Modelica_Magnetic.FluxTube.Leakage

Leakage flux tubes with position-independent permeance and hence no force generation; my_r=1

Information


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.

Package Content

NameDescription
Modelica_Magnetic.FluxTube.Leakage.PartialLeakage PartialLeakage Base class for leakage flux tubes with position-independent permeance and hence no force generation; my_r=1
Modelica_Magnetic.FluxTube.Leakage.QuarterCylinder QuarterCylinder Leakage flux from one edge to the opposite plane through a quarter cylinder
Modelica_Magnetic.FluxTube.Leakage.QuarterHollowCylinder QuarterHollowCylinder Leakage flux in circumferential direction through a quarter hollow cylinder
Modelica_Magnetic.FluxTube.Leakage.HalfCylinder HalfCylinder Leakage flux through the edges of a half cylinder
Modelica_Magnetic.FluxTube.Leakage.HalfHollowCylinder HalfHollowCylinder Leakage flux in circumferential direction through a half hollow cylinder
Modelica_Magnetic.FluxTube.Leakage.QuarterSphere QuarterSphere Leakage flux through the corners of a quarter sphere
Modelica_Magnetic.FluxTube.Leakage.QuarterHollowSphere QuarterHollowSphere Leakage flux through the edges of a qarter hollow sphere
Modelica_Magnetic.FluxTube.Leakage.EighthOfSphere EighthOfSphere Leakage flux through one edge and the opposite plane of an eighth of a sphere
Modelica_Magnetic.FluxTube.Leakage.EighthOfHollowSphere EighthOfHollowSphere Leakage flux through one edge and the opposite plane of an eighth of a hollow sphere
Modelica_Magnetic.FluxTube.Leakage.CoaxCylindersEndFaces CoaxCylindersEndFaces Leakage flux between the end planes of a inner solid cylinder and a coaxial outer hollow cylinder
Modelica_Magnetic.FluxTube.Leakage.LeakageWithCoefficient LeakageWithCoefficient Leakage reluctance with respect to the reluctance of a useful flux path (not for dynamic simulation of actuators)


Modelica_Magnetic.FluxTube.Leakage.PartialLeakage Modelica_Magnetic.FluxTube.Leakage.PartialLeakage

Base class for leakage flux tubes with position-independent permeance and hence no force generation; my_r=1

Modelica_Magnetic.FluxTube.Leakage.PartialLeakage

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.QuarterCylinder Modelica_Magnetic.FluxTube.Leakage.QuarterCylinder

Leakage flux from one edge to the opposite plane through a quarter cylinder

Modelica_Magnetic.FluxTube.Leakage.QuarterCylinder

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.1Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.QuarterHollowCylinder Modelica_Magnetic.FluxTube.Leakage.QuarterHollowCylinder

Leakage flux in circumferential direction through a quarter hollow cylinder

Modelica_Magnetic.FluxTube.Leakage.QuarterHollowCylinder

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.1Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Realratio1Constant ratio b/r_i

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.HalfCylinder Modelica_Magnetic.FluxTube.Leakage.HalfCylinder

Leakage flux through the edges of a half cylinder

Modelica_Magnetic.FluxTube.Leakage.HalfCylinder

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.1Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.HalfHollowCylinder Modelica_Magnetic.FluxTube.Leakage.HalfHollowCylinder

Leakage flux in circumferential direction through a half hollow cylinder

Modelica_Magnetic.FluxTube.Leakage.HalfHollowCylinder

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.1Depth orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Realratio1Constant ratio b/r_i

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.QuarterSphere Modelica_Magnetic.FluxTube.Leakage.QuarterSphere

Leakage flux through the corners of a quarter sphere

Modelica_Magnetic.FluxTube.Leakage.QuarterSphere

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Diameterd0.01Diameter of quarter sphere [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.QuarterHollowSphere Modelica_Magnetic.FluxTube.Leakage.QuarterHollowSphere

Leakage flux through the edges of a qarter hollow sphere

Modelica_Magnetic.FluxTube.Leakage.QuarterHollowSphere

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.01Thickness of sperical shell [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.EighthOfSphere Modelica_Magnetic.FluxTube.Leakage.EighthOfSphere

Leakage flux through one edge and the opposite plane of an eighth of a sphere

Modelica_Magnetic.FluxTube.Leakage.EighthOfSphere

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Radiusr0.01Radius of eighth of sphere [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.EighthOfHollowSphere Modelica_Magnetic.FluxTube.Leakage.EighthOfHollowSphere

Leakage flux through one edge and the opposite plane of an eighth of a hollow sphere

Modelica_Magnetic.FluxTube.Leakage.EighthOfHollowSphere

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Thicknesst0.01Thickness of sperical shell [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.CoaxCylindersEndFaces Modelica_Magnetic.FluxTube.Leakage.CoaxCylindersEndFaces

Leakage flux between the end planes of a inner solid cylinder and a coaxial outer hollow cylinder

Modelica_Magnetic.FluxTube.Leakage.CoaxCylindersEndFaces

Information


Please refer to the description of the enclosing sub-package Leakage for a description of all elements of this package.


Parameters

TypeNameDefaultDescription
Radiusr_010e-3Radius of inner solid cylinder [m]
Radiusr_117e-3Inner radius of outer hollow cylinder [m]
Radiusr_220e-3Outer radius of outer hollow cylinder [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

Modelica_Magnetic.FluxTube.Leakage.LeakageWithCoefficient Modelica_Magnetic.FluxTube.Leakage.LeakageWithCoefficient

Leakage reluctance with respect to the reluctance of a useful flux path (not for dynamic simulation of actuators)

Modelica_Magnetic.FluxTube.Leakage.LeakageWithCoefficient

Information


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.


Parameters

TypeNameDefaultDescription
LeakageCoefficientc_leak0.3Ratio leakage flux/(leakage flux + useful flux) = leakage flux/total flux [1]
Reference reluctance
ReluctanceR_mUsefulTotModelica.Constants.infTotal reluctance of useful flux path as reference [H-1]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port

Modelica definition

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;

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