Calculation of the single layer RF coil with an arbitrary winding pitch.
For the calculation of the single-layer inductors can be used Maxwell's model, then each turn represents as a circular ring. The Lorentz model is an attempt to avoid this simplification, however, and it works with good accuracy only if the turns are close to each other and in the end, both models are not suitable for the calculation of the helical coils with a large winding pitch.
How to solve the problem of calculation of such coils? First, using a semiempirical correction formula. Is the easy path.
Single-layer coil with winding pitch can be calculated as a singlelayer coil with close-winding. Only the amendment to the resulting inductance is added:
where:
- ΔL is the amendment to the inductance of the close-winding coil (always negative) (µH)
- N - number of turns
- D - diameter of winding (mm)
- d - diameter of wire (mm)
- p - winding pitch (distance between centers of neighboring turns = the gap between the turns + wire diameter) (mm)
The formula gives an error of about ±5-10% when the number of turns of at least 5 and the winding pitch of not more than two diameters of the wire. The formula was used in early versions of the Coil32 program.
Now Coil64 app uses a more advanced method of calculation of the single layer coil with an arbitrary pitch of winding with any number of turns.
In 1926 Chester Snow successfully tried to create a realistic model of the coil with a pitch of winding [reference #19]. He developed a mathematics method of calculating the mutual inductance between two infinitely thin helical conductors. Using the concept of “geometric mean distance”, which introduced J.C. Maxwell, it is possible to calculate the self-inductance of the helical coil. However, the formula of C. Snow is inaccurate, particularly in the limiting cases of a single turn of the helix, and when the helix with an endless pitch of winding degenerates into a straight wire—the formula gives the wrong result. In 2012, Robert Weaver, based on Snow's method, corrected these lacks and created a numerical calculation algorithm for calculating a coil with an arbitrary winding pitch. Snow's original formula is as follows:
where:
- M - mutual inductance;
- R2 = r12 + r22 - 2r1r2cos(θ2 - θ1)
- N - number of turns;
- r1, r2 - radii of the helical filaments;
- y1, y2 - axial offset;
- px - pitch factor (pitch p = 2πpx);
- θ1 - starting angle of the helix;
- θ2 - ending angle;
Snow defined the coil geometry in cylindrical co-ordinates. The details in the original Weaver's work.
In his work, Robert Weaver created a method for solving integral using numerical algorithm, based on Simpson's rule and on Newton - Raphson method and provided the programming algorithm, the source codes of which was ported and is used in Coil32 v for Windows XP, in the cross-platform app Coil64 and in Android version with slight modifications, allowing to accelerate the iteration.
