Modelica.Magnetic.FluxTubes.Shapes.Force

Information

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

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

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

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

Type	Name	Default	Description
Boolean	useSupport	false	= true, if support flange enabled, otherwise implicitly grounded
RelativePermeability	mu_r	1	Relative magnetic permeability [1]
Integer	dlBydx	1	Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radius	r_i		Inner radius of (hollow) cylinder [m]
Radius	r_o		Outer radius of (hollow) cylinder [m]
Variable geometry
Length	l	s	Axial length (in direction of flux) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative magnetic port
Flange_b	flange	Generated reluctance force at armature position
Support	support	Support/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

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

Type	Name	Default	Description
Boolean	useSupport	false	= true, if support flange enabled, otherwise implicitly grounded
RelativePermeability	mu_r	1	Relative magnetic permeability [1]
Integer	dlBydx	1	Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radius	r_i		Inner radius of hollow cylinder [m]
Radius	r_o		Outer radius of hollow cylinder [m]
Variable geometry
Length	l	s	Axial length (orthogonal to direction of flux) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative magnetic port
Flange_b	flange	Generated reluctance force at armature position
Support	support	Support/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

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

Type	Name	Default	Description
Boolean	useSupport	false	= true, if support flange enabled, otherwise implicitly grounded
RelativePermeability	mu_r	1	Relative magnetic permeability [1]
Integer	dlBydx	1	Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Length	a		Width of rectangular cross-section [m]
Length	b		Height of rectangular cross-section [m]
Variable geometry
Length	l	s	Axial length (in direction of flux) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative magnetic port
Flange_b	flange	Generated reluctance force at armature position
Support	support	Support/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

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

Type	Name	Default	Description
Boolean	useSupport	false	= true, if support flange enabled, otherwise implicitly grounded
RelativePermeability	mu_r	1	Relative magnetic permeability [1]
Integer	dlBydx	1	Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Length	a		Width of rectangular cross-section [m]
Length	b		Height of rectangular cross-section (in flux direction) [m]
Variable geometry
Length	l	s	Length in direction of motion (orthogonal to flux) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative magnetic port
Flange_b	flange	Generated reluctance force at armature position
Support	support	Support/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

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

Type	Name	Default	Description
Boolean	useSupport	false	= true, if support flange enabled, otherwise implicitly grounded
RelativePermeability	mu_r	1	Relative magnetic permeability [1]
Integer	dlBydx	1	Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Length	w		Width orthogonal to flux; mean circumference of flux tube in case of cylindrical poles [m]
Radius	r		Radius of leakage field [m]
Variable geometry
Length	l	s	Axial length (in direction of flux) [m]

Connectors

Type	Name	Description
PositiveMagneticPort	port_p	Positive magnetic port
NegativeMagneticPort	port_n	Negative magnetic port
Flange_b	flange	Generated reluctance force at armature position
Support	support	Support/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;