Pole Tip & Bl(x) Throw

Spatial convolution

The instantaneous force factor at cone displacement $x$ is the cross-correlation of the stationary axial flux distribution $B_g(z)$ with the moving winding's turn density $n_\ell$. Each slice of the coil contributes the flux it actually sits in:

The profile $B_g(z)$ is flat inside the gap and rolls off beyond each plate face roughly as $\exp(-(d/w)^{1.75})$, with a fringe length $w \approx 1.65$ mm. That roll-off rounds the corners of Plate VI's ideal trapezoid and stretches the tails, so measured Bl(x) curves are bells, not trapezoids.

Undercut T-pole

On a straight cylindrical pole the inner fringe (towards the back plate) is stronger than the outer fringe, because flux leaks diagonally down the pole shaft. The model widens the inner fringe by 38 % for straight poles, which tilts Bl(x): more motor strength on inward strokes than outward. Turning a relief just below the plate (the undercut T) forces flux to jump radially across the gap instead, equalising both fringes.

An asymmetric Bl(x) expands as $Bl(x) \approx Bl_0 + b_1 x + b_2 x^2$ with a nonzero linear term $b_1$, which rectifies the drive into even-order distortion and a dynamic DC cone offset. Symmetry drives $b_1 \to 0$.

Two Xmax figures

The geometric stroke $X_{max} = |h_c - h_g|/2$ marks where turns begin leaving the flat-field region; the 70 % limit $X_{Bl70}$ marks where Bl has actually fallen to 0.7 of rest value. For overhung coils the fringe stretches $X_{Bl70}$ well past geometric Xmax; for underhung coils they nearly coincide.