amplitude of 108% of the rms value, with a 9% contribution from the third
harmonic, which could be eliminated by a delta connection. There is virtually
no harmonic contribution past the ninth harmonic' .
Having only dealt with rectangular coils, the field variation at off axis
positions for circular and racetrack coils must be defined. Circular coils
require the solution of complete elliptical integrals of the first and second
kind. Racetrack off axis positions require combinations of straight line
element type formulae, with extra terms involving the solution of incomplete
elliptical integrals of the first and second kind. The overall technique for
solution of rate of change of air gap flux linkage is similar to that used for
rectangular coils, but is computationally more awkward. When working at large
air gaps an equivalent area rectangular coil can be used for first order approximation, with only a slight loss of accuracy.
Once the harmonics of the back emf are known, the force pulsations on the vehicle, and indeed the force profile can be determined by taking harmonics in
turn and applying their values to equation 15. If the current has significant
harmonic content, this too can be incorporated to produce the total pulsation of the parasitic harmonic forces.
3.3 Wavelength Optimization
For machine parameter optimization a more general model is required.
Operational optimization in choice of a minimum power factor efficiency product
working point used an equivalent circuit model. Ideally a wide track giving a
long active length per pole, and a short pole pitch, to enable a large number
of poles to be used might appear to be the best choice. However, track width
is usually determined by the likely infrastructure and compatibility
requirements with other systems, as well as vehicle width; pole pitch might be
fixed by inverter or cycloconverter upper frequency limits, and cryostat and
coil fabrication constraints. A simple model can be formed in which levitation
height and track width are the main variable parameters, which are functions of groups of track and vehicle constants.
Thornton analysed a simple model of a finite width track interacting with a two dimensional field distribution from an effectively infinite width magnet
a r r a y ^ ^ . For this case the time averaged thrust force FB is related
to a maximum fundamental thrust force F^ by the current angle. Similarly the
normal force F{j is the quadrature variation. Fg and Fjj are expressed
by
Fb “ Fvj sin« (38)
f n - -Fm cosO<
By comparison with equation 15, can be related to the back emf and phase
current by
FM - m Eg| (39)
v
Taking a J X B product with the assumption of balanced phase currents and a sinusoidal current sheet representing the magnet array, with a strength of Iy AT per pole, from N poles, then,
FM ‘ 2 *> N " Xv ITD 2 e'2*t’A (40) A
m and If are the number of stator phases and the track per phase current, w, Xand h are track width, wavelength and coll height, and D is a dimensionless
constant which expresses the meander winding end turn shape.
Two more useful parameters are PA , the power loss in the armature per unit
track length, and MA , the meander mass per unit track length. PA is
proportional to the efficiency of the machine, and hence power consumption, and
is measured in watts per metre. MA is proportional to the amount of capital
cost in installing the guideway meander conductor, and is in kilogrammes per metre. So ?• ( X +2W) n r (41) and M. ■ m ( X +2w) AT A __ X ( 4 2 ) 111
where
r
and q are Che density and resistivity of the Crack conductor.Combining 41 and 42,
M AP A “ ™2 ( > + 2 w > 2 lx2?«" (43)
and eliminating the track current in (40) with (43)
" 2 NIv (2MaPa )* H (44)
9
<r
where H - D w (4 5)
\ +2w
H embodies width, height and wavelength in a form that can be optimised.
Equation 44 contains (apart from H) items that are very often fixed by economic
constraints or material conditions. For example, working the track at a
particular implies a choice of efficiency maximum, and for M^, a track
capital repayment cost can be inferred. NIV is reflected directly into the
vehicle cost. It is reasonable, therefore, to assume that these parameters
take fixed values, and that design manipulation must be through H.
Thornton chose to maximise the thrust per pole, which implies that a vehicle thrust requirement can be met by simply adding poles. The maglev vehicle however has finite length, which is fixed at an upper limit by a tradeoff between lightness and structural stiffness in torsion and bending moment.
Abel, however chose to maximise the thrust per stator conductor length under the vehicle - stator current product, or effectively H divided by
w a v e l e n g t h ^ '. The power transferred to the finite length vehicle is
maximised, which is more reasonable since usually the full vehicle length will be required for propulsion magnets, and may in fact limit the thrust capability in very high powered vehicles.
If the thrust per unit stator length-current product is found for various values of wavelength, the resulting magnitude tends to have a broad maximum, depending
on the crack width and magnet height. Figure 33 shows the normalized thrust,