Graph plot of sinusoidal current/voltage vs displacement of speaker membrane from its resting position

For a very long time I believed that the speaker membrane just moves according to the amplitude of a electrical signal fed to it. Like, If at any instant, the current/voltage is at c% (0 <= c <= 100) of its amplitude (in positive direction), I thought speaker membrane would also be at c% displacement of its maximum displacement in the corresponding direction, which depends on how we wire it (and I believed that speaker's displacement amplitude would be constant throughout its working if the electrical signal is constant, and this displacement amplitude would be proportional to electrical signal amplitude by a constant ratio).

But after learning the speaker's inductive nature, realising that the membrane would have momentum and air would offer resistance as well, I'm thinking that there wouldn't be a great correlation between these two graphs. I'm I right?

Usually, speakers work in the frequency region above their mechanical resonance (except "woofers" in the lowest frequency part of their range, see below).

Effect A: So usually, the mass and not the spring force is dominant in the membrane movement. Therefore it is not the displacement that is proportional to the electrical signal, as you assumed, but the acceleration is proportional to the electrical signal. The displacement in that case is proportional to the electrical signal twice integrated! This also means that the displacement becomes quadratically smaller with increasing frequency, given the same electrical signal amplitude.

Effect B: So one might ask: if that is true, why don't all those speakers show a 1/f^2 downslope in their frequency response? That is because of the wave equation for sound waves. If a source of sound is small compared to the wavelength, then it acts as a "bad transmitting antenna", the same as is the case for electrical antennas. In both cases this gives a quadratic deterioration with longer wavelengths, or equivalently a quadratic increase in antenna factor with increasing frequency. This completely cancels the quadratic drop of effect A! Only for tweeters in their highest frequency range this is different, they are no longer small compared to the wavelength.

So two special cases are left:

1. Woofers somewhere below 100 Hz have their resonance frequency where the mass no longer dominates over the spring force. At the resonance the two would cancel so the displacement will become very high, that's why damping is needed. Below the resonance the spring force dominates and displacement becomes proportional to electric signal so we lose effect A, which is bad because effect B still exists (unless you are in a small room where the chamber resonances play a role, but let's assume here free space). So usually below the woofer resonance there's a quadratic frequency response drop giving a low-frequency limitation.
2. Tweeters at high frequency where their membranes are not small compared to the wavelength. Therefore effect B does not apply any more, which is bad because effect A still applies. So at that point we will have a drop in frequency response if frequency further increases. This can be somewhat mitigated because at high frequency the sound will also get more focused in the forward direction. But that's usually undesirable (and for instance dome tweeters are used to actually prevent it) so this high-frequency limitation also remains.

So the main part of the audio spectrum, say the whole 100Hz to 10kHz range, is simply described by effects A and B!

I just want to add a couple of points to Jos' excellent answer.

Fundamentally, it is the current through the voice coil that determines the force that it produces. The current has a complicated relationship to the voltage for all the reasons you mentioned (and more).

Also, it isn't the position of the membrane that matters in the end, but rather the amplitude of the pressure wave that it produces in the air, which is more closely related to force than position.

• I think we should also add that speakers usually have a bad efficiency. Most of the current is converted to heat in the resistance of the coil, instead of delivering mechanical power to the membrane. And also, most of the force that is generated is used to accelerate the mass (or to overcome the spring force if in the very low frequency region) and not to apply force on the air. For that reason the analysis can, somewhat surprisingly, ignore back-reaction forces that the air certainly provides. This only fails at the resonance frequency, where we suddenly do get a good efficiency! Apr 25 at 19:55