## Machine Sizing

Crucial to the behaviour of the VHM is the size of the airgap, which is a result of the manufacturing and assembly process. A nominal airgap of 1mm was thought to be the minimum achievable. A convenient magnet thickness was 4 mm.

There are two factors which limit the shear stress of this machine, the heat dissipation of the coils and the saturation of the iron. The former is governed by the dimensions of the coil and is dependent on the surrounding environment whereas the latter depends on the shape and properties of the iron. Consideration of the flux flow within the machine implies that the most likely place for iron saturation will be in the root of the tooth and will hence directly affect the flux density in the airgap under the tooth, Bt. The maximum flux density achievable in mild steel is around 1.9 T (see Figure 4.8 later). Equation (4.6), a combination of (3.12) and (3.14) from Chapter 3, shows the shear stress for the VHM in terms of the translator tooth-root flux density. The behaviour of the relationship is given in Figure 4.4 which, for a density of 1.9 T, shows a clear peak for a magnet width of around 17 mm and subsequent decline with further magnet width increase.

MoMr wm

root

g+tr

Consider each phase being subjected to a sinusoidal current whilst moving at a constant velocity, the resulting three phase force reacted would be of the form given in Figure 4.5. As shown, the total contribution is equal to 1.5 times the peak of a single phase.

red phase yellow phase | ||||||

blue phase Combined | ||||||

y V |
\/VV | |||||

I\l | ||||||

1 7 |
y j y | |||||

\A |
A |
Ik |
\J |

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 arbitrary time

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 arbitrary time

Figure 4.5: Example of three phase force

To remove 3 kW from the translator moving at 0.5 ms-1 requires the entire three phase machine to react an average force of 6 kN, corresponding to each phase having a peak force of this value divided by 1.5, 4 kN. Each phase has four active areas which, in combination with the predicted shear stress, implies that each area should be 0.0079 m2. For an axial length of 0.1 m, this corresponds to a pole face width of 0.079 m. The definition of shear stress consists of pairs of opposed forces, due to the interaction of a tooth and slot region, implying it only holds for an even number of magnets per stator pole. The closest number of magnets which gives the correct active area is therefore 6. These dimensions are given in Table 4-1

The coil affects two functions of the VHM, the product of current and number of turns gives mmf, NI, whereas the number of turns only affects the magnitude of emf induced. Substitution of the values obtained thus far into (4.5) gives the predicted open circuit emf per coil turn as 0.55 Volts. In line with the likely overestimation of this value, and the desire for a large output emf for use in the power factor correction equipment, 240 turns per coil are used, giving a 130 Volt peak output.

The dimensions of the coil depend on the current flowing within it and the allowed temperature rise due to power loss. The 'knee' of the B-H curve for iron, shown in Figure 4.8 later, is at around 1.5 T. Limiting the flux density through the tooth to this value hence ensures the iron remains unsaturated and behaves linearly. In order to have a flux density of 1.5 T within the tooth requires an mmf of 5.7 x 10 Ampere turns, according to (4.2), corresponding to 23 Amps flowing in the 240 turns. This will be the peak value of a sinusoidal current having an RMS of 16 Amps. If the instantaneous peak current density of the coil is limited to 4 Amm-2, the copper wire will need a diameter of 2.7 mm. Assuming a fill factor of 40%, the coil will have a cross sectional area of 0.0034 m2, with dimensions of approximately 4 cm by 8 cm. The length of wire used will be around 100 m, which corresponds to a coil resistance of 0.3 Q if it is manufactured from copper with a resistivity of 1.73 x

10-8 Qm-1. The resulting RMS power loss of 76 Watts manifests itself as a rise in temperature according to (4.7), a simplified expression of convection cooling [73].

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