Chapter 20 LINEAR INDUCTION MOTORS 20.1 INTRODUCTION For virtually every rotary electric machine, there is a linear motion counterpart. So is the case with induction machines. They are called linear induction machines (LIMs). LIMs directly develop an electromagnetic force, called thrust, along the direction of the travelling field motion in the airgap. The imaginary process of “cutting” and “unrolling” rotary counterpart is illustrated in Figure 20.1. Figure 20.1 Imaginary process of obtaining a LIM from its rotary counterpart The primary usually contains a three phase winding in the uniform slots of the laminated core. The secondary is either made of a laminated core with a ladder cage in the slots or of an aluminum (copper) sheet with (or without) a solid iron back core. Apparently the LIM operates as its rotary counterpart does, with thrust instead of torque and linear speed instead of angular speed, based on the principle of travelling field in the airgap. In reality there are quite a few differences between linear and rotary IMs such as [1 - 8] • The magnetic circuit is open at the two longitudinal ends (along the travelling field direction). As the flux law has to be observed, the airgap field will contain additional waves whose negative influence on performance is called dynamic longitudinal end effect (Figure 20.2a). • In short primaries (with 2, 4 poles), there are current asymmetries between phases due to the fact that one phase has a position to the core longitudinal ends which is different from those of the other two. This is called static longitudinal effect (Figure 20.2b). • Due to same limited primary core length, the back iron flux density tends to include an additional nontravelling (ac) component which should be considered when sizing the back iron of LIMs (Figure 20.2c). • In the LIM on Figure 20.1 (called single sided, as there is only one primary along one side of secondary), there is a normal force (of attraction or © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… repulsion type) between the primary and secondary. This normal force may be put to use to compensate for part of the weight of the moving primary and thus reduce the wheel wearing and noise level (Figure 20.2d). • For secondaries with aluminum (copper) sheet with (without) solid back iron, the induced currents (in general at slip frequency Sf 1 ) have part of their closed paths contained in the active (primary core) zone (Figure 20.2c). They have additional-longitudinal (along OX axis)-components which produce additional losses in the secondary and a distortion in the airgap flux density along the transverse direction (OY). This is called the transverse edge effect. secondary current density paths at zero slip primary a.) dynamic longitudinal effect 120 0 120 0 i i i a b c b.) static longitudinal effect x ττ 0 airgap flux lines S=1 c.) back core flux density distribution iron Al secondary current density x repulsion normal force (F ) nr attraction normal force (F ) na z y primary current primary F =F -F nrnan d.) nonzero normal force F n J y J x zero secondary currents nonzero secondary currents y B z 0 e.) transverse edge effect xx ∆ primary ejection lateral force (F ) le y realigning lateral force (F ) lra F =F -F lralee 0 f.) nonzero lateral force F e Figure 20.2 Panoramic view of main differences between LIMs and rotary IMs • When the primary is placed off center along OY, the longitudinal components of the current density in the active zone produce an ejection © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… type lateral force. At the same time, the secondary back core tends to realign the primary along OY. So the resultant lateral force may be either decentralizing or centralizing in character (Figure 20.2.f). All these differences between linear and rotary IMs warrant a specialized investigation of field distribution and performance in order to limit the adverse effects (longitudinal end effects and back iron flux distortion, etc.) and exploit the desirable ones (normal and lateral forces, or transverse edge effects). The same differences suggest the main merits and demerits of LIMs. Merits • Direct electromagnetic thrust propulsion (no mechanical transmission or wheel adhesion limitation for propulsion) • Ruggedness; very low maintenance costs • Easy topological adaptation to direct linear motion applications • Precision linear positioning (no play (backlash) as with any mechanical transmission) • Separate cooling of primary and secondary • All advanced drive technologies for rotary IMs may be applied without notable changes to LIMs Demerits • Due to large airgap to pole pitch (g/τ) ratios–g/τ > 1/250–the power factor and efficiency tend to be lower than with rotary IMs. However, the efficiency is to be compared with the combined efficiency of rotary motor + mechanical transmission counterpart. Larger mechanical clearance is required for medium and high speeds above 3m/s. The aluminum sheet (if any) in the secondary contributes an additional (magnetic) airgap. • Efficiency and power factor are further reduced by longitudinal end effects. Fortunately these effects are notable only in high speed low pole count LIMs and they may be somewhat limited by pertinent design measures. • Additional noise and vibration due to uncompensated normal force, unless the latter is put to use to suspend the mover (partially or totally) by adequate close loop control. long (fix) secondary aluminum sheet short (moving) primaries short (moving) secondary element double sided long (fix) primary a.) b.) Figure 20.3 Double sided flat LIMs a.) double sided short (moving) primary LIM; b.) double sided short (moving) secondary LIM for conveyors © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… As sample LIM applications have been presented in Chapter 1, we may now proceed with the investigation of LIMs, starting with classification and practical construction aspects. 20.2 CLASSIFICATIONS AND BASIC TOPOLOGIES LIMs may be built single sided (Figure 20.1) or double sided (Figure 20.3a), with moving (short) primary (Figure 20.1) or moving (short) secondary (Figure 20.3b). As single sided LIMs are more rugged, they have found more applications. However (Figure 20.3b) shows a double sided practical short-moving secondary LIM for low speed short travel applications. LIMs on Figure 20.1-20.3 are flat. For flat single sided LIMs the secondaries may be made of aluminum (copper) sheets on back solid iron (for low costs), ladder conductor in slots of laminated core (for better performance), and a pure conducting layer in electromagnetic metal stirrers (Figure 20.4a, b, c). In double sided LIMs, the secondary is made of an aluminum sheet (or structure) or from a liquid metal (sodium) as in flat LIM pumps. conducting sheet solid back iron a.) conducting ladder laminated secondary core with slots b.) fixed primary liquid metal c.) Figure 20.4 Flat single-sided secondaries a.) sheet on iron ; b.) ladder conductor in slots; c.) liquid metal © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… Besides flat LIMs, tubular configurations may be obtained by rerolling the flat structures along the transverse (OY) direction (Figure 20.5). Tubular LIMs or tubular LIM pumps are in general single sided and have short fix primaries and moving limited length secondaries (except for liquid metal pumps). The primary core may be made of a few straight stacks (Figure 20.5a) with laminations machined to circular stator bore shape. The secondary is typical aluminum (copper) sheet on iron. The stator coils have a ring shape. While transverse edge effect is absent and coils appear to lack end connections, building a well centered primary is not easy. An easier solution to build is obtained with only two-size disk shape laminations both on primary and secondary (Figure 20.5b). The secondary ring shape conductors are also placed in slots. primar y stacks conductor sheet secondary back iron primary circular coils 2p τ movin g secondar y a.) disk shape laminations ring shape secondary conductors in slots b.) primary thermal insulation liquid sodium channel stainless steel shell secondary back iron (yoke) center return pipe (if any) c.) Figure 20.5 Tubular LIMs a.) with longitudinal primary lamination stacks; b.) with disk-shape laminations c.) liquid metal tubular LIM pump the secondary Better performance is expected by the fact that, in the back cores, the magnetic field goes perpendicular to laminations and tends to produce © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… additional core losses. The interlamination insulation leads to an increased magnetization m.m.f. and thus takes back part of this notable improvement. The same rationale is valid for tubular LIM liquid metal pumps (Figure 20.5c) in terms of primary manufacturing process. Pumps allow for notably higher speeds (u = 15m/s or more) when the fixed primaries may be longer and have more poles (2p 1 = 8 or more). The liquid metal (sodium) low electrical conductivity leads to a smaller dynamic longitudinal effect which at least has to be checked to see if it is negligible. 20.3 PRIMARY WINDINGS In general, three phase windings are used as three phase PWM converters and are widely available for rotary induction motor drives. Special applications which require only 2p 1 = 2 pole might benefit from two phase windings as they are both placed in the same position with respect to the magnetic core ends. Consequently, the phase currents are fully symmetric at very low speeds. 1 4 2 5 3 6 A ZB XC Y 2p =2; q = 1 1 A 1 C' 2 B 3 A' 4 C 5 B' 6 A B C X,Y,Z 1 4 2 5 3 6 7 10 8 11 9 12 A Z B X C Y A 1 C' 2 B 3 A' 4 C 5 B' 6 2p =4; q = 1 1 A 7 C' 8 B 9 A' 10 C 11 B' 12 a.) 1 6 2 5 3 8 4 7 A BXY 2p =2; q = 1 1 b.) A B X,Y A 1 A 2 B 3 B 4 A' 5 A' 6 B' 7 B' 8 Figure 20.6 Single layer windings a.) three phase; b.) two phase © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… LIM windings are similar to those used for rotary IMs and ideally they produce a pure travelling m.m.f However, as the magnetic circuit is open along the direction of motion, there are some particular aspects of LIM windings. A B' B C' A' B A' C B' C A B' A C' B C' A' B A' C B' C A C' 2p =4, q = 1 1 Figure 20.7 Triple layer winding with chorded coils (y/τ = 2/3) (very short end connections) A C B X ZY A 1 C' 2 B 3 A' 4 C 5 B' 6 A 7 C' 8 B 9 A' 10 C 11 B' 12 A' C B' A C' B A' C B' A C' B 13 14 2p +1 = 5; q = 1, y / = 2/3 τ 1 2p +1 = 7; q = 2, y/ = 5/6 τ 1 Figure 20.8 Double layer chorded coil windings with 2p 1 + 1 poles Among the possible winding configurations we illustrate a few • Single layer full pitch (y = τ) windings with an even number of poles 2p 1 , Figure 20.6 three phase and two phase. • Triple layer chorded coil (y/τ = 2/3) winding with an even number of poles 2p 1 (Figure 20.7). • Double layer chorded coil (2/3 < y/τ < 1) coil winding with an odd number of poles 2p 1 + 1 (the two end poles have half-filled slots)-Figure 20.8. • Fractionary winding for miniature LIMs (Figure 20.9). © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… 1 2 3 4 5 6 AC B X Z Y 2p =2, q = 1/2, m = 3phases 1 a.) A B X Y 2p =2, q = 1/2, m = 2phases 1 b.) Figure 20.9 Fractionary single layer winding (low low end connections) a.) three phase; b.) two phase A few remarks are in order • The single layer winding with an even number of poles makes better usage of primary magnetic core but it shows rather large coil end connections. It is recommended for 2p 1 = 2, 4. • The triple layer chorded coil windings is easy to manufacture automatically; it has low end connections but it also has a rather low winding (chording) factor K y = 0.867. • The double layer chorded coil winding with an odd total number of poles (Figure 20.8) has shorter end connections and is easier to build but it makes a poorer use of primary magnetic core as the two end poles are halfwound. As the number of poles increases above 7, 9 the end poles influence becomes small. It is recommended for large LIMs (2p 1 + 1>5). • The fractionary winding (Figure 20.9)-with q = ½ in our case-is characterised by very short end connections but the winding factor is low. It is recommended only in miniature LIMs where volume is crucial. • When the number of poles is small, 2p 1 = 2 especially, and phase current symmetry is crucial (low vibration and noise) the two phase LIM may prove the adequate solution. • Tubular LIMs are particularly suitable for single-layer even-number of poles windings as the end connections are nonexistent with ring-shape coils. In the introduction we mentioned the transverse edge and longitudinal end effects as typical to LIMs. Let us now proceed with a separate analysis of transverse edge effect in double sided and in single sided LIMs with sheet- secondary. 20.4 TRANSVERSE EDGE EFFECT IN DOUBLE-SIDED LIM A simplified single dimensional theory of transverse edge effect is presented here. The main assumptions are © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… • The stator slotting is considered only through Carter coefficient K c . os 2 os s c b g 5 b g ; /g1 1 K γ+ =γ τγ− = (20.1) • The primary winding in slots is replaced by an infinitely this current sheet traveling wave J 1 (x, t) () τ == τ π −ω 1 11w1 m xtSj m1 p IKW23 J ;eJt,xJ 1 (20.2) 2p 1 -pole number, W 1 -turns/phase, K w1 -winding factor; τ-pole pitch, I 1 -phase current (RMS). Coordinates are attached to secondary. • The skin effect is neglected or considered through the standard correction coefficient. − + ≈ ss ss s skin d d cos d d cosh d d sin d d sinh d2 d K (20.3) 2 S d 1 Al10 s σωµ ≈ (20.4) For single sided LIMs, d/d s will replace 2d/d s in (20.2); d s -skin depth in the aluminum (copper) sheet layer. Consequently, the aluminum conductivity is corrected by 1/K skin , skin Al Als K σ =σ (20.5) • For a large airgap between the two primaries, there is a kind of flux leakage which makes the airgap look larger g l [1]. leakagecl KgKg = 1 2 g 2 g sinh K leakage > τ τ = (20.6) © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… • Only for large g/τ ratio K leakage is notably different from unity. The airgap flux density distribution in the absence of secondary shows the transverse fringing effect (Figure 20.10). g a a e a a e Figure 20.10 Fringing and end connection flux considerations The transverse fringing effect may be accounted for by introducing a larger (equivalent) stack width 2a e instead of 2a (20.7) ( g0.22.1a2a2 e ÷+= ) For large airgap in low thrust LIMs this effect is notable. As expected, the above approximations may have been eliminated provided a 3D FEM model was used. The amount of computation effort for a 3D FEM model is so large that it is feasible mainly for special cases rather than for preliminary or optimisation design. • Finally, the longitudinal effect is neglected for the time being and space variations along thrust direction and time variations are assumed to be sinusoidal. In the active region (|z| ≤ a e ) Ampere’s law along contour 1 (Figure 20.11) yields τ π −ω == ∂ ∂ xtj m 1x2 y e 1 eJJ ;dJ z H g (20.8) Figure 20.11 shows the active and overhang regions with current density along motion direction. The same law applied along contour 2, Figure 20.11.b, in the longitudinal plane gives () dJJHH x g z2 m 0x e +=+ ∂ ∂ − (20.9) © 2002 by CRC Press LLC [...]... secondary On the other hand, the transverse edge effect may be reduced, when needed, by making the overhangs of a larger cross-section or of copper (Figure 20.14) In general, the larger the value of SGi, the larger the transverse edge effect for given ae/τ and c/τ In low thrust (speed) LIMs as the pole pitch τ is small, so is the synchronous speed us u s = 2τf1 = τ ω1 π (20.32) Consequently, the goodness... called the exit and entry end-effect waves respectively The real parts of γ1,2 (γ1r, γ2r) determine the attenuation of end-effect waves along the direction of motion while the imaginary part jγi determines the synchronous speed (use) of end effect waves u se = ω1 π ; τe = γi γi (20.67) The values of 1/γ1r and 1/γ2r may be called the depths of the end effect waves penetration in the (along) the active... γ 2 r γ1r (20.68) Consequently, the effect of backward (exit) end effect wave is negligible Not so with the forward (entry) end effect wave which attenuates slowly in the airgap along the direction of motion The higher the value of goodness factor Ge and the lower the slip S, the more important the end effect waves are High Ge means implicitly high synchronous speeds The pole pitch ratio of end-effect... zero slip, there is a certain value of the realistic goodness factor Geo, for which the end effect force is zero This value of Ge is called the optimum goodness factor • For large values of Ge, the end effect force changes sign more than once as the slip varies from 1 to zero • The existence of the end effect force at zero slip is a distinct manifestation of longitudinal end effect Further on the airgap... results for the same LIM as above are given in Figure 20.13 The lateral force Fz decreases as ae/τ decreases or the overhangs c-ae, b-ae > τ/π In fact it is of no use to extend the overhangs of secondary beyond τ/π as there are few currents for |z| > |τ/π + ae| The transverse edge effect correction coefficients In the absence of transverse edge effect, the magnetization reactance Xm has the conventional... iteratively computed Then σe and ge are calculated from (20.40)-(20.42) Finally Ge, is determined The primary phase current I1 is computed from (20.44) All above data serve to calculate the LIM thrust and other performance indices to be dealt with in the next paragraph in a technical longitudinal effect theory of LIMs 20.6 A TECHNICAL THEORY OF LIM LONGITUDINAL END EFFECTS Though we will consider the double... (20.102) From (20.102), the net normal force becomes repulsive if SG e > τ geπ (20.103) which is not the case in most low speed LIMs (LIAs) even at zero speed (S = 1) Core losses in the primary core have been neglected so far but they may be added as in rotary IMs Anyway, they tend to be relatively smaller as the airgap flux density is only Bgn = (0.2-0.45)T, because of the rather large magnetic airgap... 1) and the thrust and goodness factor for given (constant) stator current I1 = 3A (RMS) are shown in Figure 20.23 Figure 20.23 summarizes the influence of skin effect, saturation, and eddy currents in the secondary back iron As the frequency increases Ge slowly deriorates (decreases) because the skin depth δi in iron decreases and so the back iron contribution to the equivalent airgap increases The airgap... both the secondary efficiency and power factor Typical numerical results for a super-high speed LIM are shown in Figure 20.19 So far, we considered the primary core as infinitely long In reality, this is not the case Consequently, the field in the exit zone decreases more rapidly (Figure 20.18) and thus the total secondary power losses are in fact, smaller than calculated above However, due to the same... which is solely dependent on the number of poles 2p1 (Figure 20.20) Geo is a rather intuitive compromise as higher Ge leads to both conventional performance enhancement and increase in the longitudinal effect adverse influence on performance LIMs where the dynamic longitudinal end effect may be neglected are called low speed LIMs or linear induction actuators while the rest of them are called high speed . machines. They are called linear induction machines (LIMs). LIMs directly develop an electromagnetic force, called thrust, along the direction of the. LINEAR INDUCTION MOTORS 20.1 INTRODUCTION For virtually every rotary electric machine, there is a linear motion counterpart. So is the case with induction