## Calculating the approximate inductance of a coil

## Calculating the approximate inductance of a coil

(OP)

Engineers,

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?

Thank you very much,

David Albertson

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?

Thank you very much,

David Albertson

## RE: Calculating the approximate inductance of a coil

## RE: Calculating the approximate inductance of a coil

Thank you very much for your response!

Yes, I agree with you; normally those forces would totally dominate the balance. My posting doesn't go into the valve body design, but the kicker is that the nitrogen source is regulated at 1 kPa (about 0.15 psi gauge). The valve itself is a simple plate and orifice, with an extruding o-ring on the orifice serving as a seal to the plate. The contribution of the 1 kPa gas pressure axially to the actuator's force balance is approximately 0.15 N (0.034 lbf), and is discontinuous: it exists only when the valve is closed.

Again, I thank you very much for your help. I believe I have written the basic force balances correctly, with the caveat that I may have errors in my FEMM programming. It is a new package for me, but Dr. Meeker seems to have made it as simple as possible to use!

I re-calculated the inductance expression in my OP, and I still get 426 Henrys. It still seems high . Can anyone see what I have done wrong?

Best Regards,

David Albertson

## RE: Calculating the approximate inductance of a coil

## RE: Calculating the approximate inductance of a coil

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.

## RE: Calculating the approximate inductance of a coil

Thank you both for your insights and suggestions. I've spent the last week looking in the directions you've suggested, and I would like to tell you where the valve part of the project is right now, and ask some further questions if I may.

1. The approximation formula (shown in OP) I was using is apparently for single-layer coils; this explains the huge error on the 28-layer coil in my model. David Meeker's FEMM program calculates the L value to be a much saner 3mH.

2. Per your suggestion, I've done some reading on mechanical resonance. I see its significance, and have an understanding of the analog between it and electrical resonance frequencies. However, short of an FEA program approach or a build-and-measure, I cannot find heuristics or ROT approaches for calculating mechanical resonance frequencies in little structures. Can someone provide a pointer to a good source?

3. I've purchased 4 examples of small solenoid valves and taken them completely apart. I've gotten several ideas, but the examples I have are really not very close to what the project calls for:

A) They are all spring-return, where we need a permanent magnet return.

B) They are all designed to be piped or tubed to, and we need a valve internal to a large manifold, that is, the valve mechanics all reside in the high-side of a manifold and open to allow flow out of the manifold into a fitting.

C) They are all designed for much higher pressures than we need. The manifold, which is sized as a large integrating accumulator, is only pressured up to 1 kPa (about 0.15 psi gauge).

I've looked through Thomas Register and made some phone calls, but not gotten very close. if anyone can suggest a product I can buy to take apart that is a bit closer to our end goal I'd surely appreciate it!

Thank you very much for your comments and assistance!

David Albertson

## RE: Calculating the approximate inductance of a coil

The effect would be the return magnet would tend to "stick" and then fly apart. A coil spring will offer a much more linear force versus displacement characteristic.

The only solenoids I have seen that use permanent magnets are latching bistable relays, they snap from one position to the other, and then hold (latch) magnetically, without requiring any electrical energisation. That is the least desirable sort of characteristic for the linear positioning of a solenoid.

Commercial solenoid valves will have a valve body, with threaded pressure fittings, just as you suggest. These usually dismantle into at least two major sections. It is possible to fit just the solenoid and actuator part directly to your own manifold, provided your manifold is machined to duplicate the original valve seat. That can eliminate much external pipework, and greatly reduce gas volumes and the possibility of leaks.

I have seen this done in medical equipment, where a machined manifold has solenoid bodies attached directly to a common aluminium manifold block. It makes for a very simple and compact assembly. Other fittings such as needle valves and pressure transducers thread directly into the same solid manifold block.

## RE: Calculating the approximate inductance of a coil

Thank you very much for your response.

Regarding the PM return action, I agree with you completely: I think it is an unnecessary complication. The customer absolutely insists on it. A nice, (linear!) Hooke's law approach with an understressed spring seems to me to be far more practical. The customer is convinced that a PM return-action is significantly more reliable than a spring. Within reason, we like to do what the customer wants; a sad state of affairs that binds all engineers together!

I really like what you said about using 1/2 the SOV and mounting it inside the manifold, with appropriate machining on the manifold inside face. That's very clever! If I could find an SOV cheap enough, and low enough power requirements, and magnetic return, I'd run it by the team and see if we could reach agreement.

On the magnetic resonance issue you mentioned earlier:

determining the resonant frequencies of the valve, especially the moving armature. Would you please point me towards a good method for finding the frequencies? For structures more complex than, say, a cylinder, is a structural FEA program the defacto method? If so, I would really appreciate a pointer to a good quality GNU/shareware/freeware program. (The FEMM program has me spoiled; now I don't want to pay for software!

Thank you again for all the help!

David Albertson

## RE: Calculating the approximate inductance of a coil

It is far easier to just measure resonance, and probably more accurate. Just connect your solenoid to a high voltage switching transistor driven by a variable frequency oscillator. As the frequency is varied through resonance, the solenoid will chatter violently at one particular specific frequency. You will definitely hear and feel it vibrate. Anything from 5Hz to 30 Hz may be fairly typical.

Try testing the solenoids you already have to get a feel for it. You can also work out the L/R time constant, but I bet you will find it is well above the mechanical resonance frequency, and therefore not the significant limiting factor for speed of response.

One thing to realise is that when switching off the solenoid, the voltage across the coil should be allowed to rise reasonably high for fast magnetic decay. But it will need to be clamped somehow to prevent damage to the switching transistor (or FET). The amount of power to be clamped could be several watts when the solenoid is switched continuously at a fairly fast rate, so things may become rather hot.

## RE: Calculating the approximate inductance of a coil

I may be wrong but I get the feeling that your team has no prior experience with electropneumatic valve design. It is all wrong to analyse the coil separately from the complete magnetic circuit of the valve combined with the mechanical response of the valve coupled with the pneumatic forces inside the valve.

You have to put down the complete static and dynamic magnetic circuit to be able to analyse the valve electrical and mechanical response time. All this in the confined space you allow for the valve.

From my experience you should put down the magnetic and mechanical/pneumatic equations and solve them numerically for the magnetic flux density, forces, currents, response time etc. Then and only then you can check the complete valve behavior.

## RE: Calculating the approximate inductance of a coil

Thank you very much for your very helpful responses!

Israelkk, you have found me out! I don't know a blessed thing about valve design, electropneumatic or otherwise, and the other 3 guys on the team know even less. Our (very small) company employs two engineers and two scientists; we design and manufacture specialized gas chromatography equipment. The scientists are both chemists, and the other engineer is a ChemE. I'm an EE; I do the signal processing and the controls work. The conversation went like this: "Hey! SOVs use coils, and electrons and stuff, right? Must be a job for our tame EE!" Honestly, I think the chemical guys gang up on me sometimes just for fun!

Your point about writing the complete static & dynamic systems and approaching the problem from a total perspective is very well taken indeed. My original post was the result of trying to size up the problem with some back-of-envelope calculations, and noticing that the approximation for coil inductance was generating a colossal, quite impossible number. This problem has been cleared up (the approximation was only intended for single-layer coils) and I'm gradually collecting enough information to begin a real design, which, as you point out, must take into account the entire system.

WarpSpeed, I connected the test jig you suggested to one of the SOVs I bought. Your prediction was correct: as I swept the function generator's 0::100 Hz range the SOV went berserk at 17 Hz. An impedance bridge showed that 1/(L/R) = 190 Hz... a full order of magnitude beyond the mechanical resonance frequency. Nice call!

Your comment on transistor protection from large back-voltages as the coil field collapses, combined with quick magnetic decay, and putting the waste heat outside of the manifold, left me scratching my head a bit. I had thought to simply put a diode across the coil, back-biased relative to the supply current, and with an appropriately large PIV rating. Well, that will protect the output transistor, but is pretty bad in terms of delaying the magnetic decay, as well as leaving all the waste heat in the manifold, heating up the gas. Since the energy dissipation will be largely by resistive copper losses, I could take part of the losses outside the manifold with a resistor in series with the diode mounted in a junction box. That would, however, put a larger resistance (coil + resistor) to the coil discharge circuit, thereby slowing the magnetic field collapse even more. I was wondering if you had a better idea for dissipating the coil energy?

Thank you very much for all the help!

David Albertson

## RE: Calculating the approximate inductance of a coil

Select drive transistor first. Say 1000 volt if coil can operate at 500 volt.

Select transient suppressors to clamp at less than 500 volt at coil current.

Calculate energy stored in coil.

J=1/2*L*I*I

J=Joule

L=inductance in henries

I=current in amp

Calculate watts in surge suppressors.

Watt=J*PPS

PPS=pulses per second

Do whatever is necessary to get rid of heat in surge suppressors.

A resistor (+ back based diode) instead of transient suppressors will work but valve operating speed will be slower.

MOV's are probably not a good choice due to their poor operating voltage tolerance.

## RE: Calculating the approximate inductance of a coil

Another approach to this is to see what other gas chromatograph manufacturers are using. It is not really cheating, but it will give you an idea of the current state of the art, and what is realistic.

It is many years since I worked on any gas chromatography equipment, and that was mainly with the detection end of things, and oven temperature control of the column, rather than the gas control system. It should be possible to slow the elution time down somewhat by using a much longer column, but that is a problem for the scientists to figure out.

Getting back to coils. A diode would continue to conduct current for on full L/R period after the transistor switches completely off. That is definitely going to slow down the solenoid release time. By allowing the coil back EMF to rise fairly high, the current fall time through the winding can be made MUCH shorter. Just as the current rise time would be much faster with a higher applied voltage when turning on. It is exactly the same effect in reverse.

There are many ways to do this, a diode directly across the coil being the slowest responding. Definitely use a high voltage switching transistor, 1Kv rating may be overkill, but something rated at 400v minimum would be fairly easy and economical, along with some sort of hard voltage clamping device such as a tranzorb.