Interior Pm Motor

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Torque Optimization of the Interior-Permanent Magnet Synchronous Motors using Design Sensitivity Analysis
Mohammadreza Hassan Zadeh1, Arash Kiyoumarsi2

Electrical Engineering Department,
Abhar Islamic Azad University, Abhar, 22, Iran,
phone: +98 281 3349816, e-mail:,

Electrical Engineering Department, Isfahan University
Isfahan, Iran,


This paper presents a shape optimal design
approach to reduce the torque ripple of the interior-permanent magnet synchronous motors. The shape design sensitivity
formula and the finite element method were employed for shape optimization of the machine. The numerical results show that the optimized motor has lower torque ripple and more average torque.

Key words
Design sensitivity analysis, shape optimization, 2D finite
element method, torque ripple.

1. Introduction
Until recently, design sensitivity analysis for shape
optimization was widely used [1]-[3]. In conjunction with
the two-dimensional finite element method, the
sensitivity analysis reported in [1] has been successfully
applied to some optimization problems in magnetostatic
systems. The shape optimization of the permanent
magnet synchronous (PMS) motors was the subject of
many papers. It is indicated the back-EMF waveform has
an important role to produce the smooth torque. Lee and
Park employed the shape design sensitivity formula and
the finite element method for calculating the sensitivity
of flux-linkage to the design variables determining the
shape of iron pole piece [2].
Shape design sensitivity analysis in electromagnetic
systems can be developed using two fundamentally
different approaches. The one is the discrete approach,
where design derivatives of a discretized system equation
are taken to obtain sensitivity information. The other
called the continuum approach, where design derivatives
of the variational governing equation of the
electromagnetic system are taken to obtain explicit
design sensitivity formula in an integral form.
In the sensitivity analysis there are some design variables
that are independent quantities are varied in order to
achieve the optimum design. Upper and lower limits are

specified to serve as constraints on the design variable.
These limits define the range of variation if design
variable. The sensitivity analysis method uses gradients
of the optimization function with respect to design
variables. For each iteration, gradient calculations (which
may employ a steepest descent method) are performed in
order to determine a search direction, and a line search
strategy is adopted to minimize the optimization function.
Thus each iteration is composed of a number of
subiterations that include search direction and gradient
These iterations continue until convergence is achieved.
The problem is said to be converged if the change in
objective function from the best design to the current
design is less than the objective function tolerance or the
change in objective function from the previous design to
the current design is less than the objective function

Figure 1 shows the algorithm used for the shape
optimization in 2D finite element model of the interior
permanent magnet (IPM) motor.
For permanent magnet synchronous motors one of the
most important parameters is the torque ripple when
motor works in the steady-state situation. To improve the
torque ripple, the motor sizes of magnet length, pole arcs,
slot opening and skew have been controlled or the
number of slots and pole arc chosen, but a substantial
ripple still remains. In this paper, we want to obtain the
torque ripple as low as possible, so the torque ripple
selected as optimization function and minimized. For this
purpose, we must obtain the torque waveform for each
iteration when the motor rotates at synchronous speed.

2. Optimal shape design of IPM motor

Fig. 3. The initial proposed IPM...
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