Please have a look at Reluctance forces in the Users Guide for an explanation of the different flux tube categories and resulting sub-packages.
Except for the element LeakageWithCoefficient, the permeances of all elements of this package are calculated from their geometry. These flux tube elements are intended for modelling of leakage fields through vacuum, air and other media with a relative permeability my_r=1.
All dimensions are defined as parameters. As a result, the shape of these elements will remain constant during dynamic simulation of actuators and reluctance forces will not be generated in these flux tube elements. A simple leakage flux tube with reluctance force generation is provided with the element Force.LeakageAroundPoles. In cases where the accuracy of that element is not sufficient, the leakage elements of this package can be adapted and extended so that they are able to change their shape with armature motion and to generate reluctance forces. This requires an extension of the partial model Force.PartialForce, a higher variability of the variables representing the flux tube's dimensions, definition of a relationship between armature position and these dimensions and determination of the analytic derivative dG_m/dx of the flux tube's permeance G_m with respect to armature position x.
Name | Description |
---|---|
![]() | Base class for leakage flux tubes with position-independent permeance and hence no force generation; my_r=1 |
![]() | Leakage flux from one edge to the opposite plane through a quarter cylinder |
![]() | Leakage flux in circumferential direction through a quarter hollow cylinder |
![]() | Leakage flux through the edges of a half cylinder |
![]() | Leakage flux in circumferential direction through a half hollow cylinder |
![]() | Leakage flux through the corners of a quarter sphere |
![]() | Leakage flux through the edges of a qarter hollow sphere |
![]() | Leakage flux through one edge and the opposite plane of an eighth of a sphere |
![]() | Leakage flux through one edge and the opposite plane of an eighth of a hollow sphere |
![]() | Leakage flux between the end planes of a inner solid cylinder and a coaxial outer hollow cylinder |
![]() | Leakage reluctance with respect to the reluctance of a useful flux path (not for dynamic simulation of actuators) |
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
partial model PartialLeakage "Base class for leakage flux tubes with position-independent permeance and hence no force generation; my_r=1" extends Modelica_Magnetic.Interfaces.TwoPortComponent; SI.Reluctance R_m "Magnetic reluctance"; SI.Permeance G_m "Magnetic permeance"; equation V_mag = Phi * R_m; R_m = 1/G_m; end PartialLeakage;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.1 | Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane) [m] |
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Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model QuarterCylinder "Leakage flux from one edge to the opposite plane through a quarter cylinder" extends PartialLeakage; parameter SI.Thickness t = 0.1 "Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane)"; equation G_m = my_0 * 0.52 * t; end QuarterCylinder;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.1 | Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m] |
Real | ratio | 1 | Constant ratio b/r_i |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model QuarterHollowCylinder "Leakage flux in circumferential direction through a quarter hollow cylinder" extends PartialLeakage; parameter SI.Thickness t = 0.1 "Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i)"; parameter Real ratio = 1 "Constant ratio b/r_i"; equation G_m = 2* my_0 * t * Modelica.Math.log(1 + ratio) /pi; end QuarterHollowCylinder;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.1 | Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges) [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model HalfCylinder "Leakage flux through the edges of a half cylinder" extends PartialLeakage; parameter SI.Thickness t = 0.1 "Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges)"; equation G_m = my_0 * 0.26 * t; end HalfCylinder;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.1 | Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m] |
Real | ratio | 1 | Constant ratio b/r_i |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model HalfHollowCylinder "Leakage flux in circumferential direction through a half hollow cylinder" extends PartialLeakage; parameter SI.Thickness t = 0.1 "Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i)"; parameter Real ratio = 1 "Constant ratio b/r_i"; equation G_m = my_0 * t * Modelica.Math.log(1 + ratio) /pi; end HalfHollowCylinder;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Diameter | d | 0.01 | Diameter of quarter sphere [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model QuarterSphere "Leakage flux through the corners of a quarter sphere" extends PartialLeakage; parameter SI.Diameter d = 0.01 "Diameter of quarter sphere"; equation G_m = my_0 * 0.077 * d; end QuarterSphere;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.01 | Thickness of sperical shell [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model QuarterHollowSphere "Leakage flux through the edges of a qarter hollow sphere" extends PartialLeakage; parameter SI.Thickness t = 0.01 "Thickness of sperical shell"; equation G_m = my_0 * 0.25 * t; end QuarterHollowSphere;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Radius | r | 0.01 | Radius of eighth of sphere [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model EighthOfSphere "Leakage flux through one edge and the opposite plane of an eighth of a sphere" extends PartialLeakage; parameter SI.Radius r = 0.01 "Radius of eighth of sphere"; equation G_m = my_0 * 0.308 * r; end EighthOfSphere;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Thickness | t | 0.01 | Thickness of sperical shell [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model EighthOfHollowSphere "Leakage flux through one edge and the opposite plane of an eighth of a hollow sphere" extends PartialLeakage; parameter SI.Thickness t = 0.01 "Thickness of sperical shell"; equation G_m = my_0 * 0.5 * t; end EighthOfHollowSphere;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Type | Name | Default | Description |
---|---|---|---|
Radius | r_0 | 10e-3 | Radius of inner solid cylinder [m] |
Radius | r_1 | 17e-3 | Inner radius of outer hollow cylinder [m] |
Radius | r_2 | 20e-3 | Outer radius of outer hollow cylinder [m] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model CoaxCylindersEndFaces "Leakage flux between the end planes of a inner solid cylinder and a coaxial outer hollow cylinder" extends PartialLeakage; parameter SI.Radius r_0 = 10e-3 "Radius of inner solid cylinder"; parameter SI.Radius r_1 = 17e-3 "Inner radius of outer hollow cylinder"; parameter SI.Radius r_2 = 20e-3 "Outer radius of outer hollow cylinder"; equation assert(Modelica.Math.log(r_2/r_1) >= 2*(r_2-r_1)/(r_1+r_0), "No proper values assigned to Radii r_0...r_2!"); G_m = 2*my_0 * sqrt( ((r_1+r_0)/2 * Modelica.Math.log(r_2/r_1))^2 - (r_2-r_1)^2); end CoaxCylindersEndFaces;
Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.
Differently from the other flux tube elements of this package that are calculated from their geometry, this leakage reluctance is calculated with reference to the total reluctance of a useful flux path. Please refer to the Parameters section for an illustration of the resulting magnetic network. Exploiting Kirchhoff's generalized current law, the leakage reluctance is calculated by means of a leakage coefficient c_leak.
Attention:
This element must not be used for dynamic simulation of electro-magneto-mechanical actuators, where the shape of at least one flux tube element with reluctance force generation in the useful flux path changes with armature motion (e.g. air gap). This change results in a non-zero derivative dG_m/dx of those elements permeance G_m with respect to armature position x, which in turn will lead to a non-zero derivative of the leakage element's permeance with respect to armature position. This would result in a reluctance force generated by the leakage element that is not accounted for properly. Instead, use the leakage element for static analyses of magnetic networks only as illustrated in Examples.ElectrodynamicActuator.MagneticCircuitModel.
Type | Name | Default | Description |
---|---|---|---|
LeakageCoefficient | c_leak | 0.3 | Ratio leakage flux/(leakage flux + useful flux) = leakage flux/total flux [1] |
Reference reluctance | |||
Reluctance | R_mUsefulTot | Modelica.Constants.inf | Total reluctance of useful flux path as reference [H-1] |
![]() |
Type | Name | Description |
---|---|---|
PositiveMagneticPort | p | Positive magnetic port |
NegativeMagneticPort | n | Negative magnetic port |
model LeakageWithCoefficient "Leakage reluctance with respect to the reluctance of a useful flux path (not for dynamic simulation of actuators)" extends PartialLeakage; parameter SI.LeakageCoefficient c_leak = 0.3 "Ratio leakage flux/(leakage flux + useful flux) = leakage flux/total flux"; SI.Reluctance R_mUsefulTot = Modelica.Constants.inf "Total reluctance of useful flux path as reference"; //Default value to ensure proper assignment of reference reluctance in magnetic network model equation assert(R_mUsefulTot < Modelica.Constants.inf, "No proper value assigned to R_mUsefulTot!"); c_leak * R_m = R_mUsefulTot * (1 - c_leak); // Generalized Kirchhoff's current law end LeakageWithCoefficient;