Office of Technology Transfer – University of Michigan

A Harmonic Balance Finite Element Method for the Steady-State Simulation of Synchronous Machines

Technology #5115

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Heath Hofmann
Managed By
Drew Bennett
Associate Director - Software Licensing 734-615-4004


A key challenge in designing electric machines for vehicular applications is achieving high power density and efficiency while maintaining the ability to operate over a wide speed range. This requires the electric machine to be designed and optimized by characterization of its steady state behavior over its entire range of torque-speed operating points. Finite element analysis (FEA) is a powerful and highly accurate tool for determining electric machine characteristics. The harmonic balance finite element method (HBFEM) is a potential strategy to determine the steady-state behavior of electric machines, but is not widely used due to storage cost, computational complexity and modeling difficulties of the existing methods. Development of accurate and computationally cost effective harmonic balance finite element methods can open up opportunities in the FEA software market that is greater than $2.2M in size.


A computationally efficient harmonic balance finite element method for the steady-state analysis of electric machines has been presented by the researchers at the University of Michigan. An important contribution comes from the development of a model of the airgap magnetic vector potential to capture the periodic motion of the rotor; making the problem effectively stationary. In addition, a new frequency domain boundary condition is introduced which allows the stator model to be reduced to one third of a pole. This reduced the storage cost and computational complexity of the method which is at least an order of magnitude faster than existing methods. The method also yields better results for a given simulation time in a way that it is more effective at resolving low frequencies which determine the fundamental behavior of a permanent magnet machine. Proven by numerical simulations, the proposed technique is an accurate harmonic analysis tool that is significantly faster than standard transient analysis as well as the state of art steady analysis techniques.

Applications and Advantages


  • Electromagnetic simulation software
  • Design and analysis of electric machines


  • Faster method
  • Reduced computational complexity
  • Improved accuracy