Modelica.Magnetic.FluxTubes.Shapes.Force

Flux tubes with reluctance force generation; constant permeability

Information


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

Flux tube elements with generation of a reluctance force are intended for modelling of position-dependent air gap sections and permanent magnet sections respectively of translatory actuators. By default, the position co-ordinate of the mechanical connector flange.s is identical with the dimension l of the package's flux tube elements. l is the dimension changes with armature motion. If needed, the identity l=flange.s can be replaced by an actuator-specific equation, for example, when a flux tube length increases with decreasing armature position. The position co-ordinate of an element's translatory connector flange.s in turn will be identical with the armature position x in most cases, as the examples illustrate.

The derivative of each element's permeance with respect to armature position dGmBydx is calculated from the derivative of the flux tube's permeance with respect to its varying dimension dGmBydl and the derivative of this dimension with respect to armature position dlBydx:

    dG_m   dG_m   dl
    ---- = ---- * --
     dx     dl    dx

The parameter dlBydx must be set in each flux tube element to +1 or -1 according to the definition of the armature co-ordinate and the position of the element in a device's magnetic circuit. Proper match between armature motion and resulting variation of the flux tube length assures that the element's reluctance force acts in the right direction.

The shapes of the flux tubes defined in this package are rather simple. Only one dimenion varies with armature motion. Flux tubes with more complex variations of dimensions with armature motion can be defined by extending the base class Interfaces.PartialForce, if needed. Determination of the analytic derivative dGmBydl could become more complex for those flux tubes.

Package Content

NameDescription
Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderAxialFlux HollowCylinderAxialFlux (Hollow) cylinder with axial flux; constant permeability
Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderRadialFlux HollowCylinderRadialFlux Hollow cylinder with radial flux; constant permeability
Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidParallelFlux CuboidParallelFlux Cuboid with flux in direction of motion, e.g. air gap with rectangular cross-section; constant permeability
Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidOrthogonalFlux CuboidOrthogonalFlux Cuboid with flux orthogonal to direction of motion; constant permeability
Modelica.Magnetic.FluxTubes.Shapes.Force.LeakageAroundPoles LeakageAroundPoles Leakage flux tube around cylindrical or prismatic poles


Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderAxialFlux Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderAxialFlux

(Hollow) cylinder with axial flux; constant permeability

Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderAxialFlux

Information


Please refer to the enclosing sub-package Force 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.PartialForce (Base class for flux tubes with reluctance force generation; constant permeability).

Parameters

TypeNameDefaultDescription
BooleanuseSupportfalse= true, if support flange enabled, otherwise implicitly grounded
RelativePermeabilitymu_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radiusr_i Inner radius of (hollow) cylinder [m]
Radiusr_o Outer radius of (hollow) cylinder [m]
Variable geometry
LengthlsAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position
SupportsupportSupport/housing of component

Modelica definition

model HollowCylinderAxialFlux 
  "(Hollow) cylinder with axial flux; constant permeability"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialForce;

  SI.Length l = s "Axial length (in direction of flux)";
  parameter SI.Radius r_i( start = 0) "Inner radius of (hollow) cylinder";
  parameter SI.Radius r_o( start = 0.01) "Outer radius of (hollow) cylinder";

  SI.MagneticFluxDensity B "Homogeneous flux density";

protected 
  parameter SI.Area A = pi*(r_o^2 - r_i^2) 
    "Cross-sectional area orthogonal to direction of flux";

equation 
  G_m = mu_0*mu_r * A /l;

  dGmBydx = -1 * mu_0*mu_r * A /l^2 * dlBydx;

  B = Phi/A;

end HollowCylinderAxialFlux;

Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderRadialFlux Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderRadialFlux

Hollow cylinder with radial flux; constant permeability

Modelica.Magnetic.FluxTubes.Shapes.Force.HollowCylinderRadialFlux

Information


Please refer to the enclosing sub-package Force 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.PartialForce (Base class for flux tubes with reluctance force generation; constant permeability).

Parameters

TypeNameDefaultDescription
BooleanuseSupportfalse= true, if support flange enabled, otherwise implicitly grounded
RelativePermeabilitymu_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radiusr_i Inner radius of hollow cylinder [m]
Radiusr_o Outer radius of hollow cylinder [m]
Variable geometry
LengthlsAxial length (orthogonal to direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position
SupportsupportSupport/housing of component

Modelica definition

model HollowCylinderRadialFlux 
  "Hollow cylinder with radial flux; constant permeability"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialForce;

  SI.Length l = s "Axial length (orthogonal to direction of flux)";
  parameter SI.Radius r_i( start = 0.01) "Inner radius of hollow cylinder";
  parameter SI.Radius r_o( start = 0.015) "Outer radius of hollow cylinder";

  SI.MagneticFluxDensity B_avg 
    "Average flux density (at arithmetic mean radius)";

protected 
  SI.Area A_avg 
    "Average cross-sectional area orthogonal to direction of flux (at arithmetic mean radius)";

equation 
  G_m = mu_0*mu_r * 2 * pi * l /Modelica.Math.log(r_o/r_i);

  dGmBydx = mu_0*mu_r * 2 * pi/Modelica.Math.log(r_o/r_i) * dlBydx;

  A_avg = pi*(r_i + r_o) * l;
  B_avg = Phi/A_avg;

end HollowCylinderRadialFlux;

Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidParallelFlux Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidParallelFlux

Cuboid with flux in direction of motion, e.g. air gap with rectangular cross-section; constant permeability

Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidParallelFlux

Information


Please refer to the enclosing sub-package Force 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.PartialForce (Base class for flux tubes with reluctance force generation; constant permeability).

Parameters

TypeNameDefaultDescription
BooleanuseSupportfalse= true, if support flange enabled, otherwise implicitly grounded
RelativePermeabilitymu_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Lengtha Width of rectangular cross-section [m]
Lengthb Height of rectangular cross-section [m]
Variable geometry
LengthlsAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position
SupportsupportSupport/housing of component

Modelica definition

model CuboidParallelFlux 
  "Cuboid with flux in direction of motion, e.g. air gap with rectangular cross-section; constant permeability"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialForce;

  SI.Length l = s "Axial length (in direction of flux)";
  parameter SI.Length a( start = 0.01) "Width of rectangular cross-section";
  parameter SI.Length b( start = 0.01) "Height of rectangular cross-section";

  SI.MagneticFluxDensity B "Homogeneous flux density";

protected 
  parameter SI.Area A = a*b 
    "Cross-sectional area orthogonal to direction of flux";

equation 
  G_m = mu_0*mu_r * A /l;

  dGmBydx = -1 * mu_0*mu_r * A /l^2 * dlBydx;

  B = Phi/A;

end CuboidParallelFlux;

Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidOrthogonalFlux Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidOrthogonalFlux

Cuboid with flux orthogonal to direction of motion; constant permeability

Modelica.Magnetic.FluxTubes.Shapes.Force.CuboidOrthogonalFlux

Information


Please refer to the enclosing sub-package Force 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.PartialForce (Base class for flux tubes with reluctance force generation; constant permeability).

Parameters

TypeNameDefaultDescription
BooleanuseSupportfalse= true, if support flange enabled, otherwise implicitly grounded
RelativePermeabilitymu_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Lengtha Width of rectangular cross-section [m]
Lengthb Height of rectangular cross-section (in flux direction) [m]
Variable geometry
LengthlsLength in direction of motion (orthogonal to flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position
SupportsupportSupport/housing of component

Modelica definition

model CuboidOrthogonalFlux 
  "Cuboid with flux orthogonal to direction of motion; constant permeability"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialForce;

  SI.Length l = s "Length in direction of motion (orthogonal to flux)";
  parameter SI.Length a( start = 0.01) "Width of rectangular cross-section";
  parameter SI.Length b( start = 0.01) 
    "Height of rectangular cross-section (in flux direction)";

  SI.MagneticFluxDensity B "Homogeneous flux density";

protected 
  SI.Area A "Cross-sectional area orthogonal to direction of flux";

equation 
  A = a*l;
  G_m = mu_0*mu_r * A /b;

  dGmBydx = mu_0*mu_r * a /b * dlBydx;

  B = Phi/A;

end CuboidOrthogonalFlux;

Modelica.Magnetic.FluxTubes.Shapes.Force.LeakageAroundPoles Modelica.Magnetic.FluxTubes.Shapes.Force.LeakageAroundPoles

Leakage flux tube around cylindrical or prismatic poles

Modelica.Magnetic.FluxTubes.Shapes.Force.LeakageAroundPoles

Information


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

Leakage flux around a prismatic or cylindric air gap between to poles can be described with this model. Due to its constant radius of the leakage field r, the model is rather simple. Whereas in reality the leakage radius is approximately constant for air gap lengths l greater than this radius, it decreases with air gap lengths less than the leakage radius. This decrease for small air gaps is neglected here, since the influence of the leakage flux tube compared to that of the enclosed main air gap (connected in parallel) decreases for decreasing air gap length l.

Note that in [Ka08] the equation for G_m is accidentally swapped with that of a similar element.

Extends from Modelica.Magnetic.FluxTubes.Interfaces.PartialForce (Base class for flux tubes with reluctance force generation; constant permeability).

Parameters

TypeNameDefaultDescription
BooleanuseSupportfalse= true, if support flange enabled, otherwise implicitly grounded
RelativePermeabilitymu_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Lengthw Width orthogonal to flux; mean circumference of flux tube in case of cylindrical poles [m]
Radiusr Radius of leakage field [m]
Variable geometry
LengthlsAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortport_pPositive magnetic port
NegativeMagneticPortport_nNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position
SupportsupportSupport/housing of component

Modelica definition

model LeakageAroundPoles 
  "Leakage flux tube around cylindrical or prismatic poles"

  extends Modelica.Magnetic.FluxTubes.Interfaces.PartialForce;
  SI.Length l = s "Axial length (in direction of flux)";

  parameter SI.Length w( start = 0.1) 
    "Width orthogonal to flux; mean circumference of flux tube in case of cylindrical poles";
  parameter SI.Radius r( start = 0.01) "Radius of leakage field";

equation 
  //adapted from [Ka08], but corrected
  //(in [Ka08] equation accidentally swapped with that of a similar element)
  G_m = mu_0 * w /pi * Modelica.Math.log(1 + pi * r/l);

  //derivative at full length:
  //  dGmBydx = mu_0 * w /pi * 1/(1 + pi * r/l) * (-1)*pi*r/l^2  * dlBydx;
  //simplified:
  dGmBydx = - mu_0 * w * r * dlBydx / (l^2 *(1 + pi * r/l));

end LeakageAroundPoles;

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