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.
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.