## B

Ftop <2CgFgap

Where:

This result assumes negligible leakage and fringing and that the flux density is uniform in the air gap. Substituting from Eq. 4 into Eq. 19 gives

If the saturation flux density, Bsat, of the iron material is used for B, Eq. 20 defines the load capacity as a function of pole area for a magnetic bearing.

d. Linearization of the force/current characteristicâ€”The air gap flux density in the homopolar bearing is the superposition of the bias and control flux. The control coils and permanent magnet polarity are arranged such that when the control flux adds to the bias in the top air gaps, the control subtracts from the bias in the bottom air gaps. Thus the net vertical force is

CgA 2 2

Fy = Ftop ~ Fbot = ' Ufibias + Bcon)top ~ (Bbias ~ Bcon~]2ot] (21)

This can be reduced in a similar manner as before to produce

Substituting from Eq. 16

4CgAN

Thus the control force is proportional to the control current as desired.

e. Force constant and negative stiffnessâ€”The expression given in Eq. 10 for the heteropolar bearing also applies to the homopolar bearing:

The definition of the force constant is

-4Cg ANBbias kf =-= force constant or current stiffness (25)

The expression for negative stiffness is not easily reducible to analytical form due to the complexity of the bias circuit. However, the existence of the permanent magnet in the bias flux path as a fixed and large reluctance improves the linearity of this bearing for offcenter operation. Eqs. 13 and 14 for the heteropolar bearing apply to the homopolar bearing as well.

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