Table of Contents
Closed-form equations for force between cylindrical magnets
| Stan Zurek, Closed-form equations for force between cylindrical magnets, Encyclopedia Magnetica, https://e-magnetica.pl/doku.php/closed_form_equations_for_force_between_cylindrical_magnets |
Closed-form equations for force between cylindrical magnets - simplified analytical equations (not requiring integration) which can be used for calculations of approximate magnetic force acting between cylindrical permanent magnets.
Equations as a function of distance
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Complex 3D integrals can be used for such calculation and they can provide good agreement between the calculations and experimental results or finite-element modelling (FEM).1)2)3)4)5)6)7)
However, closed-form equations are considerably simpler to use and the forces can be calculated by assuming the magnets to be equivalent to point-like magnetic dipoles m or surfaces with hypothetical magnetic charges.8)9)10)
The value of magnetic moment m is related to magnetisation M of the magnet and its volume V. However, M is not a value which is typically available for commercial magnets, but instead the remanence flux density Br is typically provided instead, so the equations can be expressed either by m, M or Br.
| (1) Magnetic moment m is related to magnetisation M and remanence flux density Br | |
|---|---|
| $$ m = M·V = \frac{B_r}{μ_0}·π·R^2·L $$ | (A·m2) |
| where: $m$ - magnetic dipole moment (A·m2), $M$ - magnetisation (A/m), $V$ - magnet volume (m3), $B_r$ - remanence flux density (T), $μ_0$ - permeability of vacuum (H/m), $R$ - radius (m) of the cylindrical magnet, $L$ - length (m) of the cylindrical magnet | |
There are several analytical equations derived in the literature, as listed below. However, they all have limitations as far as their applicability is concerned. For example, equations (2) and (3) are applicable only if the distance between the magnets is much larger than their size.11)12)13)
| (2) Castañer, Medina, and Cuesta-Bolao 14) | |
|---|---|
| $$ F_{Castaner}(x) = \frac{3·μ_0·m^2}{2·π·x^4} = \frac{3·π·{B_r}^2·L^2·R^4}{2·μ_0·x^4} $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $μ_0$ - absolute permeability of vacuum (H/m), $m$ - magnetic dipole moment (A·m2), $B_r$ - remanence flux density (T), $L$ - length (m) of the cylindrical magnet, $R$ - radius (m) of the cylindrical magnet | |
| (3) Furlani 15) | |
|---|---|
| $$ F_{Furlani}(x) = \frac{π·B_r^2·R^4}{4·μ_0}· \left( \frac{1}{x^2} + \frac{1}{(2·L+x)^2} - \frac{2}{(L+x)^2} \right) $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $B_r$ - remanence flux density (T), $R$ - radius (m) of the cylindrical magnet, $μ_0$ - absolute permeability of vacuum (H/m), $L$ - length (m) of the cylindrical magnet | |
Equation (4) was published16) showing seemingly good agreement with the experimental data, but only because the value of $B_r$ was fitted to the data. Independent verification shows that this equation does not perform well, especially for magnets with a small aspect ratio (“flat”, like a coin).17)
| (4) Cheedket and Sirisathitkul 18) | |
|---|---|
| $$ F_{Cheedket}(x) = \frac{π·B_r^2·R^2}{2·μ_0}· \left( \frac{2·(L+x)}{ \sqrt{(L+x)^2+R^2} } - \frac{2·L+x}{\sqrt{(2·L+x)^2 + R^2} } - \frac{x}{\sqrt{x^2+R^2}} \right) $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $B_r$ - remanence flux density (T), $R$ - radius (m) of the cylindrical magnet, $μ_0$ - absolute permeability of vacuum (H/m), $L$ - length (m) of the cylindrical magnet | |
Equation (5) was published for cuboidal magnets (rather than cylindrical), but it was included here because if it was correct then it should also be applicable for large distances, but it is not because of the lack of 1/x4 characteristics for large distances.19)
| (5) Schomburg, Reinertz, Sackmann, and Schmitz 20) | |
|---|---|
| $$ F_{Schomburg}(x) = \frac{F_0 · {d_e}^2}{(x+{d_e})^2} $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $F_0$ - contact force (N) at x=0, $d_e$ - distance (m) at which $F(d_e) = F_0/4$ (the value of $F_0$ must be known from some other method) | |
Another equation was published21) as a modification of equation (2). This equation includes a correction of the separation distance for the aspect ratio of the magnet, which reduces to a constant “c” added to the distance between the magnets, as in equation (6).
| (6) Zurek,22) based on eq. (2) by Furlani 23) | |
|---|---|
| $$ F_{Zurek}(x) = \frac{π·B_r^2·R^4}{4·μ_0}·\left( \frac{1}{(x+c)^2} + \frac{1}{(2·L+x+c)^2} - \frac{2}{(L+x+c)^2} \right) $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $B_r$ - remanence flux density (T), $R$ - radius (m) of the cylindrical magnet, $μ_0$ - absolute permeability of vacuum (H/m), $L$ - length (m) of the cylindrical magnet, $c = 0.8·R$ | |
Equations for contact force
There are also examples of closed-form equations for a contact force $F_0$ (x=0, i.e. the two magnets touching).24)25)26)
| (7) Agashe and Arnold 27)28) | |
|---|---|
| $$ F_{Agashe}(x=0) = \frac{π·{B_r}^2·R^2·L}{64·μ_0·\sqrt{R^2+L^2}}·\left(32 + \frac{3·R^4}{(R^2+L^2)^2} + \frac{9·R^8+12·R^6·L^2}{(R^2+L^2)^4} \right) $$ | (N) |
| where: $x$ - distance or gap (m) between the magnets, $B_r$ - remanence flux density (T), $R$ - radius (m) of the cylindrical magnet, $L$ - length (m) of the cylindrical magnet, $μ_0$ - absolute permeability of vacuum (H/m) | |
The original equation by Vokoun et al. requires solving elliptical integrals of the first and second kind, so technically does not provide a closed-form solution, but such integrals can be approximated with closed-form equations with accuracy better than 0.5 %. However, even such small errors lead to much larger errors of the calculated force,29) as shown below.
| (8) Vokoun, Beleggia, Heller, and Šittner 30) | |||
|---|---|---|---|
| $$ F_{Vokoun}(x=0) = \frac{2·π·{B_r}^2·R·L}{μ_0}·\left(\frac{E(l_1)-K(l_1}{l_1} - \frac{E(l_2)-K(l_2)}{l_2} \right) $$ | (N) | ||
| $ l_1 = 1/\sqrt{1+(L/R)^2} $ and $l_2 = 1/\sqrt{1+0.25·(L/R)^2}$ | $K_a(l)=\frac{π}{2·(1-l)^{0.19}}-0.17·(l+0.015)^{0.8}$ | $E_a(l)=\frac{π}{2}-0.567·l^{2.4+(l+0.1)^{5.8}}$ | |
| where: $x$ - distance or gap (m) between the magnets, $B_r$ - remanence flux density (T), $R$ - radius (m) of the cylindrical magnet, $L$ - length (m) of the cylindrical magnet, $μ_0$ - absolute permeability of vacuum (H/m), $K(l)$ and $E(l)$ - elliptical integrals of the first and second kind respectively, $K_a(l)$ and $E(l)$ - approximations which can be used instead of elliptical integrals $K(l)$ and $E(l)$ respectively | |||
Other tools
There are other tools which can be used for quick estimation of the force between magnets, as listed below. They do not represent calculations by the means of closed-form equations, but provide at least order-of-magnitude information about the expected force values:
- K&J Magnetics online calculator - uses a proprietary algorithm based on experimental data.31)32) (This calculator shows good agreement with FEM calculations in COMSOL Multiphysics.33))
Additional data can be calculated by using finite-element modelling software such as FEMM.38)
Performance of equations
The comparison of the data from the equations listed above is shown in Fig. 2.
Forces calculated from $F_{Castaner}$ and $F_{Furlani}$ give correct values only for large distances so the 1/x4 characteristics is represented correctly, but diverge to infinity when approaching contact (x=0).
If the real value of $B_r$ is taken into account, $F_{Cheedket}$ does not behave correctly for smaller distances, and this incorrect behaviour exacerbates for lower aspect ratios (“flat” magnets). $F_{Schomburg}$ can be made to fit the data for small gaps, but fails to predict the correct force for large distance, as can be expected from the equation.
S. Zurek, E-Magnetica.pl, CC-BY-4.0
S. Zurek, E-Magnetica.pl, CC-BY-4.0
On the other hand, $F_{Zurek}$ provides reasonable agreement over the whole range of distances, as compared to the experimental data and finite-element modelling (FEM).39)
Similar behaviour can be expected for all the major types of magnets: NdFeB, SmCo, and hard ferrites.40)
For the contact force there is a significant scattering of the calculated values, as shown in Fig. 3. The discrepancies for equation (8) (Vokoun et al.) approximated with the closed-form version of elliptic integrals increase for magnets which are “flatter”. This is due to the shortcomings of the approximation of the elliptic integrals, not the equation itself.41)
Equation (4) (Cheedket and Sirisathitkul) significantly underestimates the contact force, especially for “flat” magnets (coin-like).
Difference between attracting and repelling
All the closed-form equations listed above assume that the force does not depend on the polarity of the magnets, and the measurements were made with the magnets repelling.
In reality, there can be significant differences between the configuration for attracting (magnets aligned N-S) or repelling (aligned N-N). This is because the relative permeability of the magnets is greater than unity, and thus for the attraction the force due to magnetisation is aided with the force due to ferromagnetic permeability. However, in the repelling configuration the force due to permeability acts against the main repelling force, so smaller values can be expected.42)