Modified Bertotti model identification tool

Introduction

This document presents the modified Bertotti model identification tool available in Flux. This tool is provided to the user as a Microsoft Excel file, and uses iron loss measurements to determine the coefficients and exponents in that model that best fit the measurement set. The identification of these parameters is required to create a material containing a Bertotti model for iron loss computations in Flux.

The following subjects are covered in the next sections of this documentation:

how to find the modified Bertotti model identification tool in Flux installation directory;

a reminder of the modified Bertotti model for iron loss computations in Flux;
a brief discussion on the data fitting approach implemented by the identification tool;
how to use the identification tool.

Location of the identification tool

The Bertotti model identification tool is available in the following path:

INSTALLATION_FOLDER\Flux\DocExamples\Tools\BertottiLossesCoefficients

in which INSTALLATION_FOLDER represents the Flux installation directory in your system.

To run the modified Bertotti model identification tool, open the file BertottiLossesCoefficients.xls available in that location in Microsoft Excel.

The modified Bertotti model

The modified Bertotti model implemented in Flux separates iron losses in three different parts: hysteresis losses, classical eddy current losses and excess losses accordingly with the equation below:

$d P = k_{1} B_{m a x}^{α_{1}} f + {k_{2} {(B}_{m a x} f)}^{α_{2}} + {k_{3} {(B}_{m a x} f)}^{α_{3}}$

Flux uses the expression above to evaluate the power density in AC steady state applications. In that equation:

dP represents the total iron losses per unit volume (in W/m³);
k₁ is the hysteresis losses coefficient;
k₂ is the classical Foucault losses coefficient;
k₃ is the excess losses coefficient;
α₁ is the hysteresis losses exponent;
α₂ is the classical Foucault losses exponent;
α₃ is excess (or supplementary) losses exponent;
f is the frequency (in Hz) and
B_max is the maximum magnetic flux density in an electrical period (in T).

Input data

Table 1 contains an example of iron losses measurements provided by an electrical steel supplier. Such measurements are usually performed on an Epstein frame, by imposing a sinusoidal magnetic flux density to the material.

Table 1. Specific iron losses measured at various frequencies, for several magnetic flux density values.
B_max(T)	Losses at 50 Hz (W/kg)	Losses at 100 Hz (W/kg)	Losses at 200 Hz (W/kg)	Losses at 2500 Hz (W/kg)
0.1	0.05	0.04	0.08	3.89
0.2	0.06	0.14	0.32	14.3
0.3	0.11	0.30	0.73	29.6
0.4	0.20	0.49	1.21	50.2
0.5	0.23	0.71	1.78	76.7
0.6	0.38	0.97	2.44	110
0.7	0.50	1.25	3.19	153
0.8	0.62	1.57	4.03	205
0.9	0.77	1.92	4.97	270
1.0	0.92	2.31	6.01	349
1.1	1.10	2.75	7.19
1.2	1.31	3.26	8.54
1.3	1.56	3.88	10.1
1.4	1.92	4.67	12.2
1.5	2.25	5.54	14.4
1.6	2.53
1.7	2.75
1.8	2.94

The input data required by the identification tool is similar in format to Table 1. It consists of a list of triplets (B_max, L, f) relating the peak magnetic flux density B_max (in teslas) in the material to both specific iron losses L (in W/kg) and frequency f (in Hz). The user also needs to provide the density or specific mass of the material ρ (in kg/m³).

The goal of the identification tool is to find exponents (α₁, α₂, α₃) and coefficients (k₁, k₂, k₃) that best fit the input data, as discussed in the next section.

Note: The input data format described above assumes a sinusoidal steady-state operation at each frequency. However, the parameters (k₁, k₂, k₃) and (α₁, α₂, α₃) identified by the tool are valid for both AC steady state and Transient applications in Flux.

Data fitting approach

To determine coefficients (k₁, k₂, k₃) and exponents (α₁, α₂, α₃) of the modified Bertotti model for a given set of measurements, the identification tool employs the least squares minimization approach presented below.

Let r_ij be the squared residual between specific loss measurement m_ij and the specific loss predicted by the modified Bertotti model b_ij = dP(k₁, k₂, k₃, α₁, α₂, α₃, Bmax_i , f_j ) / ρ , both evaluated at frequency f_j . This residual is computed by the following expression:

r_ij = (m_ij - b_ij)².

The identification tool uses an initial guess of parameters (k₁, k₂, k₃, α₁, α₂, α₃) and Excel's GRG Nonlinear solver (available in the Solver add-in) to minimize the total weighted residual:

R = Σ_j { w_j Σ_i [ r_ij ] },

in which w_j are weights attributed to each frequency f_j in the measurement set. These weights and parameters (k₁, k₂, k₃, α₁, α₂, α₃) are constrained to real values greater or equal than zero.

How to use the modified Bertotti model identification tool

As already mentioned, the modified Bertotti model identification tool is provided in the form of a Microsoft Excel file. This file contains four spreadsheets, namely:

Identification Tool: a spreadsheet containing empty input cells, and ready to be filled by the user.
Single Frequency Example: an example of identification using measurements performed at a single frequency.
Multi-Frequency Example: an example of identification using measurements performed at several frequencies.

The following steps are required to identify the coefficients k₁, k₂, k₃ and the exponents α₁, α₂, α₃ using the Identification Tool spreadsheet:

Fill the "Measured Losses" table with measurements performed at one or several frequencies.
Adjust the weights of the measurements for each frequency in the cell range C48:H48;
Provide the density of the material in cell M5.
Set an initial guess of the Bertotti coefficients and exponents in the cell range M11:M16.
Open the Excel solver menu: [Data] → [Solver] (requires enabling the Solver Add-in in Excel).
Configure and run the Excel solver as follows:
- "Set objective" field: set to $L$19
- Choose "To Min" option
- "By changing variable cells" field: set to $M$11:$M$16
- "Subject to the Constrains" field: set to $M$11:$M$16 >= 0
- "Select a Solving Method" field : choose the "GRG Nonlinear" solver.
- Click on "Solve"
Find the fitted/updated coefficients in the "Modified Bertotti Model Parameters" table.
Scroll down to verify the plots comparing the fitted Bertotti model to the measurements at each frequency.

Note: The user may reinforce or diminish the relative "importance" of the measurements performed at each frequency by adjusting the frequency weights. Setting the weight of a frequency to zero means disregarding the corresponding measurements in the data-fitting process.

Note: It may be useful to run the identification process more than once, by using the parameters identified in trial n as an initial guess for the trial n+1. The user may also find useful to adjust the weights iteratively between trials, by analyzing the partial residuals obtained for each frequency in the "Least Squares Fitting" table.