Determining Inductance
To demonstrate the concept and interactions, an iron yoked single coil
solenoid of the following design is considered:
- Yoke: 99.8% Iron, Relative Permeability – \(u_{\text{rs}}\): 5000
- Permeability of a vacuum – \(u_{0}\): 4 π x 10-7
- Coil wire: PIT Nb3Sn – Cored Rutherford Cable
- Wire current density capability - Is: 3000
A/mm2
- Coil turn count – \(N_{S}\): 200
- Coil length – \(l_{s}\): 0.5 m
- Coil inner radius – \(a_{\text{si}}\): 0.0225 m
- Coil outer radius - \(a_{\text{so}}\): 0.0361 m
- Coil b factor – \(b_{s}\) = ½ \(l_{S}\) = 0.25
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.
- (1) Solenoid Self-Inductance (S-I) - \(L_{s}\): 0.0002561
- (2) Yoke Enhanced Solenoid S-I – \(L_{s}\): 0.7994379
- (1+) Yoke Enhanced Solenoid S-I – \(L_{su}\): 1.2803811
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.