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Formulae: Magnetic field inside simple coils |
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all lengths (L, R, A, B, Z): current I The magnetic induction B = µH will be calculated in |
[cm] [mA] [nT] |
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1. Ring coil |
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R = radius of the ring coil N = number of windings Z = distance from the coil centre along the coil axis  |
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2. Double ring coil (Helmholtz - coil) |
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The distance between two co-axial ring coils has to be the radii of one ring coil. R = radius of ring coils N = number of windings of one ring coil Z = distance from the coil centre along the coil axis  The field in dependency of the distance from the coil centre (Z) is shown in Figure 1 for single and double ring coils. |
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3. Rectangle (square) coil |
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2A, 2B = dimensions of the rectangle N = number of windings Z = distance from the rectangle centre perpendicular to the rectangle  |
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4. Double square coil |
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Similar to the HELMHOLTZ coil system two square coils can be used to improve the field homogeneity of the field around the coil centre. The distance A0 between the two square coils has to be 1.089 of the half dimension of the square (2A) 2A = dimensions of the square A0 = distance between square coils (A0 = 1.089 A) N = number of windings Z = distance from the coil system centre perpendicular to the square planes |
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The field in dependency of the distance from the coil centre (Z) is shown in Figure 1 for single and double square coils. |
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5. Cylinder coil |
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L = length of the coil (Z-direction) R = radius of the coil (A0 = 1.089 A) n = number of windings per centimetre Z = distance from the coil centre along the coil axis  Figure 3 shows the homogeneity of cylinder coils in dependency of the length/ radii ratio. |
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6. Compensated cylinder coil |
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To improve the field homogeneity around the coil centre two cylinder coils with equal radius, different length and opposite current direction can be combined. L1 = length of the first coil (longer coil) L2 = length of the second coil R = radius of both coil. ( L1 / R > 1.5 ) n1 = number of windings per centimetre of the first coil n2 = number of windings per centimetre of the second coil Z = distance from the coil centre along the coil axis If R, L1 and n1 are given L2, n2 have to be calculated as follow:  |
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The axis field of an optimised compensated coil in dependency of the distance from the coil centre (Z) is shown in comparison to a non compensated cylindrical coil in Figure 3. |
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 Figure 1: Coil axis field in single and double ring and square coils |
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 Figure 2: Coil axis field in cylinder coils in dependency from the L/R ratio |
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 Figure 3: Coil axis field in compensated and non-compensated cylinder coils |