From: dpierce@cartchunk.org   
      
   UnsteadyKen wrote:   
   > One small question. Would it take a similar amount of power to halt the   
   > driver as it approaches the end of travel ready for the return journey?   
   > It seems to me that more would be required as we now have the mass of   
   > the driver plus the velocity imparted by the acceleration?   
      
   We can look at a steady-state analysis of the situation.   
   While that may initially confine us to single-tone sine-wave   
   conditions, Fourier allows us to extend that to stationary   
   signals of finite energy and and arbitrary complexity.   
      
   The reason I mention "finite energy" abnove is that there   
   is this mistaken cocnept of transients as involving instantaneous   
   starts and stops. To go from a dead stop to some finite velocity   
   in an instant has all sorts of real implications that reduce   
   such a scenario to one of absurdity. For instance it requires a   
   system with infinite bandwidth. A direct corralary of that is   
   that involves the availability of infinite energy. All this   
   REGARDLESS of the masses that might or might not be involved.   
      
   The mere fact that all the systems we deal with have limited   
   bandwidths makes these "inconveniences" completely disappear.   
      
   So forget any notion whatsoever of "instantaneous" or similar   
   ideas: they cannot happen in real systems, and expending   
   any energy on concept requiring infinite energy are, well,   
   of infinitessimal efficacy.   
      
   That being said, let's proceed with a brief steady state   
   analysis.   
      
   One concept that it's useful to keep in mind is that above   
   the fundamental mechcanical resonant frequency of the system   
   and if we pick our 500 Hz example, that's well above such a   
   resonance, speaker systems are what are called mass-controlled.   
   What t6hat means is that the forces we exert to move the cone   
   are dominated by those needed to accelerate the combined   
   effective mass of the cone.   
      
   A diversion: below resonance, speaker are stiffness controlled.   
   That means that the forces applied are primarily those required   
   to change the position of the cone, working against the spring   
   provided by the stiuffness of the speaker's suspension   
   components are the compression and rarification of the air inside   
   the box and the like.   
      
   Two direct implications of this are that above resonance, all   
   other things being equal, ABOVE resonance, acceleration is   
   constant with frequency and excursion, being the second integral   
   of acceleration, goes as the inverse square of frequency (double   
   the frequency, excursion is one quarter). BELOW resonance,   
   excursion is constant with frequency, and acceleration,   
   being the second derivative of excursion, goes as the square   
   of frequency (double the frequency, acceleration quadruples).   
      
   Now, in this context, "steady state" is taken to mean single-   
   frequency sine wave excitation.   
      
   We take the input signal as a sinusoidal drive voltage   
   whose value is   
      
       Et = Epeak * sin(2 pi F t)   
      
   where Et is the voltage at time t, Epeak is the peak voltage,   
   and F is the frequency. Assuming a resistive load presented   
   be the driver, the amplifier is able to impart a current   
   through the speaker innphase with the voltage whose value is:   
      
       It = Et / Rs   
      
   where Rs is the effective resistance of the speaker.   
      
   Now, assuming we're talking about a normal electrodynamic   
   direct-radiator loudspeaker (and that assumption is only   
   to make this brief analysis simpler), the combination of   
   the voice coil winding and the magnetic field in the gap   
   results in an electrical-to-mechanical transduction equal to   
   the length of the wire immersed in the magnetic field times   
   the desnity of that field.   
      
   Suchn  a figure is in units of flux density times length,   
   "Tesla meters" which, quite conveniently, has the same units   
   as force per unit current, e.g., Newtons per Amp.   
      
   And what that means is that we can now calculate the force   
   on the driver by a given input curreent.   
      
   For example, take a very typical 8" mid-woofer with a very   
   typical 1.25" voice coil in a moderate-sized magnet, and we   
   typically have a transduction factor of about 9 N/A. Put   
   8 volts DC across our 8 ohm driver, and the voice coil will   
   exert a force of 6 Newtons (since 8 volts across 8 ohms   
   imparts a current of 1 amp).   
      
   By, more significantly, we can go in the opposite direction:   
   if I know what acceleration I need, I can compute how much   
   current is needed to make the cone behave in such a way as   
   to produce the required motion.   
      
   We saw earlier that our hypothetical 9" woofer would   
   have an excursion of about 1/1000 of an inch to produce   
   100 dB SPL at 500 Hz. Assume the cone has an effective   
   moving mass of 25 grams, how much peak force is needed   
   to make the cone move that way?   
      
   Well, we also figured an eccleration of about 325 m/s^2.   
   since f = mA, and m = .025 kg and a = 325 m/s^2, the force   
   ends up being:   
      
       f = 325 m/s^2 * 0.025 kH   
      
   or about 8 Newtons.   
      
   And in our hypothetical driver would require all of 1 amp   
   peak to accelrate it and decelerate it to move properly to   
   produce 100 dB SPL at 500 Hz.   
      
   And, it would require the same current to produce the same   
   sound pressure level at pretty much all frequencies above   
   resonance.   
      
   > Or have I misunderstood how amplifiers control the drivers?   
   > Which is very likely so.   
      
   You wouldn't be the first. And you'd be in the same league   
   as many loudspeaker "designers."   
      
   --   
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