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.
- Cargo plate construction material: Steel, undefined.
- Plate Side Length: 2 m
- Supporting Satellites Per Side: 2
- Plate ERC Maximum Radius – apo: 1m
- Sub-ERC Outer Radius – aeo: 0.5m
- Plate Thickness - \(l_{p}\): 0.05m
- Sub-ERC Inner Radius – aei: 0.45m
- Sub-ERC Fabry Factor Alpha - \(\alpha_{e}\): 1.11
- Sub-ERC Fabry Factor Beta - \(\beta_{e}\): 0.056
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)
- Carbon Steel, Permeability – \(u_{\text{ec}}\):
1.26x10-4
- Annealed Stainless Steel -\(\ u_{\text{ea}}\):
1.26x10-3
- (1) Sub-ERC Self-Inductance (S-I) – \(L_{e1}\):
1.76x10-5
- (1) Zero-width (\(\alpha_{e}\):1,\(\ \beta_{e}:\)0.05) - \(L_{e01}\):
1.97x10-5
- (1) Wide
(\(\alpha_{\text{ei}}:\)0.05,\(\alpha_{e}:\)10,\(\ \beta_{e}:\)0.5) -\(L_{ew1}\): 1.97x10-6
- (6) Sub-ERC S-I (\(u_{\text{ec}})\)– \(L_{e6}\):
1.67x10-5
- (6) Sub-ERC S-I (\(u_{\text{ea}})\)– \(L_{e6}\):
1.51x10-4
- (7) Sub-ERC S-I – \(L_{e7}\): 2.29x10-6
- (6) Zero-width ERC S-I – \(L_{e06}\): ln(x/0) t.f. null
- (7) Zero-width ERC S-I – \(L_{e07}\): ln(x/0) 0 t.f. null
- (8) Wide ERC SI – \(L_{ew8}\): 3.14x10-7
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.
- Solenoid midline radius - \(a_{s}\) = 0.0293m
- Sub-ERC midline radius - \(a_{e}\) = 0.475m
- Solenoid to cargo plate distance - \(\rho\) = 0.01m
- (12) Solenoid Elliptic Moduli – \(k_{s}\): 0.32897
- (13) Sub-ERC Elliptic Moduli – \(k_{e}\): 0.46777
- (14) Dimensionless Elliptic Moduli – \(c^{2}\): 0.2189
- Solenoid First Elliptic Result – K( \(k_{s})\): 1.6161
- Sub-ERC First Elliptic Result – K( \(k_{e})\): 1.6692
- Solenoid Second Elliptic Result – E( \(k_{s})\): 1.5273
- Sub-ERC Second Elliptic Result – E( \(k_{e})\): 1.481
- Solenoid Third Elliptic Result –\(\gamma\)( \({c^{2},k}_{s})\): 1.8318
- Sub-ERC Third Elliptic Result – \(\gamma\)( \({c^{2},k}_{e})\):
1.8958
- (11) Mutual Inductance – \(M_{\text{se}}\left(\rho\right)\):
0.00031
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)
- Plate Thickness - \(l_{p}\) = 0.05m
- Circular Wire Cross Section (c-s) - \(a_{c}\) = 1963.5
mm2
- Rectangular (Square) Wire C-S - \(a_{r}\ \)= 2500
mm2
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.
- Solenoid midline radius - \(a_{s}\) = 0.0293m
- Sub-ERC midline radius - \(a_{x}\) = 0.0294m
- Solenoid to cargo plate distance - \(\rho\) = 0.01m
- Solenoid Elliptic Moduli (12) – \(k_{s}\): 0.11434
- Sub-ERC Elliptic Moduli (13) – \(k_{x}\): 0.98576
- Dimensionless Elliptic Moduli (14) – \(c^{2}\): 0.999997
- Solenoid First Elliptic Result – K( \(k_{s})\): 1.57596
- Sub-ERC First Elliptic Result – K( \(k_{x})\): 3.1861
- Solenoid Second Elliptic Result – E( \(k_{s})\): 1.5656
- Sub-ERC Second Elliptic Result – E( \(k_{x})\): 1.03797
- Solenoid Third Elliptic Result –\(\gamma\)( \({c^{2},k}_{s})\): 912.87
- Sub-ERC Third Elliptic Result – \(\gamma\)( \({c^{2},k}_{x})\):
5367.32
- Mutual Inductance – \(M_{\text{sx}}\left(\rho\right)\): 0.001383
- Scenario 1 sub-ERC induced current – \(I_{x1}\): 3,333 A/s
- Scenario 1 current density - \(I_{x1}/a_{c}\): 1.70
A/mm2
- Scenario 3 sub-ERC induced current – \(I_{x3}\): 555 A/s
- Scenario 3 current density - \(I_{x3}/a_{c}\): 0.28
A/mm2
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.