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

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-D magneto quasi-static (MQS) 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
$\vec {I}$
and the helical tangent vector
$\vec {T}$
, to express the magnetic vector potential
$\vec {A}$
. The model is used to estimate the magnetic flux density vector
$\vec {B}$
and the flux linkage
$\Psi $
anywhere in 3-D space, excluding the (source) regions of conducting wire. A constant scaling coefficient, based on the measured terminal inductance, is proposed for calibration of the model’s magnetic field. Error analysis is presented on 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 MQS 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 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.