Improved Modeling and Analysis of High-Frequency Copper Loss in Planar Transformers: The Influence Mechanism of Magnetizing Current and Circular Windings
Wei Jiwen, Yang Xu, Li Qingzheng, Li Bingyang, Chen Wenjie
School of Electrical Engineering Xi’an Jiaotong University Xi’an 710049 China
Abstract：The winding design and copper loss calculation of planar transformers are generally based on conventional one-dimensional models. However, for most planar transformers with toroidal windings, the electromagnetic field distribution is inherently non-one-dimensional. Traditional models assume a uniform electromagnetic field along the winding width. However, in toroidal windings, the field distribution is noticeably non-uniform due to differences between the inner and outer radii. Furthermore, conventional models often neglect the influence of the magnetizing current and assume that the primary and secondary magnetomotive forces (MMFs) cancel completely. In topologies such as flyback and LLC converters, magnetizing current cannot be ignored. The MMF does not fully cancel out, resulting in significant limitations in copper-loss estimation and structural design using traditional approaches. This paper derives an analytical solution for the electromagnetic field in toroidal windings by establishing the magneto-quasistatic field equations in cylindrical coordinates and applying appropriate boundary conditions. Expressions for the AC resistance and the AC-to-DC resistance ratio are derived when the magnetic field on the winding's upper and lower surfaces is known. The proposed model shows that, under the condition that the DC resistance per turn is equal within the same layer, the AC-to-DC resistance ratio of the toroidal winding matches that of the conventional one-dimensional model. By representing the electromagnetic field in phasor form, the model incorporates the magnetizing current's influence on AC resistance. Then, multilayer winding stacking structures and the impact of magnetizing current on copper loss are analyzed. Both the magnitude and phase of the magnetizing current significantly affect copper loss, which increases with increasing magnetizing current. Moreover, a copper-loss-optimization strategy for magnetizing current is proposed. By analyzing the phasor diagram of layer currents in multilayer windings, the influence of different stacking configurations on proximity loss can be conveniently evaluated. Specifically, the path with the shortest distance to the origin in the phasor connection corresponds to the stacking structure with the minimum proximity loss. In contrast, the path with the farthest distance corresponds to the structure with the maximum proximity copper loss. Finally, the proposed model is validated through finite element simulations and experiments. Using COMSOL 2D analysis, simulations under ideal conditions agree with the established copper-loss model. Different proximity-loss behaviors of traditional interleaved structures can be distinguished when magnetizing current is included. Under non-ideal conditions, such as winding gaps and core air gaps, the conventional model's error reaches 18%, whereas the proposed model's error remains below 2.5%. Two planar transformer prototypes with different stacking structures were fabricated, and the primary and secondary resistances were measured through open-circuit and short-circuit small-signal tests. The experimental data are highly consistent with the model predictions, with a maximum error of less than 4.2%.
魏吉文, 杨旭, 李清正, 李冰洋, 陈文洁. 平面变压器高频铜损的改进建模与分析：励磁电流与环形绕组的影响机理[J]. 电工技术学报, 2026, 41(12): 3951-3967.
Wei Jiwen, Yang Xu, Li Qingzheng, Li Bingyang, Chen Wenjie. Improved Modeling and Analysis of High-Frequency Copper Loss in Planar Transformers: The Influence Mechanism of Magnetizing Current and Circular Windings. Transactions of China Electrotechnical Society, 2026, 41(12): 3951-3967.
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