derive the dynamic equations of motion of the ... · generated magnetic flux to seek the minimum...

Solenoid Modeling K. Craig 1

Derive the dynamic equations of motion of the electromechanical

system.

Solenoid Modeling


Operating Principles of Solenoids

• A solenoid is a translational motion actuator with a rather limited motion range and is used in fluid-flow control valves and small-range translational displacement applications.

• A solenoid is made of a coil, a frame which is a material with high permeability to guide the magnetic flux, a plunger, a stopper (and a centering spring in most cases), and a bobin.

• A bobin is a plastic or nonmagnetic metal on which the coil is wound. It is nonmagnetic so that there is no short circuit for the flux between the coil and plunger.


Vi

R

V iR

c

Ni magnetomotive force (At)

reluctance (At/Wb)A

1 permeance

RA

Electrical / Magnetic Circuit Analogy


• The operating principle of solenoids is based on the tendency for the ferromagnetic plunger and coil-generated magnetic flux to seek the minimum reluctance point. As a result, when the coil is energized, the plunger is pulled in towards the stopper.

• The higher the magnetic field strength (number of turns in the coil times the current, i x ncoil) and the better the magnetic permeability, μ, of the medium that guides the flux to the plunger, the higher the force generated.

• The plunger works on the pull principle. However, by mechanical design we can obtain pull or push motion from the solenoid. The mechanical connection between the plunger, made of a ferromagnetic material, and the tool must be via a nonmagnetic material.


• The quality of the magnetic circuit and its ability to guide magnetic flux lines depend on the permeability of the coil frame, the air-gap between the plunger and coil (fixed gap), and the air gap between the plunger and stopper (variable gap).

• For a given current, the force generated by the solenoid varies as a nonlinear function of the air gap between the plunger and the stopper. The smaller this gap is, the smaller the effective reluctance of the magnetic flux path, and hence, the higher the force generated.

• As the plunger closes the air gap between itself and the stopper, the force capacity increases. This is called the holding force, which is generally higher than the average pull-push force.


• Note that the shape of the force-displacement curve for a constant current can be affected by the geometric shape of the plunger and stopper head.

• A solenoid is a single-acting type device; that is, when current is applied the plunger moves in one direction in order to minimize the reluctance, regardless of the direction of the current. Force is always generated in one direction, pull or push.

• A double-acting solenoid is basically two solenoids with one plunger, two coils, and two stops. It can move in both directions by generating force in both directions, both pull and push directions.


• A double-acting solenoid can have three positions:– center position when both solenoids are deenergized– left position when the left solenoid is energized– right position when the right solenoid is energized

• If the current in each solenoid is controlled proportionally, instead of fully on or fully off, then the displacement of the plunger can be controlled proportionally instead of two or three discrete positions. This is the method used in proportional valves.

• Solenoids are rated in terms of their coil voltage, maximum plunger displacement, and maximum force.

• The coil acts as the electromagnet in all electric actuators. Current (i), number of turns (ncoil), and the effective permeance of the magnetic medium (core material, air gap, etc.) determine the electromagnetic field strength generated.


• At the same time, there are mechanical size and thermal considerations. Rated current determines the minimum diameter requirement of the conductor wire. The wire diameter and the number of turns determine the mechanical size of the coil. In general, the insulation material increases the effective conductor diameter by about 10% and different insulation materials have different temperature ratings.

• Once the coil wire diameter, number of turns, and the mechanical size are known, the resistance of the coil is determined. Hence, the resistive heat dissipation is known. In order to make sure the temperature of the coil stays within the limits of its coil-insulation rating, the thermal heat conducted from the coil should balance the resistive heat generated.


• The coil design requires the balancing of electrical capacity (current and number of turns), mechanical size, and thermal heat.

• The force generated by a solenoid is a function of the current in the coil (i), the number of turns in the coil (ncoil), the magnetic reluctance (a function of the plunger displacement, the design shape, and the material permeability), and the temperature.

• For a given solenoid, the number of turns in the coil is fixed, and the magnetic reluctance varies with the displacement of the plunger and the air gap between the winding coil and the plunger.

• The main effect of the temperature is to change the resistance of the coil. This leads to a change in current for a given terminal voltage.


• If the control system regulates the current in the coil, the effect of temperature on force other than its effect on resistance is negligible.

• Therefore, for a given solenoid, the generated force is a function of plunger displacement and current, and this relationship is nonlinear.

• Note that there is always a residual left in the core when the current is turned off due to the hysteresis nature of electromagnetism.

• In the basic mode of operation, a solenoid is driven by DC voltage Vt at its coil terminals, and the current developed (neglecting the inductance of the coil) is given by Vt / Rcoil.

• The physical size of the solenoid determines the maximum amount of power it can convert from electrical power to mechanical power.


• The continuous power rating of the solenoid should not be exceeded in order to avoid overheating.

• In some applications, it is desirable to provide a larger current, called the in-rush current, and after the initial movement, the current is reduced to a lower value, called the holding current.

• One way to reduce the in-rush current to holding current is to divide the coil winding into two resistor sections and provide an electrical contact between the two series resistors. When large in-rush current is needed, short circuit, via an electronic transistor switch, the second resistor section to increase the current. When it is desired to reduce the current, turn off the switch to include the second part of the resistor in the circuit in series.

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Electromagnet

Infrared LED

PhototransistorVsensor ≈ 2.5 VAt Equilibrium

Levitated Ballm = 0.008 kgr = 0.0062 m

Equilibrium Conditionsgap0 = 0.0053 mi0 = 0.31 A

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Magnetic Levitation System

Emitter

Detector

Review


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m m m core gap object return pathm

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N iN N L i

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x A NN NL xA A xA

A1 1W L x i2 2

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0 gap gap

2 2

2 2e 0 gap 1

0 gap gap 2 gap

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1 dL(x) 1 1 if i A N K2 dx 2 A x K x

Magnetic Levitation System Derivation

2

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Derive the dynamic equations of motion of the electromechanical

system.


• The figure shows in cross section a cylindrical solenoid magnet in which the cylindrical plunger of mass M moves vertically in brass guide rings of thickness g and mean diameter d. The permeability of brass is the same as that of free space.

• The plunger is supported by a spring whose spring constant is K. Its unstretched length is ℓ0. A mechanical load force ft is applied to the plunger from the mechanical system connected to it.

• Assume that the frictional force is linearly proportional to the velocity and that the damping coefficient is B. The coil has N turns and resistance R. Its terminal voltage is et, and its current is i. The effects of magnetic leakage and reluctance of the steel are negligible.


• Reluctance of the Magnetic Circuit– Reluctance of the magnetic circuit is that of the two

guide rings is series, with the flux directed radially through them

– Assume constant flux density in the guide rings with respect to the radial distance since g << d

– g = length of the flux path in the direction of the field– (πxd, πad) = (upper, lower) area of flux path

perpendicular to the field– assume that, for the upper gap reluctance expression,

the field is concentrated in the area between the upper end of the plunger and the lower end of the upper guide ring

upper gap lower gap0 0

g g xd ad


– Total Reluctance

– Inductance

– Magnetic Force acting upward on the plunger in the positive x direction

upper gap lower gap

0 0 0 0

g g g 1 1 g a x+xd ad d a x d x

2 220 0adN x adNN xL x L ' where L 'g a x a x g

2 2e 2

W i, x 1 dL 1 aF i i L 'x 2 dx 2 a x


– Induced voltage in the coil

– Equations of Motion• Newton’s 2nd Law (total spring stretch = ℓ1-x)

2

d di dL dx x di ai dxe Li L i L ' L 'dt dt dx dt a x dt dta x

2

x 2

2 2

t 1 22

d xF Mdt

d x dx 1 aif M B Mg K x L 'dt dt 2 a x


• KVL applied to the circuit

• The second term in the equation is the self-inductance voltage term. The third term is the speed-voltage term and is common to all electromechanical-energy-conversion systems; it is responsible for energy transfer to and from the mechanical system by the electrical system.

– These two equations of motion are valid as long as the upper end of the plunger is well within the upper guide ring, e.g., 0.1a < x < 0.9a, which is the normal working range of the solenoid.

20

t 2adNx di a dxe iR L ' iL ' where L '

a x dt dt ga x

derive the dynamic equations of motion of the ... · generated magnetic flux to seek the minimum...

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