CHAPTER 2.  PASSIVE MAGNETO-DYNAMIC SUSPENSION (MDS)

The possibility of creating a stable suspension system was proven more than two centuries ago and supported by the Lagrange-Direchlet theorem [1]. Applied to our case it states that equilibrium of a MDS levitator is stable if the potential energy of the MDS magnetic field has its local minimum in the equilibrium position.

The wrong conclusions that a stable magnetic suspension based on permanent magnets and steel cores is a physical impossibility have been stemmed from misreading the Earnshaw’s theorem. This theorem is true for systems containing bodies with constant permeability. Meanwhile permeability of steel suspension components is highly dependant on field intensity. This peculiarity of saturated steel suggests that to create a stable magnetic suspension (MS) based on moving permanent magnets and rigidly fixed steel cores is in fact possible.

Magnetic suspension utilizing permanent magnets, steel cores and rigid constrains is a conservative system, i.e. one that conserves its potential magnetic energy E_p. The potential magnetic energy is that part of energy that can not be transformed into kinetic energy because of the rigid constrains. Two parts of MS: stator and levitator - are separated in space, interacting with each other through magnetic field. Internal forces in a the conservative system are derivatives of the potential magnetic energy E_p with respect to coordinates of a shift between its parts.

2.1  Some considerations on building MS.  

(1) As we said above that analyzing Earnshaw theorem one can come to the conclusions that it is true only for the objects with constant magnetic permeability. However, permeability of steel parts of the MS is very dependable on the intensity of the field. This peculiarity of saturated steel makes it possible to build a stable MS based on the permanent magnets and steel cores. The deduction method was applied to the Lagrange‑Dirichlet theorem that in our case states that position of MS levitator magnets is stable if potential energy E_pof magnetic field produced by permanent magnets has a local minimum E_pm, not coinciding with a cores surface. It is impossible to reveal or to measure potential energy of magnetic field. Therefore it is expedient to proceed from the opposite premise: if MS levitator is in equilibrium position, and any its small shift produces a stabilizing force then MS is stable. Such interpretation of Lagrange‑Dirichlet theorem prompts the way of creating a stable MS.

(2) For almost 100 years it was considered impossible to build a stable MS. To prove the opposite let us see into the essentials of MS. We will note one peculiarity of the Lagrange-Dirichlet theorem.

The levitator magnets are fixed rigidly on a Maglev vehicle and together with it form a solid body of cylindrical shape. Thus rigid constrains assemble the levitator into a solid body. As a result: (a) a shift δ of any point of the levitator causes the same shifts of the rest of its points; (b) the levitator summarizes all forces applied to it in total equivalent force. When proving Lagrange‑Dirichlet theorem these peculiarities of a solid body (in our case, levitator) were considered: a real‑conservative system comprising bodies of different configurations was substituted by a system of mass points. In theoretical mechanic this procedure is called “ transition into space of configuration”. By this approach the real distribution of forces acting on each body is substituted by an equivalent force. Therefore in order to employ Lagrange‑Dirichlet theorem for creating a stable suspension of real bodies of definite shapes, the equivalent force must be expanded into components in such a way to satisfy all necessary equations for static body equilibrium. Speaking of Maglev system we must ensure stable equilibrium of two bodies only: an immovable stator and a flying vehicle with a levitator. To attain this it is necessary to situate levitator magnets and stator steel cores in a specific manner.

(3) Interaction between the levitator magnets and stator steel cores produces stabilizing forces. To give the flying vehicle maximum stability the torques of these forces must be as big as possible. It is known that a torque of the couple of forces is proportional to a distance between these forces. Therefore the magnets must be fixed to the bottom and walls along the whole vehicle close to its walls. Correspondingly, the stator cores must be located along the guideway and be parallel to flying magnets at a small distance from them. The stator cores together with two sets of magnets located in series on each side of the vehicle are designated for producing stabilizing forces. In this case the shapes of both MS magnets and cores must be cylindrical with their generatrices parallel to the Axis OX of the guideway. Hence the vehicle has just one degree of freedom directed along the Axis OX. The other shifts and turns should produce stabilizing forces and torques.

(4) In order to make MS stable the interaction between the levitator magnets and the stator steel cores must produce stabilizing forces only.

Next it will be shown how to attain this.

It is known that a permanent magnet is always attracted to a steel body , i.e. it produces a destabilized force F_d , tending to increase its shift. This force is oriented contrarily to a stabilizing force F_s. Let us think what configuration a steel body must have to make a rectangular magnet to produce a stabilizing force in its vicinity. Fig. 2.1 shows a magnet with its poles situated symmetrically between tips of a C-shaped steel core. The destabilizing forces of attraction of the magnets to both the right and left core tip are balanced and only a vertical stabilizing force F_s acts on the magnet opposing its vertical shift. When the magnet is shifted downwards it stretches Faraday’s pipes of magnetic flux (similar to an arrow stretching a bowstring) thus producing the stabilizing force F_s that tends to bring it back to the initial position. Therefore it will be logical to build MS of rectangular magnets assembled into a levitator and C-shaped steel cores assembled into a stator. This is the first step toward creating a stable suspension based on permanent magnets.

(5)  Derivative of E_p (system potential energy ) with respect to levitator shift d equals magnetic force ,directed to E_p lessening. Therefore in the vicinity of equilibrium, where potential energy is minimum E_p =E_pm., forces of interaction between levitator magnets and the stator cores must be stabilizing.  It follows from the above that speaking of the real Maglev system minimum of potential energy we mean that it must be achieved simultaneously in the entire volume of the levitator- vehicle assembly in the regular three-dimensional space or (that is the same) just in one massive point Q of the space of configuration.

Let’s assume that design of MS ensures local minimum of potential energy E_pm. It means that when the levitator is in the position corresponding to minimum of potential energy (point Q in Fig.2.2) then the sum of all forces applying to it is zero, i.e., the levitator is in equilibrium. However each magnet produces its own force. Consequently, equilibrium is a result of action of counter forces balancing each other. Hence we come to the following conclusions:

(a) the situation of magnets and steel cores in MS must be mirror- symmetrical (Figs. 2.4 and 2.5);

(b) the force produced at magnets shift d from the equilibrium should be stabilizing opposing the shift.

If to expand this force in Maclaurin’s series it can be expressed by

(2-1)

where  is stiffness of the stabilizing force.

One can see from formula (2-1) that value of stabilizing force F_s is proportional to the product of shift d and value of its stiffness . The less the volume of magnetic field of energy E_p in the air gap g (Fig.2.5) between the levitator and stator comparing to g²  the bigger the stiffness. Since magnetic field in MS system is produced by extended magnets of the levitator flying along the stator cores it can be considered plane-parallel and calculations of its parameters are performed per one meter of vehicle length.

As distinguished from any regular function that has a minimum just in a single point, in our case the potential energy of MS has a minimum in the whole volume of the levitator body with the stabilizing force as a function of the shift having the same value and direction in all the points. Therefore the levitator-vehicle assembly may be substituted by a massive point Q (Fig.2.2) coinciding with equilibrium (as if we have compressed the real body in a single point Q ). At this approach function of potential energy in the vicinity of Q looks like regular function of two variables (y, z).

Considering this peculiarity of a solid body we can determine whether or not the MS possesses a minimum or maximum of potential energy.

When the levitator is in stable equilibrium, the resulting stabilizing force  equals zero. Therefore in close vicinity [δ] of the equilibrium it can be expanded in Maclaurin's’ series and expressed by:

            (2-2)

where is a dominant term of the series.

It is possible to find out if MS has a minimum of potential energy. All the MS stator parts are rigidly connected with each other and form an immovable solid body. The levitator magnets are affixed rigidly to a vehicle and together form a movable solid body. Considering above peculiarity of a solid body first we will determine whether the MS possesses an extreme of potential energy.

 By integrating formula (2-1) with respect to the shift δ we obtain

, (2-3)

that is the equation of a parabola. Hence the distribution of E_p(y,z) in a small vicinity of the point Q is similar to a paraboloid surface (Fig.2.2).

Let us draw a circle with a radius δ around point the Q on the plane YQZ (it is shown on the top and bottom of Fig.2.2 for the upper and the lower paraboloids respectively) and then displace the levitator from the equilibrium (point Q) in any direction along the radius by shift δ.. Next we will find an internal magnetic force (2-2) that is a derivative of potential energy with respect to δ. If it is directed away from the center (the lower paraboloid) we have a maximum of energy. In this case Q=Q_max, and all the forces are destabilizing. If the internal force is directed toward the center (the upper paraboloid) we have a minimum of energy. In this case Q= Q_min , and all the forces are stabilizing. Thus a radial pattern of the vectors of the internal forces may indicate maximum or minimum of the potential energy.

The presence of the equilibrium for the levitator flying along the Axis OX supposes that the shape of levitator and stator parts are symmetrical with respect to the plane XOZ (Fig.2.11). In addition the levitator–vehicle assembly is in homogenous gravity. Therefore MS should compensate its weight.

We will show that it is possible to ensure extreme of potential energy E_p(y,z) in the entire volume of moving levitator by varying its shape, physical characteristics of substance, and position of the levitator and stator parts with respect to each other

2.2  How to build a stable MS.

Following the Lagrange-Dirichlet theorem we are searching for a stable system only among those that have equilibrium. Next we will consider the simplest magnetic devices having an equilibrium position and analyze if they posses an extreme of potential energy.

Fig.2.1 shows a unit containing a permanent magnet located symmetrically in equilibrium (dotted line) between the ends of a C-shaped steel core. Vertical shift δ_V of the magnet downward produces stabilizing force –F_s :

,

directed upwards which tends to move the magnet back to equilibrium. Horizontal shift of the magnet δ_H  produces destabilizing force F_d:

from the equilibrium towards any core tip increases the shift. It means that there is no extreme of energy in the magnetic field of this device.

Fig. 2.3 Magnetic devices in equilibrium for analyzing if they possess an extreme of potential energy. (a) Extreme is absent. (b) Graph at the left-hand side   shows  maximum,  graph at the right-hand side shows minimum.

Fig.2.3 shows a device consisting of two identical units which are reciprocally perpendicular to each other. The device have rigid constrains connecting magnets with each other and separately steel cores in two different assemblies . The rigid constrains can restrict the movements of magnets and steel cores thus conserving their potential energy. It means that the above assembly may be a conservative system.

In this case vertical shift δ_V downward of both magnets from their equilibrium produces an contrarily directed stabilizing force –F_s in the upper unit. However the same shift produces a destabilizing force F_d in the lower unit. Rigid constrain between the magnets summarizes the forces geometrically. The resulting force is F_Σ = F_d – F_s. It may be either destabilizing (if F_d > F_s) or stabilizing (if F_s > F_d).

Shift δ_α of the magnets by any angle α (in the plane of Fig.2.3) produces both destabilizing and stabilizing forces in each unit that are proportional to projections of the shift on Axes Y and Z. Summarizing geometrically each pair of stabilizing forces and each pair of destabilizing forces separately we obtain:

, (2-4)

 (2-5)

where f(α) = (cos² α+ sin² α)^1/2 = 1.

One can see from (2-4) and (2-5) that both kind of forces are directed along the radiuses δ of the circle drawn around the equilibrium Q in plain YQZ and do not change their values no matter how the angle α changes (Fig 2.3, bottom). Depending on which one is bigger we can judge whether the system of two units has maximum or minimum of its potential energy.

Hence we make the conclusion: the assembly shown in Fig.2.3 proved that its potential energy may have a local extreme. This is the second step towards achieving stable suspension.

As a rule the destabilizing force in such a device is bigger than stabilizing. It means that it has a maximum of potential energy and therefore is not stable.

Regardless of the fact that above magnetic device is unstable, its consideration was beneficiary because it provided us a clue what direction we should keep on in our research. It means that in order to make this device stable we should assemble units in such a way to satisfy inequality F_s > F_d in each of them. Substituting (2-4) and (2-5) in this inequality we obtain the condition of stability:

(2-6)

Conclusions:

1. a stable magnetic device must be build of two reciprocally perpendicular units;
2. the stiffness of the stabilizing force in each unit must exceed the stiffness of the destabilizing force.

Considering inequality (2-6), we can see that little can be done to increase its left part because this leads to proportional increase of its right part respectively. However it is possible to reduce its right part without changing the left part. Let us show how to achieve this in the MS.

2.3  Design of the MS unit and the system.

The idea to apply saturated steel to lessen the right side of inequality (2-6) has been mentioned briefly in paragraph 1.7. Now we will show this in detail. 

To employ saturation of steel C-shaped cores (mentioned in paragraph 2.1) for reducing destabilizing forces we will build a magnetic unit (Fig.2.4) where:

1. four levitator magnets together with a steel insert between each pair of magnets are assembled in a quadrupole;
2. two C-shaped cores are assembled into a stator by connecting them with a rigid non-magnetic constrain and then situating it mirror-symmetrically with respect to the quadrupole.

Fig.2.4 Cross-sectional view of the unit

Fig.2.5 Perspective view of the unit

Similar to the magnetic device shown in Fig.2.1 a vertical shift δ_v of the quadrupole in the unit of such design produces a stabilizing force F_s. However the unit is essentially distinguished from the mentioned magnetic device by that it has four magnets assembled into the quadrupole with the steel insert between each pair of the magnets and the magnets are situated between core tips of two mirror-symmetrical C-shaped steel cores rigidly connected with each other. The insert serves for splitting the magnetic circuit in two independent contours with each containing a steel core and two magnets. Saturating a steel core backs we make its reluctance non-linear.

It is known that non-linearity employed in electric circuits is common used in electro- radio-, and computer technology for stabilization. We will utilize non-linearity in magnetic circuit for lessening the destabilizing force F_d at a horizontal shift δ_h of the quadrupole.

Let us consider the magnetic circuit of the unit (Fig.2.8). As we mentioned above it consists of two independent left-hand and right-hand. flux contours:. At shifts δ_h of the quadrupole the fluxes in both counters are different. The differences in fluxes get into the insert Therefore the quadrupole’s insert is always unsaturated and its magnetic reluctance is zero. Now we will show how to make core back magnetic reluctances non-linear.

Employing magnetization curve B_f(H_f) (Fig.2.6) for core steel we can build dependencies of relative permeability μ_f  and  specific magnetic reluctance r_f on induction in the core B_f for the fragment of the curve where steel is saturated (Fig. 2.7):

and .

One can see from curve  r_f (B_f)  that at B_f >2T  magnetic reluctance of the core increases rapidly with the growth of B_f  approaching inclined straight line described by equation , where  B and N  are constants.

This peculiarity can be employed for analytical calculation of magnetic fluxes and forces in the unit.

However there is a leakage magnetic flux from the lateral surfaces of the core-backs  when they are saturated because of there are no natural magnetic insulating materials for a magnetic flux similar to those for electric current. Therefore it is very difficult to employ non-linearity in magnetic circuits.

Let us assume that lateral surfaces of the cores are covered with a layer of magnetic insulation and there is no leakage flux. Then by varying dimensions of the magnets and core back in the unit at symmetrical position of the quadrupole we can obtain induction  in core back cross-section equaled to its value in the middle of the linear fragment of the inclined straight line of Fig. 2.7 , where and .

In this case if quadrupole shifts to the right then the right-hand gap reduces, the flux in the right-hand core grows up thus increasing induction in there. But as a result its non-linear magnetic reluctance grows as well and thus restricts the growth of the attraction force at the right side. At the same time the left-hand gap increases, reducing its magnetic reluctance and thus restricting decrease of the attraction force at the left side. Therefore the difference between the right and left attraction forces (destabilizing force F_d) falls abruptly. Thus saturation stabilizes the difference of fluxes and forces. In addition the value of stabilizing force depends on the shape of unsaturated core tips. Hence there is necessary to find their optimum profile.

Fig.2.5 illustrates cross-sectional view of the modified MS unit. One can see that the unit consists of two components: movable and immovable. The movable component (a part of the levitator) contains four long permanent magnets of rectangular cross-section assembled in a quadrupole with the help of a steel insert. The immovable component (a part of the stator) consists of two steel cores of C- shaped cross-sections extended along the entire guideway. They are located mirror-symmetrically and rigidly affixed to each other (by non-magnetic constrain) and to the guideway. Each core has a long back and two thickened tips. A constant air gap is maintained between the left and right-hand core tips. The quadrupole is inserted into this air gap and can move freely in all directions in there.

Magnetic fluxes in the unit (Fig. 2.5) penetrating into core tips produce forces attracting the quadrupole to the cores.

A specific force acting on a unitary surface of unsaturated steel (core tip) is oriented perpendicularly to the surface and proportional to the square of magnetic flux density [8]. Proceeding from this basis, it will be proven that:

1. a quadrupole located symmetrically between the core tips is in equilibrium;
2. a lateral shift of the quadrupole from symmetrical position produces destabilizing force F_d tending to increase the shift and to attract the quadrupole to the nearest core tips;
3. a vertical shift of the quadrupole produces stabilizing force F_s tending to decrease the shift and to bring the quadrupole back to equilibrium. The mode of the unit action at the vertical shift is similar to a bow. Tubes of magnetic flux closed through air are resilient. They are permanently coupled with their source – permanent magnets – and attracted by the tips of the steel cores. If the quadrupole shifts up or down, the flux tubes bend and stretch, thus creating the stabilizing force opposing the shift, which grows proportionally to the shift increase.

Hence the peculiarity of the unit is that quadrupole shifts produce not only a destabilizing force but also a stabilizing force.

The fluxes in the unit follow Ohm’s law for magnetic circuit. As we said above if the quadrupole is shifted to the right by small lateral shift Dy then the right-hand air gaps and their reluctance are reduced, and the flux increases (at the left side everything is vice versa). In this case parity of the fluxes is violated and destabilizing force F_d arises, attracting the quadrupole to the right-hand core tips. Equilibrium of the quadrupole is unstable in this direction.

At small vertical shift Dz of the quadrupole in a symmetrical plane XOZ with respect to the cores, parity of the fluxes penetrating into the right- and left-hand core tips is retained. However in each side the parts of the flux entering the top and bottom halves of the core tip surfaces are redistributed producing a vertical force opposite to the vertical shift. This force is a stabilizing force F_s which is perpendicular to F_d. Hence the vertical shift of the quadrupole stretches and bends the magnetic flux pipes and produces the stabilizing force proportional to the shift that brings the magnet back to equilibrium position. Equilibrium of the quadrupole is stable in this direction.

At a longitudinal shift of the quadrupole the fluxes will not change and forces do not appear. The quadrupole is at neutral equilibrium in this direction.

Let us situate two identical units of Fig.2.4 in such a manner that their axes of symmetry would be reciprocally- perpendicular. Then we fix separately their quadrupoles (levitator) and their cores (stator) by rigid non-magnetic constrains as it is shown on Fig.2.10

If the quadrupoles of each pair of units are inserted in the air gap between the core tips of the corresponding pair of cores, then the stabilizing force of the supporting unit (at the bottom left side) compensates for the destabilizing forces of the guiding unit (at the top right side) and vice versa, the stabilizing force of the guiding unit compensates for the destabilizing force of the supporting unit. It has been proven above (2-6) that if the stiffness of a stabilizing force exceeds stiffness of a destabilizing force in each unit, then MS is stable and self-regulating. It means that any small shift of the levitator as well as its small turn will result in instantly arising an internal force or torque stabilizing it.

We will show how to build a stable MS consisting of several pairs of units connecting as shown on Fig 2.11. It follows from the stability condition (2-6) that in each unit a lateral shift produces a force that is less than this produced by a vertical shift. In terms of the magnetic circuit (Fig.2.8) that is equivalent to the unit of Fig 2. 5 it means that at a lateral shift of the quadrupole, fluxes penetrating into right- and left-hand core tip are just slightly dependent on changing the unit air gap reluctance. After all, the flux in each magnetic circuit loop is determined by its total reluctance. Therefore if non-linear magnetic reluctances (connected in series with air gap reluctance and increasing with the growth of the flux) are present in the circuit, then the total loop reluctance is almost independent on the lateral shift of the quadrupole.

We will show (2-12) that magnitude of the destabilizing force is proportional to the difference of magnetic fluxes Ψ in the right –hand and left-hand core tips and therefore the less the difference of fluxes in both core tips at horizontal shift the less destabilizing force (and its stiffness). Hence in order to decrease stiffness of the destabilizing force it is necessary to reduce dependence of the difference of magnetic fluxes at the shift.

It was said above that we utilize a quadrupole with an insert between its magnets and situate it between the core tips to separate magnetic fluxes and obtain two independent magnetic contours (Fig.2.8). The magnetic flux in each contour is determined by the value of magneto motive force - mmf and magnetic reluctances both linear and non-linear. The sources of mmf are permanent magnets. The linear reluctances are: reluctance of magnet body itself - r_in plus reluctance of the air gap between the steel insert and core tips- r_gR and r_gL. The non-linear are: reluctances of saturated steel core backs covered by screens (which are non-conductive for magnetic flux) - r_fR and r_fL.

If the core backs are saturated then we can employ ρ_f (B) - the non-linear dependency of specific magnetic reluctance of steel on the flux density B (Fig.2.7) for calculation of the fluxes in this circuit.

Fig. 2.8 The upper half of equivalent magnetic circuit of a unit.

We have mentioned that if quadruple magnets are in equilibrium and core backs are evenly saturated along their entire length then shift of the quadruple to the right decreases the right air gap, the flux in the right back increases too and its magnetic reluctance grows up thus compensating for the decrease of the right gap. The opposite effect takes place in the left contour. Thus each contour with the non-linear reluctance performs the function of a stabilizer of magnetic flux. Fig 2.8 shows just upper symmetrical part of the whole equivalent scheme.

The value of compensation for changing linear magnetic reluctances and  by non-linear reluctances:

        (2-7)

depends on the angle θ of inclination of the straight fragment on the curve ρ_f (B_f) in Fig.2.7, B_f >2.01T.. The full compensation would be possible at  . In fact  .

The dependency of the flux difference on the magnets shift decreases and the value of the derivative  drops sharply, thus ensuring inequality (2-6). During a vertical shift of the quadruple, saturation of the core backs and therefore  changes negligibly. It means that saturation of core backs radically reduces just only the stiffness of the destabilizing force but does not change much the stiffness of the stabilizing force.

Fig.2.9 Quadrupole’s magnets divided into equal fragments

Fig. 2.10 Two identical units connected by non-magnetic constrains with reciprocally-perpendicular axes of symmetry.

As we mentioned above magnetic flux flowing in the above circuit has leakage because a substance for isolation of magnetic fluxes does not exist. Therefore saturation of the core back falls with the distance from the core tip increasing To maintain saturation at the required level along the whole core back we propose the following:

1) The unit permanent magnets extending along the whole levitator length would be divided into n equal fragments situated with respect to each other as shown in Fig.2.9 thus making alternating polarity. During the levitator motion these magnets produce alternating magnetic flux in the steel cores.

2)   The steel cores would be made of laminated electrical steel M-5 with thin sheets (with big specific electrical resistance ρ). In this case the laminated steel with isolated sheets suppresses electrical part of electromagnetic field of traveling wave produced by magnets of moving levitator and turns it into pure magnetic one.

3)   Each core back would be covered with an aluminum screen from both sides (Fig.2.12). Then a part of magnetic flux of the wave traveling along the screen surface where (leakage flux) would penetrate inside and induce a layer of eddy currents there. Following the Lentz’s Law of Electromagnetic Inertia the eddy currents produce its own magnetic field opposite in direction and almost equaled in value to the leakage flux from the core suppressing it thus performing function of electromagnetic barrier (i.e., magnetic insulator).

At the levitator speed V>10m/s the electromagnetic barrier almost completely suppress the leakage flux, maintaining core saturation at the required level. Therefore the stability of the MS levitator may be attained just during its motion along the stator cores. By this reason MS is called Magneto-dynamic suspension (MDS).  

It will be shown that employing saturation of the steel core back together with applying electromagnetic barrier make it possible to reduce the value of destabilizing force by up to 17 times while the stabilizing force reduces by 1.7 times only.. to add!

As it was illustrated in Fig 2.10 we rigidly connected two identical units in such a way that their planes of symmetry were reciprocally perpendicular, similar to Fig.2.3. We will remind that the vertical unit is supporting, the horizontal unit is guiding. The reduced destabilizing force of the supporting unit is suppressed by the stabilizing force of the guiding unit and vice versa: the reduced destabilizing force of the guiding units is suppressed by the stabilizing force of supporting unit. Thus we obtain a double magnetic device with its equilibrium Q coinciding with the minimum of potential energy. Therefore, following the Lagrange-Dirichlet theorem this device is stable with respect to any shift of the magnets.

Fig. 2.11  Schematic cross-sectional view of magneto-dynamic suspension system

Two such double devices located mirror symmetrically are rigidly affixed to the vehicle at both its sides and extending along the whole of its length (Fig.2.11). They form an MDS where both external forces and torques are compensated by internal stabilizing forces and torques. Therefore MDS is completely stable. Each shift and turn of

the vehicle instantly and faultlessly produce the stabilizing force and torque necessary to bring it to the assigned trajectory.

If to cover the core surface by aluminum screen, eddy currents induced there during the vehicle motion produce oppositely directed magnetic field which performs function of an electromagnetic barrier suppressing density B_n of the leakage flux from saturated core. This is the third step toward attaining stable suspension.

2.4  Forces acting on the levitator.

As said above any shift of the magnet assembly (levitator) with respect to the stator results in stabilizing and destabilizing forces appearing. The condition of stability (2-6) of the MDS system has been found analytically. When this condition is fulfilled, potential energy of the system has a strict local minimum, the resulting force is stabilizing and the system is stable. It has been proven that:

a) in order to satisfy the condition of stability it is necessary to ensure a certain level of saturation in the steel cores;

b) during magnets motion the level of saturation can be achieved with the help of aluminum screens covering the core backs and suppressing leakage fluxes from their lateral surfaces.

It has been also proven that the MDS system reacts to any shift and/or turn of the vehicle by producing internal magnetic forces and/ or torques which return the vehicle to its trajectory.

The conclusion has been done that it is possible to build an engineering system ensuring a stable flight of the magnet assembly (levitator) along the extended steel cores (stator) by exploiting rigid constraints between magnetized bodies and dependence of their permeability on field intensity.

Now we consider a magnetic device "unit" (Figs. 1.2 and 2.12) comprising two immovable steel laminated cores having unlimited length and C-shaped cross-section, and four rectangular permanent magnets No.1 – No.4 (with dimensions 2h ´ W ´ l_M) assembled into a quadrupole by unsaturated steel insert. The cores consist of thin backs (with cross-section 2l_s ´ t_s) and thickened core tips. The distance between the cores equals 2W + t_w + 2g, where t_w is thickness of the steel insert, g indicates the air gaps between the magnet poles and the ends of the core tips.

Similar to the scheme of Fig.2.8 the quadrupole magnets and cores in the unit form a closed magnetic circuit containing: four permanent magnets with each magneto-motive force mmf = e = μ_oW×J (where the magnetizing vector J indicates coercive force H_c) and four internal reluctances  r_i, four air gaps of total length 4g with magnetic reluctances r_gR and r_gL at the right and left sides respectively, two C-shaped steel cores with magnetic reluctances 2r_fR and 2r_fL at the right and left sides respectively (when the core backs are saturated).

The unsaturated steel insert with its magnetic reluctance equaled to zero separates the right-hand and left-hand loops of the magnetic circuit. This allows to determinate magnetic fluxes Ψ_wR and Ψ_wL, following the Ohm's law.

Any force f = f_n n^o + f_t t^o  acting per unit surface S of the steel core in the magnetic field may be determined by formula [8]:

(2-8)

where n⁰ and t⁰ are unitary vectors: normal and the tangential to the surface S respectively; B_n and H_t are respectively normal magnetic flux density and tangential magnetic intensity to S (from outside); μ_f is relative permeability.

If the steel is unsaturated, then  μ_f » ¥ and H_t » 0. In this case  f_t = 0, and, therefore (2-9)

Magnetic field of the quadrupole is attenuating quickly as distance grows. Then, assuming, that the magnetic forces act only on the lateral surface S of the four unsaturated core tips we obtain the formula:

_(2-10)

According to the Newton's Third Law a force of the same value but of opposite direction acts on permanent magnets in the unit. All the four tips and four magnets are identical and symmetrical with respect to the planes Y = 0, Z = 0 (Fig. 2.12).

Hence, the first integral in formula (2-10) can be rewritten in a form :

(2-11)

where the letters "R" and "L" in subscripts indicate the right side and left side core tips respectively.

We can express the above integrand as a product of two multipliers and assign them as:

At even cylindrical surfaces S_R= S_L of the core tips the values of the magnetic flux densities B_R(Q_R) and B_L(Q_L) as well as  are continuous. In addition we have:  (a) at the shift of the magnets to the right-hand side: B_R(Q_R) > B_L(Q_L), and  (b) at the shift of the magnets to the left-hand side: B_L(Q_L) > B_R(Q_R). Consequently, functions g(Q) and f(Q) satisfy all the conditions of the generalized theorem about mean value [9]. Hence, a point Q^* exists at S_R , where

However   and 

Therefore (2-12)

Similarly, we can transform the other integral in formula (2-10) and prove that:

(2-13)

 
Fig 2.12  Cross-sectional and perspective view of MDS unit

where the letters "b" and "t" in subscripts identify the bottom and upper halves of the core tip surface respectively.

The constants in the formulae (2-12) and (2-13) are expressed as follows:

Consequently, the force F_y is proportional to the difference between the working fluxes. This difference appears at the shift +Δy of the magnets because of imbalance between the reluctances of the right- and left-hand parts of the air gaps.

In the right-hand part of (2-13) we can see the parts of the same working fluxes:

and

penetrating into the tips from the bottom and top respectively. Force F_z is proportional to the sum of the difference between these parts caused by the vertical shift Δz and the redistribution of the permeances of upper and lower parts of the air gaps (Fig. 2. 15).

It follows from formulae (2-12) and (2-13) that an equilibrium is achieved at symmetrical position of a quadrupole between the cores since

,  ,   and

In this case the equilibrium is vertically stable because any vertical shift δ = Δz of the quadrupole causes stabilizing force F_s = -F_z counteracting the shift and tending to bring the quadrupole back to its symmetrical position. However, the equilibrium is unstable with respect to horizontal direction because any horizontal shift δ = Δy of the quadrupole causes destabilizing force F_d = F_y increasing the shift and tending to move the quadrupole off the symmetrical position.

The differences between magnetic fluxes in the right-hand parts of formulae (2-12) and (2-13) grow with increasing shifts Δy and Δz of the quadrupole from the equilibrium. Therefore, the internal forces in a small vicinity of the equilibrium can be expanded in Maclaurin's series and expressed by

, (2-14)

with accuracy up to higher order derivatives.

The experiments on physical models of separate units together with accurate analytical calculations of the forces  F_s  and  F_d    indicated  that within the interval  0 ≤ Δz,  Δy ≤ g / 2  the stiffness of the destabilizing  ∂F_d / ∂y  and  stabilizing  ∂F_s / ∂z forces are almost constant.  Forces  F_s  and  F_d   are reciprocally perpendicular. Therefore,  if four identical units are assembled in a manner shown in  Fig.2.13  they form Magneto-dynamic Suspension (MDS) system. In this system  a force  F_s  acting on a quadrupole compensates for a force F_d acting on another. Depending on whether F_s > F_d  or  F_s < F_d, any shift of the levitator will produce only stabilizing or destabilizing force acting on the quadrupole's assembly. Then we can say  whether the MDS is stable or not.

2.5  Conditions for stable levitator flight.

As was said the MDS system consists of two independent parts: the freely movable levitator (the quadrupole assembly) and immovable stator (steel cores assembly). The shape of all parts, both the stator and levitator, is cylindrical, with the generetrix parallel to the trajectory of the levitator movement. Permanent magnets and the magnetized adjoining fragments of the steel cores produce a magnetic field causing internal magnetic force F and torque M which move and turn the levitator.   The stiff constrains connecting the magnets of all the units into a solid levitator (Fig.2.13) perform the following functions

1. make the shifts Δy and Δz of the magnets interdependent: Δy_i = Δz_i+1,        Δz_i = Δy_i+1;

2. create proportional relation between shifts Δy, Δz of the magnets and turning angles α_x, α_y, α_z (of the whole levitator around the coordinate Axes  0X, 0Y, 0Z);

3. summarize all the external forces and torques applied to the levitator.

Fig 2.13 MDS assembled from four identical units

All above functions make it possible to express the resulting force F and moment M acting on the levitator in terms of the forces F_si and F_di acting on the quadrupoles 1, 2, 3, 4 (Fig.2.11).

Let us consider a levitator of length L_x (along Axis 0X) and width L_y (along Axis 0Y). The forces opposing a displacement of the levitator from its equilibrium are assumed to have a positive direction. At the small shift  δ = jΔy + kΔz of the levitator eight forces acting on them (stabilizing forces F_s1 = F_s4 , F_s2 = F_s3 and destabilizing forces F_d1 = F_d4 , F_d2 = F_d3) may be summated geometrically. Therefore, the value of the resulting force is:

Substituting the forces by the expressions from formula (2-14) we obtain

When turning  the front  (x = x₀) and rear (x = x₀ - L_x) ends of the levitator around Axes 0Y or 0Z by a small angle  , both ends of the levitator will be shifted by δ = ± α∙L_x /2 thus producing a pair of forces ± F_Σ (δ) with an the arm L_x /2 ( i.e., torque M (α) ):

When turning the left side (y = - y_o) and right side (y = + y_o) of the levitator around Axis 0X by an angle α_x the both sides of the levitator will be shifted by δ_z = ± α_x∙L_y/2 thus producing a pair of forces ± F_Σ (δ_z)  with arm L_y/2 ( i.e., torque M (α_x)), where L_y is the width of the levitator:

It means that if the conditions of stability are satisfied (2-6), the MDS system produces only stabilizing forces and torques at any small shift δ and any turn α . In other words, it is a self-regulating stable system.

Thus, in order to achieve stability of MDS system it is both necessary and sufficient to ensure just one condition: the stiffness of the stabilizing force ∂F_s/∂z in a unit should exceed the stiffness of the destabilizing force ∂F_d/∂y in a vicinity  [ δ < g ] at symmetrical position of quadrupoles.

The greater the difference between the left-hand and right-hand sides of the inequality (2-6), the more stable the MDS system.

As we said above, in theoretical mechanics similar systems are considered as conservative (since they conserve their magnetic field energy) with their stability obeying the Lagrange-Dirichlet theorem [1]. According to this theorem a conservative system has a stable equilibrium if its potential energy has a local (i.e., not coinciding with boundaries of magnetized bodies) minimum. It was also shown above that when condition (2-6) is satisfied in all the MDS units, the system potential energy possesses a strict local minimum along the whole track of the levitator flight.

2.6  How to reduce destabilizing force with the help of steel saturation.

In order to reduce the destabilizing forces in a unit it should contain saturated steel parts in it’s the closed magnetic contour ( see paragraph 2.3). For this purpose quadrupole magnets are situated between unsaturated tips of C-shaped cross-section stator cores. Saturation of the core backs by increasing field induction up to B= (2.05 to 2.06)T  at μ_f ≈ (40 to 20) can be attained by selection of its dimensions, such as :  l_s and t_s  - of the core backs, W and 2h -of the quadrupole's magnets, and g – air gaps between magnets and core tips. Then non-linear magnetic reluctances r_fR and r_fL compensate changing linear magnetic reluctances  r_gR and r_gL at qudrupoles shifts δ < g .

The profile of core tip cross-sections should coincide with the branches of hyperbola (with Axes “a” and “b”) , Fig.2.14. In this case a conformal mapping function transforming the domain between core tips into the canonical domain ( the upper semi-plain ) is known, and the precise calculation of forces F_s and F_d can be easily obtained.

The expression for a destabilizing force (2-12) is rather simple. It suggests a way how to suppress the destabilizing force F_d and its stiffness ∂F_d/∂y by employing non-linear magnetic characteristic B (H) of the steel core.

It is known that core magnetic reluctance grows rapidly as a magnetic flux increases. Having the magnetization curve B_f (H) (Fig.2.6), it is easy to build function  ρ_f (B_f), where ρ_f = μ_oH_f ¤ B_f is a relative unitary magnetic reluctance of the core steel. Fig.2.7 illustrates function ρ_f (B_f) for M-5 grain-oriented electrical steel (of thickness = 0.012").  One can see that if  B_f ³ 2.02T, then the curve ρ_f (B_f) may be approximated with a sufficient accuracy.by straight line equation

(2-15)

where  B_s = 1.996 T , N = 1.916.

This peculiarity is inherent to all magneto-soft steels. As a consequence of this characteristic the magnetic density B_f is uniformly distributed in a cross section of saturated steel cores  and is determined by:

The magnetic reluctance r_f of the saturated core back (B_f > 1.01B_s) of length l_s  and thickness t_s equals:

(2-16)

where b = l_s ¤ (N∙t_s) and R_f is absolute magnetic reluctance of the core back.

If quadrupole is in equilibrium and the unit core backs are saturated, then a small shift Δy < g increases the flux Ψ_WR ( caused by the reduction of reluctance r_gR) and at the same time increases magnetic reluctance r_fR of the core by value Δr_fR. Vice versa, a decrease in the flux Ψ_WL ( caused by the growth of the reluctance r_gL) will lessen the reluctance  r_{f L} by value  Δr_fL. Thus, in a unit with saturated cores the horizontal shift of the magnets Δy changes the full reluctances  r_m = r + r_f  of the magnetic circuit loops: in the right-side loop  r_mR= (r - Δr_g) +(r_f + Δr_f),  and in the left-side loop r_mL= ( r + Δr_g) + (r_f - Δr_f)   by value r_mL - r_mR = 2(Δr_g - Δr_f). In these expressions r = r_i + r_g is a linear part of the loop reluctances (spaces between the steel insert  and the tips  in Fig.2.12) which is less than in case of unsaturated cores, i.e., r_gL - r_gR = 2Δr_g > 2×(Δr_g - Δr_f). It follows that the difference between the working fluxes and therefore the destabilizing force F_d= C_d (Ψ_WR- Ψ_WL) produced by this difference in the unit can be reduced by employing core steel saturation.

To employ core steel saturation more effectively, i.e., to raise levitator stability it is necessary to ensure optimum correlations between the dimensions of magnets and cores (W, h, l_s, t_s, L, l_t, g) and characteristics (J, N, B_s) of the permanent magnets and steel.

It is worthwhile to mention that we use the linear fragment of the curve ρ_f (B_f) (Fig.2.7) as its working part for calculation of magnetic forces in levitator since non-linearity of steel at this fragment comes up to the required level and in addition the working magnetic flux there can be calculated analytically. Therefore flux density B_f in core backs should vary just at the linear fragment between B_f1=2.02T and B_f2=2.09T (for electrical steel M5).

In this case it necessary to fulfill the following conditions:

1. At the shift Δy_m < g of the levitator magnetic flux density in the core back (at the left side in Fig. 2.7) is restricted from the bottom:   B_fL ³ 1.01B_s;

2. Saturation of core backs should not lessen magnetic flux density B_n on the core tip surfaces and working fluxes in the cores more than by  Ψ ⁰_W / Ψ_W = ε times(ε < 1.4). Assuming that  Ψ ⁰_W = e ¤ r and Ψ_W = e / (r + r_f (B_f) ), where e is magneto-motive force (mmf) expressed  by e = μ_o×W×J  we can find  ε from the  following expression  r_f = r(ε - 1) = b(B_f - B_s) and hence: B_fR ≤ B_s + r(ε -1) ¤ b  that corresponds to restriction of magnetic flux density from the top (at the left side in Fig. 2.7).

In order to determine r_i and r_g separately it is necessary to find magnetic field distribution in the space between the insert and core tips of the qudrupole. However, in our case it is sufficient to find their sum  r = r_i + r_g, which is independent on magnetic field intensity. Therefore, the values of r,  r_R = r - Δr_g and r_L= r + Δr_g can be easily determined by calculating the fluxes in the circuit loops with unsaturated cores by the formulae:

where Ψ ⁰_WR and  Ψ ⁰_WL are the magnetic fluxes in the right- and left-hand unsaturated cores at the shift  ±Δy  of the unit's magnets.

When quadrupole is in equilibrium position (Δy = 0, Δ z= 0) the magnetic flux Ψ_W in the core with saturated back and magnetic reluctance r_f is: Ψ_W = e / (r+r_f).

Assuming that saturation ratio is ε = Ψ ⁰_W ¤ Ψ_W in the interval (1.1 £ ε £ 1.4),  we substitute the expressions for fluxes in the above formula and thus obtain:

Magnetic flux density in the saturated core back is:

Substituting the expressions for B_f  and  r_f in formula (2-16), we obtain the quadratic equation with respect to parameter b:

    (2-17)

The solution of the equation is:

    (2-18)

Then, we can find the thickness of a core back:

    (2-19)

that ensures automatic fulfillment of the above conditions 1 and 2.

After determining t_s  we can find working fluxes in the cores:

(2-20)

Substituting the expressions for   from (2-16) into formula (2-20) we obtain two square equations with respect to Ψ_WR and Ψ_WL:

, (2-21)

Solving them we find fluxes Ψ_WR and Ψ_WL in the right- and left-hand saturated cores:

	(2-22)

Thus, we have found the accurate link between fluxes  Ψ _WR,L in saturated steel cores with non-linear magnetic permeability μ _f  and fluxes  Ψ ⁰_W  penetrating in unsaturated  (μ=∞) core tips from the air. Therefore formulae (2-18, 19, 22) essentially simplify calculation of the forces produced by the quadrupole magnets.

After magnetic fluxes are determined, we can find their densities B_fR and B_fL in the core backs, magnetic reluctances r_fR and r_fL of the core backs and also the counter magneto-motive forces (that is mmf drops) : C_R and C_L which appear on length l_s of the saturated core backs reducing mmf of each magnet by the value:

at the right,

 at the left.

Thus, saturation of the core backs reduces mmf in the right- and left-hand loops and also lessens working fluxes and their densities B_nR and B_nL on the core tip surfaces (Figs.2.15 (a),(b) ).

, (2-23)

Fig.2.15(c) and formula (2-13) show that the ratio  Y ^b_W ¤Y ^t_W does not depend on saturation of the core back at the shift D z of the quadrupole.

Dimensions l_s and t_s of the stator core backs and also W and 2h of the quadrupole magnets (Fig.2.12) are related to each other as shown in formulae (2-18,19).

Fig.2.15  Illustration of lines of force in the gap between core tips and magnet poles

They are determined by saturation ratio ε = Ψ ⁰_W / Ψ_W and dependence ρ_f (B_f) (Fig.2.7) and ensure uniform saturation in a unit core back (B ≈ 2.05T, μ_f ≈ 40) when a quadrupole is in the equilibrium position. As a result at the small lateral shifts δ of the quadrupole magnets the increments of non-linear reluctances r_fR and r_fL of the core backs will compensate the increments of linear reluctances r_gR and r_gL of the air gaps thus essentially reducing the difference of magnetic fluxes  (2-12) and, hence, destabilizing force F_d =F_y (Fig.2.15)

2.7  Suppression of leakage fluxes from a saturated steel core.

When the core backs in the unit get saturated, their magnetic reluctances r_f (B_f) increase, and so does the tangential component of magnetic field intensity H_t on the lateral surfaces of the core backs. In this case, leakage fluxes grows as well, causing reduction of core backs magnetic flux density B_f , that, in turn, leads to that saturation falls.

Now we will show that it is possible to suppress the leakage fluxes and to maintain a required level of saturation (B_f) over the whole core back length 2l_s (Fig.1.12) by covering the core backs by aluminum screens and at the same time making magnetic fluxes in the cores to alternate periodically. In order to achieve this we split each levitator magnet into n equal parts and change polarity of all the even parts (turning them by angle 180^°). When such levitator moves with speed V_v magnetic flux alternates with frequency ω = 2πV_v /λ, where λ = 2L_l / v is the length of magnetic field traveling wave, L_l is the length of the levitator,  V_v  is vehicular speed.

To make the analysis more comprehensible let us simplify the magnetic unit design without changing its working characteristics. Cross-section of such a unit is shown in Fig.2.16. The unit is assembled from the parts of simple configuration. The lateral part of the unit containing a core back is presented as three-layer medium symmetrical with respect to the core back (parallel to Axis 0Z).

The levitator’s magnets of alternating polarity, flying over the stator’s core tips at small distance g , magnetize the core backs. The cores are recommended to be made of thin laminated steel M5 of thickness 0.3 mm (Fig.2.17). Eddy-currents induced in such cores are negligibly small, and therefore magnetization happens practically instantaneously over the entire back length  2l_s  producing there magnetic wave traveling along the guideway (Axis OX) with velocity V_v coinciding with the vehicle speed.. Both sides of each core back are covered by aluminum screens (Fig.2.16) of specific conductivity σ_a=3.7∙10⁷[1/(Ohm∙m)]. It is necessary that the core back has such a thickness to ensure magnetic flux density  B ≥2.05 T distributed over the entire length that corresponds to  μ_f ≤ 40μ₀ (Fig.2.7). In this case dropping magnetic reluctance r_g of the air gap g will be substantially compensated by growing magnetic reluctance r_f of the saturated core back.

The propagation process of a flat electromagnetic wave traveling over a metallic surface of high electrical conductivity σ has been investigated in detail in electrodynamics It has been proved that a part of the wave touching the surface (aluminum screen plane) decelerates its own velocity and changes its movement direction penetrating into the screen almost perpendicularly to the screen plane. Intensity of that part of magnetic field H and electrical field E attenuate exponentially as it penetrates into metal.

At the depth  ∆^o determined by simplified formula

      (2-24)

The values of H and E decrease by e = 2.7183 times.

In the formula (2-24)  λ is wave length,  V_v is its velocity in the core back (equaled to vehicular speed), is oscillation frequency, σ_a=3.7∙10⁷[1/(Ohm∙m)] is specific aluminum conductivity, , . The velocity V_em of the electromagnetic wave penetration into aluminum screen and its length inside the screen λ_em are determined by formulas:

, λ_em= 2π ∆^o , where σ ₀ = 3.7 . (2.25)

2.8  How to eliminate leakage flux from a saturated core back.

In our case the electrical part of electromagnetic wave has been vanished and only a non-sinusoidal magnetic part of the wave travels along the core back.

Fig. 2.17 Cross-sectional view of laminated steel core sheets with eddy current closed contours performed in large scale.

Its working magnetic flux Ψ_W is produced by two rows of identical rare-earth magnets of rectangular shape of alternating polarity (Figs. 2.9, 2.16) are rigidly affixed to the wider side of a steel insert. The narrow side of the insert is rigidly affixed to the bottom of the vehicle ( or to its external wall ) as shown in Fig. 2.13. Each group of four magnets form a quadrupole (Fig.2.16). Each pair of magnets affixed to the same side of the insert produces its own part of the working magnetic flux ∆Ψ_W closed through a separate loop including: the laminated steel core (made of thin sheets of electrical steel M-5),  two air gaps g (between the magnet poles and core tips), and the insert body (Figs. 2.4, 2.5).

Since all the steel sheets of the laminated core are electrically isolated from each other the alternative flux in each sheet induces its own eddy current contour ∆ i (Fig.2.17). The eddy currents flow in the cross section of the sheet: on the right and left its side in opposite direction (+y and –y) closing on the ends adjacent to aluminum screens as shown in Fig. 2.17 and 2.16. Since the width t_s of the sheet is much bigger than its thickness h_l (t_s>30h_l) and the directions of the current in both sides of the sheet are opposite to each other the resistance of the contour is big and the current is very small in value. In addition electrical fields of contrary currents in the sheet compensate each other and therefore the resulting electrical field outside of the sheet is negligibly small. As a result only magnetic wave travels in the core along axis OX .with velocity V_v equaled to the speed of the moving vehicle. In this case magnetic leakage fluxes ∆Ψ_l with their density B_n =B / µ_f = 2.05/40 = 0.05125 T will penetrate into the screen from the saturated core. The lines of forces of the leakage fluxes induce contours of eddy-currents on the screen surface. According to Lentz’s Electromagnetic Inertia Law magnetic field produced by the eddy currents in the screen is almost equaled and directed contrarily to the leakage fluxes thus suppressing them. Approximate estimations have shown that thickness of aluminum screen t_scrs =1cm is quite sufficient for suppressing leakage fluxes. In addition it insures durability of unit assembly. Consequently a metallic screen of high conductivity performs a function of an insulator of alternative magnetic flux. Hence the level of saturation of steel core may be maintained on required level that insures a necessary value of the stabilizing force F_s at a vehicle shift ( in range ± g/2) and its stable flight.

We assume that magnetic flux density there is B ≈ 2.05T (at μ_F £ 40). Then the amplitude of magnetic field intensity H_{f 0} in the core back is

    (2.26)

where K₀ is the amplitude of magnetic field intensity in the core back.

Alternating magnetic fluxes ∆Ψ_l  penetrating into aluminum screens induce eddy-currents which in turn produce a counter magnetic field suppressing amplitude of leakage flux ∆Ψ_l oscillations and, thus maintaining saturation in the core backs along the whole lengths.

Plane magnetic wave traveling in the laminated core back along Axis 0X crosses the planes of its steel sheets (Fig.2.16) in three-layer medium (Fig.2.18) containing two flat aluminum screens. The lateral surface of the core back of Fig. 2.16 is the upper boundary (plane y=a) of the layer No.1. The plane y=−a is the bottom boundary of the first layer of the laminated steel core back.

The layers No.2 ( from the top and bottom of the core back) are aluminum screens of thickness b-a, conductivity σ₂ = 3.7 ×10⁷[1/(ohm×m)],  and permeability μ₂ = μ_{0 .}  At the very top and bottom of the medium is the air (y > b, y <− b) with conductivity σ₃ = 0, and permeability μ₃ = μ₀  (layers No.3).

The main part of the wave in the layer No.1 travels along Axis 0X with constant speed V_v of the vehicle not spending much energy. However its lateral parts, touching aluminum screen, interact with free (mobile) electrons and produce eddy-currents. Energy  of these parts of the wave is spent for heating screens. Their velocities V_em drop abruptly and direction of their movement becomes almost perpendicular to the screen surface.

MDS is an example of a conservative system comprising permanent magnets and steel cores covered by aluminum screens. Its potential energy has a local extreme. However when the MDS levitator is still, its equilibrium is unstable because in this case the extreme appears to be maximum. As levitator speed increases, leakage flux decreases, energy maximum is suppressed and then, at the speed of V>10 m/s, will turn into minimum. When the levitator speed exceeds 100 m/s the stiffness of its stabilizing force per 1 m of the levitator length reaches 2.0×10⁶ N/m². Such stiffness ensures stable flight of the vehicle along the core tips without touching and friction at the shifts from Axis OX less than 3 mm. It means that it is possible to make an air gap between a magnet and core tips less than 1 cm.

Conclusions: it is possible in principle to build stable MDS based on rare-earth permanent magnets and steel laminated cores employing electromagnetic barrier.

2.9  Design of magneto-dynamic stable suspension system.

It has been shown above that at speed V>V₀ the levitator of the MDS system is capable to fly stably along the extended stator steel cores.

Next will show that the value of the magnetic levitation forces F_s produced by permanent magnets can be sufficient for suspension of a high speed transportation vehicle. Moreover, their stiffness ∂F_s / ∂d considerably exceeds the values achieved by the other types of suspension – EMS and EDS

During levitator motion, interrelated and interdependent mechanical and electromagnetic processes appear in MDS system. The intensity of these physical processes depends completely on the shapes and sizes of the parts of magnetic devices assembled into MDS system, conductivity and permeability of their materials, and also environment.

Stable flight of the levitator is possible if the values of all parameters, both geometric (such as: W, h, l_m , l_s , t_s ,l_t ,g) and physical (such as: μ, J, σ, N, B_s) correlate. Thus, the internal stabilizing force F_s (counteracting any shift of the levitator from the assigned trajectory of its flight) and its stiffness ∂F_s /∂d essentially depends on these twelve parameters. Therefore, the search in multi-dimensioned space for a domain where the force F_s would exceed a necessary value F_sn is a complex problem.

The existing numerical methods are not capable to ensure a necessary accuracy for solving this problem. Therefore, we propose an analytical method [3] for calculation of magnetic field and internal forces in magnetic devices (units) of MDS. The essence of this method is to find a preliminary information about the integral properties of the sought solution (in our case magnetic field distribution), then to utilize it both in formulation and algorithm of solution. This allows narrowing down the rank of functions where the solution exists, thus reducing the volume of calculations and distortion of the information.

The above mentioned method is based on some knowledge which may simplify the analysis of magnetic field in the unit such as:

1. Magnetic field in a unit is assumed plane-parallel.

2. Each quadrant of the unit magnetic field may be calculated (Fs.2.12, 2.8) separately because:

a) an unsaturated (μ=∞) steel insert eliminates mutual influence of one loop of  the unit magnetic circuit on another. It means that magnetic field of right-hand and left-hand permanent magnets are independent;

b) at levitator speed V≥V₀, aluminum screens become non-conductive for alternating magnetic flux thus eliminating interaction between upper and bottom fields of permanent magnets in a quadrupole that also become independent.

3. Magnetic field in the unit is a result of superposition of two magnetic fields:

a) primary magnetic field, produced by permanent magnets in unlimited space (partial solution of Poisson's equation), and

b) secondary magnetic field, produced by magnetization currents in steel cores (solution of Dirichlet's boundary problem for Laplace equation) [12].

4. The primary field is determined by an integral of the product of the magnetization vector J and the Green's function over the magnet cross-section. In our case, the magnet cross-section is a rectangle with sides 2Wx2h (Fig.2.19), and the above integral can be evaluated analytically. Scalar magnetic potential j₁ of the primary field and the components of the vector of primary magnetic intensity H_1y and H_1z are determined by the following formulae:

(2-27)

	(2-28)

These formulae are accurate at μ_r=1. Actually the relative magnetic permeability μ_r of rare-earth magnets Crumax equals 1.07. Consequently, the above formulae provided more conservative results the primary magnetic field by (2 to 5%)

5. The secondary magnetic field is determined by solution of Dirichlet's problem for Laplace equation with respect to potential j₂ on the domain boundary. The accurate analytical solution of this problem in terms of Poisson's integral is known only for canonical domains such as: upper semi-surface, circle, or infinite strip [13]. However, this solution can be expanded over domains D (Fig.2.21) of more complicated profiles, if an analytical function, conformably transforming domain D into a canonical domain, is known.

6. To simplify search for conformal mapping function we exploit a peculiarity of the sought field.

If fragment D₀ between two magnetic lines Y=C₁ and Y=C₂ is restricted by two cylindrical non-conductive (m=0) surfaces (Fig.2.20) then magnetic field outside this domain will not change. Actually, on the surface Y=const. normal components of intensity are H_nº 0 and H=H_t, and on the boundary of impenetrable medium H_nº0 and H=H_t as well. Consequently, by superposing the boundary of the impenetrable medium with the surface of Y=const. we will not change the boundary conditions for the magnetic field in domain D, which is equivalent to preserving the same field.

7. The value of the ratio of the forces F_s / F_d  in the unit depends on the core tip profile. We have to find the profile ensuring maximum of the above ratio. To simplify the search we will seek the tip cross-section profile within hyperbolic family (Fig.2.21). The sizes of “a” (real) and “b” (imaginary) of the half-axis of hyperbola are parameters which unequivocally determine its two branches. The function conformally transforming an infinite strip into a domain bounded by branches of a hyperbola is known [14]

8. It follows from 6. and 7. that the profile of the internal screen surfaces (Fig.2.12), covering the core tips of length l_t from the top and bottom, should coincide with the surface Y=const. formed by the magnetic lines. As was said above at speed V³V₀ of the moving unit magnets the immovable screen becomes impenetrable for the magnetic flux produced by the moving magnets with alternating polarities. In this case, the distribution of magnetic flux density normal to the surface of the real tip (of length l_t) is the same as on a tip of hyperbolic profile (of infinite length) when the magnet is immovable. Then it becomes possible to determine the conformal mapping function for such a domain.

9. For evaluation of the forces F_s and F_d it is sufficient to find the distribution of the magnetic flux density B_n normal to the cross-section profile of the core tips within distance l_t (Fig.2.20).

10. Formulae 2-16 to 23 contain the information that has been found as a result of analysis of a magnetic circuit of the unit with saturated cores. This information is sufficient for evaluation of the forces.

2.10  Evaluation of magnetic field in a unit with unsaturated cores.

Hyperbolic profile of the core tip cross-sections makes it possible to calculate all the forces acting on the levitator at any shift Δy and Δz from its equilibrium.

Let us assume that the stator core backs are not saturated (μ_r=¥). In this case, the potential j_S=j₁+j₂=0 in the whole surface and volume of the cores. Therefore,  j₂=  – j₁

As a consequence of quadrupole symmetry at any shift Δy or Δz of the levitator, the potentials j₁ and j₂ on the steel insert remain equaled to zero. Thus, the boundary conditions of Dirichlet's problem are determined, and we can obtain its solution. Fig.2.20 is the illustration for calculation of magnetic field in the unit air gaps. The boundaries of the conformally transformed domain (core tips and screens) are outlined by solid lines. The profile of a magnet cross-section at equilibrium position and at shifts Δy and z_d are shown by dotted lines.

The analytical function Z(W) conformally transforming a strip of width 2a into a domain outlined by two branches of the hyperbola with its semi-axis “a” and “b” (Fig.2.21) is:

 (2-29)

where .

On the right-hand branch of the hyperbola v=0, W=u.  Hence,

Z(u)= a× chk₀ u +ib× shk₀u = y(u)+iz(u). (2-30)            

Formula (2-29) is derived from the known formulae [6,(4.3, p361 and 1.1.1, p347)] by two steps:

Step1.  Domain D on plain Z= y + iz  (Fig.2.21(a)) bounded by the profile of the core tips (branches of hyperbola V=0  and V=2a ) is conformally transformed into upper semi- plain D₁  (η ≥ 0 , Fig. 2.21b) with the help of conformal mapping function

where 

Step 2.  The upper semi- plain D₁  (Fig.2.21b) is conformally transformed into infinitive strip of width 2a (Fig2.21c) bounded by contours of the straightened core tips with the help of function (2-31)

The derivative of the conformal mapping function on the right-hand branch of hyperbola equals: , (2-32) where

    (2-33)

Formula (2-30) expresses the correspondence of the points (u_i,0) on the bottom boundary of the infinite strip (Fig. 2.21 c) with their images (y_i,z_i) on the right-hand branch of the hyperbola (Fig. 2.21a). Respectively the points (u_i,2a) on the upper boundary of the infinite strip have their images (-y_i,z_i) on the leftt-hand branch of the hyperbola. To make sure that formula (2-30) is correct it is sufficient to calculate values   of function Z_i (W_i) at some range of values u_i on the bottom(v=0) and upper (v=2a) boundaries of the infinite strip. Coordinates Z_i will coincide with branches of hyperbola.

Next we select series of points u_i. on the bottom boundary of the strip with a constant interval starting from u₁=0 in rather large vicinity to embrace the whole core tip contour. Then, applying formula (2-30), we calculate the corresponding coordinates y(u_i)=y_i and z(u_i)=z_i on the right-hand branch of hyperbola. The following step is to find module M_0i and arguments j _0i of the derivatives of conformal mapping functions in these points by formula  (2-32). Then, we substitute the values of coordinates y_i and z_i in formulae (2-27) and (2-28) and, thus, find the values of the potential j_1i and intensities H_1y, ,H_1zi in the points of the right-hand branch of hyperbola. We obtain the same values of potential, but with the opposite sign in the corresponding points of the bottom boundary of the strip:   j_2i= - j_1i. because of, as it was said at the beginning of this paragraph, j_S=j₁+j₂=0 for unsaturated steel.

Formula (11.14) presented in [13] determines normal magnetic flux density of the secondary field on the bottom boundary of the strip of width h=2a. As was mentioned above, each unit has unsaturated steel (m_r=¥) insert with potential j₂º0. The above function (2-30) transforms straight line y=0 of the insert boundary into straight line v=a. Thus, it conformally transforms the domain between the right-hand branch of the hyperbola and insert boundary into the strip of width h=a.

Considering all said above and applying formula (11.14) from [13] we obtain the following formula for calculation of the resulting flux density B_n(y_i,z_i) on the right-hand core tip Fig.2.22)

(2-34)

where α_R=α-Δy,   α_L=α+Δy,   α=W+g.

The value n₁ (the number of points on the bottom boundary of the strip) in the integral limits of (2-34) should satisfy an inequality    n₁ ≥2· n₂, where n₂=α/Δu. In this case, we neglect the tails of the integral with limits from ±n₁Δu to ¥ because its contribution in magnetic flux density is very small. As me mentioned above, aluminum screens   (Figs.2.12) together with unsaturated steel insert split the magnetic field of the unit into four independent quadrants which are numbered clockwise. When the levitator magnets are shifted in a horizontal direction (Δy≠0, Δz=0) magnetic fields produced in quadrants No1 and No 4 are the same, and so are in quadrants No 2 and No 3. When the levitator magnets are shifted vertically (Δy=0, Δz≠0) magnetic fields produced in quadrants No 1 and No 2 are the same, and so are in quadrants No 3 and No 4. The results of above calculation is illustrated by Fig.2.22.

Consequently, the calculation of magnetic field in a unit with unsaturated cores is reduced to series calculations: starting from determination of values y_i , z_i ,M_0i, j_0i by formulae (2-30), (2-32), then, potential j₁ and primary field intensities H_1y ,H_1z by formulae (2-27), (2-28), and, finally, distribution of normal flux density B_n on the tip contours by formula (2-34). Knowing flux densities B_n , it is easy to determine fluxes in the core tips and forces acting on the magnets.

Calculating densities of B_n we have to take into account that quadrupoles of the guiding units (Fig.2.13) are shifted from the equilibrium by a lateral force T_y<3t for the whole vehicle. Meanwhile, quadrupoles of four supporting units are shifted downward by  z_d £ 6 mm by vehicle weight G_v=25t.

2.11  Calculation of forces and stability of the MDS.

Characteristics of MDS may be determined by evaluation of the magnetic field and forces acting in a levitator at different dispositions of its quadrupoles.

It was shown above that calculation of the distribution of magnetic flux densities B_nR and B_nL over the core tip surfaces in a unit with saturated core backs is reduced to calculation of B^o_nR and B^o_nL over the same core tip surfaces with unsaturated core backs.

Fig. 2.23 Graph of the distribution of the normal magnetic flux density (solid line) and specific magnetic force (dotted line over the core tip surface).

Quadruples in the different types of units have different equilibrium position. For guiding units Δy=0; Δz=0, and for supporting units  Δy=0; Δz=z_d (Fig.2.20). Therefore, to achieve the same level of saturation in their core backs, it necessary firstly to calculate  Y⁰_W and Y⁰_Wz in the guiding and supporting units correspondingly, while assuming their core backs unsaturated (μ_f=∞). Then, employing formula   (2-19) we calculate thicknesses t_s and t_sz of the core backs and then reluctances r (for the guiding unit) and r_z (for the supporting unit) by the same formula (2-16). Finally, we can determine the force F⁰_z which attracts the steel cores to each other in the guiding units and the levitation force F_z in the supporting units.

One can see from Fig.2.7 that curve ρ_f (B_f) coincides with the straight line ρ_f =(B_f-B_s)/N only at B_f ≥2.02. Since  B_fr >B_fl the formulae (2-15) are true only at B_fl ³ 2.02T.      Consequently, if it is necessary to increase permissible shift Δy, we have to increase the level of saturation ε up to value sufficient to fulfill the above condition: 1.1 ≤ ε ≤1.4      The value of forces acting on each magnet is determined by distribution of magnetic flux density B_n on the bottom – “b” and upper – “u” parts of the right – “R” and left – “L” core tip profiles. When the magnets are in equilibrium (Δy=0; Δz=0) and the cores are unsaturated, the distribution of B ⁰_n and magnetic fluxes Y ⁰ are identical for all parts of core tip profiles.

At the simultaneous shifts Δy≠0 and Δz≠0, the distributions of B⁰_n are not symmetrical. Therefore, it is necessary to find distribution of B⁰_nR(,L), and fluxes for all four parts of the core tip profiles by formula (2-12), then to evaluate distribution of densities B_nRs and B_nLs when the core back is saturated by formulae (2-22, 23)

Calculation of the eight coordinated components of forces F⁰_Rby , F⁰_Lby , F⁰_Ruy , F⁰_Luy , F⁰_Rbz , F⁰_Lbz , F⁰_Ruz , F⁰_Luz acting on all four parts are:

(2-35)

where the magnetic flux density B_nR is calculated by formula (2-34) for unsaturated core backs.

One can see from formula (2-35) that specific densities of magnetic forces f(l_Q) acting on unit core tip surface is proportional to the square of the normal magnetic flux density distributed over this surface. Therefore as moving away from the peak of core tip profile (point 0 of Fig. 2.23)) towards its base the relative density of forces      f ⁰(l_Q) = f(l_Q )/ f(0) is reducing much faster than normal magnetic flux density (Fig.2.23). To recalculate the same forces F_Rby , F_Lby , F_Ruy , F_Luy ,F_Rbz ,F_Lbz,, F_Ruz, F_Luz when Δy=0 and the core backs are saturated it is sufficient to divide the forces F⁰ by ε², when Dy¹0. The recalculation is performed by formulae (2-22, 2-23).

The forces F_s(Δz), F_d(Δy) acting on the quadrupoles of the unit at its shift Δz and/or Δy are determined by the following formulae:

                F_s(Δz₁) = (F_Rbz+F_Lbz)-(F_Ruz+F_Luz),

F_d(Δy₁) = (F_Rby+F_Ruy)-(F_Lby+F_Luy).                           2-36)

The stiffness of these forces is determined as follows:

(2-37)

The stable flight of a vehicle is ensured by six units (Fig.2.24). Levitation force F_4lv (equaled to stabilizing force) of four supporting units should counterbalance weight G_v of the vehicle and also compensate for destabilizing force F_2dv of the two guiding units. At the same time stabilizing force F_2sv of two guiding units should compensate for destabilizing force F_4dv of four supporting units and counterbalance an external lateral  force T acting on the vehicle. In addition, the stiffness of forces F′_4Lv of four supporting units should exceed stiffness of destabilizing forces F′_2dv of two guiding units and stiffness of stabilizing forces F′_2sv of two guiding units should exceed stiffness of the destabilizing forces F′_4dv of four supporting units.

;  ;  ;  ,   (2-38)

where

F_4Lv is levitation force of four supporting units; 

F_4dv  is destabilizing force of four supporting units;

F_2sv  is stabilizing force of two guiding units;

F_2dv  is destabilizing force of two guiding units;

G_v    is vehicle weight;  T is lateral force acting on the vehicle;

To ensure safe flight it is necessary to know permissible value of external lateral force, acting on the vehicle (T_max) and corresponding value of deviations Δy, Δz of the vehicle from the assigned trajectory. The permissible deviations determine a flat domain D_st that we will term as “Domain of stability”: Design of the units should ensure fulfillment of above inequalities (2-38) in the whole domain, including its boundary.

2.12  Calculation of a working example of MDS system.

A. Technical requirements for MDS system (Fig.2.24) are:

• Vehicle is carrying 100 passengers;  
• The maximum speed of a vehicle flight V_max=150m/s;
• Dimensions of the vehicle: length L_v=20 m; width L_W = 3.0 m; height H_v=3.0m;
• The weight of the loaded vehicle  G_v= 25 t; 
• The permissible lateral load on the vehicle T =3 t:
• Minimum radius of curve = 1000 m;

Fig.2.24 Cross-sectional view of stable suspension - Magneto-Dynamic System containing four supporting and two guiding units
(Click image to enlarge)

Calculation has been performed by computer program MDLS-CLC following the accurate analytical formulas:

a)   (2-9 to 13) and (2-15 to 23) for magnetic fluxes  ,  penetrating into core tips and also their values reduced by ε times - Ψ_W  for saturated core backs;

b)   (2-27 to 28), for φ₁, H_1Y and H_1Z of primary magnetic field produced by a permanent magnet of cylindrical shape and rectangular cross-section;

c)   (2-29 to 32), for analytical function Z(W) conformally transforming a strip of width 2a into a domain outlined by two branches of hyperbola with its half- axis  “a” and “b” (Fig.2.21) and its derivative ;

d)   (2-34), for calculation of flux density  on core tips (Fig.2.22);

e)   (2-35), calculation of components of forces  and .

Then employing the same computer program we calculated stabilizing F_s(∆z) and destabilizing F_d(∆y)  forces acting on one pair of opposite core tips of one unit at shifts ∆z and ∆y of its quadrupole in the range 1 mm to 6 mm through 1mm . The results of calculations are shown in Table 1, (columns No.1 and No.2). The values of F_s and F_d  are expressed in [N/m] (per one meter of a pair of core tips length).

Each unit has two pairs of core tips. As was said above, a vehicle is levitated by four supporting units and guided by two guiding units. All MDS units, both guiding and supporting, are identical. Therefore the forces acting on 1 m of vehicle length are: levitation (stabilizing) force F_4L and destabilizing force F_4d , (both produced by four supporting units) and also guiding (stabilizing) force F_2s and destabilizing force F_2d , (both produced by two guiding units).  They are calculated in terms of obtained above values of  F_s(∆z) and F_d(∆y) as the products:

F_4L= 2∙ 4∙ F_s(∆z) = 8 F_s(∆z) ; F_4d = 8 F_d(∆y);  F_2s = 4 F_s(∆z); F_2d = 4 F_d(∆y)

To satisfy conditions (2-38) the lengths of quadrupoles are chosen different for supporting and guiding units. For each supporting unit the length equals 15 m, i.e. 5λ, for each guiding unit it equals 22m > 7λ, where  λ =3 m is the length of magnetic wave. Therefore to find forces acting on entire vehicle it is necessary to multiply the value of each force (F_4L ,  F_4d , F_2s  and  F_2d ) by the length of corresponding magnet:

F_4Lv= 15∙8 F_s(∆z) =120 F_s(∆z);  F_4dv = 120 F_d(∆y);

F_2dv = 88 F_s(∆z); F_2dv = 88 F_d(∆y)

Table 1