Inductance calculation for helical conductors

Abstract

The self-inductance for an infinitely long helical conductor and the mutual inductance between two coaxial helical conductors are investigated over the whole range from solenoids with an infinitesimal pitch length to straight conductors with an infinite pitch length. The principal terms, apart from the logarithmically divergent term due to the infinite length of the self-and mutual inductances, are rigorously obtained, evaluating the double integrals of Neumann's formula, using analytical expressions for the vector potential of a single helical thin conductor. The relations between the conventional approximate and the rigorously obtained expressions for the self-and mutual inductances of solenoids are also compared. This analytical method is applied for calculations of the self-inductance of a twisted bifilar lead and for the current distribution of a twisted superconducting 6 around 1 strand cable with insulated strands, using the cancellation of the logarithmically divergent term. As a result, it is shown that the analytical method for the inductance calculation for infinitely long helical conductors is useful, by obtaining results consistent with the magnetic energy calculation.

Magnetic field of complex helical conductors

“…An analytical method for calculating the low-frequency magnetic field of an infinitely long helical line current using the magnetic vector potential has been derived in the pioneering work [1] and the problem has afterwards been revisited in [2][3][4][5][6], [8][9][10][11][12]. Few papers take up the problem of finite length helical conductors [13][14][15][16].…”

Magnetic field calculation of an infinitely long solenoid

“…Furthermore, the asymptotic forms for these vector potentials as k → 0 (or L p → ∞) are obtained using the asymptotic forms for the modified Bessel functions [9,10]. For both the interior (r < a) and exterior (r > a) regions of a helical thin conductor,…”

“…As a result, it is revealed that the asymptotic form for these vector potentials as k → 0 (or L p → ∞) becomes the expression of an infinitely long straight conductor, as shown in equation ( 14). Similarly, the asymptotic forms for these vector potentials as k → ∞ (or L p → 0) are obtained using the asymptotic forms for the modified Bessel functions [9,10]. For the interior region of a helical thin conductor, r < a, A r (r, θ, z) ≈ 0…”

Calculations using the circuit equation for current and field distributions of type II superconductors

“…The current distribution among whole divided elements of a superconductor during a current sweep and under an external field can be calculated fundamentally by electric circuit theory as well as by the methods used for eddy current problems for normal conductors [11,12]. Furthermore, the self-and mutual inductances for two parallel straight conductors with crosssectional areas of S p and S q and a length of l can be expressed using the geometrical mean distance R pq as follows [9]: where r is the distance between two parallel filaments within each conductor, L pq,r is the principal term without the logarithmically divergent term of the self-or mutual inductance among parallel straight conductors and L pq,r is called the 'reduced' self-or mutual inductance in this article [13]. It should be noted that the expression of equation ( 1) is rigorously correct only for infinitely long straight conductors.…”