Modelica.Magnetic.FluxTubes.Shapes.Leakage

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

Information


Please have a look at UsersGuide.ReluctanceForceCalculation for an explanation of the different flux tube categories and resulting sub-packages.

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 mu_r=1. Basic.LeakageWithCoefficient accounts for leakage not by the geometry of flux tubes, but by a coupling coefficient c_usefulFlux.

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 Interfaces.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.FluxTubes.Shapes.Leakage.QuarterCylinder QuarterCylinder Leakage flux from one edge to the opposite plane through a quarter cylinder
Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowCylinder QuarterHollowCylinder Leakage flux in circumferential direction through a quarter hollow cylinder
Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfCylinder HalfCylinder Leakage flux through the edges of a half cylinder
Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfHollowCylinder HalfHollowCylinder Leakage flux in circumferential direction through a half hollow cylinder
Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterSphere QuarterSphere Leakage flux through the corners of a quarter sphere
Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowSphere QuarterHollowSphere Leakage flux through the edges of a qarter hollow sphere
Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfSphere EighthOfSphere Leakage flux through one edge and the opposite plane of an eighth of a sphere
Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfHollowSphere EighthOfHollowSphere Leakage flux through one edge and the opposite plane of an eighth of a hollow sphere
Modelica.Magnetic.FluxTubes.Shapes.Leakage.CoaxCylindersEndFaces CoaxCylindersEndFaces Leakage flux between the end planes of a inner solid cylinder and a coaxial outer hollow cylinder


Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterCylinder Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterCylinder

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

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterCylinder

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengthl Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

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

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length l( start = 0.1) 
    "Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane)";

equation 
  G_m = mu_0 * 0.52 * l;

end QuarterCylinder;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowCylinder Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowCylinder

Leakage flux in circumferential direction through a quarter hollow cylinder

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowCylinder

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengthl Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Realratio Constant ratio t/r_i

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

model QuarterHollowCylinder 
  "Leakage flux in circumferential direction through a quarter hollow cylinder"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length l( start = 0.1) 
    "Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i)";
  parameter Real ratio( start = 1) "Constant ratio t/r_i";

equation 
  G_m = 2* mu_0 * l * Modelica.Math.log(1 + ratio) /pi;

end QuarterHollowCylinder;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfCylinder Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfCylinder

Leakage flux through the edges of a half cylinder

Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfCylinder

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengthl Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

model HalfCylinder 
  "Leakage flux through the edges of a half cylinder"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length l( start = 0.1) 
    "Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges)";

equation 
  G_m = mu_0 * 0.26 * l;

end HalfCylinder;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfHollowCylinder Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfHollowCylinder

Leakage flux in circumferential direction through a half hollow cylinder

Modelica.Magnetic.FluxTubes.Shapes.Leakage.HalfHollowCylinder

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengthl Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Realratio Constant ratio t/r_i

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

model HalfHollowCylinder 
  "Leakage flux in circumferential direction through a half hollow cylinder"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length l( start = 0.1) 
    "Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i)";
  parameter Real ratio( start = 1) "Constant ratio t/r_i";

equation 
  G_m = mu_0 * l * Modelica.Math.log(1 + ratio) /pi;

end HalfHollowCylinder;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterSphere Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterSphere

Leakage flux through the corners of a quarter sphere

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterSphere

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Radiusr Radius of quarter sphere [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

model QuarterSphere 
  "Leakage flux through the corners of a quarter sphere"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Radius r( start = 0.005) "Radius of quarter sphere";

equation 
  G_m = mu_0 * 0.077 * 2*r;

end QuarterSphere;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowSphere Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowSphere

Leakage flux through the edges of a qarter hollow sphere

Modelica.Magnetic.FluxTubes.Shapes.Leakage.QuarterHollowSphere

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengtht Thickness of sperical shell [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

model QuarterHollowSphere 
  "Leakage flux through the edges of a qarter hollow sphere"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length t( start = 0.01) "Thickness of sperical shell";

equation 
  G_m = mu_0 * 0.25 * t;

end QuarterHollowSphere;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfSphere Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfSphere

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

Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfSphere

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Radiusr Radius of eighth of sphere [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

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

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Radius r( start = 0.01) "Radius of eighth of sphere";

equation 
  G_m = mu_0 * 0.308 * r;

end EighthOfSphere;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfHollowSphere Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfHollowSphere

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

Modelica.Magnetic.FluxTubes.Shapes.Leakage.EighthOfHollowSphere

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Lengtht Thickness of sperical shell [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

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

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Length t( start = 0.01) "Thickness of sperical shell";

equation 
  G_m = mu_0 * 0.5 * t;

end EighthOfHollowSphere;

Modelica.Magnetic.FluxTubes.Shapes.Leakage.CoaxCylindersEndFaces Modelica.Magnetic.FluxTubes.Shapes.Leakage.CoaxCylindersEndFaces

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

Modelica.Magnetic.FluxTubes.Shapes.Leakage.CoaxCylindersEndFaces

Information


Please refer to the enclosing sub-package Leakage for a description of all elements of this package and to [Ro41] for derivation and/or coefficients of the equation for permeance G_m.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage (Base class for leakage flux tubes with position-independent permeance and hence no force generation; mu_r=1).

Parameters

TypeNameDefaultDescription
Radiusr_0 Radius of inner solid cylinder [m]
Radiusr_1 Inner radius of outer hollow cylinder [m]
Radiusr_2 Outer radius of outer hollow cylinder [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port

Modelica definition

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

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialLeakage;

  parameter SI.Radius r_0( start = 10e-3) "Radius of inner solid cylinder";
  parameter SI.Radius r_1( start = 17e-3) 
    "Inner radius of outer hollow cylinder";
  parameter SI.Radius r_2( start = 20e-3) 
    "Outer radius of outer hollow cylinder";

  final parameter SI.Distance l_g = r_1 - r_0 
    "Radial gap length between both cylinders";
  final parameter SI.Length t = r_2 - r_1 
    "Radial thickness of outer hollow cylinder";

equation 
  // [Ro41], p. 139, Eq. (22)
  G_m = if t <= r_0 then 
        2 * mu_0 * (r_0 + l_g/2) * Modelica.Math.log(1 + 2*t/l_g) else 
        2 * mu_0 * (r_0 + l_g/2) * Modelica.Math.log(1 + 2*r_0/l_g);

end CoaxCylindersEndFaces;

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