In the magnetic vector potential formulations discussed in previous sections, the source current density {Js} is either known and input as BFE,,JS (stranded conductor analysis) or determined from a fully-coupled electromagnetic solution as a sum of eddy Je (Equation 5–124) and DC conduction Js (Equation 5–125) current densities (massive conductor analysis).
A stranded coil refers to a coil consisting of many fine turns of conducting wires. Because in the low-frequency approximation the cross-section of the wires is small compared to the skin depth, the eddy currents in the wire can be neglected and the magnitude of the current density within the wires can be considered constant. The coil can be energized by an applied voltage or current, or by a controlling electric circuit. The stranded coil analysis can be viewed as a special case of electromagnetic field analysis where the current flow direction in wires is known and determined by the winding, while the magnitude of the current in the coil or voltage drop across the coil can either be imposed or determined from the solution.
The following sections describe the magnetic and electric equations that govern the stranded coil and the different finite element formulations used by the electromagnetic elements. The formulations apply to static, transient and harmonic analysis types.
In a stranded coil analysis, the magnetic field equation
 | (5–157) |
is coupled to the electric circuit equation
 | (5–158) |
by the following expressions for the electric current density {J} and the magnetic flux Φ in the coil:
 | (5–159) |
 | (5–160) |
where:
{A} = magnetic vector potential,
ν = magnetic reluctivity,
ΔV = voltage drop across the coil,
I = total electric current,
R = total DC resistance of the coil winding,
Sc = coil cross-sectional area,
Nc = number of coil turns,
Vc = coil volume,
{t} = winding direction vector (direction of {J}).
The coupled set of electromagnetic equations governing the stranded coil is obtained by eliminating the current density {J} and magnetic flux Φ from Equation 5–157 and Equation 5–158 using Equation 5–159 and Equation 5–160, respectively. If the voltage drop across the coil ΔV is considered to be an independent electric variable in the stranded coil analysis, the A-VOLT-EMF formulation results.
This formulation is available with the stranded coil analysis option (KEYOPT(1) = 2) of the current-technology electromagnetic elements PLANE233, SOLID236 and SOLID237.
In addition to the magnetic vector potential ({A}) and the voltage drop across the coil (ΔV) unknowns, the coil electromotive force (E) is introduced as a degree of freedom to prevent the distribution of eddy currents in the wire:
 | (5–161) |
The corresponding finite element matrix equation can be written as:
 | (5–162) |
where:
[KAA] = element magnetic reluctivity matrix defined in Equation 5–112,
Sc = coil cross-sectional area (input as R2 (2D) or R1 (3D) on R command),
Nc = number of coil turns (input as R3 (2D) or R2 (3D) on R command),
Vc = coil volume (calculated (2D) or input as R3 (3D) on R command),
{t} = current direction vector (input as R5 (2D) or R4-R6 (3D) on R command),
R = total DC resistance of the coil (input as R6 (2D) or R7 (3D) on R command),
s = symmetry factor of the coil (input as R7 (2D) or R8 (3D) on R command),
{N} = vector of element scalar (node-based) shape functions,
[W] = {N} (2D) or matrix of vector (edge-based) shape functions (3D),
vol = element volume,
{Ie} = nodal current vector (input/output as AMPS),
{Ae} = nodal magnetic vector potential - Z-component of magnetic vector potential at element nodes (2D) or edge-flux at element midside nodes (3D) (input/output as AZ),
{ΔVe} = nodal voltage drop across the coil (input/output as VOLT),
{Ee} = nodal electromotive force in the coil (input/output as EMF).
Note that the VOLT and EMF degrees of freedom should be coupled for each coil using the CP,,VOLT and CP,,EMF commands.
Equation 5–162 that strongly couples magnetic
and electric degrees of freedom in a stranded coil is nonsymmetric.
It can be made symmetric by either using the weak coupling option
(KEYOPT(2) = 1) in static or transient analyses or, provided the symmetry
factor s = 1, by using the time-integrated voltage drop and emf formulation
(KEYOPT(2) = 2) in transient or harmonic analyses. In the latter case,
the VOLT and EMF degrees of freedom have the meaning of the time-integrated
voltage drop across the coil
and time-integrated
electromotive force
, and Equation 5–162 becomes:
 | (5–163) |
With the A-VOLT-EMF formulation where VOLT and EMF are 'true' (not time-integrated) voltage drop and emf respectively, the stranded coil can be node-coupled coupled to the current-based circuit elements (CIRCU124) as they share the same VOLT degree of freedom.
The stranded coil analysis results include:
Nodal magnetic flux density calculated as in Equation 5–121 (output as B)
Nodal magnetic field intensity calculated as in Equation 5–122 (output as H)
Element electric current density calculated as in Equation 5–159 (output as JT or JS)
Nodal magnetic forces calculated as in Equation 5–140 (output as FMAG)
Element Joule heat generation rate calculated as in Equation 5–146 (output as JHEAT)
Note that the calculated current density JT (or JS) and the Joule heat generation rate JHEAT are effective in the sense that they are calculated based on the coil cross-sectional area (SC) and coil volume (VC), respectively, and those real constants include the wire and the non-conducting material filling the space between the winding.