**6. Conclusions**

The ability to carry out rapid validation of the angular distribution of the magnetic properties of electrical steel is an important practical issue. Therefore, the purpose of this article was to investigate the possibility of using the Barkhausen effect and its time-frequency representation as a non-destructive tool to develop a procedure that allows retrieval of information about the resultant magnetic anisotropy in a classic grain-oriented electrical steel sheet. By assumption, this method is to be an alternative to the classic MBN approach, and ought to enable broad assessment of changes in the dynamics of the MBN phenomenon synchronously considering both time and frequency characteristics. For this purpose, tests were performed for a sample of 3% SiFe grain-oriented electrical steel. The measurements were made for 16 angles of transducer orientation, equally spaced within the range corresponding to full rotation. The paper presents the procedure of signal transformation into the *TF* domain, presentation of *TF* spectrograms and quantification of information contained therein. The obtained *TF* parameters expressing the change of the spectrogram allow the angle corresponding to the easy and hard magnetization axis to be determined. The possibility of extensive analysis of properties and observation of changes in the MBN activity makes this method possible to detect even small variations in the dynamics of the phenomenon over time resulting from the changes in magnetic properties. This creates a chance to obtain complementary information about the properties of the material, which can allow the formation of more complete knowledge.

The proposed method, based on the analysis of time-frequency characteristics in the full period of the MBN signal, allows detailed observation of the relationship between MBN properties expressed in time and in frequency for subsequent time moments. In consequence, it enabled the three characteristic MBN activity areas to be noticed. The occurrence of these areas in the MBN period has already been widely analyzed in many other works, and the angular characteristics of *TF* parameters obtained in this work are consistent with the other results reported. According to observations, these areas were associated with nucleation of reverse domains and, furthermore, the movement of the 180◦ and 90◦ DWs. The course of the first two was associated respectively with the magnetocrystalline anisotropy MCA and roll magnetic anisotropy RMA. Considering this, the angular distributions obtained by the authors underline the key importance of RMA for the alignment of resultant easy and hard magnetization axes in the tested steel. The angular characteristics of Barkhausen noise energy and numbers of events can be treated as confirmation of the effectiveness of the proposed *TF* method. These parameters, being used many times by other authors in the classic analysis, show convergent results with the *TF* ones. However, the results of both methods show some asymmetries and disturbances in the characteristics. The relatively small number of utilized measurement steps may have a decisive impact on the smoothness of the transitions of the characteristics obtained between successive measurement angles. The step applied in the paper makes it possible to determine general trends of the characteristic. On the other hand, errors for individual angles have a significant impact on the local course of the characteristics. Nevertheless, due to the sequencing of calculations procedures and the use of a sliding window over the analyzed signal, the STFT transformation leads to generalization of the phenomenon characteristics. In results, it can affect the increase of the robustness of the method for interfering external factors. This could be an explanation for obtaining slightly smoother courses of approximated *TF* characteristics.

In addition, the paper presents a detailed analysis of measurement errors and assesses the repeatability of the method in the context of the dispersion of *TF* parameters values within individual subsets of measurement signals. The choice of calculation parameters of the transformation procedure enables optimization of resolution in time and frequency, in reference to the nature of the observed changes. Based on the analysis conducted, it is crucial to emphasize the importance of the concentration measure parameter (carrying convergent information to MBN energy), which, regardless of the calculation parameters used, was characterized by high repeatability. The study also showed that the frequency value of the high-pass filter used for signal conditioning before *TF* analysis should not be lower than the value of the first step of the applied *TF* computational grid, which in turn allows for a significant increase in *TF* parameter robustness.

The presented method of time-frequency analysis is largely derived from the development of commonly applied methods of time or frequency analysis and is intended to present a possible path of development. The parameters defining the time-frequency characteristics are for the most part an extension of the classically used statistical coefficients. In principle, classical methods of analysis, in a single time or frequency domain, can be expressed as a generalization of the trends of changes presented in the time-frequency characteristics. Of course, at the same time it is not said that obtaining more detailed information will significantly improve the effectiveness of the MBN method in the future. However, the possibility of sequential data analysis itself (due to the potential to supplement the knowledge about the tested material) affects the validity of *TF* analysis usage and should be further examined. In the context of the presented arguments and already published papers, it can be seen that the use of time-frequency methods certainly does not affect the loss of information. On the contrary it can provide significant arguments to the discussion, as it gives a much wider perspective and brings a broader analysis of complex relationships of time and frequency characteristics.

In addition to the content of the work, one more aspect should be noted related to the development of future measurement systems. The classic method of analysis, despite many years of use, is still not standardized, and often the calculation procedures differ between researchers. The problematic element is, among others, the way of determining the MBN signal's background level or procedure of counting the number of events. Therefore, the time-frequency calculation procedures are not showing greater complexity in terms of the practical implementation of the MBN method, especially currently when the advancement and miniaturization of computing units is common. It should be also considered that today's diagnostic systems are increasingly based on monitoring many parameters and formulating multi-variate rules and correlation dependencies. Under these circumstances the results presented here show good potential. However the investigation with a large number of angular steps for steels having various textures and under influence of various anisotropy factors should be reaped. The issue, which also requires further research, is the synthesis method of the knowledge contained in *TF* representations, enabling greater efficiency to generalize and detailed analysis of information. The authors will present work on multiple samples and on multi-threaded analysis and synthesis procedures in the future.

**Author Contributions:** Conceptualization, G.P. and M.M.; methodology, G.P. and M.M.; software, G.P. and M.M.; validation, G.P. and M.M., formal analysis, G.P. and M.M.; investigation, G.P. and M.M.; resources, G.P.; data curation, M.M.; writing—original draft preparation, M.M.; writing—review and editing, G.P. and M.M.; visualization, G.P. and M.M.; supervision, G.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** The APC was funded by Research Funds of Faculty of Electrical Engineering of West Pomeranian University of Technology, Szczecin, Poland.

**Conflicts of Interest:** The authors declare no conflict of interest.
