING The basic idea of the proposed resonant power transfer scheme is shown in Fig. 2. This scheme involves three coils (windings): 1) a sending (S) coil, 2) a receiving (R) coil, and 3) a flux-guide (G) coil. The result is essentially a three-winding transformer. The G-coil can be viewed as the tertiary winding of the transformer, which is short-circuited through a tunable capacitor. Fig. 3(a) shows the model of a three-winding transformer used to explain the power transfer concept of the proposed idea. If one connects a suitable capacitor to the tertiary windings, labeled as the resonant winding in Fig. 3(a), the capacitor can be tuned to create a parallel resonance between the magnetizing impedance and the tertiary winding. The resulting total equivalent magnetizing impedance could theoretically be infinity, as shown in Fig. 3(b), if the resistance of the tertiary winding is neglected. Consequently, no shunt branch bypasses the power (or current) flowing between the primary and secondary windings. This feature makes the operation of the air-core transformer possible without a large magnetizing current. The main thrust of the proposed Fig. 2. Proposed resonant power transfer idea. Fig. 3. Three-winding transformer model for the proposed idea. idea is, therefore, to increase the magnetizing impedance by using a resonant structure. Physically, the G-coil appears to behave like a flux guide that channels (or focuses) most of the fluxes produced by the S-coil to the R-coil, thereby creating an efficient power transfer structure. In actual applications, the resistance of the resonant winding must be taken into account. This resistance conducts the resonance currents, causing power loss and creating an obstacle to obtaining a magnetizing impedance with infinite value. Based on Fig. 3, the condition for resonance can be derived as follows: where is the resonant frequency. By neglecting the resistance of the resonant winding, (1) can be simplified as shown in (2), so that the resonant capacitor can be designed to maximize the equivalent magnetizing impedance (1) (2)

3 BAGHERI et al.: A RESONANT TERTIARY WINDING-BASED NOVEL AIR-CORE TRANSFORMER CONCEPT 1521 prototype transformer by tests in the power frequency of 60 Hz and in the operation frequency of 5 khz. Then, an appropriate capacitor size and a frequency range were selected based on the obtained model to perform the main performance tests in order to evaluate the proposed idea. Fig. 4. Schematic view of the built transformer core shape. A. Model Parameters in Low Frequency First, open-circuit (o/c) tests in 60 Hz were carried out on the transformer. Three o/c tests were conducted, one for each winding. For each one of them, a 60-Hz power source supplied one winding, while the other two windings were open circuited. During each test, the instantaneous values of the voltages and currents for each winding were measured and recorded. Since the supply voltage contained some harmonic distortion, the fundamental frequency component of the recorded waveforms of the voltages and currents was computed and used for the studies. As the result of the o/c tests, the following impedance matrix of the transformer in 60 Hz was obtained: Fig. 5. Photograph of the transformer built and the experimental setup for tests and measurements. By using this method, the power circuits and the resonant circuit of the transformer are electrically isolated, unlike the circuits in [6] [8],. Therefore, the resonant-tertiary winding can be designed with different specifications than those of the power primary and secondary windings. For example, one can use superconducting materials in only the conductor of the resonanttertiary winding. This feature is one of the main advantages of the proposed method. III. EXPERIMENTAL PROTOTYPE A simple hand-made air-core transformer with three windings and a toroidal core was built for the experimental tests. A toroidal shape was chosen in order to maximize the magnetic coupling between the concentric windings. The core was made of Polyvinyl chloride (PVC), which is a nonmagnetic material, so that the transformer can be considered an air-core one. Fig. 4 shows a schematic view of the designed core and its dimensions. Eighteen American wire gauge (AWG) wires were used in the three windings. The three concentric windings: the inner one, the middle one, and the outer one, have 278, 249, and 238 turns, respectively. For the experimental tests, the inner, middle, and outer windings were used as the resonant, primary, and secondary windings of the transformer, respectively. This prototype was handmade, so the number of windings and core dimensions were not optimized. Only the capacitor size was designed, since the main goal was to test the concept. Fig. 5 shows a photograph of the transformer built and the experimental setup for the tests and measurements. IV. ANALYTICAL MODEL AND PARAMETER DETERMINATION In this section, an analytical model is derived based on the o/c (open circuit) and s/c (short circuit) tests applied to the prototype transformer. The conventional model was verified for the where each entry of this matrix is defined as (3) for (4) Thus, for example, by applying the o/c test to the primary winding of transformer, the following Z matrix elements are achieved:, and. Since an air-core transformer is a purely linear device, the corresponding Z matrix should be diagonal symmetric. The absence of diagonal symmetry for the obtained Z matrix is probably due to measurement errors. In addition, as all windings in the transformer have a pure inductive coupling, the nondiagonal entries are expected to be imaginary. Thus, the real parts of those entries in the obtained matrix are negligible since their values are very small (in order of m ) and it should also be due to the measurement errors. This matrix will be referred to as the measured matrix since it is derived based on measurements. The following step is carried out to verify if the Z matrix associated with the conventional model of three-winding transformers with general parameters (as shown in Fig. 6), which is defined by (5), can be used to analytically represent the prototype The method used to verify the conventional model can be described as follows: 1) obtain the model parameters based on the measured Z matrix, and use the known turn ratios, 2) build a Z matrix for the model with the obtained parameters and the known turn ratios, and 3) compare this calculated Z matrix with the measured Z matrix. (5)

4 1522 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 TABLE I COMPARISON OF THE PARAMETERS IN 60 Hz AND 5 khz Fig. 6. Conventional model with general parameters. All components are transferred to the resonant-winding side (a = N =N, a = N =N, and N, N, and N are the numbers of turns for the primary, secondary, and resonant windings). For the prototype, the turn ratios are The following equations are the obtained parameters for the model and the matrix built based on them, respectively: In matrices (3) and (8), the elements of the lower triangle, including the diagonal elements, exactly match each other, except for the entry. It is explained in Appendix A why this element never exactly matches in the conventional model, and a more accurate revised version of this model, in which all elements match, is also presented. The elements of the superior triangle (excluding the diagonal entries) are not exactly equal for the two matrices, although they are similar. This result occurs because of the diagonal nonsymmetry of the measured impedance matrix. Based on these results, the conventional model of a threewinding transformer can be applied to the experimental air-core transformer in 60 Hz with suitable accuracy. B. Model Parameters in High Frequency The experimental transformer has large winding resistance values compared to its leakage reactance values, since this transformer is handmade. Such a transformer cannot represent typical power transformers with a low X/R ratio. Since reducing the winding resistances for a small handmade transformer is difficult, one can increase the winding reactance by running the tests at high frequencies. Thus, for experimental investigations, the proposed transformer is tested at high frequencies around 4000 Hz. The objective is to mimic the X/R ratio of a large power transformer, so that the resulting X/R ratio becomes comparable to those of common power transformers. The transformer parameters in any frequency can be simply obtained by using the test results in the 60 Hz presented in the previous subsection. However, some phenomena, such as the (6) (7) (8) skin effect and stray capacitances, occur in high frequencies and affect the model parameters. Therefore, short-circuit (s/c) and open-circuit (o/c) tests were also conducted on the transformer in 5 khz as a sample high frequency to obtain the model parameters. Table I compares the results of 60 Hz and 5 khz. As expected, the inductance values are lower in 5 khz because of the effect of stray capacitances, and the resistance values are higher in 5 khz because of the skin effect. C. Capacitor Design for Different Frequencies In this section, the effects of the resonance idea on the magnetizing impedance in different frequencies and the required capacitor are calculated and discussed. The required capacitor is calculated in order to maximize the equivalent magnetizing impedance shown in Fig. 3(b). All of the calculations are based on the model parameters obtained in the previous subsections. The results presented in Table II show that the method can be very effective in increasing the equivalent magnetizing impedance. However, the effect of the capacitor on the increase of the magnetizing impedance is not significant in low frequencies. Thus, the decision was made to perform the main tests of the proposed idea in relatively high frequencies (e.g Hz). In high frequencies, the magnetizing impedance, which is purely inductive before adding the resonant-capacitor, becomes almost resistive with the capacitor. This resistive characteristic represents the resistive power loss in the resonance winding. V. PERFORMANCE TESTS In this section, the experimental results of the proposed idea are presented and discussed. A 5- F capacitor was connected to the resonant winding. The resonant frequency with this capacitor is around 3753 Hz. A variable frequency voltage source was connected to the primary winding. Experiments were performed for a wide range of frequencies ( Hz) in order to achieve a better understanding of the resonance phenomenon. For all of the results shown in this section, the analytical results from the calculations based on the model are also presented to provide more insight into the transformer performance and to verify the analytical model. Tests were done for two cases as follows. In the first case, the transformer was tested with no load; in the second case, a 10- resistor was connected to the secondary winding as a load. In addition, the tests were conducted with

5 BAGHERI et al.: A RESONANT TERTIARY WINDING-BASED NOVEL AIR-CORE TRANSFORMER CONCEPT 1523 TABLE II REQUIRED CAPACITORS AND THEIR EFFECTS ON THE MAGNETIZING IMPEDANCE IN DIFFERENT FREQUENCIES defined as the ratio of the equivalent magnetizing impedance with the capacitor over the magnetizing impedance without the capacitor. Fig. 8. Magnitude of o/c input impedance versus frequency. Fig. 9. Efficiency versus frequency. Fig. 7. Experimental schematics. (a) Case 1-without capacitor. (b) Case 1-with capacitor. (c) Case 2- without capacitor. (d) Case 2-with capacitor. and without the resonant capacitor, as schematically shown in Fig. 7(a) (d). For the no-load case, the behavior of the input transformer impedance ( in Fig. 7(a) and (b)) with the frequency is shown in Fig. 8, where the curves that were obtained by using the analytical model are compared with the measured curves. Due to the no-load condition, this impedance is approximately equal to the magnetizing branch impedance. The capacitor in the resonant winding had a significant effect on the increase of the magnetizing impedance, especially in the resonant frequency ( 3800 Hz). The only significant discrepancy between the analytical and experimental results in Fig. 8 is the maximum value of the impedance in the resonant frequency. In the second case (with a load), the following four indices were chosen to quantify the capacitor effect on the transformer performance: 1) power efficiency; 2) voltage regulation; 3) primary to secondary current ratio; and 4) power factor. These four indices were analyzed before and after adding the capacitor to the resonant winding. The behavior of these indices with the frequency was measured by experiments and is shown in Figs. 9 12, respectively, where the curves obtained analytically based Fig. 10. Input power factor versus frequency. on the mathematical model are also presented. Appendix B explains how to derive these indices based on the transformer model parameters. The analytical results are generally in good agreement with the experimental results. The observed discrepancies between the analytical and experimental results are most likely caused by the inaccuracy of the model used for deriving the analytical results. As expected, the results show that the capacitor in the resonant winding has a positive effect on the transformer operation indices, especially for the input current and power factor, as Figs. 10 and 11 illustrate. Indeed, with the resonant capacitor, the transformer can practically achieve a unity power factor and a supply current equal to the load-side current. This capability demonstrates that the transformer magnetizing current is quite low on the resonant frequency, resulting in equal primary and secondary transferred current values. The capacitor effect

6 1524 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 Fig. 11. Ratio of supply to load current I =I (where all current values are transferred to the resonance winding I = I a, I =I a ). Fig. 13. Voltage regulation versus load resistance. (The load resistance value is transferred to the resonance winding side R =R =a.) Fig. 12. Voltage regulation (V 0 V )=V (all voltage values are transferred to the resonance winding: V = V =a, V = V =a ). on the voltage regulation is small (Fig. 12) because most of the capacitor impact is on the magnetizing impedance of the transformer, while the voltage regulation depends mainly on the series impedance. Fig. 9 shows that the proposed method improves the operation efficiency. However, the efficiency still may not be considered satisfactory. This issue will be further discussed in the next section by using a more suitable industrial design. The voltage and current stress on the capacitor were also measured during the experiments. On the resonant operating point, the capacitor to the load voltage and current ratios were approximately 1.1 and 2.5, respectively. VI. FURTHER STUDIES AND DISCUSSIONS Since the results from the mathematical model suitably match those from the experimental tests, in this section, the transformer performance is further investigated by using the analytical model in order to determine the transformer behavior under variable loading. Due to the experimental setup limitations, only the analytical model is used to extrapolate the results presented in this section. In addition, some suggestions to improve the transformer performance are also discussed and analyzed based on this model. A. Performance Under Different Loading For the experimental results presented in the previous section, a constant power load was used for the transformer. In this section, the transformer performance is investi- Fig. 14. Efficiency versus load resistance. gated under different loading conditions. The four transformer operation indices introduced in the previous section were calculated based on the transformer model for different values of a resistive load. The results are shown for the following three cases. The first one is a transformer without the capacitor added to the resonant winding. The second case has the capacitor in the resonant winding. For the third case, an ideal situation is assumed by ignoring the resonant-winding resistance to achieve an infinite equivalent magnetizing impedance. The operation frequency used for all three cases is the resonant frequency determined for the second case (where 3763 Hz). Figs. 13 and 16 show the results for the three cases previously mentioned. As Fig. 13 reveals, the capacitor improves the voltage regulation for all values of the loading. Fig. 14 shows that the capacitor usage always increases the efficiency. However, the maximum efficiency is limited ( 64%) and for low loading (high-resistive loads), the obtained efficiency is small because of the power loss in the resonant winding. Since this power loss is almost constant for different loadings, low output power (which means a load with a large resistance) results in poor efficiency. For the ideal situation, the efficiency reaches almost 100%, showing that the resonant-winding resistance is one of the main obstacles to obtain higher efficiency levels for this small transformer. Figs. 15 and 16 demonstrate that the capacitor always has a significant effect on the supply to load current ratio and input power factor. The results obtained with the capacitor are close to the ideal mode results (where the supply current is equal to the load current value). The same conclusion is valid about the power factor. As a result, the resonant-winding resistance

7 BAGHERI et al.: A RESONANT TERTIARY WINDING-BASED NOVEL AIR-CORE TRANSFORMER CONCEPT 1525 So the question is: Can the equivalent magnetizing impedance be increased by designing the air-core transformer with different core size, number of turns, and wire sizes? This question is investigated here by using the analytical model. The equivalent magnetizing impedance can be calculated as a function of the transformer specifications in the resonant condition as follows: (9) Fig. 15. Supply to load current ratio versus load resistance. (Current values are transferred to the resonance winding side.) Applying the resonance condition described by (2), neglecting, and assuming, one can obtain (10) By expressing and as functions of transformer size dimensions, one has (11) Fig. 16. Input power factor versus load resistance. where and are the radius of the cross-section and the main diameter of the transformer core, respectively, and is the perlength resistance of the resonance winding wire. By substituting (11) into (10), the equivalent magnetizing impedance can be determined based on the transformer design as follows: has a low adverse impact on the supply current ratio and transformer power factor when compared with the issues affecting the efficiency. B. Possibilities of Further Improvements in Transformer Performance The experimental and analytical results presented in the previous sections demonstrated that the idea proposed in this paper improved the air-core transformer performance significantly. However, the performance was still lower than that obtained with commercial transformers. For example, the maximum efficiency achieved by the proposed method was around 70%, but is usually more than 95% for commercial iron-core transformers. In this section, the improvement of these performance indices based on a better design is investigated. The main challenging issue adversely affecting the transformer efficiency is the power loss associated with the resonant-winding resistance. Mathematically, this resistance makes achieving infinite magnetizing impedance by using the resonance method impossible. The negative impact of this resistive power loss is even worse for operations under power system frequencies, such as 60 Hz (as shown in Table II). Ideally, if almost infinite magnetizing impedance was obtainable, then the proposed air-core transformer could have a performance similar to that of ordinary iron-core ones. Therefore, the goal is to maximize the magnetizing impedance to achieve higher performance. (12) Equation (12) reveals that a higher magnetizing impedance can be achieved by implementing a larger size core with a higher number of turns and lower per-length resistance. Based on this fact, a new design of the air-core transformer is presented here. The core size dimensions and number of turns were chosen to be high enough to obtain acceptable performance results even for a 60 Hz operation. The core size dimensions were twelve times those of the built prototype transformer (i.e., the radius of the cross-section and main diameter of the transformer core were increased twelve times). Instead of an 18 AWG wire size, a ticker wire such as 4/0 AWG (with a diameter times the one of the 18 AWG) was used for the windings. Instead of a single layer, eight layers were used for each winding. Consequently, the number of turns for the windings could be increased up to eight times. Since the suggested new design is in fact a larger scale of the experimental prototype, its model parameters can be estimated from the prototype s measured ones. Therefore, the relationships among the resistance and reactance values associated with the new and the old transformer, based on the number of turns, size dimensions and wire resistance, are (13) (14)

8 1526 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 Fig. 17. Voltage regulation versus load resistance for the new design in 60-Hz operation. Fig. 18. Efficiency versus load resistance for the new design in 60-Hz operation. By applying the above expressions to the parameters measured in 60 Hz of the prototype (7) to determine the model parameters of the new transformer in the 60 Hz, one obtains (15) The capacitor was designed in order to operate the transformer in the resonant frequency of 60 Hz F). Based on the similar analytical procedure discussed in Section VI-A, the achieved efficiency and voltage regulations of the new designed transformer under different loadings at the rated power frequency (60 Hz) were calculated. The previous three cases (i.e., without the capacitor, with the capacitor, and for the ideal mode) were analyzed. The results are shown in Figs. 17 and 18. For the new design of the transformer, when compared to the prototype, the resonant idea is quite effective in improving the voltage regulation and efficiency. Fig. 17 shows that the voltage regulation values obtained with the capacitor are close to the ones obtained with the ideal model. Fig. 18 demonstrates that the transformer can achieve efficiencies up to 95% with the capacitor for a wide range of resistance values from 400 to This result roughly indicates that for an operation in a voltage level, such as 25 kv, the transformer will be able to supply loads ranging from 500 to 1500 kw with efficiencies close to those of conventional transformers. The results related to the other two indices discussed in Section VI-A, the primary to secondary current ratio and the power factor, are not presented here since they were already satisfactory for the experimental prototype and they are almost the same for the new design. Another possibility for improving the transformer performance is to use superconductor wires for only the resonance winding. Then, the winding resistance is almost eliminated so that the equivalent magnetizing impedance can become as large as desired. VII. CONCLUSION This paper proposed a new resonant method for wireless power transfer and tested its application to the development of air-core transformers. The scheme is based on the usage of three-winding transformers with a resonant-tertiary winding. This paper relates, therefore, the results of a proof-of-concept research based on experimental and theoretical results. The results showed that the idea is technically feasible, opening the doors for more advanced research. The resonant-winding resistance was identified as the most challenging issue for the proposed method. The main contributions of the paper are: 1) presenting a new resonant structure for power transfer through air; indeed, the concept of flux-guide could have other applications such as energy harvest and wireless charger; 2) proposing a new type of air-core transformer based on the flux-guide concept and establishing the circuit models for the transformer; and 3) proving the feasibility of the proposed air-core transformer through experimental studies. It is important to note that the proposed transformer concept is not limited to multi-mw power transmission applications. In fact, it is expected that the most promising applications, after further development, involve power levels less than 1 MW, such as aviation, transportation, military, and utility distribution systems. APPENDIX A. Revised Version of the Conventional Model for Three-Winding Transformers The conventional model for three-winding transformers assumes that the three windings share the same mutual flux, which is represented as a single magnetizing inductance in the model ( in Fig. 6). However, this assumption may not be exactly true, and all of the windings may not exactly share the same magnetic flux. In a toroidal transformer, one portion of the magnetic flux is not shared by all of the windings: the mutual flux between the middle and outer one is not shared with the inner winding because of the small gap between the middle and inner winding wires, as shown in Fig. 19. This magnetic flux can be represented by adding an extra leakage inductance to the conventional model ( in Fig. 20). The procedures for verifying the conventional model used in Section IV-A can also be applied for this revised model. Thus,

9 BAGHERI et al.: A RESONANT TERTIARY WINDING-BASED NOVEL AIR-CORE TRANSFORMER CONCEPT 1527 current ratio, and input power factor) were derived analytically based on the transformer conventional model parameters (Fig. 3) for both cases (i.e., with and without the capacitor in the resonant winding). Let us define Fig. 19. Cross-sectional view of transformer windings, showing the gap between the wound wires that cause a magnetic flux that is not shared by all of the windings. For the voltage regulation without capacitor with capacitor (18) (19) Without the capacitor With the capacitor Fig. 20. Revised version of the conventional model, created by adding an extra leakage inductance (L ). (All parameters are transferred to the resonant-winding side: a = N =N ; a = N =N.) For efficiency the impedance matrix based on the model parameters is shown in (16), at the bottom of the page. The impedance matrix of the revised model based on the o/c tests in the power frequency is Without the capacitor (20) With the capacitor (17) As (17) reveals, all of the left side and diagonal entries of the model matrix are equal to the matrix entries derived from the measurements (3). Even the entry is also matched for this model. Therefore, the revised model can be considered a more accurate representation of the air-core toroidal transformer. B. Analytical Derivation of the Performance Indices Based on Model Parameters Four indices were used in this paper for quantifying the transformer performance. Besides the experimental results, the analytical results were also provided. This Appendix presents how the four indices (voltage regulation, efficiency, supply to load For supply to load current ratio: Without the capacitor With the capacitor Finally for obtaining the input power factor: Without the capacitor (16)

10 1528 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 3, JULY 2012 With the capacitor [7] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, Wireless power transfer via strongly coupled magnetic resonances, Science, vol. 317, no. 5834, pp , Jul [8] S. C. Bell and P. S. Bodger, Equivalent circuit for high-voltage partialcore resonant transformers, Inst. Eng. Technol. Elect. Power Appl., vol. 2, no. 3, pp , May REFERENCES [1] N. Okada, H. Kamijo, T. Ishigohka, and M. Yamamoto, Fabrication and test of superconducting air-core autotransformer, IEEE Trans. Magn., vol. 28, no. 1, pp , Jan [2] R. V. Harrowell, Feasibility of a power transformer with superconducting windings, Proc. Inst. Elect. Eng., vol. 117, no. 1, pp , Jan [3] S. P. Mehta, N. Aversa, and M. S. Walker, Transforming transformers [superconducting windings], IEEE Spectrum, vol. 34, no. 7, pp , Jul [4] H. Yamaguchi, Y. Sato, and T. Kataoka, Conceptual design of air-core superconducting power transformer for cable transmission system, IEEE Trans. Power Del., vol. 11, no. 2, pp , Apr [5] Z. Hamici, R. Ltti, and J. Champier, A high-efficiency power and data transmission system for biomedical implanted electronic devices, Meas. Sci. Technol., vol. 7, pp , Nov [6] D. Schneider, Wireless power at a distance is still far away [Electrons Unplugged], IEEE Spectrum, vol. 47, no. 5, pp , May Pooya Bagheri received the B.Sc. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2010 and is currently pursuing the M.Sc. degree in electrical engineering at the University of Alberta, Edmonton, AB, Canada. His research interests are power quality and distribution systems. Wilsun Xu (M 90 SM 95 F 05) received the Ph.D. degree in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in From 1989 to 1996, he was an Electrical Engineer with BC Hydro, Vancouver. Currently, he is a Research Chair Professor with the University of Alberta, Edmonton, AB, Canada. His research interests are power quality, information extraction from power disturbances, and power system measurements. Walmir Freitas (M 02) received the Ph.D. degree in electrical engineering from the University of Campinas, Campinas, Brazil, in Currently, he is an Associate Professor at the University of Campinas, Campinas, Brazil. His areas of research are mainly related to distribution systems and distributed generation.