## 6 Conclusions  
Dynamic hysteresis is the primary research area affecting the accuracy of TMR sensors. However, there are varieties of related research, but it is still a research focus.  
Based on the quasi-static probability model, this paper further regards the variation of the same input amplitude with different frequencies as the time distribution of the hysteresis operator. Then, the calculation method of the time distribution is deduced by adding some limiting conditions. Finally, the dynamic hysteresis model is constructed by using the static model and time distribution modeling. However, deviation propogation causes a shift in the results. Therefore, a cured method using the characteristic of equal periodic extremum output was proposed. Two sensors are used to verify the model at 1 kHz, 3 kHz, and 5 KHz, respectively. The results show that the maximum error can be reduced by 46.7%, the variance can be reduced by 77.8%, and the average value can be reduced by 65.1%. This shows that the model can effectively reduce errors caused by hysteresis and proves its effectiveness. The performance improvement of this model may be due to the following reasons. First, this model adopts a high-frequency triangular wave signal, while the Preisach model adopts a measuring step relaxation time constant. Second, the model can calculate more detailed time distributions, which helps reduce errors. In addition, this model mainly regards the sensor as a set of hysteresis operators and then calculates the probability and time distribution of hysteresis operators, which has little to do with the specific sensor type; therefore, it should have better generalization abilities. Interpolation is mainly used in the solution process after the model is established. The algorithm is relatively simple, and the calculation speed is swift. This model is relatively simple, easy to implement, and is irreversible. Still, suppose that the appropriate hysteresis operator is selected. In that case, it may also be used to model the inverse problem directly in order to solve the reversible problem, which will be the next attempt.  
The innovations of this paper are as follows: (1) The dynamic hysteresis characteristics of TMR are studied and applied to dynamic measurements; (2) the time distribution characteristic is introduced into the static probabilistic hysteresis model, and the probabilistic model is applied to the dynamic hysteresis model; (3) compared with the Preisach model, the maximum error of this method can be reduced by 46.7%, the variance can be reduced by 77.8%, and the average value can be reduced by 65.1%.  
Y.L.: Conceptualization, methodology, experiment, and writing--original draft. L.W.: Experiment and writing--review and editing. Z.Q.: Conceptualization, supervision and writing--review and editing. H.Y.: Writing--review and editing. J.A.: Writing--review and editing. Y.P.: Conceptualization and writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: Not applicable.  
**Conflicts of Interest**: The authors declare no conflict of interest.