Pot-Magnet Gap Flux Density

In an electrodynamic loudspeaker the field coil sits in a soft-iron pot; its flux crosses a narrow annular gap where the voice coil moves. This plate solves the magnetic circuit, including leakage flux and the gradual saturation of the centre pole.

Parameters

Field ampere-turns

NI

A·t

Radial gap length

g

Pole diameter

D_p

Top-plate thickness

h_g

Iron path length

\ell_i

Iron relative permeability

\mu_r

Leakage factor

\sigma

Total pole flux ÷ useful gap flux

Iron saturation

B_{sat}

Fig. 5Section through a field-coil pot. Dashed blue lines trace the flux path; the gap glows hotter as B rises. Gap width is exaggerated for clarity.

Magnetic circuit

Gap flux density

B_g

0.774 T

Gap flux

\Phi_g

378 µWb

Pole flux density

B_p

1.31 T

Ideal (no iron)

\mu_0 NI/g

1.4 T

Gap area

A_g

488.2 mm²

Gap reluctance

\mathcal{R}_g

1.47 MA/Wb

Gap energy105 mJ

Gap MMF share55.4 %

☞ 55.4 % of the MMF is spent driving the gap; the rest is lost in the iron.

Gap flux density vs. ampere-turns (magnetisation curve)

with saturationlinear iron

How it works

Theory · assumptions · formulas

The magnetic circuit

Treat the pot as a series circuit of two reluctances driven by the coil's magnetomotive force $\mathcal{F} = NI$ . The gap is an annulus of radial length $g$ , mean diameter $D_p + g$ and height equal to the top plate thickness $h_g$ :

A_g = \pi (D_p + g)\,h_g, \qquad \mathcal{R}_g = \frac{g}{\mu_0 A_g}

Not all flux leaving the pole crosses the gap; some leaks from the pole sides straight to the top plate and shell. The leakage factor $\sigma$ (typically 1.5–2.5 for pot magnets) is the ratio of pole flux to gap flux, so the iron carries $\sigma\Phi_g$ . Ampère's circuital law around the loop gives

NI = \Phi_g\,\mathcal{R}_g + H_i(B_p)\,\ell_i, \qquad B_p = \frac{\sigma \Phi_g}{\pi D_p^2/4}

Iron saturation

The iron's B–H curve is modelled with the Fröhlich relation, which starts with slope $\mu_0\mu_r$ and bends over towards $B_{sat}$ :

B = \frac{H}{a + b\,H} \;\Longleftrightarrow\; H = \frac{aB}{1 - bB}, \qquad a = \frac{1}{\mu_0\mu_r},\; b = \frac{1}{B_{sat}}

The equation for $NI(\Phi_g)$ is monotonic, so the calculator inverts it by bisection. The magnetisation curve shows the result: linear at first (gap-dominated), then flattening as the pole saturates and the gap flux approaches its ceiling $B_{sat}A_p/(\sigma A_g)$ .

Design hints

Halving the gap nearly doubles $B_g$ for the same NI, but the voice coil must still fit with clearance.
A thicker top plate spreads the same flux over more height: lower $B_g$ but a longer linear region (Plate VI).
Classic field-coil speakers reached 0.8–1.3 T with 1000–3000 A·t; beyond that the pole saturates and extra field power is wasted.