[Paper Review] Scaling Conception of Energy Loss' Separation in Soft Magnetic Materials

K. Sokalski, J. Szczygłowski+1 more|arXiv (Cornell University)|Nov 3, 2011

Magnetic Properties and Applications17 references3 citations

TL;DR

This paper proposes a Partial Data Collapse (PDC) framework to enable universal comparison of energy loss data in soft magnetic materials (SMMs) beyond the two-component model. By scaling energy loss data into dimensionless, sample-independent representations across two-dimensional subspaces spanned by consecutive frequency powers, the method allows data collapse even when higher-order losses (e.g., third and fourth order) are significant. The key contribution is a universal, scalable method to separate and compare hysteresis, classical, and excess losses across different SMMs, revealing that crystalline SMMs exhibit positive excess losses while amorphous SMMs show negative excess losses in the $L_{1,2}$ representation.

ABSTRACT

Data collapse enables comparison of measurement data measured in different laboratories on different samples. In the case of energy losses in Soft Magnetic Materials (SMM) the data collapse is possible to achieved only if the measurement data can be described by the two components formula. For more complicated cases we propose to perform data collapse's sequence in the two-dimensional subspaces $L_{i,i+1}$ spanned by the appropriate powers of frequency $\{f^{i},f^{i+1}\}$. Such approach enables the data comparison in the different two-dimensional subspaces. This idea has been tested with measurement data of the four SMM-s: amorphous alloy extrm{Fe}_{78} extrm{Si}_{13} extrm{B}_{9}$, amorphous alloy $ extrm{Co}_{71.5} extrm{Fe}_{2.5} extrm{Mn}_{2} extrm{Mo}_{1} extrm{Si}_{9} extrm{B}_{14}$, crystalline material -- oriented electrotechnical steel sheets 3% Si--Fe, iron--nickel alloy $79% extrm{Ni}- extrm{Fe}$. Intermediate calculations revealed interesting property of the energy losses in the cristalline and amorphous SMM-s which lead to the following hypothesis. Let $P_{tot\,1,2}=f_{1,2}(1+f_{1,2})$ be scaled two-components formula for the energy loss in SMM, where $f_{1,2}$ is the corresponding scaled frequency. Then the scaled energy losses' values in amorphous SMM are below the second order universal curve $P_{tot\,1,2}=f_{1,2}(1+f_{1,2})$, whereas the scaled energy losses' values in crystalline SMM are above that universal curve.

Motivation & Objective

• To address the limitation of traditional data collapse methods, which fail when higher-order energy loss terms (beyond $f$ and $f^2$) are significant in soft magnetic materials (SMMs).
• To develop a scalable, universal framework for comparing energy loss data across different SMMs, laboratories, and experimental setups, even when data do not collapse under standard two-component models.
• To enable separation and quantification of hysteresis, classical, and excess energy losses in a dimensionless, sample-independent manner through multi-dimensional scaling and gauge transformations.

Proposed method

• Proposes a two-step scaling and gauge transformation to map raw energy loss data into dimensionless, sample-independent representations in two-dimensional subspaces $L_{j,j+1}$ spanned by $\{f^{j}, f^{j+1}\}$.
• Derives a generalized scaling formula $P_{totackslash, j,j+1} = f_{j,j+1}^{j}(1 + f_{j,j+1}) + \psi_{j,j+1} + \chi_{j,j+1}$, where $f_{j,j+1}$ and $P_{totackslash, j,j+1}$ are scaled frequency and loss, respectively.
• Applies gauge transformation to isolate the universal term $f_{j,j+1}^{j}(1 + f_{j,j+1})$, which forms the basis for partial data collapse (PDC) in each subspace.
• Uses power series expansion of total energy loss $P_{tot} = B_m^\beta \sum_{k=1}^n \Gamma_k \left( \frac{f}{B_m^\alpha} \right)^k$ to model losses, with $\Gamma_k$ estimated from experimental data.
• Applies the method to four SMMs: amorphous Fe78Si13B9, Co71.5Fe2.5Mn2Mo1Si9B14, 3% Ni-Si-Fe, and 79% Ni-Fe, validating the approach across material types.
• Establishes a theorem proving that for any $n$-th order generalized homogeneous loss model, $n-1$ such scaling+gauge transformations exist, enabling PDC across multiple subspaces.

Experimental results

Research questions

• RQ1 Can energy loss data from different soft magnetic materials be universally compared when higher-order loss terms (beyond $f$ and $f^2$) are significant?
• RQ2 How can data collapse be extended beyond the two-component model to enable sample-independent comparison of energy losses in SMMs?
• RQ3 What is the role of the $f_{j,j+1}^j(1 + f_{j,j+1})$ term in enabling partial data collapse across different two-dimensional subspaces of frequency powers?
• RQ4 Why do amorphous and crystalline SMMs exhibit distinct behaviors in the $L_{1,2}$ representation, with amorphous materials below and crystalline materials above the universal curve?
• RQ5 Can a universal, scalable framework be constructed to separate and quantify hysteresis, classical, and excess losses in a dimensionless, sample-independent form?

Key findings

• The Partial Data Collapse (PDC) framework successfully enables data collapse in two-dimensional subspaces $L_{j,j+1}$, even when higher-order loss terms are non-negligible, overcoming limitations of traditional two-component models.
• The term $f_{j,j+1}^j(1 + f_{j,j+1})$ is identified as the universal, sample-independent component in each $L_{j,j+1}$ subspace, forming the core of the PDC method.
• For the $L_{1,2}$ representation, amorphous SMMs exhibit scaled excess losses $P_{ex\,1,2}^{amorphous} \leq 0$, while crystalline SMMs show $P_{ex\,1,2}^{crystall} \geq 0$, indicating a fundamental difference in loss mechanisms.
• The method is universally applicable to any experimental data governed by a generalized homogeneous scaling law, with a proven theorem showing existence of $n-1$ such transformations for an $n$-th order model.
• The PDC framework allows comparison of data from different laboratories and samples by introducing a common, dimensionless measure of error, resolving prior inconsistencies in inter-laboratory comparisons.