New Fusion Reactor Design Function

The Fusion Energy Toolbox for MATLAB is a toolbox for designing fusion reactors and for studying plasma physics. It includes a wide variety of physics and engineering tools. The latest addition to this toolbox is a new function for designing tokamaks, based on the paper in reference [1]. Tokamaks have been the leading magnetic confinement devices investigated in the pursuit of fusion net energy gain. Well-known tokamaks that either have ongoing experiments or are under development include JET, ITER, DIII-D, KSTAR, EAST, and Commonwealth Fusion Systems’ SPARC. The new capability of our toolboxes to conduct trade studies on tokamaks allows our customers to take part in this exciting field of fusion reactor design and development.

The Fusion Reactor Design function checks that the reactor satisfies key operational constraints for tokamaks. These operational constraints result from the plasma physics of the fusion reactor, where there are requirements for the plasma to remain stable (e.g., not crash into the walls) and to maintain enough electric current to help sustain itself. The tunable parameters include: the plasma minor radius ‘a’ (see figure below), the H-mode enhancement factor ‘H’, the maximum magnetic field at the coils ‘B_max’, the electric power output of the reactor ‘P_E’, and the neutron wall loading ‘P_W’, which are all essential variables to tokamak design and operation. H-mode is the high confinement mode used in many machines.

Illustration of the toroidal plasma of a tokamak. R is the major radius while a is the minor radius of the plasma. The red line represents a magnetic field line which helically winds along the torus. Image from [2].

This function captures all figure and table results in the original paper. We implemented a numerical solver which allows the user to choose a variable over which to perform a parameter sweep. A ‘mode’ option has been incorporated which allows one to select a desired parameter sweep variable (‘a’, ‘H’, ‘B_max’, ‘P_E’, or ‘P_W’) when calling the function. Some example outputs of the function are described below.

As an example, we will consider the case of tuning the maximum magnetic field at the coils ‘B_max’. The figure below plots the normalized operation constraint parameters for a tokamak as functions of B_max from 10 Tesla to 25 Tesla. The unshaded region, where the vertical axis is below the value of 1, is the region where operational constraints are met. We see that for magnetic fields below about 17.5 Tesla there is at least one operation constraint that is not met, while for higher magnetic fields all operation constraints are satisfied, thus meeting the conditions for successful operation. This high magnetic field approach is the design approach of Commonwealth Fusion Systems for the reactor they are developing [3].

Operational constraint curves as a function of B_max. Successful operation occurs if all of the curves are in the unshaded region. Note, f_B/f_NC, a ratio of the achievable to required bootstrap current, is set equal to 1. In this case P_E = 1000 MW, P_W = 4 MW/m2, and H = 1. For more details on the plotted parameters and how they function as operational plasma constraints, see reference [1].

Note, however, that there is a material cost associated with achieving higher magnetic fields, as described in reference [1]. This is illustrated in the figure below, which plots the cost parameter (the ratio of engineering components volume V_I to electric power output P_E) against B_max. There is a considerable increase in cost at high magnetic fields due to the need to add material volume that can structurally handle the higher current loads required.

Cost parameter (units of volume in cubic meters per megawatt of power, m3/MW) as a function of B_max.

In this post we illustrated the case of a tunable maximum magnetic field at the coils, though as mentioned earlier, there are other parameters you can tune. This function is part of release 2022.1 of the Fusion Energy Toolbox. Contact us at info@psatellite.com or call us at +01 609 276-9606 for more information.

Thank you to interns Emma Suh and Paige Cromley for their contributions to the development of this function.

[1] Freidberg, Mangiarotti, and Minervini, “Designing a tokamak fusion reactor–How does plasma physics fit in?”, Physics of Plasmas 22, 070901 (2015); https://doi.org/10.1063/1.4923266
[2] http://www-fusion-magnetique.cea.fr/gb/iter/iter02.htm
[3] https://news.mit.edu/2021/MIT-CFS-major-advance-toward-fusion-energy-0908

Hohmann Transfer Simulation with the Spacecraft Control Toolbox

Hohmann transfers are a well-known maneuver used to change the semi-major axis of an orbit. The Spacecraft Control Toolbox allows you to compute the required velocity changes, and integrate them into a full simulation.

In this demonstration, we create a 6U CubeSat that has 3 orthogonal reaction wheels and a single hydrazine thruster. The thruster is aligned with the body x-axis and must be aligned with the velocity vector to do the maneuver. An ideal Hohmann maneuver is done with impulsive burns at two points in the orbit. In reality, with a thruster, we have to do a finite burn.

The Hohmann transfer is computed with the following Spacecraft Control Toolbox code:

rI = [-7000;0;0];
vI = [0;-sqrt(mu/Mag(rI));0];
OrbMnvrHohmann(Mag(rI),rF);
[dV,tOF] = OrbMnvrHohmann(Mag(rI),rF);

The first time OrbMnvrHohmann is called, it generates the plot below of the planned Hohmann transfer. The function computes the delta-V and also the time of flight, which will be used to determine the start time of the second thruster burn.

We create a short script with numerical integration to implement the maneuver using a thruster. The burn durations are computed based on the thrust and the mass of the spacecraft. In this case, they are about three minutes long. The maneuver is quite small, so the mass change is not important. The attitude control system uses the PID3Axis function which is a general-purpose attitude control algorithm. The simulation is a for loop, shown below. The ECI vector for the burn is passed to the attitude control system, which updates every step of the simulation.

% Simulation loop
for k = 1:n

  % Update the controller
  dC.eci_vector = uBurn(:,kMnvr);
  [tRWA, dC]    = PID3Axis( x(7:10), dC );

  % Start the first burn
  inMnvr = false;
  if( t(k) > tStart(1) && t(k) < tEnd(1) )
    inMnvr = true;
  end

  % Switch orientation
  if( t(k) > tEnd(1) )
    kMnvr = 2;
  end

  % Start the second burn
  if( t(k) > tStart(2) && t(k) < tEnd(2) )
    inMnvr = true;
  end
  
  if( inMnvr )
    dRHS.force = thrustE*QTForm(x(7:10),dC.body_vector)*nToKN; % kN
  else
    dRHS.force = [0;0;0];
  end
  el = RV2El(x(1:3),x(4:6));
  xP(:,k) = [x;tRWA;Mag(dRHS.force)/nToKN;el(1);el(5)];

  % Right hand side
  dRHS.torqueRWA = -tRWA;
  x = RK4(@RHSRWAOrbit,x,dT,0,dRHS);
end

The maneuver logic just waits a quarter orbit then performs the first burn, by applying the thrust along the body vector. It then waits for the time of flight and then starts the next burn. The start and stop times are pre-computed. RK4 is Fourth Order Runge-Kutta, a popular numerical algorithm included with the toolbox.

At the final orbit radius an attitude maneuver is needed to reorient for the final burn.

The spacecraft body rates, in the body frame, during the maneuver are shown below.

The reaction wheel rates are shown below. The simulation does not model any particular wheel. Friction is not included in the simulation, although the right-hand-side function can include friction.

The wheel torques and rocket thrust are shown below. The thruster is a 0.2 lbf hydrazine thruster that is based on the Aerojet-Rocketdyne MR-103. The PID controller does not demand much torque.

The semi-major axis and eccentricity are shown below. The middle portion is during the transfer orbit.

The eccentricity is zero at the start and finish. Note the slope in both eccentricity and semi-major axis due to the finite acceleration. At the end of the simulation, we print the achieved orbital elements:

Final SMA        7099.72 km
  SMA error         0.28 km
Final e          1.3e-05

The result is very close to the ideal solution!

This post shows how you can easily integrate attitude and orbit control. Email us for more information! We’d be happy to share the script. We can also offer a 30 day demo to let you explore the software.

Nuclear Space Propulsion Zoom Talk for the Foundation for the Future

Michael Paluszek of Princeton Satellite Systems, will talk about nuclear propulsion for space at the Foundation for the Future Zoom meeting on Thursday, May 19 at noon EDT.

A nuclear fusion powered spacecraft near Mars.

The talk will discuss space propulsion and how fusion and fission power will revolutionize space exploration.

The Foundation for the Fusion has many other excellent speakers on both Wednesday and Friday of this week. Please join in!

ARPA-E 2022 Fusion Annual Meeting

I attended the ARPA-E 2022 Fusion Annual Meeting at the Westin St. Francis hotel in San Francisco. Here is the poster for our Princeton Field Reversed Configuration ARPA-E OPEN 2018 grant.

Here is our ARPA-E GAMOW poster on power electronics. It includes work by Princeton Fusion Systems, Princeton University, Qorvo and the National Renewable Energy Laboratory.

The meeting had two days of interesting talks by distinguished speakers. Dr. Robert Mumgaard of Commonwealth Fusion Systems talked about their work on advanced high-temperature superconducting magnets and the theory behind high field Tokamaks. Dennis Stone of NASA discussed NASA COTS programs. Dr. Wayne Sullivan of General Atomic talked about their research programs. General Atomics has been operating a Tokamak possibly longer than anyone else. We heard talks on the Centrifugal Mirror at the University of Maryland and WHAM, the high field mirror, at the University of Wisconsin. Andrew Holland of the Fusion Industry Association said FIA had verified 31 companies that were developing fusion power technology.

We talked to several organizations in need of high voltage and high current power electronics. We plan to pivot our GAMOW work to meet the needs of these potentially near-term customers.

The meeting had breakout sessions in which we discussed funding for fusion research and how to help gain social acceptance for nuclear fusion power. Both are challenging.

A Third Planet Discovered Orbiting Proxima Centauri

Introduction

A third planet, as large as 26% of the mass of Earth, has been discovered orbiting our nearest stellar neighbor, Proxima Centauri .Astronomer João Faria and his collaborators detected Proxima Centauri d using the Echelle Spectrograph for Rocky Exoplanets and Stable Spectroscopic Observations.

It would be exciting to send a spacecraft to enter the Alpha-Centauri system and orbit this planet. At Princeton Satellite System we’ve looked at interstellar flight using the Direct Fusion Drive nuclear fusion propulsion system.

Interstellar Fusion Propulsion

At the 2021 Breakthrough Energy Conference we presented findings for both flyby and orbital missions. Flyby missions are easier, but orbit entry would allow detailed study of the planet. A flyby gets your spacecraft close, but it is moving really fast!

The following charts give an outline of our talk. The first shows the optimal exhaust velocity based on sigma, the ratio of power to mass. Our designs have a sigma from 0.75 to 2 kW/kg. With 2 kW/kg, the optimal exhaust velocity is 4000 km/s. The mission would take about 800 years. Our current designs can’t get exhaust velocities higher than 200 km/s. We’d need another method to produce thrust.

Mission Analysis

The next plot shows a point mission that reaches Alpha Centauri in 500 years. This requires a sigma of about 20. The spacecraft accelerates and decelerates continuously. The mission could be improved by staging, much like on a rocket that launches from the Earth into orbit.

Selected Mission

The next figure shows how the starship would enter the Alpha Centauri system.

Alpha Centauri System Insertion

The final plot shows the orbital maneuvers that lower the orbit and rendezvous with the planet.

Lowering the orbit to rendezvous with the planet.

Even 500 years is a long time! This is over ten times the lifetime of Voyager, but much less than some engineering marvels built on the Earth.

We hope to someday be able to build fusion powered spacecraft that will head into interstellar space!

Writing about Fusion

Hi! I’m Paige, and I’m an undergraduate at Princeton interested in physics and science communications. This January, I got to work as an intern here at Princeton Satellite Systems. These past few weeks, I’ve been writing about the fusion-related projects PSS is working on, such as their Princeton Field-Reversed Configuration (PFRC) fusion reactor concept and plans for a space propulsion engine.

My first task was to write a four-page report on the PFRC, including its design, market demand, and development timeline. I knew very little about fusion coming into this internship, so first I had to learn all I could about the process that powers the sun and has the potential to supply the earth with clean, practically limitless energy.

Various types of fusion reactors are under development by companies and coalitions all over the world; they differ in the reactors they use and their methods of confining and heating plasma. ITER, for instance, is an example of a tokamak under construction in France; it uses superconducting magnets to confine plasma so that the reaction of tritium and deuterium can occur. 

The PFRC, currently in the second stage of experiments at the Princeton Plasma Physics Laboratory, uses radio frequency waves to create a rotating magnetic field that spins and heats the plasma inside, which is contained by closed magnetic field lines in a field-reversed configuration resulting from the opposition of a background solenoidal magnetic field to the field created by the rotating plasma current. The fusion reaction within the PFRC is that of helium-3 and deuterium, which offers multiple advantages over reactions involving tritium. Compared with other fusion reactors, the PFRC is incredibly compact.  It will be about the size of a minivan, 1/1000th the size of ITER; this compactness makes it ideal for portable or remote applications.

After learning about the design and market applications of the PFRC, I created a four page brochure about PFRC, writing for a general audience. I included the basics of the reactor design and its advantages over other reactors, as well as market estimates and the research and development timeline. I went on to write four page brochures about PSS’s Direct Fusion Drive engine, which will use PFRC technology for space propulsion purposes, and GAMOW, the program under which PSS is collaborating on developing various power electronics for fusion reactors.

These past few weeks have been quite informative to me, and I realized how much I loved writing about science and technology! I learned all about fusion, and I especially loved learning the details of the PFRC reactor design. To summarize the design, research, and development of the PFRC and other technologies within four page flyers, I had to learn how to write about technology and research comprehensively and engagingly for a general audience, which improved my science communication skills.

Moonfall – The Movie

Moonfall is a movie coming out in 2022. It creates a scenario where the Moon’s orbit is changed and set on a collision course with the Earth. It is fun to work out the orbital mechanics.

Let us assume that the Moon is in a circular orbit around the Earth. It is actually more influenced by the Sun than the Earth, but the circular orbit approximation is sufficient for our purposes. A mysterious force changes the orbit from circular to elliptical so that at closest approach it hits the Earth. The transfer orbit has an eccentricity of 0.9673 and a semi-major axis of 195000 km. The new orbital period is 9.9 days so it will hit the Earth in 5 days!

What kind of force is needed? The required velocity change is 0.83 km/s so a force of 6 x 1016 N applied over 10 seconds is required. To get an idea of how large that force really is, the Space Launch System (SLS) Block 2 vehicle produces about 10 million pounds of thrust [1], which is approximately 50 x 106 N (50 MN). Hence it would take 1.2 billion SLS rockets firing for 10 seconds to perform such a re-direction of the Moon! An image of the SLS is shown below (image from [1]).

As the Moon approaches the Earth it is going to raise the tides. A simple formula (really only valid when the Moon is far from the Earth) is

where is the gravitational constant for the moon, is the gravitational constant for the Earth, r is the distance between the Earth and Moon and a is the radius of the Earth. The distance during the approach and the wave height are shown in the following plot.

By around 3 days the tides started getting really big! We’d expect the Moon’s gravitational force also to pull on the solid part of the Earth’s surface, causing all sorts of trouble.

References:

[1] https://www.nasa.gov/sites/default/files/atoms/files/0080_sls_fact_sheet_10092018.pdf

Millisecond Pulse Load Switch Design

This summer I worked on the design of a millisecond pulse generator as part of the ARPA-E GAMOW grant. The goal of this project was to supply pulses of very high current to a fusion reactor’s plasma control antenna using solid-state power electronics. Some key design considerations were the ability to parallelize the pulse generator to scale to many power levels, to operate at high voltages, and to minimize the current rise time through the load antenna. I spent most of my time working in LTspice XVII to simulate the circuit and its response to rapid pulses of current. I wanted to make sure the circuit performed as desired, while also remaining safe for both the devices in the circuit and the operators controlling the circuit.

We based the design of the circuit on a load switch developed previously at Princeton Satellite Systems by Cindy Li and Eric Ham. Our biggest progress was in the selection of switching devices and the improvement of the gate drive circuitry. Because our goal is to switch current as quickly as possible, we did not want to rely on outdated klystrons as our switches. We decided to use many parallel MOSFETs to switch the current. The device we chose was a 650 V silicon carbide cascode JFET from UnitedSiC. This FET has low on-state resistance, meaning that it does not heat up as much when pulling large amounts of current. By using many devices in parallel, we can pull more total current while keeping each individual FET below its current capacity. Then by using multiple boards in parallel, we can reach different power levels for different fusion reactors. 

Implementing safe gate driver circuitry was another important step. The power MOSFETs used to switch the current cannot be turned on directly from a computer control signal, due to both power and safety concerns. A logic-level signal is not powerful enough to activate the FET, and directly connecting the computer to the high-voltage circuitry is unsafe. To solve both problems, I designed a galvanically-isolated gate drive circuit based on an Infineon gate driving IC. The intermediate circuit takes the computer control signal and steps it up to an intermediate voltage level high enough to activate the MOSFET, while keeping the both sections electrically isolated from each other. Each MOSFET has its own gate drive circuit, enabling independent control.

Working on the millisecond pulse generator was a great experience as an intern. I gained lots of practice working with team members across organizations and disciplines. I became much more proficient in LTspice, and I learned how to rigorously approach challenging engineering problems.

1U CubeSat Structural Design and 3D Print

As a Summer 2021 intern, my first project was to complete the structural design of a 1U CubeSat that will fly in orbit with and observe the NASA Solar Cruiser. The 1U CubeSat needed to follow the CubeSat design specifications set by the California Polytechnic State University; it needed to have specific dimensions, needed to weigh a certain amount, and needed to be able withstand structural loads and natural frequency/vibrational loads. In order to design and test the CubeSat, I used Fusion360’s design and simulation softwares. I based my design of the CubeSat off of the engineering drawing provided by the California Polytechnic State University’s “CubeSat Design Specification” manual.

I designed the initial model in Fusion360 as one part made up of different components, as shown below:

The top of the CubeSat faces the positive z-direction, while the front faces the negative y-direction and the right side faces the positive x-direction. The CubeSat also needed four deployable solar panels attached by hinge mechanisms to the four edges of the top face. The panels needed to start parallel to the walls and then, when deployed by some mechanism, needed to swing upward in the positive z-direction. 

After designing the idealized CubeSat, I ran multiple modal frequency analyses and structural analyses in order to make sure the CubeSat could withstand the proper loads. First, I ran modal frequency analyses, with fixed boundary conditions for a cantilever beam. The natural frequencies for the first four modes of the CubeSat are shown in the following table:

The above natural frequencies calculated in Fusion360 are very similar to the theoretical natural frequencies of a cantilever beam, given by the formula

Where “E” is the modulus of elasticity (also known as Young’s Modulus), and “I” is the area moment of inertia. This formula can be used to find the natural frequencies of a cantilever beam for any mode of vibration, “n”.

My results were also similar to other experimental results. For example, a study titled “Design, Analysis, Optimization, Manufacturing, and Testing of a 2U CubeSat” published in the International Journal of Aerospace Engineering performed a modal frequency analysis of a 2U CubeSat and found the following natural frequencies for the first four modes: 

These results are similar to the ones I found in my modal frequency analyses. 

After running the modal frequency analyses, I ran a few structural load analyses. The CubeSat frame had a honeycomb structure, which I modeled in Fusion360, and was made up of Aluminum 7075 material. The CubeSat needed to be able to withstand a maximum pressure differential of 15.2 psi (0.104 MPa) created by the Space Launch System (SLS) ascent into space, according to NASA’s Space Launch System Program’s White paper.

The maximum displacement of the CubeSat’s structure due to the applied force was 0.009 m, which is very low. Fusion360 calculates Von Mises stresses, and the maximum stress was 16.08 MPa, which is well under Young’s Modulus of Aluminum 7075 (71.7 GPa). The safety factor of the structure was 8+ everywhere on the structure, meaning the structure is much stronger than the  15.2 psi (0.104 MPa) load applied. 

After running the modal frequency and static stress analyses in Fusion360 and getting the desired results, the CubeSat was ready to be modeled as 3D printable parts and 3D printed with PLA on the FlashForge Creator Pro printer: 

The initial CubeSat design in Fusion360 had to be modified and broken up into different parts that were each 3D printable; after printing each part, I assembled them to form the whole CubeSat. I decided to break the initial design up into the following 3D printable parts: the top, the bottom, four separate side walls, four separate side rails, four deployable solar panels; finally, I needed to add four hinges plus four rods to attach each of the solar panels to the main structure (similar to a door hinge mechanism). This allowed the solar panels to rotate on a hinge from their initial position up to 180 degrees upward and back. Photos of the different 3D printed parts are shown below: 

After 3D printing all the necessary parts, I needed to assemble them. I modeled screw holes in Fusion360 on each 3D printed part in specific locations, so that I would not need to bore holes manually after the parts were printed. I ordered screws for plastic from McMaster Carr, so I knew the correct diameter and length for the screw holes I modeled in Fusion360. This way, the parts were ready to be assembled immediately after 3D printing. Images of the final assembled 1U CubeSat are shown below:

3D printing the final product was an iterative process, so I ended up assembling two different CubeSats entirely and printing a multitude of different versions of each part until I assembled the final product correctly. During the printing process I ran into many problems with the design of the parts, as well as issues with the printer itself. Some design problems included incorrect part sizes, incorrect screw hole placement, incorrect screw hole tolerancing/sizing, and incorrect dimensions of the overall assembled cube. Some printer issues included warping and two nozzle clogs. Some of my parts warped due to a lack of adhesion between the printer bed and the filament coming out of the nozzle, meaning the corners of these parts bent upward and were no longer usable. I solved this problem by reducing the heat of the 3D print bed to make sure the filament could cool down correctly on the bed. On a couple of occasions, parts would not print at all or the filament would come out tangled and would not stick to the bed; I solved this problem by taking apart the nozzle and manually unclogging it so that the filament could come out correctly. I also re-leveled the bed, to make sure the nozzle was close enough to the printer bed so that when the filament initially came out of the nozzle it would stick to the bed immediately. Photos of intermediate designs are shown below: 

Overall, this project was educational, challenging, and fun! I learned a new CAD software, Fusion360, which will be useful in the future, and I practiced my engineering design and 3D printing skills!

Overall, this project was educational, challenging, and fun! I learned a new CAD software, Fusion360, which will be useful in the future, and I practiced my engineering design and 3D printing skills!

Nuclear Fusion Power and Propulsion in the News

We just started our latest project for ARPA-E under the ARPA-E GAMOW program in which we will be build power amplifiers for fusion reactors. The goal is to lower the cost and increase the reliability of fusion reactor power electronics. We currently have grants under the DOE INFUSE program and another ARPA-E project that is part of the ARPA-E OPEN 2018 program. We just finished a NASA STTR grant to study the effects of plasma pulses on low temperature superconducting coils.

For those who have been following our work, you know that there are many articles and videos about our work. For your convenience, we’ve collected many of the URLs for them in this blog post.

2020 NJ Edison Patent Award (You’ll need to look for our award on this page. The others are interesting too!)

Popular Mechanics: The Direct Fusion Drive that Could Get Us to Saturn in Just 2 Years, Oct 21, 2020

Universe Today – Titan mission paper from Polito, Oct 19, 2020

ITER article: Space Propulsion: Have Fusion, Will Travel, July 15, 2019

Space Q, August 29, 2019 – Podcast replay of May 29, 2019 FISO

Space.com: Fusion-Powered Spacecraft Could Be Just a Decade Away, Mike Wall, June 11, 2019

PPPL press release: PPPL physicist receives funding to research improvements to unique fusion device, March, 2019

Could Tiny Fusion Rockets Revolutionize Spaceflight? June 12, 2017

Will Mini Fusion Rockets Provide Spaceflight’s Next Big Leap? Charles Choi, June 9, 2017

Video of DFD talk from DPSS 2017 (Facebook)

Essay by John G. Cramer, NIAC external council, June 30, 2016

NASA 360 video on Facebook, Pluto mission, June 13, 2017

Futurism: NASA-Funded Company Wants to Redefine Space Travel With Fusion Rockets, June 13, 2017

Princeton Field Reversed Configuration Fusion Reactor for Space Rocket Propulsion, Federal Lab Consortium award, 2018

Time.com: Going to Mars via Fusion Power, 2013