Linear Reactor Design

A method of design for linear reactors is based on three assumptions which are justified in the foregoing:

(a)   The air gap is large compared to l_c/μ, μ, being the d-c permeability.

(b)   A-c flux density depends on alternating voltage and frequency.

(c)  A-c and d-c fluxes can be added or subtracted arithmetically.

From (a) the relation B = μH becomes B = H. Because of fringing of flux around the gap, an average of 0.85B crosses over the gap. Hence B_dc = 0.4πNI_dc/0.85l_g. With l_g in inches this becomes

Bdc = 0.6 NIdc / lg   gauss

[35]

Transposing equation 34

Bac = (3.49E106) / fAcN   gauss

[36]

The sum of B_ac and B_dc is B_max, which should not exceed 11,000 gauss for 4% silicon steel, 16,000 gauss for grain-oriented steel, or 10,000 gauss for a 50% nickel alloy. Curves are obtainable from steel manufacturers which give incremental permeability μ_Δ for various combinations of these two fluxes. Figure 70 shows values for 4% silicon steel.

Fig. 70. Incremental permeability for 4% silicon steel with high a-c induction.

By definition, inductance is the flux linkages per ampere or, in cgs units,

[37]

But

If this is substituted in equation 37

[38]

provided that dimensions are in inches. The term A_c in equation 38 is greater than in equation 36 because of the space factor of the laminations; if the gap is large A_c is greater still because the flux across it fringes. With large gaps, inductance is nearly independent of μ_Δ, at least with moderate values of B_max. With small gaps, permeability largely controls. There is always a certain amount of gap even with punchings stacked alternately in groups of 1. Table IX gives the approximate gap equivalent of various degrees of interleaving laminations for magnetic path l_c of 5.5 in.

Table IX. Equivalent Gaps with Interleaved Laminations

0.014-in. Laminations Alternately Stacked	Equivalent. Air Cap in Inches (Total) with Careful Stacking
In groups of 1	0.0005
In groups of 4	0.001
In groups of 8	0.002
In groups of 12	0.003
In groups of 16	0.004
Butt stacking with zero gap	0.005