Design and simulation of electrical engineering applications containing ferromagnetic parts require the accurate modeling of hysteresis characteristics and the implementation of the models and solve the nonlinear partial differential equations derived from Maxwell’s equations.
The partial differential equations are generally nonlinear because of the nonlinear characteristics of ferromagnetic materials.
The numerical analysis of electromagnetic fields can be characterized by the electric and magnetic field intensities and flux densities formulated by Maxwell’s equations, which are the collection of partial differential equations of the electric field intensity \(\mathbf{E}\), the magnetic field intensity \(\mathbf{H}\), the electric flux density \(\mathbf{D}\) and the magnetic flux density \(\mathbf{B}\).
Constitutive relations between the above quantities are defined to take into account the macroscopic properties of the medium.
We consider a constitutive relation between the magnetic field intensity and the magnetic flux density which is nonlinear , given by the operator \(\mathbf{B} = \mathcal{B}(\mathbf{H})\) or equivalently \(\mathbf{H} = \mathcal{B}^{−1}(\mathbf{B})\). These nonlinear characteristics can be reformulated by introducing the polarization method and the resulting system of nonlinear equations can be solved using fixed point iterations.
According to the polarization method, the magnetic flux density can be split in two parts as
\[\mathbf{B} = \mu \mathbf{H} + \mathbf{R}\]
where \(\mu \mathbf{H}\) is a linear term, \(\mu\) is constant and the nonlinearity is in the second term \(\mathbf{R}\). It is a magnetic flux density like quantity.
The question is the appropriate value of the parameter \(\mu\). This representation can be reformulated as
\[\mathbf{R} = \mathbf{B}-\mu \mathbf{H} = \mathbf{B}-\mu \mathcal{B}^{-1}(\mathbf{B})\]
Here, the inverse type hysteresis characteristics are used.
It is important to note that the nonlinear equations are solved by iterative methods, the \(n\)-th iteration is denoted by the superscript \((n)\) in the following.
By using the various formula in the constitutive relations defined in Maxwell’s equations, the solution of the nonlinear partial differential equations with appropriate boundary conditions can be formulated as
\[\mathbf{B}^{(n)} = \mathcal{M}(\mathbf{R}^{(n-1)})\]
where the operator \(\mathcal{M}\) represents the set of Maxwell’s equations and the boundary conditions. The starting \(\mathbf{R}^{(0)}\) is arbitrary. The value of electromagnetic field quantities in the \(n\)-th step are depending on the value of \(\mathbf{R}\) in the \((n − 1)\)-th step and they represent source like quantities.
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The source of electromagnetic fields (e.g. the electric current density \(\mathbf{J}\)) is not changing during fixed point iteration, i.e. the fixed point iteration has no particular physical meaning.
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It may be better to use the reluctivity \(\nu\) when applying the inverse hysteresis model, because \(\nu(B) = \frac{\partial H}{\partial B}\).
Let us now multiply the relation above by \(\nu_{\mathrm{o}} = 1/\mu_{\mathrm{o}}\), we have
\[\nu_{\mathrm{o}} \mathbf{B} = \mathbf{H} + \nu_{\mathrm{o}}\mathbf{R},\]
\[\mathbf{H} = \nu_{\mathrm{o}} \mathbf{B} - \nu_{\mathrm{o}}\mathbf{R} = \nu_{\mathrm{o}} \mathbf{B} + \mathbf{I}\]
where \(\mathbf{I}= -\nu_{\mathrm{o}}\mathbf{R}\), \(\nu_{\mathrm{o}}=\frac{\nu_{\mathrm{min}}+\nu_{\mathrm{max}}}{2}\) and \(\nu_{\mathrm{min}}\) and \(\nu_{\mathrm{max}}\) are the minimum and maximum slope of the inverse of the hysteresis characteristics.
We have \(\nu_{\mathrm{min}} = 1/\mu_{\mathrm{max}}\) and \(\nu_{\mathrm{max}}=1/\mu_{\mathrm{min}}\).
The nonlinear fixed point iteration can be summarized as follows.
The iteration can be started by an arbitrary value of \(\mathbf{I}^{(0)}\), then for \(n > 0\) we do,
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the magnetic flux density \(\mathbf{B}^{(n)}\) can be calculated by solving the partial differential equations obtained from Maxwell’s equations and using \(\mathbf{I}^{(n−1)}\), in other words,
\mathbf{B}^{(n)} =\mathcal{M}({\mathbf{I}^{(n−1)}}),
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the magnetic field intensity \(\mathbf{H}^{(n)}\) can be calculated by applying the inverse type
hysteresis model, \(\mathbf{H}^{(n)} = \mathcal{B}^{−1}(\mathbf{B}^{(n)}),\)
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the nonlinear residual term can be updated by using the magnetic flux density and
magnetic field intensity,
\mathbf{I}^{(n)} = \mathbf{H}^{(n)} − \nu_{\mathrm{o}} \mathbf{B}^{(n)} = \mathcal{B}^{−1}({\mathbf{B}^{(n)}}) − \nu_{\mathrm{o}} \mathbf{B}^{(n)}
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and the sequence defined by steps (i)–(iii) must be repeated until convergence.
Convergence criterion can be \(||\mathbf{I}^{(n)} − \mathbf{I}^{(n−1)}|| < \varepsilonstem:\), where \(\varepsilon\) is the tolerance.
Convergence criterion can be also defined by using the magnetic field intensity or the magnetic flux density.
\[\nabla \times (\nu_{\mathrm{o}} \nabla \times \mathbf{A} ) = \mathbf{J} - \nabla \times \mathbf{I}\quad \text{ in } \Omega\]
associated to proper boundary conditions for \(\mathbf{A}\).
The fixed point iteration algorithm reads
We start with arbitrary \(\mathbf{I}^{(0)}\) and then for \(n > 0\)
\nabla \times (\nu_{\mathrm{o}} \nabla \times \mathbf{A}^{(n)} ) = \mathbf{J} - \nabla \times \mathbf{I}^{(n-1)}\quad \text{ in } \Omega
\mathbf{B}^{(n)}=\nabla \times \mathbf{A}^{(n)}
\mathbf{I}^{(n)} = \mathcal{B}^{−1}({\mathbf{B}^{(n)}}) − \nu_{\mathrm{o}} \mathbf{B}^{(n)}
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repeat steps i-iii until convergence, e.g.
||\mathbf{I}^{(n)}-\mathbf{I}^{(n-1)}|| < \varepsilon
