## 1 Introduction  
The giant magnetoresistance effect (GMR) was discovered by [PERSON] et al. [1] and [PERSON] et al. [2] in 1988. Because of its excellent characteristics, it is applied in many fields, such as medicine [3], aviation [4], nondestructive testing [5], magnetic storage [6], electric power [7, 8], and automobiles [9], etc. Its discoverers also won the 2007 Nobel Prize. There are four kinds of GMR, namely magnetic tunnel structure, spin valve structure, multilayer structure, and granular alloy structure. The tunnel magnetoresistive sensor (TMR) has attracted widespread attention because of its high sensitivity, suppression of temperature drift, self-compensation, and wide linear range. One of the limitations of TMR sensors in the high-frequency field is their low accuracy, and one of the main reasons for the low accuracy is hysteresis. Hysteresis can be divided into static (quasi-static) hysteresis and dynamic hysteresis. Static hysteresis mainly refers to the hysteresis that occurs when working at an extremely low frequency regardless of time. Dynamic hysteresis refers to the hysteresis at a higher frequency, which is related to frequency or input change rate. Currently, there are mainly two methods to reduce the influence of hysteresis. One is to reduce the hysteresis by designing the sensor's manufacturing process, structure, materials, or hardware circuit. For example, AlOx barrier [10], spin direction and shape [11], self-bias sandwich structure [12], external bias structure [13], and so on. However, most of these methods are complex, expensive, involve high power consumption, and are not easy to apply. Another method is to model the hysteresis characteristics of the sensor and to compensate for the measurement results in order to reduce the influence of hysteresis. Currently, there are mainly two kinds of models. One is the black-box model, which does not explain the input-output relationship but directly fits it. It includes the models based on empirical formulas and those based on machine learning. Empirical formula models mainly include the use of spline functions [14], rational polynomial [15], arctangent function [16], etc. However, the scope of the application of empirical formulas models is relatively small. The machine learning method includes the Recurrent Neural Network Model [17], artificial neural network [18], NARMAX [19], etc. These methods can solve the problem of model mapping to a certain extent, but it is difficult to realize and the cost is high. The other is the white-box model. A series of models based on the Preisach model is the most representative. For example, the JA model [20] and KP model [21]. However, the accuracy of these models is not high, and the measurement is complex, which cannot effectively solve the hysteresis problem. Therefore, studying the hysteresis problem further is necessary to obtain a more efficient and easy-to-implement modeling method. In the previous study of this research group, we thought that the change of the first-order rotation curve of TMR has a strong correlation with the probability distribution of the hysteresis operator. We analyzed the probability distribution characteristics of the hysteresis operator. Then, we regarded the output of the sensor as the ratio of the hysteresis operator satisfying certain conditions of all hysteresis operators. Moreover, we obtained the fast and accurate calculation of the static hysteresis characteristics of the TMR sensor [22]. We consider that dynamic hysteresis in dynamic measurements is related to the transition time of the hysteresis operator. Therefore, we introduced the conversion time distribution characteristics of the hysteresis operator into the static model. Then, we constructed a dynamic hysteresis model based on the probability estimation of the conversion time of the hysteresis operator.