In this chapter, we consider widely commercialized electrostatic and electromagnetic transducers. In sensing and low-power actuator applications, the electrostatic devices may ensure desired performance and capabilities. The mini- and microscale electrostatic transducers, actuators, and sensors are fabricated using bulk and surface micromachining, which complies with microelectronic technologies. The commercialized microelectromechanical systems (MEMS) technology guarantees affordability, high yield, robustness, etc. The electromagnetic devices ensure high force, torque and power densities. There are variable-reluctance (solenoids, relays, electromagnets, levitation systems, etc.) [1,2,3,4 and 5], permanent-magnet, and other electromagnetic transducers. The stored electric and magnetic volume energy densities ρ _{ We } and ρ _{ Wm } for electrostatic and electromagnetic transducers are ρ W e = 1 2 ε E 2 and ρ W m = 1 2 μ − 1 B 2 = 1 2 μ H 2 ,
In this chapter, we consider widely commercialized electrostatic and electromagnetic transducers. In sensing and low-power actuator applications, the electrostatic devices may ensure desired performance and capabilities. The mini- and microscale electrostatic transducers, actuators, and sensors are fabricated using bulk and surface micromachining, which complies with microelectronic technologies. The commercialized microelectromechanical systems (MEMS) technology guarantees affordability, high yield, robustness, etc. The electromagnetic devices ensure high force, torque and power densities. There are variable-reluctance (solenoids, relays, electromagnets, levitation systems, etc.) [1,2,3,4 and 5], permanent-magnet, and other electromagnetic transducers. The stored electric and magnetic volume energy densities ρ _{ We } and ρ _{ Wm } for electrostatic and electromagnetic transducers are
where ε is the permittivity, ε = ε_{0}ε_{ r }; ε_{0} and ε_{ r } are the permittivity of free space and relative permittivity, ε_{0} = 8.85 × 10^{−12} F/m; E is the electric field intensity; μ is the permeability, μ = μ_{0}μ_{ r }; μ _{0} and μ _{ r } are the permeability of free space and relative permeability, μ_{0} = 4π × 10^{−7} T-m/A; B and H are the magnetic field density and intensity.
The maximum energy density of electrostatic actuators is limited by the maximum field (voltage), which can be applied before an electrostatic breakdown occurs. In mini- and microstructures, the maximum electric field cannot exceed E _{max} resulting in the maximum energy density ${\rho}_{We\text{max}=\frac{1}{2}}{\epsilon}_{0}{\epsilon}_{r}{E}_{\text{max}}^{2}$ . In ~100 × 100 μm to millimeter size structures with a few micrometers air gap, E _{max} may reach ~3 × 10^{6} V/m. Depending on the device, the relative permittivity ε_{ r } may vary from 1 to ~10. One estimates ρ _{ We } _{max} to be less than 100 J/m^{3}. For electromagnetic actuators, the maximum energy density ρ _{ Wm } _{max} is limited by the saturation flux density B _{ sat } (which is ~1 T), material permeability μr, (BH)_{max}, etc. The resulting magnetic energy density ρ _{ Wm } _{max} may reach ~100,000 J/m^{3}. Hence, ρ_{ Wm }/ρ_{ We } ≫ 1. However, in many applications, electrostatic MEMS are an effective, affordable, and consistent solution. To actuate any mechanism, the electrostatic force or torque must be greater than the load force or torque. The load forces and disturbances can be small. For example, ~1,000,000 electrostatic micromirrors (each ~10 × 10 μm) are repositioned in the Texas Instruments digital light processing (DLP) module, as shown in Figure 3.1a. In the Texas Instruments DLP5500, there are 1024 × 768 aluminum ~10.8 × 10.8 μm mirrors that are actuated by a small electrostatic force. These DLPs are used in high-definition displays and projection systems. Electrostatic microactuators are individually controlled ensuring ~8000 Hz bandwidth. Images of the ADMP401 MEMS electrostatic microphone (100–15,000 Hz) are illustrated in Figure 3.1b. Using the MEMS technology, various electrostatic actuators and sensors can be fabricated, diced, bonded, and packaged as illustrated in Figure 3.1c.
Figure 3.1 (a) Texas Instrument DLP with ~1,000,000 electrostatic torsional micromirrors that are individually controlled; (b) ADMP401 MEMS microphones with −42 dBV sensitivity, 250 μA current consumption, 100 Hz to 15 kHz frequency response; (c) Rochester Institute of Technology MEMS transducers and devices on the silicon wafer.
Using the bulk and surface micromachining, the fabricated diced structures, sensors, electrostatic and electromagnetic actuators are reported in Figures 3.2. Figures 3.2 document images of various electromagnetic, variable-capacitance electrostatic transducers, and multifunctional sensors. All MEMS must be interfaced with electronics and packaged. For example, 15 μm gold wire–bonded packaged devices are illustrated in Figure 3.2b. These MEMS devices are interconnected with microelectronics to perform sensing and actuation. The Texas Instruments MEMS photodiode with on-chip transimpedance amplifier OPT101 and visible light sensors OPT3001DNPR are illustrated in Figure 3.2c. The physical quantities are measured. For example, for the micromachined actuators, documented in Figure 3.2e, the deflection of the diaphragm, see Figures 3.2d and e is measured by using the variations of resistances of four polysilicon resistors that form the Wheatstone bridge. By measuring the varying capacitance C(x), the displacement can also be measured. The energy stored in the electric field between two surfaces in capacitors is ${W}_{e}=\frac{1}{2}QV=\frac{1}{2}{CV}^{2}$ . The energy stored in the inductor is $\frac{1}{2}{Li}^{2}$ . The electrostatic energy variations, due to the varying capacitance C(x) or C(θ _{ r }), is the foundation of the device physics. The MEMS transducers are controlled by applying the voltage as depicted in Figure 3.2f. The electrostatic force and torque are
The physics of electromagnetic devices is based on the following principles:
Permanent-magnet electromechanical motion devices usually surpass induction and variable-reluctance devices. However, there are fundamental, technological, and market limits and constraints on permanent-magnet motion devices. Various variable-reluctance devices are used in many systems. In this section, we cover different radial and axial topologies for translational and rotational electrostatic variable-capacitance actuators and electromagnetic variable-reluctance transducers.
Figure 3.2 (a) Diced MEMS with coils and structures deposited on ~30 μm silicon diaphragm; (b) Wire-bonded packaged MEMS devices on evaluation boards for testing and characterization; (c) The Texas Instruments OPT101 monolithic photodiode with on-chip transimpedance amplifier and OPT3001DNPR visible light sensor; (d) Etched silicon structure and cross section of a ~35 μm silicon diaphragm; (e) Micromachined ~3 × 3 mm MEMS actuators with displacement sensors. The displacement is measured using the changes in four polysilicon resistors that form the Wheatstone bridge; (f) Micromachined electrostatic actuator with the suspended movable plate: voltage is applied to displace the top plate by the electrostatic force F e (x).
The following sequential steps ensure a consistent design flow:
Consider the translational and rotational electrostatic actuators. These affordable actuators are fabricated using cost-effective and high-yield micromachining, thin film, electroplating, and other technologies. The images of electrostatic MEMS are documented in Figures 3.1 and 3.2. We perform electrostatic and electromechanical analyses.
Consider the parallel-plate capacitor charged to a voltage V. The separation between two plates is x. The dielectric permittivity is ε. We neglect the fringing effect at the edges and assume that the electric field is uniform, such that E = V/x. The stored electrostatic energy is
where C(x) = ε(A/x); A is the effective area.
Using C(x) = ε(A/x), one finds the electrostatic force as a nonlinear function of the voltage applied V and displacement x
The resulting equations of motion are
where the air friction, elastic, and load (perturbation) forces F _{ air }, F _{ elastic }, and F _{ L } are device- and application-specific. For example, in vacuum, there is no air friction force on the movable suspended plate, cantilever, diaphragm, or membrane, and F _{ air } = 0.
Consider a movable 250 × 250 μm silicon plate with the thickness ~30 μm. Deposited aluminum thin films with the thickness ~0.5 μm on the movable plate and substrate form a parallel plate capacitor. The voltage 0 ≤ u ≤ u _{max} is applied to actuate a micromirror. The image of the micromachined electrostatic actuator is reported in Figure 3.3a. The repositioning from the equilibrium (x _{ e } = 0) to the final position must be accomplished with high bandwidth and minimal settling time. The top plate is suspended, and the separation between the capacitor surfaces is x _{0} = 10 μm. From $C\left(x\right)=\epsilon \frac{A}{x+{x}_{0}}$ , the electrostatic force is ${F}_{e}=\frac{1}{2}\epsilon A\frac{1}{(x+{x}_{0})}{\mu}^{2}$ . The experimental open-loop dynamics is reported in Figure 3.3b when the voltage pulses ~9 and 13.5 V are applied to ensure ~3 μm and 5 μm repositioning. The actuator repositions to the equilibrium x _{ e } = 0 as a result of the restoration of the elastic force when the applied voltage is u = 0. The electrostatic actuator ensures only one-directional active control capabilities. The settling time is ~0.002 sec.
Figure 3.3 (a) Electrostatic microactuator; (b) Open-loop system response: Dynamics of x(t) if the voltage pulses 9 and 13.5 V are applied to reposition actuator for ~3 and 5 μm.
The elastic and air friction force are modeled as F _{ air } = a _{1} v + a _{2} v|v| and F _{ elastic } = b _{1} x + b _{2} x|x|. Using the Newtonian mechanics m a = ΣF and F_{elastic} = b _{1} x + b _{2} x|x|. Using the Newtonian mechanics ma = ΣF and ${F}_{e}=\frac{1}{2}\epsilon A\frac{1}{{(x+{x}_{0})}^{2}}{\mu}^{2}$ , we have
The Simulink^{®} model and simulation results are reported in Figures 3.4. One concludes that the simulated dynamics corresponds to the experimental results.
We study rotational electrostatic transducers. These transducers are used only as the limited displacement angle devices. One must interconnect two conducting surfaces, including the rotating rotor. In actuators, the electromagnetic torque can be developed only in one direction. Figure 3.5 shows ~500 μm diameter limited-angle electrostatic micromachined actuators.
As the voltage V is applied to the parallel conducting rotor and stator surfaces, the charge is Q = CV, where C is the capacitance, C = ε(A/g) = ε(WL/g); A is the overlapping area of the plates, A = WL; W and L are the width and length of the plates; g is the air gap between the plates.
The energy associated with the electric potential is ${W}_{e}=\frac{1}{2}{CV}^{2}$ . The electrostatic force at each overlapping plate segment ${F}_{el}=\frac{\partial {W}_{e}}{\partial g}=-\frac{1}{2}\frac{\epsilon WL}{{g}^{2}}{V}^{2}$ is balanced by the opposite segment. We assume an ideal fabrication for which W, L, and g are the same for all conducting surfaces.
The electrostatic tangential force due to misalignment is ${F}_{t}=\frac{{\partial W}_{e}}{\partial y}=\frac{1}{2}\frac{\epsilon}{g}\frac{\partial \left(WL\right)}{\partial y}{V}^{2}$ , where y is the direction in which misalignment potentially may occur.
The capacitance of a cylindrical capacitor is found to derive the electrostatic torque. The voltage between the cylinders are obtained by integrating the electric field. The electric field at a distance r from a conducting cylinder has only a radial component, E _{ r } = ρ/2πɛr, where ρ is the linear charge density, and Q = ρL. The potential difference is $\mathrm{\Delta}V={V}_{a}-{V}_{b}={\int}_{a}^{b}\overrightarrow{E}\xb7d\overrightarrow{l}={\int}_{a}^{b}{E}_{r}\xb7dr=\frac{\rho}{2\pi \epsilon}\mathrm{ln}\frac{{r}_{2}}{{r}_{1}}$ , where r _{1} and r _{2} are the radii of the rotor and stator where the conducting plates are positioned. Thus, $C=\frac{Q}{\mathrm{\Delta}V}=\frac{2\pi \epsilon L}{\mathrm{ln}\frac{{r}_{2}}{{r}_{1}}}$ . The capacitance per unit length is $\frac{C}{L}=\frac{\rho}{\mathrm{\Delta}V}=\frac{2\pi \epsilon}{\mathrm{ln}\frac{{r}_{2}}{{r}_{1}}}$ .
Figure 3.4 (a) Simulink® model; (b) Open-loop system response x(t) if the voltage pulses 9 and 13.5 V are applied to reposition actuator.
Figure 3.5 Electrostatic actuators.
Using the stator–rotor conducting surfaces overlap, the capacitance is a function of the angular displacement $C\left({\theta}_{r}\right)=N\frac{2\pi \epsilon}{\mathrm{ln}\frac{{r}_{2}}{{r}_{1}}}{\theta}_{r}$ , where N is the number of overlapping stator–rotor surfaces. The electrostatic torque is ${T}_{e}=\frac{1}{2}\frac{\partial C\left({\theta}_{r}\right)}{{\partial \theta}_{r}}{V}^{2}=N\frac{\pi \epsilon}{\mathrm{ln}{r}_{2}-\mathrm{ln}{r}_{1}}{V}^{2}$ . Other expressions for C(θ _{ r }) can be found. Assuming that viscous friction is T _{ friction } = B _{ m }ω_{ r }, the torsional–mechanical equations of motion are
The actuator rotates in the specified direction if T _{ e } > (T _{ L } + T _{ friction }). The motor must develop the electrostatic torque ${T}_{e}=N\frac{\pi \epsilon}{\text{ln}\text{}{r}_{2}-\mathrm{ln}{r}_{1}}{V}^{2}$ higher than the rated or peak load torque. Estimating the load torque T _{ L } _{max} and assigning the desired acceleration capabilities yields T _{ e }. One evaluates the effect of N, r _{1}, r _{2}, and J. The motor sizing estimates can be found and the fabrication technologies (processes and materials) asserted. The applied voltage V is bounded. The fabrication technologies and processes affect the motor dimensions and parameters. For example, one may attempt to minimize the air gap to attain the minimal value of (r _{2} − r _{1}). By minimizing ln(r _{2}/r _{1}), one maximizes T _{ e }. The moment of inertia J can be minimized by reducing the rotor mass using cavities, polymers, etc. There are physical limits on the maximum V and E. The technologies affect and define materials, tolerance, r _{2}/r _{1} ratio, etc. Another critical issue is the need for a contact with the conducting rotor surfaces. This fact limits the application of rotational electrostatic actuators as limited-angle actuators.
Variable-reluctance electromagnetic devices are widely used. We consider the translational (solenoids, relays, electromagnets, and magnetic levitation systems) and rotational (variable-reluctance synchronous motors) electromagnetic devices. The varying reluctance results in the electromagnetic force and torque.
Solenoid and relay usually consist of a movable member (called plunger or rotor) and a stationary member [1–6]. High-permeability ferromagnetic materials are used. The windings are wound in a helical pattern. The electromagnetic force is developed due to the varying reluctance. The performance of variable-reluctance devices is defined by the magnetic system, materials, relative permeability, friction, etc. Solenoids and relay (electromagnet) with a movable member are shown in Figures 3.6a and b. Magnetic levitation systems with the suspended ferromagnetic ball are depicted in Figure 3.6c. The electromagnetic system is formed by stationary and movable members. When the voltage is applied to the winding, current flows in the winding, magnetic flux is produced within the flux linkages path, and the electromagnetic force is developed. The movable members (plunder and ball) move to minimize the reluctance. When the applied voltage becomes zero, the plunger resumes its equilibrium position due to the returning spring. The undesirable phenomena such as residual magnetism and friction must be minimized. Different materials for the central guide (nonmagnetic sleeve) and plunger coating (plating) are used to minimize friction and wear. Glass-filled nylon and brass (for the guide), copper, aluminum, tungsten, platinum, or other low-friction plunger coatings are used. Depending on the surface roughness and material composition, the friction coefficients of lubricated (solid film and oil) and unlubricated materials are tungsten on tungsten 0.04–0.1 and ~0.3; copper on copper ~0.04 to 0.1 and ~1.2; aluminum on aluminum 0.04–0.12 and ~1; platinum on platinum 0.04–0.25 and ~1.2; titanium on titanium 0.04–0.1 and ~0.6.
Figure 3.6 (a) Schematic and images of solenoids; (b) Schematic of a relay or electromagnet. The spring placement depends on device applications. Note: Helical springs are designed for compression and tension. The tension spring is designed to operate with a tension load, and the spring stretches as the force is applied. The compression spring operates with a compression load, and the spring contracts as the force is applied. In variable-reluctance devices, both springs usually have a helical coil, flat, or V-spring designs ensuring robustness, strength, and elasticity; (c) Schematics and image of a magnetic levitation system with the suspended ferromagnetic ball.
To analyze variable-reluctance devices, we apply laws of electromagnetics and mechanics. Consider the electromagnet reported in Figure 3.6b. The current i in N coils produces the flux Φ. Assume that the flux is constant. The displacement (the virtual displacement is dx) changes the magnetic energy stored in two air gaps. From ${W}_{m}=\frac{1}{2}{\int}_{v}\mu {\left|\overrightarrow{H}\right|}^{2}dv=\frac{1}{2}{\int}_{v}{\mu}^{-1}{\left|\overrightarrow{B}\right|}^{2}dv$ , we have ${dW}_{m}={dW}_{m\text{}airgap}=2\frac{{B}^{2}}{{2\mu}_{0}}A\text{\hspace{0.17em}}dx=\frac{{\mathrm{\Phi}}^{2}}{{\mu}_{0}A}dx$ , where A is the cross-sectional area, A = l _{ w } l _{ t }.
The flux Φ is constant if i = const. Hence, the increase in the air gap dx leads to increase in the stored magnetic energy. Using ${F}_{e}=-\nabla {W}_{m}=-\frac{{\partial W}_{m}}{\partial x}$ , one finds the electromagnetic force as ${\overrightarrow{F}}_{e}=-{\overrightarrow{a}}_{x}\frac{{\mathrm{\Phi}}^{2}}{{\mu}_{0}A}$ . The force tends to reduce the air gap length. That is, the reluctance is minimized. The movable member, for which the gravitational force is mg, is attached to the restoring spring.
The reluctances of the ferromagnetic materials of stationary and movable members are ${\Re}_{1}=\frac{{l}_{1}}{{\mu}_{0}{\mu}_{r1}A}$ and ${\Re}_{2}=\frac{{l}_{2}}{{\mu}_{0}{\mu}_{r2}A}$ . The total air gap reluctance with two air gaps in series is ${\Re}_{g}=\frac{2x}{{\mu}_{0}A}$ .
Using the fringing effect, the air gap reluctance is ${\Re}_{g}=\frac{2x}{{\mu}_{0}({k}_{g1}{l}_{w}{l}_{t}+{k}_{g2}{x}^{2})}$ , where k _{ g1} and k _{ g2} are the nonlinear functions of the ferromagnetic material, l _{ t }/l _{ w } ratio, B–H curve, electromagnetic load, etc.
The magnetizing inductance is $L\left(x\right)=\frac{{N}^{2}}{{\Re}_{total}\left(x\right)}=\frac{{N}^{2}}{{\Re}_{g}\left(x\right)+{\Re}_{1}+{\Re}_{2}}$ .
The electromagnetic force is ${F}_{e}=-\nabla {W}_{m}=-\frac{{\partial W}_{m}}{\partial x}=\frac{1}{2}{i}^{2}\frac{\partial}{\partial x}L\left(x\right)=-\frac{1}{2}{i}^{2}\frac{\partial}{\partial x}\left(\frac{{N}^{2}}{{\Re}_{g}\left(x\right)+{\Re}_{1}+{\Re}_{2}}\right)$ .
Using ℜ_{ total }(x) and L(x), one finds F _{ e } and emf. The governing differential equations result.
Example 3.1
Figure 3.7a documents a cross-sectional view of a variable-reluctance actuator (solenoid or relay) with N turns. The equivalent magnetic circuit with the reluctances is illustrated in Figure 3.7b. The source of the flux linkages in the ferromagnetic members and air gaps is the magnetomotive force (mmf), $mmf=Ni={\mathrm{\Sigma}}_{f}{H}_{j}{l}_{j}={H}_{1}{l}_{1}+{H}_{2}{l}_{2}+2{H}_{x}x,{\oint}_{l}\overrightarrow{H}\xb7d\overrightarrow{l}={\int}_{s}\overrightarrow{J}\xb7d\overrightarrow{s}$ .
The air gap (separation) between the stationary and movable members is x(t). The movable member evolves in x ∈[x _{min} x _{max}], x _{min} ≤ x ≤ x _{max}, x _{min} ≤ 0. The stroke is Δx = (x _{max} − x _{min}). The mean lengths of the stationary and movable members are l _{1} and l _{2}, and the cross-sectional area is A. One can find the force exerted on the movable member as a function of the current i(t) and displacement x(t). The permeabilities of stationary and movable members are μ _{ r1} and μ _{ r2}.
The electromagnetic force is ${F}_{e}=-\nabla {W}_{m}=-\frac{\partial {W}_{m}}{\partial x}$ , where ${W}_{m}=\frac{1}{2}{Li}^{2}$ .
The magnetizing inductance is $L=\frac{N\mathrm{\Phi}}{i\left(t\right)}=\frac{\psi}{i\left(t\right)}$ , where the magnetic flux is $\mathrm{\Phi}=\frac{Ni\left(t\right)}{{\Re}_{1}+{\Re}_{x}+{\Re}_{x}+{\Re}_{2}}$ . The reluctances of the ferromagnetic stationary and movable members ℜ_{1} and ℜ_{2} and the air gap reluctance ℜx are ${\Re}_{1}=\frac{{l}_{1}}{{\mu}_{0}{\mu}_{r1}A},{\Re}_{2}=\frac{{l}_{2}}{{\mu}_{0}{\mu}_{r2}A}$ , and ${\Re}_{x}=\frac{x\left(t\right)}{{\mu}_{0}A}$ .
Using reluctances, we find $\psi =N\mathrm{\Phi}=\frac{{N}^{2}i\left(t\right)}{\frac{{l}_{1}}{{\mu}_{0}{\mu}_{r1}A}+\frac{2x\left(t\right)}{{\mu}_{0}A}+\frac{{l}_{2}}{{\mu}_{0}{\mu}_{r2}A}}$ .
Figure 3.7 (a) Schematic of a variable-reluctance actuator (relay or solenoid). The plunger displaces in x ∈[x min x max], x min ≤ x ≤ x max, x min ≤ 0. The stroke is Δx = (x max − x min); (b) Equivalent magnetic circuit.
The magnetizing inductance is a nonlinear function of the displacement, and
From ${F}_{e}=-\nabla {W}_{m}=-\frac{\partial {W}_{m}}{\partial x}=-\frac{1}{2}\frac{\partial}{\partial x}L\left(x\left(t\right)\right){i}^{2}\left(t\right)$ , the one-directional electromagnetic force in the x direction is ${F}_{e}=\frac{{N}^{2}{\mu}_{0}{\mu}_{r1}^{2}{\mu}_{r2}^{2}A}{{({\mu}_{r2}{l}_{1}+{2\mu}_{r1}{\mu}_{r2}x+{\mu}_{r1}{l}_{2})}^{2}}{i}^{2}$ .
The voltage u(t) is applied changing i(t). We use the Kirchhoff voltage law $u=ri+\frac{d\psi}{dt}$ , where the flux linkage is ψ = L(x)i. From $u=ri+L\left(x\right)\frac{di}{dt}+i\frac{dL\left(x\right)}{dx}+i\frac{dL\left(x\right)}{dx}\frac{dx}{dt}$ , one finds $\frac{di}{dt}=\frac{1}{L\left(x\right)}[-ri-\frac{{2N}^{2}{\mu}_{0}{\mu}_{r1}^{2}{\mu}_{r2}^{2}A}{{({\mu}_{r2}{l}_{1}+2{\mu}_{r1}{\mu}_{r2}x+{\mu}_{r1}{l}_{2})}^{2}}iv+u]$ .
The force, acceleration, velocity, and displacement are vectors. The electromagnetic force F _{ e }, which is a vector, is developed to minimize the reluctance. If voltage u is applied, F _{ e } is developed to minimize the air gap x, and x _{min} ≤ 0. The tension spring exhibits the force when it stretches, and, using an ideal Hooke’s law we have F _{ spring } = k _{ s } x _{ spring }. At the steady state, F _{ e } = F _{ spring }. In differential equations, the movable member displacement is used as a variable x.
Consider a tension spring that exhibits zero force at zero-length when spring is relaxed. Using the plunger displacement x(t), x _{min} ≤ x ≤ x _{max} and the zero-length (x _{0} − x _{max}), we have F _{ spring } = k _{ s }(x _{0} − x), x _{0} = x _{max}. One has F _{ spring } _{max} = k _{ s }(x _{0} − x _{min}), x _{min} ≤ 0. At the zero-length (x _{0} − x _{max}), the spring is relaxed and exhibits zero force F _{ spring } _{min} = k _{ s }(x _{0} − x _{max}) = 0.
The motional emf opposes the voltage applied. We have a set of three differential equations
There are limits on the plunger displacement x _{min} ≤ x ≤ x _{max} and x ∈[x _{min} x _{min}], x _{min} ≤ 0. As the voltage u is applied, the plunger moves to the left minimizing the air gap until x becomes x _{min}, which is a mechanical limit. For u = 0 V, the return spring restores the plunger to the equilibrium position x _{max} at which F _{ spring } = k _{ s }(x _{0} − x) = 0.
Example 3.2
Figure 3.8a illustrates a solenoid with a stationary member and a movable plunger.
The magnetizing inductance is $L\left(x\right)=\frac{{N}^{2}}{{\Re}_{f}+{\Re}_{x}}=\frac{{N}^{2}{\mu}_{0}{\mu}_{r}{A}_{f}{A}_{x}}{{A}_{x}{l}_{f}+{A}_{f}{\mu}_{r}(x+2d)}$ , where ℜ_{ f } is the total reluctances of the ferromagnetic movable members (stator and plunger are made using the same electric steel grade with μ _{ r }); ℜx is reluctance of the effective air gap, which is (x + 2d) due to thickness d of the nonmagnetic sleeve; A _{ f } and A _{ x } are the cross section areas; l _{ f } is the equivalent lengths of the magnetic path in the ferromagnetic stationary member and plunger; d is the nonmagnetic sleeve thickness, which is usually ~1 μm. In general, l _{ f plunger } varies if x changes. However, l _{ f plunger }(x) variations do not affect the results due to high μ _{ r }. Hence, we let l _{ f } = const.
Figure 3.8 (a) Solenoid schematics; (b) Dynamics of x(t); (c) Plot for F e (x, i).
Assuming that the magnetic system is linear, the coenergy is ${W}_{c}(i,x)=\frac{1}{2}L\left(x\right){i}^{2}$ . The electromagnetic force is ${F}_{e}(i,x)=-\frac{\partial {W}_{c}(i,x)}{\partial x}=-\frac{1}{2}{i}^{2}\frac{dL\left(x\right)}{dx}$ , where $\frac{dL}{dx}=-\frac{{N}^{2}{\mu}_{0}{\mu}_{r}^{2}{{A}_{f}}^{2}{A}_{x}}{{[{A}_{x}{l}_{f}+{A}_{f}{\mu}_{r}(x+2d)]}^{2}}$
Using Kirchhoff’s law $u=ri+\frac{d\psi}{dt},\psi =L\left(x\right)i$ , one has $u=ri+L\left(x\right)\frac{di}{dt}+i\frac{dL\left(x\right)}{dx}\frac{dx}{dt}$ .
We apply Newton’s second law for translational motion $m\frac{{d}^{2}x}{{dt}^{2}}={F}_{e}-{B}_{v}\frac{dx}{dt}-{k}_{s}({x}_{0}-x)-{F}_{L}$ .
The resulting nonlinear differential equations are
The back emf opposes the applied voltage. The friction and spring forces act against the electromagnetic force.
Example 3.3
Consider a solenoid depicted in Figure 3.8a with the stroke 5 cm. Let the relative permeabilities of the stationary member and plunger be different, μ _{ rs } ≤ μ _{ rp }. We obtain the mathematical model and perform simulations with the following parameters: r = 8.5 ohm, L _{ l } = 0.001 H, N = 700, m = 0.095 kg, μ _{ rs } = 4500, μ _{ rp } = 5000, l _{ fp } = 0.055 m, l _{ fs } = 0.095 m, A _{ f } = A _{ x } = 0.00025 m^{2}, B _{ v } = 0.06 N-sec/m, k _{ s } = 10 N/m, and x _{0} = 0.05 m. The subscripts p and s stand for the plunger and stationary member.
Variations of l _{ fp }(x) result in very minor overall changes while complicating the equations and obscuring the expression for F _{ e }. The assumptions l _{ fp } = const and d = 0 result in error less than 0.1%. The magnetizing inductance is $L\left(x\right)=\frac{{N}^{2}}{{\Re}_{fs}+{\Re}_{fp}+{\Re}_{x}}=\frac{{N}^{2}}{\frac{{l}_{fs}}{{\mu}_{0}{\mu}_{rs}A}+\frac{{l}_{fp}}{{\mu}_{0}{\mu}_{rp}A}+\frac{x}{{\mu}_{0}A}}=\frac{{N}^{2}{\mu}_{0}{\mu}_{rs}{\mu}_{rp}A}{{\mu}_{rp}{l}_{fs}+{\mu}_{rs}{l}_{fp}+{\mu}_{rs}{\mu}_{rp}x}$
The electromagnetic force is ${F}_{e}=-\frac{\partial}{\partial x}\frac{1}{2}L\left(x\right){i}^{2}=\frac{{N}^{2}{\mu}_{0}{\mu}_{\mathrm{rs}}^{2}{\mu}_{\mathrm{rp}}^{2}A}{2({\mu}_{\mathrm{rp}}l\text{fs}\text{}\text{+}\text{}{\mu}_{\mathrm{rs}}l\text{fp}\text{+}\text{}{\mu}_{\mathrm{rs}}{\mu}_{\mathrm{rp}}x{)}^{2}\text{\hspace{0.17em}}}{i}^{2}$ . The electromagnetic force is developed to minimize the air gap. Using the leakage inductance L _{ l }, and applying an ideal Hooke’s law, one finds the following nonlinear differential equations
The displacement of the plunger x(t) is constrained by hard mechanical limits, and x ∈[0.0005 0.05] m. As the voltage u is applied, the plunger moves to the left, minimizing the air gap until x becomes 0.0005 m. The simulations are performed. The limits on x(t) can be set in the integrator, or using the Saturation block. For u = 0 V, the return spring restores the plunger to the left equilibrium position 0.05 m to the zero-length spring position at which F _{ spring } = k _{ s }(x _{0} − x) = 0. Figures 3.8b and c illustrate the evolution of x(t) as well as the three-dimensional plot F _{ e }(x, i) for x ∈[0.005 0.025] m and i ∈[0 2] A. In catalogs, the plots for F _{ e }(x) are reported for different currents or voltages. The statement is N=700; mu0=4*pi*1e−7; mus=4500; mup=5000; ls=0.095; lp=0.0275; A=2.5e−4; x=linspace(0.005,0.025,50); i=linspace(0,2,50); [X,Y]=meshgrid(x,i); Fe=(0.5*N^2*mu0*mup^2*mus^2*A*Y.*Y)./(lp*mus+ls*mup+mup*mus*X).^2 ; surf(x,i,Fe); xlabel('Displacement, {\itx} [m]’,’FontSize’,18); ylabel('Current, {\iti} [A]’,’FontSize’,18); zlabel('Electromagnetic Force, {\itF_e} [N]’,’FontSize’,18);
The equations for physical quantities and governing differential equations were found based on assumptions. Nonlinear magnetization, nonlinear magnetic system, secondary effects, cross section area variations, nonuniformity, and other effects were not considered or were simplified. Using the experimental data, one obtains consistent and high-fidelity models using the results reported. For example, the experimental L(x) and F _{ e }(x) can be measured.
Example 3.4
The magnetizing inductance L(x) and electromagnetic force F _{ e } can be experimentally measured. Inductance L(x) can be measured using the solenoid or relay as a transformer adding additional coils on the plunger. The electromagnetic force F _{ e } is measured by loading a device. The use of reluctances and coenergy result in the following expressions
The unknown coefficients a _{1}, b, and c were found using the solenoid parameters such as lengths, area, μ _{ r }, N, etc. For a solenoid, the experimental data for the measured F _{ e }(x) if i = 1 A is depicted in Figure 3.9a. The maximum plunger displacement is ~5 cm. We find the unknown constants a, b, and c for the following approximations of the electromagnetic force
The aforementioned expressions for F _{ e }(x) correspond to the device physics recalling that ${F}_{e}=\frac{\partial}{\partial x}\frac{1}{2}L\left(x\right){i}^{2}=\frac{a}{{(b+cx)}^{2}}{i}^{2}$ .
Figure 3.9 (a) Experimental data for the measured F e (x), i = 1 A; (b) Plots of the measured F e (x) (dots) and F e = a ( b + c x ) 2 i 2 , a = 52.77, b = 0.737, and c = 96.1 (solid line); (c) Plots of the measured F e (x) (dots) and F e = ae −bx i 2, a = 58.7, and b = 105 (solid line); (d) Plots of the measured F e (x) (dots) and F e = ae −bsincx i 2, a = 59.8, b = 6.31, and c = 17.2; (e) Plots of the measured F e (x) (dots) and approximation F e = a e − d x ( b + c x ) 2 i 2 , a = 62.5, b = 1, c = 27.1, and d = 64.3.
Example 3.5
Consider a solenoid if the measured electromagnetic force is F _{ e }(x) = ae ^{−bx } i ^{2}, 0 ≤ x ≤ 0.05, a > 0 and b > 0. Our goal is to derive the resulting equations of motion.
From ${F}_{e}(i,x)=-\frac{{\partial W}_{c}(i,x)}{\partial x}=-\frac{1}{2}{i}^{2}\frac{dL\left(x\right)}{dx}$ , we conclude $\frac{dL}{dx}=-{2ae}^{-bx}$ .
The integration gives $-\int 2{ae}^{-bx}\text{}dx=2\frac{a}{b}{e}^{-bx}$ . The positive-definite magnetizing inductance is $L\left(x\right)=2\frac{a}{b}{e}^{-bx}$ . Using Kirchhoff’s law, we have $u=ri+\frac{d\psi}{dt}=ri+L\left(x\right)\frac{di}{dt}+i\frac{dL\left(x\right)}{dx}\frac{dx}{dt}$ . Applying Newtonian mechanics, one obtains
We perform simulations using the parameters found in Example 3.4. In particular, a = 58.7, b = 105, r = 15 ohm, m = 0.1 kg, B _{ v } = 0.06 N-sec/m, and k _{ s } = 10 N/m. The Simulink model is developed as depicted in Figure 3.10a. The following parameters are uploaded >> a=58.7; b=105; r=15; m=0.1; x0=0.05; Bv=0.06; ks=10;
The plunger displacement is constrained, x _{min} ≤ x ≤ x _{max}, 0_{min} airgap _{} ≤ x ≤ 4 cm_{max stroke }. The voltage u is applied as steps $u=\{\begin{array}{cc}3& \text{V}\\ 2& \text{V\u2019}\end{array}$ , see Figure 3.10b. The displacement is limited, 0 ≤ x ≤ 0.04 m.
The dynamics of the current i(t), velocity v(t), and position x(t) are reported in Figures 3.10c and d for different initial plunger displacement x _{ t=0}. Consider the plunger at x _{ t=0} = 4 cm and x _{ t=0} = 0. As the voltage is applied, the F _{ e } is developed. As illustrated in Figures 3.10b and c, for the specified voltages $u=\{\begin{array}{cc}3& \text{V}\\ 2& \text{V\u2019}\end{array}$ , the plunger is repositioned to the equilibrium positions x _{ e } at which F _{ e } = F _{ spring }.
Figure 3.10 (a) Simulink® model; (b) Applied voltage pulses u = { 3 V 2 V . (c) Open-loop system dynamics for the current i(t), velocity v(t), and x(t) if x t = 0 = 0.04 m, 0 ≤ x ≤ 0.04 m; (d) Open-loop system dynamics for the current i(t), velocity v(t), and displacement x(t) if x t = 0 = 0.
Example 3.6
We derive the equations of motion for a solenoid with the magnetizing inductance $L\left(x\right)=\frac{{ae}^{-bx}}{(c+dx)}$ , where a, b, c, d > 0.
One applies ${F}_{e}(i,x)=\frac{\partial {W}_{c}(i,x)}{\partial x}=-\frac{1}{2}{i}^{2}\frac{dL\left(x\right)}{dx}$ . Using the quotient rule, for a function f/g, the derivative is (f′_{ g–g }′f)/g ^{2}. Hence, ${F}_{e}=\frac{1}{2}{ae}^{-bx}\frac{b(c+dx)+d}{{(c+dx)}^{2}}{i}^{2}$ .
The maximum electromagnetic force at x = 0 is
Using the Faraday law of inductance, the total emf is
where the motional emf is ${ae}^{-bx}\frac{b(c+dx)+d}{{(c+dx)}^{2}}iv$ .
The resulting nonlinear differential equations are
Consider a solenoid illustrated in Figures 3.6a and 3.11a. The equivalent magnetic circuit is reported in Figure 3.11b. One finds the reluctances of the stationary member ${\Re}_{fs}=\frac{{l}_{fs}}{{\mu}_{0}{\mu}_{r}{A}_{1}}$ , the stationary member which faces the plunger ${\Re}_{fp}=\frac{{l}_{fp}}{{\mu}_{0}{\mu}_{r}{A}_{2}}$ , the air gap ${\Re}_{x}=\frac{x}{{\mu}_{0}{A}_{2}}$ , and the plunger ${\Re}_{fp}=\frac{{l}_{fp}}{{\mu}_{0}{\mu}_{r}{A}_{2}}$ .
The equivalent magnetic circuit, as depicted in Figure 3.11b, yields
From Ni = ℛ_{ fs }(Φ_{1} + Φ_{2}) + 2(Φ_{ fsp } + ℛx + ℛ_{ fp })Φ_{3}, using Φ_{1} + Φ_{2} = Φ_{3}, we obtain Ni = (ℛ_{ fs } + 2ℛ_{ fsp } + 2ℛ_{ x } + 2ℛ_{ fp })Φ_{3}. The magnetic flux Φ_{3} and flux linkages ψ = NΦ_{3} are
The magnetizing inductance is $L\left(x\right)=\frac{{N}^{2}}{{\Re}_{fs}+{2\Re}_{fsp}+{2\Re}_{x}+{2\Re}_{fp}}=\frac{{N}^{2}{\mu}_{0}{\mu}_{r}{A}_{1}{A}_{2}}{{I}_{fs}{A}_{2}+{2l}_{fsp}{A}_{1}+{2A}_{1}{\mu}_{r}x+{2l}_{fp}{A}_{1}}$ . Using the coenergy, the electromagnetic force is ${F}_{e}=-\frac{\partial}{\partial x}\frac{1}{2}L\left(x\right){i}^{2}=\frac{{N}^{2}{\mu}_{0}{\mu}_{r}^{2}{A}_{1}^{2}{A}_{2}}{{({l}_{fs}{A}_{2}+{2l}_{fsp}{A}_{1}+{2A}_{1}{\mu}_{r}x+{2l}_{fp}{A}_{a})}^{2}}{i}^{2}$ .
The Kirchhoff voltage law and Newton law yield
Figure 3.11 (a) Solenoid schematics; (b) Equivalent magnetic circuit for a solenoid; (c) The force-displacement characteristics of the Ledex B11M-254 solenoid when u max is 12, 17, 24, and 38 V for 100%, 50%, 25%, and 10% duty cycle operations, r = 17.3 ohm, and 0 ≤ x ≤ 2.2 cm.
Numerical analysis can be performed using the model developed and by simulating the solenoid. The experimental studies are of great interest. We examine a Ledex B11M-254 solenoid, a driving circuit, and a proportional–integral control law. For a “pull” operation solenoid, the parameters are as follows: u _{max} = 12, 17, 24 and 38 V (at 100%, 50%, 25%, and 10% duty cycle operation), 15.5 N holding force, A-class coil insulation, 105°C maximum temperature, the plunger weight is 17 g. The F _{ e }(x) characteristics are depicted in Figure 3.11c. The maximum stroke is 2.2 cm. The solenoid parameters are r = 17.3 ohm, L _{ self } = 0.0064 H, L _{ l } = 0.001 H, N = 1780, m = 0.017 kg, μ _{ rs } = μ _{ rp } = 5500, l _{ fp } = 0.02 m, l _{ fs } = 0.048 m, l _{ fsp } = 0.08 m, A _{ f } = A _{ x } = 2 × 10^{−4} m^{2}, B _{ v } = 0.25 N-sec/m, and k _{ s } = 5 N/m.
A closed-loop system includes solenoid, a position sensor, a one-quadrant PWM regulator, filters, and a proportional–integral controller. The solenoid displacement x(t) is measured and compared with the reference displacement r(t) to obtain the tracking error e(t) = r(t) − x(t). Using pulse width modulation (PWM), we control the duty cycle of the MOSFET transistor, thereby changing the average voltage u(t) applied to a winding. The control voltage u _{ c }(t) is compared with a periodic (triangular or sinusoidal) signal u _{ t }. The electronics schematics and images of the system are depicted in Figures 3.12. Notations, definitions, subcircuitry, components, and signals are labeled and accentuated correspondingly.
The first component of the circuit is the error circuit to obtain e(t). The error circuit adds the reference voltage signal r(t) with the inverted linear potentiometers output signal which corresponds to the measured plunger displacement x(t). The error circuit is implemented using a unit-gain instrumentation amplifier INA128. The tracking error e(t) = r(t) − x(t) is fed into the controller circuit, which implements a proportional-integral control law u _{ c } = k _{ p } e + k _{ i }∫ edt, where k _{ p } and k _{ i } are the proportional and integral feedback gains. The proportional term k _{ p } e is implemented by using an operational amplifier TLC277 in an inverting configuration. The proportional gain k _{ p } is realized by using the input and feedback resistors, k _{ p } = −R _{ P2}/R _{ P1}. To implement the integral feedback k _{ i } ∫ edt, an integrator is implemented using an inverted operational amplifier with an input resistor R _{ I } and a feedback capacitor C _{ I }. The integral feedback gain is k _{ i } = −1/R _{ I } C _{ I }. Due to the inverting configuration for the proportional and integral terms, the outputs are inverted and summed to yield the control voltage u _{ c }. To perform the inversion and summation, instrumentation amplifiers are used due to their robustness, tolerance, low noise, linearity, etc. By fixing the positive input of the instrumentation amplifier to ground and feeding the respective signal to the negative input, a simple inverter is implemented. By feeding the output of an inverting amplifier to the reference node of the following circuitry, a summing circuit is implemented.
Figure 3.12 (a) Closed-loop system: Solenoid with power electronics, sensor, controller, and filters; (b) Images of closed-loop solenoid hardware.
The control signal u _{ c } is supplied to a comparator that compares two inputs to produce an output, see Figure 3.12a. The comparator is implemented using an operational amplifier with the control signal u _{ c } supplied to its inverting terminal and a periodic near-triangular waveform u _{ t } applied to its noninverting terminal. The comparator outputs a positive rail voltage V _{ cc } for the duration of time when its positive input is greater than its negative input. The comparator will output −V _{ cc } when its negative input is greater than its positive input. The comparator develops the PWM waveform changing the duty cycle of a square wave, whose amplitude is equal to the comparators rail, with respect to control signal u _{ c } = k _{ p } e + k _{ i }∫ edt. The periodic triangular waveform is established by a function generator.
As depicted in Figure 3.12a, the first LM324 operational amplifier in the function generator circuit is the comparator. The positive input of the comparator is the output of the second operational amplifier in the circuit, which is an integrator. The oscillation of the comparator produces a square waveform, which is integrated by the second operational amplifier with a capacitor in the negative feedback path to produce a near-triangular waveform u _{ t }. The frequency of u _{ t } is determined by the time constant defined by the input resistance and feedback capacitance of the integrator. The remaining two LM324 operational amplifiers in the function generator circuit are the buffer/attenuator amplifier for u _{ t } and an adjustable DC offset. For the reported circuitry, the maximum frequency of robust operation is ~6.8 kHz.
Various waveform generating and switching circuits can be used. The oscillating frequency of the n-stage ring pulse-oscillator is $f=\frac{1}{n({\tau}_{1}+{\tau}_{2})}$ , where τ_{1} and τ_{2} are the intrinsic propagation delays; n is the number of inverters (n is odd). The oscillators, wave generation and shaping circuit design using a Schmitt trigger, the Wien bridge oscillator, the Colpitts oscillator, the Hartley oscillator, the RC phase-shift concept, and other solutions are well known and can be used.
The PWM waveform, produced by comparing the control signal u _{ c } and near-triangular waveform u _{ t }, allows one to control the average voltage u supplied to the solenoid. The TLC277 operational amplifier that produces the PWM waveform can output a maximum current and voltage of ~30 mA and 18 V. For the studied solenoid, the rated current and voltage are ~1 A and up to ~40 V. The TLC277 comparator cannot ensure the necessary current. We use a one-quadrant power electronics stage, as reported in Figure 3.12a. A power MOSFET switch is controlled by applying the voltage to a gate. The comparator output is connected to the MOSFET gate to control the switching activity. A low pass filter is implemented using an inductor L _{ F } and capacitor C _{ F }. The filter corner frequency ${f}_{c}=\frac{1}{2\pi \sqrt{{L}_{F}{C}_{F}}}$ . Let f _{ c } = 3.4 kHz, with L _{ F } = 500 μH, and C _{ F } = 4.7 μF. The plunger of the solenoid is connected to a linear potentiometer to measure displacement x(t). One finds e(t). A second-order low pass filter is implemented to attenuate the noise. The low pass filter is documented in Figure 3.12a between the potentiometer and error circuit input. Our goal is to ensure a high signal-to-noise ratio within the operating solenoid bandwidth. For an unity-gain filter, the cutoff frequency is ${f}_{c}=\frac{1}{2\pi {R}_{f}{C}_{f}}$ . To ensure f _{ c } = 3 Hz, R _{ f } = 1 Mohm and C _{ f } = 47 nF.
The control voltage u _{ c } (output of the proportional–integral controller) and the near-triangular signal u _{ t } are illustrated in Figure 3.13a. These signals are compared by the comparator, and a PWN signal drives a MOSFET. The MOSFET gate voltage and the voltage applied to the solenoid are shown in Figure 3.13b.
The transient dynamics for the closed-loop system are reported in Figure 3.14. The comparison of the reference r(t) and plunger displacements x(t) provide an evidence that: (1) fast nonoscillatory repositioning is accomplished; (2) stability is guaranteed; (3) the steady-state tracking error is zero within the positioning sensor accuracy; (4) the disturbances are attenuated (F _{ L } is applied when x reaches the steady-state value at the second bottom plot for x); (5) robustness to parameter variations is accomplished. One-directional control of the displacement x(t) is accomplished. The returning spring restores the plunger position at the equilibrium if u = 0. Consistent analysis and design are accomplished and verified using fundamental results, circuits design, and experimental studies.
Figure 3.13 (a) Near-triangular waveform u t and control voltage u c for three reference r plunger positions: r = 0.5, r = 1, and r = 1.5 cm; (b) MOSFET gate voltage and voltage applied to a solenoid winding u when r = 0.5 cm (3.2 and 2.6 V), r = 1 cm (8.4 and 6.8 V), and r = 1.5 cm (12.2 and 9.8 V).
We examine the radial topology limited-angle reluctance actuators. The synchronous reluctance motors are examined in detail in Section 6.2. Our goal is to study and analyze these transducers, which are used in many applications, such as limited-angle rotational actuators, relays, etc. The considered problem is relevant to variable-reluctance stepper motors. A single-phase four-pole synchronous reluctance actuator (rotational relay) is illustrated in Figure 3.15a. The path for ψ_{ as } is illustrated. If rotor rotates, the positive-definite reluctance ℜ (θ_{ r }) > 0 varies with the period π/2, ℛ_{min} ≤ ℛ(θ_{ r }) ≤ ℛ_{max}. The magnetizing inductance L(θ _{ r }) > 0 is a periodic function with the period π/2, and L _{min} ≤ L(θ _{ r }) ≤ L _{max}. The variations of inductance L _{ m }(θ _{ r }) is studied in Section 6.2 using the quadrature and direct magnetic axes. Having found the quadrature (q) and direct (d) axes reluctances, ℛ_{ mq } and ℛ_{ md }, ℛ_{ mq } > ℛ_{ md }, the inductances are ℛ_{ mq } and L _{ md } = 1/ℛ_{ md }. For the direct axis, reported in Figure 3.15a, the reluctance ℛ_{ md } is minimum because the air gap is minimum; The reluctance in the quadrature axis ℛ_{ md } is maximum due to the maximum air gap. Using the magnetizing inductances L _{ mq } and L _{ md }, ${\overline{L}}_{m}=\frac{1}{2}({L}_{mq}+{L}_{md})$ and ${L}_{\mathrm{\Delta}m}=\frac{1}{2}({L}_{md}-{L}_{mq})$ .
Let the stator and rotor magnetic system and geometry be designed such that ${L}_{m}\left({\theta}_{r}\right)={\overline{L}}_{m}-{L}_{\mathrm{\Delta}m}{\text{cos}}^{3}({40}_{r}-\frac{\pi}{16})$ , where ${\overline{L}}_{m}$ , is the average inductance; L _{Δm } is the magnitude of inductance variations.
Using the coenergy ${W}_{c}({i}_{as},{\theta}_{r})=\frac{1}{2}({\overline{L}}_{m}-{L}_{\mathrm{\Delta}m}{\text{cos}}^{3}({40\theta}_{r}-\frac{\pi}{16})){i}_{as}^{2}$ , the electromagnetic torque T _{ e }, developed by a single-phase reluctance actuator, is
Figure 3.14 Reference inputs r (top plots, r = 1.25 and 2.2 cm) and the plunger displacement x (bottom plots) with the plunger at its initial state (x = 2.2 cm, u = 0 V). The bottom figure reports the maximum stroke repositioning.
By applying a DC voltage u _{ as }, one has a DC current i _{ as }. Within a limited angle θ _{ r } _{min} ≤ θ _{ r } ≤ θ _{ r max}, the electromagnetic torque is developed. The variable-reluctance actuator (relay) rotates to minimize the reluctance path. The reluctance ℛ(θ_{ r }) is minimum when the rotor assumes a position when the stator–rotor air gap is minimum. We apply the Kirchhoff law ${u}_{as}={r}_{as}{i}_{as}+\frac{{d\psi}_{as}}{dt}$ , where the total emf is
The Newton law for rotational motion is $\frac{{d\omega}_{r}}{dt}=\frac{1}{J}({T}_{e}-{T}_{friction}-{T}_{sprong}-{T}_{L}),\frac{{d\theta}_{r}}{dt}={\omega}_{r}$ .
Figure 3.15 (a) Radial topology limited-angle reluctance actuator with a torsional spring; (b) Simulink® model to simulate a limited-angle reluctance actuator; (c) Open-loop dynamics for i(t), ωr(t) and θ r (t) if u a s = { 0 5 V , f = 0.25 Hz , θ r 0 = − 0.25 rad .
One obtains a set of three first-order nonlinear differential equations
In this chapter, electrostatic and variable-reluctance electromechanical motion devices were covered. We examined basic physics and functionality of electromechanical transducers. Modeling, simulation, and analysis were accomplished. The use of MATLAB and experimental data in evaluation of performance and capabilities were reported.
Figure 3.16 (a) Electrostatic actuator; (b) Magnetic levitation system.