. The Bell System technical journal . y of the magnetic field at a point P with theshield removed and Hi is the density of the magnetic field at thatpoint when enclosed by the shield, fj, is the permeability of thematerial at the frequency in question. We have qi = nV-Rl^ wi = 1 - gi, 52 = ^2V^2^ «2 = 1 - §2, ** 53 = rsIR,\ n,= I - §3, (4) 212 = -RlV^2^ W12 = 1 — §12, §23 = Ri^lri, fiiz = \ — q23, where n, Ri, r^, etc., are the various radii of the cylinders as shownby Fig. 2. By making qz = \ (or Uz = 0), in (2), we get the relation for twoconcentric cylinders. g = 1/4/i(l - qiq2 + l/Annifiif
. The Bell System technical journal . y of the magnetic field at a point P with theshield removed and Hi is the density of the magnetic field at thatpoint when enclosed by the shield, fj, is the permeability of thematerial at the frequency in question. We have qi = nV-Rl^ wi = 1 - gi, 52 = ^2V^2^ «2 = 1 - §2, ** 53 = rsIR,\ n,= I - §3, (4) 212 = -RlV^2^ W12 = 1 — §12, §23 = Ri^lri, fiiz = \ — q23, where n, Ri, r^, etc., are the various radii of the cylinders as shownby Fig. 2. By making qz = \ (or Uz = 0), in (2), we get the relation for twoconcentric cylinders. g = 1/4/i(l - qiq2 + l/Annifiifin) + 1. (5) By making 52 = 1 (or W2 = 0), (5) changes into an equation for oneshell only g = 1/4 Ml -g) + 1. (6) MAGNETIC SHIELDING OF TRANSFORMERS 421 It has been shown (A. P. Wills, Phys. Rev., vol. 24, page 243,February 1907) that for a shield of predetermined size, that is whenthe smallest and the largest radii (ri and R^ in the case of three cylin-ders. Fig. 2) are specified, the radii of the surfaces of the successive. I Fig. 2. cylinders should be in a geometric progression to give the most efficientshield. That is, we should have qi — qi — qz = qn = §23 = q- Equa-tion (2) then becomes g = 1/4 m(1 - g + 1/16 fx^n -f M« + 3/4 m«) + 1. (7) For two cylinders we get g = 1/4 m(1 -2^ + 1/4 M«^) + 1. (8) In these equations, the following relations hold between n and Rsfor (7) and ri and R2 for (8) Rs = rJ^[i^ R, = rif^\ (9) The effect upon the shielding efficiency of varying 5 (Fig. 3) from 422 BELL SYSTEM TECHNICAL JOURNAL zero to P keeping rijRi = r^jRi can be obtained by means of thefollowing equation g- 1/4m where 1 -^ + 1/4m(i --^Xn,,912 \ Vgi2/ R2 + 1> (10) If Ri/r2 (that is Vgi2) is varied from 1 to R^/ri the desired result isobtained. Assume in Fig. 3 the thickness of the two cylinders to be the same,that is, R\ — r\ = R2 — ^2- The variation in shielding efficiency vs. 5
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