Determining Inductance

To demonstrate the concept and interactions, an iron yoked single coil solenoid of the following design is considered:
Inductance must be found first [52] to determine the solenoids current creation capability in the equivalent resistance circuit (ERC) of the object above, whether satellite or cargo plate, then the resultant field interaction force. If this exceeds the inertial force requirement of the proposed 2000kg freight mass then maglev cargo acceleration is a success. Inductance is a measure of influence that an electromagnetic field has on the object above’s surface and to a skin penetration depth relative to the applied field strength. The field lines intersect with the conductive material and create a circular current around their intercept, the pancake coil ERC. In more conductive materials and contexts, there is a lower electrical resistance so a greater current is induced. Finding the current created in the object above’s ERC is thus the key to validating the interaction.
To find the current created by the solenoid in the object above’s ERC, the self inductance of each component is found then used to determine their mutual inductance as a system. Two comprehensive treatments of solenoid analysis are [52] & [31] however neither completely addresses the proposed design.
\(L_{s}=u_{0}N_{s}^{2}\text{\ a}_{\text{si}}(\frac{\text{πα}}{2\beta}\))
Solenoid self inductance [31 (3.81)] (1)
\begin{equation} L_{s}=\frac{u_{\text{rs}}u_{0}N_{s}^{2}\text{\ π}a_{\text{si}}^{2}}{l_{S}}\ \nonumber \\ \end{equation}
Solenoid self inductance [52 (13, 21) ] (2)
The self inductance formulae above do not distinguish between resistive or superconductive material selection but rather by the inclusion of a yoke rod’s enhancement of the magnetic relative permeability (\(u_{\text{rs}}\)=\(\ u_{s}/u_{0}\)) in the centre of the coil and their treatment of the solenoid winding influence. The multiplicative effect of enhancing magnetic permeability within the coil’s yoke rod is visible in the inductance and force generation formulae detailed in [52 (22)] and discussed further in the force derivation section below.
Magnet designers use the Fabry factors \(\alpha,\beta\) to describe the solenoid shape and classify coil design subtypes [31]. The coil radii and length characteristics determine the Fabry factors and later elliptic integral results as shown below, classifying the proposed solenoid as a thin walled solenoid.
\begin{equation} \alpha_{s}=\ \frac{\text{\ \ }a_{\text{so}}}{\text{\ \ a}_{\text{si}}}\ =1.6\nonumber \\ \end{equation}
Fabry coil design factor - Alpha [31, P115] (3)
\begin{equation} \beta_{s}=\ \frac{b_{s}}{\text{\ \ a}_{\text{si}}}=11.1\nonumber \\ \end{equation}
Fabry coil design factor - Beta [31, P115] (4)
Equating the above self-inductance calculations to find their differences and focus on the coils alone gives the following equivalence when removing the yokes multiplicative influence:
\begin{equation} \frac{\pi a_{\text{si}}^{2}}{l_{S}}\equiv a_{\text{si}}\frac{\text{πα}}{2\beta}=a_{\text{si}}\frac{\pi\frac{\text{\ \ }a_{\text{so}}}{\text{\ \ a}_{\text{si}}}}{2\frac{b}{\text{\ \ a}_{\text{si}}}}\ =\ \frac{\pi\text{\ a}_{\text{so}}\text{\ a}_{\text{si}}}{l_{S}}\nonumber \\ \end{equation}
(1) \(\equiv\) (2) Coil design factor comparisons (5)
The reduction of the comparison above to\(a_{\text{si}}\equiv\ a_{\text{so}}\) results in agreement of the formulae on a hypothetical coil width of zero where\(a_{\text{si}}=\ a_{\text{so}}\). This is not unreasonable for the object above’s theoretical ERC but gives an appreciable difference of 1.6 when comparing the self inductance of [52] to [31] for the tape wound solenoid. Given the latter’s topic is case studies in superconductive magnet design, Iwasa’s [31] formulae will be preferred. Despite this variance in the literature, it is evident that the yoke rods enhancement of magnetic permeability within a coil is a linear multiplier, though this effect diminishes and requires numerical methods once the yoke is saturated [31]. In quadrupole accelerator magnets with fields well above 1T the exterior yoke is a minor field component [31] thus the factor is actually a function and design specific modelling is required to determine the realistic effect between minor enhancement and linear multiplication. Despite this source of variance, it is clear that any conductive yoke enhances an electromagnetic field [51] thus inductance and force applied. This gives a rational basis for inclusion of \(u_{\text{rs}}\ \)in (1) for the proposed design despite the variance in magnitude and potential function substitution.
To determine the inductance of the flat metal launch plate \(L_{P}\), the ERC is considered as a single wire pancake coil, acting as the second component of the two coil interaction. The proposed context can be analysed using [31 p112]’s assessment of thin solenoid to pancake coil interactions in Section 3.8.1 as there is no material specification factors between the superconductive solenoid coil or resistive metal plate.
The surficial ERC in the object above acts as a simple circuit of a single coil of wire as seen below in Figure 4. In the proposed context where a cargo plate is suspended above a square array of satellites, the voltage source represents the net effect of the induced fields. To decompose this problem, the interaction of an individual satellite’s solenoid and their sub-ERC’s area of effect is analysed. The inductive voltage generation and material resistance of the plate provides the inputs to resolve the circuit. The sub-ERC’s and plate-wide ERC are identical, their recombination and the holistic result is discussed further below.