To frame this problem analytically, determine the launch pulse strength, the inductance in each satellite above and the mass of a satellite, launch plate and cargo. This system can be modelled in MS Excel by dimensioning each component of the satellite design, creating a swarm array table and applying the mechanics. Results can be compared quickly by building component selection as a series of drop down menus to define system level functions.

Inductance equations are based on the surface geometry of objects in the field and coil design had a significant impact. Multicoil solenoids use coil inductance calculations to find the resulting total inductance of the electromagnet design, which is a key component for pulse wave calculations and overall swarm reactions. Each component of the multicoil design must have its self-inductance determined then the effect of this component on the others within the design. This creates a system of mutual inductances between the multiple coils and yoke rod in the centre of the solenoid. When one coil’s electromagnetic field induces a current in the next coil, it reduces the energisation required to bring that coil up to full power — creating a cyclic effect from the system of mutual inductances. This is the key to multicoil designs generating 100T+ field strengths in pulsed high magnetic field research.

The table below lays out the component self and mutual inductance elements of the multifactor time variant problem that models a launch pulse:

Table 1. Inductance Calculations & Substitutions

Launch Pulse Power

The launch pulse’s field strength is determined in three distinct stages, giving an approximately trapezoidal shape when overall power consumption is graphed against time. The first time step is the triangular ramp-up stage as the magnet is energised by the capacitors, second is the rectangular flat top of consistent power output for the duration of the launch pulse and the third is the down-ramping depowering of the solenoid.

Each stage can be modelled as simple power expenditure over time line graphs generated based on the satellite design. The size of the satellite determines the available space for the superconductive solenoid, power storage and refrigeration systems which in turn define the thrust capability. The solenoid materials and dimensions define the peak power capacity of the coil and the energisation requirement for peak field generation. The capacitor system & pulse transformer peak throughput rates determine the energisation rate of the coil and thus ramp up time to peak field strength.

Due to high earth orbit operation, a satellite is essentially sitting in a 10C ambient temperature that fluctuates moderately during the orbital cycle. The solar radiation input rate defines the edge boundary of the satellite thermal partial differential equation, where containment chambers and refrigeration must be capable of maintaining the solenoid at 2K, drawing energy that could otherwise be used in the launch pulse. In this way, a number of dependent components & systems reduce the available electrical power for the launch pulse, so a depth of discharge factor must be used to limit the power drain. Each of these systems can be dimensioned in Excel from component designs and series of equations that model the physical behaviour of the satellite and create a picture of the launch pulse dynamics.

The ramp-up stage energises the superconductive coil with power, creating the electromagnetic field up to a peak rate of continuous flat power consumption. This power consumption over time graph is a right angle triangle, with the peak at the pulse waves rectangular flat-top power output once the magnet is energised. The gradient of the ramp-up triangle is limited by the energisation rate of the pulse power transformer and capacitor system.

Breaking each stage of the pulse down to simple linear results in three time steps gives the inputs to the electromotive swarm model and allows the field interaction force results to be found. The field interaction force is created from the field strength of the pulse applied at the spatial position of the next satellites wing frame. This result applied to the inductance of the material in the frame creates a resultant induced field strength along with a mechanical force moment and defines the direction of the resulting thrust vector.

Figure 2. Launch Mechanics Force Diagram

Orbital Maintenance Method

The swarm’s orbital vector is impacted with each launch, if there was no way to reset the system, it would be a glass cannon. Given the objective of operating this system as a service business, building a glass cannon doesn’t present a viable business case. Instead, the orbital maintenance issue has several solution components, both in satellite design and launch system operation. To explain the orbital maintenance method, design elements must be detailed first as this explains the granular swarm control. The launch reset method to maintain swarm cohesion and return to formation is key to how the swarm maintains its orbital vector.

The central Propulsion Core design is scaled down to create Positioning Propulsion Cores, with several mounted below the satellite and in the wings to give granular intraswarm movement. The PPC’s allow the swarm to manipulate units and organise them using the larger tethered magnetic mass to control smaller untethered magnetic masses, ie singular units within the swarm. The periodic engagement of the PPC’s to create magnetic tethers & maintain buffer spacing works in the same way as the launch pulse field interaction, just like a maglev train. By engaging magnetic tethers across the swarm to anchor the majority of units, singular units and layers can be manipulated using the PPCs of surrounding neighbours. This constant push / pull adjustment of horizontal PPC’s or the strong pull of the main PC is what allows units to pull against the larger magnetic mass without moving other units out of place.

Engagement of the magnetic tethering following the launch pulse is a matter of timing, with the first retraction pulse rapidly following the launch pulse. This is critical to stopping the escape of layers moving beyond effective reset range. The strong retraction pulse is followed by a series of graded reset pulses that are used to bring the swarm back to it’s prelaunch spacing. Removing the majority of the outwards velocity in a singular pulse requires coordination between base mounted PPC’s and the lower layers PC engagement. This interaction is repeated several times with decreasing power to minimise structural stresses and risk of satellite collision.

As seen in the Ring Launcher diagram above, the greater magnetic mass of the lower satellite layer is the key to driving the next layer forwards. For the base of the swarm, a non magnetic thrust offset is required in the form of fuel based thruster. This gives each satellite a number of launches at a swarm base layer position before the fuel is expended and the unit must be cycled up in the swarm to another position.

The spatial reference system defines its zero point at the centre of the cargo transfer plate atop the swarm launch pad. This allows the swarm architecture to be modelled in -z depth while the vector mechanics are generated in a positive reference frame from the zero point. Defining cargo vector calculations from a zero position allows the orbital vector to be easily combined with the launch pulse vector, to chart a course for cargo arrival and positional changes required for orbital maintenance. The flip side of this is that it becomes easier to calculate the swarm reset vector algorithm.

The force reaction pushing against the lower satellites from the induced field is negligible, the force on the base layer of moving the combined magnetic mass above is not. The orbital impact of the launch pulsewave is intended to be zeroed by the fuel based thruster assembly. This provides the foundation for the rapid series of retraction pulses from each layer to be anchored to. The finer details of PPC design, launch pulse energy consumption, depth of capacitor discharge and swarm anchor force values are retained for IP protection.

By modelling a singular satellite in detail, the swarm representative table can be modelled as a spatially defined array to apply electromotive equations and calculate satellite unit location changes over time from the vector result. The combination of detailed unit modelling with higher level electromotive time function modelling of the swarm during launch pulses can be used to empirically prove the novel thrust method is functional and achievable. As above, this system can be built and modelled in excel using existing components for validation.

Thanks for reading, @h.mjw

https://h-industries.io

References

Algarni, A, Gleason, F & Mohanakumaran, A, 2014. Electromagnetic Ring Launcher.

Han, K, Ishmaku, A, Xin, Y, Garmestani, H, Toplosky, V, Walsh, R, Swenson, C, Lesch, B, Ledbetter, H, Kim, S, Hundley, M and Sims, J, 2002. Mechanical properties of MP35N as a reinforcement material for pulsed magnets.

Hurley, W, Duffy, M, Zhang, J, Lope, I, Kunz, B & Wölfle, W, 2015. A Unified Approach To The Calculation Of Self And Mutual Inductance For Coaxial Coils In Air.

Shi, J, Han, X, Xie, J & Li, L, 2016. Analysis and Design of a Control System for the 100T Pulsed High Magnetic Field Facility at WHMFC.

Zhou, Y, Ghaffari, M, Lin, M, Xu, H, Xie, H, Koo, C & Zhang, M, 2018. High Performance Supercapacitor Under Extremely Low Environmental Temperature.

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