Drawing No. EH–EE–011 // Electrical Engineering

Electromagnetic Solenoid Force Simulator

Reviewed August 2026

A solenoid is a magnetic circuit with a deliberate weak point — the air gap it is trying to close. Change current, turns, gap and core and watch force, saturation, the pull-in point against the return spring, and the electrical time constant that decides how fast any of it actually happens.

Educational model. One-dimensional magnetic circuit with air-gap and iron reluctance, a smooth illustrative saturation cap, and Maxwell stress on the pole face. Leakage, fringing, hysteresis, eddy currents, detailed B–H behavior and plunger dynamics are not solved, so real actuator force can differ substantially from this idealized result.

Force against air gap

the inverse-square wall every solenoid has to climb

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Parameters

Live readout

Model limits. The force and inductance calculations are stationary-gap magnetic-circuit estimates. The saturation curve is a teaching approximation, not a material B–H dataset. The response-time view applies a DC voltage step while holding the gap fixed, and the heating view uses a user-set lumped thermal resistance rather than predicting enclosure-specific temperature rise.

What to watch for

The gap owns the magnetic circuit

With μ_r = 1800, a 12 cm iron path is worth only 0.07 mm of air. Even a 1 mm gap therefore carries more than 90 % of the reluctance. Improving the steel barely moves the force; closing the gap transforms it.

Current helps until the magnetic circuit saturates

At a fixed gap in the approximately linear region, magnetic force and copper loss both scale roughly with I². Near saturation, extra current produces progressively less additional flux while I²R heating keeps increasing, so force-per-watt deteriorates.

Pull-in is a crossing, not a threshold

On the pull-in tab the magnetic curve rises steeply as the gap closes while the spring line rises linearly with travel. Where magnetic force sits below the spring line the plunger simply does not move — and if the curves cross twice, the solenoid stalls part way.

Turns change force, resistance and response together

At fixed current and before saturation, doubling turns can roughly quadruple magnetic force and inductance. With the same wire size and mean turn length, coil resistance also doubles, so the stationary-gap electrical time constant L/R rises by about a factor of two rather than four. Under fixed-voltage drive, the final current also falls as resistance rises.

Holding needs a fraction of pulling

Once closed, the gap is almost zero and the same force needs far less current. Peak-and-hold drivers exploit this: full current to pull in, then a small holding current, which is often the difference between a continuous rating and a short-time one.

Thermal rating is installation-specific

The heating tab separates copper I²R loss from the assumed coil-to-ambient thermal resistance. Real allowable duty depends on insulation class, ambient temperature, housing, airflow, switching cycle and the rise of copper resistance with temperature.

Sources and technical basis

Magnetic-circuit, Maxwell-stress and RL response treatment; the saturation and thermal submodels are explicitly simplified.

MIT OpenCourseWare — magnetic actuators, materials and magnetic circuitsMagnetostatic force, solenoids, magnetic materials, magnetic circuits and energy methods. MIT OpenCourseWare — Maxwell stress and air-gap magnetic machinesForce-density and stress-tensor basis for magnetic pressure in an air gap. Texas Instruments — peak-and-hold solenoid drive reference designPractical current control, pull-in/hold operation and reduced steady power after plunger motion. NIST — 2022 CODATA fundamental constantsCurrent reference set for the vacuum magnetic permeability and other physical constants.