5. Mathematical Model for Analysis of Series Parallel Compensated Wireless Power Transfer System for Different Misalignments
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Conclusion and Future Works
Contents
6.1 Summary of the Work . . . . 116 6.2 Limitations of the Work . . . . 118 6.3 Scope for Future Research . . . . 119
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6. Conclusion and Future Works
6.1 Summary of the Work
For the designing WPT system, an accurate modelling of the magnetic field is required. The 3-D finite element analysis (FEA) is used to design a coil system because of its high accuracy. However, high mesh-density in the 3-D FEA model requires high computational power. An iterative design process, to achieve the desired coil system, requires numerous modifications in the 3-D FEA model, which is a tedious process. Moreover, many FEA packages do not provide a straight relationship between the performance and the parameter of the WPT system.
To address this issue, the objective of this thesis is to present an alternative FEA modelling for the designing of the WPT system. For this, an accurate, comprehensive analytical model to predict parameter variation during misalignment for the WPT system has been given. The analytical model consists of electromagnetic and steady-state models. The electromagnetic model is used to compute the mutual inductance between the coils for different misalignments. On the other hand, the steady- state model calculates the voltage, current, power, and efficiency of the SP compensated WPT system (by using the mutual inductance information from the electromagnetic model). This model can be used to analyze the component stress of the SP compensated WPT system. The results of the math- ematical model are verified experimentally. Thus, the proposed method can be adopted in the initial design process of the SP compensated WPT system.
The electromagnetic model has been developed for different coil systems. First, the modelling has been done for the coil system without a magnetic shield. For the modelling, various modelling method has been discussed, and it is found that harmonic model found suitable to describe the magnetic field distribution and mutual inductance between the coils of the coil system of WPT system. The harmonic model divides the three-dimensional coil system into different regions in the non-periodic direction (Here, z-direction) according to the material property. For each region, either three-dimensional mag- netic vector potential or magnetic scalar potential, by solving Poisson or Laplace equation using the method separation of variables, has been formulated in the cartesian coordinates system. These magnetic potentials contain unknown coefficients, which is solved by applying boundary conditions between the regions. The boundary conditions are defined by the continuity of tangential magnetic field intensity and normal component of the magnetic flux density at the edge of each region in the TH-2341_126102029
6.1 Summary of the Work
z-direction. Applying the various boundary conditions gives a set of linear equations, which is solved to obtain the value of unknown coefficients of magnetic potentials. The number of linear equations is equal to the number of unknowns. These magnetic potentials have been used to calculate the magnetic flux density distribution in the air-gap and mutual inductance between the coils. The coil is modelled by four current overlapping bars and mathematically represented in similar for magnetic vector po- tential. The mutual inductance from the proposed analytical model has been calculated for different misalignments and validated with a 3-D FEA model and measurements. Moreover, the variation in mutual inductance for different misalignments (such as horizontal, angular, planar, etc.), which are the practical case of wireless charging for EVs, are also discussed. The results of the analytical model are compared with the 3-D FEA model and the measurements, and a good agreement has been found.
Furthermore, the analytical model for the coil system having magnetic below the primary coil has been developed. The developed analytical model for the coil system without a shield with some modification has been successfully applied to the coil system having magnetic below the primary coil.
With the addition of a magnetic shield underneath of primary coil, as compared to the previous model, a region representing a magnetic shield has been added in the modelling. The magnetic scalar poten- tial is used to describes the magnetic field in this region. This magnetic potential contains unknown coefficients, which is calculated by applying boundary conditions at the boundary of the magnetic shield region. The boundary conditions have been defined by the continuity of magnetic scalar poten- tial and normal component of the magnetic flux density at the edge of the magnetic shield region. This model has the flexibility to change the width and permeability of shielding material. The presented analytical model is verified for different misalignments with FEA results and measurements.
The proposed analytical model considers the length of the magnetic shield is infinite, which is impractical. In practice, the magnetic shielding of finite permeability and finite dimensions is practi- cally used with the primary and secondary coils. Therefore, a 3-D analytical model that considers the magnetic shields of finite permeability and finite dimension has been developed. The magnetic vector potential formulation in the cartesian coordinate system has been used in subdomain modelling. For the subdomain modelling, the continuity of magnetic vector potential and magnetic field intensity is applied in two directions (i.e., x- and y-edges). The comparison in magnetic field distribution obtained TH-2341_126102029
6. Conclusion and Future Works
from the subdomain model and 2-D FEA has been made. The magnetic field distributions obtained from the 2-D subdomain model coincide with the magnetic field obtained from 2-D FEA, which con- firms the accuracy of the 2-D subdomain model. A 3-D subdomain model has been formulated by the superposition of two 2-D subdomain models to obtain the result close to measurement results. For limiting the 2-D subdomain model in the third direction, a correction factor has been introduced. The proposed subdomain model is validated by the computation of magnetic field density distribution in the air gap and compared with FEA and experiment. It shows a good agreement. The mutual induc- tance from the 3-D subdomain model has been compared with FEA results and measurement. The proposed subdomain model facilitates to change the parameters (length, width, and permeability) of magnetic shields. The results from the proposed subdomain model are acceptably accurate for initial designing and optimization of the WPT system having magnetic shielding of finite dimension and permeability.
The change in the mutual inductance impacts the power transfer capability and the efficiency of the entire WPT system. To investigate the variation in voltage, current, power, and efficiency of the SP compensated WPT system for different misalignments, such as horizontal, angular, planar, etc. , a steady-state model of SP compensated WPT system has been developed. The mutual inductance calculated by electromagnetic model is used as direct input to the steady-state model. In order to get an accurate result, harmonics effect is considered in the steady-state model. Moreover, an experimental setup of 300 W for SP compensated WPT system with different misalignments is developed in the laboratory to validate the efficacy of the model. The results of the mathematical model are compared with the simulation and the measurement results, and a good agreement has been found. Therefore, the proposed model can be used for analyzing the component stress of the SP compensated WPT system for different misalignments, which helps in the initial design process of such a WPT system.