Coil Type: Stranded Coil¶

A stranded coil is a coil wound from a conductor consisting of many thin insulated strands. Electrically, all strands are connected in series along the winding path and carry the same current.

In numerical models the individual strands are not resolved geometrically. Instead, the entire winding pack is replaced by a homogenized conducting body with equivalent electromagnetic, thermal, and electrical behavior.

Stranded coils are the standard model for motor and transformer windings, small inductors, actuator solenoids, MRI gradient coils, and any multi-turn winding where the strand diameter stays smaller than the skin depth.

Figure 1: Homogenization of a coil with individual windings, which significantly reduces the computational cost. Instead of modeling each individual wire we combine them to form a solid block.

Although a stranded coil may geometrically resemble a solid coil, the physics are fundamentally different:

The current density is amplified by the number of turns.
The current density is uniform over each cross-section of the winding pack.
Eddy currents are absent (no closed loops inside the conductor).
Only a fraction of the cross-section conducts current → fill factor.
Thermal and electrical properties become orthotropic (strong along the wire, weak across it).

Validity¶

This approximation is valid when the strand diameter is small compared to the skin depth. For example, consider a wound coil with a wire diameter of $d=0.25\,\mathrm{mm}$ and an electric conductivity of $5.8 \times 10^7\,\mathrm{S/m}$. At a frequency of $f=1000\,\mathrm{Hz}$ the skin depth is $$ \delta_s = \sqrt{\frac{2}{\omega \mu \sigma}} = \sqrt{\frac{2}{ \left(2\pi \cdot 1000\,\mathrm{Hz}\right) \left(4\pi \cdot 10^{-7}\,\mathrm{H/m}\right) \left(5.8 \times 10^7\,\mathrm{S/m}\right) }} \approx 2.1\,\mathrm{mm}, $$ which is larger than the diameter of the wire. This means that the current remains uniform inside each strand to a very good approximation. This reduces the computational cost by several orders of magnitude while preserving correct global behavior.

Electric Current Homogenization¶

The homogenization of the coil requires certain adaptations. Consider the front face of the stranded coil shown in Figure 1.

Figure 2: Cross-section of the homogenized stranded coil used to derive the averaged current density and effective conductivity.

If each strand carries a current $I$, then across the coil cross-section we have an averaged electric current density given by $$ j_z = \frac{N_t I}{S_c} \quad, $$ where $I\,[\mathrm{A}]$ is the applied current, $N_t$ is the number of turns, and $S_c\,[\mathrm{m}^2]$ is the cross-section of the coil.

We further introduce the fill factor $n_f$ of a stranded coil given by $$ n_f = \frac{S_\text{ws}}{S_c} \quad, $$ where $S_\text{ws}$ is the total wire cross-section area. The conductivity assigned to the homogenised region is the raw conductor conductivity $\sigma_\text{ws}$; the fill-factor correction enters the Ohmic-loss integrand, so the effective dissipation density is $$ P_\Omega = \frac{|\mathbf{J}|^2}{n_f\,\sigma_\text{ws}} \quad, $$ where $\mathbf{J}$ is the homogenized electric current density. The correction is supplied separately through the wire specification; without one it defaults to $n_f = 1$ (no correction).

Wire Specification¶

The fill factor $n_f$ — the fraction of the bundle cross-section actually occupied by copper — is supplied to CoilTypeStranded via the optional wire_specification argument. Three equivalent forms are available; pick whichever is most natural given the data on hand (manufacturer datasheet, CAD geometry, or measured copper fraction). Figure 3 shows the geometric picture behind the three forms.

Figure 3: Three bundles of $N_t$ wires with the same bundle cross-section $S_c$. Each wire is individually insulated (enamel coating, plus air or epoxy filling the interstitial space), so only the copper area conducts current and the rest of the bundle is dead weight in the fill factor $n_f = N_t\,A_\text{wire}/S_c$. Left: closely packed round wires (high $n_f$). Middle: round wires of smaller diameter with more insulation, lower $n_f$. Right: rectangular conductors pack more tightly than round ones for the same bundle size. Typical magnet windings reach $n_f \approx 0.5$–$0.8$; well-stacked rectangular bar conductors can exceed $0.9$.

Class	Argument	Effective $n_f$	When to use
`StrandedWireSpecificationFillRatio`	`fill_ratio` (dimensionless, $0 < n_f \le 1$)	$n_f$ given directly	When only the bulk copper-to-bundle ratio is known (datasheet, measured) and the bundle cross-section is roughly uniform along the coil.
`StrandedWireSpecificationWireDiameter`	`wire_diameter` $[\mathrm{m}]$	$n_f(x) = N_t\,\pi(d/2)^2\,/\,S_c(x)$	For round wires of known diameter — the common case for magnet wire. Stays accurate when the bundle tapers along the coil.
`StrandedWireSpecificationWireCrossSection`	`wire_cross_section` $[\mathrm{m}^2]$	$n_f(x) = N_t\,A_\text{wire}\,/\,S_c(x)$	For non-circular wires (rectangular bar conductors, custom strand shapes) where the diameter is not a meaningful input. Stays accurate when the bundle tapers along the coil.

where $N_t$ is the number of turns and $S_c(x)$ the local bundle cross-section.

If no wire specification is given, $n_f \equiv 1$ and no fill-factor correction is applied.

Note

The fill factor enters the coil resistance and therefore the Ohmic heating: lower $n_f$ means less copper carrying the current, so the resistance rises as $1/n_f$ and the dissipated power $P_\Omega = |\mathbf{J}|^2 / (n_f\,\sigma_\text{ws})$ rises with it. Without a wire specification both quantities are reported as if the bundle were fully copper.

Example¶

A bare stranded coil with no fill-factor correction:

from mufem.electromagnetics.coil import CoilTypeStranded

coil_type = CoilTypeStranded(number_of_turns=280)

The same coil with a 70 % copper fill ratio:

from mufem.electromagnetics.coil import (
    CoilTypeStranded,
    StrandedWireSpecificationFillRatio,
)

coil_type = CoilTypeStranded(
    number_of_turns=280,
    wire_specification=StrandedWireSpecificationFillRatio(fill_ratio=0.7),
)

Same coil, but specified through the physical wire diameter ($d = 0.5\,\mathrm{mm}$):

from mufem.electromagnetics.coil import (
    CoilTypeStranded,
    StrandedWireSpecificationWireDiameter,
)

coil_type = CoilTypeStranded(
    number_of_turns=280,
    wire_specification=StrandedWireSpecificationWireDiameter(wire_diameter=0.5e-3),
)

Class	Argument	Effective \(n_f\)	When to use
`StrandedWireSpecificationFillRatio`	`fill_ratio` (dimensionless, \(0 < n_f \le 1\))	\(n_f\) given directly	When only the bulk copper-to-bundle ratio is known (datasheet, measured) and the bundle cross-section is roughly uniform along the coil.
`StrandedWireSpecificationWireDiameter`	`wire_diameter` \([\mathrm{m}]\)	\(n_f(x) = N_t\,\pi(d/2)^2\,/\,S_c(x)\)	For round wires of known diameter — the common case for magnet wire. Stays accurate when the bundle tapers along the coil.
`StrandedWireSpecificationWireCrossSection`	`wire_cross_section` \([\mathrm{m}^2]\)	\(n_f(x) = N_t\,A_\text{wire}\,/\,S_c(x)\)	For non-circular wires (rectangular bar conductors, custom strand shapes) where the diameter is not a meaningful input. Stays accurate when the bundle tapers along the coil.