Modelica_Magnetic.FluxTube.Force

Flux tubes with reluctance force generation; constant permeability

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.

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:

    dGm   dGm   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 PartialForce, if needed. Determination of the analytic derivative dGmBydl could become more complex for those flux tubes.

Package Content

NameDescription
Modelica_Magnetic.FluxTube.Force.PartialForce PartialForce Base class for flux tubes with reluctance force generation; constant permeability
Modelica_Magnetic.FluxTube.Force.HollowCylinderAxialFlux HollowCylinderAxialFlux (Hollow) cylinder with axial flux; constant permeability
Modelica_Magnetic.FluxTube.Force.HollowCylinderRadialFlux HollowCylinderRadialFlux Hollow cylinder with radial flux; constant permeability
Modelica_Magnetic.FluxTube.Force.CuboidParallelFlux CuboidParallelFlux Cuboid with flux in direction of motion, e.g. air gap with rectangular cross-section; constant permeability
Modelica_Magnetic.FluxTube.Force.CuboidOrthogonalFlux CuboidOrthogonalFlux Cuboid with flux orthogonal to direction of motion; constant permeability
Modelica_Magnetic.FluxTube.Force.LeakageAroundPoles LeakageAroundPoles Leakage flux tube around cylindrical or prismatic poles


Modelica_Magnetic.FluxTube.Force.PartialForce Modelica_Magnetic.FluxTube.Force.PartialForce

Base class for flux tubes with reluctance force generation; constant permeability

Modelica_Magnetic.FluxTube.Force.PartialForce

Information


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


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

partial model PartialForce 
  "Base class for flux tubes with reluctance force generation; constant permeability" 
  
  extends Modelica_Magnetic.Interfaces.TwoPortComponent;
  
  parameter SI.RelativePermeability my_r = 1 "Relative magnetic permeability";
  
  SI.Force F_m "Reluctance force";
  
  SI.Reluctance R_m "Magnetic reluctance";
  SI.Permeance G_m "Magnetic permeance";
  Real dGmBydx "Derivative of permeance with respect to armature position";
  parameter Integer dlBydx = 1 
    "Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1";
  
  
Modelica.Mechanics.Translational.Interfaces.Flange_b flange 
    "Generated reluctance force at armature position";
equation 
  
  V_mag = Phi * R_m;
  flange.f = -F_m;
  
  R_m = 1/G_m;
  
  F_m = 0.5 * V_mag^2 * dGmBydx;
  
end PartialForce;

Modelica_Magnetic.FluxTube.Force.HollowCylinderAxialFlux Modelica_Magnetic.FluxTube.Force.HollowCylinderAxialFlux

(Hollow) cylinder with axial flux; constant permeability

Modelica_Magnetic.FluxTube.Force.HollowCylinderAxialFlux

Information


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


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radiusr_i0Inner radius of (hollow) cylinder [m]
Radiusr_o0.01Outer radius of (hollow) cylinder [m]
Variable geometry
Lengthlflange.sAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

model HollowCylinderAxialFlux 
  "(Hollow) cylinder with axial flux; constant permeability" 
  
  extends Modelica_Magnetic.FluxTube.Force.PartialForce;
  
  SI.Length l = flange.s "Axial length (in direction of flux)";
  parameter SI.Radius r_i = 0 "Inner radius of (hollow) cylinder";
  parameter SI.Radius r_o = 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 = my_0*my_r * A /l;
  
  dGmBydx = -1 * my_0*my_r * A /l^2 * dlBydx;
  
  B = Phi/A;
  
end HollowCylinderAxialFlux;

Modelica_Magnetic.FluxTube.Force.HollowCylinderRadialFlux Modelica_Magnetic.FluxTube.Force.HollowCylinderRadialFlux

Hollow cylinder with radial flux; constant permeability

Modelica_Magnetic.FluxTube.Force.HollowCylinderRadialFlux

Information


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


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Radiusr_i0.01Inner radius of hollow cylinder [m]
Radiusr_o0.015Outer radius of hollow cylinder [m]
Variable geometry
Lengthlflange.sAxial length (orthogonal to direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

model HollowCylinderRadialFlux 
  "Hollow cylinder with radial flux; constant permeability" 
  
  extends Modelica_Magnetic.FluxTube.Force.PartialForce;
  
  SI.Length l = flange.s "Axial length (orthogonal to direction of flux)";
  parameter SI.Radius r_i = 0.01 "Inner radius of hollow cylinder";
  parameter SI.Radius r_o = 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 = my_0*my_r * 2 * pi * l /Modelica.Math.log(r_o/r_i);
  
  dGmBydx = my_0*my_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.FluxTube.Force.CuboidParallelFlux Modelica_Magnetic.FluxTube.Force.CuboidParallelFlux

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

Modelica_Magnetic.FluxTube.Force.CuboidParallelFlux

Information


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


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Breadtha0.01Breadth of rectangular cross-section [m]
Heightb0.01Height of rectangular cross-section [m]
Variable geometry
Lengthlflange.sAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

model CuboidParallelFlux 
  "Cuboid with flux in direction of motion, e.g. air gap with rectangular cross-section; constant permeability" 
  
  extends Modelica_Magnetic.FluxTube.Force.PartialForce;
  
  SI.Length l = flange.s "Axial length (in direction of flux)";
  parameter SI.Breadth a = 0.01 "Breadth of rectangular cross-section";
  parameter SI.Height b = 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 = my_0*my_r * A /l;
  
  dGmBydx = -1 * my_0*my_r * A /l^2 * dlBydx;
  
  B = Phi/A;
  
end CuboidParallelFlux;

Modelica_Magnetic.FluxTube.Force.CuboidOrthogonalFlux Modelica_Magnetic.FluxTube.Force.CuboidOrthogonalFlux

Cuboid with flux orthogonal to direction of motion; constant permeability

Modelica_Magnetic.FluxTube.Force.CuboidOrthogonalFlux

Information


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


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Breadtha0.01Breadth of rectangular cross-section [m]
Heightb0.01Height of rectangular cross-section [m]
Variable geometry
Lengthlflange.sAxial length (in direction of motion, orthogonal to flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

model CuboidOrthogonalFlux 
  "Cuboid with flux orthogonal to direction of motion; constant permeability" 
  
  extends Modelica_Magnetic.FluxTube.Force.PartialForce;
  
  SI.Length l = flange.s 
    "Axial length (in direction of motion, orthogonal to flux)";
  parameter SI.Breadth a = 0.01 "Breadth of rectangular cross-section";
  parameter SI.Height b = 0.01 "Height of rectangular cross-section";
  
  
  SI.MagneticFluxDensity B "Homogeneous flux density";
  
protected 
  SI.Area A "Cross-sectional area orthogonal to direction of flux";
  
equation 
  A = a*l;
  G_m = my_0*my_r * A /b;
  
  dGmBydx = my_0*my_r * a /b * dlBydx;
  
  B = Phi/A;
  
end CuboidOrthogonalFlux;

Modelica_Magnetic.FluxTube.Force.LeakageAroundPoles Modelica_Magnetic.FluxTube.Force.LeakageAroundPoles

Leakage flux tube around cylindrical or prismatic poles

Modelica_Magnetic.FluxTube.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_leak, 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.


Parameters

TypeNameDefaultDescription
RelativePermeabilitymy_r1Relative magnetic permeability [1]
IntegerdlBydx1Derivative of flux tube's varying dimension with respect to armature position; set to +1 or -1
Lengtht0.1Depth orthogonal to flux; mean circumference of flux tube in case of cylindrical poles [m]
Radiusr_leak0.01Radius of leakage field [m]
Variable geometry
Lengthlflange.sAxial length (in direction of flux) [m]

Connectors

TypeNameDescription
PositiveMagneticPortpPositive magnetic port
NegativeMagneticPortnNegative magnetic port
Flange_bflangeGenerated reluctance force at armature position

Modelica definition

model LeakageAroundPoles 
  "Leakage flux tube around cylindrical or prismatic poles" 
  
  extends Modelica_Magnetic.FluxTube.Force.PartialForce;
  SI.Length l = flange.s "Axial length (in direction of flux)";
  
  
  parameter SI.Length t = 0.1 
    "Depth orthogonal to flux; mean circumference of flux tube in case of cylindrical poles";
  parameter SI.Radius r_leak = 0.01 "Radius of leakage field";
  
equation 
  G_m = 2 * my_0 * t /pi * Modelica.Math.log(1 + pi/2 * r_leak/l);
  
//derivative at full length:
//  dGmBydx = 2 * my_0 * t /pi * 1/(1 + pi/2 * r_leak/l) * (-1)*pi/2*r_leak/l^2  * dlBydx;
//simplified: 
  dGmBydx = - my_0 * t * r_leak * dlBydx / (l^2 *(1 + pi/2 * r_leak/l));
  
end LeakageAroundPoles;

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