Analytical Methodology for Modelling of Circulating Current Loss in Synchronous Electrical Machines With Permanent Magnets

Abstract

This paper describes a novel analytical methodology for modelling winding copper losses as a result of circulating currents (CC) in permanent magnet synchronous machines (PMSM). CCs are important to be considered at an early design stage especially in high frequency and high performance machines where sub-optimal design choices can lead to significant alternating current (AC) losses. Nowadays mainly FEM method is used for precise calculation of circulating currents. However, it suffers on significant computational time required for building models and simulation of circulating current effect that makes it inapplicable for optimization purpose. The paper demonstrates the computationally efficient methodology through comparison with FEM based models for high power density PMSM with concentrated winding and validation against experimental results for a stator section at the fundamental frequency from 500 Hz to 2000 Hz. The methodology allows simulation of CC and AC Ohmic losses when machine supplied by any current waveforms, for arbitrary size and location of the conductors in the slots. A key novelty of the proposed method is the utilization of subdomain (SDM) approach in conjunction with solution of a system of ordinary differential equations (ODE) for an equivalent electrical circuit of machine windings. This approach is precise and fast but has never been used before for circulating current loss evaluation in windings of electrical machines. The model is intended to be used at the design stage of an electrical machine where multiple geometric dimensions, winding configurations and conductor placement in the slot are considered towards an optimal design. Index Terms-Analytical model, circulating current loss, AC loss in electrical machines NOMENCLATURE ܴ ௌଵ , ܴ ௌଶ . . ܴ ௌ Strands resistance ‫ܮ‬ ௌଵ , ‫ܮ‬ ௌଶ . . ‫ܮ‬ ௌ Strands inductance ‫ܧ‬ ଵ , ‫ܧ‬ ଶ . . ‫ܧ‬ bEMFs induced in strands ‫ܯ[‬ ௦௧ ] Square n×n strands inductance matrix ൣ‫ܯ‬ ൧ Rectangular n×x matrix of mutual inductances between machine phases and each strand of investigated coil, x -number of phases [ܴ] Resistance matrix ‫ܫ‬ ଵ . . ‫ܫ‬ Current through each strand ‫ܫ‬ _ଵ . . ‫ܫ‬ _௫ Phase current ݊ Number of parallel strands per coil ܷ Voltage across the terminals of stranded coil ‫ܣ‬ Vector potential in the air gap ‫ܣ‬ ௦ Vector potential in the slot opening ‫ܣ‬ ௦ Vector potential in the slots with bulk coils ‫ܣ‬ ெ Vector potential in the PM subdomain ‫ܣ‬ ௧ , ‫ܣ‬ , ‫ܣ‬ ௧ Vector potentials in the slots with stranded coils ‫ܣ‬ ெ , ‫ܤ‬ ெ , ‫ܥ‬ ெ , ‫ܦ‬ ெ Coefficients for the function of vector potential in PM subdomain ‫ܣ‬ , ‫ܤ‬ , ‫ܥ‬ , ‫ܦ‬ Coefficients for the function of vector potential in the air gap ‫ܣ‬ ௦ , ‫ܤ‬ ௦ , ‫ܣ‬ , ‫ܤ‬ Coefficients for the function of vector potential in the slot opening ‫ܣ‬ ௦ , ‫ܤ‬ ௦ , ‫ܣ‬ , ‫ܤ‬ Coefficients for the function of vector potential in the slots with bulk coils

Designing for Conductor Lay and AC Loss Variability in Multistrand Stator Windings

“…Simplified models are valuable at a system level in estimating the loss behaviour of an electrical machine across its operating envelope. A large number of analytical methods have been presented in the literature [17], [26]- [28]. However, analytical methods are often constructed for a defined conductor lay (e.g.…”

Short‐circuit current difference of parallel strands in winding turns and its influence on the distribution of electromagnetic force

“…In order to further verify the accuracy of the calculation method, the leakage inductance between the high and lowvoltage (LV) windings was calculated using the energy method, as shown in Equation (17). And the calculation results were compared with experiment result, as shown in Table .4.…”

“…To achieve fast and accurate current calculation in parallel strands, equations of the frequency domain were introduced in the calculation. To calculate circulation losses in electrical machines and transformers, literature [17][18][19] proposed field-circuit hybrid methods, respectively. The computation time is significantly reduced.…”

“…The computation time is significantly reduced. However, the methods proposed in [17][18][19] associate the FEM with the circuit equation at the very beginning, increasing the difficulty of programming. Moreover, this kind of method only analysed the impact of current differences among parallel strands on transformer losses and did not investigate the spatial distribution of winding electromagnetic force under external shortcircuit conditions.…”

Novel ABC—Variable Based Closed-Form Analytical Model of Surface Mounted Permanent Magnet AC Machines With Sinusoidal and Non-Sinusoidal Back-EMF

“…Sub-domain models were also applied for analyzing electric machines in a number of research work [13]- [17]. In these methods, the machine is divided into sub-regions where Maxwell's equations are developed in each and boundary conditions are imposed to take the interfaces into account.…”