I need to design several coils for electromagnetic actuators for small nitrogen valves.  One is a iron-core electromagnet, another is an air coil for a moving-magnet style "voice coil motor".  Both are two-position, bang-bang actuators operating on DC current (100mA @ 10VDC).  

The requirements have not locked down the actuator stroke, although 8mm seems very likely.  The unusual (to me) part is the actuator's maximum full-cycle frequency: 30Hz.  For an open/close valve, that's fairly quick action.  

As a first pass, I want to make sure that the L/R constant is small enough to permit the coil to cycle so quickly.  A full cycle contains an energize and a de-energize period, so: 1/(2*30s^-1) = 0.0167s, so choosing L/R <= 0.005s seems reasonably cautious.  As mentioned above, the requirements call for 100mA at 10VDC, so the DC resistance of the coils seems to have been choosen for me at 100 ohms.  Therefore, L <= 0.005s * 100 ohms, or 500 mH as a maximum choice for the coil's inductance.  

If any of this so far seems screamingly funny, I'd like to mention that I take criticism well.

Here, however, is where I demonstrate how little EMag I can recall.  A brief FEA (thank you, FEMM and author David Meeker!) showed that I could probably squeak by with a coil of 275 A*T.  On a 3mm diameter core, 10mm in length, using 28 layers of 98 turns of 38 AWG yields 274 A*T and 100 ohms resistance.  Using an insulation factor of 10% and a winding irregularity factor of 15% results in a coil outside diameter of 10.7mm
 
Using the iron-cored electromagnet as an example, I want to calculate the impedance, L, in Henries:

L=N^2 * u * A / d

Where N = the total number of turns = 28*98 = 2744 Turns.
      u = absolute permeability of iron (using 6.29E-3 H/m).
      A = Cross-sectional area of the coil (including the
          core) = (10.7/2)^2 * pi = 8.99E-5 m^2
      d = axial length of the coil, for the purpose of
          this approximation ignoring the continuation of
          the core out both ends of the coil = 10mm

This yields L = 426 H.  That's H, not mH or uH.  Whoa!  Seems a little high!  

If you kind people don't mind, I've got a short list of questions that I should know the answers to, but don't:

1.  Please tell me where I went wrong in the inductance calculation above.  I'm REALLY hoping that my little coil doesn't possess 426 H of inductance.  

2.  Is 28 layers on an electromagnet simply too many?  

3.  Are there heuristics for either optimal or maximum values or ratios of core length / core diameter / coil OD / number of layers / etc?  I recall a physical geometry ratio for high-frequency coil design called the Brooks Ratio (?) that claimed optimality; does anything like that exist in the DC world of magnetostatics? 

If the electromagnet is for nitrogen valves what are the forces that the coil needs to open/close the valve at the various valve positions and as the fuction of the PWM duty cycle command? This is where you should start. First the coil has to be able to close/open the valve against the nitrogen pressure and the resulting forces on the valve orifices including friction etc.

Possibly the most significant time constant will be a mechanical resonance. At some frequency the mass and stiffness will team up to create a natural "bounce" frequency for the moving valve mechanism.

That is going to set a definite limit how fast the whole thing can move, and is a fundamental constraint. Operating the PWM frequency anywhere near mechanical resonance will just create uncontrollable chatter, and possibly rapid wear and eventual failure. It will not control gas flow very predictably either.

If the PWM frequency is set at perhaps ten times the mechanical resonant frequency, operation will be smooth. There will be a slight amount of "dither" or "flutter", and that can actually be a good thing.  It will go a long way to eliminating any sticking or friction. Mechanical inertia then largely filters the PWM switching frequency.

Surprisingly the L/R ratio has more to do with the mass of iron and copper than the number of turns, so making it as small as possible will guarantee a fast rise and fall in the current waveform.

The last problem will probably be the gas dynamics of your system. Volumes and flows, filling and emptying rates, and so on.  Many of these systems that require very fine control often have a needle valve in series with the control solenoid. That can be used as a final tuning aid to get the required response from the system.

I honestly feel that testing and experimenting with some commercial solenoid valves will give you a much better feel for what is going on. Trying to do it all from scratch from first principles will likely be a very long and difficult task.

By all means engineer your own solenoid valve, but It may be best to test a few commercially available valves first and try to get your head around all the problems. 

If the electromagnet is for nitrogen valves what are the forces that the coil needs to open/close the valve at the various valve positions and as the fuction of the PWM duty cycle command? This is where you should start. First the coil has to be able to close/open the valve against the nitrogen pressure and the resulting forces on the valve orifices including friction etc.

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