Voice-Coil Force Factor Bl

The motor strength of a moving-coil loudspeaker is the product of gap flux density and the length of voice-coil wire immersed in it. Drag the excursion slider and watch the coil leave the gap: Bl falls, and with it the driving force.

Parameters

Gap flux density

B

From Plate VT

Voice-coil diameter

D_{vc}

Voice-coil turns

N_{vc}

Coil winding height

h_c

Gap (plate) height

h_g

Cone excursion

x

Signal current

i

Fig. 6Overhung voice coil. The bright copper band is the portion of winding inside the gap at x = 0 mm.

Motor

Bl at rest

Bl_0

3.77 T·m

Bl at x

Bl(x)

3.77 T·m

Relative100 %

Linear stroke

X_{max}

±3 mm

Wire length

\ell

7.18 m

Force at x

F

3.77 N

Winding typeoverhung

☞ Bl stays constant to ±3 mm and falls to 70 % at ±4.88 mm.

Force factor Bl(x) vs. excursion

How it works

Theory · assumptions · formulas

Lorentz force

A conductor of length $\ell$ carrying current $i$ perpendicular to a field $B$ feels a force $F = B\,\ell\,i$ . For a voice coil all turns are radial-field perpendicular, so only the length inside the gap counts:

\ell = N_{vc}\,\pi D_{vc}, \qquad Bl_0 = B\,\ell\,\frac{\min(h_c,\,h_g)}{h_c}

Excursion

When the cone moves by $x$ , the coil occupies $[x - h_c/2,\ x + h_c/2]$ while the field occupies $[-h_g/2,\ h_g/2]$ . The immersed fraction is the overlap of these intervals:

Bl(x) = B\,\ell\,\frac{\max\!\bigl(0,\ \min(x + \tfrac{h_c}{2}, \tfrac{h_g}{2}) - \max(x - \tfrac{h_c}{2}, -\tfrac{h_g}{2})\bigr)}{h_c}

The flat top extends to $X_{max} = |h_c - h_g|/2$ . An overhung coil (longer than the gap) keeps constant Bl because turns leaving one edge are replaced by turns entering the other; it wastes some wire in the fringe field. An underhung coil sits wholly in the gap: very linear, but heavier magnet for the same Bl.

SimplificationReal gaps have fringe fields that extend roughly one gap length beyond each plate face, rounding the corners of the trapezoid and stretching the tails. The ideal model here is the usual starting point for a design.

Field-coil speakers of the 1930s typically used short overhung coils with Bl around 5–10 T·m: modest excursion, but high sensitivity thanks to strong electromagnet fields.