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d2BZA.m
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function d2BZA = d2BZA(l,a,b,z) % % function d2BZA = d2BZA(l,a,b,z) % % Returns the SECOND derivative of the on-axis magnetic field % associated to a rectangular coil centered on z=0 % in which a unit current density 1/(a*b) [A/m^2] is flowing % The analytic formula is exact on the axis and is based on % Martin Wilson: 'Superconducting Magnets', Oxford Science publications, 1990, p20. % plus some cumbersome derivation. % % % Version: 1.0 J.-P. Hogge Sept 25, 2001 % % a : Minor radius % b : major radius % l : longitudinal extent of the coil (from -l/2 to +l/2) % z : a one-dimensional array mu0 = 4*pi*1e-7; T1 = T(l,a,b,-z); T2 = T(l,a,b,z); T3 = T(l,a,a,-z); T4 = T(l,a,a,z); T5 = (T1-b/a).^2; T6 = (T2-b/a).^2; T7 = (T3-1).^2; T8 = (T4-1).^2; d2BZA = mu0/16/a^4 * ( ... (l^3 +12*l*z.^2) .* ( -1./(T1.*T5.^1.5) -1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... +1./(T3.*T7.^1.5) +1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ) ... + (8*z.^3 + 6*l^2*z) .* ( +1./(T1.*T5.^1.5) +1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... -1./(T3.*T7.^1.5) -1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ) ... + 24*z*a^2 .* ( -1./(T1.*sqrt(T5)) +1./(T2.*sqrt(T6)) +1./(T3.*sqrt(T7)) -1./(T4.*sqrt(T8))) ... + 12*l*a^2 * ( +1./(T1.*sqrt(T5)) +1./(T2.*sqrt(T6)) -1./(T3.*sqrt(T7)) -1./(T4.*sqrt(T8))) ... ); % d2BZA = mu0/16/a^4 * ( ... % l^3 * ( -1./(T1.*T5.^1.5) -1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... % +1./(T3.*T7.^1.5) +1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ) ... % + 6*l^2 * ( +1./(T1.*T5.^1.5) +1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... % -1./(T3.*T7.^1.5) -1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ).*z ... % + 8*z .* ( a^2 * ( -3./(T1.*sqrt(T5)) +3./(T2.*sqrt(T6)) +3./(T3.*sqrt(T7)) -3./(T4.*sqrt(T8)) ) ... % +z.^2 .* ( +1./(T1.*T5.^1.5) +1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... % -1./(T3.*T7.^1.5) -1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ) ) ... % + 12*l * ( a^2 * ( +1./(T1.*sqrt(T5)) +1./(T2.*sqrt(T6)) -1./(T3.*sqrt(T7)) -1./(T4.*sqrt(T8)) ) ... % +z.^2 .* ( -1./(T1.*T5.^1.5) -1./(T1.^2.*T5) -1./(T2.*T6.^1.5) -1./(T2.^2.*T6) ... % +1./(T3.*T7.^1.5) +1./(T3.^2.*T7) +1./(T4.*T8.^1.5) +1./(T4.^2.*T8) ) ) ... % ); return
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