Modelica.Magnetic.FluxTubes.Shapes.Leakage

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.

Extends from Modelica.Icons.VariantsPackage (Icon for package containing variants).

Package Content

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

Type	Name	Default	Description
Length	l		Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edge and plane) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Length	l		Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Real	ratio		Constant ratio t/r_i

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Length	l		Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>distance between edges) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Length	l		Axial length orthogonal to flux (=2*pi*r for cylindrical pole and r>>r_i) [m]
Real	ratio		Constant ratio t/r_i

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Radius	r		Radius of quarter sphere [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Length	t		Thickness of sperical shell [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Radius	r		Radius of eighth of sphere [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Length	t		Thickness of sperical shell [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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

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

Type	Name	Default	Description
Radius	r_0		Radius of inner solid cylinder [m]
Radius	r_1		Inner radius of outer hollow cylinder [m]
Radius	r_2		Outer radius of outer hollow cylinder [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative 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;