With length and flux density configurations, how to calculate the magnetic field strength?

To calculate the magnetic field strength (Magnetic Field Strength, $\mathrm{<br>}$H) based on length and magnetic flux density (Magnetic Flux Density, $\mathrm{<br>}$B), it's essential to understand the relationship between these two quantities. The magnetic field strength $\mathrm{<br>}$H and magnetic flux density $\mathrm{<br>}$B are typically related through the magnetization curve (B-H curve) or permeability ( $\mathrm{<br>}$μ).

1. Basic Formula

• The relationship between magnetic field strength $\mathrm{<br>}$H and magnetic flux density $\mathrm{<br>}$B can be expressed by the following formula:

• Where:

• B is the magnetic flux density, measured in teslas (T).

• $\mathrm{<br>}$H is the magnetic field strength, measured in amperes per meter (A/m).

• $\mathrm{<br>}$μ is the permeability, measured in henries per meter (H/m).

• Permeability $\mathrm{<br>}$μ can be further broken down into the product of the permeability of free space ${\mathrm{<br>}}_{}$μ0 and the relative permeability ${\mathrm{<br>}}_{}$μr:

• Where:

• μ0 is the permeability of free space, approximately  $\mathrm{<br>}$4π×10−7H/m.

• μr is the relative permeability of the material, which is approximately 1 for non-magnetic materials (like air, copper, aluminum) and can be very high (in the hundreds to thousands) for ferromagnetic materials (like iron, nickel).

2. Calculating Magnetic Field Strength  $\mathrm{<br>}$H Given  $\mathrm{<br>}$B and  $\mathrm{<br>}$μ

If you know the magnetic flux density $\mathrm{<br>}$B and the permeability $\mathrm{<br>}$μ, you can directly use the above formula to calculate the magnetic field strength $\mathrm{<br>}$H:

For example, suppose you have an iron-core transformer with a magnetic flux density B=1.5T and a relative permeability μr=1000. Then:

3. Considering Nonlinear Magnetization Curves

For ferromagnetic materials, permeability $\mathrm{<br>}$μ is not constant but varies with the magnetic field strength H. In practice, especially at high field strengths, permeability may decrease significantly, leading to slower growth in magnetic flux density $\mathrm{<br>}$B. This nonlinear relationship is described by the material's B-H curve.

• B-H Curve: The B-H curve shows how magnetic flux density $\mathrm{<br>}$B changes with magnetic field strength $\mathrm{<br>}$H. For ferromagnetic materials, the B-H curve is typically nonlinear, especially as it approaches the saturation point. If you have the B-H curve for your material, you can determine the magnetic field strength $\mathrm{<br>}$H by finding the corresponding $\mathrm{<br>}$H value for a given $\mathrm{<br>}$B.

• Using the B-H Curve:

1. Locate the given magnetic flux density  $\mathrm{<br>}$B on the B-H curve.

2. Read the corresponding magnetic field strength H from the curve.

4. Considering Magnetic Circuit Length

If you also need to consider the geometry of the magnetic circuit (such as the length $\mathrm{<br>}$l of the core), you can use the magnetic circuit law (analogous to Ohm's law in electrical circuits) to calculate the magnetic field strength. The magnetic circuit law can be expressed as:

Where:

• $\mathrm{<br>}$F is the magnetomotive force (MMF), measured in ampere-turns (A-turns).

• $\mathrm{<br>}$H is the magnetic field strength, measured in A/m.

• $\mathrm{<br>}$l is the average length of the magnetic circuit, measured in meters (m).

The magnetomotive force $\mathrm{<br>}$F is typically determined by the current $\mathrm{<br>}$I and the number of turns $\mathrm{<br>}$N in the coil:

Combining these two equations, you get:

This formula is useful when you know the magnetic circuit length $\mathrm{<br>}$l and the parameters of the coil (number of turns N and current $\mathrm{<br>}$I).

5. Summary of Steps

1. Determine Magnetic Flux Density  $\mathrm{<br>}$B: Use the given magnetic flux density  $\mathrm{<br>}$B.

2. Select the Appropriate Permeability  $\mathrm{<br>}$μ: For linear materials (like air or non-magnetic materials), use the permeability of free space  ${\mathrm{<br>}}_{}$μ0. For ferromagnetic materials, consider the relative permeability μr, or use the B-H curve.

3. Calculate Magnetic Field Strength H: Use the formula H=μB or read the corresponding  $\mathrm{<br>}$H value from the B-H curve.

4. Consider Magnetic Circuit Length (if applicable): If you need to account for the geometry of the magnetic circuit, use the magnetic circuit law H=lN⋅I for further analysis.

Conclusion

To calculate the magnetic field strength given length and magnetic flux density, first determine the permeability $\mathrm{<br>}$μ, then use the formula $\mathrm{<br>}$H=μB. For ferromagnetic materials, it's advisable to use the B-H curve to handle the nonlinear relationship. If you need to consider the geometry of the magnetic circuit, use the magnetic circuit law $\mathrm{<br>}$H=lF for further analysis.