Iron loss modelling in finite-element analysis
Modelling iron losses is generally regarded as difficult. So far, no generally-accepted best solution has emerged.
The different modelling approaches can be divided into two categories:
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Post-processing models
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Online models
The following subsections briefly introduce these two, and summarize their implementation in EMDtool.
Post-processing models
In post-processing approaches, the actual finite-element solution is computed without considering the iron losses. Indeed, the nonlinear behaviour of ferromagnetic materials is represented by their anhysteretic (single-valued) BH-curve only, independent of their rate of change or past behaviour.
The iron losses are then estimated after the solution, from the computed flux density waveforms. Currently, there are two approaches in EMDtool.
Frequency-domain Bertotti model
In the frequency-domain Bertotti model, the computed flux density waveforms are then decomposed into their different frequency components, using the fast fourier transform (fft).
The total iron loss power density (in W/m^3 or W/kg) is then computed with the following formula:
\[p = \sum\limits_{n=1}^N c_\text{hysteresis} f_n B_n^2 + c_\text{eddy} (f_n B_n)^2 \quad \left( + c_\text{excess} (f_n B_n)^{1.5} \right) .\]Of this, the actual losses are then computed by integrating the loss density over the problem volume.
Note that depending on the author/context, the excess loss term may or may not be included. Indeed, even most EMDtool materials are modelled without it, as better fits with measured losses are often obtained this way.
NOTE: This model pretty much requires that a full electrical period is simulated. Otherwise, the model typically ‘sees’ a very large jump in flux density. This is because FFT essentially thinks that the signal given to it is periodic - that after reaching its end, it continues again from the beginning.
NOTE: Machines with concentrated windings often have significant subharmonics, meaning that several electrical periods have to be analysed, to simulate an entire electrical period in the rotor frame.
The loss coefficients are by default stored in the array material_properties.coeffs, as \([c_\text{hysteresis}, c_\text{eddy}, c_\text{excess}]\)
Time-domain Steinmetz model
The Steinmetz model propably means different things to different authors. Without further discussion, the approach adopted in EMDtool is the following:
By default, the hysteresis loss exponents \(a, b\) are both equal to 1.
A common question regarding this model is whether or not it includes minor loops, PWM effects, or DC-bias sensitivity. The short answer is yes.
The longer answer is related to the hysteresis loss term, which is normally understood as being the most influenced by the aforementioned factors. Specifically, the effect of DB-bias is seen via the \(B(t)\) term. It’s probably not a very accurate model, but the effect is considered.
Likewise, minor loops and other distortions are seen thanks to the \(\left| \frac{\text{d}B(t)}{\text{d}t} \right|^b\). In fact, in the author’s experience, the approach adopted here tends to agree rather well with the commonly-seen approach where the hysteresis losses are first computed for the fundamental harmonic only, and then multiplied by a distortion factor, be it based on THD or some sum-of-minor-loops approach.
Indeed, it must be stressed that all approaches are phenomenological, and not exactly relying on an underlying physical model.
EMDtool details and notes
By default (see the Material class), the Bertotti model loss coefficients are stored in the .coeffs field of the .material_properties property-struct of the Material object. These coefficient denote the loss density per kilogram, at 50 Hz frequency and sinusoidal flux density at 1 T peak value.
The .iron_loss_density_frequency_domain_Bertotti method is then used to compute the loss density in (W/m^3), using the Bertotti approach.
The .iron_loss_density_time_domain_Steinmetz then returns the losses from the Steinmetz approach. The same loss coefficients are used as for the Bertotti model, plus suitable scaling.
The intended way of computing the losses is the MaterialBase.losses, which then calls the .loss_density method of each Material in the problem. For the basic Material class, the behaviour of the losses method can be toggled by using the iron_loss_computation_method property.
2-dimensional flux densities
Perhaps the most significant difference between the textbook Bertotti/Steinmetz model is that they are typically given in 1D form, while finite element analysis results are very much 2D (or 3D).
Long story short and all discussion aside, the approach taken by EMDtool (at the time of writing at least) is to sum the contributions from the different axes together.
That means, using
\[B_n := \sqrt{ B_{x,n}^2 + B_{y,n}^2 }\]in the Bertotti model, and
\[\left( \frac{\text{d}B(t)}{\text{d}t} \right)^2 := \left( \frac{\text{d}B_x(t)}{\text{d}t} \right)^2 + \left( \frac{\text{d}B_y(t)}{\text{d}t} \right)^2\] \[\left| B(t) \right|^a \left| \frac{\text{d}B(t)}{\text{d}t} \right|^b := \left| B_x(t) \right|^a \left| \frac{\text{d}B_x(t)}{\text{d}t} \right|^b + \left| B_y(t) \right|^a \left| \frac{\text{d}B_y(t)}{\text{d}t} \right|^b\]for the Steinmetz model.
However, suggestions for improvements are more than welcome.
DC-flux bias
Losses under DC-biased flux density (hello rotors) are another tricky issue.
The Bertotti model ignores the DC-bias altogether, which is know to underestimate the losses to some degree.
By contrast, the Steinmetz model tends to overestimate the hysteresis losses, due to the B-term in the equation. The remove_DC argument can be given to the MaterialBase.losses method, to remove the DC-flux component from either the rotor, stator, or all domains of the model, to (over)compensate for this issue.
Online methods
In online computation methods, the hysteretic behaviour of magnetic materials is directly taken into account while running the simulation. More specifically, the BH-behaviour of the material now depends on its past behaviour.
Thanks to it object-oriented behaviour, EMDtool does support drop-in replacement of more advanced material models. Currently (2022/12), an online hysteresis model based on the stop hysteron is shipped with EMDtool. As typical, suitable material data is needed for fitting the models.