Air gap vs. solenoid force: why it is closer to inverse square than linear

Force in an electromagnetic actuator falls with the square of the air gap over most of its working range, not with the gap itself. A change from 0.30 mm to 0.35 mm removes 27 % of the force at the same current, and a change to 0.40 mm removes 44 %. That single relationship explains more weak-solenoid cases than any material question, because gaps grow in assembly and nobody re-measures them after the drawing is approved.

Why this happens

The magnetic circuit is a chain of reluctances in series. Each element is a length divided by a permeability and an area, and the element with the largest ratio controls the flux. In a small actuator the air gap wins by a wide margin, so the iron becomes almost a bystander.

The arithmetic is worth doing once. Take a magnetic path of 40 mm through 1215 steel at a permeability of 1000 and a working gap of 0.5 mm with the same cross-section. The iron contributes 0.040 divided by 1000, which is 40 micrometres of equivalent air. The gap contributes 500 micrometres. The iron is 7 % of the total and the gap is the other 93 %. Put a better steel in with a permeability of 3000 and the iron falls to 13 micrometres, the total drops from 540 to 513, and force rises by 10 %. Put a 0.05 mm burr on the mounting face instead and the total rises from 540 to 590, and force falls by 16 %. The tolerance is worth more than the material, in the opposite direction.

Why the square and not the cube or the line. Reluctance rises linearly with gap length. Flux equals magnetomotive force divided by reluctance, so flux falls as one over the gap. Force at a pole face goes with flux density squared, so force falls as one over the gap squared. Every step is linear, and the square arrives at the last one.

The square holds until two things stop it. At small gaps the iron ceases to be negligible and force flattens out; this is where saturation at the pole face sets the ceiling. At large gaps, fringing spreads the flux beyond the pole perimeter, adding effective area in proportion to the gap. A round pole of radius r gains roughly one gap length of fringing all around, which is an area increase of about two times the gap divided by the radius. At a 0.3 mm gap on a 3 mm radius that is 20 %, at 1.5 mm it is 100 %, and by then the simple inverse-square rule is describing something else.

Saturation explains why more current stops helping near contact. To reach 1.5 T across a 0.3 mm gap requires about 358 ampere-turns if the gap were the only reluctance. Across 0.5 mm it requires 597. A winding designed for the far end of the stroke therefore drives the iron well past saturation near contact, where the extra ampere-turns go into heat rather than flux. This is the mechanism behind the flat left end of most force-stroke curves, and it is set out from the winding side in ampere-turns and why equal magnetomotive force is not equal force.

Temperature moves the current, which moves the flux. Copper rises 0.393 % per kelvin, so a winding settling 70 K above ambient carries 27.5 % more resistance and draws 78 % of its cold current at fixed voltage. Force follows current squared in the unsaturated region, so 61 % of the cold force survives. Combined with a gap that grew 0.05 mm in assembly, a nominal 10 N actuator delivers about 4.5 N at end of stroke. The derivation is in predicting hot-state force from cold resistance.

Plating is gap. Nickel at 20 micrometres on each pole face contributes 40 micrometres of non-magnetic material in series. Against a 0.30 mm gap the effective length becomes 0.34 mm and force drops 22 %. Zinc, epoxy and phosphate coatings all behave the same way, and the effect is invisible on a drawing that dimensions the steel rather than the finished surface.

The assembly adds two more gaps nobody draws. A mounting face that is flat to 0.05 mm over its span adds up to that much when the housing is bolted down, and a shoulder that is square to 0.03 mm tilts the plunger so the gap is uneven around the circumference. In the tilted case the effective gap is the largest local value, not the average, which is why a face that measures 0.30 mm on a feeler gauge can behave like 0.36 mm in the magnetic circuit. The installed-versus-bench pattern is covered in works on the bench, fails in the machine.

Check these in order

1. Measure the gap on the assembled unit, not the parts. Hold the plunger against its mechanical stop and use feeler gauges at three points around the pole face. Record all three. A spread above 0.02 mm between readings means the faces are not parallel, and the magnetic circuit sees the largest reading rather than the mean.

2. Subtract the coatings. Ask what the finished plating thickness is on each face and double it. Electroless nickel at 20 micrometres per face costs 40 micrometres of gap. On a design with a 0.25 mm nominal gap that is 16 % of the magnetic length before anything has moved.

3. Separate the gap that exists from the gap that is used. A curve starting at 0 mm assumes pole faces in contact. If the mechanism has a residual stop of 0.15 mm, the working stroke begins at a point where the force has already fallen 30 % from its peak. Drawing the two together usually explains a “weak” unit without any parts being wrong.

4. Check the current the coil actually receives. Measure voltage at the coil terminals while energised, not at the power supply. A 0.5 mm² cable 3 m long carrying 1 A drops about 20 mV per metre, which is small, but a shared return path or a driver with 1.5 V of saturation loss on a 12 V rail removes 12 % of the voltage and 23 % of the force.

5. Compare the same unit hot and cold before touching geometry. Run it for the thermal time constant, usually 5 to 15 minutes for a small coil, then re-measure. If the loss matches the resistance rise, the gap is fine and the problem is thermal. If the loss exceeds it, the gap is growing with temperature, which points at a thermoplastic bobbin or a housing with a different expansion coefficient.

6. Look for the gap that opens under load. Mount the actuator in its mechanism, apply the working load, and measure again. A bracket that deflects 0.1 mm under load opens the gap by that much at the worst possible moment, and a bench test with a rigid fixture will never show it. The vending lock case in latch geometry and solenoid force is this failure in its usual form.

What actually to change

Attack the stack of tolerances, not the parts inside them. The gap is the sum of the drawing dimension, the plating, the mounting face flatness, the shoulder squareness and any residual shim. Each of those is a tolerance somebody already accepted, and the sum is what the magnetic circuit sees. Tightening one from 0.05 mm to 0.02 mm is usually cheaper than any redesign of the coil.

Set the nominal gap from the force requirement, not from the stroke. If the mechanism needs 4 mm of travel and the actuator can produce the required force only up to 3.5 mm, the answer is not a bigger coil. The answer is a different topology or a linkage that converts force into stroke, as the crossover behaviour in open frame versus tubular designs shows at 1 to 3 mm.

Use a non-magnetic shim deliberately, and dimension it. A 0.1 mm brass shim at the stop is a legitimate way to prevent pole faces from sticking on residual magnetism. It is also 0.1 mm of gap, worth 52 % of the force against a 0.30 mm design. Put it in the tolerance stack rather than adding it during commissioning, because a shim added late is a force reduction nobody has analysed.

Treat coating choice as a magnetic decision. If corrosion resistance requires a thick coating, move the coating off the pole faces or recess the faces so the finished surface sits at the designed dimension. A design that specifies 25 micrometres of nickel on an active face and dimensions the steel underneath has already given away 50 micrometres of gap in the drawing.

Measure force at the working gap, with the real load. A supplier who quotes peak force is quoting the value at or near contact, which is the one operating point the mechanism never uses. Ask for force at the end of the working stroke, at the stabilised winding temperature, at the low end of the supply tolerance. Those four conditions turn a number into a specification.

When it IS the harder problem

When the gap is already at its minimum and force is still short. Below about 0.15 mm the iron stops being negligible, saturation at the pole face sets the ceiling, and further gap reduction buys almost nothing. At that point the limits are pole area and the flux the iron can carry, and the fix is a larger pole or a different material, not tighter tolerances.

When the gap changes during the stroke in a way the curve does not describe. A plunger that tilts as it travels, a bore that is not concentric, or a load that applies a side force all make the effective gap a function of position and of load. A single-number gap measurement cannot diagnose this. Measuring force at three strokes with and without the load applied will show it.

When the loss appears only after thermal cycling. That pattern points at creep in a thermoplastic bobbin or relaxation in a spring rather than at the gap, and the diagnosis runs through force dropping after thermal cycling rather than through geometry. The distinguishing test is whether the gap measured cold, after the unit has been run hot and allowed to cool, is larger than it was when new.

When the customer’s drawing and the customer’s mechanism disagree. This happens often enough to be worth a step in the process. The gap on the drawing is the one the designer intended; the gap in the mechanism is the one the magnetic circuit sees. Measuring the second one, on a unit built from production parts and installed in the real bracket, is the only measurement that settles the question.

FAQ

Is force really inverse square, or is that an approximation? It is exact while the gap dominates the reluctance and the iron is unsaturated, which covers most of the working range of a small actuator. It fails at both ends. Near contact, iron reluctance and pole-face saturation flatten the curve. At large gaps, fringing adds area and the fall becomes slower than inverse square. For a 4 mm stroke on a 6 mm pole, the middle of the range is well described by the square and both ends are not.

Does a larger pole area fix a gap problem? It raises the force at every gap, but it does not change the sensitivity to gap. Doubling pole area doubles force at 0.3 mm and doubles it at 0.4 mm as well, so the 44 % loss between those two points remains 44 % of a larger number. Area is the right answer when the absolute force is short at the design gap; it is the wrong answer when the force is adequate on the bench and short in the machine.

Can I compensate a large gap with more ampere-turns? Only until the iron saturates. Below saturation, doubling ampere-turns doubles flux and quadruples force, so a gap problem can be bought out with power. The cost is heat: the same coil at twice the current dissipates four times the power, and the resulting temperature rise raises resistance and takes back part of the gain. In a continuously energised application the trade is usually a loss, which is why the gap is worth fixing first.

Why does the same design work in one customer’s machine and not another’s? Because the gap is a property of the assembly, not of the actuator. A bracket with a different flatness, a housing in a different material, a screw torque that differs, or a stop that wears in differently all change the magnetic length. When one customer reports weak force and another does not, the first measurement to request is the assembled gap from both.