Development and verification of printed circuit board toroidal transformer model

Development and verification of printed circuit board toroidal transformer model Jens Pejtersen, Jakob Døler Mønster and Arnold Knott DTU Electrical Engineering, Technical University of Denmark Ørsteds Plads, Building 349, 8 Kgs. Lyngby, Denmark Tel.: 45 455349 Fax.: 45 45887 Abstract An analytical model of an air core printed circuit board embedded toroidal transformer configuration is presented. The transformer has been developed for galvanic isolation of very high frequency switch-mode dc-dc power converter applications. The theoretical model is developed and verified by comparing calculated parameters with 3D finite element simulations and experimental measurement results. The developed transformer model shows good agreement with the simulated and measured results. The model can be used to predict the parameters of printed circuit board toroidal transformer configurations with a maximum deviation of approximately %. I. INTRODUCTION The continuing quest for increasing the power density of power converters has focused research on the potential of very high frequency VHF) switch-mode power conversion, where resonant converter topologies are operated at switching frequencies in the VHF range. VHF power converters reduces the requirements for the inductance and capacitance values required for a desired output power []. Galvanic isolated dc-dc converters are typically implemented using an isolation transformer. Designing a galvanic isolated dc-dc power converter for VHF operation poses a new problem in the design of the transformer: The core loss of conventional core materials is significant in the VHF range []. Air core magnetics are a suitable alternative, due to the low inductance requirements in VHF converters. The air core magnetics benefits from no core loss as well as no saturation effects. The drawback of air core magnetics is the lower permeability and that conventional air core planar magnetics are prone to cause EMI problems, due to the unenclosed flux. A printed circuit board PCB) embedded air core toroidal transformer topology was proposed in [3] as a possible solution, which is the result of combining the low loss properties of the air core magnetics with the enclosed magnetic flux of conventional transformers. This paper presents an analytical inductive model of the air core toroidal transformer topology. First the concept of the PCB embedded toroidal inductors and transformers is presented. Then an analytical inductive model of the transformer is developed. Finally the developed model is verified by comparison with 3D finite element method FEM) simulations and experimental measurements. II. TRANSFORMER CONCEPT The PCB embedded toroidal transformer concept is based on placing two toroidal inductors in a coaxial configuration such that one toroidal inductor is placed inside the other. A 3D model illustrating the concept is shown in Fig. a. This configuration allows magnetic coupling between the toroidal magnetic fields of the two inductors. Thus energy is coupled magnetically from one inductor to the other and the configuration is equivalent to a two winding transformer. The number of turns in each winding can be arbitrarily selected and is limited only by the design rules of a four layer PCB. This transformer topology was presented by the authors in [3]. The inner toroid is defined as the primary winding and the outer toroid is defined as the secondary winding. Parameters related to each of these windings are designated with subscript p and s respectively. The geometric parameters of the toroidal transformer are defined in Fig. b. The transformer windings are implemented as toroidal inductors. Before an analytical inductive model for the transformer can be developed it is necessary to look at the analytical model of the toroidal inductors. A. Toroidal Inductors A toroidal air core inductor can be implemented as an embedded PCB component where the winding is wound using planar copper layers and via interconnects to create a 3D toroid structure, as presented in [4]. The inductance L of the toroidal inductors is modeled as two series connected inductors L n and L, where L n models the inductance contribution of the toroidal magnetic field and L models the inductance contribution of the magnetic field flowing through the center hole of the inductor structure [5]. The inductive model of the toroidal inductor is L = L n L L n = µ Nh π ro r i L = r i r o µ [ ) 8 r ) ] o r i r o r i where µ is the free space permeability, N is the number of turns of the winding, h is the height of the toroidal core window, r o and r i are the outer and inner radius of the toroidal core window. III. TRANSFORMER MODEL A two winding transformer can be modeled as an inductive two port [6], see ), where the primary and secondary ) 978--4673-4355-8/3/$3. 3 IEEE 654

r i,s r i,p h p h s r o,p Fig.. r o,s Center a) b) Conceptual drawings of the PCB toroidal transformer. a) shows a 3D model of the transformer. b) shows the cross section of the transformer. inductance parameters are designated with p and s. The selfinductance of the windings are L pp and L ss, and the mutual inductance between the two windings is L ps. vp t) v s t) = Lpp L ps d L ps L ss dt ip t) i s t) The two winding transformer model only accounts for one coupling mechanism and can therefore not be used directly to model the magnetic coupling in toroidal air core transformers. The coupling between the single-turn inductances is not negligible compared with the toroidal inductance coupling. The model must be extended to take both the toroidal field and the single-turn field coupling into account. The toroidal inductor model ) assumes that the two inductance contributions are connected in series, and therefore must have the same current flowing through them. This is equivalent to stacking two transformers as shown in Fig. where the upper transformer models the mutual coupling between toroidal inductances and the lower transformer models the mutual coupling between the single-turn inductances. The upper transformer models the coupling of the toroidal fields. L is the toroidal inductance contribution of the primary winding and L is the toroidal inductance contribution of the secondary winding. The lower transformer models the coupling between the single-turn fields where L 33 and L 44 are the single-turn inductance contributions of the primary and secondary winding respectively. The mutual inductances of the upper and lower transformer are designated L and L 34. Both of the upper and the lower transformer are modeled using the two port model from ). The two inductance matrices are added such that the series connection requirement of ) is met. The resulting inductive two port of the toroidal transformer is vp t) v s t) = A. Self-Inductances L L 33 L L 34 d L L 34 L L 44 dt [ ip t) i s t) The self-inductances L pp and L ss of the inductive two port model are determined for a specific transformer geometry and ] ) 3) number of turns of the windings from the equations below. L pp = L L 33 4) L ss = L L 44 5) L = µ Np ) h p ro,p 6) π r i,p L = µ Ns ) h s ro,s 7) π r i,s L 33 = r i,p r o,p µ [ 8 r ) ] o,p r i,p 8) r o,p r i,p L 44 = r i,s r o,s µ [ 8 r ) ] o,s r i,s 9) r o,s r i,s B. Mutual Inductances The self-inductances of the two transformer windings can readily be determined for a defined set of geometric and material parameters using 4) and 5). The mutual inductances L and L 34 are on the other hand not readily available but is approximated as decribed below. Assuming that the geometry and the number of turns of both the primary and the secondary is fixed, there exists an additional degree of freedom in selction of the relative winding directions. The two windings can either be wound in the same or in the opposite direction relative to each other. If the windings are wound in the same direction, the toroidal and the single-turn field coupling mechanisms are in phase. This configuration is shown with the black dots in Fig.. L ps,same = L L 34 ) If the windings are wound in opposite directions, the mutual single-turn coupling opposes the mutual toroidal coupling. This configuration is shown with the grey dots in Fig.. L ps,opp = L L 34 ) The mutual toroidal inductance L arises from the flux flowing through the mutual toroidal core area of the two windings. The primary winding is defined as the inner toroid which is entirely surrounded by the secondary toroid. The mutual core area is therefore the same as the cross sectional 655

L ps N p : N s v p v v 3 Fig.. L L 33 L : L L 44 L 34 Inductive model of toroid transformer. area of the primary toroid. Thus L can be approximated by ), where N p is the number of primary turns, and N s is the number of secondary turns, and the core area is based on the geometric parameters of the primary winding: r i,p and r o,p and h p. L = µ N p N s h p π ro,p r i,p ) v v 4 v s ) The cross sectional area through which the single-turn flux couples between the primary and the secondary is approximated by the center-hole area of the secondary toroid. As the single-turn inductances per definition only have one turn, the mutual inductance L 34 of the single-turn coupling is assumed to be equal to the single-turn inductance of the secondary toroid L 44. L 34 = L 44 = r i,s r o,s µ [ 8 r ) ] o,s r i,s r o,s r i,s IV. MODEL VERIFICATION 3) The proposed model is verified by comparing model calculations with 3D FEM simulations and two port measurements of prototype transformers. The prototypes was implemented in 4-layer PCB with FR4 as dielectric material [3]. A prototype transformer is shown in Fig. 3. The vertical interconnects of the windings were implemented using multiple vias. Buried vias were used for the layer to layer 3 interconnects. The prototypes have only been produced with windings wound in the same direction. The two port measurements were performed using a Rhode&Schwarz ZVC 6.9 6 vector network analyzer. The two port S-parameters acquired over the frequency range MHz GHz [3]. The inductance parameters of the prototype transformers were derived from the measured S-parameters [7]. The inductance values referred to in this paper are an average of the measured inductances in the range from MHz MHz in order to remove measurement imperfections. Fig. 3. Prototype transformer with N p = N s = and r o = 9 mm, r i = 3 mm. TABLE I GEOMETRIC PARAMETERS OF PROTOTYPE TRANSFORMERS Parameter Value [mm] r i,p 3.5 r i,s 3 r o,p 8.5 r o,s 9 h p.7 h s.5 Multiple 3D CAD models of the transformers, similar to that shown in Fig. a, were generated for the 3D FEM simulations. These are similar to the PCB prototypes with the exception of the vertical interconnects between the layers, which are approximated by a solid copper connection instead of the multiple vias. The CAD models and the prototype transformers have the same dimesions as listed in Table I and defined in Fig. b. CAD models were generated for both winding configurations in order to verify the winding directions influence on mutual inductances and coupling. The FEM simulations was simulated using Ansoft Maxwell over the same frequency range as the prototype measurements. The magnetic coupling coefficient k between the primary and the secondary of a two winding transformer is defined as: k = L ps Lpp L ss 4) The coupling coefficient will be used in the evaluation of the transformer model below. A. Field Distribution The main idea behind the toroidal transformer concept is to utilize the coupling between the toroidal magnetic fields enclosed within the toroid structures, and thus minimize the generated external field. Figure 4a and Fig. 4b show the flux density of a transformer with windings wound in the same direction and N p = N s =. It is seen that the magnitude of the toroidal flux density inside the toroid is larger than that of the single-turn flux density outside the toroid. The flux crowds towards the center of the toroid, which is in good agreement with the analytical model for flux distribution in a toroid. 656

a) b) Fig. 4. 3D FEM simulation of the flux density in a PCB embedded toroidal transformer. a) shows a vector representation with a vertical and horizontal cut, and b) shows the magnitude with a cross sectional cut. B. Fixed Geometry In this section the model is compared with simulations and measurements for transformers with the same geometry and turns ratio but different number of turns. The calculated, simulated and measured inductances L pp, L ps, L ss and the coupling coefficient k are plotted as a function of the number of turns N p = N s = N. See Fig. 5. Since the self-inductances L pp and L ss are independent of the winding directions, the calculated values are plotted only once in in Fig. 5a and Fig. 5c. The primary self-inductance in Fig. 5a shows good agreement between the results of the calculated, simulated and measured values. The simulated values also shows as expected that the self-inductance does not change with winding direction. The secondary self-inductance shows the same consistency, but has a larger inductance as expected due to the larger cross sectional area of the secondary toroid. The mutual inductance in Fig. 5b shows a good consistency between the model, simulation and measurements. It is seen that the mutual inductance of the opposing winding configuration is smaller than for the configuration with windings wound in the same direction. This results in a lower coupling as seen in Fig. 5d. Note that in the opposing winding configuration, a lower number of turns results in a negative mutual inductance and coupling. This originates from the fact that the singleturn inductance is dominant in these situations, which causes the voltage across the winding to change sign with respect to the dot reference. It does not make any sense to design a toroidal transformer for all practical purposes with dominant single-turn coupling, as the majority of the magnetic field is generated outside the toroidal core, which defeats the purpose of the toroidal transformer configuration. The calculated, simulated and measured inductances of a : transformer and a : transformer are listed in Table II and Table III. The calculated values are smaller than the simulated and measured values. The primary self-inductance is predicted by the model with a maximum deviation of approximately % with respect to the measured values. The self-inductance of the secondary and mutual inductance is TABLE II MODEL VERIFICATION DATA FOR : TRANSFORMER Method L pp [nh] L ps [nh] L ss [nh] k Calculations 9.8 8.4 4.98.647 5.38.75 43.7.65 Simulations.86.84 44.4.656 Max %.97. 6.5.99 TABLE III MODEL VERIFICATION DATA FOR : TRANSFORMER Method L pp [nh] L ps [nh] L ss [nh] k Calculations 5.34 3.97 56.45.664 3.34 6.7 57.4.663 Simulations 8.9 6.84 6.8.643 Max % 6.48 8.75 8.63 4.47 predicted with a smaller maximum deviation of approximately %. These numbers are not corrected for production tolerances of the PCB prototypes or measurement uncertainties. C. Arbitrary Configurations The model is compared with simulation results for different turns ratios and geometric parameters in order to investigate the precision of the model over a range of different design parameters. The coupling coefficient is used for this verification as it incorporates all of the inductances for a given configuration. The coupling coefficient is plotted as a function of the turns ratio n for transformers with the same geometry. See Fig. 6. In Fig. 6a N p = 4 and N s is the swept parameter. In Fig. 6b N s = 4 and N p is the swept parameter. The calculated and simulated values are very similar. For this specific transformer geometry the coupling is best at n =. The coupling coefficient k of the transformers is plotted as a function different geometries in Fig. 7. In Fig. 7a the inner radius r i is swept for a fixed value of r o r i. Thus the core area of toroid is kept constant with an increasing mean radius. Figure 7b shows the coupling coefficient plotted as a function of outer radius r o where the inner radius r i is fixed. Thus the core area of the toroid is increased with increasing r o. 657

9 9 Inductance [nh] 8 7 6 5 4 3 Calculated 4 5 6 7 8 9 3 4 5 6 7 8 9 Inductance [nh] 8 7 6 5 4 3 Caluclated - opposing 4 5 6 7 8 9 3 4 5 6 7 8 9 a) Primary self-inductance L pp b) Mutual inductance L ps Inductance [nh] 8 6 4 8 6 4 Calculated.8.6.4.. 4 5 6 7 8 9 3 4 5 6 7 8 9.4 4 5 6 7 8 9 3 4 5 6 7 8 9 c) Secondary self-inductance L ss d) Coupling coefficient k Fig. 5. Comparison between calculated, measured and simulated inductances and coupling of transformers as a function of number of turns, but with the same geometric specifications and N p = N s..8.8.6.4...6.4...4.4..5.3.35.4.45.5.55.6.65.7.75.8.85.9.95 Turns ratio [n]..4.6.8..4.6.8 3 3. 3.4 3.6 3.8 4 4. 4.4 4.6 4.8 5 Turns ratio [n] Fig. 6. a) Comparison between calculated and simulated coupling coefficient as a function of the turns ratio n for transformers with the same geometry. b).8.8.6.4...6.4...4.4 4 6 8 4 6 8 4 6 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Secondary inner diameter [mm] Secondary outer diameter [mm] a) b) Fig. 7. Comparison between calculated and simulated coupling coefficient as a function of geometric parameters with n = and a fixed height. In b) the inner radius r i is swept for a fixed value of r o - r i and in a) the outer radius r o is swept with the inner radius r i fixed. 658

V. CONCLUSION An analytical two port inductive model of the printed circuit board embedded toroidal air core transformer has been developed, that takes both the single-turn and the toroidal coupling into account. Two simple approximations have been made in order to predict the mutual inductances of the transformer. The model has been verified by comparison of calculated results with 3D finite element method simulations and experimental measurement results. The model shows good consistency with both the simulations and measurements. The calculated inductance parameters showed a maximum deviation of % compared with measurements performed on a prototype transformer. The model generally underestimates the calculated inductances compared with both the simulations and the measurements. Furthermore the model has been verified for multiple turn ratios and geometry combinations using simulations only, with good consistency. The proposed analytical model is suitable to predict the inductive behavior of the toroidal transformer structure. REFERENCES [] D. Perreault, J. Hu, J. Rivas, Y. Han, O. Leitermann, R. Pilawa-Podgurski, A. Sagneri, and C. Sullivan, Opportunities and challenges in very high frequency power conversion, in Applied Power Electronics Conference and Exposition, 9. APEC 9. Twenty-Fourth Annual IEEE, feb. 9, pp. 4. [] Y. Han, G. Cheung, A. Li, C. Sullivan, and D. Perreault, Evaluation of magnetic materials for very high frequency power applications, in Power Electronics Specialists Conference, 8. PESC 8. IEEE, june 8, pp. 47 476. [3] J. Pejtersen and A. Knott, Design and measurement of planar toroidal transformers for very high frequency power applications, in IEEE 7th International Power Electronics and Motion Control Conference - ECCE Asia,, pp. 688 69. [4] S. Orlandi, B. A. Allongue, G. Blanchot, S. Buso, F. Faccio, C. A. Fuentes, M. Kayal, S. Michelis, and G. Spiazzi, Optimization of Shielded PCB Air-Core Toroids for High-Efficiency DC DC Converters, Power Electronics, IEEE Transactions on, vol. 6, no., pp. 837 846, May. [5] C. Sullivan, W. Li, S. Prabhakaran, and S. Lu, Design and fabrication of low-loss toroidal air-core inductors, in Power Electronics Specialists Conference, 7. PESC 7. IEEE, june 7, pp. 754 759. [6] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, Second Edition. Kluwer Academic Publishers,, ISBN 978-793777. [7] D. M. Pozar, Microwave Engineering, Third Edition. Wiley, 4, p. 87, ISBN 978-47448785. 659