How strong can a magnetic field be before you get hurt or die?

[SME]

EMF = -(BL * v) with v being the velocity of the moving coil.

 

Ohm's Law applies to the sum of that and the voltage from the amp, that is if you ignore inductance.  This can be solved simultaneously with a mechanical momentum balance (Newton's Laws, really) to determine the current and the cone motion.  For low frequencies where radiation is omni-directional, output is proportional to cone acceleration times surface area.

 

Of course, the above approach neglects inductance, which is usually substantial for middle and high frequencies and is unfortunately very hard to model accurately.  Inductance may be an even more critical parameter impacting the "control" an amp has over the driver.  I haven't dug in enough to find convincing analytical or empirical evidence of substantial, audible distortion from inductance at low-to-moderate playback levels, but I have a hunch that inductance behavior may have a big impact on the perceived "transient response" of a driver.

[mwmkravchenko]

Your equation is correct for a one way travel.  Keep in mind that there is an AC voltage as expressed in the signal sent to the driver and the resulting back EMF.  So the voltage swings positive and negative on both sides sending and receiving.    

 

I tried looking for a comprehensive calculator that will allow an input of the drivers Thiel/Small parameters and some voice coil information to calculate a coil velocity and facilitate an EMF computation.  This is not necessary to describe what is going on.  And I cannot find something that gives me a comprehensive answer.  It's a pretty big request when I think of it.  We are working on  a very comprehensive driver design suite and I don't think we have ever discussed these calculations.  What is awesome is stumbling upon this website:

 

http://www.arcavia.com/kyle/Equations/index.html

 

I'd have to look through quite a few text books and papers to get all these equations.

 

 

Looks like setting up a proper calculation spreadsheet is in order.

 

Inductance is one of the causes of the momentary storage of the the back EMF.  It is a time difference described in terms of phase angle.  It cannot be ignored when you are trying to calculate this properly.  There are a number of papers that go in depth into what inductance does to a drivers frequency response.  Some of the most interesting ones are the papers that combine inductance compensation with the effects of inductance.

  As another example inductance compensation rings do not remove inductance.  They moderate the differences seen in the broad frequency and impedance range that a driver functions in.   

 

Nor can a simple model of the closed box properly describe what actually happens in side a closed box.  Air is a non linear medium.  It's it's attributes change under pressure and under temperature variations.  The ideal gas model is not a good way to calculate what goes on inside a sealed loudspeaker enclosure.  The term air spring is correct.  But it is not a linear spring.  It is a nonlinear spring.  And a drivers suspension is just as important in the free air as it is inside a sealed or vented enclosure.  It is not dominated by the enclosure in any way shape or form.  The suspension serves two purposes in it's most basic application.  Sealing the air pressure difference from the front of the cone from the rear of the cone.  And centering the voice coil in the magnetic gap.

[SME]

Of course the equation works for two-way travel.  If v is positive, then the EMF is negative.  If v is negative, then the EMF is positive.

 

You can easily calculate v for sine waves if you know the excursion x:

 

    x(t) = x_peak * exp(i*omega*t)

    v(t) = dx / dt = i *omega * x_peak * exp(i*omega*t)

 

=>  v(t) = i * omega * x(t) = i * 2 * pi * f * x(t)

 

The exp(i*omega*t) is a sine wave represented in a more mathematically convenient form.  The variable omega = 2*pi*f is, where f is the frequency. 

 

The i is the imaginary number.  Using complex math (with both "real" and "imaginary" quantities) is a clever trick that allows both magnitude and phase to be accounted for throughout the calculations.  The i in the last equation indicates that the velocity oscillation will be 90 degrees ahead in phase compared to displacement oscillation.

 

So velocity will tend to peak (in either direction) when displacement is at zero, and velocity will go to zero at the extremes of displacement.  This should make intuitive sense.  Another thing that should make intuitive sense is that if you keep RMS excursion the same and increase the frequency, the RMS velocity increases.

 

The ideal gas model is perfectly suitable for modeling the sealed box air spring, but the air spring is not linear and does create some distortion.  In practice however, I believe this is almost always an insignificant contributor.  Smaller boxes tend to make other forms of distortion rise, so by the time the box is small enough for the air spring to matter, it probably still won't.

 

Also, back EMF is totally different from inductance, even though both phenomena involve energy storage and release.  They also can interact with one another.  With back EMF, the energy is stored in the mechanical sub-system, where it oscillates between kinetic (motion of the driver) and potential (air + suspension spring) forms.  With inductance, the energy is stored in the magnetic field around the coil.  They act in very different ways.

 

I won't argue that a quality suspension is a critical part of a good driver.  I'm just pointing out that suspension linearity may be less important in smaller sealed boxes and with drivers that have stronger motors.  I believe suspension non-linearity distortion primarily manifests with frequencies below resonance.  And while shrinking box size tends to increase most kinds of distortion, it doesn't affect suspension on-linearity nearly as much.  Meanwhile, increasing motor strength helps reduce almost all kinds of distortion below resonance.

 

Other than distortion though, the properties of the suspension can be largely dominated by the air spring in a sealed box system.  I often see people note "it has high Vas, so I shouldn't use it in a small box".  That's wrong.  If low frequency performance is preferred, higher Vas (assuming same Sd, of course) will almost always be better, even in the small box.  Although the smaller the box, the less it even matters.

 

And no, I'm not a driver designer.  A driver designer would argue my point that higher Vas will always be better because the suspension has to be stiff enough to adequately support the moving parts. 

[mwmkravchenko]

Steve you are much faster (better) at math than I am.  30 years rusty is I. It takes me a while but I eventually get there. I'm the guy all the egg heads in the physics classes would ask how the formulas applied to real life.  And also the guy who regularly struggled big time with remembering them. Fun watching you run rings around me.   

 

It's interesting talking to you via this discussion. You have a slightly different perspective on the same subject and I appreciate it.  Plus I enjoy a good discussion.

 

We agree on the fundamentals. I'm fleshing out the differences. The ideal gas model does not take into account non-linearity. That's my point. And it has been demonstrated mathematically and empirically through test and measurement that the closed box heating of the air internally changes the properties of the air spring. If you are well read enough to have a great conversation so far I'm guessing you have at least touched a bit on this.  The heat sources are two fold the compression of the air as described in the ideal gas model. This is insignificant really. The main contributor is the driver motor inefficiency. Even a 100 db/watt driver is over 93% inefficient. As in for 100 watts input 93 watts are turned into heat. Like a toaster!

 

http://www.sengpielaudio.com/calculator-efficiency.htm

 

So with increase in temperature the driver suspension parameters and DCR are now being modified due to changes in materials temperature and stiffness goes down.  Generally a desirable thing in a sealed enclosure.  But it does change the distortion products.  They go up.

 

"A driver designer would argue my point that higher Vas will always be better because the suspension has to be stiff enough to adequately support the moving parts.  :)"

 

High Vas is a softer suspension.

 

Never said back EMF is the same as inductance.  So I agree there to.  Simply said that you canna ignore it captain!

 

 "With back EMF, the energy is stored in the mechanical sub-system, where it oscillates between kinetic (motion of the driver) and potential (air + suspension spring) forms.  With inductance, the energy is stored in the magnetic field around the coil.  They act in very different ways."

 

OK now you are pulling a Mark.  Looks like you were tired and thought one thing and typed another.

 

I'm sure you mean this statement a little differently than I'm reading it.