Patent Grant
9024487
The present disclosure is related to electro-magnetic devices used to create forces, including levitation and alignment forces. In particular, the present disclosure is related to conserving, storing, and/or harvesting energy by translational movement of magnetic arrays proximal to switchable inductive elements.
Levitation is the process by which an object is suspended without physical contact. There are a number of types of levitation, including static levitation in the absence of motion of the suspended object, as well as dynamic levitation in which the suspended object undergoes some form of movement, such as rotational or translational movement.
Levitation can be achieved by applying a levitation force between an electromagnetic device and one or more magnetic materials, such as paramagnetic materials, diamagnetic materials, or superconducting materials. Dynamic levitation forces and other forces, such as alignment forces, can improve the efficiency and speed of the transportation of materials and people (e.g., via levitating trains or vehicles).
Conventional levitation systems can employ a moving magnetic array translating above immobile inductive elements. U.S. Pat. Nos. 5,722,326, 6,629,503, 6,633,217, 6,664,880, 6,758,146, 7,096,794, and 7,478,598 disclose one such system referred to as “Inductrack.” One drawback to the Inductrack system is that the magnetic arrays in this system induce favorable and disfavorable directions of current flow in the inductive elements. Current that flows in the disfavorable direction produces magnetic forces that counteract levitation forces. This is especially the case at low translational speeds. Accordingly, there remains a need for improved levitational systems.
Specific details of several embodiments of magnetic levitation systems are described herein along with related methods, devices, and systems. The terms “magnetic levitation system” and “magnetic levitation device” are used interchangeably herein to refer to any of a myriad of public, private, commercial, consumer, or recreational systems or devices that employ magnetic levitation. For example, a magnetic levitation system can include a public transportation system (e.g., a passenger train) or a freight delivery system. A magnetic levitation system or device can also include a car, bus, boat, or any other suitable vehicle that carries a person, animal, and/or material. Further, a magnetic levitation system or device can include recreational equipment, safety equipment, and sports equipment, to name a few examples. Such equipment can include shoes, skis, snowboards, bicycles, skates, or other suitable equipment. In some embodiments, this equipment can be worn by or attached to a user of the equipment. In other embodiments, the equipment does not require user intervention. A person skilled in the relevant art will understand that the present technology may have additional embodiments and that the technology may be practiced without several of the details of the embodiments described below with reference to FIGS. 1A-C87.
As illustrated, the units 82 are carried by a support structure 84 that includes supports 86 carrying a rail 88. The support structure 84 can be a material (such as concrete, wood, plastic, or fiberglass) that has low electrical conductivity, and high magnetic permeability. As described in further detail below, to allow a high magnetic permeability, the supporting structure 84 may have air-gaps (not shown) between individual SIEs or within the interior areas of the SIEs 14. A person skilled in the art will appreciate that the support structure 84 can have any of a myriad of configurations. As shown, the SIEs 14 can be located within the rail 88 of the support structure 84. In another embodiment, the SIEs 14 can be located at the unit 82 and the magnetic component 10 can be located at the rail 88 of the support structure 84. Also, while not visible in
The individual magnetic components 10 include an array of magnetic elements (not shown). As described in more detail below, the magnetic elements have differently oriented magnetic fields that are in magnetic communication with the SIEs 14.
The individual SIEs 14 provide levitation forces through magnetic communication with the magnetic fields of the magnetic components 10. As described in further detail below, the individual SIEs 14 can include a looped circuit (shown in later Figures) or other suitable inductive features that produce components of the levitation force. In accordance with the present technology, the SIEs 14 also include a switch element (shown in Later Figures) for controlling current flow in the SIE 14 based on translational motion between the SIEs 14 and the magnetic components 10.
Translational motion (and the conversion or storage thereof) can be provided by any of a variety of sources. In certain embodiments, a turbine or propeller can provide a translational force without requiring mechanical contact with the support structure 84. Additionally or alternatively, a wheel, gear, or other suitable mechanical component (driven, e.g., by an engine, motor, or actuator) can provide a lateral force by temporarily or periodically engaging the rail 88 of the support structure 84. In some embodiments, a wind force, a gravity force, and/or a hydraulic force provide the translational force or a portion of the translational force. The SIEs 14 can convert kinetic energy from these and other sources. For example, the SIEs 14 can convert energy from the motion provided by motorized or engine-powered locomotion (e.g., trains, subways, automobiles, buses, trucks, and motorcycles) or human or animal powered locomotion (e.g., bicycles) or forces of nature (e.g., gravity powered sleds, snowboards, skis, and/or various other sports and recreational devices). In these various applications, the available translational power available for conversion to electrical power can be in the range of several watts to many megawatts.
Although translational motion is described herein in the context of movement of a magnetic component relative to a stationary SIEs, in some embodiments, translational movement can include movement of SIEs relative to a stationary magnetic component. In some embodiments, both the magnetic components and the SIEs can undergo translational motion with respect to a common stationary location. In other embodiments, the magnetic components and the SIEs can be rotated. Also, in certain embodiments, magnetic components and SIEs can be positioned on a sloped or curved surface. In particular embodiments, the magnetic components and SIEs can be in relative movement on a sloped or curved surface.
As shown in
For purposes of clarity and to facilitate discussion of their various properties, the first through fourth magnetic elements 102-105 are shown as generally cubic in shape having a first length m1 in the x-axis direction, a second length m2 in the y-axis direction, and a third length m3 in the z-axis direction. In other embodiments, the first through fourth magnetic elements 102-105 can have other dimensions and/or shapes (e.g., wedge or circular), can be spaced apart from one another, and/or can include additional or alternative magnetic elements with field oriented differently than shown or described herein.
Referring to
The individual switch elements 118 can include, for example, a magnetic switch, an inductive switch, an optical switch or other suitable gating device that can be controlled to complete or interrupt a circuit path or otherwise regulate current flow. Although shown in the Figures as containing two switch elements, an SIE can include more or fewer than two switch elements. For example, a single switch element can be aligned with the center of the second portion 131b of the inductive element 130. In some embodiments, multiple switch elements can align and maintain the SIE in desired orientation during operation. For example, the first and second switch elements 118a and 118b can align the second portion 131b of the magnetic element 130 with the y-axis of
The switch elements 118 may operate in any of a variety of suitable ways. In one embodiment, the magnetic array 111 (
In some embodiments, the switch elements 118 can be commonly coupled to the inductive elements of two or more SIEs. For example, every fourth SIE 114 (e.g., each of the SIEs 114 below the fourth magnetic element 105 shown in
In general terms, the switch elements of the various embodiments of the individual SIEs described herein provide for efficient “harvesting” of the kinetic energy associated with the translational movement of the magnetic array (or translational movement of the SIE in alternate embodiments). In particular, the switch elements control current flow (or voltage differentials) in the inductive element (e.g., the inductive element 130) such that the magnetic field variations in the vicinity of the SIEs induce an electrical current that provides favorable magnetic field responses (or forces). For example, switch elements can be set ON (or ramp up current flow) when the induced current provides a favorable force component for levitation, alignment, and/or other purposes. Alternatively, the switch elements can be set OFF (or ramp down current flow) when the induced current would provide a disfavorable force component. For example, referring to the SIE 114, the switch elements 118 can be set OFF when the induced current would cause an attractive force between the SIE 114 and one or more magnetic elements adjacent the SIE 114. Accordingly, the SIE 114 operates to provide levitation forces that reduce friction and thus reduce energy required to provide translational motion. In various other embodiments, the SIEs can have different configurations for providing various other forces in addition to the levitation forces. In addition, as used herein the term “ON” can refer to a state in which current flow is immediately initiated in the SIE; however, the term can also refer to a state where the current flow is gradually increased. Similarly, as used herein the term “OFF” can refer to a state in which current flow is immediately ceased in the SIE; however, the term can also refer to a state where the current is gradually decreased.
As discussed above, SIEs can also store kinetic energy in electrical form and use the stored energy for purposes other than for providing levitation or alignment forces, such as for purposes that include powering components (e.g., electrical motors, lights, control systems, etc.) For example, as shown in
Referring again to
Referring again to
As discussed above, the switch elements control the induced current I1. The switch elements 118 can be switched ON (e.g., automatically or via a controller circuit or component) at appropriately favorable times when the current flow in the inductive elements induces magnetic fields that allow for intended levitation forces. The switch elements can be switched OFF (e.g., automatically or via a controller circuit or component) in circumstances when the induced current I1 would have induced magnetic fields that are not favorable for intended levitation.
Referring again to
In some embodiments, the switch elements can be set ON or OFF based on the velocity of the translational motion between the magnetic array and the SIE 114 (e.g., automatically or via a controller circuit or component). In particular, operation at high velocities can provide fewer disfavorable induced current in the SIE.
Example Section A (below) provides further electromagnetic and force analyses of the magnetic array moving over the SIE, including a determination of approximation formulas for the amounts of inductive current in the SIE 114, the induced magnetic flux 128 (
As shown in
Similar to the SIE 114 described above, the SIE 214 can have a variety of shapes and configurations (including coils or laminate structures) and can include any of a variety of materials, including aluminum, copper, or other suitable materials. As shown, the SIE 214 can be L-shaped and have the first inductive element 230a in a plane aligned with the x-y plane and the second inductive element 230b in a plane aligned with y-z plane. However, the first and second inductive elements 230a and 230b can be positioned in different planes, including planes that are non-parallel and not orthogonal. As illustrated, the first and second inductive elements 230a and 230b have a rectilinear shape; however, in other embodiments, the first and second inductive elements 230a and 230b can have different shapes. For purposes of clarity and to facilitate discussion of its various properties, the first inductive element 230a is referenced with the dimensions d0-d2 described above with reference to the first inductive element 130 of the SIE 114 (and defining a first air gap 232a of the SIE 214) and the second inductive element 230b is referenced with these dimensions and a dimension d4 in the z-axis (which define a second air gap 232b of the SIE 214). A person skilled in the art will appreciate, however, that the SIE 214 can have other dimensions.
As shown in
Referring to
In operation, the first inductive element 230a uses the induced electrical current to create magnetic fields (and induced magnetic flux 128) that provides the levitation force 129. The second inductive element 230b uses the induced electrical current to create magnetic fields (and induced magnetic flux 228) that provide alignment forces (shown by an arrow 135). For example, the second inductive element 230b can generate an alignment force when at a position where there is considerable external magnetic flux in the right-to-left horizontal direction along the positive x-axis (although the internal flux of a magnetic element may be in the opposite direction, left-to-right, the second inductive element 230b is well below that magnet's internal flux). As discussed in Example Section B, the horizontal alignment forces produced can have considerable magnitude even at low velocities. This alignment force induces an alignment torque force which increases as tan(θ), where θ is the angle of deviation of the magnetic array 111 from the x-axis.
As illustrated in
Example Section B (below) provides further electromagnetic and force analysis of the magnetic array moving over the SIE 214, including determination of approximation formulas for the amounts of inductive current in the SIE, induced magnetic flux 128, and induced drag forces, levitation forces, and horizontal alignment forces along the x-axis. Again, these formulas depend on the physical dimensions, material properties (including the magnet's magnetic flux density, the SIE's electrical resistivity, magnetic permeability of the air-gaps 232a and 232b of the SIE 214), as well as the speed of the magnetic array. The analysis indicates that considerable levitation and horizontal alignment forces along the x-axis are produced even at low to moderate velocities.
Referring to
In general, the magnetic elements of the magnetic arrays 311 and 411 can be rectilinearly shaped, and assembled to form rectilinear arrays. As discussed above, however, magnetic arrays and magnetic elements within the arrays can have any of a variety of shapes, arrangements, or configurations. In various embodiments, 2D magnetic arrays can provides a 2D arrangement of magnetic flux. Although shown as having a specific orientation in the Figures, a person skilled in the art will appreciate that magnetic patterns can depend on the particular configuration of magnetic elements of a 2D magnetic array. For example, either of the magnetic arrays 311 and 411 can include additional or alternative magnetic elements with flux directed at other angles, for example at 45 degrees to the x-, y-, and/or z-axis.
Referring to
Referring to
In many aspects, the inductive current induced in the SIE 714s can have a waveform similar to the current I1 described with reference to
Referring to
Referring to
In some embodiments, the first and second inductive elements 1145 and 1146 may be powered by external electrical power, or may be powered by external induction, or may be powered by inductive effects from relative movement of the EM arrays 1111 with respect to individual SIEs (shown, for example, as SIEs 114 in
As shown in
In operation, a current within the EM Array 1111 is initially present and induces a magnetic flux within the first inductive element 1145. (the current within the EM Array 1111 will diminish due to resistive loss, unless it is supplemented by some source of power, which can power harvested from the from relative movement of the EM Array 1111 with respect to the SIEs 114. The magnetic flux of the second inductive element 1146 then induces a current within the SIEs 114, from relative movement of the EM Array 1111 with respect to the SIEs 114 (as discussed above). This current within the SIEs induces a magnetic flux from the SIEs. The induced magnetic flux of the SIEs in turn induces, from relative movement of the EM Arrays 1111 with respect to SIEs, an inductive current within the first inductive element 1145 of the individual EM Arrays 1111. Hence, ultimately the current and induced flux within both the SIEs and the individual EM Arrays 1111 are induced from the relative movement of the EM Arrays 1111 with respect to SIEs.
As discussed above, an inductive device can include a controller communicatively coupled to an inductive component, including an individual SIE and the switches of an SIE.
The controller 1280 can be a computer-based controller that includes hardware and software for executing computer-based instructions. Accordingly, certain embodiments may take the form of computer-executable instructions, including routines executed by a processor or programmable computer. Accordingly, representative examples of the controller 1280 can be housed in a single unit or distributed over multiple interconnected units, e.g., through a communication network. The components of the controller 1280 can accordingly include local and/or remote memory storage devices and any of a wide variety of suitable computer-readable media. In some embodiments, the controller 1280 can include computers and/or other data processors, such as desktop computers, laptop computers, Internet appliances, or hand-held devices.
From the foregoing, it will be appreciated that specific embodiments of the technology have been described herein for purposes of illustration, but that various modifications may be made without deviating from the technology. For example, although particular embodiments were described above with respect to particular planes in an x-, y-, z-axis and/or as having a vertical or horizontal orientations, in other embodiments, similar components can have other orientations. In some embodiments the switch elements operate autonomously so as to open and close in direct response to the relative movement between a magnetic element and an inductive component. In other embodiments, a controller directs the operation of the switch elements based on the foregoing motion. In addition, certain aspects of the technology described in the context of particular embodiments may be combined or eliminated in other embodiments. SIEs and magnetic arrays described in the context of particular embodiments may be combined with other SIEs and magnetic arrays devices in other embodiments. Further, while advantages associated with certain embodiments of the technology have been described in the context of those embodiments, other embodiments may also exhibit such advantages, and not all embodiments need necessarily exhibit such advantages to fall within the scope of the technology. Accordingly, the disclosure and associated technology can encompass other embodiments not expressly shown or described herein.
This Appendix provides an electromagnetic and force analysis of a representative magnetic array moving over a representative levitation-inducing SIE (“L-SIE”), including determination of approximation formulas for the amounts of inductive current in the L-SIE, induced magnetic flux, and induced drag forces and levitation forces. These formulas depend on the L-SIE's and HA's physical dimensions, their material properties (including the magnet's magnetic flux density, the L-SIE's electrical resistivity, magnetic permeability of the media within the air-gap of the L-SIE), as well as the speed of the HA. The analysis indicates that considerable levitation forces along the z-axis are produced even at low to moderate velocities.
Throughout the analysis, certain assumptions and approximations are made. Such assumptions and approximations may be appropriate for some but not all of the embodiments of the technology disclosed herein. Accordingly, at least some embodiments of the disclosed technology may not fit within or be fully characterized by the following analysis.
The HA is aligned parallel to the x-axis and is composed of a series of magnets with internal magnetic orientation alternatively vertical and horizontal, as indicated in the various figures. The magnetic flux density is the amount of magnetic flux through a unit area taken perpendicular to the direction of the magnetic flux, gauged in units of Tesla.
For purposes of analysis assume the magnets of the HA with vertically aligned internal magnetic flux have near their lower surfaces external magnetic flux density B directed vertically along the z-axis, but have external magnetic flux density close to 0 elsewhere. For purposes of analysis assume the magnets of the HA with horizontally aligned internal magnetic flux have near their surfaces external magnetic flux density 0, but somewhat below their lower surfaces there is external magnetic flux density B directed horizontally along the x-axis.
For purposes of analysis will to the size dimensions of the magnetic elements of the magnetic array of
The levitation-inducing L-SIE will be assumed to be composed of conductive nonmagnetic material (e.g., copper, aluminum or similar material) with electrical resistivity ρ in SI units of ohm meter (Ωm). For purposes of analysis refer to the dimensions of the L-SIE shown in
The air-gap of the of the L-SIE's coil is the rectilinear region of size d′1×d′2×d3 enclosed by the L-SIE, where
The inductive element of the L-SIE is the rectangular section of size d0×d′2×d3 that borders left side of the air-gap of the L-SIE and is parallel to the y-axis. The two electrogeneration parts of the L-SIE are the rectangular sections of size d0×d′1×d3 that border the air-gap of the L-SIE and are parallel to the x-axis.
For purposes of analysis assume that d′1=m1, d1=m1+2d0, d′2=m2, d2=m2+2d0, and d3=m3, so the air-gap of the L-SIE has the same dimensions (m1×m2) in the y-axis and z-axis direction as the magnets of the HA. Note that as a consequence, the overall dimensions of the air-gap is (m1×m2×d3).
To simplify the discussion below, define α0=2/((d1−2d0)/d2+1).
Also, define the following factors:
The L-SIE switches are set ON only when the electrogeneration part of the SIE 214 is under a magnet of the HA with internal flux directed vertically, either downward or upward. Assume the resistance across the switches is near 0, the electrical resistance and magnetic inductance of the L-SIE when the L-SIE switches are ON allowing induced current flow within L-SIE are estimated below.
To estimate the resistance of L-SIE one can view the L-SIE as a single coil with wire cross-section area approximately d0d3. To estimate the length to this coil, one can use the perimeter of an (d1−d0)×(d2−d0) rectangle, which has length 2(d1−d0)+2(d2−d0)=4α0d2 where α0=((d1−2d0)/d2+1)/2. Let ρ be the L-SIE's electrical resistivity. The electrical resistance R of the L-SIE is estimated as the product of electrical resistivity ρ times the length 4α0d2 divided by the cross-section area d0d3. Hence R=4α0ρd2/(d0d3).
To estimate the magnetic inductance of L-SIE, the L-SIE can be viewed as a single coil solenoid. Let μ be the magnetic permeability of the media within the air-gap of the L-SIE (which typically will be air or a supporting material such as concrete). The air-gap of the L-SIE when cross-sectioned by a plane perpendicular to the z-axis has area A=d′1d′2. The magnetic inductance L of the L-SIE when viewed as a single coil solenoid is approximately the product of μ times its interior area A=d′1d′2=α1α2d1d2 divided by its height d3. Hence L=μA/d3=α1α2μd1d2/d3.
Let v denote the velocity magnitude of a translational movement of the HA in the positive direction of the x-axis from left to right over the immobile L-SIE. First analyzed is the levitation force at a position where L-SIE is below an individual magnetic element with internal flux directed downward, in the negative direction of the z-axis, and the L-SIE switches are ON allowing induced current flow.
Observe that the magnetic flux of the HA crosses the L-SIE coil at velocity v, across the electrogeneration part of the L-SIE of distance d′2 in the y-axis direction. Recall the electrogeneration part of the L-SIE is the rectilinear section of size d0×d′2×d3 that borders left side of the air-gap L-SIE and is parallel to the y-axis. When the L-SIE switches are ON, a resulting induced current ION runs across the electrogeneration part of the L-SIE in the negative direction of the y-axis. This induced electric current magnitude is approximately ION=vBd′2/R. By substituting the previously determined estimate for R=4α0ρd2/(d0d3), it can be shown that ION=vBd′2/R=vBd′2/(4α0ρd2/(d0d3)) which simplifies to ION=vBd0d3α2/(4ρα0). Hence it can be concluded that the induced current ION has magnitude growing linear both in the velocity v and magnetic flux density B. (Note that due to inductive resistance of the L-SIE coil, the current running entirely around the coil would initially be less than this.).
The generation of the induced electric current ION across the electrogeneration part of the L-SIE when the L-SIE's switches are ON induces a reverse drag force Fdrag per L-SIE in the opposite direction of the HA, that is in the right to left direction that is a negative direction in the x-axis. (Note that the drag force is near 0 when the L-SIE's switches are OFF.) This drag force is Fdrag=−IONd′2B, which when substituted with ION=vBd′2/R, gives Fdrag=−IONd′2B=−(vBd0d3α2/(4ρα0))d′2B, which simplifies to Fdrag=−v(Bd2α2)2d0d3/(4ρα0). Hence the magnitude of the drag force grows linearly with the velocity v. Since the drag force is near 0 when the switches are OFF, which occurs half the time, the average drag force is Favg-drag=Fdrag/2=−v(Bd2α2)2d0d3/(8ρα0). Since the interior area of each L-SIE is A=d′1d′2=α1α2d1d2, the average drag force on all L-SIEs per square meter of area is Favg-drag/A=−v(Bd2α2)2d0d3/(8ρα0(α1α2d1d2)), which simplifies to Favg-drag/A=−vB2α2d0d2d3/(8ρα0α1d1).
Recall the electromagnetic parts are the rectilinear sections of size d0×d′1×d3 that border the air-gap of the L-SIE and are parallel to the x-axis. The induced magnetic flux density per L-SIE generated by the current ION in each electromagnetic part of the L-SIE is the product of μION times a factor that drops off by a factor of 1/(1+yπ/d3), where y is the distance from the electromagnetic part. Hence the induced magnetic field has flux density approximately B′y=μION/(1+yπ/d3). The induced magnetic flux extends from y=0 to y=d′2. The integral of 1/(1+yπ/d3) from y=0 to y=d′2 is the same as the integral of 1/y′ from y′=1 to y′=1+πd′2/d3, which is β=ln(1+πd′2/d3)=ln(1+πα2d2/d3). Hence the integral of B′y from y=0 to y=d′2 is μIONβ.
That induced magnetic field B′y has flux directed upward, in the positive direction of the z-axis. This is in the opposite direction of the magnetic flux of the HA magnet directly above the L-SIE, whose flux is directed downward, in the negative direction of the z-axis. Hence these two opposing magnetic fields generate a levitation force.
Let x be the distance the magnet M in the HA has moved in the x-axis direction over the L-SIE. Note that when x=0, the left-most part of the magnet M is just overhead the electrogeneration part (the left-most part) of the L-SIE, and the L-SIE's switches are just turned ON. Also note that when x=d1, the left-most part of the magnet M is just overhead the electrogeneration part of the L-SIE, and the L-SIE's switches are turned just turned OFF. Further that when x=2d1, the right-most part of the magnet M is just overhead the electrogeneration part of the L-SIE.
The levitation forces induced by the L-SIE's two elements (the electrogeneration part and the section to its right) parallel to the y-axis, averaged over x, can be seen to cancel each other. Hence the analysis can include the levitation forces induced by the L-SIE's two electromagnetic parts.
For each x between 0 and d1, note that the L-SIE extends under the magnet M distance x. Hence the levitation force Fx per L-SIE electromagnetic part varies with x, and is given by xB/(2μ) times the integral of B′y from 0 to d2. Hence the levitation force Fx for both of its two L-SIE electromagnetic parts is Fx=2(xB/(2μ))(μIONβ), which simplifies to Fx=xBβION.
At x=d1 one gets the maximum levitation force Fmax=d1BβION. Substituting into this the current magnitude ION=vBd0d3α2/(4ρα0), one gets Fmax=d1Bβ(vBd0d3α2/(4ρα0)), which can be written as Fmax=vB2d0d1d3α2β/(4ραd0).
The average levitation force of Fx per L-SIE over the range of x from 0 to d1 is F′avg=Fmax/2=vB2d0d1d3α2β/(8ρα0).
Let the time t=0 at the time the L-SIE switches are turned ON. Since the HA moves in the positive x-axis direction at velocity v, and x is the length in the x-axis of the overlap between the L-SIE and the magnet M, it follows that at time t, x=tv. From x=0 to x=d1 there is a linear increase of the levitation force.
Let t0=d1/v be the time when x=d1 and the L-SIE switches are turned OFF. Let t′=t−t0 be the time duration after L-SIE switches are turned OFF. After time t0 when the switches are set OFF, the current I=ION is within the coil of the L-SIE, but then the L-SIE still has magnetic field energy is I2L/2 but suffers a power loss rate of I2R, so in the absence of further electrical power, the induced magnetic flux, and levitation force decrease with time t by exp(−t′/τ), where the time constant is τ=L/R.
Substituting the previously determined estimates of R=4α0ρd2/(d0d3) and L=α1α2μd1d2/d3, one gets a time-delay constant τ=L/R=(α1α2μd1d2/d3)/(4ρα0d2/(d0d3)) which simplifies to τ=d0d1μα1α2/(4α0ρ).
For a valid analysis, can assume the HA (and hence the magnet M) moves no further than τv<εd1 within a time constant τ for some small ε where 0<ε<<1. Note this implies the velocity v should not exceed vmax=εd1/τ, which can be written as vmax=4εα0ρ/(d0μα1α2).
Since it is assumed that τ<<v/d1, it follows that exp(−t′/τ) is approximately 0 when t′=vd1 and x=2d1 so at that time electrical power, the current, induced magnetic flux, and levitation force are all approximately 0.
Since τ<<v/d1, the integral of exp(−t′/τ) from t′=0 to t′=vd1 is approximately τ. Hence, averaging over all the x over the interval from x=d1 to x=2d1, the average remaining current, induced magnetic flux, and levitation force are all a factor of approximately τ of their values when x=d1. In particular, since the levitation force is Fx=d1BβION/2=2F′avg when x=d1, it follows that the average levitation force over the interval x=d1 to x=2d1 is F″avg=τ2F′avg.
Combining these averages, one can conclude the average levitation force Favg-lev per L-SIE over the full interval x=0 to x=2d1 is Favg-lev=(F′avg+F″avg)/2=(τ+½)F′avg. Substituting the estimate for F′avg=vB2d0d1d3α2β/(8ρα0), one can derive the estimate for the overall average levitation force (averaged over all x positions) per L-SIE: Favg-lev=(τ+½)F′avg=(τ+½)vB2d0d1d3α2β/(8ρα0).
There is a similar levitation force at a position of the HA where L-SIE is below a magnetic element with internal flux directed upward (rather than downwards, as considered above), in the negative direction of the z-axis, and the L-SIE switches are ON allowing induced current flow within L-SIE. In this case, the induced current is the same magnitude, but in the opposite direction (that is in the positive direction of the y-axis). The resulting induced magnetic flux B′y is directed downward, in the negative direction of the z-axis. This is in the opposite direction of the magnetic flux directed upward of the HA magnet directly above the L-SIE. Hence these two opposing magnetic fluxes again generate a levitation force on the HA directed upward. Also, the drag forces in this case remain the same.
Since the interior area of each L-SIE is A=d′1d′2=α1α2d1d2, the overall average levitation force for all L-SIEs per square meter of area is Favg-lev/A=(τ+½)vB2d0d1d3α2β/(8ρα0(α1 α2d1d2)), which simplifies to Favg-lev/A=(τ+½)vB2d0d3β/(8ρα0α1d2).
For purposes of analysis the levitation frictional coefficient can be defined as the ratio of the magnitude of the average drag force divided by the magnitude of the average levitation force. (Note that the levitation frictional coefficient is defined here is similar to the usual definition of rolling friction coefficient, which is the ratio of the magnitude of the horizontal rolling friction drag force divided by the magnitude of the vertical load force.) Recalling that the average drag force per L-SIE is Favg-drag=−v(Bd2α2)2d0d3/(8ρα0), one can determine that the levitation frictional coefficient=|Favg-drag|/|Favg-lev|=(v(Bd2α2)2d0d3/(8ρα0))/((τ+½)vB2d0d3β(8ρα0α1d2)), which simplifies to |Favg-drag|/|Favg-lev|=(d2)3α1(α2)2/((τ+½)β).
One can define a typical class C of examples: where (i) τ is near 0 and (ii) there are relatively small side-wall thickness d0<<min(d1,d2)), so α1, and α2 are both near 1, and (iii) and where d1>>d2, so α0=2/((d1−2d0)/d2+1) is approximately 2d2/d1).
Summarizing the estimates, it can be shown:
For the following further examples in the table below, make the following further assumptions:
This Appendix provides an electromagnetic and force analysis of a representative HA moving over a representative alignment-force-inducing SIE (“AL-SIE”), including determination of approximation formulas for the amounts of inductive current in the AL-SIE, induced magnetic flux, and induced drag forces, levitation forces, and alignment forces. These formulas depend on the AL-SIE's and HA's physical dimensions, their material properties (including the magnet's magnetic flux density, the AL-SIE's electrical resistivity, magnetic permeability of the media within the air-gap of the AL-SIE), as well as the speed of the HA.
Throughout the analysis, certain assumptions and approximations are made. Such assumptions and approximations may be appropriate for some but not all of the embodiments of the technology disclosed herein. Accordingly, at least some embodiments of the disclosed technology may not fit within or be fully characterized by the following analysis.
The horizontal-force-inducing AL-SIE will be assumed to be composed of conductive nonmagnetic material (e.g., copper or similar material) with low electrical resistivity ρ. The HA will also be assumed to be sized as in the magnetic array 110 illustrated in the figures. For simplicity, it will be assumed that almost all the external magnetic flux of the HA (below magnets of the HA with sidewise internal flux) extends no further than distance m1 below the HA.
For purposes of analysis of the horizontal-force-inducing AL-SIE, refer to the dimensions illustrated in
To simplify the discussion below, again define the following factors:
The air-gaps of the AL-SIE's coil are the following two rectilinear regions:
Let μ again be the magnetic permeability of the media within the air-gaps of the AL-SIE (which typically will be air).
In this example's analysis, assume that d′1=m1, d1=m1+2d0, d′2=m2, d2=m2+2d0, d3=m3, and d4=m1 so the first air-gap of size (d′1×d′2×d3) has the same dimensions (m1×m2) in the y-axis and z-axis direction as the magnets of the HA, so both air-gaps have overall dimensions (m1×m2×d3).
The AL-SIE switches are set ON only when the electrogeneration part of the AL-SIE is under a magnet of the HA with internal flux directed vertically, either downward or upward. Assume the resistance across the switches is near 0, the electrical resistance and magnetic inductance of the AL-SIE when the AL-SIE switches are ON allowing induced current flow within L-SIE are estimated below.
To estimate the resistance of AL-SIE, the AL-SIE will be viewed as a single coil with wire cross-section area approximately d0d3. To estimate the length to this coil, one can use the perimeter of an (2d1−d0)×(d2−d0) rectangle, which has length 2(2d1−d0)+2(d2−d0)=4α′0d2 where α′0=((2d1−2d0)/d2+1)/2. Let ρ be the AL-SIE's electrical resistivity. The electrical resistance R′ of the AL-SIE is estimated as the product of electrical resistivity ρ times the length 4α′0d2 divided by the cross-section area d0d3. Hence R′=4α′0ρd2/(d0d3).
To estimate the magnetic inductance of AL-SIE, one can view the AL-SIE as a coil solenoid. Let μ be the magnetic permeability of the media within the air-gap of the AL-SIE (which typically will be air). Each of the horizontal and vertical air-gap regions of the AL-SIE have area A=d′1d′2, so the total area is A′=2d′1d′2. The magnetic inductance L′ of the AL-SIE when viewed as a single coil solenoid is approximately the product of μ times area A′=2d′1d′2 divided by its height d3. Hence L′=μA′/d3=2α1α2μd1d2/d3.
Let v denote the velocity magnitude of a translational movement 122 of the HA in the positive direction of the x-axis from left to right over the immobile AL-SIE. First analyzed is the case of a position where AL-SIE is below a HA magnet M (e.g., magnetic element 104 in the figures) with internal flux directed downward, in the negative direction of the z-axis, and the AL-SIE switches are ON allowing induced current flow.
Observe that the magnetic flux of the HA crosses the AL-SIE coil at velocity v, across the electrogeneration part of the AL-SIE of distance d′2 in the y-axis direction. When the AL-SIE switches are ON, a resulting induced current I′ON runs across the electrogeneration part in direction parallel to the y-axis in the negative direction of the y-axis. This induced electric current magnitude is approximately I′ON=vBd′2/R′. Substituting the previously determined estimate for R′=4α′0ρd2/(d0d3), one arrives at I′ON=vBd′2/R′=vBd′2/(4α′0ρd2/(d0d3)) which simplifies to I′ON=vBd0d3α2/(4ρα′0). Again the induced current I′ON has magnitude growing linear both in the velocity v and magnetic flux density B. (Note again that due to inductive resistance of the AL-SIE coil, the current running entirely around the coil would initially be less than this.)
The generation of the induced electric current I′ON across the electrogeneration part of the AL-SIE in direction parallel to the y-axis when the AL-SIE's switches are ON induces a reverse drag force F′drag per AL-SIE in the opposite direction of the HA, that is in the right to left direction that is a negative direction in the x-axis. (Note that the drag force is near 0 when the AL-SIE's switches are OFF.) This drag force is F′drag=−I′ONd′2B, which when is substituted with I′ON=vBd0d3α2/(4ρα′0), gives F′drag=−IONd′2B=−(vBd0d3α2/(4ρα′0))d′2B, which simplifies to F′drag=−v(Bd2α2)2d0d3/(4ρα′0). Since the drag force is near 0 when the switches are OFF, which occurs half the time, the average drag force is F′avg-drag=F′drag/2=−v(Bd2α2)2d0d3/(8ρα′0). Since the area of the horizontal air-gap of each AL-SIE is A=d′1d′2=α1α2d1d2, the average drag force on all AL-SIEs per square meter of area is F′avg-drag/A=−v(Bd2α2)2d0d3/(8ρα′0(α1α2d1d2)), which simplifies to F′avg-drag/A=−vB2α2d0 d2d3/(8ρα′0α1d1).
First the levitation forces produced by the levitation-electromagnetic parts of the AL-SIE are considered. These have exactly the same shape as those of Example A. However, the overall resistance and inductance of the levitation-inducing AL-SIE is R′ and L′ (instead of R and L), resulting in some slight changes that can be observed. From the analysis of Example A, if x is the horizontal overlap in the x-axis direction of the magnet M over the AL-SIE, then from x=0 to x=d1, while the switches are ON, there is a linear increase of the levitation force to a maximum of vB2d0d1d3α2β/(8ρα′0), where β=ln(1+πα2d2/d3).
But from x=d1 to x=2d1 the switches are set OFF and in absence of further electrical power, the induced magnetic flux, and levitation force decrease with time t by exp(−t′/τ), where the time constant τ′ is in this case L′/R′=2(α1α2μd1d2/d3)/(4ρα′0d2/(d0d3)) which simplifies to τ′=d0d1μα1α2/(2α′0ρ). For this analysis of Example A to be valid, one needs to assume the HA moves at velocity v not exceeding v′max=εd1/τ′=2εα0ρ/(d0μα1α2), for some small ε where 0<ε<<1. Since τ′<<v/d1, the integral of exp(−t′/τ′) from t′=0 to t′=vd1 is approximately τ′. Hence the average levitation force, over all values of x, is F′avg-lev=(τ′+½)vB2d0d1d3α2β/(8ρα′0).
Again, since the area of the horizontal air-gap of each AL-SIE is A=d′1d′2=α1α2d1d2, the average levitation force on all AL-SIEs per square meter of area is F′avg-lev/A=(τ′+½)vB2d0d1d3α2β/(8ρα′0(α1 α2d1 d2)), which simplifies to F′avg-lev/A=(τ+½)vB2d0d3β/(8ρα′0α1d2).
The levitation frictional coefficient can be the ratio of the magnitude of the average drag force divided by the magnitude of the average side-wise-alignment force. Recalling that the average drag force per AL-SIE is F′avg-drag=−v(Bd2α2)2d0d3/(8ρα′0), one can determine that the levitation frictional coefficient=|F′avg-drag|/|F′avg-lev|=(v(Bd2α2)2d0d3/(8ρα′0))/(τ′+½)vB2d0d1d3α2β/(8ρα′0) which simplifies to |F′avg-drag|/|Favg-lev|=(d2)2α2/(τ+½)d1β, which is the same formula as in Example A for the levitation-inducing AL-SIE.
Next is considered the alignment-electromagnetic parts of the AL-SIE, which form a vertical U that, when current flows through them, generate an electromagnetic field that induces the intended alignment along the x-axis. Recall two of these electromagnetic parts (which will be referred to as type 1) are of size (d0×d4×d3), extending downward vertical distance d4 in the negative z-axis direction. Recall these connect at their upper ends end to the electromagnetic part (which will be referred to as type 2) of size (d0×d2×d3), extending across distance d2 in the y-axis direction.
As in Example A, the induced magnetic flux density generated by the current I′ON in each type 1 alignment-electromagnetic part of the AL-SIE at distance y from it is approximately B′y=μI′ON/(1+yπ/d3). Also as in Example A, the integral of B′y from y=0 to y=d′2 is μI′ONβ, where again β=ln(1+πα2d2/d3).
Also, the induced magnetic flux density generated by the current I′ON in the type 12 alignment-electromagnetic part of the AL-SIE at distance z from it is approximately B′z=μI′ON/(1+zπ/d3), and the integral of B′z from z=0 to y=d′1 is μI′ONβ, where β′=ln(1+πα1d1/d3).
As shown in
These two similarly-directed magnetic flux induces an alignment torque force that grows proportionally to tan(θ), where θ is the angle of deviation of the HA from the x-axis. In particular, the alignment torque force is |Falign tan(θ)| where the side-wise alignment force Falign is approximated below.
The side-wise alignment force Falign1 per each type 1 alignment-electromagnetic part is given by d′1β/(2μ) times the integral of B′y from y=0 to y=d2. Hence the side-wise alignment force Falign1 for each of its two type 1 alignment-electromagnetic parts is Falign1=2(d′1B/(2μ))(μI′ONβ), which simplifies to Falign1=d′1BβI′ON/2.
The side-wise alignment force Falign2 for the type 2 alignment-electromagnetic part is given by d′2B/(2μ) times the integral of B′z from z=0 to z=d1. Hence the side-wise alignment force Falign2 for the type 2 alignment-electromagnetic part is Falign2=d′2B/(2μ))(μI′ONβ′), which simplifies to Falign2=d′2Bβ′I′ON/2.
Hence the side-wise alignment force Falign in the direction of the x-axis, totaled for all the alignment-electromagnetic parts is Falign=2Falign1+Falign2=d′2Bβ′I′ON+d′1BβI′ON/2=(d′2β+d′1β′/2)BI′ON, which can be substituted with I′ON=vBd0d3α2/(4ρα′0), is Falign=(d′2β+d′1β′/2)B(vBd0d3α2/(4ρα′0))=vB2γd0d3α2/(4ρα′0), where γ=(d′2β+d′1β′/2).
This side-wise alignment force Falign remains approximately the same from x=0 to x=d,. However, recall from Example A that from x=d1 to x=2d1 the switches are set OFF and in absence of further electrical power, the induced magnetic flux, and induced force decrease with time t exponentially with the time constant τ′=d0d1μα1α2/(2α′0ρ). The integral of these forces from x=d1 to x=2d1 are approximately a factor of τ′ times their value at x=d1. Hence the average (averaged over all x positions) side-wise alignment force Favg-align is a factor (τ′+½) times its value at x=d1 and hence Favg-align=(τ′+½)Falign=(τ′+½)vB2γd0d3α2/(4ρα′0).
There is a similar side-wise alignment force at a position of the HA where the AL-SIE is below a HA magnet M (e.g., magnetic element 104 in the figures) with internal flux directed upward (rather than downwards, as considered above), in the negative direction of the z-axis, and the AL-SIE switches 17 are ON allowing induced current flow within AL-SIE. As shown in
Since the interior horizontal region of each AL-SIE is A=d′1d′2=α1α2d1d2, the overall average alignment force for all AL-SIEs per square meter of area is |Favg-align|/A=(τ+½)vB2γd0d3α2/((4ρα′0)(α1α2d1d2)), which simplifies to |Favg-align|/A=(τ+½)vB2γd0d3α2/(4ρα′0d1d2).
The levitation-alignment frictional coefficient=|Favg-drag|/([F′avg-lev|+|Favg-align| is defined as the ratio of the magnitude of the average drag force divided by the sum of magnitudes of the average levitation force and side-wise-alignment force. Recalling that the average drag force per AL-SIE is Favg-drag=−v(Bd2α2)2d0d3/(8ρα′0), one can determine that the levitation-alignment frictional coefficient=|Favg-drag|/([F′avg-lev|+|Favg-align|=(v(Bd2α2)2d0d3/(8ρα′0))/(((τ+½)vB2d0d1d3α2β/(8ρα′0))+(τ′+½)vB2γd0d3α2/(4ρα′0)) which simplifies to |Favg-drag|/([F′avg-lev|+|Favg-align|)=8α2(d2)2/((τ+½)d3(2d1β+γ)).
Again, define a typical class C of examples: where (i) τ′ is near 0 and (ii) there is a relatively small side-wall thickness d0<<min(d1,d2)), so α1, and α2 are both near 1, and (iii) and where d1>>d2, so α0=2/((d1−2d0)/d2+1) is approximately 2d2/d1, and α′0=((2d1−2d0)/d2+1)/2 is approximately d2/d1.
Summarizing the estimates:
In the following further examples, make the following further assumptions:
Illustrations of a linear Halbach Array (HA) aligned on the x-axis are given in
(a) Magnet 2 has internal magnetic flux 6 directed vertically upward, in the positive direction of the z-axis; it has a magnet 3 to its right in the positive x-axis direction.
(b) Magnet 3 has internal magnetic flux 7 directed horizontally left to right, in the positive direction of the x-axis; it has a magnet 4 to its right in the positive x-axis direction.
(c) Magnet 4 has internal magnetic flux 8 directed vertically downward, in the negative direction of the z-axis; it has a magnet 6 to its right in the positive x-axis direction.
(d) Magnet 6 has internal magnetic flux 9 directed horizontally right to left, in the negative direction of the x-axis; it has a magnet 2 to its right in the positive x-axis direction.
Illustrations of a 2D array of magnets consisting of multiple linear HAs are given in
Illustrations of a 2D array of magnets arranged as a two dimensional Halbach Array (2D-HA) are given in
Illustrations of a dynamically configured SIE are given in
Illustrations of a dynamically configured linear array of SIEs are given in
Illustrations of a dynamically configured 2D array of SIEs are illustrated in
Illustrations of a pair of HAs, the lower one vertically flipped, positioned above each other aligned with the x-axis, are given in
Illustrations of a Electro-Magnetic Halbach Array (EM-HA) aligned on the x-axis are given in
Illustrations of an EM-HA of
The EM-SIE 56 is illustrated in
The present application claims priority to and the benefit of U.S. Patent Application No. 61/561,198, filed on Nov. 20, 2011 and titled HARVESTING TRANSLATIONAL ENERGY FOR MAGNETIC LEVITATION VIA SWITCHED INDUCTIVE ELEMENTS, which is incorporated herein by reference. To the extent the foregoing application and/or other materials incorporated herein by reference conflict with the present disclosure, the present disclosure controls.
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