Machine Design

Toward the Instant-On Actuator

It is important to understand what factors can delay the turn-on of devices that employ electromagnetic action.

Typical magnetic-solenoid-actuator geometry cross section (Components not shown to scale)

A typical magnetic-solenoid actuator is axisymmetric, because it has a central axis around which all its parts are revolved, including its cylindrical plunger armature made of solid steel. An electric current flowing in its coil produces a magnetic flux and magnetic force which move the plunger toward a stationary stopper.

Application of a dc step voltage to a simple actuator coil gives rise to a coil current following the classic V/R curve with time constant τe, which therefore is an initial delay. Typical approximate values are V = 75 V, R = 150 Ω and L = 3 H, giving τe = 20 msec. In actuality, the real current waveform differs from the ideal case. One reason is because coil inductance changes as the plunger moves and the air gap (between the plunger and the stopper) is reduced. The other reason is that a "motional" voltage is produced in the coil proportional to the plunger speed that causes a dip in the coil current. Exact predictions of coil current waveforms generally make use of electromagnetic finite element software.

Designers account for the magnetic diffusion turnon time by adding a diffusion resistor RDin parallel with the actuator coil, along with a small leakage inductor in series. Typical values for diffusion resistors and leakage inductors respectively are RD= 600 Ω and Lleak = 65 mH. There are approximate formulas for RDderived for simple actuator geometries. In one example calculation of the air gap versus position, a Simplorer CAD model accounted for all four of the delays and computed a 101-msec time to close a 10-mm air gap, agreeing with the measured time.

A typical magnetic sensor includes a permanent magnet with north and south poles and a solid-steel pole onto which a pickup coil is wound. A steeltoothed wheel rotated near the sensor changes the magnetic flux through the coil, thus inducing a voltage proportional to the speed of the wheel.

John R. Brauer
Director Electrical Engineering and Computer Programs
Milwaukee School of Engineering
Milwaukee, Wis.

Edited by Leland Teschler

Most engineers are familiar with the magnetic actuators and sensors applied in systems such as automotive antilock brakes, computer disk drives, and factory automation. In many cases their motion must be rapid, which means the magnetic actuators and/or sensors must respond quickly when turned on.

Turn-on time can be important for these devices. So it is useful to understand the basic steps for how most magnetic actuators energize: First, the energizing voltage turns on and the coil current rises, delayed in part by an electrical time constant. Then magnetic flux rises, delayed in part by a time constant called the magnetic diffusion time. Next the actuator force rises as the magnetic flux rises. Finally, the force produces acceleration and motion of the armature and/or load mass attached to the actuator.

Each of the above four steps may produce a time delay, and these delays can add up to a significant total delay. Such a delay is often undesirable because it slows the speed of response. Thus it is worth examining each step in turn-on and how to reduce delay there.

To analyze how an electrical time constant delays the rise in coil current, it helps to remember that a typical magnetic solenoid actuator is axisymmetric. It has a central axis around which all its parts are revolved, including its cylindrical plunger armature made of solid steel. An electric current flowing in its coil produces a magnetic flux and magnetic force that makes the plunger move toward the stationary stopper.

Note there is an inductance L associated with the magnetic flux produced by the coil. This inductance and the resistance R of the coil means the actuator is basically a series L-R circuit. The electrical time constant of such a circuit is τe = L/R.

Assuming this simple circuit sees a dc step voltage (such as 12 V from an automobile) on its input, then the coil current rise looks like the classic L/R curve. It reaches 63% of its final value IF = V/R in the time constant τe, which therefore is an initial delay.

Thus, we must reduce-L to reduce this delay. But L usually must be kept high to keep magnetic flux and force high. Increasing R to produce a high IF as quickly as possible won't help. Though it would reduce the electrical time constant, higher R will also reduce the coil current. It might also be possible to apply more dc voltage during this time period to raise the coil current.

In actuality, the waveform of electrical current differs from that of the classic L/R curve for two reasons. One is that the inductance L changes as the plunger moves and there is less of an air gap between the plunger and the stopper. The other reason is that a "motional" voltage is produced in the coil proportional to the plunger speed. This voltage causes a dip in the coil current. It's possible to predict how the coil current waveform will look using electromagnetic finiteelement software such as Maxwell from Ansoft Corp.

So much for the delays stemming from the electrical time constant. Now consider those arising when the magnetic flux rises, delayed in part by a time constant called the magnetic diffusion time.

Diffusion is a common phenomenon. One example is that of a sponge picking up water; the water gradually diffuses from the outside of the sponge to deep within. A fluid diffusion time is proportional to the time it takes for the fluid to diffuse throughout the sponge. Similarly, magnetic flux produced by a typical current-carrying actuator coil diffuses into the steel plunger and other solid-steel parts according to a magnetic diffusion time. When the steel part is axisymmetric as with typical plungers and stoppers, it can be shown that the magnetic diffusion time caused by a step turn-on of current is:

τm = k μ σ r 2

where k = proportionality constant = 0.173 for SI (metric) units; μ = magnetic permeability of the steel (assumed a constant value), H/m; σ = electrical conductivity of the steel, S/m; and r = plunger radius, m. Thus, electrical condutivity should be as low as possible to reduce the magnetic diffusion time. Laminations can sometimes be a practical way of reducing the effective steel conductivity. Otherwise, the steel part should be high-silicon steel or other steel with the lowest possible conductivity.

The magnetic diffusion time t m is analogous to the electrical time constant τe. Thus there is a delay in flux turn-on proportional to τm. For a typical actuator, with a plunger made of steel with σ = 1.7E6 S/m and permeability μ = 630 × 12.57E-7 H/m, the above equation gives τm = 93 msec.

The above simple equation for t m is approximate in that it is based on only one dimension (the radial direction) and ignores 2D and 3D effects such as air gaps and return paths. Software such as Maxwell 3D can carry out 2D and 3D electromagnetic finite-element analysis that gives more accurate-predictions of magnetic diffusion effects.

The permeability of steel μ is only a constant value when the steel operates at low magnetic flux density B in its linear B-H range, well below saturation. For compact and low-cost designs, however, the steel is often saturated, and the diffusion time varies with the steel B-H curve. Thus, the above diffusion time equation must be modified to account for nonlinearity.

Equations for nonlinear magnetic diffusion show that saturation can greatly reduce t m, typically by as much as a factor of eight or more. Again, electromagnetic FEA software such as Ansoft's Maxwell can accurately predict magnetic diffusion.

To account for the magnetic diffusion turn-on time t m, a parallel diffusion resistor RD can be added across the actuator coil along with a small leakage inductor in series with it. Approximate formulas for RDhave been derived for simple geometries.

Now consider the delay that takes place when the actuator force rises as the magnetic flux rises. The magnetic force in a typical solenoid actuator is approximated by:

where A = area of the plunger face, m2; B = flux density assumed uniform over the face, T; and μo = permeability of air (12.57E-7 H/m). The force is proportional to the square of B, so its rise is definitely related to the rise in B caused by the current producing the flux. Thus the force delay is roughly proportional to the sum of the time constants τe + τm.

Finally, consider the delay as 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. The mass must include the mass of the actuator armature, such as the plunger, plus any load mass. Thus the total mass should be as small as possible for rapid acceleration and quick motion, to produce the shortest possible mechanical delay.

Software such as Maxwell, Simplorer, Spice, and Matlab can model motion produced by magnetic forces. Models can accurately predict position versus time and can include springs and dampers, if appropriate.

Consider a typical magnetic sensor that has a permanent magnet with north (N) and south (S) poles with a solid-steel pole on which a pickup coil is wound. When a steel-toothed wheel or gear rotates near the sensor, the action changes the magnetic flux through the coil, which induces a voltage according to Faraday's Law. The voltage is proportional to the speed of the wheel. Thus such a sensor can gauge speed for automotive antilock brake systems and other automotive electronic stability control systems.

The sensor usually operates with the coil open-circuited, so the coil current is assumed zero. Thus of the four delays discussed above, the only one that applies for this sensor is its magnetic diffusion time t m. As a tooth approaches the sensor pole, the magnetic field intensity H grows in the steel sensor pole. The solid-steel tooth of width w acts essentially as a planar (not axisymmetric) steel slab, which has a diffusion time for constant permeability given by:

This simple formula for t m, like the previous one for axisymmetric steel, is based on onedimensional assumptions for steel operated in its linear B-H region. As before, it's possible to greatly reduce t m under typical nonlinear B-H operation. Thus, the above linear diffusion times can serve as estimates for the longest possible delays caused by magnetic diffusion.

Of course, electromagnetic FEA software coupled to electromechanical software can precisely predict the electromechanical behavior of magnetic actuators and sensors, including time delay.


MSOE in cooperation with MACHINE DESIGN is hosting a Webcast covering electromagnetic actuators and sensors. The event takes place Sept. 6. Go to more details.

MSOE will also be giving a course on "Magnetic Actuators and Sensors" in Milwaukee on Nov. 8 and 9. Further details are available at

Milwaukee School of Engineering,


  1. J.R. Brauer — Magnetic Actuators and Sensors, Wiley IEEE Press, 2006.
  2. J. R. Brauer and I. D. Mayergoyz —"Finite element computation of nonlinear magnetic diffusion and its effects when coupled to electrical, mechanical, and hydraulic systems," IEEE Trans. Magnetics, March 2004.
  3. J. R. Brauer — "Finite element computation of magnetic diffusion times in nonlinear steel with surface field turned on and off," Proc. Applied Computational Electromagnetics Society Conference, Miami, March, 2006.
  4. Ansoft Corp. Web site,
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