Magnetic Design for Three-Phase Dynamic Wireless Power Transfer With Constant Output Power

This paper details the design of a three-phase transmitter-receiver topology for use in a dynamic wireless power transfer (DWPT) system for heavy-duty electric vehicles. The proposed approach eliminates the power oscillation and voltage/current sharing problems that plague planar three-phase DWPT systems. Multi-objective optimization is employed to maximize magnetic coupling and minimize undesired positive–negative sequence coupling. Selected designs are simulated for a 200-kW/m DWPT system, verifying the elimination of power oscillation and phase imbalance with minimal impact on magnetic performance. A prototype DWPT system is designed, and low-ripple power output is demonstrated at 52-kW operation.


I. INTRODUCTION
S IGNIFICANT environmental benefits lie with deploying dynamic wireless power transfer (DWPT) technology for heavy-duty vehicle (HDV) fleets. However, it is difficult to accommodate the power needs of HDVs with single-phase topologies; therefore, three-phase solutions are of interest. The arrangement of the three-phase transmitter (tx) and receiver (rx) coils considered herein is shown in Fig. 1. In this planar topology, the magnetic poles travel transversely to the road. An advantage of this configuration is that it can be readily scaled lengthwise to meet a range of power ratings. However, a noted shortcoming is magnetic imbalance, leading to power fluctuation and poor current/voltage sharing among phases [1].
In general, power fluctuation arises from four sources: (1) segmentation of the transmitter windings, (2) ac-dc rectification, (4) interphase-mutual-coupling imbalance. Source 1 depends on a wide range of high-level, multidisciplinary system architecture choices, where the frequency depends on coil spacing and vehicle velocity. Since the focus in this work is the mitigation of higher-frequency oscillations originating from the electrical system, this is not considered further, although it is noted that the topology in Fig. 1 readily lends itself to either a short-track or long-track paradigm. Considering source 2, polyphase topologies are inherently capable of delivering lower-ripple power at the output of a passive rectifier compared to single-phase topologies [2], [3]. With regard to source 3, [2], [3], [4] demonstrate that polyphase designs can reduce variation in mutual coupling for DWPT systems with magnetic poles oriented along the road; whereas topologies with poles across the road (single-phase and polyphase) naturally achieve good flux continuity during vehicle travel [5]. Unfortunately, polyphase DWPT systems reduce power fluctuation due to sources 2 and 3 at the cost of introducing source 4. In [1], [6], several options to mitigate source 4 are discussed, primarily focused on reducing coupling or adding external compensation to balance the phases. Herein, it is shown that reducing or uniformly balancing mutual couplings in a three-phase DWPT system is not necessary to achieve good current/voltage sharing and constant power throughput, leading to simpler, more compact designs. Specifically, a symmetrical components (SC)-based transformation of the inductance matrix for a generic pair of three-phase coils yields a T-equivalent circuit with explicit voltage source terms responsible for phase imbalance. It is shown that these sequencecoupling terms may be eliminated through the design of the tx/rx coil layouts. The design process involves a multi-objective optimization, which yields a set of tx/rx designs representing the tradeoff between maximizing magnetic coupling and minimizing sequence interaction for the given topology. For illustration purposes, a baseline design and three optimized designs are This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ simulated at an average output of 200 kW/m (kW per m of rx length) to verify the expected reduction in power fluctuation and phase imbalance. Finally, a laboratory prototype DWPT system is designed, and low-ripple power output is demonstrated at 52 kW.

II. TRANSMITTER-RECEIVER EQUIVALENT CIRCUIT MODEL
The flux linkage equations for a generic three-phase tx-rx pair (in a magnetically linear system) may be expressed as in which the uppercase flux linkage and current variables indicate phasors. The inductance matrix is symmetric positivedefinite; otherwise, matrix elements are independent.
A classical approach to analyzing imbalance in three-phase ac circuits is the SC transformation. By introducing the SC transformation matrix A, unbalanced phase quantities in a three-phase system are decomposed as a linear combination of balanced zero-sequence, positive-sequence, and negative-sequence phasor sets (denoted by a 012 superscript): ( To apply SC to the flux linkage equations (1), we left-multiply the tx/rx equations by A −1 , and utilize (2) Because the inductance matrix in (1) is real, the transformed diagonal submatrices are Hermitian, and the off-diagonal submatrices are conjugate-transposes of one another. Thus, the inductance matrix transformed into SC is also Hermitian. We note the following regarding (3). First, for any real matrix P transformed as A −1 PA, it can be shown that the positive-and negative-sequence diagonal elements are complex conjugates: e.g., M 22 tr = M 11 * tr . Furthermore, since the self-inductance submatrices L 012 t and L 012 r are Hermitian, we can define real-valued tx/rx sequence self-inductances Third, it can be shown that the positive-negative-sequence offdiagonal elements are complex conjugates: i.e., L 21 t = L 12 * t , L 21 r = L 12 * r , and M 21 tr = M 12 * tr . Fourth, it is convenient to refer the rx-side quantities to the tx side as is typical for transformers, using the "turns ratio" factor: n L R /L T [7]. Finally, it is assumed that the zero sequence can be neglected for the purpose of analyzing power transfer. Under these conditions and simplifications, the positive-sequence equations of (3) can The negative-sequence equations are identical in form, except that positive-and negative-sequence variables exchange places, and all elements in the second matrix are conjugated.
To avoid interaction between positive-and negativesequences, it is necessary to eliminate the inductances in the second matrix of (6). To this end, we derive the sequence T-equivalent circuits and express the undesired couplings in a normalized form. Let leakage inductance be defined as Splitting the tx/rx self-flux linkages in (6) into leakage and magnetizing components using the definitions in (5) and (7) yields the equations corresponding to a typical T-equivalent circuit with additional undesired terms representing sequence interaction. These sequence-coupling terms may be expressed in terms of voltages with normalized coefficients by defining leakage and magnetizing flux linkages as Finally, replacing the negative-sequence currents of (6) with corresponding expressions from (8) yields: Equations (9) and (10) imply a T-equivalent circuit of the form shown in Fig. 2, where the dependent voltage sources correspond to the last two terms of the equations, with V = jωΛ.
(Resistances are not included for clarity.)

III. OPTIMIZATION OF COIL GEOMETRY
A design optimization problem is posed to show that it is practicable to reduce sequence interaction for planar windings without significant magnetic performance degradation. Specifically, to eliminate the sequence coupling in the magnetizing branch of Fig. 2, the coefficient M 12 / tr M must vanish. Assuming this is feasible, (9) and (10) indicate that the remaining factors to minimize are L 12 t /L l and L 12 r /L l . To aid optimization convergence, it is helpful to define a unitless "sequence-coupling factor" to be minimized, which represents the aggregate effect of these undesirable terms: A common metric for DWPT magnetic performance, which we wish to maximize, is the coupling factor: which is based on the definitions of (4) and (5). Let x be a vector of geometric parameters defining a candidate design, as indicated in Fig. 1. Then, the two-objective design problem is given by arg max x {σ −1 , k}. As an illustrative case study, an optimization was performed to design tx-rx pairs on a per-turn, per-length basis. The coil distance d tr was set to 21 cm, and the rx/tx widths were constrained so as not to exceed 0.9 m and 1.2 m, respectively. The receiver core was specified to be MN60 ferrite with thickness t rc = 1 cm. This optimization problem was solved using an evolutionary computing toolbox [8]. The tx-rx inductance matrix in (1) was computed for each candidate design (assuming perfect alignment) using the boundary element method [9].
The optimization produced the Pareto-optimal front shown in Fig. 3. The performance metrics of a "uniform" conductor arrangement (spaced equally across the full widths of the tx/rx) are also plotted therein. It can be shown that σ vanishes if The inductance matrix for the "low-σ" design, reported below for reference, meets these constraints within the tolerance corresponding to the sequence-coupling factor: This result is achieved for (x B , x A , x C ) = (50, 279, 594), (x b , x a , x c ) = (52, 147, 445), w rc = 900, and d rc = 11 (all in mm) as defined in Fig. 1.

IV. VERIFICATION OF OPTIMIZED DESIGNS
Simulations were performed for each of the selected designs highlighted in Fig. 3 using the elementary system detailed in Fig. 4 to verify that the optimized designs reduce undesirable effects due to sequence interaction. LCC and series compensation schemes were used to condition the tx and rx, respectively, although the sequence-interaction property of each tx/rx design is independent of the compensation circuit. The compensation circuit elements and load were sized to achieve equal source/load voltages and equal tx/rx currents at f 0 = 85 kHz: where ω 0 = 2πf 0 . The input voltages were an ideal, three-phase set with amplitudes specified to achieve 200 kW/m (kW per m of rx length): {189, 179, 172, 167} V rms for the Uniform, Hi-σ, Mid-σ, and Low-σ design simulations, respectively. The steady-state simulated output power for the selected designs is plotted in Fig. 5, and key performance data are listed  in Table I. The results indicate that the low-σ tx/rx design does indeed transfer power with minimal output ripple by virtue of balancing the rx currents (and source currents, by extension), which is a vast improvement over the higher-σ and uniform designs. Also, it can be seen in Table I that σ correlates well with the output power ripple ΔP load .
A laboratory prototype of the proposed tx/rx topology was constructed as shown in Fig. 6. The tx/rx were instantiated with two turns each and sized lengthwise to achieve a sequence mutual inductance M = 2.37 µH; the characterized value was 2.41 µH. The measured tx and rx sequence self-inductances were 23.4 and 14.6 µH, respectively. The characterized sequencecoupling factor was σ = 6.7%, primarily due to imperfect alignment and imbalance in end turns and leads. LCC and series compensation circuits were designed for the tx and rx, respectively, tuned for 40-kHz operation. A SiC-based three-phase inverter employing 180 • -switching converted a regulated dc supply to a positive-sequence excitation; a SiC-based three-phase passive rectifier at the rx output supplied a resistive load bank. The DWPT system was operated with a dc input of 750 V, achieving output power P load = 52 kW. The measured tx and rx phase currents and load power are plotted in Fig. 7. It is observed therein that good balance is achieved between phase currents, and the measured 7.2% power ripple agrees well with the characterized value of σ.

V. CONCLUSION
This paper has presented an approach to design three-phase planar tx/rx coils for DWPT that mitigates the deleterious impacts of phase imbalance. The resulting designs are strikingly simple, while maintaining sufficient magnetic performance to meet the power targets for HDVs.