WO2025010502A1 - Split primary winding for common mode emi reduction - Google Patents

Abstract

Systems and methods for reducing common-mode (CM) conducted electromagnetic interference (EMI) noise in full-bridge (FB) resonant converters is proposed. The FB resonant converter has a bridge circuit, a transformer, a resonant tank and a rectifier. The transformer has a primary winding and a secondary winding which are magnetically coupled. The primary winding is split into a first primary winding section and a second primary winding section. The first and second primary winding section contains winding turns P1 and P2, respectively. The resonant tank has a capacitor and inductor and is electrically coupled between the first and second primary winding section. The secondary winding is electrically coupled to the rectifier. The bridge circuit has two or more switches operating with a duty cycle, and is coupled to the primary winding in series with the first and second primary winding section.

Description

SPLIT PRIMARY WINDING FOR COMMON MODE EMI REDUCTION CROSS REFERENCE [0001] This application relies for priority on U.S provisional Patent Application Serial No. 63/525,944, entitled “SPLIT PRIMARY WINDING COMMON MODE EMI REDUCTION”, filed July 10, 2023, the entire content of which is hereby incorporated by reference. FIELD [0002] Embodiments of the present disclosure generally relate to the field of electrical circuit topology, and more specifically, embodiments relate to devices, systems and methods for improved common mode noise reduction. INTRODUCTION [0003] The Full Bridge (FB) LLC resonant converter has been widely used in medium-to- high power applications because of its simple structure, high cost-effectiveness, and soft- switching capability. The FB LLC converter possesses a natural symmetrical structure, which yields a pair of switching nodes having complementary electrical potentials with respect to the ground. When the associated parasitic capacitances of these nodes are equal, the generated Common Mode (CM) noise displacement currents can be completely cancelled by each other. However, the voltage across the resonant tank affects the electric potentials of the transformer’s primary winding terminals, which generates a large CM noise displacement current in the interwinding capacitance of the transformer. This issue is particularly severe in planar transformers, which have a larger parasitic interwinding capacitance than traditional wire-wound transformers due to the larger overlapping area between adjacent winding layers. The combination of large parasitic interwinding capacitance and large dv/dt difference leads to severe CM noise problems. Various methods have been proposed to suppress CM noise in the transformer of isolated power converters, including the use of shielding techniques, Y-capacitors, and additional passive components. However, these approaches have drawbacks such as conduction losses, large leakage currents, and decreased power density. SUMMARY

- 1 - CAN_DMS: \1006164620 [0004] There is a need for an improved solution to reduce common-mode (CM) conducted electromagnetic interference (EMI) noise in full-bridge (FB) LLC resonant converters that is simple, cost-effective, and does not require additional components. In conventional FB LLC resonant converters, the position of the resonant tank can affect the symmetry of the entire circuit, leading to different dv/dt values at the two winding terminals of the transformer. This results in non-cancelable displacement currents generated in the parasitic interwinding capacitance of the transformer, leading to severe CM noise issues. [0005] The embodiments presented addresses this need by proposing a Split Primary Winding Transformer (SPWT) configuration for the FB LLC converter. By placing a resonant tank between two or more split primary windings, the voltage across the original primary winding is redistributed and the two or more separate primary winding branches have complementary dv/dt characteristics; thus, the CM noise current generated in the transformer can be canceled completely with the symmetrical winding structure. Compared with the conventional FB LLC converter, the influence of the resonant tank on the CM noise current generated in the transformer can be eliminated and the total CM noise current can be minimized without any additional cost. [0006] The use of complementary couple-turns is proposed to ensure a symmetrical winding arrangement for the planar transformer in the Printed Circuit Board (PCB) layout stage before it is fabricated physically. Practical considerations have been discussed when implementing the SPWT configuration, including leakage inductance, transformer winding loss, and voltage conversion ratio. DESCRIPTION OF THE FIGURES [0007] In the figures, embodiments are illustrated by way of example. It is to be expressly understood that the description and figures are only for the purpose of illustration and as an aid to understanding. [0008] Embodiments will now be described, by way of example only, with reference to the attached figures, wherein in the figures:

- 2 - CAN_DMS: \1006164620 [0009] FIG. 1 is a circuit diagram of a CM noise propagation paths of conventional FB LLC resonant converter, according to some embodiments. [0010] FIG. 2 is a circuit diagram of a lumped interwinding capacitance model of transformer in conventional FB LLC resonant converter, according to some embodiments. [0011] FIG. 3A is a block circuit diagram of a circuit topology of a resonant converter, where the center-tapped transformer uses the split transformer winding configuration, according to some embodiments. [0012] FIG.3B is block circuit diagram of a circuit topology of a resonant converter, where the two-winding transformer uses the split transformer winding configuration, according to some embodiments. [0013] FIG. 3C is circuit diagram of a FB LLC resonant converter with proposed SPWT configuration, according to some embodiments. [0014] FIG. 3D is circuit diagram of a FB LLC converter where a SPWT configuration based matrix transformer is implemented with two magnetic cores, according to some embodiments. [0015] FIG. 3E is circuit diagram of a FB LLC converter where a SPWT configuration based matrix transformer is implemented with three magnetic cores, according to some embodiments. [0016] FIG. 4 is circuit diagram of a lumped interwinding capacitance model of transformer in proposed FB LLC resonant converter, according to some embodiments. [0017] FIG.5A is a circuit diagram of a CM noise equivalent circuits of proposed FB LLC resonant converter with SPWT configuration, according to some embodiments. [0018] FIG.5B is a circuit diagram of a decoupled CM noise equivalent circuit with current noise sources, according to some embodiments.

- 3 - CAN_DMS: \1006164620 [0019] FIG.5C is a circuit diagram of a decoupled CM noise equivalent circuit with voltage noise sources, according to some embodiments. [0020] FIG. 6 is a 3D model of couple-turn when the layout is symmetrical, according to some embodiments. [0021] FIG.7A is a circuit diagram of a one-capacitor model of couple-turn having a C_struct between corresponding terminals a’ and c’, according to some embodiments. [0022] FIG.7B is a circuit diagram of a one-capacitor model of couple-turn having a C_struct between corresponding terminals b’ and d’, according to some embodiments. [0023] FIG.8 is a 3D model of couple-turn when the layout is asymmetrical, according to some embodiments. [0024] FIG. 9A is an exemplary winding arrangement of a planar transformer, according to some embodiments. [0025] FIG. 9B is a circuit diagram of a planar transformer, according to some embodiments. [0026] FIG. 9C is a circuit diagram of a CM noise current propagation path of a planar transformer, according to some embodiments. [0027] FIG.10A is an exemplary winding arrangement of a planar transformer, according to some embodiments. [0028] FIG.10B is a circuit diagram of a propagation path of CM noise currents generated in couple-turns P1_3-S1_2 and P2_3-S2_2, according to some embodiments. [0029] FIG.10C is a circuit diagram of a propagation path of CM noise currents generated in couple-turns P1_4-S1_2 and P2_4-S2_2, according to some embodiments. [0030] FIG. 11A is a circuit diagram of complementary couple-turns having a complementary couple-turns P1_m-S1_n and P2_m-S2_n, according to some embodiments.

- 4 - CAN_DMS: \1006164620 [0031] FIG. 11B is a circuit diagram of complementary couple-turns having a complementary couple-turns P1_m-S2_n and P2_m-S1_n, according to some embodiments. [0032] FIG.11C is a circuit diagram of complementary couple-turns having a propagation path of CM noise currents generated in P1_m-S1_n and P2_m-S2_n, according to some embodiments. [0033] FIG.11D is a circuit diagram of complementary couple-turns having a propagation path of CM noise currents generated in P1_m-S2_n and P2_m-S1_n, according to some embodiments. [0034] FIG. 12 is a circuit diagram of a SPWT configuration based FB LLC resonant converter with FB rectifier on secondary side, according to some embodiments. [0035] FIG.13A is a circuit diagram of a two-winding planar transformer when secondary winding is even in number, according to some embodiments. [0036] FIG. 13B is a circuit diagram of a complementary couple-turns for a two-winding planar transformer when secondary winding is even in number, according to some embodiments. [0037] FIG.13C is a circuit diagram of a two-winding planar transformer when secondary winding is odd in number, according to some embodiments. [0038] FIG. 13D is a circuit diagram of a complementary couple-turns for a two-winding planar transformer when secondary winding is odd in number, according to some embodiments. [0039] FIG. 14 Is a transformer winding arrangement of example planar transformer #3, according to some embodiments. [0040] FIG. 15 is a circuit diagram of a FB series resonant converter with SPWT configuration, according to some embodiments.

- 5 - CAN_DMS: \1006164620 [0041] FIG. 16 is a circuit diagram of a FB parallel resonant converter with SPWT configuration, according to some embodiments. [0042] FIG. 17 is a circuit diagram of a FB LCC resonant converter with SPWT configuration, according to some embodiments. [0043] FIG. 18A is a transformer winding arrangement and terminal connections of proposed FB LLC resonant converter, according to some embodiments. [0044] FIG. 18B is a graph showing the distribution of H(x)² for both proposed and conventional planar transformers, according to some embodiments. [0045] FIG. 18C is a transformer winding arrangement and terminal connections of conventional FB LLC resonant converter, according to some embodiments. [0046] FIG.19 is a technical diagram of a structure and dimensions of ECW34C magnetic core, according to some embodiments. [0047] FIG. 20A is a Maxwell 3D model of a proposed transformer, according to some embodiments. [0048] FIG.20B is a Maxwell 3D model of a conventional transformer, according to some embodiments. [0049] FIG. 21A is a circuit diagram of a mutual inductance model of conventional transformer employed in simulation, according to some embodiments. [0050] FIG. 21B is a circuit diagram of a π model of conventional transformer, according to some embodiments. [0051] FIG. 21C is a circuit diagram of an equivalent model used for calculating total leakage inductance, according to some embodiments. [0052] FIG. 22A is a circuit diagram of a mutual inductance model of proposed transformer employed in simulation, according to some embodiments.

- 6 - CAN_DMS: \1006164620 [0053] FIG. 22B is a circuit diagram of a π model of proposed transformer, according to some embodiments. [0054] FIG. 22C is a circuit diagram of a equivalent model used for calculating total leakage inductance, according to some embodiments. [0055] FIG.23 is a circuit diagram of a complete interwinding capacitance model of SPWT configuration based planar transformer, according to some embodiments. [0056] FIG. 24 is a graphical representation of variation of Z_{ru_n} concerning frequency changes, according to some embodiments. [0057] FIG.25A is a circuit board for a proposed SPWT daughter card, according to some embodiments. [0058] FIG. 25B is a circuit board for a conventional PCB transformer daughter card, according to some embodiments. [0059] FIG. 26A is a component diagram of primary windings of a conventional planar transformer, according to some embodiments. [0060] FIG. 26B is a component diagram of primary windings of a proposed planar transformer, according to some embodiments. [0061] FIG. 26C is a component diagram of secondary daughter cards for both conventional and proposed planar transformers, according to some embodiments. [0062] FIG. 27 is a graph of experimentation results for a measured voltage conversion ratio, according to some embodiments. [0063] FIG.28 is a graph of experimentation results for measured efficiency, where Vin = 190 V. Vo = 12 V, according to some embodiments. [0064] FIG.29A is a circuit diagram of circuit connections for measuring C_{P1_S} and C_{P2_S}, according to some embodiments.

- 7 - CAN_DMS: \1006164620 [0065] FIG. 29B is a circuit diagram of an equivalent capacitance network for measuring C_{P1_S} and C_{P2_S}, according to some embodiments. [0066] FIG. 30A is a circuit diagram of circuit connections for measuring K₁ and K₂, according to some embodiments. [0067] FIG. 30B is a circuit diagram of an equivalent capacitance network for measuring K₁ and K_2, according to some embodiments. [0068] FIG. 31A is a graph of experimental results of measured waveforms for v_AB and v_EB, according to some embodiments. [0069] FIG. 31B is a graph of experimental results of measured waveforms for v_CD and v_ED, according to some embodiments. [0070] FIG. 32A is a waveform diagram of transformer terminals in conventional FB LLC resonant converter under full load conditions in boost mode (fs < fr), Vin = 180V, fs = 135 kHz, according to some embodiments. [0071] FIG. 32B is a waveform diagram of transformer terminals in conventional FB LLC resonant converter under full load conditions in buck mode (fs > fr), Vin = 210V, fs = 165 kHz, according to some embodiments. [0072] FIG. 33A is a waveform diagram of transformer terminals in proposed FB LLC resonant converter under full load conditions in boost mode (f_s < f_r), V_in = 180V, f_s = 135 kHz, according to some embodiments. [0073] FIG. 33B is a waveform diagram of transformer terminals in proposed FB LLC resonant converter under full load conditions in buck mode (f_s > f_r), V_in = 210V, f_s = 165 kHz, according to some embodiments. [0074] FIG. 34A is a waveform diagram of key waveforms during dead-time period when f_s = f_r for a conventional FB LLC resonant converter under full load conditions, according to some embodiments.

- 8 - CAN_DMS: \1006164620 [0075] FIG. 34B is a waveform diagram of key waveforms during dead-time period when f_s = f_r for a proposed FB LLC resonant converter under full load conditions, according to some embodiments. [0076] FIG.35A is a graph of measured CM noise spectra when in boost mode (f_s < f_r), V_in = 180V, f_s = 135 kHz, according to some embodiments. [0077] FIG.35B is a graph of measured CM noise spectra when in boost mode (f_s < f_r), V_in = 180V, f_s = 135 kHz, according to some embodiments. [0078] FIG. 35C is a graph of measured CM noise spectra when f_s = f_r = 152 kHz, V_in = 195V, according to some embodiments. [0079] FIG. 36A is a circuit diagram of a lumped interwinding capacitance models of P1_4-S1_1 and P1_5-S2_1, according to some embodiments. [0080] FIG. 36B is a circuit diagram of a lumped interwinding capacitance models of P2_4-S2_1 and P2_5-S1_1, according to some embodiments. [0081] FIG.37 is a 3D models of P1_8 and P2_8, according to some embodiments. DETAILED DESCRIPTION [0082] A system and method to address the issue of common-mode (CM) conducted electromagnetic interference (EMI) noise in full-bridge (FB) LLC resonant converters is proposed. In conventional FB LLC resonant converters, the position of the resonant tank can affect the symmetry of the entire circuit, leading to different dv/dt values at the two winding terminals of the transformer. This results in non-cancelable displacement currents generated in the parasitic interwinding capacitance of the transformer, leading to severe CM noise issues. In the proposed embodiments, a low CM noise FB LLC resonant converter with the Split Primary Winding Transformer (SPWT) configuration is proposed. The SPWT configuration can be applied to both wire-wound and planar transformer applications. The transformer primary winding is split into two or more windings and the resonant tank is connected between these two or more windings. With a symmetrical winding structure, the CM noise current generated in the transformer can be canceled completely. For wire-wound

- 9 - CAN_DMS: \1006164620 transformers, the sandwich winding structure can be used to meet the symmetrical condition. For planar transformers, the use of complementary couple-turns is proposed to ensure a symmetrical winding arrangement for the planar transformer in the Printed Circuit Board (PCB) layout stage before it is fabricated physically. Practical considerations are discussed when implementing the SPWT configuration, including leakage inductance, transformer winding loss, and voltage conversion ratio. The proposed SPWT configuration and the use of complementary couple-turns can be extended to various FB resonant converters, including Series Resonant Converters (SRC), Parallel Resonant Converters (PRC), and LCC resonant converters, among others. [0083] FIG.1 depicts the CM noise propagation paths of a conventional FB LLC converter 100. During the conducted EMI measurement, the Secondary Ground (SG) is connected to the Protective Earth (PE). The CM noise currents generated by the switching nodes will couple into the PE via parasitic capacitances, which can be detected by the Line Impedance Stabilization Network (LISN) 102. The LISN 102, as a passive network, serves the purpose of isolating the testing system with a reference impedance and providing measurement points to the Electro-Magnetic Interference (EMI) receiver. A noise separator 104 is utilized to effectively separate the original conducted EMI noise into its CM noise component. [0084] As shown in FIG.1, on the secondary side, i_{CM_SR1} and i_{CM_SR2} denote the CM noise currents generated by the secondary-side voltage pulsation nodes on the circuit-to-PE parasitic capacitors (C_SR1 and C_SR2). Since SG is connected to PE, they circulate back through SG instead of LISN 102. Hence, i_{CM_SR1} and i_{CM_SR2} do not contribute to the total CM noise and can be ignored. [0085] Typically, the magnitude of the current passing through the LISN 102 is in the range of microamperes to milliamperes. In particular, a CM current of 40 μA at 150 kHz (which translates to 66 dBμV when flowing into 50 Ω) exceeds the limits specified by EN55032 Class B (quasi-peak value). Therefore, the dv/dt of the LISN 102 can be considered negligible compared to that of the converter's 100 voltage pulsation nodes, and the primary ground (PG) can be treated as equivalently connected to PE from the perspective of CM noise coupling. On the primary side, C_Q2 (C_Q4) is the parasitic capacitor between the drain of MOSFET Q₂ (Q₄) and PE. By using PE as the reference point, the

- 10 - CAN_DMS: \1006164620 electric potentials of primary phase-leg midpoints are denoted by v_Q2 and v_Q4. The CM noise currents generated in C_Q2 and C_Q4 are denoted by i_{CM_Q2} and i_{CM_Q4}, respectively. Given the symmetrical layouts of the two primary phase legs and the consistent packaging of the MOSFETs, C_Q2 can be considered as equal to C_Q4. Since Q₂ and Q₄ are switched at 50% duty and 180 degrees out of phase with each other, v_Q2 and v_Q4 have complementary dv/dt characteristics. When v_Q2 has a positive dv/dt, v_Q4 will have a negative dv/dt with the same amplitude. Therefore, i_{CM_Q2} and i_{CM_Q4} will be canceled by each other. i_{CM_Q2} and i_{CM_Q4} are removed from the following CM noise analysis. [0086] According to the above analysis, the total CM noise current i_{CM_conv_LLC} is dominated by i_{CM_TX}. i_{CM_TX} represents the total CM noise displacement current flowing through the distributed interwinding capacitance C_ps of the transformer 106. i_{CM_TX} is related to the electric potentials of the transformer winding terminals and parasitic interwinding capacitance of the transformer 106. In order to quantitatively analyze i_{CM_TX}, the parasitic interwinding capacitance model of the transformer 106 has been developed. By ignoring the effect of the transformer’s leakage inductance, the two-capacitor model can be used to characterize the interwinding capacitance of a center-tapped three-winding transformer 106. As shown in FIG.2, C_ae and C_be are used to model the lumped interwinding capacitors of the transformer 106. Since the winding terminal e is connected to the dc output, the corresponding dv/dt can be treated as zero. Thus, i_{CM_TX} can be calculated by i CM _{_} TX ^ _C ^dv a ae ^ _C ^dv b be (1) [0087] where v

a and v_b terminals a and b with respect to PE. When a transformer is constructed, the values of C_ae and C_be are then determined. v_a and v_b will vary with different operation conditions of the LLC converter 100, and then influence the CM EMI performance of the whole system. [0088] In FIG.1, since the winding terminal b is connected to the drain of MOSFET Q4, v_b is consistent with v_Q4 which is a typical trapezoidal wave. v_a is equal to v_Q2 + v_Zr, where v_Zr denotes the voltage across the resonant tank. So, Equation (1) can be rewritten as

- 11 - CAN_DMS: \1006164620 _d ( _v 2 ^{^} _v ) _dv 4 _dv ^dv i _^ C ^{Q Zr} _^ C ^Q _^ ^Q 4 ^{v Zr} C_M _ _{TX ae be} ( C _{be ^} C _ae ) _^ C _ae [0089]

dv/dt characteristics. If the transformer 106 is made symmetrically, the values of C_ae and C_be can be treated as equal; then the influence of dv_Q4/dt on i_{CM_TX} can be eliminated. However, the influence of v_Zr still exists. [0090] In conclusion, the placement of the resonant tank between the midpoint of the primary phase leg and transformer winding terminal a introduces an additional CM noise voltage variable, v_Zr. v_a and v_b will exhibit different dv/dt characteristics because of the influence of v_Zr. The CM noise displacement currents generated by v_a and v_b cannot be canceled with a symmetrical transformer winding arrangement. And CM noise is not minimized. In order to reduce the CM noise in the conventional FB LLC converter 100, the displacement currents generated by v_a and v_b via the corresponding parasitic interwinding capacitors should be eliminated. [0091] Proposed SPWT configuration for FB LLC resonant converter [0092] FIGs. 3A and 3B show the circuit topology of the proposed FB resonant converters, 300A and 300B. In FIG. 3A a center-tapped transformer 301 uses the Split Transformer Winding Configuration. In FIG.3B a two-winding transformer 301 uses the Split Transformer Winding Configuration. The FB resonant converter 300A and 300B contains an FB circuit 308 which is coupled in parallel with a resonant tank 302. By using different resonant tanks 302, various FB resonant converters can be constructed, including LLC resonant converters, Series Resonant Converters (SRC), Parallel Resonant Converters (PRC), and LCC resonant converters, among others. The primary winding of the converter 300A is split into a plurality of winding sections, in the embodiment shown, there are two primary winding sections P1304 and P2306. In some embodiments, there can be three or more primary winding sections. P1304 is connected between terminals A and B, and P2306 is connected between terminals C and D. Terminals B and C are coupled to the resonant tank 302 such that the resonant tank is connected between (i.e., splits) the two primary winding sections P1304 and P2306. In FIG. 3A, the secondary transformer windings 310

- 12 - CAN_DMS: \1006164620 are a center tapped transformer 301 having two transformer windings S1 310A and S2 310B. In some embodiments, the primary winding section P1304 and secondary winding section S1, and the primary winding section P2304 and secondary winding section S2 may have complementary couple turns to enable the reduction in EMI noise. In FIG. 3B, the secondary transformer windings 310 contains a single winding section S. In some embodiments, the primary winding section P1304 and secondary winding section S, and the primary winding section P2304 and secondary winding section S may have complementary couple turns to enable the reduction in EMI noise. As can be seen from a comparison of FB resonant converters 300A and 300B against FB LLC converter 100, no additional components are necessary as in manufacturing the proposed converters 300A and 300B in order to achieve the lower EMI noise levels provided by the proposed converters 300A and 300B. [0093] In some embodiments, the connections between the resonant tank 302 and transformer terminals A and D are not needed for all resonant converters, such as an LLC resonant converter and a Series Resonant Converters (SRC). [0094] FIG. 3C shows the circuit diagram of the proposed FB LLC converter 300C with the proposed SPWT configuration. Compared to conventional FB LLC converters 100, the transformer’s 301 primary winding is split into two separate windings P1304 and P2306. The split two primary windings P1304 and P2306 have the same number of turns. The resonant inductor L_r and resonant capacitor C_r are placed between the primary windings P1 304 and P2 306. L_m1 (L_m2) denotes the magnetizing inductance seen from the primary winding P1304 (P2306). M denotes the mutual inductance of L_m1 and L_m2. The total primary side magnetizing inductance L_{m_SPWT} is calculated as: L^{m_ SPWT ^ L m 1 ^ L m 2 ^ 2 M} ₍₃₎ [0095] It should be noted that to ensure the same voltage gain characteristics as the conventional FB LLC converter 100, L_{m_SPWT} should be equal to the transformer 301 magnetizing inductance of the conventional FB LLC converter 100. The operation principle of the proposed converter is the same as that of the conventional FB LLC converter 100.

- 13 - CAN_DMS: \1006164620 [0096] The CM noise model of the proposed FB LLC converter 300C has been developed to better illustrate the cancellation mechanism of CM noise displacement currents. [0097] The input and output DC capacitors are treated as short circuits within the conducted EMI (Electro-Magnetic Interference) frequency range. The LISN 102 is modeled as a 25-Ω resistor. C_AE and C_BE (C_CE and C_DE) are used to model the lumped interwinding capacitors between the secondary windings and the primary winding P1304 (P2306), as shown in FIG.4. It is noted that C_AE, C_BE, C_CE, and C_DE are equivalent parasitic capacitances of the transformer 301. They reflect the distributed interwinding capacitance between the transformer primary 304, 306 and secondary windings 310, allowing designers to conveniently analyze the equivalent model for CM noise in the transformer 301. [0098] It should be pointed out that the interwinding capacitance between P1304 and P2 306 does not introduce the CM noise current as the displacement current generated in that capacitance is confined within the transformer primary side. Therefore, the interwinding capacitance between P1304 and P2306 is not shown in FIG.4. [0099] The remaining circuit elements in FIG.3C are replaced with CM noise sources by using substitution theory. Based on substitution theory, when any circuit branch in FIG.3C is substituted with a voltage source or current source that has the same voltage or current waveform as the original circuit branch, the currents and voltages in other parts of the circuit remain unchanged. Two rules should be followed when applying substitution theory to CM noise analysis: [00100] 1) The substitution should avoid creating voltage loops and current nodes. [00101] 2) Although various substitutions are possible, those most convenient for CM noise analysis are preferred. [00102] Q₁, Q₃, SR₁, SR₂, and resonant tank 302 are substituted with current sources with their own current waveforms, which are denoted by i_Q1, i_Q3, i_SR1, i_SR2, and i_Zr respectively. Q₂, Q₄, and primary winding P1304 are substituted with voltage sources with their own voltage waveforms, which are denoted by v_Q2, v_Q4, and v_P1 respectively. Based on the transformer 301 turns ratio, all other transformer windings are substituted with voltage-controlled voltage

- 14 - CAN_DMS: \1006164620 sources v_P2, v_S1, and v_S2. Since L_m1 and L_m2 are in parallel with voltage sources v_P1 and v_P2, they are ignored in the CM noise analysis. Finally, the CM noise model of the proposed FB LLC converter 300C is obtained, as shown in FIG.5A. [00103] In FIG. 5A, i_{CM_SPWT} represents the total CM noise displacement current flowing 5 through the interwinding capacitance of the transformer 301. To calculate i_{CM_SPWT}, superposition theory is used to simplify the circuit. When analyzing one noise source, the other voltage sources are considered as short circuits and current sources are considered as open circuits. FIG.5B and 5C give the decoupled CM noise equivalent circuits with current and voltage noise sources, respectively. 10 [00104] According to FIG. 5B, the current paths of i_Q1, i_Q3, and i_Zr are confined within the primary side. The current paths of i_SR1 and i_SR2 are confined within the secondary side. Hence, i_Q1, i_Q3, i_SR1, i_SR2, and i_Zr do not contribute to i_{CM_SPWT}, which are ignored. In FIG.5C, v_S1 and v_S2 are open, which do not generate any CM noise currents. [00105] Four CM noise sources remain: v_Q2, v_Q4, v_P1, and v_P2. Then, i_{CM_SPWT} can be 15 calculated by: i ^ _C ^dvA ^{dv dv dv} AE ^ _C D DE ^ B C CM _ SPWT _C BE ^ _C CE [00106] In the

, transformer primary winding terminals with respect to PE (Protective Earth), which are given by: ^ v A ^{^} v Q 2 ^ ₂₀ [00107] As v and v_Q4 are

, complementary (i.e., v_A = -v_D). Furthermore, since the split two primary windings have the same number of turns, by ignoring the leakage inductance, v_P1 = v_P2. Thus, v_B = v_Q2 - v_P1 = -

- 15 - CAN_DMS: \1006164620 v_Q4 - v_P2, which implies v_B and v_C are also complementary (i.e., v_B = -v_C). Based on this, Equation (4) can be rewritten by: i ^ ( _{C AE} ^ _{C DE} ) ^dvA ^ ( _{BE CE} ^{dv B} C_{M _ SPWT C} ^ _C ) [00108] It is

symmetrically, which means C_AE = C_DE and C_BE = C_CE. As a result, the CM noise current generated in C_AE can be canceled by the CM noise current generated in C_DE, and the CM noise current generated in C_BE can be canceled by the CM noise current generated in C_CE, thereby resulting in a net CM noise current of zero, as shown in Equation (7). i ^{dv d} WT ^ ( _C AE ^ _C DE ) A ^v B CM _{_} SP ^ ( _C BE ^ _C CE ) [00109] In some

inductance is typically realized by utilizing the leakage inductance of the transformer 301. This, however, does not compromise the effectiveness of the proposed SPWT configuration in reducing CM noise. As shown in FIG. 5C, for winding P1 304 (P2 306), its leakage inductance is in series with the voltage source v_P1 (v_P2). When the transformer 301 is symmetrically wound, the leakage inductances of P1304 and P2306 can be considered equal. Consequently, the voltage drop caused by the leakage inductances of P1304 and P2 306 is the same, as they carry identical resonant currents. As a result, v_B and v_C remain complementary, allowing CM noise currents in C_BE and C_CE to mutually cancel each other. [00110] As discussed above, achieving a symmetrical winding arrangement may effectively reducing the CM noise displacement current generated in the transformer 301 interwinding capacitance when utilizing the SPWT configuration. Specifically, the symmetrical condition refers to the equality of C_AE and C_DE as well as C_BE and C_CE.

- 16 - CAN_DMS: \1006164620 [00111] In a further potential embodiment, the SPWT may be applied to a matrix transformer 303, as shown in FIGs.3D and 3E. Matrix transformers are capable of achieving voltage sharing on the primary side and current sharing on the secondary side, making them widely used in applications with high input voltage, low output voltage, and large output current. For example, in electric vehicles, the DC bus voltage can reach up to 800 V and is expected to increase further. Additionally, the 1-kV bus is an emerging technical trend likely to see widespread use in warships and drones. Furthermore, the characteristic of low output voltage and high output current is common in modern energy storage systems, such as Tesla’s™ Powerwall, portable power stations, and residential battery storage systems. [00112] Matrix transformer 303 may use multiple (two or more) magnetic cores to construct the transformer system of a resonant converter. Each individual transformer in the matrix transformer 303 uses the same construction method to achieve optimal efficiency. This includes using identical core sizes, identical core materials, identical turn ratios, and identical winding arrangements, among other factors. FIG. 3D shows an FB LLC converter 300D where the SPWT configuration based matrix transformer 303 is implemented with two magnetic cores, containing P1304 and P2306. In some embodiments, a third winding may be included within one of the two magnetic cores. The third winding may be placed within either of the first or second cores and in series with P1304 or P2306. FIG.3E shows an FB LLC converter 300E where the SPWT configuration based matrix transformer 303 is implemented with three magnetic cores, containing P1304, P2306 and P3314, and P4316 respectively. [00113] In FIG. 3D and FIG. 3E, the transformers are connected in series on the primary side to withstand high voltage and in parallel on the secondary side to allow for large load current. This configuration achieves voltage sharing on the primary side and current sharing on the secondary side. [00114] When applying the SPWT configuration to a matrix transformer 303, the number of magnetic cores will impact the placement of the resonant tank 302. When the matrix transformer 303 uses an even number of magnetic cores (as seen in FIG.3D), such as 2n magnetic cores (n ≥ 1), the resonant tank should be placed between the primary windings of the nth and the (n+1)th magnetic cores (see in FIG. 3D, P1304 and P2306). When the

- 17 - CAN_DMS: \1006164620 matrix transformer 303 uses an odd number of magnetic cores (as seen in FIG.3E), such as 2n+1 magnetic cores (n ≥ 1), the primary winding of the (n+1)th magnetic core should be split into two (see in FIG.3E, P2306 and P3314), and the resonant tank should be placed between these two split windings. [00115] As shown in Fig.3D, there are two magnetic cores, namely Core #1 and Core #2. L_m1 (L_m2) is the magnetizing inductance of Core #1 (Core #2). The total magnetizing inductance is calculated as L_m1 + L_m2. L_r and C_r are the resonant inductor and resonant capacitor, respectively. The resonant inductor Lr can be realized by the leakage inductances of Core #1 and Core #2, or by a physical magnetic core with windings. P1304 (P2306) is the primary winding of Core #1 (Core #2). By connecting the resonant tank (L_r and C_r) between P1304 and P2306, complementary dv/dt distribution within P1304 and P2306 can be achieved. As long as the transformer 303 windings within Core #1 and Core #2 are symmetrically wound, the CM noise displacement currents within Core #1 and Core #2 can be completely canceled. [00116] As shown in FIG. 3E, there are three magnetic cores, namely Core #1, Core #2, and Core #3. L_m1 (L_m4) is the magnetizing inductance of Core #1 (Core #3). P1304 (P4316) is the primary winding of Core #1 (Core #3). Core #2’s primary winding is split into two separate windings P2306 and P3314. The resonant inductor L_r and resonant capacitor C_r are placed between the primary windings P2 306 and P3 314. L_m2 (L_m3) denotes the magnetizing inductance seen from the primary winding P2306 (P3314). M denotes the mutual inductance of L_m2 and L_m3. The total primary side magnetizing inductance is calculated as L_m1+L_m4+L_m2+L_m3+2M. The resonant inductor L_r can be realized by the leakage inductances of Core #1, Core #2, and Core #3, or by a physical magnetic core with windings. By connecting the resonant tank (L_r and C_r) between P2306 and P3314, complementary dv/dt distribution within P2306 and P3314, as well as P1304 and P4316, can be achieved. As long as the transformer 303 windings within Core #1, Core #2, and Core #3 are symmetrically wound, the CM noise displacement currents within Core #1, Core #2, and Core #3 can be completely canceled. [00117] Design of symmetrical transformer winding arrangement

- 18 - CAN_DMS: \1006164620 [00118] In a preferred embodiment, a planar transformer may be used to achieve a symmetrical transformer design. In further embodiments, other types of transformers, such as wire-wound transformers, could be used. For wire-wound transformers, a sandwich winding structure can be implemented to meet the symmetrical condition, which is a common winding structure used in switch-mode power supplies. The primary focus here is on methods for achieving symmetrical winding arrangement in planar transformers. [00119] The planar transformer structure may be suited for LLC resonant converters due to its low height, low leakage inductance, high repeatability, and excellent thermal characteristics. The planar transformer typically features a large interwinding capacitance between overlapping primary and secondary winding turns. Therefore, it is crucial to take this large parasitic interwinding capacitance into consideration to ensure the symmetrical winding arrangement for the planar transformer. However, the parasitic interwinding capacitance model of the transformer 200 in FIG.2 is a lump sum model. Therefore, it does not reflect the effects of parasitic capacitances between each winding turn and thus cannot guide the symmetrical winding arrangement of the planar transformer. The use of couple- turn and its one-capacitor model, and an example analysis of symmetrical winding arrangement for the planar transformer are discussed below. The use of complementary couple-turns implemented within LLC resonant converter 300C, 300D and 330E may achieve the symmetrical winding arrangement for the planar transformer to achieve EMI noise reduction. [00120] Review of Couple-Turn and Its One-Capacitor Model [00121] FIG.6 shows the 3D model of a couple-turn in the planar transformer. Couple-turn denotes a pair of overlapping winding turns that belong to the primary and secondary winding respectively. It is assumed that the overlapping primary and secondary winding turns have a symmetrical layout. Hence, the parasitic structural interwinding capacitance C_struct of the couple-turn can be calculated by Equation (8), where ε₀ denotes the permittivity of vacuum, ε_Insul denotes the relative permittivity of the insulation material, d_PS denotes the distance between two winding turns, L and w denote the length and width of the winding turns.

- 19 - CAN_DMS: \1006164620 _C ruct ^{^ ^ w ^ L} st 0 ^{^} Insul ₍₈₎

[00122] From the CM noise perspective, the parasitic structural interwinding capacitance of the couple-turn can be equivalently placed between a pair of corresponding terminals. There are two types of one-capacitor couple-turn models as shown in FIGs.7A and 7B. [00123] Four terminals of the couple-turn are denoted by a’, b’, c’, and d’, which correspond to the terminal sequence letters in FIG.6. Terminals a’ and c’ (b’ and d’) on the couple-turn are a pair of corresponding terminals. i_{CM_couple-turn} denotes the CM noise current generated in the couple-turn. Either of the two types of models shown in FIGs.7A and 7B can be used to analyze and characterize the CM noise current in the couple-turn. The selection criterion aims to acquire a CM noise model of the planar transformer that enables straightforward analysis. [00124] Asymmetric layouts of couple-turns may also be used in the design of planar transformers. For example, when the secondary winding of the transformer has one turn per layer while the primary winding has multiple turns per layer, the couple-turns formed by the turns of the primary and secondary winding will inevitably have an asymmetric layout. It should be noted that in this case, the one-capacitor couple-turn model is still valid. When calculating the structural capacitance of the couple-turn, the non-overlapping area needs to be removed. [00125] FIG. 8 gives a typical example when the layout of the couple-turn is asymmetrical and C_struct can be recalculated as shown in Equation (9), where w_P denotes the width of the primary winding turn. It is noted that the edge effect may introduce additional fringing capacitance. Equation (9) maintains its accuracy when most of the electric field energies are confined within the overlapping areas of the two winding turns. C_{truct ^ ^ 0 ^ Insul} ^{w P} s ^{^ L} (9)

- 20 - CAN_DMS: \1006164620 [00126] Example Analysis of Symmetrical Winding Arrangement for Planar Transformer [00127] A Printed Circuit Board (PCB) winding based planar transformer with a turns ratio of 8:2:2 is used here as an example to demonstrate the effectiveness of the proposed SPWT configuration. Two examples are analyzed which are both using the sandwich winding structure shown in FIGs. 9A, 9B and 9C but different implementations of primary winding turns. [00128] 1) Example planar transformer #1: The sandwich winding arrangement and schematic of example planar transformer #1 902 are shown in FIGs. 9A and 9B. The connections of winding terminals (A, B, C, D, E, F, and G) are shown in FIG. 3C. This transformer includes two couple-turns: P1_4-S1_2, (the couple-turn between primary winding P1_4 and secondary winding S1_2) and P2_4-S2_2, (the couple-turn between primary winding P2_4 and secondary winding S2_2), as shown in FIG.9A. Each turn in both the primary and secondary windings is implemented using a single layer. It is assumed that all turns have the same length and width, and the structural interwinding capacitances of the couple-turns P1_4-S1_2 and P2_4-S2_2 are considered equal by using the same insulation material with equal thickness, which are both denoted by C_cell. [00129] Utilizing the one-capacitor couple-turn model, a lumped interwinding capacitance model of transformer #1902 is shown in FIG. 9C. Here, v_B and v_{S1_2} (v_C and v_{S2_2}) denote the electric potentials of corresponding terminals for the couple-turn P1_4-S1_2 (P2_4- S2_2). The CM noise currents generated in couple-turns P1_4-S1_2 and P2_4-S2_2 are denoted by i_{CM#1_P1_4-S1_2} and i_{CM#1_P2_4-S2_2}, which can be calculated using Equations (10) and (11), respectively. d ( v_B ^{^} v _S 1 ) i CM # 1 _ P 1 _ 4 ^{^} S 1 _ 2 _^ C _ 2 cell ₍₁₀₎

[00130] As the value of the winding turn resistance is significantly smaller than that of the inductance, the voltage difference between any two adjacent winding turns can be

- 21 - CAN_DMS: \1006164620 considered as constant. Consequently, v_{S1_2} and v_{S2_2} are calculated as v_F/2 and v_G/2 which are complementary (i.e., v_{S1_2} = -v_{S2_2}), and similarly, v_B and v_C are also complementary (i.e., v_B = -v_C). As a result, i_{CM#1_P1_4-S1_2} = -i_{CM#1_P2_4-S2_2}, indicating the CM noise current generated in couple-turn P1_4-S1_2 can be completely canceled by the CM noise current generated in couple-turn P2_4-S2_2, as shown in Equation (12). The CM current path shown in FIG.9C illustrates that there is no additional CM current coupling into the secondary side, leading to a net CM noise current of zero. In conclusion, transformer #1902 is symmetrical from the CM noise perspective. d ( v C ^{^} v S 2_ 2 ) d ( v B ^{^} v S 1 ) i _ P 1_ 4 ^ S 1_ 2 _^ i CM #1_ P 2_ 4 ^ S 2_ 2 _^ _ 2 CM #1 C cell _^ C cell

[00131] 2) Example planar transformer #2: FIG.10A illustrates the winding arrangement of example planar transformer #2 1002. Different from transformer #1 902, the primary winding turns are implemented with two turns per layer. There are four couple-turns which are P1_4-S1_2, P1_3-S1_2, P2_4-S2_2, and P2_3-S2_2. It is noted that couple-turns P1_4- S1_2 and P2_4-S2_2 have the same structural interwinding capacitance since they have the same overlapping area and the same isolation material with equal thickness. The conclusion is the same for the couple-turns P1_3-S1_2 and P2_3-S2_2. The structural interwinding capacitances of couple-turns P1_4-S1_2 and P2_4-S2_2 (P1_3-S1_2 and P2_3-S2_2) are denoted by C_outer (C_inner). [00132] FIG.10B gives the selected one-capacitor models of couple-turns P1_3-S1_2 and P2_3-S2_2. v_{P1_3} and v_{S1_2} (v_{P2_3} and v_{S2_2}) are the electric potentials of corresponding terminals for the couple-turn P1_3-S1_2 (P2_3-S2_2). The CM noise currents generated in these two couple-turns are denoted by i_{CM#2_P1_3-S1_2} and i_{CM#2_P2_3-S2_2}, which can be calculated as shown in Equations (13) and (14), respectively.

- 22 - CAN_DMS: \1006164620 d ( v_P 1_3 ^{^} v _S 1_ 2 ) i CM #2_ P 1_3 ^{^} S 1_ 2 _^ C inner ₍₁₃₎

[00133] Assuming a constant voltage difference between any two adjacent primary winding turns, v_{P1_3} and v_{P2_3} are calculated as shown in Equation (15). ^_{v = v} ^{^} ₃ ( ^v A ^{^ v} B ) ^ ^v A ^{^ 3 v} B ^ ^{^} P 1 ^ 3 A

[00134] As a result of the complementary electric potentials of v_A and v_D, v_B and v_C, the potentials of v_{P1_3} and v_{P2_3} are complementary (i.e., v_{P1_3} = -v_{P2_3}). Additionally, since v_{S1_2} and v_{S2_2} are complementary as well (i.e., v_{S1_2} = -v_{S2_2}), i_{CM#2_P1_3-S1_2} and i_{CM#2_P2_3-S2_2} can be completely canceled by each other, as shown in Equation (16). d ( v P 1_3 ^{^} v S 1_ 2 ) d ( v ^{^} v ) i _^ i _^ C _^ C P 2_3 S 2_ 2 CM #2_ P 1_3 ^ S 1_ 2 CM #2_ P 2_3 ^ S 2_ 2 inner inner

[00135] FIG.10C gives the selected one-capacitor models of couple-turns P1_4-S1_2 and P2_4-S2_2. v_B and v_{S1_2} (v_C and v_{S2_2}) are the electric potentials of corresponding terminals for the couple-turn P1_4-S1_2 (P2_4-S2_2). The CM noise currents generated in these two couple-turns are denoted by i_{CM#2_P1_4-S1_2} and i_{CM#2_P2_4-S2_2}, which can be calculated as shown in Equations (17) and (18), respectively.

- 23 - CAN_DMS: \1006164620 d ( v_B ^{^} v _S 1_ ) i 2 CM #2_ P 1_4 ^{^} S 1_2 _^ C outer ₍₁₇₎ ) (₁₈₎

[00136] Owing to the complementary electric potentials of v_B and v_C, v_{S1_2} and v_{S2_2}, i_{CM#2_P1_4-S1_2} and i_{CM#2_P2_4-S2_2} can be completely canceled by each other, as shown in Equation (19). Therefore, the total CM noise displacement current generated in transformer #21002 is zero, which indicates that transformer #21002 is symmetrical from the CM noise perspective. d ( v B ^{^} v S ) d ( v ^{^} v ) i _^ 1_ 2 C S 2_ 2 CM #2_ P 1_ 4 ^ S 1_ 2 i CM #2_ P 2_ 4 ^ S 2_ 2 _^ C outer _^ C outer

[00137] Concept of Complementary Couple-Turns [00138] The use of complementary couple-turns is proposed as a beneficial solution to achieve asymmetrical winding arrangement for the planar transformer. Complementary couple-turns refer to a pair of couple-turns that generate complementary CM noise currents, which can be completely canceled by each other. For instance, in example planar transformer #1902, P1_4-S1_2 and P2_4-S2_2 are a pair of complementary couple-turns. Similarly, as shown in example planar transformer #2 1002, P1_4-S1_2 and P2_4-S2_2 (P1_3-S1_2 and P2_3-S2_2) are also a pair of complementary couple-turns. [00139] Center-tapped planar transformer: In SPWT configuration based center-tapped planar transformer 1100, there are two types of complementary couple-turns, as shown in FIGs.11A and 11B. [00140] FIG. 11A shows the complementary couple-turns P1_m-S1_n and P2_m-S2_n. FIG. 11B shows the complementary couple-turns P1_m-S2_n and P2_m-S1_n. P1_m

- 24 - CAN_DMS: \1006164620 (P2_m) is defined as the mth primary winding turn of P1 (P2) from terminal A (D). S1_n (S2_n) is defined as the nth secondary winding turn of S1310A (S2310B) from terminal E. N_s denotes the turns number of secondary windings S1310A and S2310B. N_P1 and N_P2 denote the turns number of primary windings P1304 and P2306, respectively. [00141] In FIGs. 11A and 11B, the corresponding terminals of all complementary couple- turns are denoted by black dots. It should be noted that the difference between couple-turn P1_m-S1_n in Fig. 11A and couple-turn P1_m-S2_n in Fig. 11B lies in the distribution of their corresponding terminals. Specifically, in P1_m-S1_n, the corresponding terminals of the primary and secondary winding turns are oriented in the same direction, while in P1_m-S2_n they are oriented in opposite directions. In other words, the dv/dt phase of the primary and secondary winding turns are the same in P1_m-S1_n, but opposite in P1_m-S2_n. As a result, there will be different CM noise behaviors for the couple-turns P1_m-S1_n and P1_m- S2_n, which is a reason for the existence of two types of complementary couple-turns. [00142] Based on the one-capacitor couple-turn model, FIGs. 11C and 11D give the lumped interwinding capacitance model of the couple-turns shown in FIGs. 11A and 11B, respectively. In FIG. 11C, v_{P1_m} and v_{S1_n} (v_{P2_m} and v_{S2_n}) denote the electric potentials of corresponding terminals for the couple-turn P1_m-S1_n (P2_m-S2_n). In FIG. 11D, v_{P1_m} and v’_{S1_n} (v_{P2_m} and v’_{S2_n}) denote the electric potentials of corresponding terminals for the couple-turn P1_m-S2_n (P2_m-S1_n). By ignoring the impact of the leakage inductance and assuming a unit coupling coefficient, the voltage difference between any two adjacent winding turns is considered as a constant value, denoted as Δv, which can be calculated by Equation (20). ^_v =^{vA ^ v B} _^ ^{v C ^ v D} _^ ^{v F ^ v E} _^ ^{v E ^ v G} N ₍₂₀₎ P 1 N P 2 N S N S [00143] v_{P1_m}, v_{P2_m}, v_{S1_n}, v_{S2_n}, v’_{S1_n}, and v’_{S2_n} are calculated accordingly by Equations (21)-(26). v_{P 1_ m} ^{^ v} _A ^{^ m ^ v} ₍₂₁₎

- 25 - CAN_DMS: \1006164620 v_{P 2_ m} ^ v _D ^ m ^ v (22) ^ _{^ ^}

[00144] The CM noise currents generated in couple-turns P1_m-S1_n and P2_m-S2_n are denoted by i_{CM_P1_m-S1_n} and i_{CM_P2_m-S2_n}, which can be calculated as shown in Equations (27) and (28), by assuming a constant parasitic structural interwinding capacitance C₀ of each couple-turn. i ^{d ( v} _P 1_ _m ^{^ v} _S 1_ _n ^{) d} ₍ ^v _A ^{^} ₍ ^{m ^ n ^} ₁₎ ^{^ v} ₎ CM _ P 1_ m ^{^} S 1_ n _{^ C} 0 _{^ C} 0 ₍₂₇₎

[00145] Based on Equations (27) and (28), since v_A = -v_D, it can be concluded that i_{CM_P1_m- S1_n} = -i_{CM_P2_m-S2_n}, implying that couple-turns P1_m-S1_n and P2_m-S2_n have complementary CM noise displacement currents that can be canceled by each other. Therefore, P1_m-S1_n and P2_m-S2_n are a pair of complementary couple-turns. [00146] Likewise, based on FIG. 11D, the CM noise displacement currents generated in couple-turns P1_m-S2_n and P2_m-S1_n are given in Equations (29) and (30). i M _ P1 _ m ^{^} S 2 _ n _^ ^{d ( v} _P 1 _ _m ^{^ v '} _S 2 _ _n ^{) d} ₍ ^v _A ^{^} ₍ ^{m ^ n} ₎ ^{^ v} ₎ C C 0 _^ C 0 ₍₂₉₎

[00147] Since v_A = -v_D, it can be concluded that i_{CM_P1_m-S2_n} = -i_{CM_P2_m-S1_n}, implying that P1_m-S2_n and P2_m-S1_n are also a pair of complementary couple-turns.

- 26 - CAN_DMS: \1006164620 [00148] The above analysis is based on full-wave rectifier, where two secondary windings are used. It is noted that the use of complementary couple-turns remains applicable even with a Full-Bridge (FB) rectifier on the secondary side. FIG.12 shows the circuit diagram of the LLC converter 1204 with FB rectifier 1206, where the transformer 1202 has only one secondary winding (i.e., a two-winding transformer). [00149] Two-winding planar transformer: FIG. 13A illustrates the schematic of a two- winding planar transformer 1300 based on the SPWT configuration. The transformer winding ratio is N_P1:N_P2:2N_S (assuming an even number of turns in the secondary winding). The secondary winding terminals F and G are connected to the FB rectifier, implying that their respective electric potentials, v_F and v_G, are complementary. Assuming an even distribution of dv/dt across the winding, the middle node E of the secondary winding maintains a constant electric potential, resulting in a dv/dt of zero at middle node E. Consequently, from the perspective of CM noise coupling, the middle node E can be considered equivalently connected to PE. By dividing the secondary winding of the transformer 1300 from the middle node E into two segments, the CM current of this two-winding planar transformer 1300 can be analyzed by following the analysis approach of the center-tapped planar transformer. As shown in FIG. 13B, the secondary winding is divided into two segments, S1310A and S2 310B. By defining S1_n (S2_n) as the nth secondary winding turn of S1310A (S2310B) from the middle node E, the CM noise currents generated in the couple-turns P1_m-S1_n and P2_m-S2_n can also be calculated by using Equations (27) and (28). Therefore, P1_m- S1_n and P2_m-S2_n form a complementary couple-turn pair. The same conclusion can be drawn when analyzing the couple-turns P1_m-S2_n and P2_m-S1_n. [00150] When the number of turns in the secondary winding is an odd number (2N_S + 1), the middle node E is positioned at the midpoint of the (N_S + 1)th turn of the secondary winding, as shown in FIG. 13C. In this scenario, when dividing the secondary winding into segments S1310A and S2310B, the (N_S + 1)th turn of the secondary winding serves as both the first turn of S1310A and S2310B, as illustrated in FIG.13D. In this figure, v_{S1_n} and v_{S2_n} can be calculated using Equations (31) and (32) respectively. v^S 1_ ^{n ^} ₍ ^{n ^ 3} ) ^{^ v} 2 ₍₃₁₎

- 27 - CAN_DMS: \1006164620 ^vS 2_ ^{n ^ ^} ₍ ^{n ^ 3} ) ^{^ v} ₍₃₂₎

[00151] The CM noise currents generated in the couple-turns P1_m-S1_n and P2_m-S2_n can be calculated by Equations (33) and (34). By comparing Equations (33) and (34), it can be concluded that i_{CM_P1_m-S1_n} = -i_{CM_P2_m-S2_n}, implying that P1_m-S1_n and P2_m-S2_n form a complementary couple-turn pair. The same conclusion can be drawn when analyzing the couple-turns P1_m-S2_n and P2_m-S1_n. d^{( v ^ ( m ^ 3} d_{( vP m} ^{^} _{v S n ) A} ^{n ^ ) ^ v )} ₎

[00152] The use of complementary couple-turns enables the effective reduction of CM noise in planar transformers. The cancellation of CM noise is achieved by designing the winding arrangement and interwinding capacitance in such a way that the CM noise currents generated in the complementary couple-turns are equal in magnitude and opposite in phase. [00153] Symmetrical Winding Arrangement Based on Complementary Couple-Turns [00154] Based on the previous analysis, the planar transformer winding arrangement can be symmetrical from the CM noise perspective and the total CM noise displacement current generated in the planar transformer can be fully canceled when the following two conditions are satisfied: [00155] 1) Overlapping layers should be selected based on the concept of complementary couple-turns. There are two types of complementary couple-turns: P1_m-S1_n and P2_m- S2_n, as well as P1_m-S2_n and P2_m-S1_n. Definitions for P1_m-S1_n, P2_m-S2_n, P1_m-S2_n and P2_m-S1_n are provided in FIG. 11 for the center-tapped transformer structure 1100 and in FIG.13 for the two-winding transformer 1300 structure.

- 28 - CAN_DMS: \1006164620 [00156] As shown in FIG. 11, for the center-tapped transformer structure 1100, P1_m (P2_m) is defined as the mth primary winding turn of P1304 (P2306) from terminal A (D). S1_n (S2_n) is defined as the nth secondary winding turn of S1 310A (S2 310B) from terminal E. [00157] As shown in FIG.13, for the two-winding transformer structure 1300, P1_m (P2_m) is defined as the mth primary winding turn of P1 304 (P2 306) from terminal A (D). By dividing the secondary winding of the transformer from the middle node E into two segments, S1310A and S2310B, S1_n (S2_n) is then defined as the nth secondary winding turn of S1 310A (S2310B) from the middle node E. [00158] If the mth winding turn of P1304 is overlapped with the nth winding turn of S1 310A, the mth winding turn of P2306 needs to be overlapped with the nth winding turn of S2 310B so as to generate the complementary CM noise current. Likewise, If the mth winding turn of P1304 is overlapped with the nth winding turn of S2310B, the mth winding turn of P2 306 needs to be overlapped with the nth winding turn of S1310A. [00159] 2) The complementary couple-turns should be designed to have the same parasitic structural interwinding capacitances. This can be achieved through the symmetrical layout of complementary couple-turns. It should be noted that the thicknesses of insulation layers in most multi-layer PCBs are not consistent. When constructing planar transformers using a single PCB board, attention should be given to the interlayer FR4 distance to ensure the same parasitic structural interwinding capacitances of complementary couple-turns. [00160] Complementary couple-turns provide the advantage of utilizing the inherent coupling capacitance between the primary and secondary side of a planar transformer to cancel out CM currents generated within the transformer. This eliminates the need for additional shielding layers or extra transformer windings, which results in reduction in manufacturing costs and transformer winding losses. Furthermore, the utilization of complementary couple-turns is entirely compatible with the interleaved winding structure. This compatibility arises from the fact that the overlapping layers introduced by the interleaved winding structure can be harnessed to create complementary couple-turns. For conventional transformers, the interleaved winding structure can reduce leakage inductance

- 29 - CAN_DMS: \1006164620 and AC winding losses. However, the interleaved winding structure increases the transformer's parasitic interwinding capacitance, leading to deteriorated CM noise performance. By using the proposed SPWT configuration and the symmetrical winding method with complementary couple-turns for planar transformers, the trade-off between interleaved winding structure and CM noise performance is eliminated. That is, smaller transformer leakage inductance and AC winding losses, as well as minimized CM noise, can be achieved. [00161] The above analysis assumes that the primary winding has even number of turns. When the primary winding has odd number of turns, proposed complementary couple-turns can still be utilized to ensure a symmetrical transformer winding arrangement when using the SPWT configuration. A planar transformer with a turns ratio of 3:1:1 is used as an example. [00162] FIG. 14 illustrates the winding arrangement of example planar transformer #3 1400. The connections of winding terminals (A, B, C, D, E, F, and G) are shown in FIG.3C. In FIG. 14, the secondary windings S1310A and S2310B of the transformer 1400 each have only one turn, namely S1_1 and S2_1. The primary winding P2306 has one turn, namely P2_1. The primary winding P1304 has two turns, namely P1_1 and P1_2. P1_2 is split into two parallel winding turns that are placed on the top and bottom layers, respectively. As shown in Fig.14, P1_1-S1_1 and P2_1-S2_1 represent a complementary pair of couple- turns. Based on Equations (27) and (28), the CM noise currents generated in these two couple-turns can be calculated as shown in Equations (35) and (36), respectively. Δv represents the voltage difference between any two adjacent winding turns in the transformer 1400, which is considered as a constant value. i ^d ( ^v _A ^{^ ^ v} ) CM #3_ P 1_1 ^{^} S 1_1 _{^ C} P 1_1 ^{^} S 1_1 ₍₃₅₎

[00163] The parasitic structural capacitances of P1_1-S1_1 and P2_1-S2_1 are denoted by C_{P1_1-S1_1} and C_{P2_1-S2_1}, which are considered equal by using the same insulation material with equal thickness. Since v_A and v_D are complementary (i.e., v_A = -v_D), it follows that

- 30 - CAN_DMS: \1006164620 i_{CM#3_P1_1-S1_1} and i_{CM#3_P2_1-S2_1} are also complementary. In conclusion, the CM noise currents generated in these two couple-turns can mutually cancel each other. [00164] P1_2 in transformer #31400 is split into two parallel winding turns that are placed on the top and bottom layers, respectively. P1_2 does not overlap with the turns of the secondary winding. As a result, P1_2 does not generate any additional CM noise current, and transformer #31400 remains symmetrical from the CM noise perspective. [00165] When the number of turns in the original primary winding of a planar transformer is odd, it is not possible to have an equal number of turns in both P1 304 and P2 306. Specifically, if the number of turns in P1304 is one more than that of P2306 (i.e., N_P1 = N_P2 + 1), an additional CM noise displacement current will be induced by the primary winding turn P1_N_P1 if it overlaps with the secondary winding turns. To avoid this issue, it is necessary to ensure that P1_N_P1 does not overlap with the secondary winding turns. In the above example (see FIG. 14), P1_2 does not overlap with any secondary winding turns. P1_2 is implemented with two parallel turns. One turn is placed on the top layer and the other is placed on the bottom layer. [00166] Extending to Other Resonant Converters [00167] The proposed Symmetrical Primary Winding Transformer 301 (SPWT) configuration can be applied to various Pulse-Frequency-Modulation (PFM) based FB resonant converters, including Series Resonant Converters (SRC), Parallel Resonant Converters (PRC), and LCC resonant converters. In some embodiments, the SPWT configuration may be applied to phase-shift control based FB resonant converters. FIGs.15, 16 and 17 depict the topology diagrams of the SRC, PRC, and LCC converters with the implemented SPWT structure. In the proposed SRC converter 1500 (FIG. 15), the primary winding of the transformer 301 is split into two windings 304 and 306, with the resonant tank 1502 (L_r and C_r) placed between the separated windings, distinguishing it from the conventional SRC converter. Similarly, in the proposed PRC converter 1600 (Fig.16), both the primary winding of the transformer 301 and the parallel resonant capacitor 1604 connected in parallel to it are split into two parts, with the resonant inductor 1602 (L_r) placed between the separated windings, distinguishing it from the conventional PRC converter. If

- 31 - CAN_DMS: \1006164620 windings P1304 and P2306 have the same number of turns, then both C_r1 and C_r2 are equal to 2C_r, where C_r represents the required resonant capacitor value. Finally, in the proposed LCC converter 1700 (FIG. 17), the primary winding of the transformer 301, along with the parallel resonant capacitor 1702 that is connected in parallel to the primary winding, are split into two parts, with the resonant tank 1704 (L_r and C_r) placed between the separated windings, distinguishing it from the conventional LCC converter. If windings P1 304 and P2306 have the same number of turns, then both C_p1 and C_p2 are equal to 2C_p, where C_p represents the required value of the parallel resonant capacitor. [00168] The CM noise equivalent circuits of FIGs. 15, 16 and 17 are exactly the same as the circuits illustrated in FIG. 5, which will not be described herein. Consequently, the CM noise currents generated in the transformers can be completely canceled with the implementation of a symmetrical winding structure. [00169] Practical Considerations [00170] This section discusses the influences of SPWT configuration on the FB LLC converter 300C, including transformer leakage inductance, transformer winding loss and voltage conversion ratio. [00171] In FIG.18A, 18B and 18C, two center-tapped planar transformers with a turns ratio of 16:1:1 are presented. Fig. 18A corresponds to the transformer 301 designed for the proposed FB LLC converter 300C as shown in FIG. 3C, while Fig. 18C corresponds to the transformer 106 designed for the conventional FB LLC converter 100 as shown in FIG.1. [00172] For a fair comparison, apart from the different terminal connections of the primary windings, the construction methods for these two transformers 301, 106 may be identical. It should be pointed out that the proposed planar transformer 301 can be rendered identical to the conventional planar transformer 106 by short-circuiting terminals B and C, and subsequently connecting the resonant tank 302 to terminal A. [00173] As shown in Fig. 18C, for the conventional planar transformer 106, there is only one primary winding which is denoted by P1’. P1’_1 is the first primary winding turn which is connected to the resonant tank. P1’_16 is the last primary winding turn which is connected to

- 32 - CAN_DMS: \1006164620 Q₃ (Source) in FIG.1. As shown in FIG.18A, for the proposed planar transformer 301, there are two primary windings which are P1304 and P2306. P1_1 is the first primary winding turn of P1304, which is connected to Q₁ (Source) in FIG. 3C. P2_1 is the first primary winding turn of P2306, which is connected to Q₃ (Source) in FIG.3C. The resonant tank 302 is connected between P1_8 and P2_8, which are the last primary winding turns of P1304 and P2306, respectively. The remaining specifications for these two transformers 301, 106 are identical and can be summarized as follows: [00174] 1) DMR96 ferrite core material from DMEGC is selected and the core size is ECW34C (customized from DMEGC). The specific structure and dimensions of ECW34C are provided in FIG. 19. There is an approximately 0.22 mm air gap in the middle of the center column and the two side columns of the magnetic core. Three air gaps are all filled with Kapton tapes. [00175] 2) The transformer windings are designed with a single turn per layer, each having the same width of 8 mm. Instead of the sandwich winding structure shown in FIG. 9A and FIG. 10A, a partial interleaved winding structure is employed by stacking six PCBs. The choice of a partial interleaved winding structure in the final design aims to achieve lower AC losses for the transformer windings, which provides higher system efficiency. Specifically, the primary windings are constructed by using four 4-layer 4 OZ PCBs with a PCB board thickness of 1.63 mm, while the secondary windings are constructed by using two 8-layer 4 OZ PCBs with a PCB board thickness of 2.41 mm. Detailed layer stack-up structures of the primary and secondary winding PCBs are provided in Table 0.1 and Table 0.2, respectively. The insulation layers between different PCBs are implemented by 0.25 mm electrical insulation papers with a relative permittivity of 2.6. [00176] In further embodiments, different planar transformer parameters can be used, including different core materials from different manufacturers, varying transformer air gap lengths, different PCB winding widths, and different PCB board thicknesses. The construction of complementary couple-turns imposes no restrictions on these transformer parameters, providing designers with significant flexibility in designing planar transformers. The limitation introduced by complementary couple-turns is that they should be designed to have the same parasitic structural interwinding capacitances. For example, as shown in FIG.

- 33 - CAN_DMS: \1006164620 18A, P1_4-S1_1 and P2_4-S2_1 are a pair of complementary couple-turns. To meet the requirement of having the same parasitic structural interwinding capacitances, the insulation layers within P1_4-S1_1 and P2_4-S2_1 may use the same type of material with equal thickness. Table 0.1: Stack-up structure of primary winding PCBs Layer Stack up Thickness (mm) Solder Mask 0.02 Layer1 3 OZ+ Plating 0.14 Prepreg (FR4) 0.28 Layer2 4 OZ 0.14 Core (FR4) 0.47 Layer3 4 OZ 0.14 Prepreg (FR4) 0.28 Layer4 3 OZ + Plating 0.14 Solder Mask 0.02 Table 0.2: Stack-up structure of secondary winding PCBs Layer Stack up Thickness (mm) Solder Mask 0.025 Layer1 3 OZ + Plating 0.14 Prepreg (FR4) 0.16 Layer2 4 OZ 0.14 Prepreg (FR4) 0.12 Layer3 4 OZ 0.14 Prepreg (FR4) 0.28 Layer4 4 OZ 0.14 Core (FR4) 0.12

Layer6 4 OZ 0.14 Prepreg (FR4) 0.12 Layer7 4 OZ 0.14 Prepreg (FR4) 0.16 Layer8 3 OZ + Plating 0.14 Solder Mask 0.025

- 34 - CAN_DMS: \1006164620 [00177] For the proposed planar transformer 301, there are two pairs of complementary couple-turns which are P1_4-S1_1 and P2_4-S2_1, P1_5-S2_1 and P2_5-S1_1. Since 6 PCBs are stacked by using the same type of insulation material with the same thickness, the complementary couple-turns have the same parasitic structural interwinding capacitances. Thus, this planar transformer 301 is symmetrical from the CM noise perspective and the total CM noise displacement current is significantly reduced. [00178] Transformer Leakage Inductance and Winding Loss [00179] The leakage inductance referred to the primary side can be calculated by Equation (37): E ^{^} ₂ 1 ₂ e^{nergy ^} ₀ ^ ^{H dV ^ L lk I P}

[00180] where L_lk is the leakage inductance in primary side, I_P is the current of primary winding, and H is the magnetic field intensity in the window. [00181] Equation (37) demonstrates that the energy stored in the leakage inductance equals the leakage magnetic field energy. By calculating the total energy of the leakage magnetic field, the theoretical value of the leakage inductance can be determined. Here, the total energy of the leakage magnetic field refers to the summation of energies stored in the primary winding layers, secondary winding layers, and insulation layers. [00182] For the proposed planar transformer 301, because the primary windings P1304 and P2306 are connected in series, the currents in these windings are equal and denoted as I_P. Assuming the magnetic field intensity is constant in the horizontal direction, the leakage inductance can be calculated as shown in Equation (38). ^ ^{^ b l} x L _{0 H} ( _x ) ² ₀ w w ² lk ^{^} 2 ^ _dx ^{^} 2 ^ _H ( _x ) _dx ⁽³⁸⁾

- 35 - CAN_DMS: \1006164620 [00183] where V is the total volume of window area, b_w is the width of magnetic core window, l_w is the depth of the magnetic core, H(x) is the magnetic field intensity in the direction the geometric position x (see FIG. 18B). Based on Equation (38), the leakage inductance is proportional to the integral of the square of the magnetic field intensity. Hence, the distribution of H(x)² provides an intuitive reflection of the magnitude of the leakage inductance. The graphical interpretation of the H(x)² distribution is shown in Fig. 18B, in accordance with the Ampere's circuital theorem. It's noted that secondary windings S1310A and S2310B conduct alternately for half of the switching period, and Fig.18B is based on the operating scenario where S1310A conducts. Due to the identical winding arrangement in both the proposed SPWT and conventional planar transformers 301, 106, they exhibit the same H(x)² distribution. In other words, their leakage inductance performance is the same. The same conclusion can be drawn when analyzing the operating scenario where S2310B conducts. [00184] It should be pointed out that the previous discussion does not take into account the effects of eddy current when the transformer 301 operates under high-frequency (HF) conditions. In HF range, both the skin effect and proximity effect cause the winding current to concentrate near the conductor's surface. This may lead to an inhomogeneous distribution of H(x)², consequently changing the total leakage magnetic field energy. [00185] On the other hand, the skin effect and proximity effect also contribute to increased AC resistance of the transformer windings. This can be explained by Ferreira's formula provided in J. A. Ferreira, "Improved analytical modeling of conductive losses in magnetic components”, as shown in Equation (39). R_{tx _ ac} ^{^ R} _{tx _ dc} ^{^ ( F} _skin ^{^ F} _proximity ⁾ ₍₃₉₎

[00186] where F_skin and F_proximity represent the skin effect ratio and proximity effect ratio, respectively. Their expressions are given by Equations (40) and (41). F ^{^ sinh( ^ ) ^ sin( ^ ) h skin ^ ^} ^ _{^ ^} ^， _{^ = ^ (40)}

- 36 - CAN_DMS: \1006164620 _{F =} ^ ^ ₍ ^{2 F ( h ) ^} _{1) 2} ^{^ sinh(} ^ ^{) ^ sin(} ^ ^{) ^ proximity} _^ ^{^} _{^ ^} cosh( ^ ) _^ cos( ^ ) _{^ ^} ⁽⁴¹⁾

[00187] where h denotes the thickness of conductors, and δ stands for the skin depth at the operating frequency. F(h) and F(0) correspond to the Magnetomotive Forces (MMFs) at the borders of the conductor, which depend on the transformer's winding arrangement. To reduce the influence of the skin effect, the general design criterion is to ensure that the conductor thickness remains less than twice the skin depth. To reduce the influence of the proximity effect, an interleaved structure is commonly employed in planar transformer applications. This structure reduces the MMF that drives lower F_proximity. Both the proposed SPWT 301 and conventional planar transformer 106 have equal conductor thickness for their primary and secondary windings, resulting in the same F_skin value under the same operating frequency. Furthermore, their winding arrangements are identical, theoretically yielding the same MMF distribution and the same F_proximity value under the same operating frequency. Consequently, these two transformers 301, 106 have equal additional losses introduced by the skin and proximity effects. [00188] To quantitatively analyze the leakage inductance and winding loss, a 3D Finite Element Analysis (FEA) simulation is conducted using Maxwell software. FIGs. 20A and 20B presents 3D models of the proposed and conventional transformers 301, 106. [00189] The interlayer distances of transformer winding turns are selected based on Table 0.1 and Table 0.2. To save simulation time, PCB vias are removed and the rounded corners of the magnetic core are replaced by right angles. The Maxwell solver is eddy current type and the excitation is 150 kHz current. As mentioned earlier, secondary windings S1310A and S2 310B conduct alternately during half of the switching period. Therefore, each simulation model in FIGs. 20A and 20B is executed twice to determine the transformer's leakage inductance and AC/DC resistance ratio. [00190] FIG. 21A illustrates the mutual inductance model of the conventional transformer 106 employed in the simulation, corresponding to the operating scenario where S1310A conducts. Here, i_P and i_S1 represent the current excitations for the primary winding P1’ and

- 37 - CAN_DMS: \1006164620 secondary winding S1310A, respectively. L₁₁, L₂₂ and M₁₂ denote the values of self and mutual inductances. [00191] FIG.21B presents the π model of the conventional transformer 106. L_lkP1’ and L_lkS1 denote the leakage inductances for the primary winding P1’ and secondary winding S1 310A, respectively. L_m denotes the magnetizing inductance. The total leakage inductance referred to the primary side is denoted by L_{lkP_S1_conv}, which can be measured from the primary winding terminals when the secondary winding S1310A is short-circuited. Notably, L_{lkP_S1_conv} includes both the primary leakage inductance L_lkP1’ and secondary reflected leakage inductance. When S1310A is short-circuited, the secondary leakage inductance L_lkS1 is reflected to the primary side, resulting in L_lkS1’, as illustrated in FIG. 21C. Consequently, L_{lkP_S1_conv} is approximated as L_lkP1’ + L_lkS1’. [00192] In FIG.21A, the overall relationship is given by Equation (42). ^ ^v ^ P 1 ^{’ ^} ^ ^{^ ^ L} ^ 11 ^M 12 ^{^} _{d ^} ^{^ i} ^ P ^{^} ^ ₍₄₂₎

[00193] By using Laplace Transform, the relationship shown in Equation (42) can be rewritten by Equation (43) with matrix operation. By setting v_S1 = 0 and v_P1’ = 1, L_{lkP_S1_conv} can be obtained, which is equal to 1/G₁₁. ^ ⁱ ^ _P ^{^} 1 ^ ^{^ ^ ^G} ^ 11 ^G 12 ^{^} _{d ^} ^{^ v} ^ _P 1 ^{’ ^} ^ ₍₄₃₎

[00194] Aforementioned method can be utilized to obtain the value of L_{lkP_S2_conv}, which represents the total leakage inductance referred to the primary side when the secondary winding S21304 is short-circuited. [00195] FIG. 22A illustrates the mutual inductance model of the proposed SPWT transformer 301 employed in the simulation, corresponding to the operating scenario where S1 310A conducts. Here, i_P and i_S1 represent the current excitations for the primary and

- 38 - CAN_DMS: \1006164620 secondary windings, respectively. L₁₁, L₂₂, and L₃₃ denote the self inductances for transformer windings P1304, P2306, and S1310A. M₁₂, M₁₃, and M₂₃ denote the mutual inductances. FIG.22B presents the π model of the proposed transformer 301. L_lkP1 and L_lkP2 denote the leakage inductances for the primary windings. The total leakage inductance referred to the primary side is denoted by L_{lkP_S1_prop}, which can also be measured from the primary winding terminals when the secondary winding S1310A is short-circuited. Notably, L_{lkP_S1_prop} includes both the primary leakage inductances and secondary reflected leakage inductances. When S1310A is short-circuited, the secondary leakage inductance L_lkS1 is reflected to the primary side, resulting in L_lkS’ and L_lkS’’, as illustrated in FIG. 22C. Consequently, L_{lkP_S1_prop} is approximated as L_lkP1 + L_lkP2 + L_lkS1’+ L_lkS1’’. [00196] In FIG.22A, the overall relationship is

by Equation (44). ^ v P 1 ^{^ ^} L 11 M 12 M 13 ^{^ ^} i ^{^ d} P ^ ^{^ ^ ^ ^ ^}

[00197] By using Laplace Transform, the relationship shown in Equation (44) can be rewritten by Equation (45) with matrix operation. ^ i ^{^ ^} G 11 G 12 G 13 ^{^ ^} v 1 ^{^ ^ P ^} 1 ^{^ ^ ^ P ^}

[00198] By setting v_S1 = 0, the relationship between v_P1, v_P2, and i_P is expressed as shown in Equation (46). S ^{^} G ^ 11 G ^{^} 1 12 ^{^} ^ ^{^} i ^ P ^{^} ^ ^{^ ^} v ^ P 1 ^{^} ^ (46)

[00199] By solving Equation (46), L_{lkP_S1_prop} can be derived as shown in Equation (47).

- 39 - CAN_DMS: \1006164620 _L ^G 11 ^{^ G} 22 ^{^} 2 ^G lkP ^_ S ^1_ prop ^{^} 12 ^ _{2 (47)}

[00200] Aforementioned method can be utilized to obtain the value of L_{lkP_S2_prop}, which represents the total leakage inductance referred to the primary side when the secondary winding S21304 is short-circuited. [00201] The self-inductance and mutual-inductance matrices for both conventional and proposed planar transformers 106, 301 are computed using Maxwell and presented in Table 0.3-Table 0.6. It is noted that in these tables, some mutual inductance values are negative. For instance, in Table 0.3, the mutual inductance between the winding P1’_conv and the winding S1_Conv is -10.92 μH. This indicates that the magnetic fluxes generated by the currents flowing through the P1’_conv and S1_Conv windings are in opposite directions. [00202] It should be pointed out that all inductance values in Table 0.3-Table 0.6 are deliberately retained to five significant figures. This ensures the accuracy of the leakage inductance calculation results. In practical calculations, the fewer significant figures retained for the inductance values in Table 0.3-Table 0.6, the greater the deviation in the calculated leakage inductance results. Using inductance values with at least five significant figures in simulations is a good practice. Table 0.3: Simulated self and mutual inductance matrix for conventional transformer 106 at 150 kHz switching frequency: S1310A conducts while S2310B is OFF (Unit: μH) P1’_Conv S1_Conv

S1_Conv -10.915 0.68625

- 40 - CAN_DMS: \1006164620 Table 0.4: Simulated self and mutual inductance matrix for conventional transformer 106 at 150 kHz switching frequency: S2310B conducts while S1310A is OFF (Unit: μH) P1’_Conv S2_Conv P1’_Conv 175.13 -10.939 S2_Conv -10.939 0.6877 Table 0.5: Simulated self and mutual inductance matrix for proposed transformer 301 at 150 kHz switching frequency: S1310A conducts while S2310B is OFF (Unit: μH) P1_Prop P2_Prop S1_Prop

P2_Prop 43.057 44.204 -5.4468 S1_Prop -5.4546 -5.4468 0.68539 Table 0.6: Simulated self and mutual inductance matrix for proposed transformer 301 at 150 kHz switching frequency: S2310B conducts while S1310A is OFF (Unit: μH) P1_Prop P2_Prop S2_Prop P1_Prop 44.265 43.107 -5.4541 P2_Prop 43.107 44.255 -5.46 S2_Prop -5.4541 -5.46 0.68614 [00203] Subsequently, the total leakage inductances for the conventional and proposed planar transformers 301 are calculated and provided in Table 0.7. Table 0.7: Simulated leakage inductances (Unit: μH)

- 41 - CAN_DMS: \1006164620 _Conventional ^{LlkP_S1_conv 1.14} Proposed

[00204] It can be observed that, whether the secondary winding S1310A is conducting or the secondary winding S2 310B is conducting, the conventional and proposed planar transformers 106, 301 exhibit the same total leakage inductance. This observation validates the analysis that the utilization of the SPWT configuration does not affect the planar transformer's leakage inductance. [00205] As shown in Equation (39), the AC/DC resistance ratio refers to the summation of F_skin and F_proximity, which arises from the skin and proximity effects and determines transformer winding losses. The benchmark for calculating the AC/DC resistance ratio is established through a 60-Hz simulation, during which the skin and proximity effects are negligible. A comparison of solid losses between simulation results at 150 kHz and 60 Hz enables a convenient determination of the ac/dc resistance ratio. The 3D FEA simulation results are presented in Table 0.8-Table 0.11. Table 0.8: Simulated solid losses for conventional transformer 106: S1310A conducts while S2310B is OFF Frequency P1’_Conv S1_Conv 60 Hz 80.9 mW

150 kHz 143.7 mW 321.5 mW Table 0.9: Simulated solid losses for conventional transformer 106: S2310B conducts while S1310A is OFF Frequency P1’_Conv S2_Conv 60 Hz 80.9 mW 149.0 mW

- 42 - CAN_DMS: \1006164620 150 kHz 143.7 mW 312.4 mW Table 0.10: Simulated solid losses for proposed transformer 301: S1310A conducts while S2310B is OFF Frequency P1_Prop P2_Prop S1_Prop 60 Hz

150 kHz 70.8 mW 72.6 mW 321.2 mW Table 0.11: Simulated solid losses for proposed transformer 301: S2310B conducts while S1310A is OFF Frequency P1_Prop P2_Prop S2_Prop 60 Hz

150 kHz 70.5 mW 73.0 mW 312.9 mW [00206] As shown in Table 0.8, the solid losses for the primary winding P1' are approximately 80.9 mW at 60 Hz and 143.7 mW at 150 kHz. Hence, the AC/DC resistance ratio for P1' at 150 kHz can be calculated as 143.7 mW / 80.9 mW ≈ 1.78. Similarly, the AC/DC resistance ratio for secondary winding S1310A at 150 kHz can be calculated as 321.5 mW / 150.2 mW ≈ 2.14. Table 0.12 summarizes the calculation results for the AC/DC resistance ratios based on Table 0.8-Table 0.11. It is observed that the conventional and proposed planar transformers 106, 301 exhibit the same AC/DC resistance ratios for both primary and secondary windings. This observation validates the analysis that the utilization of the SPWT configuration does not introduce extra winding loss. Table 0.12: Simulated AC/DC resistance ratios at 150 kHz Primary Secondary winding Secondary winding

- 43 - CAN_DMS: \1006164620 winding(s) S1 S2 Conventional 1.78 2.14 2.10 Proposed 1.77 2.14 2.10 [00207] In summary, the SPWT configuration based planar transformer 301, when compared to the conventional planar transformer 106, demonstrates similar performance in terms of leakage inductance and winding loss. This similarity arises from the fact that the two primary windings in the proposed planar transformer 301 are connected in series, ensuring identical conduction currents in both primary windings. By utilizing the same construction approach as the conventional planar transformer 106, the proposed transformer 301 presents the equivalent leakage magnetic field energy and eddy current effects. [00208] Voltage Conversion Ratio [00209] The voltage regulation of LLC converters can be affected by interwinding capacitances when employing the SPWT configuration. As shown in FIG. 23, these capacitances include not only those between the primary and secondary windings but also capacitances between the primary windings P1304 and P2306. [00210] Here, C_AD and C_BC are used to model the interwinding capacitors between P1304 and P2306. It can be observed that C_BC, C_BE and C_CE will participate in the resonance of L_r and C_r, potentially affecting the gain performance of the FB LLC converter 300C under HF operation conditions. [00211] The impedance of the resonant unit consisting of L_r, C_r, C_BC, C_BE, and C_CE is calculated in Equation (48) as: Z ^{C L} 2 r r ^{s ^ 1} ru_{( s )} ^{^} 2 ⁽⁴⁸⁾

[00212] Where, as shown in Equation (49):

- 44 - CAN_DMS: \1006164620 _s ^{^} _j ^{^} _s , ^{^} _s ^{^} 2 ^{^} _{f s} , _{C e} ^{^} _{C BC} ^{^ C BE C CE} ^ ₍₄₉₎

[00213] C_e represents the equivalent capacitance in parallel with the resonant tank 302. The impedance zero and pole of Z_ru are given by Equations (50) and (51), respectively. 1 f^{r ^} ^ ⁽⁵⁰⁾

[00214] When f_s = f_r, Z_ru is zero, equivalent to a short circuit, resulting in a voltage gain of 1 for the FB LLC converter 300C. In contrast, when f_s = f_p, Z_ru is infinite, equivalent to an open circuit, yielding a voltage gain of 0. By defining, as shown in Equation (52): k_e ^{^ Ce} _{, f} ^{f s} n ^{^} _{, Z r} ^{^} _j ^{^} _{s L r} ^{^ 1} ^ ₍₅₂₎

[00215] Z_ru can be normalized and rewritten as: Z ^Z 1 ru ^_ n ^{^} ru ^{^} ^ _{^ 2} ⁽⁵³⁾

[00216] where Z_r represents the base impedance, corresponding to the series impedance of L_r and C_r. [00217] FIG.24 illustrates the variation of Z_{ru_n} concerning frequency changes. Notably, as K_e increases, the deviation between Z_ru and Z_r becomes more pronounced, particularly at switching frequencies higher than f_r. In other words, for the same range of switching frequency variation, Z_ru can provide a larger range of impedance variation compared to Z_r.

- 45 - CAN_DMS: \1006164620 This implies that the FB LLC converter 300C can achieve a wider voltage gain variation range, which is advantageous for wide-voltage-range applications, such as electric vehicle battery chargers and AC-DC power adapters. For instance, to support the use of universal AC-DC power adapters worldwide, a wide input voltage range (90 to 264 Vac) is necessary. In these scenarios, the capability to accommodate a wide voltage gain variation is essential to meet the demand. The capacitors 2302 in parallel with the resonant tank 302 are not additional components. C_e represents the parasitic interwinding capacitance of the transformer 301 introduced by the SPWT configuration. [00218] In the proposed planar transformer 301, C_e is estimated to be around 93 pF. The derivation process is detailed at the end of the detailed description. Combined with the designed C_r value of 17.6 nF, k_e is calculated as 0.005, corresponding to the red curve in FIG. 24. With this small k_e, the influence of C_e on the LLC voltage gain can be ignored. Consequently, the voltage gain of the proposed planar transformer 301 based FB LLC converter 300C should be similar to the conventional one. [00219] Experimental verification [00220] A 360 W FB LLC converter with 180–210 V input and 12 V/30 A output is used as a typical example. This converter is originally designed for an open-frame two-stage AC-DC power supply with a 90-264 Vac input. When the input AC voltage is in low line range (90- 136 Vac), the DC bus ranges from 180 to 210 V and the LLC converter operates in FB mode. [00221] FIG. 25A shows an embodiment of the proposed FB SPWT LLC converter 300C. The main PCB in FIG. 25A is also compatible for building the conventional FB LLC embodiment 100 by replacing the SPWT daughter card with the conventional PCB transformer daughter card as shown in FIG.25B. The transformer winding arrangement and terminal connections for the proposed and conventional planar transformers 301, 106 are shown in FIGs. 18A and 18C, respectively. The electrical connections within the layers of the multi-layer PCB board are established through PCB vias. Meanwhile, connections between different PCB boards are accomplished through soldering.

- 46 - CAN_DMS: \1006164620 [00222] FIGs.26A, 26B and 26C provide detailed views of the transformer windings of the proposed FB LLC transformer 301 and a conventional transformer 106. In FIGs. 26A and 26B, the solder joints are indicated by red rectangles, while the paths of resonant currents are highlighted by red arrows. To minimize the secondary current loops, the secondary windings are constructed using two daughter cards, each having its own SR (Synchronous Rectification) MOSFETs. The secondary daughter cards for the conventional and proposed transformers 106, 301 are identical, as shown in FIG.26C. Except for the planar transformer, other specifications of the conventional transformer 106 are the same as the proposed FB LLC transformer 301, which can be found in Table 0.13. Table 0.13: Specifications of FB LLC resonant converter Parameter Value 62 μH (3C97, PQ26/20, Ferroxcube) Resonant inductor (L_r) 24 turns (Litz 160x0.06 mm) Winding structure: 8-8-8 Resonant capacitor (C_r) 17.6 nF Resonant frequency (f_r) (C2225C222KZGACAUTO, 8 in parallel) Primary switches (Q₁-Q₄) 152 kHz Primary driver OSG65R125JF Secondary SRs (SR₁, SR₂) EG3116D SR driver NTMFS5C604NLT1G [00223] Measured Leakage Inductance [00224] An LCR meter (FLUKE PM6306) is used to measure the leakage inductances at 150 kHz. The total primary leakage inductances are measured by following the methods presented in the FEA simulations discussed above. For the conventional transformer 106, L_{lkP_S1_conv} is measured across the primary winding by short-circuiting secondary winding S1 310A while leaving S2 310B open. Similarly, L_{lkP_S2_conv} is measured across the primary winding by short-circuiting secondary winding S2310B while leaving S1310A open. The same method is employed to measure the leakage inductances of the proposed SPWT transformer 301, wherein two separate primary windings, P1 304 and P2 306, were connected in series, and L_{lkP_S1_prop} and L_{lkP_S2_prop} are measured between winding terminals A and D.

- 47 - CAN_DMS: \1006164620 [00225] Measured leakage inductances are provided in Table 0.14. Table 0.14: Measured Leakage Inductances (Unit: μH) Measured Simulated Conventional ^{LlkP_S1_conv 1.18 1.14} L_{lkP_S2_conv 1.19 1.13} P^{roposed LlkP_S1_prop 1.18 1.14} L_{lkP_S2_prop 1.19 1.13} [00226] It can be observed that, whether the secondary winding S1310A is conducting or the secondary winding S2 310B is conducting, the conventional and proposed planar transformers 106, 301 exhibit the same total leakage inductance. Furthermore, the experimental results match well with the simulation results, with a discrepancy of only 0.04- 0.06 μH, or 3%-5%. This discrepancy mainly arises from the parasitic inductance of the short-circuit connecting wires between the secondary winding terminals. This parasitic inductance can be reflected to the transformer primary side and increase the measured value of the total leakage inductance. [00227] Measured Voltage Conversion Ratio and Efficiency [00228] By sweeping the switching frequency from 135 kHz to 165 kHz, the measured voltage gain curves of the proposed and conventional transformer 301, 106 are shown in FIG. 27. In FIG. 27, it can be observed that they exhibit similar voltage gain performance under both 30 A full load and 1 A light load conditions. The maximum voltage gain difference occurs at 135 kHz/1 A testing condition, where the voltage gain of the conventional transformer 106 is only 1% lower than that of the proposed transformer 301. These experimental results validate the analysis presented above that the proposed transformer and conventional FB LLC transformer 301, 106 should exhibit similar voltage gain performance. [00229] The measured efficiency curves of the proposed transformer 301 and conventional transformer 106 are illustrated in FIG.28. In FIG.28, it can be observed that, under testing conditions ranging from 5 A light load to 30 A full load, they exhibit similar efficiency

- 48 - CAN_DMS: \1006164620 performance. The maximum efficiency difference occurs at 5 A load current condition, where the efficiency of the proposed transformer 301 is 0.07% higher than that of the conventional transformer 106. These experimental results validate the analysis presented above that the use of the SPWT configuration does not introduce extra winding loss, and proposed and conventional FB LLC transformer 301, 106 should exhibit similar efficiency performance. [00230] Symmetry Verification of Proposed Planar Transformer [00231] The symmetry of the proposed planar transformer 301 can be validated by extracting the capacitances of the four parasitic interwinding capacitors (C_AE, C_BE, C_CE, and C_DE) associated with the proposed transformer 301. The precise locations of these capacitors are shown in FIG.4. The extraction process involves two steps: [00232] 1) Measure the capacitances of C_{P1_S} and C_{P2_S}. C_{P1_S} denotes the structural capacitance between P1304 and the secondary windings (S1310A and S2310B), which corresponds to the summation of C_AE and C_BE. C_{P2_S} denotes the structural capacitance between P2306 and the secondary windings 310, which corresponds to the summation of C_CE and C_DE. [00233] 2) Determine the capacitance ratio K₁ (K₂) between C_AE and C_{P1_S} (C_CE and C_{P2_S}) by using a signal generator. Then, C_AE and C_BE, as well as C_CE and C_DE, can be obtained by solving Equations (54) and (55). ^ C P 1_ S ^{^} C AE ^{^} C BE ^

[00234] FIG. 29A illustrates the circuit connections during the measurement of C_{P1_S} and C_{P2_S}. The transformer winding terminals are short-circuited to ensure uniform electric

- 49 - CAN_DMS: \1006164620 potential within each winding. FIG. 29B gives the equivalent capacitance network. Notably, C_{P1_S} and C_{P2_S} cannot be directly measured because of the existence of C_{P1_P2} (interwinding capacitor between P1304 and P2306). In FIG. 29A, the measured capacitances between M1 and M2, M1 and M3, and M2 and M3 are denoted by C_{M1_M2}, C_{M1_M3}, C_{M2_M3}, respectively. Then, the capacitances of C_{P1_S} and C_{P2_S} can be obtained by solving Equations (56)-(58), which are both around 113 pF. C C P 1_ S C P M 1_ M 2 ^{^} C P 1_ P 2 ^{^} 2_ S ^ ⁽⁵⁶⁾

[00235] FIG. 30A illustrates the method for extracting the capacitance ratios K₁ and K₂. A signal generator is applied to P1304 winding terminals A and B, while the other winding terminals are left open. This method simulates the generation of the CM noise displacement currents via the interwinding capacitors. The equivalent circuit is given by FIG. 30B. By adding the sinusoidal excitation signal v_ac at 150 kHz, the voltage distribution along transformer windings is established. Then in Fig.30B, the voltage from E to B (v_EB), and the voltage from A to B (v_AB) are measured to calculate K₁, as shown in Equation (59). In Equation (59), C_probe represents the parallel capacitance introduced by the voltage probe, which is 3.9 pF for Tektronix TPP0250. V_{EB_peak} (V_{AB_peak}) is the peak value of v_EB (v_AB). Similarly, the voltage from E to D (v_ED), and the voltage from C to D (v_CD) are measured to calculate K₂, as shown in Equation (60). V_{ED_peak} (V_{CD_peak}) is the peak value of v_ED (v_CD). V ^{EB _ peak ^} _{C AE} ^{^} _{K 1 ^ ^} ⁽⁵⁹⁾ S (⁶⁰⁾ S

- 50 - CAN_DMS: \1006164620 [00236] FIGs.31A and 31B give the measured waveforms of v_AB, v_EB, v_CD, and v_ED. In FIG. 31A, V_{EB_peak} and V_{AB_peak} are measured as 3.6 V and 7.28 V, respectively. Then, the value of K₁ can be determined by solving Equation (59), which is around 0.512. In FIG.31B, V_{ED_peak} and V_{CD_peak} are measured as 3.52 V and 7.2 V, respectively. Then, the value of K₂ can be determined by solving Equation (60), which is around 0.506. [00237] Substituting the values of K₁, K₂, C_{P1_S}, and C_{P2_S} into Equation (54) and Equation (55) allows for determining the capacitances of C_AE, C_BE, C_CE, and C_DE, which are provided in Table 0.15. Table 0.15: Measured Capacitances of C_AE, C_BE, C_CE, and C_DE C_AE C_BE C_CE C_DE Experimental results 58 pF 55 pF 57 pF 56 pF [00238] In Table 0.15, it can be observed that the capacitances of C_AE and C_BE, as well as C_CE and C_DE, are very close. This observation verifies the symmetry of the proposed planar transformer 301 from the CM noise perspective. In addition, Table 0.16 provides the calculated values for these four capacitances, all of which are 65 pF. These calculations match the experimental results in Table 0.16, showing a minimal discrepancy of 8 to 10 pF, or 12% to 15%. This discrepancy mainly arises from the imperfect stacking of different PCBs and insulation papers, resulting in distributed air gaps between them. Consequently, C_{P1_S} and C_{P2_S} are slightly overestimated during the calculation process, leading to the subsequent overestimations of C_AE, C_BE, C_CE, and C_DE. [00239] Measured Voltage Waveforms of Transformer Winding Terminals [00240] FIGs. 32A and 32B provide the resonant tank current waveform (i_res), as well as the electric potential waveforms of the transformer terminals (v_a and v_b), for the conventional FB LLC transformer 106 operating under full load conditions. [00241] In FIGs.32A and 32B, the reference ground for v_a and v_b is PG. It is worth noting that v_a and v_b exhibit different dv/dt characteristics in both boost (f_s = 135 kHz < f_r) and buck (f_s = 165 kHz > f_r) mode operation range. Specifically, as shown in Fig.32A, negative dv/dt

- 51 - CAN_DMS: \1006164620 on v_a is observed during the interval [t_{C_a}, t_{C_b}], which induces CM noise current from the secondary side to the primary side through the corresponding parasitic interwinding capacitance. During the interval [t_{C_b}, t_{C_c}], positive dv/dt on both v_a and v_b is observed, inducing CM noise currents from the primary side to the secondary side through the corresponding parasitic interwinding capacitance. As a result, the different dv/dt characteristics of v_a and v_b generate CM noise displacement currents with different magnitudes and phases that cannot cancel each other, thereby deteriorating the CM EMI performance of the conventional LLC transformer 106. The conclusion is the same when the conventional LLC transformer 106 operates above f_r. [00242] FIGs. 33A and 33B provide the voltage waveforms of the transformer terminals with respect to PG in the proposed FB SPWT LLC transformer 301 under full load conditions. In FIGs. 33A and 33B, it can be observed that v_B and v_C (v_A and v_D) have complementary dv/dt characteristics in both boost and buck mode operation range, which is consistent with the disclosure above. [00243] When f_s = f_r, the impedance of the resonant tank 302 is zero. For the conventional FB LLC converter 100, v_Q2 (see FIG.1) and v_a are in phase. However, due to the influence of v_Lm (voltage across the magnetizing inductor), the dv/dt of v_a and v_b exhibit different slew rates during the dead time. FIGs.34A and 34B respectively show the key waveforms of the conventional 100 and proposed FB LLC converters 300C during the dead-time when f_s = f_r. [00244] As illustrated in FIG. 34A, the dv/dt of v_a is slower than v_Q2. Consequently, the complementary nature of v_a and v_b is significantly weakened. In contrast to the conventional FB LLC converter 100, the proposed FB LLC converter 300C demonstrates enhanced complementary characteristics in transformer winding terminals when f_s = f_r, as illustrated in FIG.34B. [00245] The complementary dv/dt characteristics introduced by the proposed SPWT configuration enable the CM noise cancellation in the transformer with a symmetrical winding structure, resulting in low CM EMI noise. [00246] Measured CM Noise Spectra

- 52 - CAN_DMS: \1006164620 [00247] The conducted CM noise is initially measured using a LISN 102 named LI-125C, and then separated into its CM component using a noise separator named ZSC-2-2+. The conducted CM EMI spectrum is scanned by an EMI receiver named RSA306B in accordance with the EN55032 class B standard. FIGs. 35A and 35B illustrates the measured CM noise spectra (quasi-peak value) of the conventional and proposed FB LLC transformers 106, 301under full load conditions. The enveloping curves of the CM noise for the conventional and proposed LLC transformers 106, 301 are depicted by dashed and solid lines, respectively. Some peaks are not connected to the enveloping curves to ensure the clear readability of the amplitude trend of the CM noise. In FIGs.35A and 35B, some noise peaks occur at frequencies that are not integer multiples of the switching frequency, which is attributed to background noise caused by the layout of the PCB. These peaks do not affect CM noise reduction and can be resolved by appropriate PCB layout design. For example, the auxiliary power supply circuit in the converter should be kept as far away as possible from the DC input traces of the main converter. Additionally, the copper area for high dv/dt nodes should be minimized during copper pouring. [00248] The results presented in FIGs. 35A and 35B show that the proposed SPWT configuration and symmetrical winding arrangement can achieve an approximately 11 dBμV reduction in CM noise for the fundamental frequencies in both boost and buck mode operation ranges of the FB LLC transformer 301. At the resonant frequency, a reduction of approximately 16 dBμV in CM noise for the fundamental frequency can be achieved. It should be noted that this CM noise attenuation is achieved without increasing the Bill of Materials (BOM) cost due to no further components being necessary for the FB LLC transformer 301 in comparison to the conventional layout 100. However, for frequencies above 5 MHz, the reduction in CM noise is reduced due to the dominant impedance of the transformer leakage inductance. In this range, the transformer 301 should be modeled using a complex parasitic inductance and capacitance network for CM noise analysis. Nevertheless, the CM noise remains low across most of the frequency spectrum. Therefore, the proposed SPWT configuration is effective for reducing the size of the CM EMI filter since the corner frequency of the EMI filter is determined by the noise magnitude at the fundamental frequency.

- 53 - CAN_DMS: \1006164620 [00249] Derivation of interwinding capacitances (C_e) for proposed planar transformer [00250] The principle of displacement current conservation is used to derive the interwinding capacitances of the proposed planar transformer 301 shown in FIG. 18A, including C_AE, C_BE, C_CE, C_DE, C_BC, and C_AD. The precise locations of these capacitors are shown in FIG.23. In FIG.18A, there are distributed interwinding capacitors within five pairs of overlapping layers, which are P1_4-S1_1, P1_5-S2_1, P2_4-S2_1, P2_5-S1_1, and P1_8-P2_8. The structural capacitance within each pair of overlapping layers remains the same. This structural capacitance is denoted as C_{strut_prop} in the following analysis. [00251] Derivation of C_AE, C_BE, C_CE, and C_DE: Based on the one-capacitor couple-turn model, the lumped interwinding capacitance models of P1_4-S1_1, P1_5-S2_1, P2_4-S2_1, and P2_5-S1_1 are illustrated in FIGs. 36A and 36B. In FIG. 36A, black dots denote the corresponding terminals that are used to place C_{strut_prop}. It is noted that C_{strut_prop} in P1_4- S1_1 and P1_5-S2_1 are placed at the same position and the total capacitance is 2C_{strut_prop}. The same applies to FIG.36B. In FIG.36A, the displacement current from primary winding P1304 to secondary windings is calculated in Equation (61) as dv i P 1_4 P S ^{^}2 C strut prop ⁽⁶¹⁾

[00252] where v_{P1_4} can be derived in Equation (62) as v _{^ v ^} ⁴ _{^ v ^ v ^} ^v A ^{^ v} B _^ ^v A ^{^ v} B P ^{1_ 4} A A _{2 2 (62)}

[00253] In Equation (62), Δv is the voltage gradient on each turn. Substituting Equation (62) into Equation (61), i_{P1_S} can be rewritten as Equation (63): i ^d ( ^v _A ^{^ v} _B ) P 1_ S _{^ C} strut _ prop ₍₆₃₎

- 54 - CAN_DMS: \1006164620 [00254] According to the principle of displacement current conservation, the displacement current generated in C_AE and C_BE should be equal to i_{P1_S}, which can be expressed in Equation (64) as: i P 1_ S ^{^} _C ^dv A AE ^{^} _C ^dv B BE ₍₆₄₎

[00255] By comparing Equation (63) and Equation (64), C_AE and C_BE can be derived as C_AE ^{^ C} _BE ^{^ C} _{strut _ prop (65)} [00256] The method used to derive C_AE and C_BE can similarly be applied to derive C_CE and C_DE. The specific calculation process is omitted, and C_CE and C_DE are directly provided as shown in Equation (66): C_AE ^{^ C} _BE ^{^ C} _{strut _ prop (66)}

[00257] Derivation of C_AD and C_BC: C_AD and C_BC primarily arise from the structural capacitance between primary winding turns P1_8 and P2_8. The winding-to-core capacitances are ignored due to their relatively small values. FIG.37 provides 3D models of P1_8 and P2_8. [00258] In FIG.37, x denotes the winding length direction and the overall winding length is denoted by L. The voltage distributions of P1_8 and P2_8 are denoted by v_{P1_8}(x) and v_{P2_8}(x) respectively. By assuming that the voltage of the winding turn is evenly distributed, v_{P1_8}(x) and v_{P2_8}(x) can be calculated in Equation (67) by: ^ ^{^ ^} _v P 1_8 ( _x ) ^{^} _v B ^{^ ^} _v ^{^ ^ v ^ x} , _x ^{^} (0, _L )

- 55 - CAN_DMS: \1006164620 [00259] The displacement current flowing from P1_8 to P2_8 can be calculated in Equation (68) as: i ^L ^⁰ ^ ^{insul w PCB d} v ^{d (2 v ^} 8_{( x ) v P 2_8 ( x ) d} ^{B ^ v )} P_{1_8} ^ _{P 2_8} ^{^} _{^ P 1_} ^{^} _{^ x} ^{^} _{C strut _ prop} ⁽⁶⁸⁾

[00260] Since Δv is equal to (v_A - v_B) / 8, Equation (68) can be rewritten as shown in Equation (69). i ^{d v} A ^{+15 v} P 1_8 ^{^} P 2_8 ^{^} _C strut _ prop ₍ B _{) (69)}

[00261] Based on the principle of displacement current conservation, the displacement current generated in C_BC and C_AD should be equal to i_{P1_8-P2_8}, which is expressed in Equation (70) as i _{^ C} ^{d( v} _A ^{^ v} _D ⁾ ^ _C ^{d ( v} _B ^{^ v} _C ⁾ ^ _{2 C} ^dv _{A ^} ^dv _{B P} 1_8 ^{^} _P 2_8 _{AD BC AD 2 C BC (70)}

[00262] By comparing Equation (69) and Equation (70), C_AD and C_BC can be derived as ^ C C strut _ pro ^ ^{^} p ^ AD (₇₁₎

[00263] The capacitance C_e can be calculated in Equation (72) by C ^C _B ^C 23 ^{C strut} _{_} ^prop e ^{^} _C BC ^{^} _{E CE ^} ^{^ (72)}

[00264] The calculation results are summarized in Table 0.16.

- 56 - CAN_DMS: \1006164620 Table 0.16: Interwinding capacitances of proposed planar transformer 301 C_AE, C_BE, C_CE, C_DE C_AD C_BC C_e Calculation results 65 pF 4 pF 61 pF 93 pF [00265] Applicant notes that the described embodiments and examples are illustrative and non-limiting. Practical implementation of the features may incorporate a combination of some or all of the aspects, and features described herein should not be taken as indications of future or existing product plans. Applicant partakes in both foundational and applied research, and in some cases, the features described are developed on an exploratory basis. [00266] The term “connected” or "coupled to" may include both direct coupling (in which two elements that are coupled to each other contact each other) and indirect coupling (in which at least one additional element is located between the two elements). [00267] Although the embodiments have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the scope. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the process, machine, manufacture, composition of matter, means, methods and steps described in the specification. [00268] As one of ordinary skill in the art will readily appreciate from the disclosure, processes, machines, manufacture, compositions of matter, means, methods, or steps, presently existing or later to be developed, that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein may be utilized. Accordingly, the appended embodiments are intended to include within their scope such processes, machines, manufacture, compositions of matter, means, methods, or steps. [00269] As can be understood, the examples described above and illustrated are intended to be exemplary only.

- 57 - CAN_DMS: \1006164620

Claims

WHAT IS CLAIMED IS: 1. A full bridge resonant converter circuit for cancelling common mode noise in a primary winding, the circuit comprising: a bridge circuit having two or more switches operating with a duty cycle and electrically coupled to the primary winding at a first and a second terminal; a transformer having the primary winding and a secondary winding, the primary winding and secondary winding being magnetically coupled; a first primary winding section of the primary winding having winding turns P1 and electrically coupled between the first terminal and a third terminal; a second primary winding section of the primary winding having winding turns P2 and electrically coupled between the second terminal and a fourth terminal; a rectifier electrically coupled to the secondary winding; and a resonant tank having at least one inductor and at least one capacitor and electrically coupled to the primary winding at the third and the fourth terminal.

2. The circuit of claim 1, wherein the secondary winding has a first secondary winding section having winding turns S1 and a second secondary winding section having winding turns S2.

3. The circuit of claim 2, wherein the secondary winding is connected as a full wave rectifier.

4. The circuit of claim 1, wherein the secondary winding is connected as a full bridge rectifier.

5. The circuit of claim 2, wherein the first primary winding section and the first secondary winding section have at least one turn within P1 and S1 which is overlapping, and the second primary winding section and the second secondary winding section have at least one turn within P2 and S2 which is overlapping. - 58 - CAN_DMS: \1006164620

6. The circuit of claim 5, wherein each of the winding turns of P1, P2, S1 and S2 are a single layer.

7. The circuit of claim 2, wherein two turns within P1 of the first primary winding section overlap with one turn within S1 of the first secondary winding section, and two turns within P2 of the second primary winding section overlap with one turn within S2 of the second secondary winding section.

8. The circuit of claim 2, wherein the first primary winding section and the first secondary winding section have one turn within P1 and S1 which is overlapping, the first primary winding section and the second secondary winding section have one turn within P1 and S2 which is overlapping, the second primary winding section and the first secondary winding section have one turn within P2 and S1 which is overlapping, and the second primary winding section and the second secondary winding section have one turn within P2 and S2 which is overlapping.

9. The circuit of any one of claims 5 to 8, wherein the structural interwinding capacitances of the overlapping turns are equal.

10. The circuit of any one of claims 5 to 8, wherein the overlapping turns are enclosed by an isolation material.

11. The circuit of claim 1, wherein the primary and secondary winding each comprise two or more magnetic cores.

12. The circuit of claim 11, wherein each magnetic core of the secondary winding has a first secondary winding section having winding turns S1 and a second secondary winding section having winding turns S2.

13. The circuit of claim 12, wherein the at least two magnetic cores of the primary winding are connected in series, and the at least two magnetic cores of the secondary winding are connected in parallel.

14. The circuit of claim 11, wherein the primary winding comprises two magnetic cores, a first magnetic core housing the first primary winding section and a second - 59 - CAN_DMS: \1006164620 magnetic core housing the second primary winding section, the first and second primary winding sections having identical winding arrangements.

15. The circuit of claim 14, wherein the two magnetic cores of the transformer have identical core sizes, identical core materials, identical turn ratios, and identical winding arrangements.

16. The circuit of claim 15, wherein the resonant tank is coupled in series between the primary winding sections of the two magnetic cores.

17. The circuit of claim 16, wherein the resonant tank comprises one inductor and one capacitor.

18. The circuit of claim 11, wherein the transformer comprises three magnetic cores, the first magnetic core housing a first primary winding section, a second magnetic core housing a third and a fourth primary winding section, and a third magnetic core housing the second primary winding section, and wherein: the first and third magnetic core have identical core sizes, identical core materials, identical turn ratios, and identical winding arrangements; and the third and fourth primary winding section have identical winding arrangements.

19. The circuit of claim 11, wherein when there is an even number of magnetic cores, the magnetic cores are equally split into two sections, the magnetic cores within each of the two sections are coupled in series, and the resonant tank is electrically coupled between the two sections of the magnetic cores.

20. The circuit of claim 11, wherein when there is an odd number of magnetic cores, the magnetic cores are separated into a first group, a second group and a central magnetic core, the first group and second group of magnetic cores have an equal number of magnetic cores, and wherein: the first group of magnetic cores are connected in series; the second group of magnetic cores are connected in series; - 60 - CAN_DMS: \1006164620 the central magnetic core contains a third and a fourth primary winding section; the resonant tank is electrically coupled between the third and fourth primary winding section; and the third primary winding section is coupled in series to the first group of magnetic cores, and the fourth primary winding section is coupled in series to the second group of magnetic cores.

21. The circuit of claim 1, wherein when the primary winding has an even number of turns, P1 and P2 have the same number of turns.

22. The circuit of claim 1, wherein when the primary winding has an odd number of turns, P1 and P2 differ by one winding turn.

23. The circuit of claim 1, wherein the resonant tank is connected in parallel to the primary winding and is electrically coupled to the first, second, third and fourth terminal of the primary winding.

24. The circuit of claim 1, wherein the resonant tank has a first capacitor electrically coupled to the first and third terminal, a second capacitor electrically coupled to the fourth and second terminal, and an inductor electrically coupled to the third and fourth terminal.

25. The circuit of claim 1, wherein the resonant tank has a first capacitor connected in parallel to the first primary winding section and electrically coupled to the first and third terminal, a second capacitor and an inductor connected in series and electrically coupled to the third and fourth terminal, and a third capacitor connected in parallel to the second primary winding section and electrically coupled to the third and fourth terminal.

26. The circuit of any one of claims 23 to 25, wherein the at least one inductor of the resonant tank is a physical inductor with magnetic cores.

27. The circuit of any one of claims 23 to 25, wherein the at least one inductor of the resonant tank comprises utilizing the leakage inductance of the transformer. - 61 - CAN_DMS: \1006164620

28. The circuit of claim 1, wherein the bridge circuit has four switches.

29. The circuit of claim 1, wherein the duty cycle is a 50% duty cycle.

30. The circuit of any one of claims 1 to 29, wherein the circuit is used for electric vehicle battery charging or AC-DC power adapters.

31. A method for cancelling common mode noise in a primary winding of a resonant converter circuit, the circuit containing a transformer having the primary winding and a secondary winding, a bridge circuit having two or more switches operating with a duty cycle and electrically coupled to the primary winding at a first and a second terminal, a first primary winding section of the primary winding having winding turns P1 and electrically coupled between the first terminal and a third terminal, and a second primary winding section of the primary winding having winding turns P2 and electrically coupled between the second terminal and a fourth terminal, the method comprising: separating the first primary winding section and second primary winding section with a resonant tank electrically coupled to the third and fourth terminal; balancing, through at least one capacitor and at least one inductor within the resonant tank, a primary inductance of the first primary winding section with a secondary inductance of the secondary primary winding section.

32. The method of claim 31, wherein the secondary winding has a first secondary winding section having winding turns S1 and a second secondary winding section having winding turns S2.

33. The method of claim 32, wherein the secondary winding is connected as a full wave rectifier.

34. The method of claim 31, wherein the secondary winding is connected as a full bridge rectifier.

35. The method of claim 32, wherein the first primary winding section and the first secondary winding section have at least one turn within P1 and S1 which is - 62 - CAN_DMS: \1006164620 overlapping, and the second primary winding section and the second secondary winding section have at least one turn within P2 and S2 which is overlapping.

36. The method of claim 35, wherein each of the winding turns of P1, P2, S1 and S2 are a single layer.

37. The method of claim 32, wherein two turns within P1 of the first primary winding section overlap with one turn within S1 of the first secondary winding section, and two turns within P2 of the second primary winding section overlap with one turn within S2 of the second secondary winding section.

38. The method of claim 32, wherein the first primary winding section and the first secondary winding section have one turn within P1 and S1 which is overlapping, the first primary winding section and the second secondary winding section have one turn within P1 and S2 which is overlapping, the second primary winding section and the first secondary winding section have one turn within P2 and S1 which is overlapping, and the second primary winding section and the second secondary winding section have one turn within P2 and S2 which is overlapping.

39. The method of any one of claims 35 to 38, wherein the structural interwinding capacitances of the overlapping turns are equal.

40. The method of any one of claims 35 to 38, wherein the overlapping turns are enclosed by an isolation material.

41. The method of claim 31, wherein the primary and secondary winding each comprise two or more magnetic cores.

42. The method of claim 41, wherein each magnetic core of the secondary winding has a first secondary winding section having winding turns S1 and a second secondary winding section having winding turns S2.

43. The method of claim 42, wherein the at least two magnetic cores of the primary winding are connected in series, and the at least two magnetic cores of the secondary winding are connected in parallel. - 63 - CAN_DMS: \1006164620

44. The method of claim 41, wherein the primary winding comprises two magnetic cores, a first magnetic core housing the first primary winding section and a second magnetic core housing the second primary winding section, the first and second primary winding sections having identical winding arrangements.

45. The method of claim 44, wherein the two magnetic cores of the transformer have identical core sizes, identical core materials, identical turn ratios, and identical winding arrangements.

46. The method of claim 45, wherein the resonant tank is coupled in series between the primary winding sections of the two magnetic cores.

47. The method of claim 46, wherein the resonant tank comprises one inductor and one capacitor.

48. The method of claim 41, wherein the primary winding comprises three magnetic cores, the first magnetic core housing a first primary winding section, a second magnetic core housing a third and a fourth primary winding section, and a third magnetic core housing the second primary winding section, and wherein: the first and third magnetic core have identical core sizes, identical core materials, identical turn ratios, and identical winding arrangements; and the third and fourth primary winding section have identical winding arrangements.

49. The method of claim 41, wherein when there is an even number of magnetic cores, the magnetic cores are equally split into two sections, the magnetic cores within each of the two sections are coupled in series, and the resonant tank is electrically coupled between the two sections of the magnetic cores.

50. The method of claim 41, wherein when there is an odd number of magnetic cores, the magnetic cores are separated into a first group, a second group and a central magnetic core, the first group and second group of magnetic cores have an equal number of magnetic cores, and wherein: - 64 - CAN_DMS: \1006164620 the first group of magnetic cores are connected in series; the second group of magnetic cores are connected in series; the central magnetic core contains a third and a fourth primary winding section; the resonant tank is electrically coupled between the third and fourth primary winding section; and the third primary winding section is coupled in series to the first group of magnetic cores, and the fourth primary winding section is coupled in series to the second group of magnetic cores.

51. The method of claim 31, wherein when the primary winding has an even number of turns, P1 and P2 have the same number of turns.

52. The method of claim 31, wherein when the primary winding has an odd number of turns, P1 and P2 differ by one winding turn.

53. The method of claim 31, wherein the resonant tank is connected in parallel to the primary winding and is electrically coupled to the first, second, third and fourth terminal of the primary winding.

54. The method of claim 31, wherein the resonant tank has a first capacitor electrically coupled to the first and third terminal, a second capacitor electrically coupled to the fourth and second terminal, and an inductor electrically coupled to the third and fourth terminal.

55. The method of claim 31, wherein the resonant tank has a first capacitor connected in parallel to the first primary winding section and electrically coupled to the first and third terminal, a second capacitor and an inductor connected in series and electrically coupled to the third and fourth terminal, and a third capacitor connected in parallel to the second primary winding section and electrically coupled to the third and fourth terminal.

56. The method of any one of claims 53 to 55, wherein the at least one inductor of the resonant tank is a physical inductor with magnetic cores. - 65 - CAN_DMS: \1006164620

57. The method of any one of claims 53 to 55, wherein the at least one inductor of the resonant tank comprises utilizing the leakage inductance of the transformer.

58. The method of claim 31, wherein the bridge circuit has four switches.

59. The method of claim 31, wherein the duty cycle is a 50% duty cycle.

60. The method of any one of claims 31 to 59, wherein the method is used for electric vehicle battery charging or AC-DC power adapters.

61. A non-transitory computer readable medium, storing machine readable instructions, which when executed by a processor, cause the processor to perform a method according to any one of claims 31-60. - 66 - CAN_DMS: \1006164620