I am studying a simple magnetostatic coil arrangement and would appreciate your opinion on the achievable field homogeneity.
The geometry consists of four coaxial square loops placed along the (z)-axis. Two loops are larger primary square loops, and two smaller loops are auxiliary/support loops. Each primary loop is connected in series with one support loop, so the current is the same in the paired loops.
For the current case, I am keeping the geometry constrained as follows:
a_p = 5 cm
a_s = 3 cm
N_p = N_s = 1
where (a_p) and (a_s) are the half-side lengths of the primary and support square loops. The main design variables I am changing are only the axial spacing between the primary loops and the axial positions of the support loops.
Using numerical simulation based on the square-loop axial magnetic field equation and superposition, the best result I obtained so far gives only moderate homogeneity over a relatively wide axial region. The field improves compared with the primary pair alone, but it still does not become very flat over the full region of interest.
My question is:
With the loop sizes and turn numbers fixed, is there a known analytical method to select the axial spacings more effectively, such as using Taylor expansion or derivative cancellation of (B_z(z)) around the center?
Or is the achievable flatness physically limited under these constraints, meaning that a significant improvement would require changing the physical design, such as proportional scaling of the loop sizes, increasing the number of turns, or changing the support-loop dimensions?
I am mainly trying to understand whether spacing optimization alone can produce a near-flat axial field, or whether the geometry needs to be modified more fundamentally.
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