Ampere-turns in a solenoid: why equal magnetomotive force is not equal force

Ampere-turns is the product of turns and current, and at fixed power dissipation in a fixed winding window it does not depend on wire diameter at all. A 0.20 mm winding driven at 24 V and a 0.40 mm winding driven at 6 V deliver the same 877 ampere-turns at the same 3 W. What changes between them is the supply voltage, the current, and how the same magnetomotive force divides across the magnetic circuit, which is where two nominally identical actuators end up 40 % apart in force.

Why this happens

The winding produces magnetomotive force, and the magnetic circuit converts it into flux. Only the second step produces force, and it is lossy in ways the winding cannot see.

Turns scale with one over the diameter squared. A winding window holds a fixed area of copper, set by the bobbin and the fill factor. The number of turns that fit is that copper area divided by the wire cross-section, so halving the wire diameter quadruples the turns. A 0.20 mm wire carries four times the turns of a 0.40 mm wire in the same window.

Resistance scales with one over the diameter to the fourth. Resistance is resistivity times length over area. The length grows with turns, which is one over the diameter squared, and the area falls with the diameter squared, so the two effects multiply. Halving the diameter multiplies resistance by 16.

At fixed power the two cancel. Current is the square root of power over resistance, so halving the diameter cuts current by a factor of four. Turns went up by four, current went down by four, and the product is unchanged. This is why the gauge question is a voltage question. A 24 V system wants fine wire and many turns; a 6 V system wants coarse wire and few turns; both deliver the same magnetomotive force for the same watts. The full trade table is in wire gauge and turns for a fixed window.

What the winding cannot control is the division of that magnetomotive force. Ampere-turns divides across the gap, the iron and the leakage paths in proportion to their reluctances. A 40 mm iron path at a permeability of 1000 behaves like 40 micrometres of air. A working gap of 0.4 mm is 400 micrometres. The gap takes 91 % of the magnetomotive force and the iron takes 9 %. Improve the steel to a permeability of 3000 and the iron share falls to 3 %, the total reluctance falls 6 %, and force rises 13 %. The gap is untouched, so the ceiling is set by geometry rather than metallurgy. The gap side of this arithmetic is worked through in air gap and solenoid force.

Leakage is the other invisible division. Flux that crosses between the flanges instead of through the working gap contributes nothing to force but still consumes magnetomotive force. A tubular design with a close-fitting bore leaks less than an open frame with a long exposed flux path, which is why the same 877 ampere-turns can produce noticeably more force in one topology than the other at the same stroke. The comparison at equal copper is in open frame versus tubular.

Saturation sets the ceiling on what ampere-turns can buy. Air carries about 1.26 microtesla per ampere-turn per metre, so a 0.4 mm gap needs roughly 477 ampere-turns to reach 1.5 T if the gap were the only reluctance, and about 520 with the iron path included. Common low-carbon steels saturate between 1.6 T and 1.8 T. Past that, doubling the current raises flux by a few percent and quadruples the heat. Force goes with flux squared, so the gain is a few percent against a copper loss that has gone up fourfold.

The practical consequence is that ampere-turns has an optimum, not a maximum. A winding designed to saturate the pole face at end of stroke uses its copper efficiently. A winding designed well past that point uses the extra copper to heat the bobbin, and it does so at the worst location, because the heat is generated in the winding that is already the hottest part of the assembly. The thermal limit is treated in coil temperature limits.

Temperature takes back part of what the winding delivers. Copper resistance rises 0.393 % per kelvin. A winding that stabilises 70 K above its measurement temperature has 27.5 % more resistance and draws 78 % of the cold current, so the ampere-turns fall 22 % and the force falls about 39 % in the unsaturated region. A design that sits just above the saturation knee cold will fall below it hot, and the force loss will be larger than the resistance ratio suggests. Predicting that fall from a cold resistance measurement is covered in the copper temperature coefficient.

Check these in order

1. Compute ampere-turns from the winding, not from the label. Turns times current, using the current measured in the circuit rather than the nameplate. A 24 V coil at 192 ohms draws 125 mA, and 7016 turns gives 877 ampere-turns. If the measured current is 100 mA, the figure is 702, which is a 36 % shortfall in force terms because force follows the square.

2. Establish the copper fill the winding actually achieved. Measure the DC resistance and the mean turn length, then back out the copper volume. If the result implies a fill factor of 0.32 when the design assumed 0.45, the design has 29 % fewer turns than intended and the resistance is 40 % high. Fill factor is the most commonly overstated number in a winding specification.

3. Find the saturation knee before adding power. Plot force against current in five steps from 50 % to 150 % of nominal. A curve that is still rising steeply at 150 % is unsaturated and has room. A curve that has flattened is at the ceiling, and more current will only raise temperature.

4. Compare the same ampere-turns in both topologies. If a competitor’s unit produces more force from the same winding, the difference is in gap, pole area or leakage. Measure their working gap and pole diameter. In most cases the answer is a shorter gap or a larger face rather than a better steel.

5. Check what fraction of the magnetomotive force the gap consumes. Compute the iron path length, divide by permeability, and compare with the gap. If the iron share exceeds 15 %, the steel or the path length is worth attention. If it is under 8 %, the steel is not the problem and changing it will move force by single digits.

6. Measure the force at the temperature the application runs at. Ampere-turns quoted cold overstate the hot figure by roughly a fifth. A unit specified at 877 cold delivers about 690 hot, and the difference is enough to turn a marginal design into a failing one in the field.

What actually to change

If force is short and the iron is not saturated, add ampere-turns. The cheapest route is usually a higher supply voltage into the same winding, which raises current without changing the part. A 12 V to 24 V change on the same coil doubles current and quadruples force until saturation, and it costs nothing at the actuator. The limit is the insulation system and the driver.

If force is short and the iron is saturated, change the magnetic circuit. Larger pole area raises force at every operating point and does not change the saturation flux density. A pole diameter increase from 6 mm to 8 mm raises area by 78 % and force by the same 78 % at the same flux density, at the cost of a larger bore and more copper per turn for the same turns count.

If the unit is hot and weak, attack the thermal path before the winding. Force loss from temperature is a current loss, and current loss comes from resistance, which comes from temperature. Improving the conduction path from the winding to the housing lowers the stabilised temperature and recovers ampere-turns without changing the electrical design. The bottleneck is usually the bobbin wall between the copper and the shell, not the outside air.

If two suppliers quote the same ampere-turns and different force, ask for the gap. This is the fastest way to find out whether the difference is real or a measurement difference. A supplier quoting a force at 0.2 mm when the application needs 0.5 mm is quoting a different product, even if the winding is identical.

If you are specifying a custom unit, specify the force at the working gap rather than the ampere-turns. Ampere-turns is an input the supplier chooses; force at the working gap is the output the application needs. Specifying the input removes the supplier’s ability to trade gap against winding, which is usually the trade that produces the cheaper and better unit. The parameter list for a custom specification is in the eight parameters to define.

When it IS the harder problem

When the force is adequate cold and short hot, beyond what resistance explains. That pattern means the magnetic circuit is changing with temperature, not the winding. A thermoplastic bobbin creeping under load, a housing expanding at a different rate from the plunger, or a spring relaxing all open the gap as the unit warms. Measuring the assembled gap hot and cold distinguishes them, and the sequence is in force dropping after thermal cycling.

When ampere-turns is high and force is low at every temperature. Then the magnetomotive force is being spent somewhere other than the working gap. Leakage through a long exposed return path, a saturated narrow section in the iron, or a plunger that is not concentric enough to carry flux across the full face will all do this. A flux plot or a simple search-coil measurement around the return path will locate it, and no amount of extra copper will fix it.

When the application needs force at a stroke longer than the topology supports. Ampere-turns buys flux, and flux at a large gap is limited by the gap regardless of how much magnetomotive force is applied. Beyond a stroke of roughly the pole radius, a flat-face design loses force faster than any winding can compensate, and the answer is a different topology or a linkage rather than a larger coil.

FAQ

Is there a formula for solenoid force from ampere-turns? For a gap-dominated circuit, flux density is the permeability of air times ampere-turns divided by gap length, and force is flux density squared times pole area divided by twice the permeability of air. That gives force in newtons directly from ampere-turns, gap and area. It overstates force near contact, where iron reluctance and saturation matter, and it understates it at large gaps, where fringing adds area. For a first estimate across the middle of a stroke it is accurate to about 15 %.

Does more turns always mean more force? No, because more turns in the same window means thinner wire and higher resistance, which means less current at fixed voltage. At fixed voltage the force peaks at a specific turns count and falls on either side. At fixed power the force is independent of turns, which is the result derived above. Most real designs run at fixed voltage, so the fixed-voltage optimum is the one that matters, and it sits where the copper loss matches the thermal limit.

Why do two units with the same part number perform differently? Because the winding is the most repeatable part of the assembly and the magnetic circuit is the least. Turns count is set by a machine and varies by a fraction of a percent. The assembled gap varies with plating, flatness and screw torque, and the plunger concentricity varies with bore tolerance. A 0.05 mm difference in gap is worth 27 % of the force, which is far more than any winding variation.

Can I increase force by increasing the supply voltage? Yes, until saturation or until the insulation system limits you. Doubling voltage doubles current in the short term and quadruples force in the unsaturated region, but it also quadruples copper loss and raises the stabilised temperature, which brings resistance up and current down. In a continuously rated application the steady-state gain is much smaller than the instantaneous one, and the thermal limit is reached sooner. In an intermittent application with a short pulse, most of the gain is available.