Applied Force Derivation
[I] is a case study in superconductive magnet design and details the
interaction of superconductor’s winding style to create self-inductance,
then in multiple 2-coil system design contexts but without the presence
of yoke rods. [31]’s Example 3.5.3 below is an analytical
formulation of the proposed design context and uses only \(u_{0}\) for
calculating force as a result of inductance as the two coils in the
interaction are considered only in relation to each other in a vacuum.
[52] addresses the single solenoid context and includes the linear
enhancement of the force by the yokes relative permeability \(u_{r}\)which was demonstrated as nonrelevant from scenario one above.
[52]’s formulae for force interaction include the term but are
reliant on a materials resistance, which does not address the
superconductive context of R = 0. Despite this, the potential
enhancement effect of the yoke rod is again noted though not included,
unlike above.
[31]’s Example 3.5.3 illustrates the force experienced at the top of
an unyoked coil in a long thin solenoid when acting against the pancake
coil. That force derivation is shown below utilising the inductance and
current inputs to scenario one and three above.
\begin{equation}
F_{\text{sz}}\left(\rho\right)=\ -\ \frac{\mu_{0}}{2}\left(N_{e}I_{e}\right)\left(\frac{N_{s}I_{s}}{2b_{s}}\right)\ \times\nonumber \\
\end{equation}\begin{equation}
(\ \sqrt{\left(a_{s}+\ a_{e}\right)^{2}+\ \left(2b_{s}+\rho\right)^{2}}\ \begin{Bmatrix}2\left[K\left(k_{s}\right)-E\left(k_{s}\right)\right]\\
-k_{s}^{2}K\left(k_{s}\right)\\
\end{Bmatrix}\ \nonumber \\
\end{equation}\(-\ \ \sqrt{\left(a_{s}+\ a_{e}\right)^{2}+\ \rho^{2}}\ \left\{2\left[K\left(k_{e}\right)-E\left(k_{e}\right)\right]-k_{e}^{2}K\left(k_{e}\right)\right\}\ )\)
Force acting against a solenoid end in z [31 (3.42)] (17)
The results required in (17) include k2 terms but all
moduli are identical to the earlier mutual inductance moduli (12,13)
across the various scenarios with results from the same tables of
complete elliptic integrals.
- Scenario 1\(a_{e}\) sub-ERC current – \(I_{e1}\): 14,870 A/s
- Scenario 1\(a_{e}\) force - \(F_{e1}\left(\rho\right)=\) 293 N
- Scenario 1\(a_{x}\) sub-ERC current – \(I_{x1}\): 3,333 A/s
- Scenario 1\(a_{x}\) force - \(F_{x1}\left(\rho\right)=\) 1076 N
- Scenario 3\(a_{e}\) sub-ERC current – \(I_{e3}\): 2,478 A/s
- Scenario 3\(a_{e}\) force - \(F_{e3}\left(\rho\right)=\) 8.16 N
- Scenario 3\(a_{x}\) sub-ERC current – \(I_{x3}\): 555 A/s
- Scenario 3\(a_{x}\) force - \(F_{x3}\left(\rho\right)=\) 29.9 N
As scenario 1\(a_{e}\) has a current density of 7.57
A/mm2 there will be a reduction in solenoid current
required to ensure the selected plate steel is able to handle the
induced current. Scenario 1\(a_{x}\) is the clear preference given the
highest newton force generated, corresponding to the highest current in
the smallest ERC size. As with results above, the presented results are
for a single coil solenoid to a sub-ERC within the object above so a
number of factors such as coil count and yoke enhancement may be applied
to alter results further. It is noted that the larger the ERC, the
higher the current and the lower the corresponding force generation. As
the design context proposed is a square array of four satellites under
the cargo plate, the force result can potentially be combined linearly
for first pass analysis however as demonstrated above it can be seen
that mutual inductances between the sub-ERC’s will have an effect.
Interestingly the results of scenario’s 1\(a_{e}\) and 1\(a_{x}\)demonstrate that the six wire Rutherford cabling gives a linear multiple
of induced current compared to the single wire scenario’s 3\(a_{e}\) and
3\(a_{x}\)while the force generated increases by 36 times, or six
squared.
This final observation gives strong evidence for the use of Rutherford
cabled solenoids in the proposed design context. The use of a yoked dual
coil design with a correspondingly high total inductance between all
subcomponents will thus produce the optimal result despite the variances
noted in ERC radii. Simulink modelling is a clear necessity to determine
accurate electromagnetic field intercepts and ERC definition due to the
noted sources of variance. The results will be accepted for now and
inspected following definition of the cargo acceleration success
criterion.