Fig. 5. Thin solenoid to pancake coil interaction geometry.
To determine the Fabry factors of the sub-ERC pancake coil a theoretical wire width and coil height must be found. The square plate’s overall radius, \(a_{P},\) affected by each satellite, \(a_{E}\), is considered geometrically at first as the length of one side divided by the number of satellites that plate edge rests upon divided in half. While there may be inductive field overlaps in reality, this theoretical division gives a bound to one satellite’s area of effect and thus defines a maximum possible size of each sub-ERC. The array of values between this maximum and the minimum at \(a_{E}\) = 0 must be tested to evaluate the real intercept boundary. From the equations (1) and (3) above, it can be seen that the choice of inner radius \(a_{\text{ei}},\ a_{\text{pi}}\), has a significant effect on results by substantially increasing the\(\alpha\) Fabry design factor at lower values. The coil height is defined by the plate thickness and can also be substituted for wire thickness for a theoretically circular wire geometry. The zero-width case is thus the local minima for calculating inductance and force generation.
Equation’s (1) and (2) are specified for general solenoids while single wire pancake coils (n=1) are dependent on their \(\alpha\) result and whether the loop is circular or rectangular in cross-section. Note\(2R=a_{\text{ei}}(\alpha_{e}+1)\) for loop diameter &\({2a=a}_{\text{ei}}\)(\(\alpha_{e}-1\)) for loop wire diameter [31] while the permeability (non-relative) \(u_{e}=u_{p}\) is defined by the cargo plates selected steel.
\(L_{e}\cong u_{0}R[\ln(\ \frac{R}{a}\) ) + 0.079] + ¼\(u_{e}R\)
Maxwell’s general loop S-I (circular c-s) [31 (3.80b)] (6)
\(L_{e}\cong u_{0}R[\ln(\frac{R}{a}\)) + 0.886] =\(u_{0}\frac{a_{\text{ei}}(\alpha_{e}+1)}{2}[\ln(\ \frac{\frac{a_{\text{ei}}(\alpha_{e}+1)}{2}}{\frac{a_{\text{ei}}(\alpha_{e}-1)}{2}}\))+ 0.886]
Thin (\(\alpha_{e}\cong\)1) Pancake S-I (rectangular c-s) [31 (3.86c)] (7)
\begin{equation} L_{e}\approx{\frac{1}{2}u}_{0}\alpha_{\text{eo}}N_{e}^{2}\nonumber \\ \end{equation}
Wide (\(\alpha_{e}\gg\)1) Pancake S-I (rectangular c-s) [31 (3.86d)] (8)
Tabulation of formulations (1), (6), (7) & (8) allows easy comparison of self inductance values and their range. As with the earlier noted variance, it is important to understand the source of origin and the systemic effect of factor selection. Use of a low aeivalue is theoretically valid due to the cargo plate’s homogeneous construction and this will generate a much larger alpha with an artificial inductance as the factor tends towards zero so the wide pancake case (8) is discarded for cargo plate representation. The zero width wire case is also discarded as the natural log function is not defined at zero and the cargo plate does have a conductive cross section in reality.
It must be noted again that application of the \(u_{\text{re}}\) factor to (7) would significantly change the results as seen below in (9). The result is presented as the solid metal cargo plate effectively has a large central yoke within the theoretical wire ERC thus contextually aligns with (1) & (2) as discussed above. Each formulation is calculating a result based on the coil winding geometry, primarily affect by the number of turns and the yokes enhancement of permeability when present. As such, the results of (1) are presented above before (6) & (7). Given the similarity between the geometries and results, the relative permeability \(u_{\text{re}}\) term is applied to (7) as with (1) for inspection of \(L_{e}\). The resulting range of sub-ERC inductance values will later be used to calculate a minima and maxima in a range of scenarios.
\(L_{e}\cong{u_{\text{re}}u}_{0}R[\ln(\ \frac{R}{a}\) ) + 0.886]
Yoked Thin Pancake S-I (rectangular c-s) [31 (3.86c)] (9)
Carbon Steel Yoke Sub-ERC SI (9) – \(L_{e9}\): 2.29x10-4
Putting aside these noted sources of variance in inductance for now, to determine the induced current and resultant force pushing against the object above’s inertia, the self inductances of each component must be combined to determine the two coil systems mutual inductance. The proposed single coil design is presented to remove multicoil mutual inductance calculations for clarity however the mutual inductance between each satellite’s solenoid and the ERC in the object above is the key to concept validation. In coils that share a central axis, the mutual inductance M can be quickly estimated from the self inductance L and similarity of the Fabry coil design factors \(\alpha\ \&\ \beta\). The k factor is an approximation from 0 to 1 of coil similarity to remove elliptic moduli as a first pass design test for easier calculation by hand [31]. Concentric coils range from k = 0.3 to 0.6 and closer to 0.6 if the Fabry factors relating coil heights and diameters are similar [31].
\begin{equation} M_{\text{SP}}\equiv N_{S}\frac{\phi_{\text{SP}}}{I_{p}}\ \equiv M_{\text{PS}}\equiv N_{p}\frac{\phi_{\text{PS}}}{I_{S}}\ \ =k\ \sqrt{L_{s}L_{P}}\nonumber \\ \end{equation}
Mutual inductance approximation [31 (3.95a)] (10)
The detailed formulation below incorporates elliptic moduli to accurately assess the interaction between two differing winding geometries midline radius \(a_{s}\) & \(a_{e}\) (or total cargo plate radius \(a_{p}\)) at a distance ρ from each other. The complete elliptical integral tables of the first K , secondE & third \(\gamma\) kind that describe the two coil systems can be seen in [31]’s Example 3.8.1 (pp. 112) and Tables 3.1 (pp. 84) & 3.2 (pp. 90) or online using the following inputs for solenoid and cargo plate sub-ERC.
\begin{equation} M_{\text{se}}\left(\rho\right)=\ -\ \frac{\mu_{0}}{2}\left(\frac{N_{s}N_{e}}{2b_{s}}\right)\ \ \times\nonumber \\ \end{equation}\begin{equation} (\frac{\rho}{\sqrt{\left(a_{s}+\ a_{e}\right)^{2}+\ \rho^{2}}}\ \begin{Bmatrix}\left[\left(a_{s}+\ a_{e}\right)^{2}+\ \rho^{2}\right]\\ \ \times\ \left[K\left(k_{e}\right)-E\left(k_{e}\right)\right]-\ \gamma\left(c^{2},k_{e}\right)\\ \end{Bmatrix}\nonumber \\ \end{equation}
\(-\ \frac{2b_{s}+\rho}{\sqrt{\left(a_{s}+\ a_{e}\right)^{2}+\ \left(2b_{s}+\rho\right)^{2}}}\ \par \begin{Bmatrix}\left[\left(a_{s}+\ a_{e}\right)^{2}+\ \left(2b_{s}+\rho\right)^{2}\right]\\ \ \times\ \left[K\left(k_{s}\right)-E\left(k_{s}\right)\right]-\ \gamma\left(c^{2},k_{s}\right)\\ \end{Bmatrix})\ \)
Mutual inductance of thin solenoid to pancake coil at distance ρ – [31 (3.98)] (11)
\begin{equation} k_{s}=\ \sqrt{\frac{\ 4a_{e}\ a_{s}}{{{\ \ (a_{e}+a}_{s})}^{2}+{(2b_{s}+\rho)}^{2}\ }}\nonumber \\ \end{equation}
Solenoid Elliptic Moduli Root [31 p112] (12)
\begin{equation} k_{e}=\ \sqrt{\frac{\ 4a_{e}\ a_{s}}{{{\ \ (a_{e}+a}_{s})}^{2}+\rho^{2}\ }}\nonumber \\ \end{equation}
Pancake Coil Elliptic Moduli Root [31 p112] (13)
\begin{equation} c^{2}=\ \frac{\ 4a_{e}\ a_{s}}{{{\ \ (a_{e}+a}_{s})}^{2}\ }\nonumber \\ \end{equation}
Dimensionless Elliptic Moduli [31 p112] (14)
The cargo plate rests directly on top of the satellite’s as shown above in figure 5. The distance \(\rho\) between the outer edge of the solenoid and the ERC is minimal at first but increases over time as a function of the force applied and thus object acceleration.
Mutual inductance is an independent factor relating geometries of one coil to another object in a context specific manner. In the two coil case considered there is a vacuum between the components so no \(u_{r}\)term is present to enhance the magnetic permeability of the space between the coils. The mutual inductance is purely attributed to winding geometries with no influence of current density or material selection, unlike later formulations reliant on this relationship. In superconducting quadrupoles, the mutual inductance must be tightly controlled at the design stage to prevent unintended influence in the beam control fields and is often minimised to prevent the emergence of high current segments [31]. In the proposed design context where maximising component inductance is the goal, there will be a corresponding increase in mutual inductance as seen above in (10). The result of \(M_{\text{se}}\left(\rho\right)\ \)= 0.00031 presents a reasonable result for two coils of differing winding style resting on each other. The result is accepted for now until it is tested further below and the sample design is validated for the context.
The pulsed magnetic field created by each solenoid’s total inductance and rapidly pulsed current determines the current induced in the surface of the object above as seen below in (15). Iwasa [31] presents the case of two separate inductively coupled superconductive coils in Problem 1.2’s solution with the circuit analysis of Figure 4 shown in equation (15) below.
Once inductances are found for all components, equation (15) can be rearranged to find the pulsed time varying current induced in the plate, IP, in Amps per second. The traditional substitution of V=IR is not applicable in the superconductive context, giving a simplified circuit analysis despite creating a number of other concerns in reality such as the current persistence in the coil filaments. The circuit analysis result is linearly influenced by the available current in the power supply and limited by the transformer throughput to the solenoid. This reinforces the need for superconductive components with the highest current density possible to achieve peak pulse power.
\begin{equation} L\frac{dI_{S}(t)}{\text{dt}}+M_{\text{SP}}\frac{dI_{P}(t)}{\text{dt}}=0\nonumber \\ \end{equation}
Inductively Coupled Coils [31, S1.2b] (15)
Rutherford solenoid current pulse – \(I_{S}(t)\): 18,000 A/s Solenoid S-I Minimum (1) = 0.0002561
Sub-ERC induced current – \(I_{e1}(t)\): 14,870 A/s Sub-ERC current density - \(I_{e1}/a_{c}\): 7.57 A/mm2
Rutherford solenoid current pulse – \(I_{S}(t)\): 18,000 A/s Linearly Enhanced S-I Maximum (1) = 1.28
Sub-ERC induced current – \(I_{e2}(t)\): 74,321,889 A/s Sub-ERC current density - \(I_{e2}/a_{c}\): 37,851 A/mm2
Single wire solenoid current pulse – \(I_{s}(t)\): 3,000 A/s Solenoid S-I Minimum (1) = 0.0002561
Sub-ERC induced current – \(I_{e3}(t)\): 2,478 A/s Sub-ERC current density - \(I_{e3}/a_{c}\): 1.26 A/mm2
Three scenarios presented above are inductive minimum (three) and maximum (two), combining solenoid inductance with a six wire Rutherford cable then a single Nb3Sn wire to show the variance between results for the derived calculation path. Discarding the unrealistic maximum of 74.3m A’s induced in scenario two due to the engineering current density exceeding the capability of every known material, the reasonable result is between scenario one and three with the lower bound of 2,478 A/s induced in the cargo plate at the self-inductance minimum.
The standard inductance formula produce reasonable results in isolation however when applied to the proposed design generate strange outcomes which can only be assessed in verified and tested modelling tools. If the analytical path is correct, the results above may be regression tested and validated, confirming an overengineering of the proposed Rutherford cable. Scenario’s one and three are accepted to follow through the force generation equation and determine a launch capability of the design. When the derivation path is calculated with \(a_{p}\)instead of Ae, results diverge further indicating an\(a_{x}\) test value at a 1:1 ratio with \(a_{s}\) should be investigated to establish a system minima. It was highlighted above that the selection of radii for plate definition significantly affects results thus \(a_{x}\) is presented.
The result of testing \(a_{s}=\ a_{x}\) is null due to the dimensionless elliptic moduli (14) being 1 and thus undefined on the chart of the third complete elliptic integral. A fractionally larger value for \(a_{x}\) is then considered to maintain elliptic coherency such that \(a_{s}<a_{x}\). Results are presented to inspect the theoretical ERC in the object above directly opposing the solenoid.
At this stage, it is unclear whether results are unrealistic from a foundational error present in formulation or analysis, or if by absence of error, the derivation path is correct. Scenario’s one and three produce valid current densities across a range of cargo plate radii so their force application must be examined.