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CHAPTER 3. A SELF-REGULATING PERMANENT MAGNET LINEAR MOTOR
3.1 Peculiarities of permanent magnet linear synchronous motor (PMLM). Capacity for self-regulation and high stiffness of stabilizing forces in magneto-dynamic suspension (MDS) system ensures safe vehicular flight at small gaps. This makes it possible to work out a new design for self-regulating permanent magnet linear synchronous motor. The PMLM is designated for vehicle propulsion in high speed ground transportation systems. As distinguished from an ordinary linear synchronous motor, the PMLM has the following peculiarities:
Now we will explain the meaning of peculiarity (4) in detail. It is known that a rotating synchronous motor is also self-regulating: increasing loads on its shaft leads to increasing the shift of its magnet poles in the direction of their lagging from current half-wave in the stator winding while reducing the loads leads to reducing the shift. As a result rotation velocity does not change. The mode of PMLM operation is the same. Resilience of flux pipes (lines of force) lays in the ground of self-regulation of both types of motors. It is limited by value of the loads equaled to Lorentz force. For PMLM it is expressed by formula Fp= By ∙hg∙I , where hg is height of a lower yoke shoe equaled to a magnet shoe (Fig.3.2). A linear synchronous motor has the same kind of limited self-regulation as a rotating one. However the range of self-regulation in PMLM can be expanded in such manner to involve the whole possible diapasons of loads and speeds without falling out of synchronism. Let us consider the nature of propulsion force Fp. It is known that the Lorentz force FLor , produced by both electrical E and magnetic H fields, acting on electrical charge e that moves with velocity V , is expressed by formula: FLor = e( E+ μ0 [V∙H] ) However we
take interest just in a part of Lorentz force that is its component Fxlor
equaled to propulsion force Fp= Fxlor
produced by magnetic field μ0 Hy
=By,
acting at charge
When μ0H = B, and vectors V = Vz and B = By are reciprocally perpendicular, Lorentz formula looks like FLor = e∙[V∙B] = e ∙Vz ∙By
Fig. 3.1 Cutaway view of a half of permanent magnet linear
motor
In our case electrical current flows in vertical sections of the stator bus-bars situated in air gap gm (between lower yoke shoes and rotor magnet shoes moving with speed Vz ) where magnetic field By is homogeneous (Figs.3.1 and 3.2).
The current can be found by the following formula:
where n ≥ 2/3∙48=32 is the number of bus bars situated under magnet poles, Q is summarized charge flowing in n bus bars. Then the formula for component of FLor (i.e., propulsion force Fp ) is as follows:
All the bus bars of the stator winding are fixed and are immovable. Magnetic field By in the PMLM rotor displaces currents in the bus bars towards its back wall thus making pressure on them. According to the First Newton’s Law the same pressure acts on the PMLM magnets thus producing propulsion force Fp . When speed Vx of the rotor (affixed to the vehicle) is growing the length λ of the current wave, and also bus-bar width and rotor magnet poles length are growing too in proportion with the speed. Meanwhile voltage and thickness of isolation between adjacent bus-bars of the stator winding as well as current density j in the bus-bars remain constant. Therefore the sum current in the gap under magnet poles grows faster than vehicular speed. (The growth of the sum current similar to the growth of air resistance to flying vehicle is proportional to Vx2) All above peculiarities are very promising because they make it possible to ensure a self-regulation and stable operation of the PMLM. From Fig.3.3 one can get the detailed visualization of the rotor assembly together with its synchronizing mechanism and synchronizing devices mentioned in Paragraph 1.8.
3.2 Description of motor design. The PMLM comprises a linear stator assembly (the stationary part) and a permanent magnet rotor assembly (the moving part). As seen from Figs. 3.1 and 3.2 the linear synchronous motor contains:
All parts of the PMLM interact with each other. At first look their design may seem complicated. However both the general design and the design of every separate part obey common purpose: to ensure the PMLM stable work (without falling out of synchronism) and its self-regulation ( continuous adjustment of propulsion force to its speed). Design of PMLM stator As shown in Fig. 3.4 the stator has a common concrete beam with three toothed holders strengthening transverse segments of the turns of the three-phase winding along the whole stator length. When the propulsion winding is coupled to an electric generator, then a traveling current wave is produced in the transverse segments of the winding turns. Velocity V of the current traveling wave is given by formula: V = 2LWf, (3-1) where f = 25Hz is the current frequency, and LW is stator turn length. The conductors in the winding are aluminum buses. Each turn has a transverse and longitudinal segments. The transverse which contributes to creation of Lorentz forces, and also counter electromotive forces, are fixed in the slots of the holder with a single layer. Together with insulation they form a compact floor and lateral walls of a U- shaped traction channel. The longitudinal segments in facing bars as a monolithic multilayer of conductors agglutinated to one another by electrical insulation. The facing bars are freely inserted into the hollows of the rotor. In order to reduce its yoke weight the hollows must be as small as possible in size.
T
To meet these requirements the design of winding is accomplished in such a manner that number of longitudinal conductors situated in the facing bar would be minimal. Selected three - phase winding scheme is shown in Fig. 3.9 The PMLM stator winding is cut into fragments with each powered separately by a step-down transformer at the moment when a vehicle is passing it over. Current wave traveling in the stator winding coheres with magnetic field produced by the rotor magnets and propels a vehicle with force Fp (its maximum value is determined by Lorenz formula). This force must overcome the forces hindering the o meet these requirements the design of winding is accomplished in such a manner that number of longitudinal conductors situated in the facing bar would be minimal. Selected three - phase winding scheme is shown in Fig. 3.9 The PMLM stator winding is cut into fragments with each powered separately by a step-down transformer at the moment when a vehicle is passing it over. vehicle flight such as: inertia on its acceleration parts, gravity on slopes, air resistance along the whole track. Its value may reach up to 6-7 tons. Hence, each fragment of the stator winding should be durable enough to hold its propulsion force without deformation. Therefore it is necessary to build monolithic winding fragments, then to put them on concrete guideway foundation and to fill gaps between adjacent winding turns with durable solid isolation. The stator winding has three types of sections: an acceleration section, a constant velocity section, and a deceleration section. On the acceleration section of the stator, the length of the turns and the width of the buses smoothly increase towards the vehicle motion. As a consequence, the length λ and velocity V of the current traveling wave along the traction channel of the acceleration section increase until velocity reaches desirable speed VV of the vehicle. By contrast, on the deceleration section the lengths of the turns and the width of the bus-bars smoothly decrease towards the vehicle motion. Wave
length is λ = 2LW, and its velocity is V = f λ.
Then knowing the radius R of the curvature of the path turnings and
permissible value of centrifugal force
When the rotor is moving, the facing bars of the stator winding must move freely without contact through the hollows of the rotor C-shaped yoke core (Figs.3.1, 3.2). Accordingly, it is expedient to make a design the stator winding in such a way as to reduce the cross-section square of its facing bars as much as possible. This considerably reduces the weight of the steel yoke and, in addition, increases the durability of the winding, which the vehicle repels from. In order to reduce the cross-section square of the facing bars the winding is installed in such a manner that each winding phase consists of four parallel bus bars placed into eight toothed holder’s slots in strictly determined order alternating with two other phase bus bars, as shown in Figs.3.8 and 3.9. The peculiarity of the winding is that current in the bus bars placed into the slots of even numbers of each phase is directed from the starting point of the winding fragment (powered by a step-down feeding transformer) towards its end (Fig. 3.11) . At the end of this fragment all four bus-bars are banded backwards and placed into the slots of odd numbers with current flowing back to the feeding transformer. As a result, in all the eight transverse segments currents are summarized and produce current a wave traveling with velocity V while in its right- and left- hand facing bars the longitudinal currents travel with the same velocity (Fig 3.5). Their zero values coincide with the centers of the rotor poles. The scheme of the phase A of the winding is shown in Fig.3.8. In this case the maximum number of buses in the cross-section of the facing bars does not exceed 1/4 of the number of the transverse segments fitted in the traveling wave length.
The unfolded winding of the linear motor makes it possible to regulate traveling wave velocity by changing winding turns length. Consequently, at constant frequency it is possible to compose a strict program for regulation of current traveling wave velocity by means of non-uniform distribution of the stator winding turns length along the assigned track. On the other hand, the design of unfolded rotor poles makes it possible to change the magnets length proportionally to the stator winding turns length as the vehicle passes over. As a result, the Lorentz force grows proportionally to vehicular speed. Forces resisting vehicular motion increase along with speed. Thus, if the speed and feeding substations power distributions along the guideway are such that Lorentz force everywhere exceeds total sum of resistance forces, the propulsion motor will operate in a stable mode. The forces acting on the vehicle depend on values of its speed and acceleration. Expressing all forces (including FLor) in terms of lengths of the winding turns (i.e., vehicle speed) and equating their sum to zero we obtain a differential equation with respect to the winding turn lengths. Solving this equation, we find the distribution of the stator winding turn lengths and cross-sections, and also necessary distribution of Lorentz forces. Then we can determine capacities of all substations corresponding to this distribution. Now we will show how a propulsion force appears in the working gap of the rotor at vehicular movement when the traverse segments of the stator’s winding get into.
A view of the working gap gm of Fig. 3.2(c) is shown in Fig. 3.6 together with winding conductors and magnetic force pipes. During vehicular motion resistance force brakes the rotor and shifts its magnets back away from the current wave. But traveling wave current does not allow middle parts of the magnetic pipes to be displaced backwards together with the magnets, stretching and bending them and thus producing stretching forces tangential to the direction of the rotor motion. Summarized, the stretching forces produce a propulsion force Fp = FΣ proportionate to the displacement. The balance of the forces is restored and the rotor continues its motion with velocity V of current wave. However, if the force FΣ exceeds the Lorentz force FLor, stator winding currents are not able to hold magnetic pipes: they tear and propulsion force disappears. The rotor falls out of synchronism and stops. Consequently, an ordinary LSM is capable of self-regulation. If the force resisting vehicle movement changes, the LSM changes propulsion force, automatically maintaining rotor speed equal to the velocity of traveling current wave. Self-regulation continues until resistance force FΣ exceeds Lorentz force FLor. The above mentioned current wave traveling in the traction channel and along the working gap gm produces propulsion force:
that propels the rotor (and the vehicle). At the same time the magnetic field of the rotor magnets within the working gap crosses the transverse segments of the conductors and induces the counter-electromotive force:
which accompanies the process of transforming electromagnetic energy into mechanical work necessary to propel the vehicle. Next we will consider the PMLM rotor design that allows to regulate the propulsion force and thus ensures the Lorentz force exceeding the total resistance force at any point of the vehicular track. Design of PMLM rotor, its synchronizing mechanism and synchronizing device assembly The propulsion force can be regulated by two different manners. The first is increasing or decreasing the traveling wave current and the traveling wave length. The second is increasing or decreasing the pole pitch length. The most effective way is to apply both simultaneously. The optimum traveling wave length and permissible current have been considered in the stator winding design. Changing pole pitch length should be done according to the stator winding turn length at the point that the rotor is passing over at the moment. The rotor design allows for this. The PMLM rotor comprises two (front and rear) mirror-symmetrical halves (one of them is shown in Fig. 3.1) which are able to draw apart and together during the vehicular motion. This is necessary for changing smoothly the position of the rotor’s poles with respect to the traveling wave. We will obtain maximum propulsion force if the center of each magnet pole coincides with the middle of its traveling half-wave (Fig. 3.10). In addition to ensure increasing the propulsion force as the vehicular speed grows it is necessary to increase the length of rotor’s poles. For this purpose each half of PMLM rotor (front and rear) comprises of an assembly of several units (see Figs.3.1, 3.9 and 3.10) installed on the vehicle bottom in a row along the length of the vehicle. Each unit contains two (upper and bottom) core shoes situated opposite to each other and a permanent magnet of rectangular form rigidly attached to them. The pole shoes have cylindrical form. Their cross-section has a figured profile for evenly distributing magnetic induction. (Fig.3.2).). The units are inserted in the gap between the yoke core shoes and all of them except for a central one (which is the closest to the middle of the rotor) can be shifted between upper and bottom supporting tubes in a vertical direction with respect to the cores. Polarities of the unit magnets pertaining to one of the halves of the rotor are the same but opposite to those pertaining to another (Fig. 3.10(c)) The central unit is a special one (Fig.3.3): it is fixed at the lower position to the yoke and its upper pole shoe is a part of the yoke (Figs. 3.1 and 3.10). Its length L0 = 0.5m All the rest unit magnets have constant height hm and width wm,. However their lengths Li (i=1,2,...n) are different growing from unit to unit starting from the central one. (Figs. 3.9 and 3.10(c)). Therefore it is possible to increase the pole pitch length gradually engaging (dropping down) unit by unit thus adding their lengths to the length L0 of the central immovable unit. When the vehicular speed reduces the units are disengaged in reverse order. The lengths Li are calculated in such a manner to avoid jolt when engaging. Figs.3.2 and 3.3 illustrate the PMLM magnetic units. A working unit that can be switched on and off is shown in Fig. 3.2(b) and (c).The distance between the upper core shoes equals the width wm of the magnet, while the distance between the bottom core shoes is larger and equals wm + 2gm (Fig.3.2 (b)). Therefore, when magnet is shifted upward its magnetic field is closed through the upper core shoes and its unit is disengaged (Fig.3.2(c)). When the permanent magnet is shifted downward, the unit is engaged and then, together with its cores, the magnet creates a two-loop magnetic circuit containing two air gaps (distance gm ), (see Fig.3.2 (b)). In this case, the transverse segments of the winding are inserted into the air gaps gm of the mentioned circuit. The size of the air gaps is small and the magnetic flux density there has considerable value (about 1T). Rotor cores are rigidly connected with the vehicle bottom and can be moved only in the direction of vehicle motion (i.e., along Axis 0X). In addition, the unit magnets are connected with the yoke cores and can be moved only upward and downward with respect to the yoke cores (i.e., along Axis 0Z) and cannot move in a horizontal direction along Axis OY. Therefore, destabilizing forces Fy attracting the magnets to the yoke cores are compensated by the reactions of the constraints between the steel yoke cores and permanent magnets. It is appropriate to repeat that the movement of the rotor’s halves apart and together is performed by a synchronizing mechanism. Its simplest design is a long screw having both right and left-hand rectangular threads on both rotor’s halves (Figs. 3.1 and 3.3). An assembly of synchronizing devices is used for engaging and disengaging (moving up and down) the rotor’s units. Any electrical, hydraulic or pneumatic drive gear may be employed for this purpose. By embodying the above ideas in the design of PMLM it is possible to ensure self-regulation of the vehicle speed and propulsion force of the motor with stability condition fulfilled in every point of the track at minimum power of installed equipment.
3.3 PMLM operation. When stopping at a stop station the vehicle is suspended in the magnetic field of MDS supporting units and the horizontal wheels are pulled out from both its sides. During its departure from the station horizontal wheels are pulled out. The wheels support the vehicle during its motion along the starting section until its speed attains value V0= 25 m/s that is necessary to cohere with current traveling wave and to achieve stability. Now we will consider how a vehicle departs from a stop station and how to return a suspended vehicle to the nearest stop station in case of emergency. A vehicle is supplied with special side- wall pullout wheels supporting it during its motion on the start section (when it begins departure from a stop station) until its velocity reaches the value V0= 25 m/s, sufficient to achieve stability. The vehicle can be accelerated by induction motors (with pullout contacts) rotating the pullout supporting wheels. To power the induction motors three-phased contact bars are laid along the guideway walls. In addition special rails made of plastic or stiff rubber are fixed along the guideway walls where the pullout wheels may roll. At the acceleration section the winding turn length and its wire cross-section gradually increase, and so does velocity of the current traveling wave. As was noted above ( paragraph 3.2), acceleration of the vehicle is achieved with the help of both the synchronizing device assembly and the synchronizing mechanism. The synchronizing mechanism is designed for smooth regulation of the rotor pole pitches by drawing apart or together the magnetic unit assembly pertaining to the rotor’s front and rear halves. The synchronizing device assembly serves for step by step changing the length of the permanent magnet poles (simultaneously with changing the turn length of the stator winding during the vehicular motion) by engaging or disengaging a corresponding unit. The synchronizing mechanism starts working just after the vehicle reaches an acceleration section. It reacts on increment of the winding turn length and changes the distances between both halves of the rotor’s yoke, thus increasing propulsion force and the vehicle speed. Let us consider how the synchronizing device assembly works when the vehicle moves at the acceleration section. Each pair of mirror-symmetrical devices is switched on by a signal of its own sensor and increases gradually the length of the permanent magnet poles in compliance with increasing traveling wave velocity. Then, the propulsion force grows and this helps the vehicle to overcome the growing air resistance and vehicle inertia. The synchronizing device assembly engages and disengages the rotor magnet units in strict sequences. Each signal from the next sensor switches on/off its drive gear. In its turn that moves downward/upward two magnets of opposite polarity (pertaining to the front and rear halves of the rotor yoke) starting from the closest to the rotor central fixed magnets. Either process: electric, hydraulic, or pneumatic may be utilized for designing a drive gear. Fig. 3.3 shows the synchronized device assembly with engaged units. The last unit is disengaged.
3.4 Theoretical aspect of PMLM design. The summarized current in the cross-section of each facing bar of the stator winding equals zero (Fig.3.5). Consequently, current in the stator winding does not produce a magnetic flux either in the right or in the left rotor yoke and there is no direct armature reaction in the PMLM. The permanent magnetic flux running in the C- shaped yoke cores is produced only by the dropped rotor magnets (engaged in the magnetic circuit at the moment). Therefore it is easy to calculate precisely the electromagnetic processes in the PMLM in that way to optimize the shapes and sizes of its parts. Magnetic field in the working yoke gaps can be considered homogenous and therefore the propulsion force Fp can be expressed by Lorentz formula:
where B - is magnetic flux density (induction) in the working gap gm between the bottom core shoes and a magnet, hg - is a size of the bottom core shoes, and, I(t) - is the summarized current in the winding buses which gets into the working gaps of the front and rear rotor poles. A resisting force that brakes the vehicle motion consists of three components: inertia
gravity
and aerodynamic resistance
where mv is vehicle mass, a - is vehicle acceleration, G = 9.8m/s2 is gravity acceleration, b (x) - is an angle of slope of the track, F0 is the force of aerodynamic resistance at the vehicle speed V0. Hence, the propulsion force Fp of the PMLM rotor at any point of its track must satisfy the following equality: Fp(x) = q(Fi+FG+Fa)=q[mV a(x)+mV G·sinβ (x)+F0 V2(x)/V0 2] (3-10), where q>1 is a safety factor which guarantees stable work of the motor (i.e., the Lorentz force exceeds the total resisting force to the vehicle motion) . The forces hampering the vehicular motion grow with the growth of the vehicle speed and eventually they may exceed the Lorentz force (see paragraph 3.2). Therefore, to prevent the motor from falling out of synchronism during its acceleration, it is necessary simultaneously to increase its propulsion force. We can see from formula (3-6) that in this case it is necessary to increase the current I, by increasing the cross-section (or width Wb) of the winding buses together with the length of the pole-pitches.
The traveling current wave would have a sinusoidal profile and permanent amplitude if conductors of the three-phase winding were thin and placed in three layers (one phase per a layer) shifted with respect to each other by 2/3·LW (where LW is the length of a stator winding turn) Moreover, the density of the phase winding turns (i.e., the number of winding turns of a phase per unit of the stator length) would alternate sinuously along the stator. However in our case the winding conductors are thick and therefore all three phases must be placed in one layer in the slots of the toothed holder in order to fit into the working air gap between the magnet pole shoe and yoke lower core shoe (Fig. 3.2). In this case the profile the current wave is not sinusoidal and has configuration shown in Fig. 3.10 (a) and (b). However as we will show it below the propulsion force remains the same as if the current wave were sinusoidal. It will be also shown below the total current I(t) producing the propulsion force (3-6) differs from that of sinusoidal wave just only by 0.5%.
In order to reduce the vibration of the summarized current in the non-sinusoidal wave the transverse segments of the different winding phase coils must interchange in the order shown in Fig.3.9 where letters A ,B ,C (X, Y, Z) indicate front (or back) segments of phase turns. Each phase coil of the stator winding comprises eight winding turns which are placed into the holder slots in the strict sequence: X, C, C, X,X, C, X, X, B, X, X, B, B, X, B, B, Z, B, B, Z, Z, B, Z, Z, A, Z, Z and so on. Figs.3.10 (a),(b) illustrates two pictures of current distribution in a winding fragment length equaled to wave length λ at two adjacent moments of time:
Therefore oscillation of an amplitude of the total current in the buses in the working gap of the rotor is 14Im − 13.856Im ≈ 0.5% Im .
Fig. 3.9 Scheme of winding turns fitting in the length of traveling wave λ(V). Dimensions in [m] correspond to maximal velocity of the traveling wave Vm =150m/s From the distribution of the three-phase current in time shown in Fig.3.10 we can see that the values of currents in the phases are similar to those at moments t1 and t2 and alternate every 30o. It means that the pictures of current distribution in the winding buses also alternate periodically with frequency f0= 6f. We can also see from Fig 3.10 that the total (summarized) current grows when approaching the middle of the half-wave. We know that the length LW of the winding turns in the rotor air gap grows when the vehicle moves on an acceleration section. Thus, we can come to the conclusion that moving apart the rotor halves, we will synchronously increase the total current in the gap and the propulsion force. For evaluation of this phenomenon we will employ an utilization factor Ku for a rotor magnet. The utilization factor equals the ratio of value of the real current in the gap to its maximum value (i.e., the amplitude of the current Im in each bus multiplied by the number of the buses having entered into the gap at the moment).
Fig. 3.10 Space distribution of currents in traverse segments of stator's winding conductors at two different moments: (a) t1=90˚ ; (b) t2= t1 + 30˚ ; (c) disposition of rotor magnet with respect to a current wave traveling with speed 150 m/s.
The length L0 of the central magnets and length LW of the winding turns at the initial point of an acceleration section are equal to each other LW=L0=0.5m. The rotor starts accelerating with propulsion force Fp=B· hgIΣ just after it gets synchronization with the traveling current wave. At the same time the length and cross-section of the stator winding starts growing. At the beginning of the acceleration section all twenty four buses with the total current IΣ=14Im are situated in the air gap (Fig. 3.10(a)). At this moment Ku1=14Im/24Im= 0.5833. (3-11) At the same moment, a synchronizing mechanism starts operating, moving apart the rotor yoke halves synchronously in accordance with increasing the length and the cross-section of the stator winding turns. When the turning length increases by 1/3×L0 and attains value LW1=4/3×L0 then the rotor magnets of the length L0 will also move apart by 1/3×L0 (as shown in Fig.3.10(a)) and cover only central buses with total current IΣ=12Im. At this moment Ku2=12/16= 0.75=1.2857Ku1. (3-12)
Simultaneously the first unit of synchronizing device assembly in each yoke half is comes into action, dropping ( i.e., engaging in the magnetic circuit) the rotor magnet of the length L1 = LW1/6 = 1/6 ∙ 4/3×L0 = =0.1111m. Thus, moving apart rotor yoke halves during the vehicular motion on the acceleration section we increase the propulsion force by 28% . By this time an additional space is released for dropping down the next mirror-symmetrical pair of the rotor magnets of length L2 = LW2/6 = =0.148m. It will happen when the stator winding turns attains value LW2 = 4/3LW1 = 0.8889m and so on until the vehicle speed attains its maximum value Vmax=150m/s. The length of the engaged magnets is determined by the formula Li=2/9·L0·(4/3)i-1. (3-13)
The total length LΣm of all the magnets pertaining to one rotor pole (one rotor’s half) and corresponding to the vehicular maximum speed equals 2m. One can see from Fig. 3.11 that in spite of currents in two halves of the stator winding run in opposite directions their traveling wave directions are the same. Therefore magnetic flux in any yoke cross-section is zero that allows to eliminate armature reaction of the rotor (Fig.3.6).
3.5 Optimization of PMLM parameters. The propulsion force (3-6) is proportional to magnetic flux density B in the working gap gm of magnetic circuit and to the height hg of the gap (Fig.3.2). The scheme of the magnetic circuit of the rotor is shown in Fig.3.12. Magnetic field in the working gap is homogenous, the magnet cross-section is rectangular, but the length LΣi= ΣLj of the rotor poles varies at the different sections of the track. The magneto- motive force is determined by formula e=μ0 Hc ·hm., where Hc is the coercive force of a magnet, hm is its height. It worthwhile to note that the cost of the rotor is determined by the cost of the permanent magnets that must be considered in PMLM design. Our purpose is to ensure the necessary propulsion force at minimum volume of the magnets and we will show how to do this. The magnetic flux is Ψi=2BhgLΣi=e/(rin+rgi), (3-14) where hg is the height of the lower core shoe; The internal reluctance of the magnets is rin=hm/(μr wmLΣi), (3-15) where μr is its relative magnetic permeability. The reluctance of both working gaps switched in parallel is rg=gm/(2hgLΣi). (3-16) For permanent magnets Crumax 355 utilized in the PMLM Hc=8.9·105A/m, and μr=1.07. Substituting all values in formula (3-6) we obtain:
where
Let us consider the ratio of force (3-17) to the volume vi of the rotor magnets:
where d and b are constants:
and
The
value η is proportional to propulsion force produced by a magnet
of unitary weight. Therefore it may serve as an objective function for design
of an optimal rotor for a PMLM. To obtain it we will initially find values hm
and
Equations in (3-23) is satisfied under the condition
We can see from (3-25) that gm is proportional to hm. It follows from equation (3-24) that at given magnet weight, propulsion force Fp (produced by this magnet) grows when height hm and gm reduce. However, the values of hm and gm are limited from the bottom because of the following restrictions for design of the rotor magnets: 1) At given magnet volume, reducing its height hm results in increasing its width wm= vi /hm together with the height hg of the core shoes. Therefore, it is expedient to limit the ratio hm / wm ³ 0.25; 2) The working gap gm=Δ+2g, where Δ is thickness of the winding transverse segment buses, g ³ 0.01m is an air gap at each side of the bus required for loose insertion of the bus into the air gap. The bus must have a sufficient cross-section Sb=Δ×wb, (where wb is bus width) to endure the mechanical load of propulsion force and to avoid overheating when passing sinusoidal current with amplitude Im. The width of the bus is limited by the length of traveling wave λ: wb ≤ kt λ/48 where coefficient kt ≤ 0.75 is the ratio of bus width to bus tooth pitch (Fig. 3.3 (a)). On the other hand, at the vehicular speed 150 m/s and acceleration α=1 m/s2, aerodynamic resistance force Fαα» 5 tons, and inertia Fi=2.5 tons. Then assuming safety factor q=1.2 (see expression 3-10), the propulsion force must not be less than 9 tons.
3.6 Calculation of rotor propulsion force. The length of the maximum current wave λm= 6m . We assume the maximal bus width wbm= 0.72×6/48 = 0.09m, and the bus thickness Δ=0.04m. The width of the supporting tooth in the winding holder (Fig.3.3) equals 0.125-0.09 = 0.035m. The permissible (by heating) current density j = 4.5×106A/m2. The maximum current in the bus Im= j wbm ∙Δ =16200A. The number of buses in the working gap of the whole rotor equals 32 with the total current IΣ =24Im=388800A. The height of the pole shoes hg=gm μr wm / 2hm= 0.1284. Then magnetic flux density in the working gap is:
Substituting all the values in formula (3-6) we obtain the value of the maximum propulsion force: Fpm= 2B hgIΣ = 93054N > 9 tons. (3-27) Now we find the value of propulsion force F0 at the starting point of an acceleration section. The length of traveling current wave λ0 equals pole length of the rotor λ0=2L0=1m. The total current in the working gap is: IΣ0= ki ∙λ0= 75595.7A. ki in this formula is determined as ki=j∙Δ∙Ku1 ∙ kt. , where j is permissible current density, Δ is thickness of the winding transverse segment buses, Ku1 is determined by (3-11), kt. is the ratio of bus width to bus tooth pitch (Fig. 3.3 (a)) as was mentioned in the previous paragraph. Then, the propulsion force is: Fp0= 2B·hgIΣ0=1845.5kG=18092.9N. (3-28) This force is able to produce initial acceleration that equals 0.724m/s2 to a vehicle of mass mV=25000kg. This value may be increased if to double the height hm of each central magnet. In this case, magnetic flux density in the working gap and propulsion force Fp0 increases by 1/3 and initial acceleration equals 0.96 m/s2. It is worthwhile to remind that in order to increase the propulsion force it is necessary to ensure that the central part of a rotor’s pole would coincide with the central part of the half-wave λ/2. As we said in paragraph 3.3 we can do this by simultaneously moving apart smoothly two rotor’s yoke halves (by synchronizing mechanism) and gradually engaging rotor’s magnets into operation (by synchronizing devices). It is obvious that engaging magnets gradually causes rapid change of propulsion force that leads to vehicle jolts. So to reduce the jolts, a coordinated work of the synchronizing mechanism and the synchronizing devices is required. It can be reached by following way: the synchronizing mechanism moves both rotor’s halves apart slower than increasing the half-wave length. Then at the expense of lagging, some space appears for engaging the next pair of magnet units and so on until all the units will be switched on (Fig. 3.10). In addition the inertia plains jolts considerably. 3.7 Calculation of the stator winding turn length distribution. The forces in equation (3-10) depend on vehicle speed V= f λ, which, in turn, is a function of the length of traveling current wave λ=2LW . Then, propulsion force Fp(λ) =2Bhgkiλ(x), where ki = jΔKukt (3-29) The force of aerodynamic resistance is dependent on speed V(λ)=f λ and equals:
Force of inertia is Fi(λ) = mv ∙ α, where acceleration a is determined as:
Force Fg, which is opposed to the motion of the vehicle of mass mv along the ascending path inclined to the horizontal plane at angle b, is: Fg(λ) = mV Gsinβ (x) (3-32) After substituting all forces in equality (3-10) by the above expressions, transferring Fi(λ) into its left hand side and, then, dividing both parts by f 2mV ·λ we obtain the differential equation with respect to λ(x):
Now we will make the following designations:
Then equation (3-33) will look as: λ'x = α1 - α2 λ(x) – α0 (x) / λ(x) (3-35) Solving this equation we will find the distribution of λ(x). Then we can obtain all the parameters determining the work of the PMLM, such as: rotor speed: V(x) = f λ(x) ; (3-36) rotor acceleration: α(x) = f 2λ(x) (λi-λi-1) / Δx; (3-37) current in a windings phase: Iph(x) » λ(x) ki / 6; (3-38) voltage drop in each phase of the winding: power consumed by each
transformer substation:
3.8 Propulsion Motors (PM) and Power System (PS) in EMS and EDS and PMLM advantages. Creating self-regulating PM either in EMS or EDS of Maglev transportation was never even attempted. The stator winding turns were uniformly laid along the guideway and the rotor magnets installed on the vehicle were of constant length. Under such conditions rotor magnetic flux is limited and propulsion force is proportional to stator winding current. PM speed and propulsion force regulation are provided by electronic converters and inverters of current frequency located along the entire guideway and controlled by servo control systems. In EMS and EDS Maglev PM stator is powered according to the scheme employing the following components:
All this leads to tremendous complication of power system and lowering its reliability. Both neither EMS nor EDS Maglev are capable of self-regulation. The proposed design and mode of operation of the PMLM are advantageously distinguished from the other linear synchronous motors used in high speed ground transportation (HSGT and Transrapid) systems with magnetic suspension. Its advantages include:
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, flowing with speed
in the
vertical fragments of bus bar (of length hg ) of PMLM
stator winding. Electrical field E just pushes the charges along the bus
bars and does not contribute in producing propulsion force.
,
(3-2) proportional
to vehicle mass mV, it is easy to calculate the length of the
winding turn at any point of the path by formula:
(3-3)
(3-4)
(3-5)
(3-6)
(at
acceleration sections of the track) (3-7) 
(at the ascending fragments) (3-8) 
, (3-9)
(3-17)
(3-18)
(3-19)
,
(3-20)
, (3-21)
(3-22)
at maximum
value of η . Taking its partial derivatives and equating them to
zero, we obtain:
(3-23)
(3-24)
(3-25)
(3-26)
(3-30)
(3-31)
(3-33)
(3-34)
