Notes on Incremental AC Problems
For problems like the analysis of loudspeakers, it is useful to solve for an AC magnetic field perturbation about a magnetostatic solution. For loudspeakers, the magnetostatic solution is the field established by the loudspeaker's permanent magnet. The AC field solution is a perturbation about the field established by the permanent magnet. However, for loudspeakers, device's iron is typically highly saturated in spots. We desire an incremental solution about that highly saturated condition, but what is the permeability to be used?
Denote the nonlinear relationship between field intensity and flux density (i.e. the B-H curve) as:
\[b = F(h)\]
where \(h\) is a scalar value representing magnetic field intensity and \(b\) is a scalar value representing flux density. Both \(b\) and \(h\) are pointing in the same direction. To generalize to a 2D situation, we could decompose \(h\) into and x- directed and a y-directed component. We could further assume that the DC solution is completely directed along the x-axis such that:
\[ h = \sqrt{(h_o + h_x)^2 + h_y^2} \]
where \(h_o\) is the DC component and \(h_x\) and \(h_y\) are x- and y-directed perturbations in field intensity, respectively. Writing out the two components of flux density in terms of H gives:
\[ \left\{\begin{matrix} b_x \\ b_y \end{matrix} \right\} = \left\{ \begin{matrix} h_o + h_x \\ h_y \end{matrix} \right\} \frac{f(h)}{h} \]
where the h's that occur outside of \(f(h)\) merely serve to point the flux density in the same direction as the field intensity.
One can then linearize the RHS about \(h_x=0\), \(h_y=0\) to obtain:
\[ \left\{\begin{matrix} b_x \\ b_y \end{matrix} \right\} \approx \left[ \begin{matrix} \frac{df(h_o)}{dh} & 0 \\ 0 & \frac{f(h_o)}{h_o} \end{matrix} \right] \left\{ \begin{matrix} h_x \\ h_y \end{matrix} \right\} + \left\{ \begin{matrix} f(h_o) \\ 0 \end{matrix} \right\} \]
This equation can be interpreted as the DC field, \(h_o\), plus a perturbation \(h\) multiplied by an incremental permeability matrix. The interesting points are that even though the material is an isotropic one, the incremental permeability matrix is anisotropic. In the direction parallel to the DC field, the permeability is the incremental permeability of the the B-H curve (i.e. the derivative of the B-H curve at the steady-state operating point). However, in the direction normal to the DC field, the permeability is, instead, the apparent permeability: b/h. For linear materials, the incremental and apparent permeability are the same. However, for a highly saturated material, the apparent permeability can be much higher than the incremental permeability.