20.1 Superconducting Coil RFLS: Competitive Technologies

RFLS for MAGLEVs may be embedded in quite a few technologies. We select here a few, deemed as fully representative, [1,6]:

• With conductive sheet track (secondary)—Figure 20.1a

• With conductive ladder track (secondary)—Figure 20.1b

• Normal-flux type (Figure 20.1a,b)

• Null-flux integrated levitation, guidance and propulsion, Figure 20.2

• Coil + PM magnetic bearing (Figure 20.3a) [7]

• PM-PM magnetic bearing (Figure 20.3b) [8]

The competitive technologies in Figures 20.1 through 20.3 may be characterized as follows:

• The normal-flux topologies with sheet or ladder track are considered as basic because they reveal the principles and fundamental characteristics: the levitation and eddy currents drag forces, F_lev, F_drag versus speed for various net airgap (levitation height) values in the 50–150 mm range. A levitation goodness factor for forces at peak speed may be defined as

  GF=(FlevFdrag)Umax⁡⁢(20.1)

  The typical system dynamics behavior characterized by static stability but undamped oscillations occurs in the some conditions.

• The ladder track is used to increase force goodness factor G_F but levitation and drag force pulsations occur, due to ladder transverse currents.

• The null-flux systems (Figure 20.2) have been introduced to increase the force goodness factor (which has a notable impact on propulsion power) and to lower the speed at which the peak value of drag force occurs and its value. The neighboring propulsion system ac armature coils may provide repulsive-attractive vertical (and guidance) forces by adequate vector control with controlled i_d, for damping the levitation vertical or lateral oscillations. Controlled i_q is applied for propulsion. A secondary mechanical system is all that is required for meeting the Janeway ride comfort curve (standard).

  Note: In Chapter 8, the integrated propulsion/levitation/guidance SC-MAGLEV topology has been presented. The eight-shape stator track coils produce propulsion/levitation/guidance by proper connections and control on the two sides of the vehicle. This topology is not covered here.

• Repulsive magnetic PM bearings may be built with an air coil, with controlled current, that repels a moving PM (or vice versa). As the air-coil mmf (and losses) tends to be notable, a dual PM-PM bearing may be built where the control coils (connected together) may be controlled to regulate the system dynamics during rotor motion as well as the takeoff and soft landing at standstill. For this configuration, if the airgap target is not to be tracked with very high precision, the zero (current) power control described in Chapter 19 may be applied as well.

• A power levitation goodness factor for levitation and guidance may be defined as

  Gp=Flev⋅UFdrag⋅U+Pcontrol⁢(20.2)

  P_control is the control power

• Even for a 130 m/s SC MAGLEV, it was demonstrated that force goodness factors in excess of 50/1 may be obtained, while the control power for levitation and guidance may be within 5 kW/ton of vehicle. Zero average current/power levitation control may be attempted provided a dedicated secondary mechanical suspension system damps the vehicle oscillations and a tertiary cabin suspension system provides for ride comfort up to peak speed. The control of vehicle dynamics (except for propulsion) from the propulsion coils on ground reduces notably the power necessary on board of vehicle, which is to be produced by “electromagnetic induction” through dedicated generator coils on board placed between the SCs and guideway.

• Superconducting coils may be of low temperature (4–10 K, with liquid helium cooling) and strong or of high temperature and soft (40–70 K, with liquid nitrogen cooling). The appearance of an SC is shown in Figure 20.4a (for basics of superconductivity, see Chapter 1).

In essence, Figure 20.4a comprises the superconducting-wire coil, liquid helium (nitrogen) dewars, fiber glass evacuated insulation, supports, and electrical connections. The helium (nitrogen) dewar is isolated from the environment by a vacuum zone with stainless steel walls around it.

For strong SCs, the liquid nitrogen is first poured; after its evaporation, liquid helium is poured until the desired (4-10 K) temperature is reached. Only after that, the electric current is flown into the SC coil gradually; finally, the coil is short-circuited inside the low-temperature structure (Figure 20.4b). The SC may be replenished periodically (after 1 day or more).

20.2 Sheet Secondary (Track) Normal-Flux RFLS

The superconducting coils have large mmf (above (3–4) 10⁵ A turns) as they are supposed to produce in the guideway coils plane NSNS flux densities in the range of at least 0.2–0.3 T for a vertical (lateral) distance of a 0.2–0.3 m (center to center) between the two.

For start, let us consider a coil in air producing a magnetic field in point P (Figure 20.5a). As shown in Chapter 8, by using Biot-Sawart law, via Neumann formula, the magnetic flux density components B_x, B_y, B_z in point P may be obtained, as illustrated in Figure 20.5b.

The normal (along levitation direction) B_z component for one coil is far from a sinusoid along the vehicle motion direction. However, B_z varies closely to a sinusoid along the lateral direction. A row of SCs may produce, by proper length/width pitching and levitation heights, the desired flux density distribution along all three directions: x, y, z.

The SC field distribution can then be investigated analytically allowing for

• Cosinusoidal distribution along oy

• Fourier decomposition of longitudinal distribution (along ox) with the retention of fundamental, third, and fifth harmonic

• Dependence along oz (vertical) direction results from the Laplace equation and is essentially exponential for each longitudinal harmonic:

  ∂Hx0∂x2+∂2Hy0∂y2+∂2Hz0∂z2=0⁢(20.3)

with the solutions

To find the integration constants C_v0, we may use another expression of B_z0 = μ₀ H_z0 obtained from the Neumann formula (for a rectangular coil) [6]:

Putting together a few NSNS polarity SCs with a pitch Lv > L (Lv ≈ 1.2 L) and adding their contribution for a given z (height), and at, say, y = 0, after decomposition in Fourier series along X direction, we may identify from (20.6) the integration constants C₀v (v = 1, 3, 5). It has been shown that the exponential variation of magnetic field along z (vertical/levitation direction) produces errors within (2%–3%) for z=0.1–0.3 m, SC length L = 1–3 m, and SC width 2b = 0.5–0.8 m.

A similar Poisson field equation is valid for the reaction magnetic field of induced currents in the sheet secondary:

If in (20.9) the SC field variation in the sheet secondary is neglected, ∂² H_rz /∂z² = 0; (d_AL=20–25 mm), the solution of (20.9) simplifies to [6]

with

but Hz0ν,Hy0ν,Hx0ν are available, (20.4) through (20.6), and thus H_rz, H_rx, H_ry may be calculated.

The x, y secondary current density lines may simply be calculated as

For L=2.7 m, L_v = 3 m, I₀ = 3 × 10⁵ A turns, 2b = 0.5 m, z₀ = 0.25 m, 2c = 0.7 m, d_Al=25 mm at U = 10 m/s, and at U = 100 m/s, the variation of induced current density components in the secondary sheet along the direction of motion is as shown in Figure 20.6.

As the secondary sheet thickness is d_Al=25 mm, it seems (from Figure 20.6) that the variation of current density along its depth, at high speeds, may not be neglected.

This will be true even for ladder secondary when, to reduce skin effect, a transposed wire conductor has to be used.

Further on, the levitation and drag forces F_L and F_d (Figure 20.7) are

An attractive method to calculate the levitation force is to start with image levitation force F_Li, which is produced by the SC in interaction with its image (Figure 20.8):

The variation of levitation and drag force with speed may be approximated as [6]

Analytical 3D or FEM-3D methods may be used to calculate F_L(u), F_d(u), and then calibrate Equations 2.17 and 2.18 as practical approximations.

For L = 3 m, 2b = 0.5 m, d_AL = 25 mm, z₀ = 0.3 m, n₁ = 0.2, I₀ = 2.315 × 10⁵ A turns, F_Li = 55 ton, the results of using (2.17) and (2.18) are shown in Figure 20.7 for a 45 ton vehicle. The results look good, as the peak drag force occurs at a small speed and the levitation goodness factor at 130 m/s is about 45.

The drag power is still about 1.3 MW at 130 m/s, while the air drag of the vehicle is 3 MW.

It is thus evident that sheet guideway solution implies a very high drag power.

Doubling the levitation goodness factor is needed for a competitive MAGLEV. This is how the ladder (coil) secondary was introduced.

20.3 Normal-Flux Ladder Secondary RFLS

The schematics of a ladder secondary are shown in Figure 20.9.

Let us still consider l_T, the pole pitch of SCs, and l_R, the ladder loop length:

n is the member of ladder loops per SC pole pitch (n > 1).

Also, L_c is the self-inductance of a ladder loop, M_oj the mutual inductance of loops o and j. Only the right and left loop interactions are considered here. R_t and R_t are the resistances of longitudinal and transversal sides of the loops.

So the equation of loop o in Figure 20.9 is

where ψ_sc is the flux of superconducting coils in loop o and E its emf:

So the induced current in the loop o writes

Finally,

with

and

Now the levitation and drag forces for the superconducting coil are

where M_m is the mutual inductance between the SC and m the track loop (m > R for each SC). As expected, the discrete structure of the ladder secondary leads to levitation and drag force pulsations. They should be reduced by proper design; also the peak drag force should be forced to occur at low speeds to facilitate vehicle starting.

For l_T = 3.2 m, L = 2.2 m (SC length), 2b = 0.5 m, the harmonics of E are very small. For n = 2 and N-count SCs the forces are obtained from (20.23) to (20.27):

with M_L—the mutual inductance between an SC and its image (2.15). For n = 2 and v = 1,

So

For N = 12 SCs, and the aforementioned data, T₀₁ = 0.15 s, I₀ = 2.3 × 10⁵, when the vehicle has 45 ton results as in Figure 20.10 [5] are obtained (−M_L/(∂M/∂z) = 0.134 m).

It is clear that the larger the time constant T₀₁, the faster the image levitation force is obtained, however, at increased ladder secondary costs. For a realistic T₀₁ = 0.15 s, the image force is obtained around 200 km/h. The peak value of drag force is only 0.32 × 10⁵ N more than two times lower than for the sheet secondary. Also, the levitation goodness factor reaches a staggering value of 142 at 130 m/s. The force pulsations are small as E₁ has small time harmonics, by proper design. The drag power is much smaller than for sheet secondary, but track cost may not be smaller as the manufacturing costs of the ladder secondary tend to be larger.

For even better performance, the null-flux system was introduced.

20.4 Null-Flux RFLS

The null-flux system may be realized with horizontal or with vertical coils; the vertical coil solution is easy to build and handle by the vehicle. The figure-eight-shape secondary is a typical realization of this solution (Figure 20.11).

When the SC is vertically in the horizontal axis of the track coils, the SC flux in the latter is zero. But, as soon as the SC (the vehicle) falls by z₀ from that position, the flux is not zero anymore. A net levitation force is produced by the interaction of the SC field with the track-induced currents. Let us denote by M₁ and M₂ the mutual inductances between the SC and the two parts (upper and lower part) of the eight-shape coils in the track.

Then, proceeding as for the ladder secondary (with n = 2) we may obtain the levitation and drag forces per vehicle as

R_cd is the resistance of upper (or lower, if equal to each other) track coil and L_cd is the inductance of the same, considering the influence of neighboring track coils.

Taking the same example (l_T = 3.2 m, L = 2.2 m, 2b = 0.5 m) with z₀ = −2 × 10⁻² m, we may obtain

Following (20.31)

This way, the levitation stiffness S_z is

and may be calculated as S_z = 12 (it was about 8.0 for the sheet secondary); the peak drag force, for the 45 ton levitation force is now F_dk = 0.228 × 10⁵ N < 0.32 × 10⁵ N of the ladder secondary (so G_L > 200!). The levitation goodness factor is thus better. But to obtain the required levitation force, the SC mmf has to be notably larger than for the normal-flux systems.

Note: As shown in Chapter 8, the vertical eight-shape track coils connected to form a nonoverlapping three-phase winding (q = 1/2 slots/pole/phase), and connected across the vehicle to secure guidance higher stiffness, can provide integrated propulsion levitation/guidance.

All done so far in this chapter has paved the way with fields and forces fundamental knowledge to approach this rather complex solution; but it is beyond our scope to follow it farther.

20.5 Dynamics of RFLS

To illustrate the fundamentals of RFLS dynamics, we still use the normal-flux topology, as it is simpler to evaluate the essentials.

Let us consider the SCs of a bogie, on one side, and use the small deviation theory:

The track irregularities are neglected and no other suspension system is considered; the vertical motion equation is

Notice the ∂M/∂z < 0 with I₀ and U as constants:

In (20.39), it was assumed that F_Lo = mg.

The solution of (20.39) is evidently sinusoidal:

At high speeds, with

the system stiffness S_z (20.36) is

For z₀ = 0.3 m and the sheet secondary numerical example in Section 20.2 S_z = 8.0, γ = 8.75 rad, f₀ = γ/2π ≈ 1.4 Hz.

The oscillations are not attenuated, and thus the system is dynamically unstable while statically it is stable, as S_z > 0.

This rationale is valid mainly at high speeds. It has been proved that at same speeds [9] there is some positive damping for the ladder secondary (not so for sheet secondary).

20.6 Damping RFLS Oscillations

The oscillations in an RFLS may be damped in three main ways:

• Passive electric dampers (PED) (Figure 20.12)

• Active electric dampers (AED) (Figure 20.13)

• Secondary suspension system (SSS) (Figure 20.14)

• Combined passive electric and secondary suspension system (PED + SSS) (Figure 20.15)

A short characterization of the solutions reveals the following:

• It has been shown that PED, consisting of a conducting plate (or shorted coil) placed between the SC and the guideway, is able to produce a damping time constant τ < 1 s only if the distance between the SC and PED is at least 0.15 m [10]. This leaves 0.02–0.05 m vehicle-guideway airgap and thus imposes lower irregularity (higher costs) track, to avoid collisions. Also, below 2 Hz PED is not able to damp the SC oscillations.

• The active electric damper is supposed to be capable of damping the MAGLEV oscillations and assure enough ride comfort, but at notable energy consumption on board; this solution will be detailed later.

• The active (controlled) pneumatic SSS is also capable of damping oscillations and providing full ride comfort up to the maximum vehicle speed [11]; being fully mechanical, it is not followed here in more detail.

• The PED + SSS (passive) system may be capable of damping the vertical oscillations and providing ride comfort from 1 Hz onward [12,13] at reasonable initial system costs. This solution will also be treated here in the same detail. For a thorough synthesis on dynamics and stability of RFLS, see Ref. [13].

20.7 Repulsive Magnetic Wheel

The superconducting paddle wheel vehicle, Cryobus (Figure 20.17) [6], was earlier proposed for MAGLEVs [18].

The configuration of Cryobus as in Figure 20.19 may be characterized as follows:

• The superconducting heteropolar SC wheels are rotated over a triple ladder or triple short-circuited coil raw, placed along the track, to produce propulsion (much like a linear air-coil induction motor), levitation, and guidance by repulsive forces; a mechanical active secondary system is to be designed to damp oscillations for a safe and comfortable ride.

• The magnetic wheels produce propulsion/levitation/guidance as long as ac is produced in the guideway; it means that by spinning the wheel at, say, nonzero slip frequency the MAGLEV might develop enough force to lift the vehicle at zero speed. To provide lift force at zero propulsion force, we simply rotate half of the wheels in one direction and the rest in the opposite direction. So, theoretically the takeoff may take place from standstill.

• When the wheel peripheral speed U_wp is larger than the vehicle speed U, the propulsion force is positive while for U_wp < U regenerative braking is produced.

• As only the lower half of the wheel interacts with the guideway, the upper half may interact with arch-shape synchronous motor ac stator windings, to “paddle” the magnetic wheel, and save space and hardware. With 2p = 8 poles, 4 poles may be used to “paddle” the wheel.

• The free rotating copper cylinder, Figure 20.17b, may be used to damp the oscillations in levitation (as PED).

• As the magnetic coupling between the circular wheel and the flat guideway is weaker than for flat SCs, the overall efficiency of Cryobus is expected to be lower than for flat-coil RFLS–MAGLEVs. However, when used to integrate proportional levitation guidance functions, the Cryobus may attain a total power efficiency up to 65% and thus compete with electrodynamic flat-coil active guideway MAGLEVs in a simpler topology (see Ref. [6], Chapter 12, for preliminary design).

• The Cryobus needs all power to be transferred on board of vehicle as the track is passive, and thus it may prove to be an overall cost/performance solution worth reconsideration.

• Recently, the superconducting wheel has been proposed to be replaced by a heteropolar PM wheel [19,20]. The same concept was proposed to compensate the dynamic end effect in high-speed LIMs. If the PM wheel could duplicate the performance of the SC dual-axial-pole wheel (Figure 20.17b) with a higher than 1.5 T remanent flux density, then this latter solution might stand a practical application challenge, even if flying at a smaller height (0.04–0.06 m).

• In general, the replacement of SCs by strong PMs is so far less practical because of the lower weight of SCs at 0.15–0.2 m rated gap for given lift force. But stronger PMs of the future at smaller gap might reverse the situation soon [21,22].

20.8 Coil-PM Repulsive Force Levitation System

The experimental setup in Figure 20.20 [23] may be applied directly to an industrial MAGLEV platform or to an axial bearing of an air-core PM synchronous motor/generator inertial battery. In contrast to attraction force levitation systems, it allows long (up to 20 mm or more) excursion length.

To analyze the dynamics of this 1 DOF system with vertical position tracking control, the motion equation including coil force, gravity, friction, and external force disturbance is used:

where

If V(t) is the control effort, the levitation force may be approximated as follows [23]:

With and, after separating the time varying uncertainties, from lumped uncertainties L(x,t), Equations 20.62 and 20.63 may be put in the form

with

For robust control, the sliding mode control approach is corroborated here with a radial basis function network (RBFN), to estimate online the uncertainties in the system dynamics.

The sliding mode functional S(t) is standard:

The values of λ₁ and λ₂ represent the desired dynamics and may be calculated from given rise time and overshooting as in a second-order system. The globally asymptotic stability may be guaranteed (based on the Lyapunov method) with a control law of the form

Following the approach with the RBFN estimator, for m = 0.121 kg, B = 2.69, a = 1.65, b = 6.2, λ₁ = 10, δ = 10, λ₂ = 30, results as in Figure 20.21 and Table 20.1 [24] have been obtained.

The response is stable, the position tracking quality is good, and the control effort is reasonable for a large airgap.

The synthetic results in Table 20.1 [24] compare PID, SMC, and SMC-RBFN control systems and demonstrate that the SMC-RBFN performs best on an overall basis. An alternative adaptive back-stepping intelligent control system for the same prototype [7] has been shown to track complex position/time waveforms rigorously.

Still, the results in position tracking refer to small frequency bands, and further investigations into high-frequency band position control are needed for dynamically critical applications.

20.9 PM-PM Repulsive Force Levitation System

In the search for high rigidity (stiffness), the PM-PM RFLS has been proposed for axial bearings. To operate with no (or minimum) control, the repulsive force is beneficial as it provides static stability; it does not provide positive damping in general. However, as discussed previously in this chapter, the passive small resistance electric damper may help to produce positive damping in such applications where the axial speed is small. So, either PED or AED may be used in a PM-PM RFLS (Figure 20.22).

The multiple PM disks (12, 24, etc.) are mounted in the aluminum plates (Figure 20.22). The number of PM disks per circle, their thickness, and the diameters of their placing on stator and rotor may be used for optimal design and stability. For a simplified analytical model, the magnets in air may be replaced by their intensities F₁ = H_c A₂ and F₂ = H_cA₂, where H_c is the coercive magnetic field (say H_c = 1.78 × 10⁶ A/m), residual flux density B_r = 1.08 T (approximately μ_rec = 1 p.u.); A₁ and A₂ are the PM pole force areas. The PM pole intensity is considered as concentrated in the center of the PM pole. The force is

where r₁₂ is the distance between the PM pole centers.

The number of PM disks on rotor and stator per circle is the same; u¯12 is the direction of the force. There are quite a few components of this force (attraction and repulsive type) as the PMs may be “replaced” by air coils with an mmf of I₀ = H_c • h_PM (Figure 20.23). As it is easy to imagine, at small gap z₀, the resultant force will have an attraction character, while for larger gap it will become a repulsion force.

For a certain z/h_PM and PM radial length/z, the repulsive force will be maximum.

Concerning the radial force, it has to be negative (centralizing) for stable guidance, and this happens only for small radial displacements in combination with lower rotor than stator diameter of placing the PMs.

For h_PM = 10 mm, at z = 2.5 mm, the repulsive force, for D_r = 47.5 mm and D_s = 50 mm, is maximum at 23 N [8], while the radial force is negative up to 1 mm radial displacement, but only at –0.5 N. When the rotor rotates, the 24 PMs on both rotor and stator assume different relative positions and thus the repulsive (vertical) levitation force pulsates heavily even with D_r = 74.5 mm and D_s = 50 mm (± 6 N).

However, much like for cogging force, if the number of PMs on stator and rotor is different (say 26/24), the levitation force pulsations during rotor rotations become small; alternatively, with two out of phase rows of PMs on the stator at D_s1 = 50 mm and D_s2 = 45 mm, with D_r = 47.5 mm, and 24 PMs on both rotor or stator, the repulsion force pulsations become small. While in principle the PM-PM RFLS works, there is still a lot of work to be done to

• Analyze it by 3D-FEM

• Elaborate an optimal design code to check the performance/cost

• Include the eddy currents in the aluminum plates

• Develop a state space circuit model for dynamics and control

• Investigate static and dynamic stability

• Introduce oscillation passive damping or closed-loop control via coils for refined levitation height tracking [25]

All the aforementioned issues are beyond our scope here, but seem worthy of further consideration by the interested reader.

20.10 Summary

• Repulsive (electrodynamic) force may be developed in practical devices between

  • Conductors carrying currents of opposite polarity

  • A moving current carrying (or superconducting or a PM) coil raw and a fix secondary conducting sheet or ladder

  • Two rows of PMs of opposite polarity

  • One row of PMs and controlled current air coils

• The repulsive force tends to separate the two bodies with an intensity that increases when the distance (air gap) between the two decreases. It is thus statically stable. This is in contrast to attraction force levitation systems (AFLSs), which are statically unstable.

• Large repulsive forces may be produced with strong PMs or superconducting coils at air gaps 10 times larger than for AFLS (where 10–15 mm is typical); this way the track irregularities may be larger and thus lower costs are expected.

• As levitation means statically and dynamically stable hovering of a body (e.g., a MAGLEV), the dynamics and some control for repulsive force levitation system have to be investigated.

• For MAGLEVs, RFLS have been proven practical up to 585 km/h.

• The RFLS is in general based on the interaction of superconducting coils (SCs) on board of vehicle to produce 0.3–0.6 T flux density in the plane of secondary (track) conducting sheet, ladder, or eight-shape short-circuited coils placed along the guideway of MAGLEV.

• Normal-flux RFLS have been proposed first, but the null-flux eight-shape short-circuited vertical coil guideway has been proven the overall best solution for integrated levitation guidance and propulsion.

• To investigate the fundamentals of RFLS with SCs, the normal-flux topologies with sheet or ladder secondary have been investigated in some detail.

• As expected, when a row of SCs with NSNS polarity on board of MAGLEV moves above a sheet or ladder guideway, induced currents occur in the latter.

• As the speed increases, the frequency of these currents increases and thus their circuit becomes more and more inductive; at high speeds, the induced currents look almost opposite in polarity to those in the neighboring SC. This indicates a strong repulsive (levitation force) F_L. But a drag force F_d corresponds to the losses in the conducting guideway.

• While the levitation force grows monotonously with speed U, the drag force shows a peak value with speed, much like the torque in a dc stator–fed induction motor with cage rotor.

• Two criteria have been introduced to weigh the levitation “goodness”: levitation force goodness (F_L/F_d)_umax and levitation power goodness FL⋅U(Fd⋅U+Pcontrol), where P_controlis the dynamic (control) power needed to replenish the SCs and for motion dynamic stabilization and control. The higher the two criteria values the better for some given initial system costs.

• Using a larger airgap than AFLS, the RFLS with SCs has been so far proven, at higher SCs length, capable to produce better performance at higher speeds (up to 585 km/h); the SCs volume for same levitation force tends to be smaller than that of dc-controlled iron-core coils.

• But the levitation force increases with speed, and the critical speed at which the drag force has its peak values leads to the conclusion that the vehicle has to start and accelerate on retractable wheels (much like an aircraft, while AFLSs work as helicopters do).

• For 3 m long, 0.5 m wide SC coils, RFLS have been built to produce (F_L/F_d)_{130 m/s} above 50/1 for sheet guideway, 120/1 for ladder guideways, and above 200/1 for null-flux (eight-shape track coils) systems.

• An analytical study of RFLS showed that by careful design SC flux in the track (sheet or ladder type) may vary almost sinusoidally in time. Also, the SC flux density in the guide-way plane varies rather cosinusoidally along the transverse direction. Consequently, the vertical force flux density varies along the airgap exponentially. A practical field theory may be obtained to produce levitation and drag force dependence on speed and on system geometry. Curve fitting may be used to match these theoretical results to the image levitation force based (Neumann) formula and a proposed speed dependence.

• A similar (but based on circuit theory) model for the ladder and null-flux guideway system is obtained. In this case, however, additional pulsations in F_c and F_d occur; by a proper design, they may be reduced to negligible values.

• This way the aforementioned high levitation force goodness values have been obtained.

• Using these models in the motion equation (of a 1 DOF system), it is proven that the self-damping coefficient is negative at all speeds for the sheet guideway and above a small speed (15–25 m/s) for the ladder or null-flux guideway.

• Damping these natural (inherent) oscillations may be accomplished by

  • Passive electric damper (PED)

  • Active electric damper

  • Mechanical damper

  • Combined passive (on bogie) electric and mechanical damper

• PEDs are in general incapable of damping vehicle oscillations properly for the entire frequency spectrum or providing ride comfort up to maximum speed.

• The active electric damper with a room temperature controlled coil on board, placed between the SCs and the guideway, has been shown to be capable of producing satisfactory ride comfort up to maximum speed for a maximum of 5% of SC mmf in the control coil. An average of 5 kW/ton of control power is required at 550 km/h. This is only 40% more than the control power in AFLS at 360 km/h.

• The ride comfort is still defined by the Janeway curve, which limits acceleration versus frequency due to track irregularities (up to 20 Hz).

• Even the passive electric damper (two vertical aluminum sheets on the bogie, parallel to SC) with a secondary suspension system of SCs to bogie and cabin seems capable of providing good oscillation damping from 1 Hz upward.

• In the null-flux RFLS, when the eight-shape track coils are connected as a three-phase ac winding (with q = 1/2 slots/pole/phase), and with connections of the coils on the two sides of the vehicle, integrated levitation and propulsion are obtained. This time vector control of the of the ac three-phase windings (as in a linear synchronous motor—Chapter 8) will control propulsion by iq control and stabilize the vertical vehicle oscillations by id control, instead of AED. Zero levitation power control may be attempted for RFLS as for AFLSs.

• The heteropolar SC or PM wheel on board rotating at U_w > U peripheral speed above a track sheet or ladder can be designed to produce integrated levitation/propulsion/guidance at a smaller airgap (0.04–0.06 m). The upper half of so-called magnetic wheel is driven by a three-phase arch-shape stator much like in a rotary synchronous motor. A passive guide-way MAGLEV is thus produced. Due to the smaller region of interaction, that is, weaker interaction of magnetic wheels with the guideway, the PMs (SCs) mmf has to be stronger than for flat RFLS of the same performance. Horizontal magnetic wheels with PMs and vertical axis with partial overlapping of guideway conductive sheets or ladders have also been proposed for RFLS [26].

• For standstill (bearings) or industrial MAGLEV platforms, repulsive force levitation may be produced by moving PMs and fix controlled current coils (much like for planar motion linear electric motors). Sliding mode control has proven very effective to track airgaps around 20 mm with good precision in such systems.

• PM-PM repulsive force levitation has also been proven adequate, with limited range lateral stabilization and passive (aluminum plate) damping, in axial bearings with rotors rotating at large speeds; active (close-loop) control is required to fully control such systems.

• It is estimated here that R&D on RFLS will develop quickly in the near future, both for super high-speed transport and for industrial applications (bearings and industrial MAGLEV platforms in clean rooms).