Hollow Cylinder Radial Flux Reluctance Force

Description

The Hollow Cylinder Radial Flux Reluctance Force component models the generation of a reluctance force from a flux tube consisting of a hollow cylinder with radial flux flow.  The equations are

 ${V}_{m}=\mathrm{Φ}\cdot {R}_{m}$ ${\mathrm{Φ}}_{p}+{\mathrm{Φ}}_{n}=0$ ${R}_{m}=\frac{1}{{G}_{m}}$ ${G}_{m}=\frac{{\mathrm{μ}}_{0}\cdot {\mathrm{μ}}_{r}\cdot 2\cdot \mathrm{π}\cdot l}{\mathrm{ln}\left(\frac{{r}_{o}}{{r}_{i}}\right)}$ $f=-{F}_{m}$ ${F}_{m}=\frac{1}{2}\cdot {V}_{m}^{2}\cdot \frac{ⅆ{G}_{m}}{ⅆx}$ $\frac{ⅆ{G}_{m}}{ⅆx}=-\frac{{\mathrm{μ}}_{r}\cdot {\mathrm{μ}}_{0}\cdot 2\cdot \mathrm{π}}{\mathrm{ln}\left(\frac{{r}_{o}}{{r}_{i}}\right)}\cdot \frac{ⅆl}{\mathrm{dx}}$ ${B}_{\mathrm{avg}}=\frac{\mathrm{Φ}}{{A}_{\mathrm{avg}}}$ $s=l$ $s={\mathrm{flange}}_{s}-{\mathrm{support}}_{s}$

Connections

 Name Description Color ${\mathrm{port}}_{p}$ Positive magnetic port Solid orange ${\mathrm{port}}_{n}$ Negative magnetic port Open orange $\mathrm{flange}$ Armature connection Open green $\mathrm{support}$ Solid green

Variables

 Symbol Units Description Modelica ID ${V}_{m}$ $A$ Magnetic potential across component V_m ${V}_{{m}_{x}}$ $A$ Magnetic potential at port_x.V_m ${\mathrm{Φ}}_{x}$ $\mathrm{Wb}$ Magnetic flux into ${\mathrm{port}}_{x}$, port_x.Phi $\mathrm{Φ}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{m}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{m}$ $H$ Magnetic permeance Gm ${B}_{\mathrm{avg}}$ $\mathrm{Wb}$ Average magnetic flux density, at arithmetic mean radius B_avg ${A}_{\mathrm{avg}}$ ${m}^{2}$ Average cross-sectional area, orthogonal to flux flow, at arithmetic mean radius A_avg $f$ $N$ force on $\mathrm{flange}$ flange.f ${F}_{m}$ $N$ Reluctance force F_m $l$ $m$ Axial length, in direction of flux l $s$ $m$ Position of $\mathrm{flange}$ s

Parameters

 Symbol Default Units Description Modelica ID ${\mathrm{μ}}_{r}$ 1 - Relative magnetic permeability mu_r ${r}_{o}$ 0.01 $m$ Outer radius of hollow cylinder cross-section r_o ${r}_{i}$ 0 $m$ Inner radius of hollow cylinder cross-section r_i $\frac{\mathrm{dl}}{\mathrm{dx}}$ $+1$ integer Derivative of flux tube's varying dimension with respect to the armature position. Set to +1 or -1. dlBydx $\mathrm{useSupport}$ $\mathrm{false}$ boolean Enables conditional support flange support

Initial Conditions

 Symbol Units Description Modelica ID ${V}_{\mathrm{m0}}$ $A$ Magnetic potential across component V_m ${\mathrm{Φ}}_{0}$ $\mathrm{Wb}$ Magnetic flux through component Phi ${R}_{\mathrm{m0}}$ ${H}^{-1}$ Magnetic reluctance Rm ${G}_{\mathrm{m0}}$ $H$ Magnetic permeance Gm $\genfrac{}{}{0}{}{\frac{{\mathrm{dG}}_{m}}{\mathrm{dx}}}{\phantom{y}}|\genfrac{}{}{0}{}{\phantom{x}}{0}$ $\frac{H}{m}$ Initial change of permeance with length dGmBydx ${B}_{\mathrm{avg0}}$ $\mathrm{Wb}$ Initial average magnetic flux density B_avg ${F}_{\mathrm{m0}}$ N Initial reluctance force F_m ${l}_{0}$ m Initial length of air gap l ${s}_{0}$ m Initial position of flange s