API Reference

ABSCAB

_cwl_A_phi_f(rhoP, zP)

Normalized A_phi of Circular Wire Loop, far-field.

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop. This formulation is useful for points away from the wire (“far-field”) at rhoP < 1/2 or rhoP > 2 or \(|zP| >= 1\).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic vector potential

Return type

float

Meta private

_cwl_A_phi_n(rhoP, zP)

Normalized A_phi of Circular Wire Loop, near-field.

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop. This formulation is useful for points close to the wire (“near-field”) at 1/2 <= rhoP <= 2 and \(|zP| < 1\).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic vector potential

Return type

float

Meta private

_cwl_A_phi_v(zP)

Normalized A_phi of Circular Wire Loop, along rhoP=1, near-field.

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop. This formulation is useful for points along rhoP=1 with \(|zP| < 1\).

Parameters: zP (float) – normalized axial coordinate of evaluation location
Returns: normalized tangential component of magnetic vector potential
Return type: float
Meta private

_cwl_B_rho_f(rhoP, zP)

Normalized B_rho of Circular Wire Loop, far-field.

Compute the normalized radial component of the magnetic field of a circular wire loop. This formulation is useful for points away from the wire (“far-field”) at rhoP < 1/2 or rhoP > 2 or \(|zP| >= 1\).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized radial component of magnetic field

Return type

float

Meta private

_cwl_B_rho_n(rhoP, zP)

Normalized B_rho of Circular Wire Loop, near-field.

Compute the normalized radial component of the magnetic field of a circular wire loop. This formulation is useful for points close to the wire (“near-field”) at 1/2 <= rhoP <= 2 and \(|zP| < 1\).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized radial component of magnetic field

Return type

float

Meta private

_cwl_B_rho_v(zP)

Normalized B_rho of Circular Wire Loop, along rhoP=1, near-field.

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop. This formulation is useful for points along rhoP=1 with \(|zP| < 1\).

Parameters: zP (float) – normalized axial coordinate of evaluation location
Returns: normalized radial component of magnetic field
Return type: float
Meta private

_cwl_B_z_f1(rhoP, zP)

Normalized B_z of Circular Wire Loop, far-field (1).

Compute the normalized vertical component of the magnetic field of a circular wire loop. This formulation is useful for certain points away from the wire (“far-field”) at rhoP < 1/2 or (rhoP <= 2 and \(|zP| >= 1\)).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized vertical component of magnetic field

Return type

float

Meta private

_cwl_B_z_f2(rhoP, zP)

Normalized B_z of Circular Wire Loop, far-field (2).

Compute the normalized vertical component of the magnetic field of a circular wire loop. This formulation is useful for certain other points away from the wire (“far-field”) at rhoP > 2.

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized vertical component of magnetic field

Return type

float

Meta private

_cwl_B_z_n(rhoP, zP)

Normalized B_z of Circular Wire Loop, near-field.

Compute the normalized vertical component of the magnetic field of a circular wire loop. This formulation is useful for points close to the wire (“near-field”) at 1/2 <= rhoP <= 2, but not rhoP=1, and \(|zP| <= 1\).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized vertical component of magnetic field

Return type

float

Meta private

_cwl_B_z_v(zP)

Normalized B_z of Circular Wire Loop, along rhoP=1, near-field.

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop. This formulation is useful for points along rhoP=1 with \(|zP| < 1\).

Parameters: zP (float) – normalized axial coordinate of evaluation location
Returns: normalized radial component of magnetic field
Return type: float
Meta private

_sws_A_z_ax(zP)

Normalized A_z of Straight Wire Segment, along rhoP=0.

Compute the normalized axial component of magnetic vector potential of straight wire segment, evaluated along axis of wire segment (rho = 0).

Parameters: zP (float) – normalized axial coordinate of evaluation location; must not be in [0, 1] (on wire segment)
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_A_z_ax_f(zP)

Normalized A_z of Straight Wire Segment, along rhoP=0, far-field.

Compute the normalized axial component of magnetic vector potential of straight wire segment, evaluated along axis of wire segment (rho = 0). This is a special case for points away from the wire (“far-field”) for zP < -1 or zP >= 2.

Parameters: zP (float) – normalized vertical coordinate of evaluation location
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_A_z_ax_n(zP)

Normalized A_z of Straight Wire Segment, along rhoP=0, near-field.

Compute the normalized axial component of magnetic vector potential of straight wire segment, evaluated along axis of wire segment (rhoP = 0). This is a special case for points close to the wire (“near-field”) for -1 <= zP < 2.

Parameters: zP (float) – normalized axial coordinate of evaluation location; must not be in [0, 1] (on wire segment)
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_A_z_f(rhoP, zP)

Normalized A_z of Straight Wire Segment, far-field.

Compute the normalized axial component of the magnetic vector potential of a straight wire segment. This formulation is useful for points away from the wire (“far-field”) at rhoP >= 1 or zP <= -1 or zP > 2.

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized axial component of magnetic vector potential

Return type

float

Meta private

_sws_A_z_n(rhoP, zP)

Normalized A_z of Straight Wire Segment, near-field.

Compute the normalized axial component of the magnetic vector potential of a straight wire segment. This formulation is useful for points close to the wire (“near-field”) at rhoP < 1 and -1 < zP <= 2.

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized axial component of magnetic vector potential

Return type

float

Meta private

_sws_A_z_rad(rhoP)

Normalized A_z of Straight Wire Segment, along zP=0 or zP=1.

Compute the normalized axial component of the magnetic vector potential of a straight wire segment, evaluated radially along the endpoints of the wire segment (zP = 0 or zP = 1).

Parameters: rhoP (float) – normalized radial coordinate of evaluation location; must not be zero (on wire segment)
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_A_z_rad_f(rhoP)

Normalized A_z of Straight Wire Segment, along zP=0 or zP=1, far-field.

Compute the normalized axial component of the magnetic vector potential of a straight wire segment, evaluated radially along the endpoints of the wire segment (zP = 0 or zP = 1). This is a special case for points away from the wire (“far-field”) for rhoP > 1.

Parameters: rhoP (float) – normalized radial coordinate of evaluation location; must not be zero (on wire segment)
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_A_z_rad_n(rhoP)

Normalized A_z of Straight Wire Segment, along zP=0 or zP=1, near-field.

Compute the normalized axial component of the magnetic vector potential of a straight wire segment, evaluated radially along the endpoints of the wire segment (zP = 0 or zP = 1). This is a special case for points close to the wire (“near-field”) for rhoP <= 1.

Parameters: rhoP (float) – normalized radial coordinate of evaluation location; must not be zero (on wire segment)
Returns: normalized axial component of magnetic vector potential
Return type: float
Meta private

_sws_B_phi_f(rhoP, zP)

Normalized B_phi of Straight Wire Segment, far-field.

Compute the normalized tangential component of the magnetic field of a straight wire segment. This formulation is useful for points away from the wire (“far-field”) at rhoP >= 1 or zP <= 0 or zP >= 1 or rhoP/(1-zP) >= 1 or rhoP/zP >= 1.

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic field

Return type

float

Meta private

_sws_B_phi_n(rhoP, zP)

Normalized B_phi of Straight Wire Segment, near-field.

Compute the normalized tangential component of the magnetic field of a straight wire segment. This formulation is useful for points close to the wire (“near-field”) at rhoP < 1 and 0 < zP < 1 and rhoP/(1-zP) < 1 and rhoP/zP < 1.

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic field

Return type

float

Meta private

_sws_B_phi_rad(rhoP)

Normalized B_phi of Straight Wire Segment, along zP=0 or zP=1.

Compute the normalized tangential component of the magnetic field of a straight wire segment, evaluated radially along the endpoints of the wire segment (zP = 0 or zP = 1).

Parameters: rhoP (float) – normalized radial coordinate of evaluation location
Returns: normalized tangential component of magnetic field
Return type: float
Meta private

circularWireLoop_A_phi(rhoP, zP)

Compute the normalized tangential component of the magnetic vector potential of a circular wire loop.

This method selects the proper special case method to use for computing the normalized tangential component of the magnetic vector potential of a circular wire loop for given evaluation location (rhoP, zP).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic vector potential

Return type

float

Meta private

circularWireLoop_B_rho(rhoP, zP)

Compute the normalized radial component of the magnetic field of a circular wire loop.

This method selects the proper special case method to use for computing the normalized radial component of the magnetic field of a circular wire loop for given evaluation location (rhoP, zP).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized radial component of magnetic field

Return type

float

Meta private

circularWireLoop_B_z(rhoP, zP)

Compute the normalized vertical component of the magnetic field of a circular wire loop.

This method selects the proper special case method to use for computing the normalized vertical component of the magnetic field of a circular wire loop for given evaluation location (rhoP, zP).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized radial component of magnetic field

Return type

float

Meta private

magneticFieldCircularFilament(center, normal, radius, current, evalPos)

Compute the magnetic field of a circular wire loop.

Parameters

center (arr(float)) – [3: x, y, z] origin of loop (in meters)
normal (arr(float)) – [3: x, y, z] normal vector of loop (in meters); will be normalized internally
radius (float) – radius of the wire loop (in meters)
current (float) – loop current (in A)
evalPos (arr(float)) – [nEvalPos][3: x, y, z] evaluation locations (in meters)

Returns

[nEvalPos][3: B_x, B_y, B_z] Cartesian components of the magnetic field evaluated at the given locations (in T)

Return type

arr(float)

magneticFieldPolygonFilament(vertices, current, evalPos, useCompensatedSummation=True)

Compute the magnetic field of a polygon filament at a number of evaluation locations.

Parameters

vertices (arr(float)) – [numVertices][3: x, y, z] points along polygon; in m
current (float) – current along polygon; in A
evalPos (arr(float)) – [numEvalPos][3: x, y, z] evaluation locations; in m
useCompensatedSummation (bool) – If true, use Kahan-Babuska compensated summation to compute the superposition of the contributions from the polygon vertices; otherwise, use standard += summation.

Returns

[numEvalPos][3: x, y, z] magnetic field at evaluation locations; in T

Return type

arr(float)

magneticFieldPolygonFilamentVertexSupplier(numVertices, vertexSupplier, current, evalPos, useCompensatedSummation=True)

Compute the magnetic field of a polygon filament at a number of evaluation locations.

Parameters

numVertices (int) – number of polygon vertices to take into account
i), arr(float) vertexSupplier (callable(int) – should return points along polygon as [3: x, y, z] for i=0,1,…,(numVertices-1); in m
current (float) – current along polygon; in A
evalPos (arr(float)) – [numEvalPos][3: x, y, z] evaluation locations; in m
useCompensatedSummation (bool) – If true, use Kahan-Babuska compensated summation to compute the superposition of the contributions from the polygon vertices; otherwise, use standard += summation.

Returns

[numEvalPos][3: x, y, z] magnetic field at evaluation locations; in T

Return type

arr(float)

straightWireSegment_A_z(rhoP, zP)

Compute the normalized axial component of the magnetic vector potential of a straight wire segment.

This method selects the proper special case method to use for computing the normalized axial component of the magnetic vector potential of a straight wire segment for given evaluation location (rhoP, zP).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized axial component of magnetic vector potential

Return type

float

Meta private

straightWireSegment_B_phi(rhoP, zP)

Compute the normalized tangential component of the magnetic field of a straight wire segment.

This method selects the proper special case method to use for computing the normalized tangential component of the magnetic field of a straight wire segment for given evaluation location (rhoP, zP).

Parameters

rhoP (float) – normalized radial coordinate of evaluation location
zP (float) – normalized axial coordinate of evaluation location

Returns

normalized tangential component of magnetic field

Return type

float

Meta private

vectorPotentialCircularFilament(center, normal, radius, current, evalPos)

Compute the magnetic vector potential of a circular wire loop.

Parameters

center (arr(float)) – [3: x, y, z] origin of loop (in meters)
normal (arr(float)) – [3: x, y, z] normal vector of loop (in meters); will be normalized internally
radius (float) – radius of the wire loop (in meters)
current (float) – loop current (in A)
evalPos (arr(float)) – [nEvalPos][3: x, y, z] evaluation locations (in meters)

Returns

[nEvalPos][3: A_x, A_y, A_z] Cartesian components of the magnetic vector potential evaluated at the given locations (in Tm)

Return type

arr(float)

vectorPotentialPolygonFilament(vertices, current, evalPos, useCompensatedSummation=True)

Compute the magnetic vector potential of a polygon filament at a number of evaluation locations.

Parameters

vertices (arr(float)) – [numVertices][3: x, y, z] points along polygon; in m
current (float) – current along polygon; in A
evalPos (arr(float)) – [numEvalPos][3: x, y, z] evaluation locations; in m
useCompensatedSummation (bool) – If true, use Kahan-Babuska compensated summation to compute the superposition of the contributions from the polygon vertices; otherwise, use standard += summation.

Returns

[numEvalPos][3: x, y, z] magnetic vector potential at evaluation locations; in Tm

Return type

arr(float)

vectorPotentialPolygonFilamentVertexSupplier(numVertices, vertexSupplier, current, evalPos, useCompensatedSummation=True)

Compute the magnetic vector potential of a polygon filament at a number of evaluation locations.

Parameters

numVertices (int) – number of polygon vertices to take into account
i), arr(float) vertexSupplier (callable(int) – should return points along polygon as [3: x, y, z] for i=0,1,…,(numVertices-1); in m
current (float) – current along polygon; in A
evalPos (arr(float)) – [numEvalPos][3: x, y, z] evaluation locations; in m
useCompensatedSummation (bool) – If true, use Kahan-Babuska compensated summation to compute the superposition of the contributions from the polygon vertices; otherwise, use standard += summation.

Returns

[numEvalPos][3: x, y, z] magnetic vector potential at evaluation locations; in Tm

Return type

arr(float)

MU_0 = 1.25663706212e-06: vacuum magnetic permeability in Vs/Am (CODATA-2018)

MU_0_BY_2_PI = 2.0000000010887514e-07: vacuum magnetic permeability, divided by \(2 \pi\)

MU_0_BY_4_PI = 1.0000000005443757e-07: vacuum magnetic permeability, divided by \(4 \pi\)

MU_0_BY_PI = 4.000000002177503e-07: vacuum magnetic permeability, divided by \(\pi\)

Bulisch Generalized Complete Elliptic Integral

cel(k_c, p, a, b)

Generalized Complete Elliptic Integral by Bulirsch (1969).

The generalized complete elliptic integral is computed using an iterative algorithm presented by R. Bulirsch in “Numerical Calculation of Elliptic Integrals and Elliptic Functions. III” in “Numerische Mathematik” 13, 305-315 (1969):

\[cel(k_c, p, a, b) = \int\limits_0^{\pi/2} \frac{a \cos^2{\varphi} + b \sin^2{\varphi}}{ \cos^2{\varphi} + p \sin^2{\varphi}} \frac{\mathrm{d}\varphi}{\sqrt{\cos^2{\varphi} + k_c^2 \sin^2{\varphi}}}\]

Parameters

k_c (float) – parameter k_c of cel(); absolute value must not be 0
p (float) – parameter p of cel()
a (float) – parameter a of cel()
b (float) – parameter b of cel()

Returns

value of cel(k_c, p, a, b)

Return type

float

PI_2 = 1.5707963267948966: half of pi

SQRT_EPS = 1.4901161193847656e-08: sqrt of machine epsilon

Kahan-Babushka Compensated Summation

compAdd(contribution, compSum)

Add a single contribution to a compensated summation.

Add a single contribution to the sum. The compensated sum is obtained by summing the final values of s, cs and ccs after this method has been called for all contributions.

Parameters

contribution (float) – contribution to add to the sum
compSum (arr(float)) – [3: s, cs, ccs] target for output