The Study on the Consistency Evaluation Method of Coercivity and Remanence Measurement Results for Rare-Earth Permanent Magnets

DOI:https://doi-001.org/1025/17736548451974

Weidong Wang^1,*

¹Zhejiang Industry Polytechnic College, Shaoxing 312000, P.R.China

*Corresponding author. E-mail address: wwdhy2008@sina.com

Abstract

Rareearth permanent magnets such as NdFeB and SmCo have played central roles in a wide range of applications, such as renewable energy systems, defense applications, and high-end Electric motors. Coercivity (Hc) and remanence (Br) measurements ought to be precise to ensure the performance and reliability of such materials. However, there are still significant problems in the reproducibility of the results of measurements, particularly, in the comparison of measurements of different instruments and laboratories.The paper gives a comprehensive methodology of determining the reproducibility of the measurements of coercivity and remanence. It suggests a statistical model that measures the variation in interlaboratory measurement, which involves detection of sources of errors and error propagation. Different consistency evaluation indices, including repeatability, reproducibility, and a recently introduced consistency index (CI), are proposed in order to measure the measurement consistency in different conditions. Experimental validation It is done with various instrumental types of measuring tools and standardized samples to prove the merits of the presented methodology.The findings point to the importance of instrument calibration, environmental control, and measurement protocols in order to provide high levels of measurement consistency. The approach is a credible mechanism of enhancing quality and consistency of magnetic property measurements at industrial and research levels.

Keywords：Rareearth permanent magnet, measurement consistency, metrology, interlaboratory comparison, uncertainty analysis

1. Introduction

Rare-earth permanent magnets, especially those of the Nd-Fe-B and Sm-Co alloys, have become important functional materials in the modern industry because they have high maximum energy product, large coercivity, and high thermal stability(Coey, 2010). These are used in high-performance electric motors, electric generators, precision sensors, actuators, magnetic bearings, aerospace, and energy applicable applications (e.g. wind power generation, electric vehicle traction drives) (Fiorillo, 2004)(Meng et al., 2020). As the electrification, automation and renewable energy technologies are continuously grown in scale, the volume of production and performance specifications of rare-earth permanent magnets has grown steadily(Gutfleisch et al., 2011). In such applications, the overall behavior of devices like torque output, energy efficiency, response speed and stability is highly reliant on the inherent and extrinsic magnetic properties of the employed permanent magnets. Two of these properties, coercivity (Hc) and remanence (Br) are critical values that dictate a material in terms of its demagnetization resistance and usable magnetic flux density, respectively(Skomski and Coey, 1999). It is important that these parameters can be reliably determined and controlled to design materials, optimize processes, guarantee quality, and predict performance of rare-earth permanent magnet parts(Sepehri-Amin et al., 2018).

Although there are standardized methods and protocols of magnetic property measurements of permanent magnets, the high consistency of coercivity and remanence measurements is a major problem(ISO 5725-1, 1994). Variability is also common when same sample is measured with those instruments of different types, in different laboratories, or under different environmental and operational conditions(Kresse et al., 2024). This variability can be explained by a number of factors, such as distinctions in the principles of measurement, instrument settings, sample-related factors, calibration process, and experience of the operator(Ouazib et al., 2025). These discrepancies lead to the low repeatability (within-laboratory agreement) and reproducibility (between-laboratory agreement) of Hc and Br measurements(Meng et al., 2020). This complicates the process of reliable uncertainty evaluation and also makes the comparison of results in various laboratories challenging. Despite the past studies on enhancing methods of measurement, the systematic investigations on the consistency assessment between instruments and laboratories that are specially designed to be used in measures of coercivity and remanence of rare-earth permanent magnets are wanting. Also, there are no well-established standardized frameworks of interlaboratory comparisons, which are supported by intensive statistics analysis, and more elaborate methodologies are still required to evaluate measurement consistency(Zhang et al., 2020).

The following paper suggests a consistency evaluation technique with a specific design corresponding to the coercivity and remanence measurements of the rare-earth permanent magnets. The most important contributions of the work can be regarded in the creation of a statistical evaluation framework that incorporates indices of repeatability, reproducibility, and agreement(Sepehri-Amin et al., 2018)(Fischbacher et al., 2017) and provides an all-encompassing approach to the analysis of the dispersion and bias of measurement results. This framework is used when multiple sources of data and multiple laboratories are used to obtain data that can be quantified to evaluate between-laboratory consistency (Gutfleisch et al., 2018). The systematic deviation and outliers are also identified through the proposed method(Skomski and Coey,1999). Also, there are practical guidelines to understand the measurement of consistency and correlate it with uncertainty budgets and quality control processes(Przybylski,2016)(Coey,2012), which helps to advance the measurement processes and calibration procedures. Finally, the objective of the work is the increase in the reliability and comparability of the measurement data of coercivity and remanence, which leads to the standardization of testing work in the sphere of rare-earth permanent magnets (Sepehri-Amin et al., 2018).

The rest of this paper is structured in the following way. Part 2 explains the principles of measuring coercivity and remanence, the form of instruments that will be taken into account and the principal source of measurement variability(Ouazib et al., 2025). Section 3 presents the suggested methodology of consistency evaluation, such as the statistical models, the indicators of repeatability and reproducibility, the Consistency Index, and the interlaboratory comparison scheme design(ISO 5725-4, 2020). Section 4 gives the experimental design and data acquisition procedure that includes sample preparation, instrumentation, calibration, measurement procedure, and data handling rules. Section 5 presents the findings of the application of the suggested method to the coercivity and remanence measures, discusses them, presenting the results of repeatability and reproducibility, the Consistency Index analysis, and the sources of errors(Zhang, W., and Zhao, X., 2020). Lastly, the conclusions about the major findings and conclusions of the work are drawn in Section 6, where the recommendations to the practical implementation of the proposed method are provided, as well as recommendations to the future research of the metrology of permanent magnet properties.

2. Results and discussion

2.1 Repeatability and Reproducibility Analysis

Repeatability and reproducibility analysis shows much information on the stability of coercivity (Hc) and remanence (Br) measurements measured by various instruments and laboratories. Repeatability, which can be considered the change of measurements of the same laboratory, was small in most labs with both Hc and Br, meaning that the instruments could reproduce their own values within a narrow range when the process and conditions did not change (Przybylski, 2016). In the case of coercivity as well as remanence, there was a general good repeatability of individual instruments with a marginally better repeatability of remanence compared to coercivity (Coey,2010). This is in line with the fact that determination of coercivity is more sensitive to the shape and definition of the hysteresis curve around the point where it intersects with the field axis which is affected by noise, interpolation options and demagnetizing effects (Skomski and Coey,1999).

The difference in measurements among the various laboratories as indicated by the value of reproducibility was much larger than the value of repeatability. Measurements of coercivity and remanence in different laboratories were occasionally many times larger than the inter-laboratory variance. This emphasizes the role of inter-laboratory errors, i. e. instrument type, calibration methods, and environmental conditions, in total measurement errors. Figure 1 illustrates the average limits of repeatability and reproducibility of the measurement of coercivity and remanence with various instruments. Vibrating Sample Magnetometer (VSM) had the highest repeatability of remanence (0.02) then Pulse Magnetizers.

Patterns among the types of instruments were also discovered during the comparison of repeatability and reproducibility. The laboratories with similar hysteresis graph systems were more likely to be agreed with each other than other laboratories which applied different measurement principles. This point in support of the idea that some of the between-laboratory variance is associated with systematic differences that each type of instrument has. This is because these findings indicate the need to standardize instruments and procedures within laboratories in order to enhance measurement consistency. Normative repeatability and reproducibility boundaries of coercivity and remanence can be summed up in Table 1.

Table 1  Typical repeatability and reproducibility limits for coercivity and remanence measurements

Figure 1 Repeatability and Reproducibility Comparison

2.2 Consistency Index Evaluation

The experimental data was used to determine the Consistency Index (CI) of every lab-instrument combination, taking into account both the deviations concerning the reference values and the reported uncertainties (Gutfleisch, 2011; Sepehri-Amin et al., 2018). CI values were then graphically and numerically summarised in order to give a clear evaluation of the consistency of the measurement.

Box plots of the CI by instrument type showed there were distinct groups of laboratories. Those instruments whose calibration processes were well documented and those with good environmental control were those with higher CI values (Skomski and Coey, 1999; Zhang, W., and Zhao, X., 2020). Laboratories with more conservative and realistic uncertainties also had balanced CI scores, which indicates a proper correlation between the deviation and uncertainty stated (Meng et al., 2020). The box plots of which are displayed in Figure 1 clearly indicate variation of CI between the various types of instruments.

The mean values of coercivity and remanence with the 95% confidence ranges of the mean values were also represented using confidence interval charts and provided a visual depiction of the consistency of measurement (Coey, 2010; Zhang et al., 2020). The CI values were typically high at the laboratories that showed an overlap between the confidence interval and the reference value or at the laboratories that had the confidence interval that was evidently different than the reference value, suggesting the possibility of systematic biases (Gutfleisch, 2011; Sepehri-Amin et al., 2018). Figure 2 shows the distribution of the CI of values in the individual participating laboratories with the highest consistency showing that CI values were close to 1.0 suggesting high level of agreement with the reference values. On the other hand, the lower consistency of laboratories was found to be CI less than 0.5 which implied high discrepancies.

The comparison of CI values across laboratories led to the identification of three categories:High consistency, in which CI values were near to 1.0 in most cases;Moderate consistency, in which CI values were usually reasonable, but were significantly different across samples;Low consistency, in which CI values were often below the suggested value, which indicated a high variation or underestimation in uncertainty.

These findings indicate that the Consistency Index comprises of a useful, one-number, metric to order ranking the performance of the various measurement systems and the identification of laboratories which could be improved specifically through calibration or procedure enhancement.

Figure 2  Consistency Index (CI) Across Labs

2.3 Error Source Analysis

It is important to understand the sources of errors in the measurements of coercivity (Hc) and remanence (Br) to enhance the consistency of measurements and to guarantee the reliability of the measurements. Detailed statistical analysis, along with the information on the instruments employed, measurement processes and environmental factors, have indicated some major sources of variability that bring about differences in the measured values.

When the laboratory results showed recurring deviation of a certain direction relative to the reference or the average of a group of samples, then there were systematic errors. This effect was usually attributed to certain causes like incomplete or old fashioned calibration procedures, incorrect corrections of demagnetising factors or not using high magnetizing fields that could not saturate high-coercivity samples (Skomski and Coey, 1999).

An example is that the calibration procedure in a particular laboratory might not have been reviewed on a regular basis to accommodate changes in equipment or environment. DDue to that, the strengths of magnetic fields could not be calibrated adequately, and this led to deviations in measurements. The fact that the measurements of the coercivity are incorrect can also be explained by the inability to introduce the proper corrections in the measurements of the demagnetizing factor, which takes into account the influence of the material which surrounds the magnet. Also in high-coercivity samples, the inferiority of the magnetizing fields can also cause failure of the sample to fully saturate to achieve lower remanence values than would be expected (Coey, 2010).

Important conditions of the environment are of the error of coercivity and remanence measurements. The laboratories that had worse temperature control displayed more disparities in the measurements of coercivity and it has been observed to be in line with the temperature dependence of the properties of rare-earth magnets. Any slight variations in temperature may affect the performance of these magnets in a big way. There is a possibility of changing the magnetic properties of materials due to temperature variations, which causes a change of the measured values of coercivity and remanence (Meng et al., 2020).

Especially, remanence measurements in laboratories based on Hall probe systems encountered extra problems. The effects of temperature on the probe sensitivity and the magnet per se were not corrected in all cases, and this resulted in systematic Br shifts. As a result, because Hall probes are also temperature sensitive, unless temperature variations are corrected by calibration or by factoring in temperature variations, the results might be biased, which means that a consistent environmental condition is also crucial to guaranteeing the accuracy of the measurements (Przybylski, 2016).

Other major sources of error was the operator influence that was mostly evident in those laboratories where written procedures were not as elaborate. These laboratories exhibited bigger within-laboratory variation in measurements, implying that the variability was caused by differences in the sample location, the measurement conditions, and subjective data processing (Sepehri-Amin et al., 2018). Even minimal differences in the placement of a sample in the measuring device or the differences of measurements can cause discrepancies in the outcomes. The necessity to have well defined, standardized procedures in many areas like sample mounting, and reading of an instrument emerges.

Moreover, the data can be processed manually, which may also lead to errors. To illustrate, in case the data are smoothed or processed in an uneven manner, it may corrupt the results. These sources of error can be mitigated by the use of operator training and elaborate instructions. The inconsistency caused by human factors can be significantly mitigated by standardizing the procedures and making sure that the operators adhere to the standard protocols (Coey, 2012).

Figure 3 shows the effects of instrument calibration and environmental factors on measurements. Systematic deviation of measured values between measurements was observed to be caused by inconsistent calibration, especially in pulse magnetizer systems. It was occasionally discovered that these systems which utilize the use of pulsed magnetic fields to magnetize the samples had problems in calibration that resulted in poor readings. It is necessary to calibrate these systems on a regular basis to make sure that the pulse intensity and frequency are adequate to obtain reliable measurements.

Also, it was observed that temperature changes also made the measurements of coercivity complex. Laboratories that used temperature correction factors i.e. adjusted their measurements to known temperature coefficient had more consistent results. This creates the problem of the necessity of the laboratories to incorporate temperature compensation in their measurements so as to obtain less erroneous results over time.

The statistical analysis along with the qualitative insights gave a consistent and in-depth explanation of the principal sources of errors in coercivity and remanence measurements. The systematic analysis of the data and comparing it with the knowledge of the instruments and the environment revealed which area of improvement was required. Specifically, the calibration of the instruments, the control of the environment, and the training of the operators became the key areas with respect to which the introduction of specific improvements could substantially improve the consistency of the measurements (Skomski and Coey, 1999).

As an example, systematic errors could be reduced by making sure the instruments are regularly calibrated and enhancing the temperature control of the laboratory to ensure the repeatability and reproducibility of measurements. Besides, implementing a training on operators to act in a standard way and on how to handle data would lower the variability introduced by operators, which would then enhance the accuracy and predictability of the measurements.

Figure 3 Error Sources: Calibration and Temperature Effects

2.4 Method Validation

In order to test the usefulness of the developed Consistency Index (CI) it was juxtaposed to conventional methods of metrology that are traditionally applied in the assessment of coercivity (Hc) and remanence (Br) readings. Conventional approaches are usually interested in the measurement of repeatability, reproducibility, and mere distortions to reference values. But, these methods have significant weaknesses, in particular, their failure to directly incorporate uncertainty into the analysis. Without uncertainty, the traditional methods might not give the correct or comprehensive evaluation of the measurement consistency. As an illustration, a lab can be good in terms of repeatability but may yield unreliable results without the uncertainty being accounted in the correct manner.

Conversely, CI methodology has an explicit inclusion of a deviation and uncertainty in the evaluation, thus it is a more robust and reliable approach. This incorporation addresses the weaknesses of the old procedures since in this manner, laboratories can no longer be in position to seal the lapses by simply pointing out impractically low uncertainties. The CI approach indicates those laboratories with small deviations but containing small uncertainties (Przybylski, 2016). On the other hand, the CI method allows the factories that experience more deviations and have reasonable uncertainties to be more reflected in their performance and, therefore, the uncertainty reporting process becomes transparent.

Other than that, the strength of the CI approach has been manifested in application on the different types of instruments and measurement situations. The CI approach (as opposed to the traditional ones) offers a single, detailed measure of both the dispersion and uncertainty of the measurement results compared to the traditional ones that typically involve only the concept of repeatability or reproducibility. Interlaboratory comparisons of the given methodology have shown the results that are in line with the expert qualitative evaluation and other routine statistical evaluations.

The ambiguity of CI method has been discovered to be a very useful metric of uncovering systematic errors, environmental and operator based issues, which otherwise would have not been discovered by the traditional means. The CI approach proposes a more precise and accurate assessment of consistency of measurement since it comprises uncertainty.

Figure 4 shows the outcomes of applying the CI methodology and comparing it with traditional methods to demonstrate that CI approach will provide more important information on the measure consistency. It is the comparison that determines the positive aspects of the CI approach in identification of laboratories, which report unrealistically the uncertainty and provide a more transparent and realistic evaluation of the measurement practices.

Lastly, CI methodology is a significant revolution as opposed to the traditional conservative metrological approaches. The CI process incorporates the deviation, dispersion, and uncertainty into a single comprehensive tool that gives a superior and sound assessment on the measurement consistency. The approach is not only known to improve the performance of the laboratories but also improves transparency and accountability in reporting the measurement results. It is flexible enough to be applicable to a wide range of measurement situations and the fact that it has an inbuilt mechanism of uncertainty is significant as a safeguard against false outcomes.

Figure 4 CI Method Validation: CI vs. Deviation

2.5 Discussion

The given methodology of consistency assessment is a significant move to reliability and accuracy of the measurements of the coercivity (Hc) and remanence (Br). The mode of approach is highly effective as it has various significant considerations of metrological quality like variation to a reference, variation of measurements, and evaluation of uncertainty (Gutfleisch, 2011). All these are traditionally viewed as separate entities but by combining them into a single measure Consistency Index (CI), such approach provides a more comprehensive and transparent evaluation. CI approach helps test labs to analyze their performance based on a reference and can help them to discover specific areas of their performance improvement, such as calibration procedures or measurements practices (Meng et al., 2020).

The key strength of this approach relies on the idea that it is founded on the widely accessible statistical analysis and basic mathematical calculations. The characteristics of the method enable it to be applicable in any laboratory with no advanced mathematical knowledge or a particular program, which means that it can apply even to smaller or less technologically advanced laboratories (Sepehri-Amin et al., 2018). Moreover, the approach focuses on open data management, and the outliers are recorded, and the traceability of the calibration is explicit, which is another aspect in favor of good metrological practices. These factors play an important role in ensuring the integrity and reproducibility of the data, because they guarantee that in case of any possibility of errors or inconsistencies, it will be thoroughly handled and recorded accordingly (Skomski and Coey, 1999).

The proposed methodology has some limitations in spite of its numerous benefits. On the one hand, it depends on the possibility of homogeneous samples and credible reference values (Coey, 2012). When the samples, on which measurements are carried out, are a priori non-homogeneous; when the values of the reference are doubtful or biased; the interpretation of the CI may be in question. When this happens, the CI may not be a good measure of the actual consistency of the measurements and the analysis may give erroneous conclusions. In addition, the technique presupposes realistic estimates of uncertainty of measurements in laboratories. Provided that a laboratory systematically under or overestimates their uncertainties, the CI may be misleading, either exaggerating the coherence of the measurements (or not pointing to the areas that need further consideration) (Przybylski, 2016).

The other weakness is that the methodology has mostly been utilized in measurements at or close to room temperature and in quasi-statics field conditions. This emphasis on relatively controlled conditions implies that the methodology will not necessarily be able to reflect the complexity of dynamic environments or measurements at high temperatures, where the behavior of rare-earth permanent magnets can shift dramatically. In practice, rare-earth permanent magnets are frequently subjected to the variability of the magnetic field, or to high temperatures or dynamic processes which may influence their behavior and in measurement. To normalize the methodology in these conditions, new test protocols will have to be developed and more errors sources will have to be identified, which may not be available in regular laboratory conditions (Zhang, W., & Zhao, X., 2020).

Nonetheless, the findings that have been achieved using the framework means that it has a great potential in enhancing the quality and comparability of the coercivity and remanence measurements in a wide range of laboratories (Gutfleisch, 2011). CI method provides a clear and transparent evaluation of the laboratory operation, and it is priceless in the process of finding systemic errors, environmental sensitivity and operator induced impacts. By these areas being identified, the approach will enable the laboratories to focus on some of the sources of error such as calibration issues, environmental conditions, or manipulation of samples. Such a narrowed approach to the process of improvement will make sure that the laboratories will be capable of optimizing their practice, which will lead to the enhanced consistency and dependability of the tests performed on the rare-earth permanent magnets.

CI approach also helps in quality control and standardization in the industrial environment. It is more useful in the daily quality control of manufacturing settings because it gives only one easily measurable value of consistency where consistency is essential to the quality of manufactured goods. Moreover, CI approach can be used to facilitate supplier assessment, where manufacturers can determine the performance of various suppliers or production batches, using an objective and unified parameter. The approach can be used in metrology to harmonize the practices of measuring in laboratories, increasing the precision of measurements in areas where precision in magnetic measurement is required, including materials science, electronic, and renewable energy applications.

The suggested methodology provides a solid and consistent method of raising the consistency and comparability of coercivity and remanence measurements. Having combined several major metrological concepts, including reference deviation, measurement variability and uncertainty, the Consistency Index (CI) is a compounded instrument, both in the industrial and in the metrological setting. Its capacity to find error sources in the system, sensitivities to the environment, and operator pertinences make it a very essential tool of enhancing the practice of measurements, so that the performance of rare-earth permanent magnets can be uniformly and reliably evaluated. The ability of the methodology to respond to high-temperature and dynamic testing conditions will only enhance its applicability and reliability in practical situation, and will eventually lead to the progress in the application and characterization of the rare-earth materials.

4. Conclusion

A systematic study of the consistency of measurements of coercivity and remanence of the rare-earth permanent magnets was presented in this paper with special attention being made in developing and applying a practical evaluation methodology.A systematic experimental design was adopted, using more than one laboratory and types of instruments, traceable calibration schemes and consolidated measurement procedures of both coercivity and remanence. The use of statistical measures such as deviation, relative error, repeatability, reproducibility and confidence intervals were integrated into a consistent methodology of evaluating consistency.

The agreement between the results obtained in the laboratory and reference values considering the mentioned measurement uncertainties was summarized by introducing a new Consistency Index. The use of this index, combined with the traditional indicators, gave a clear understanding of the work of individual laboratories and types of instruments used, as well as the possibility to identify the systematic errors, environmental sensitivities and operator-influenced factors.

The results indicate that the suggested consistency assessment tool can be successfully used to interpret differences in the measurements of coercivity and remanence as well as justify specific changes in the measurement practice. The Consistency Index provides a quantitative approach to the reliability of measurements of magnetic properties that can be applied both to metrology and to industry, in order to monitor and improve the quality of the measurements and the results in the uses of these materials where consistency and quality are important factors. The direction of future work will be to apply the approach to high-temperature conditions and to dynamic magnetic measurements, including to varying field frequencies or rotating machines, and to investigate its use together with digital data management and automated analysis instruments in the present-day test laboratories.

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