# That’s a big transistor

As promised last time, a fun-for-Friday rerun from the early days of FAIC. But before we get started, a quick physics refresher.

Force is the ability to change the velocity of an object by a certain amount in a given amount of time. One newton (N) of force is the ability to change the velocity of a one kilogram object by one meter per second, in one second. The earth applies a force of gravity of 9.8N on every 1kg mass near it.

Work is the application of a force to an object as it moves a certain distance. Energy is the ability to do work. One joule (J) of energy is the ability to apply a force of one newton to an object as it moves one meter. [1. Note that the force has to be applied in the direction of the motion for it to count as work; the earth’s gravitational force does no work on a sideways-moving object. This should jibe with your intuitive understanding of work; it is a lot harder to raise an object by 1 meter than it is to slide it 1 meter, where the work is done by the force overcoming friction.]

Power is the rate at which energy is consumed in time. One watt of power is the consumption of one joule of energy per second.

Charge is, like mass, a fundamental property of matter. The easiest way to manipulate charge is by manipulating electrons. Charge is measured in coulombs (C). Current is the movement of charge and is measured in amperes (A); one ampere is one coulomb of charge moving past a given point in one second.

Electric potential, better known as voltage, is to charge as gravity is to mass, and is measured in volts. Applying a potential of one volt to one coulomb of moving charge consumes one joule of energy. A better way to think about it though is to divide both sides of that equation by time and get that one volt times one amp is one watt.[2. volts times amps equals watts is one of the funamental equations you have to know in order to wire a house safely. If your house power is providing a potential of 120V and your light bulb is consuming 120W of power then the current on the wire supplying the bulb is 1A. Since lighting circuits typically use wires that are safe for up to 15A, this puts a limitation on how many bulbs you can have on one circuit.]

Resistance is the tendancy of an electric conductor to resist the movement of charge, and is measured in ohms (Ω). If there is a conductor where the difference in voltage between the two ends is one volt, and the resistance is one ohm, then there will be a one amp current in the conductor.

A few years back a bunch of my coworkers and I got to discussing the space program over lunch. Someone asked why it is that we continue to launch devices into orbit by strapping a big old tank full of liquid oxygen to the device and then set it on fire. Why haven’t we developed better technology using magnets or something?

I did some poking around the web and wrote up a little summary of the analysis, which I present here for your amusement.

Suppose you’ve got a 1000 kilogram object that you’d like to be an arbitrary height above the surface of the earth. In order to get an object to an arbitrary height, we need to accelerate it to the escape velocity of the earth, which is 11 kilometres per second. How much energy does that take? The kinetic energy of a projectile is half the mass times the square of the velocity:

kinetic energy = 0.5 x 1000 kg x (11000 m/s)2 = about 60 gigajoules

Sixty billion joules of energy sounds like rather a lot, but really it isn’t. The electrical energy consumed by your house is measured in joules, but because joules are so tiny, the bill comes in kilowatt-hours. A kilowatt-hour (kWh) is a power consumption of 1000 joules per second for 3600 seconds, so 1 kWh = 3.6MJ. That costs about ten cents in Seattle. So that sixty billion joules would only cost about \$1700 — near what it costs to keep 20 one hundred watt light bulbs burning for a solid year. And power costs less if you buy it in bulk.

\$1700 to put a tonne in orbit is incredibly cheap. Why doesn’t NASA use electricity instead of chemical power? Turning the electricity into acceleration quickly is harder than turning it into heat and light slowly, but surely we can figure something out.

One way to do it would be to build a coilgun. Coilguns are very easy to build — as we saw last time, I’ve built one myself. A coilgun works like this: you have a metal projectile sitting in a tube. One end of the tube has a coil of wire wrapped around it. When an electric current is applied to the wire, it turns the coil into an electromagnet, which pulls the projectile towards it.

Of course, the current has to shut off before the projectile passes entirely through the coil, otherwise the electromagnet will be pulling the object back towards it, slowing it down.  The projectile sails on through the coil, out into space.

You can build multi-stage coilguns. The first coil gets the projectile moving. As the projectile leaves the first coil, that triggers a switch that turns on the next coil, and so on. Each coil makes the projectile a little bit faster.

So a coilgun stage has three basic parts: the coil itself, something which stores the electricity – usually a capacitor of some kind – and a switch to connect the power source to the coil at exactly the right moment.

I once had an old television set that didn’t work anymore. Before I took it to the dump I ransacked it for capacitors. I had everything I needed around the house — a few capacitors, a diode, some old telephone wire, a piece of spare kite spar tubing, and a light switch. Run wall power through the diode to charge the capacitors, run power from the capacitors to the coil, interrupted by a switch. Put a nail in the tube, hit the switch, and the nail goes flying. Which is cool.

The problem is that none of these pieces scale up.

Suppose we’ve got a really huge multi-stage coilgun. Just to keep the math easy, let’s say we have ten thousand coils, each a metre long with which we’re going to accelerate our projectile to 10000m/s by applying a uniform acceleration of 5000m/s2. That acceleration would squash humans like bugs, but we could use this thing to move large quantities of equipment into orbit. When the on switch is hit, a mere two seconds later the projectile will be heading into space at 10000m/s. [1. Rockets with people on them typically accelerate at 20m/s2, which of course applies a total force to the astronauts of three times regular gravity; the earth’s gravity doesn’t go away. But rockets have the entire height of the atmosphere to accelerate in; we’re assuming here that the coilgun is only ten kilometres long. Likely it is built by boring a mountain.]

Consider only the last coil of the ten thousand. The coil is one meter long. The projectile is going to be moving at almost 10km/s, so it will only be in the coil for about 100 microseconds. We can assume that this is about how long the last coil is going to be energized.

It is vitally important that the pulse of electricity delivered to the coil be short, for three reasons. First, as I mentioned before, the magnet had better be off by the time the projectile reaches the far side, otherwise the magnet will be slowing the projectile down. Second, the farther the projectile is from the coil when the coil is energized, the weaker the magnetic pull will be; you don’t want to waste power by turning the magnet on while the object is too far away to get much oomph from it. And most important, the magnetic field strength of an electromagnet is proportional to the electric current. Current is charge moved per second, so if you want high current you have to either make the number of electrons moved larger, or the amount of time you spend moving them smaller, or preferably both.

For all these reasons we can assume that the pulse is going to be extremely short, on the order of 100 microseconds.

Current is voltage divided by resistance, power is voltage times current, and the heat produced by resistance is a function of current. [2. These facts explain why we use high-voltage power lines to deliver power; we step up to high voltage so that we can move lots of power without creating too much heat. We then step it back down to more useful low voltage at local transformer stations.] The resistance in copper wire is going to be small but not zero, and worse, the magnetic fields set up in the electromagnet are going to themselves push against the electrons moving through the coil.[3. We could use superconducting materials that have no resistance, but it turns out that they often don’t take high currents easily.] Making magnetic fields produces an electric potential that pushes charge back in the direction it came from, lowering the current; this kind of induced resistance is called impedance. The closer together the coils, the stronger the electromagnet will be, but the more impedance will be produced. In order to get enough current we’re going to need absolutely huge voltages.

If the coils generate heat, or need to be cooled because they are superconductors then we can cool them with liquid nitrogen and make sure that we don’t pump so much power into each coil that they melt. I’m not too concerned about the heat in the coil.

But there is another place where heat is produced. 100% of the current for each coil has to pass through a switch. A conventional switch, where you have two pieces of metal with a gap between them, and you remove the gap by triggering a spring to slam the two metal plates together,[2. This is apparent in lightswitches found in older homes, where you can actually feel that you’re loading up a spring when you flip the switch, and hear the plates slamming into each other. Modern switches are much lighter and quieter.] isn’t going to work for the kind of power we’re talking about. We’re talking about currents way larger than arc welders use; a conventional switch would just weld itself shut. Even without the welding problem, the timing is still an issue: as the switch closes, the high voltages will cause current to leak across the channel when the switch is half closed, thereby increasing the amount of time that electrons are flowing. That will weaken the magnetic field. Mechanical switches just aren’t going to cut it.

Fortunately we have a very fast, very precisely controllable switching technology: transistors. A transistor is a chunk of silicon which has been carefully constructed so that it can act as either an electrical insulator or an electrical conductor. They are incredibly high-speed switches — that’s why we make computers out of them. Let’s just build ten thousand transistors, one for each coil switch. The switch for a coil will be triggered by the projectile interrupting a laser beam crossing the previous coil.

Hold on, a minute though. Transistors are semiconductors — they do not transmit electricity perfectly. They’re maybe 90% efficient. About 10% of the power transferred through the switch is going to be turned into heat inside the switch. How much heat can a transistor take before its wrecked? Anyone who has overclocked a Pentium knows what I’m talking about! Transistors get real hot real fast when you try to push a lot of power through them.

How much power? Well, power is energy per second, energy is the ability to do work, work is force applied through distance, and force is ability to accelerate mass, so we have:

power = mass x acceleration x distance / time

By the time we hit the last coil, the 1000kg projectile will be moving at about 9999.5m/s. We need to accelerate it up to 10000m/s, we have 100 microseconds in which to do so, and one metre. The acceleration is 5000m/s2. If we work it out, the total power required by the last stage is fifty billion watts, but of course we only need it for a ten-thousanth of a second, so that’s only five million joules of energy, or a little under 2kWh. The five million joules cost about a dime if it was on your electrical bill; it’s getting them where they need to be in 0.0001 seconds that’s the hard part.

We’re going to turn 10% of that energy directly into heat in the switch, so that’s five billion watts of heat energy to dissipate. Imagine the heat of ten million 500W spotlights concentrated in a small area, and all turned on for 100 microseconds.  It’ll be hot.

Note that we’re assuming that 100% of the force generated by the magnetic field is translated into motive force on the projectile. In reality, only about 25% of the magnetic force actually turns into acceleration.  We’d need to up the power by at least a factor of four, so really we’ve got more like 20GW of heat energy to dissipate. But let’s ignore that for now.

Let’s assume that the transistor is extremely thin and therefore the transistor is equally hot everywhere. An extremely thin transistor is also a good idea because a thin transistor has large surface area. The larger the surface area an object has, the faster you can cool it. Let’s wrap our transistor in a huge copper heat sink.

Slabs of copper do not diffuse heat infinitely quickly. The temperature of the surface of the heat sink that is touching the transistor will rise based on many factors — higher power consumption, smaller surface area and longer application of heat by the transistor all cause a larger temperature rise at the interface between the transistor and the sink. The heat capacity and thermal diffusion of copper are well known, and from these facts we can work out that the temperature rise in degrees Celcius is equal to about 0.6 times the wattage, divided by area in square centimeters, times the square root of the time in seconds:

temperature rise at surface in ℃ = 0.6 x ( Watts / cm2) x √seconds

Clearly if the heat sink is not sucking heat out fast enough, the heat sink is going to itself get hot enough to wreck the transistor. Let’s suppose that the transistor can take a rise of 100 degrees Celcius before it is destroyed. We have 5 billion watts of power to dissipate and only 100 microseconds to get rid of as much of the energy it as possible.[1. After the switch is off, we don’t really care how long it takes to cool; we only care about how cool we can keep it during the short time that current is flowing.] How much area do we need to ensure that the surface of the heat sink rises only 100 degrees?

Solve the above equation for area and you get 300 000 square centimeters. That’s a square transistor 5.5 metres on a side, about the size of a typical kitchen floor.

Transistor-grade wafer-thin silicon costs about \$2 for a square centimeter, so the transistor for the last stage alone will cost us about \$600 000 dollars. And that’s for the stage that only adds 0.01% of the total oomph, for a device that can only launch a one-tonne projectile, and we’ve neglected inefficiency in the coil. Clearly to build the whole device would cost multiple billions of dollars for the on switch alone. That’s an expensive on switch!

Of course, we could solve these problems by inventing new magic materials. If we had cheap superconductors that conducted electricity and heat with 100% efficiency at reasonable temperatures, supported large currents and fast switching, then sure, we could build mass drivers that put objects into orbit at low cost. Tragically, we do not have magic materials.

And of course I haven’t even mentioned the difficulties of generating and storing enough power to light Seattle and then discharging it all in two seconds. Generating large quantities of power spread out over lots of time and space is easy. Concentrating that power into a very tiny time and space is hard. Fifty gigawatts for the last stage alone — we’d need the Mr. Fusion from Back To The Future![3. Actually, we’d need around 40 of them, as Mr. Fusion generated the mere 1.21 gigawatts required to power the flux capacitor.] I’ve also glossed over many serious difficulties in the implementation of the coils; dealing with self-inductance, magnetic eddys and other issues is non-trivial.

Until there are radical new advances in material science and power management we’re going to be stuck with strapping big tanks of liquid oxygen onto the sides of projectiles if we want to get them into space.

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I stole this argument from this excellent, more technical description of the switching problem:

http://www.islandone.org/LEOBiblio/SPBI1SI.HTM

This site is a good one if you’re interested in exotic orbital technologies:

http://www.islandone.org/LEOBiblio/

This guy builds tabletop coilguns with switches rated to 14000 amps at 1300 Volts. I would imagine that these are not cheap.

This page has some great explanations of the magnetic and electric physics involved in coil guns.

## 16 thoughts on “That’s a big transistor”

1. I think you end up an order of magnitude or so off in your thermal figuring, since it’s not necessary for the transistor to dissipate anything near 50GW; it’s merely necessary for it to absorb the amount of energy it will have to dissipate in the brief moment it’s on; after that the energy may be dissipated at leisure.

Also, I’m curious about the need to switch off the coil quickly. While it’s true that one would want to have the coil de-energized by the time the projectile leaves, any energy left over in either the coil or capacitor at that point isn’t going to make it to the projectile, so why store it to begin with? Do you know if there’s some problem with trying to size components so as to minimize residual energy [some types of engine have inherent efficiency limits, such that it’s impossible for them to transfer power to the output without having energy left over; do you know if coilguns have such a problem?]

• It’s an amazing article for all the internet visitors; they will obtain benefit from it I am sure.

2. You are calculating the cost of the energy, but ignoring the demand charge, which in the Seattle area is about \$7/kW. That’s kW of peak demand, not kWh. You are using 30 million kW for 2 seconds which gives you a total electric bill of \$210,001,700.

• That’s what the capacitor is for. To charge up over a longer period and then release in just a short time.

3. What if instead of building a straight coilgun, a circular one is built? This would avoid the power problem by making the projectile accelerate in (say) 24 hours instead of 2 seconds (which would also make it possible to ship people with the rocket! Acceleration would be much slower).

Unfortunately, the centrifugal force experienced by the contents of the rocket at the time of “release” would be too high. If we replace the 10km long tube by a 10km permitere circle, the acceleration experienced right before opening the loop would be about 76000 ms^2. To reach astronaut-acceptable levels of 20ms^2 acceleration, the loop would have to go all the way around the equator.

• We do have a multi-km-long circular coilgun: the Large Hadron Collider! And the forces involved in keeping those particles going in a circle are considerable.

4. I think it would have been useful to explain what a coilgun is prior to the last post, where you explained how to build one.

Anyway, I think maybe the problem here is that you’re trying to use a transistor. Surely one would use a tube like a thyratron, or more likely a large bank of them, no?

I think making this change brings the problem from the physically impossible realm to the merely impractical realm.

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5. Great post. This less technical article on the same subject is pretty great too. You’ll recognize the author, I’m sure.

6. Furthermore, those numbers won’t get your projectile to orbit, because they neglect air resistance. 11 km/s at sea level doesn’t give you 11 km/s at the top of the atmosphere, or even at 10 km up.

Calculating the actual velocity needed would at minimum require knowing the shape of the projectiles, though they’re probably reasonably aerodynamic given the existing specs, and the location of the launch site and thus the air pressure at the mouth of the coilgun.

If the projectiles aren’t going straight up, which would require the ability to dig a 10 km vertical tunnel and have loading and launch facilities at the bottom of the tunnel, that increases the amount of time in relatively thick atmosphere to punch through.

This is another reason for those huge tanks of liquid oxygen: some of the acceleration can be applied in flight, rather than needing to accelerate to rather more than 11 km/s on the launchpad so the projectile will be at escape velocity when it needs to be.

• In fact, a body traveling at 11 km/s at sea level would most likely burn up and perhaps explode, much the way objects behave when they fall into our atmosphere. That’s why space travel entails putting the vehicle into low-earth orbit first, and then achieving escape velocity, if necessary, from there. See the note at the bottom of this Wikipedia section: http://en.wikipedia.org/wiki/Escape_velocity#List_of_escape_velocities. But note that the original requirement was to “launch devices into orbit”; not only does this not require *reaching* escape velocity, it requires *not reaching* it, since, at escape velocity, the object would not circle the planet at all, but fly away from it. Still, none of these considerations changes the substance of Eric’s argument; they only change the magnitudes of some of the numbers.

7. You make a claim at the beginning of the article that is relevant to coilguns, but is strictly speaking dead wrong.

You say, “Suppose you’ve got a 1000 kilogram object that you’d like to be an arbitrary height above the surface of the earth. In order to get an object to an arbitrary height, we need to accelerate it to the escape velocity of the earth”.

Sorry, but nonsense!

A space elevator (http://en.wikipedia.org/wiki/Space_elevator) would, in principle, let you walk(!) into geostationary orbit and beyond. And if you want to achieve an arbitrary height, place your rocket/coilgun/solar sail/ion drive/whatever up there, and amble to pretty well anywhere.

• Excellent point. All we need is a space elevator. However, if we had the impossible non-existing materials to build a space elevator, odds are good we’d also have the impossible non-existing materials to build a fast, cheap multi-gigawatt switch too.

8. Is anyone reading this page patient enough to recalculate Eric’s project requirements with the following changes?

a) replace transistors of 90% efficiency with micro-relays of (unknown, assume 99.9%) efficiency –
e.g. device 1 with switching speed 100 µs:

Click to access thielicke.pdf

or with device 2 with “contact resistance as low as 0.015Ω”:
“Micromachined relays with liquid-metal wetted contacts”

b) replace escape velocity at sea level (11km/s) with (unknown lower, to be calculated) escape velocity from high altitude, e.g. from Cerro Chajnantor. This should also reduce air friction, as half of atmosphere would be below you.

Looks like you might want a “Traveling wave railgun” – patent US20130015295 A1.

Update: ouch – just recalculated – looks like escape velocity from Cerro Chajnantor would be 10.99km/s.