Skin effect and magnetic field of a flattened conductor?

[Treva31]

A conductor with a large high frequency AC current causes a "Skin Effect" meaning that most of the current flows at the surface of the conductor.

What does the skin effect look like if you have a conductor with one side flattened?

And what does the resulting magnetic field look like?

See attached image.

 

[Babadag]

There is a way to calculate the skin effect of a-entirely-flattened conductor. I have not a simply answer for partial flattening. In my opinion a compromise between the two value could be fair enough. See:

Fig. 13-6. Curves for skin effect of isolated flat rectangular conductors

http://www.aluminum.org/sites/default/files/aecd13.pdf

 

[Baluncore]

Imagine many current filaments trying to keep away from each other while flowing along the outside of the 'D' shaped conductor. That means there will be a slightly greater density of current filaments near the tight corners of the 'D' than along the flat face of the 'D'. That will increase the local field near the corners.

The distance between field lines is inversely related to field strength. The one dotted magnetic line you draw will be far from the flat face of the 'D” due to the weaker field there, it will run closer to the arc face of the 'D' where more current filaments flow, while it will pinch in close to the sharp corners where the magnetic field is strongest due to the local concentration of the current filaments.

So your diagram of the flattened conductor with circular field is probably closer to reality than the flattened field.

 

[Baluncore]

Where current is concentrated at the tight external corners of your 'D' there will also be a locally steeper electric field. That follows because the intrinsic impedance of free space, Z₀, fixes the ratio of electric to magnetic field strength outside the conductor.
E/H = Z₀ = 4e-7 * Pi * c = 376.73 ohms.You have reached a precipice where most of what you have been taught about electrical energy transmission is about to be overturned. You will not fall because you are at the bottom of the precipice, but you may like to climb it. I can assure you, the view from the top is quite spectacular.

You may see that electrical energy does not travel with the electrons in the wires, but in an external EM field that is guided by the surfaces of the conductors making the circuit. The voltage between conductors sets the electric field strength. The current that flows on a wire is really just a proxy for the surrounding guided magnetic field. The cross product of the electric, E, and magnetic, H, fields gives the direction of energy flow for the circuit, which near the circuit is always in the same direction, towards the load. It seems nonsensical at first, but much more energy flows in the insulation than in the conductors that guide the wave.

Skin effect. You may also notice that any current flowing into the conductor is lost as it diffuses only very slowly into the conductor, at about walking pace. The period of the wave and the speed of diffusion decide the effective skin thickness. Anything flowing deeper is out of time and fossilised, canceled with the previous half cycle, or lost in wire resistance as heat. The better the conductor, the thinner the skin and the less the energy losses. Good conductors have very thin skins, make very good reflectors of EM waves, and so are shiny.

 

[Treva31]

You have reached a precipice where most of what you have been taught about electrical energy transmission is about to be overturned.

Wow you're not wrong, interesting stuff!

 

[Treva31]

OK following on from that, which of these shapes would induce the most current into the blue conductor?
And perhaps rank them in order?
I would assume A D B C E.The orange conductors are coiled around the blue conductor.
They are hollow with a coolant being pumped through them.

Assuming all orange conductors have the same:
Cross sectional area.
Very high AC frequency and very high current.
Gap to the blue conductor.

Only the shape is different.

 

[Treva31]

Cross sectional area.

I mean the same cross sectional area of conductor/copper, they have different internal cross sections of coolant.

 

[Baluncore]

Firstly, none of the profiles shown would induce a current across the page along the blue conductor, because the coils are all perpendicular to the conductor. A magnetic field would run parallel to the blue conductor, not a current.
Remember that a current induces a perpendicular magnetic field. That magnetic field in turn induces a perpendicular current in the original conductor that opposes the original current. It is a back EMF since two left 90° turns makes it face the other way. i² = –1. That is what makes conductors reflective.

An eddy current would be induced around the surface of the blue conductor. So if you want to build an induction heater, you are going the right way about it geometrically. For induction heating the clearance and conductor profile are not critical. There would be little improvement gained by deviating from ordinary round copper tube. More clearance would see heating spread along a greater part of the blue conductor, as would profile your E.

If you can explain the application, or the geometry of the current you want induced in the blue conductor, it may be possible to give you a better answer.

 

[Treva31]

Yes sorry I am thinking of an induction coilgun so your description is exactly what I mean. The blue conductor is a hollow aluminium projectile in a barrel of wound coils. So as you say the magnetic fields induce eddycurrents in the projectile which in turn lead to motion of the projectile.

I have read that the gap between the coils and the projectile has a large effect on efficiency.
So my idea is to concentrate as much magnetic energy as close to the projectile as possible.

 

[Baluncore]

There are many constraints.
How much does the aluminium projectile weigh ?
What minimum velocity do you need the projectile to achieve ?
What is the initial temperature of the aluminium projectile ?

 

[Jeff Rosenbury]

Though the energy flows in the fields, the losses come in the copper. The fields induce currents, causing I²R losses.

So for the loss of resistance in the flattened cable, I would estimate the loss of conductor due to the "overlapping" cross-sectional area at the two corners. Then I would derate the cable by that amount.

In field terms, this represents the crowding of the magnetic fields near the corners (which drive a higher current through those areas).

 

[Treva31]

There are many constraints.
How much does the aluminium projectile weigh ?
What minimum velocity do you need the projectile to achieve ?
What is the initial temperature of the aluminium projectile ?

The efficiency of the conductor shape depends on those factors?
Perhaps you are alluding to my suspicion that it may not be possible to put that much energy in without melting the projectile?Well anyway I would say:

How much does the aluminium projectile weigh ?
100kg total, 50kg of which is aluminium, 25mm thick walls

What minimum velocity do you need the projectile to achieve ?
11km/s

What is the initial temperature of the aluminium projectile ?
-200 degrees C using liquid hydrogen, also it contains 20kg of water which may work as a heat-sink

 

[Jeff Rosenbury]

The efficiency of the conductor shape depends on those factors?
Perhaps you are alluding to my suspicion that it may not be possible to put that much energy in without melting the projectile?Well anyway I would say:

How much does the aluminium projectile weigh ?
100kg total, 50kg of which is aluminium, 25mm thick walls

What minimum velocity do you need the projectile to achieve ?
11km/s

What is the initial temperature of the aluminium projectile ?
-200 degrees C using liquid hydrogen, also it contains 20kg of water which may work as a heat-sink

I'm assuming this is a thought project to check on rail guns for orbital insertion. In other words, don't try this at home.

You might consider using ice initially so you get the heat of fusion bonus for cooling. Even so I think you will end up venting steam. This isn't bad since any such system needs some "in space" thrust to achieve a stable orbit.

BTW, traditional explosives can achieve nearly this velocity using a two stage gas gun. There was a gun fired, low orbit project called "HARP", (High Altitude Research Project). The technical details are out there for anyone with funding.

The U.S. Navy seems to have solved the projectile heating problem since they are deploying rail guns. I have no idea how.

Me, I would use a ceramic projectile with a spiral, metal covered channel. I would then plan on flash heating the metal to get a plasma as the primary conductor.

There are lots of good ideas about getting mass into space cheaply. None of them have the military potential of missiles, thus have little funding for development.

 

[Baluncore]

My analysis is not of a rail-gun, but of a linear induction motor.

The reaction forces on the coils during a launch are proportional to the mass of the projectile being accelerated. The coils will be subjected to an impulse load as the projectile passes, while the projectile will have a continuous acceleration. The projectile will be resistively heated during launch, just as in an induction furnace. Will the aluminium capsule hold together when hot? or will it simply burst due to internal pressure?

If you are designing a single shot electromagnetic launcher, the coils will not need water cooling because it will all be over before they have time to be cooled. Their immediate thermal capacity will be very important. Internal water cooling could be dangerous as it may be flash heated faster than circulated. There is certainly a disadvantage in non-circular conductor profiles when it comes to launch flexing failure, or from the internal pressure of flash heated cooling water. External air or water cooling might be better as it cools the outer surface where the currents flow. The dielectric constant of external water may need to be turned to advantage.

The kinetic energy transferred to a projectile must get into all the mass of the projectile and payload without melting it. A high speed bullet melts or vaporises along with the target on impact, it then penetrates as a kinetic thermal lance. Getting energy into a bullet throws up the same problems as getting energy out of a bullet, without melting it. You will need a superconducting sheath on the surface of the bullet, and a very long launcher. I would expect something many kilometres long. Any special sheath material will not be recyclable as it will be ablated on contact with the atmosphere.

 

[Treva31]

Thank you both for your replies.
I think this thread has been answered but feel free to message me.
To answer your other comments in order:Jeff:

I am thinking of Linear Induction Coilguns not Railguns. Yes for orbit insertion.

The payload would start as ice and have the volume/pressure to allow it to melt. How fast it will heat up and how much pressure it will be under to expand into steam is indeed an issue to be addressed.

Yes I have looked into the HARP gun and its proposed successor the QuickLaunch system.

US Navy guns fire at "only" 2.5km/s as far as I know. I believe they simply accept some ablation from atmospheric heating.

A ceramic projectile with an embedded coil is a good idea and I am considering this as an option if the heating is too great of an issue. I was thinking of using superconducting wire cooled immediately before firing, it is surprisingly cheap.

I agree there are lots of ideas around. My focus is on creating a commercially attractive business plan based purely on simple and proven technology.Baluncore:

Again I agree internal pressure is something that needs to be addressed.

The idea is to fire salvos of projectiles having the coils powered the whole time, so cooling will be required. An alternative to water is possible or a system of having the coolant external.

Round conductors certainly seem like the way to go for their physical strength (with the bonus of low cost and simplicity).

Another idea I have is to encase the coils in a thermally conductive concrete to help with physical stress on the coils. Plus lengthening the barrel will mean less current per coil and so less stress.

As mentioned above superconductors are an option. I have read that in an Linear Induction Motor a shorted coil would actually be more effective than a solid projectile. So SC wire embedded in a ceramic projectile may be better (and cheaper since it is already commercially available) than a sheath.

Yes I am envisioning a minimum of 1km. Many times longer than that is not unreasonable if it solves significant problems.

 

[Baluncore]

An induction motor operates at roughly 95% of synchronous speed. Since the rotor rotates with the field, the rotor relative frequency of the currents induced in the rotor are much lower than the supply frequency, they are usually only a couple of Hz. That lower frequency results in slower changes and so currents have time to flow deeper in the rotor. The effective skin depth is therefore greater in the rotor than in the field windings.

100% of synchronous speed is not possible because then the torque is zero, and the frequency of current in the rotor is zero. Zero frequency precludes the virtual transformer action where the rotor can be seen as a secondary.

Where a linear induction motor is used to accelerate a projectile, there will be an optimum phase velocity that will need to increase along the track as the projectile accelerates. The pitch of the coils cannot be changed much because it is related to the length of the projectile, so the frequency of excitation will need to increase. I think it would be optimum when the same frequency eddy currents were induced in the projectile throughout the launch acceleration process. That will at all times come close to fully utilising the projectile conductor material.