Question

What is the formula for hysteresis effect?

Answer 1

The hysteresis effect refers to the lag between the input and output of a system, especially in materials where the response depends on the history of the input. In the context of magnetic materials, hysteresis is often described by the B-H curve, where \( B \) is the magnetic flux density and \( H \) is the magnetic field strength.

For magnetic materials, the hysteresis loop can be characterized by several key parameters:

1. **Coercivity (Hc)**: The value of \( H \) at which the material's magnetization \( B \) is zero after the material has been saturated in one direction. It measures the resistance of the material to becoming demagnetized.

2. **Remanence (Br)**: The value of \( B \) when \( H \) is zero after the material has been magnetized to saturation. It is the residual magnetization.

3. **Saturation Magnetization (Bs)**: The maximum value of \( B \) that the material can achieve when the magnetic field is sufficiently strong.

The hysteresis effect in magnetic materials can be expressed through the following simplified form of the B-H relationship:

\[ B = \mu_0 H + M \]

where:

- \( \mu_0 \) is the permeability of free space,
- \( H \) is the magnetic field strength,
- \( M \) is the magnetization of the material.

The hysteresis loss per cycle (energy lost per unit volume) is given by the area enclosed by the B-H loop. This can be calculated as:

\[ \text{Hysteresis Loss} = \int_{H_{\text{max}}}^{-H_{\text{max}}} B \, dH \]

where \( H_{\text{max}} \) is the maximum magnetic field strength applied during the hysteresis cycle.

In summary, the hysteresis effect is characterized by the B-H loop and can be quantified by parameters such as coercivity, remanence, and the area of the loop which represents the energy loss due to hysteresis.

Answer 2

The hysteresis effect, which is often seen in physical systems such as magnetic materials, mechanical systems, and electrical systems, does not have a single, universal formula. Instead, it can be described using various models depending on the context. Here are some common examples:

### 1. **Magnetic Hysteresis**

For magnetic materials, hysteresis is often described using the **Jiles-Atherton model**, which is a more complex and accurate representation. However, a simpler and more traditional way to describe magnetic hysteresis is using the **Preisach model** or by simply plotting the B-H loop (magnetic flux density vs. magnetic field strength) and observing the lag between the two curves.

### 2. **Mechanical Hysteresis**

In mechanical systems, such as materials experiencing cyclic loading, hysteresis can be described by **Bouc-Wen models** or **Kelvin-Voigt models**. A simplified representation can be:

\[ \sigma = E \cdot \epsilon \text{ for loading } \]
\[ \sigma = E \cdot \epsilon + H(\epsilon) \text{ for unloading,} \]

where \( \sigma \) is stress, \( \epsilon \) is strain, \( E \) is the elastic modulus, and \( H(\epsilon) \) represents the hysteresis loop shape, which can vary depending on the material and loading conditions.

### 3. **Electrical Hysteresis**

For electronic components such as capacitors and inductors with hysteresis, a common representation is the **relay model** or **Schmitt Trigger** behavior, which is used in digital circuits to handle noisy signals. The hysteresis effect can be visualized in a voltage versus current graph or in switching behaviors.

### 4. **General Hysteresis Modeling**

For a more general representation of hysteresis, especially in systems with rate-dependent behavior, you might use a **rate-independent Preisach model**. In this model, hysteresis is modeled as a distribution of elementary hysteresis operators (often modeled as simple relays or switches) that collectively produce the observed hysteresis effect.

### Practical Example: Magnetic Hysteresis Loop

In a magnetic hysteresis loop, you might plot the magnetic flux density (B) against the magnetic field strength (H). The shape of the loop indicates the material’s response to changes in the magnetic field, including residual magnetization and coercivity. The area inside the loop represents the energy lost due to hysteresis.

### Conclusion

In summary, the exact formula or model for hysteresis depends on the specific system and type of hysteresis you are dealing with. For detailed modeling and analysis, you'd typically need to refer to the specific models relevant to the field of study, such as the Jiles-Atherton model for magnetic hysteresis or the Bouc-Wen model for mechanical systems.