A Parametric Integral Formulation to Approximate the Magneto Quasi-Static Fields of 3D Cylindrical Solenoids with Helical Winding

Abstract

The solenoid has been used extensively in a wide range of electrical applications. In the present work, an analytical model is presented for estimating the 3-dimensional (3D) magneto quasi-static fields of the cylindrical solenoid with helix-shaped (helical) wire winding. The proposed integral formulation is based on the Schur (Hadamard) vector product of the current vector I and the helical tangent vector T, to express the magnetic vector potential A. The model is used to estimate the magnetic flux density vector B and the flux linkage Psi anywhere in 3D space, excluding the (source) regions of conducting wire. A constant scaling coefficient, based on measured terminal inductance, is proposed for calibration of the model's magnetic field. Error analysis is presented on the numerical integration, based on truncated series expansion of complete elliptic integrals in terms of Chebychev polynomials. The model may be used to estimate all three cylindrical components of the magneto quasi-static fields of solenoids as a function of core diameter, wire radius, the number of wire turns per winding layer, the number of winding layers, and the complex permeability of frequency-dependent linear material. Several numerical examples are provided to validate the helical model against the superposition of circular loops, and an idealized circuit model.