and normal flux superconducting magnet configurations where each coil consists of a pair of infinitely long parallel wires separated by a fixed distance. The null-.
IEEE Transactions on Magnetics, v o l . MAG-11, no. 2, March 1975
COMPARISON AND OPTIMIZATION OF LIFT AND DRAG FORCES ON VEHICLES LEVITATED BY EDDY CURRENT REPULSION FOR VARIOUS NULL AND NORMAL FLUX MAGNETS WITH ONE OR TWO TRACKS J . R. Hogan and H . J . Fink*
ABSTRACT L i f t and drag forces are compared f o r va.riousnull andnormal fluxsuperconducting magnet configurations where each coil consists of a pair of i n f i n i t e l y long parallelwiresseparated by afixeddistance. The n u l l flux configuration has a minimum i n the drag to l i f t ratio for a particular value of I ~ / W (I=magnetcurrent, W=vehicle weight) when the vehicle cruises freely at constantspeed.Thiscalculationtakesintoaccount losses due t o non-uniform eddy-current density distributioninthesolidtrack.Resultsindicatethatthe null-flux configuration is the most efficient design, followed by t h e normal-flux, single track configuration. 0 All other schemes are l'eSs e f f i c i e n t f o r a t h i n track configuration. A general method i s outlinedforcalculating the zero-torque case.
I. INTRODUCTION Several schemes f o r magnet and track arrangement on 1 evi tated high speed transportation vehicles have been proposed. Among thesethereare two e s s e n t i a l l y d i f f e r ent classes of track configuration. One uses metal 1 i c other solid metallic the In e i t h e r case,theelectromagneticprincipleinvolved is the same. Magnetic fields,generated by induced eddy currents i n the track, oppose changesintheprimary fields due t o l e v i t a t i o ? magnets mounted of the moving vehicle. Among the 'magnet arrangements there are again two classes. The f i r s t c o n s i s t s of a single olen no id''^'^ w h i c h i s constrained to travel either aboveone o r between two tracks..This i s the normal-flux method. The second method requires two solenoids which travel as a fixed pair, oneabove and the other below a metallic proguideway. In thissituationifthecurrentflow duces p a r t i a l l y c a n c e l l e d f i e l d s between the pair, we u lIlf-,fal u txh icsa l l on the other hand, t h e f i e l d components aid each other we label this thebrake-fluxconfiguration,because i t is suitable for braking the vehicle. I t i s t h e purpose of this paper t o compare the drag to lift ratio for the various magnet configurations w i t h the tracks being metallic sheets o f finite thickness. To simplify,the problem we assume t h a t t h e m e t a l l i c tracks are of i n f i n i t e e x t e n t and t h a t a magnet consists of two p a r a l l e l , i n f i n i t e l y longwirescarrying equal and opposite currents, separated by adistance2a, moving w i t h constant velocity normal t o their lengths over aconductingsheet a t height z. Hannakam6 has derived an expression for the eddy-current density i n a thin sheet due toa moving currentsource. The extension of his calculations6 t o a metallic sheet of a r b i t r a r y thickness is given i n Ref. 7 and i s used a s t h e s t a r t ing point.
I
,
11. FORMULATION OF THE PROBLEM AND SOLUTION We considerthemagnet'geometry shown i n Fig. 1. This model i s we1 1 suited for ourpurposesbecause it allows us tosimulatenull-flux,brake-flux and normalflux systems by appropriate choice of the currents 11, 7 t o thls 12,I3 and 14.Applying theresults ofRef. four wire case we can readily evaluate the surface current density in terms of the vector potential a t t h e surfaces. We distinguish between the t h i n (d6) l i m i t s f o r which we obtainexactsolutions t o our problem. Manuscriptreceived September30,1974
*
Department of ElectricalEngineering,University California , Davis, California 95616
F i g . 1 . Generalizedcoordinatesystemfor and track arrangement.
magnet
( d 4 ) we may interpolate For theintermediaterange In thethicklimit between the t h i n and thicklimits. the moving magnetic f i e l d from the lower magnet does not penetrate through the sheet to i t s upper surface and therefore I3 and I4 do not contribute to the curr e n t d e n s i t y a t t h e upper surface. In the t h i n l i m i t , t h e moving magnetic f i e l d s penetrate freely the sheet and thus I3 and 14.contribute tothecurrentdensity everywherein thesheet. These considerations are'valid for the null flux and the brake flux methods. With t h e r e s t r i c t i o n z t 22 = 2b = constantfora particular magnet configurakion, d 6
+
F,
2Ba1 [A+O(A3)]
2Bk[l+a2A2+0(A4)]
N u l lF l u x ,S i n g l eT r a c k
2Ba1 [A+O(A3)]
1
oA
63
d >> 6
BrakeFlux,SingleTrack
@lbi '-a-
I '
@
2a
I
d
I
1
2 2Bal[A+O(A
2Bk[l+a2A2+0(A4)]
I
2
I
6
4B
2Bk[l+a2A2+O(A4)]
)]
Normal Flux,DoubleTrack
@----2a
-7b
/Normal Flux,SingleTrack1
B
t h e t o t a l f o r c e on t h e magnet o f any c o n f i g u r a t i o n can be obtained by summing t h e f o r c e s on a l l t h e w i r e s . T h i s i s e a s i l y done f o r A z / b < < l .W i t ht h ed e f i n i t i o n s B = ' 1 I2 B a 2T b(a2+b2) l+k2
2
W = Mg = t h e w e i g h t c a r r i e d and t h a t o f t h e a1 = ( 1 + 3 ~ ) / ( 1 + ~ ) ,
2
1
' magnet,
2
a2 = (1+3x+6x ) / ( l + x ) , -1/2 , m=a/b-x p = 1 f o r t h e t h i c k 1 imit, p = 2 f o r t h e t h i n limit,
111.
/
k
EVALUATION OF DISPLACEMENT LOSSES
Fo may be neglected,otherwisenot. When F cc ,'A When A = 0,tRe c u r r e n t d e n s i t y components a t b o t h s u r f a c e due t o t h e upperandlower magnets independently are the same, i o ( y ) ,b u tw i t ho p p o s i t es i g n . Assuming t h a t d
