Driver Nonlinearity Lab

1 — Bl(x): force factor droop

Ideal motor gives constant $Bl$. Real gap has finite height. As coil moves out, fewer turns sit in $B$-field. Force per amp drops. If coil is centered, droop is symmetric; if offset (common after sag or assembly), one side drops earlier — even-order distortion.

$$F_{motor} = Bl(x) \cdot i$$

$$Bl(x) = Bl_0 (1 - \beta |x / X_{max}|)$$

$\beta$ is asymmetry/shape knob. In Klippel measurements you see a bell curve, not this triangle — peak $Bl_0$ at rest, down to 70% at $X_{max}$ for a good long-throw, 50% for a cheap one. Twice excursion → twice loss → current has to rise → more heat → more compression. That coupling is outside this model, but you feel it in the motor SVG when status flips to "Coil out of gap".

2 — Kms(x): spider gets angry

Suspension compliance is progressive. Small $x$, soft. Large $x$, spider fibers align, surround folds stretch — stiffness climbs quadratically. This is symmetric by nature, so it makes odd harmonics.

$$K_{ms}(x) = K_{ms0}(1 + \gamma (x/X_{max})^2)$$

$$K_{ms0} = M_{ms}(2\pi f_s)^2$$

Fixed $f_s=40$ Hz here gives $K_{ms0}$. $\gamma$ multiplies the square term. $\gamma=0.2$ is a soft progressive spider, $\gamma=1$ is a harsh bottoming surround. Because force is $K_{ms}(x) \cdot x$, you get an $x^3$ term — same math that makes a Duffing oscillator generate 3rd harmonic. Push Vrms and watch waveform flatten on peaks.

3 — Le(x): inductance modulates

Voice-coil inductance depends on how much steel it overlaps. Move out, $L_e$ drops ~$\delta$. That sounds harmless until you realize impedance changes with position, so current waveform distorts even with perfect voltage drive. Add flux modulation (current itself modulates $B$) and you get $Bl(i)$ — not modeled here but measured by Klippel $L_{e}(x,i)$.

$$Le(x) = Le_0 (1 + \delta |x/X_{max}|)$$

$$i = \frac{V_{in} - Bl(x) v}{R_e} \quad\text{DC approx, full: + Le·di/dt}$$

The back-EMF $Bl(x)v$ is velocity times force factor — it fights input voltage. When $Bl$ droops at peak $x$, back-EMF collapses, current spikes. That spike hits non-flat $K_{ms}$ and you have intermodulation. In the canvas lower half, watch $Bl(x)$ bell shrink with $\beta$ and dot trace current position.

4 — Where distortion comes from: Taylor tells all

Any memoryless nonlinearity can be expanded:

$$y = a_1 x + a_2 x^2 + a_3 x^3 + ...$$

Drive $x = \hat{X}\sin\omega t$:

$$x^2 = \frac{\hat{X}^2}{2}(1 - \cos 2\omega t) \rightarrow \text{2nd harmonic + DC shift}$$

$$x^3 = \frac{\hat{X}^3}{4}(3\sin\omega t - \sin 3\omega t) \rightarrow \text{3rd harmonic + fundamental compression}$$

So asymmetric $Bl(x)$ → even order (2nd), symmetric $K_{ms}(x)$ → odd order (3rd). Real driver does both. THD here computed from displacement spectrum via numerical harmonic extraction:

$$THD = \frac{\sqrt{V_2^2+V_3^2+V_4^2}}{V_1}$$

Klippel's measurement rig does this for real with laser and current sense, plotting $Bl(x)$, $K_{ms}(x)$, $Le(x)$ versus $x$. Golden rule: keep $Bl$ above 80% and $K_{ms}$ within 50% over intended $X$. Once outside, no motor feedback fixes mechanical bottoming.

5 — Playbook & honesty

How to use

• Set $V_{rms}=12$ V, $f=60$ Hz, $\beta=0.1$, $\gamma=0.2$ — this is a decent 6" woofer. Waveform looks sinusoidal, THD 2–5%, $Bl(x)$ dot stays near top.
• Now push Vrms to 30 V or drop $X_{max}$ to 2 mm. Excursion exceeds gap, coil dot leaves bell, displacement waveform flattens, 3rd harmonic bar pops up in spectrum.
• Isolate mechanisms: $\beta=0$, $\gamma=0.8$ → pure suspension odd distortion. $\beta=0.5$, $\gamma=0$ → even. Combine — you get real driver stew.
• Lower $M_{ms}$ raises acceleration $a=F/M$, raises $f_s$ via $K_{ms0}=M(2\pi f_s)^2$ — lighter cone distorts more at high freq? Check.

What this omits

Euler at 8 kHz timestep, no flux modulation $Bl(i)$, no eddy-current hysteresis in pole, no thermal memory, no cone breakup, no port nonlinearity, no equivalent circuit of $Q_{ms}$, $Q_{es}$. $Bl$, $K_{ms}$, $Le$ shapes are simplified polynomials — real curves have asymmetry, plateau, and hysteresis loops. THD is displacement THD via 1024-sample history, not acoustic SPL THD. Use for qualitative cause → effect, not for validating a driver design. For that, need Klippel LSI or laser vibrometer.