Magnetic actuators and sensors are used to produce and control motion. Applications include systems such as automotive antilock brakes, computer disk drives, and factory automation. In many of these and other cases, the motion must be rapid, which means that the magnetic actuators (and their sensors) must respond quickly when turned on.
Most linear magnetic actuators (such as solenoids, which we'll use as an example here) go through a fourstep turnon process:

Energizing voltage turns on and the coil current rises, delayed in part by an electrical time constant.

The magnetic flux rises, delayed in part by a time constant called the magnetic diffusion time.

Actuator force rises with the magnetic flux.

Actuator sensor delay
The force accelerates the armature as well as the attached load mass.
Each step takes a certain amount of time; the total can make or break an application. Achieving faster response is a matter of limiting each delay.

The energizing circuit turns on and the coil current rises, delayed in part by an electrical time constant. Let us consider the typical magnetic solenoid actuator in Fig. 1 above, left. It is axisymmetric, because it has a central axis around which all of its parts revolve, including its cylindrical plunger armature made of solid steel.
When an electric current flows in its coil, a magnetic flux and magnetic force are produced, which cause the plunger to move towards the stationary stopper.
Due to inductance L associated with the magnetic flux produced by the coil and its resistance R, the actuator is basically a series LR circuit. The electrical time constant of such a circuit is τe = L/R. If we apply a dc step voltage (such as 12 V in an automobile) to such a simple circuit, then the coil current rises as in Fig. 2. It reaches 63% of its final value I _{F} = V/R in the time constant τ_{e} — an initial delay.
To reduce this delay, L could be reduced. However, inductance must usually be kept high to keep magnetic flux and force high. To produce a high IF as quickly as possible, increasing R will not help, though it decreases the electrical time constant. Why? A higher R also reduces coil current. If possible, the applied dc voltage can be increased during this time period to increase the coil current.
Actual current waveforms differ from that of Fig. 2 for two reasons. First, inductance L changes as the plunger moves, and reduces the air gap between the plunger and stopper. Second, “motional” voltage is produced in the coil proportional to the plunger speed, thus causing a “dip” in the coil current as shown in Fig. 3.
Prediction of the coil current waveform can be made using electromagnetic finiteelement software such as Maxwell Electromagnetic Simulation from Ansoft Corp., Pittsburgh, Pa.

Magnetic flux rises, delayed in part by a time constant called the magnetic diffusion time. Diffusion is a common phenomenon. One example is a sponge picking up water; the water gradually diffuses from the outside of the sponge to deep inside it. A fluid diffusion time is proportional to the time it takes for the fluid to diffuse throughout the sponge. Similarly, magnetic flux produced by the currentcarrying coil of Fig. 1 diffuses into the steel plunger and other solid steel parts during a magnetic diffusion time.
When the steel part is axisymmetric, as the plunger and stopper in Fig. 1, magnetic diffusion time due to a step turnon of current is:
τ_{m} = k µσ r^{2}where k = Proportionality constant — 0.173 for metric units
µ = Steel magnetic permeability (assumed a constant value) σ = Steel electrical conductivity r = Plunger radius in Fig. 1.Therefore, to reduce magnetic diffusion time, electrical conductivity should be as low as possible. If, as in Fig. 1, lamination is not possible for reducing effective steel conductivity, then the steel part should be made of highsilicon or other steel with low conductivity.
The magnetic diffusion time τ_{m} is a time constant analogous to the electrical time constant τ_{e} and thus flux turnon is delayed proportionally to τ_{m}. For the actuator of Fig. 1, with a plunger made of steel with σ = 1.7 × 10^{6} S/m and µ = 630 × 12.57 × 10^{7} H/m, the above equation gives σ_{m} = 93 msec.
This equation for σ_{m} is approximate in that it considers only one dimension (the radial) and ignores 2D and 3D affects such as air gaps and return paths. For more accurate predictions of magnetic diffusion effects, 2D and 3D electromagnetic finite element analysis using software such as Maxwell 3D is recommended.
The permeability µ of steel is only a constant value when the steel is operated at low magnetic flux density B in its linear BH range, well below saturation. For compact and low cost designs, however, the steel is often saturated, and the diffusion time varies with the steel BH curve. For this reason, the above diffusion time equation must be modified to account for nonlinearity.
Equations for nonlinear magnetic diffusion show that saturation can reduce τ_{m} eightfold or more. Again, to accurately predict magnetic diffusion, electromagnetic finiteelement software is useful. Its display of the computed magnetic flux density B diffusing into part of the plunger armature of Fig. 1 is shown in Fig. 4 at 20% diffusion time τ _{m}.
To account for the magnetic diffusion turnon time τ_{m} in the circuit of Figs. 2 and 3, a parallel diffusion resistor RD can be added along with a small leakage inductor for the circuit of Fig. 5. Approximate formulas for RD have been derived for simple geometries.

The actuator force rises as the magnetic flux rises. Magnetic force in a solenoid actuator such as Fig. 1 is approximated:
F = A B^{2}/(2µ_{o})
where A = Plunger face area B = Flux density assumed uniform over the face µ_{o} = Permeability of air — 12.57 × 10^{7} H/mBecause force is proportional to the square of B, its rise is related to that of B due to the current producing the flux. Thus, force delay is roughly proportional to the sum of time constants τ_{e} + τ_{m}.

The force produces acceleration and motion. As for any mechanical motion, if friction and other forces are negligibly small, then acceleration is proportional to force divided by mass. (Mass includes that of the actuator armature, including plunger, plus any load mass.) For rapid acceleration and quick motion, total mass should be as small as possible, to produce the shortest possible mechanical delay.
Software such as Maxwell, Simplorer, SPICE, and Matlab can be used to model motion produced by magnetic forces. If appropriate, springs and dampers can also be included in the models, which can accurately predict position versus time. For our example actuator, airgap versus position computed by a Simplorer model accounting for all four of the above delays is shown in Fig. 6. The computed time to close the 10mm airgap is 101 msec, which matches the measured closure time.
Fig. 7 shows a typical magnetic sensor made of a permanent magnet and a solid steel pole on which a pickup coil is wound. When a steeltoothed wheel or gear rotates near the sensor, magnetic flux through the coil changes and induces a voltage according to Faraday's Law. The voltage is proportional to the speed of the wheel, and thus our sensor can detect speed for automotive antilock brake systems and other electronic stabilitycontrol systems. (Four such sensors are required in vehicles — one on each of the four wheels.) The output of each sensor is the voltage induced due to the speed, and thus Fig. 7 is a speed sensor.
Sensor voltage is usually obtained with the coil circuit open, and current assumed zero. Therefore, of the four delay types, the sensor exhibits only magnetic diffusion time τm. As a tooth approaches the sensor pole, magnetic field intensity H is turned on in the steel sensor pole. The solid pole of width w essentially acts as a planar (nonaxisymmetric) steel slab with diffusion time for constant permeability:
τ_{m} = µσ (w/π )^{2}
This simple formula for τ_{m} (like the one for axisymmetric steel) is based on onedimensional assumptions for steel operated in its linear BH region. As before, under typical nonlinear BH operation, diffusion time τ_{m} can be greatly reduced — so linear diffusion times can serve as estimates of the longest possible delays due to magnetic diffusion. Precise predictions of these electromechanical actuator and sensor behaviors — and the total time delay that they introduce — are best made with electromagnetic finiteelement software coupled to electromechanical software.
For more information or a list of other sources that cover this topic, email the author at [email protected] or visit johnrbrauer.com.