Electromagnets & Magnetic Circuits

An electromagnet is a coil coupled to a ferromagnetic path, usually with a movable armature and at least one air gap. Current creates magnetomotive force; the resulting flux stores field energy; changing the gap changes that energy and produces force. This entry keeps the approximations explicit so a magnetic-circuit sketch is not mistaken for a finished actuator design.

The magnetic-circuit approximation #

For a uniform section of length $l$, cross-sectional area $A$, and permeability $\mu$,

$$\mathcal{R}=\frac{l}{\mu A}, \qquad \Phi \approx \frac{NI}{\mathcal{R}_{\text{total}}}, \qquad B=\frac{\Phi}{A},$$

where $\mathcal{R}$ is reluctance, $NI$ is magnetomotive force, $\Phi$ is flux, and $B$ is flux density. This resembles Ohm's law, but the analogy has limits:

• permeability is nonlinear and history-dependent in a ferromagnet;
• flux leaks outside the intended path and fringes around an air gap;
• geometry makes $B$ nonuniform, especially near corners and pole faces;
• changing flux induces eddy currents that oppose the change;
• the electrical winding and mechanical armature make the system dynamic.

The air gap often dominates total reluctance because $\mu_0$ is far smaller than the core's effective permeability. That makes gap geometry the strongest force-setting dimension and also makes assembly tolerance important.

The magnetic-circuit sketch deliberately abstracts away the linkage that turns attraction into useful travel. In a real actuator, the coil surrounds or faces a fixed core, a movable armature closes the working gap, and a linkage carries motion to the driven member; those mechanical details enter the load, friction, stroke, and impact terms below.

From current to force #

For an approximately uniform field in a gap of area $A_g$, magnetic pressure is

$$p_m \approx \frac{B_g^2}{2\mu_0},$$

so the attractive force is approximately

$$F \approx \frac{B_g^2 A_g}{2\mu_0}.$$

An energy formulation is more general. At fixed current,

$$F_x = \frac{1}{2}I^2\frac{dL(x)}{dx},$$

with the force directed toward increasing inductance. Closing a gap lowers reluctance and usually raises inductance, so the armature is pulled toward the closed magnetic path.

The coupled electrical and mechanical model #

The coil obeys

$$v = Ri + \frac{d\lambda(i,x)}{dt},$$

where flux linkage $\lambda$ depends on current and armature position. If the region is linear enough that $\lambda=L(x)i$,

$$v=Ri+L(x)\frac{di}{dt}+i\frac{dL}{dx}\frac{dx}{dt}.$$

The last term is motional back electromotive force. Mechanical motion obeys

$$m\ddot{x}+b\dot{x}+F_{\text{spring}}(x)+F_{\text{load}}=F_m(i,x).$$

These equations explain the familiar sequence: current first rises against inductance; force overcomes preload and friction; motion changes inductance; the armature accelerates; impact and bounce occur at the stop. Release is a different transient because stored magnetic and mechanical energy must leave the system.

Pull-in, hold, and release #

Discrete electromagnets are described by thresholds rather than a single gain.

Using a high pull-in current followed by a lower hold current reduces heating, but the transition must account for supply tolerance, hot-coil resistance, worst-case gap, friction, and load.

Saturation, hysteresis, and residual flux #

Saturation limits how much additional flux an increase in current can produce. Past the knee of the $B$-$H$ curve, current and copper loss rise faster than useful force. Hysteresis means flux depends on magnetic history; remanent flux can delay release or leave an armature stuck. A deliberate nonmagnetic stop or residual air gap prevents fully closing the magnetic path and reduces sticking.

For alternating excitation, hysteresis and eddy-current losses heat the core. Laminations or high-resistivity magnetic materials break eddy-current paths. AC contactors also need a strategy to prevent force from collapsing twice per cycle near flux zero; a shading ring provides a phase-shifted flux component so holding force does not vanish at the same instant everywhere.

Coil and driver are part of the actuator #

At DC steady state, coil loss is $P=I^2R=V^2/R$. Copper resistance rises with temperature, so a voltage-driven coil draws less current and produces less force when hot. Current drive makes force more repeatable but does not remove the thermal limit.

For example, a nominal 24 V, 12 Ω cold coil initially draws 2.0 A and dissipates 48 W. If its resistance rises by 40% at operating temperature, current falls to 1.43 A and copper loss becomes about 34 W at the same supply voltage. In the unsaturated, gap-dominated approximation the available force at a fixed gap scales with $I^2$, so the hot-coil force could fall to roughly $(1.43/2.0)^2\approx51\%$ of the cold estimate. A real design must use the manufacturer's hot pull-in limits because permeability, saturation, supply tolerance, and motion make the simple square law only a screening calculation.

When the driver opens, winding current cannot change instantaneously. A flyback path is mandatory:

• a diode gives low clamp voltage, low electrical stress, and slow current decay;
• a Zener, TVS, or active clamp raises the allowed voltage and releases faster;
• an unbounded inductive transient damages the switch and creates electromagnetic interference.

What to measure #

Characterize coil resistance cold and hot, pull-in and release current over supply and temperature, force versus position and current, travel and release time, impact or bounce, insulation, and steady-state temperature rise. The useful model is the smallest one that predicts all of those observations within the intended operating envelope.