The derivation of the magnitude of hysteresis losses involves analyzing the energy dissipated during a complete cycle of magnetization and demagnetization of a ferromagnetic material.
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To simplify the derivation, we will consider a one-dimensional case where the material is magnetized along a single axis.
Hysteresis Losses Derivation
Let’s start by considering a small volume element of the ferromagnetic material, dV, with its magnetization, M, and the corresponding magnetic field intensity, H. The work done, dW, in magnetizing this small volume element is given by:
dW = H * dM
According to the magnetic hysteresis behavior, the relationship between M and H is nonlinear and can be represented by a hysteresis loop. To simplify the derivation, we will assume that the loop is symmetric with respect to the origin.
Now, the magnetization, M, is related to the magnetic field, B, through the magnetic permeability, μ, of the material:
B = μ * H
Taking the derivative of both sides with respect to H, we get:
dB = μ * dH
Since the loop is symmetric, the area enclosed by the loop is proportional to the energy dissipated during one complete cycle. Let’s denote this area as A.
Now, the work done in magnetizing the small volume element is equal to the change in magnetic energy, dU, within that element:
dU = B * dB * dV
Substituting the expressions for B and dB, we have:
dU = μ * H * μ * dH * dV = μ^2 * H * dH * dV
Integrating both sides over the volume of the material, V, we obtain the total magnetic energy, U, dissipated during one complete cycle:
U = ∫(μ^2 * H * dH * dV)
Now, using the fact that B = μ * H, we can express the magnetic energy in terms of B:
U = ∫(B * dB * dV)
The hysteresis losses, P_h, per unit volume can be defined as the energy dissipated per unit volume:
P_h = U / V
Substituting the expression for U, we have:
P_h = (1 / V) * ∫(B * dB * dV)
Since the integral represents the area enclosed by the hysteresis loop, denoted as A, we can rewrite the expression for hysteresis losses as:
P_h = A / V
Therefore, the magnitude of hysteresis losses per unit volume is given by the area enclosed by the hysteresis loop divided by the volume of the material.
It’s important to note that the derivation presented here provides a conceptual understanding of the hysteresis losses magnitude. In practice, more detailed calculations may involve additional considerations, such as the specific shape of the hysteresis loop and the material’s hysteresis characteristics.
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