A new approach for analytical modelling of interior permanent magnet machines

Interior permanent magnet synchronous machines (IPM) are becoming popular for some advantages such as; high torque and power density, wide constant power speed range (CPSR) and high efficiency. IPM is used in hybrid and electric vehicle applications due to the wide CPSR. IPM has reluctance torque because magnets are buried in the rotor and it needs less amount of magnet for the same torque compared with surface PM synchronous (SPM) machine. Magnetic model of the machine should be used during design. Analytical modeling is difficult due to the complex geometry and nonlinear magnetic behavior of the core. Even if finite element analysis (FEA) is the best modeling way to electric machines, it takes long computational time. Therefore, the design process is divided into two stages as early design stage that geometry is determined roughly and detailed design stage that final geometry is determined. Analytical methods are used to speed up design process at the early design stage. Lq inductance value of IPM varies depend on current because of the magnetic saturation so a nonlinear analytical method should be used for early design stage. There are many models which use iterative solutions for nonlinear modeling in literature. In this study, the proposed model is based on a mathematical equation of nonlinear behavior of the magnetic material. This equation is transformed into a magnetic equivalent circuit using geometric dimensions which are obtained from flux tube analysis and lumped parameter method (LPM). Firstly, a simple inductor is modeled and LambertW function is used to obtain the nonlinear magnetic model. The proposed model is validated by the experimental result which inductance is used in the application of DC-DC converter. After that, this method is examined to apply in IPM model but the limitation of the model does not allow it. Therefore, approximately analytic methods are investigated and perturbation theory is selected. The mathematical equation of the nonlinear magnetic behavior of the material is selected properly for perturbation theory. The solution of the magnetic equivalent circuit is tried with perturbation series but it diverges in saturation region even if it works in the linear region. A nonlinear Lq model is obtained after perturbation series are transformed to Pade approximants. An IPM is designed and manufactured as a prototype to validate the proposed model with experimental results. There are two type of winding for IPM machines like distributed winding and concentrated winding. There is not enough study for concentrated winding IPM machines in literature. MMF distribution of concentrated winding is not sinusoidal and causes sub-harmonics. Slot/pole combination affects torque density and magnetic features of the machine. So slot/pole combinations are investigated. The machine is designed according to the characteristic current in order to have wide CPSR and prototype is manufactured. Finally, inductance measurements and torque versus speed graph are obtained from experimental results.