The Preisach model is a mathematical framework used to describe hysteresis, particularly in magnetic materials, ferroelectrics, and other systems with memory effects. It is widely used in physics, engineering, and material science.
Key Concepts of the Preisach Model
- Hysteresis Behavior: The model captures the phenomenon where a system's response (e.g., magnetization) depends on its past states, not just the current input (e.g., applied magnetic field).
- Relay Operators: The model represents hysteresis as a collection of elementary bistable units called hysterons or relays, each of which switches at two threshold values (up-switching and downswitching).
- Preisach Plane: The relays are distributed over a Preisach plane, where each relay has a specific switching threshold pair $(\alpha, \beta)$ with $\alpha>\beta$.
- Superposition Principle: The overall system response is obtained by integrating the contributions of all relays weighted by a Preisach density function.
Mathematical Representation
$$
R_{\alpha, \beta}(H)= \begin{cases}+1, & H>\alpha \text { (switches on) } \\ -1, & H<\beta \text { (switches off) }\end{cases}
$$
where $H$ is the applied input (e.g., magnetic field).
- The total output $M(H)$ (magnetization) is given by:
$$
M(H)=\int_{\alpha>\beta} \Gamma(\alpha, \beta) R_{\alpha, \beta}(H) d \alpha d \beta
$$
where $\Gamma(\alpha, \beta)$ is the Preisach weight function.
Applications
- Magnetics: Describing ferromagnetic hysteresis loops.
- Ferroelectrics: Modeling polarization behavior in materials.
- Mechanical Systems: Predicting stress-strain responses in elastoplastic materials.
- Control Systems: Compensation of hysteresis in actuators.
Here’s a Python implementation of the Preisach model for hysteresis simulation. This example approximates the Preisach model using a grid of relay elements and visualizes the hysteresis loop.
https://gist.github.com/viadean/cfa4d75433057f2589247c9e62980538
Explanation of the Code: