Calculating flux density in a flat conductor

[Al (astral highway)]

Hi folks,

Background

I'm building overcurrent latch-up circuitry to fire within just 1.5 time constants (<7uS) and am comfortable with my design.

Eventually, an overcurrent in the completed circuit may be of the order of 150A or higher, and this will be detected by my current transformer, burden resistor and full-wave rectifier circuitry, with an op amp doing comparator doing the magic and sending a pulse to a flip-flop (CLR). This will disable the driver ICs at a zero-current transition. (A separate current transformer and circuitry will send pulses to the flip-flop CLK pin, but this is just to illustrate the setting, not directly related to my question).

I have home-built the overcurrent detection current transformer, along with its burden resistor and full-wave bridge using 1N4148s, and tested the current transformer at a low current (120mA), with pulses from my home-made square-wave generator at the design frequency.

However, in view of this comparatively tiny current, I wrapped three turns around the CT to ensure sufficient coupling. This reduced the turns ratio from 1:100 to 1:33 appx. I adjusted my sums and the measured results - in millivolts of output from the full-wave bridge - are within 9% tolerance of the theoretical figures, which is acceptable.

The question arises in the next few lines, so please bear with me...

My query

The final design will have a 7mm earthing braid passing through the centre of the current transformer and the turns ratio will be 1:100

I am curious about the flux density for a flat conductor like this. Specifically, I'd like to model the spatial characteristics of the lines of flux. That is to say, I'd like to know the extent (size) of the magnetic field around the flat conductor to assess how it interacts with the toroidal transformer /current transformer. And in turn, I'd like to model the lines of flux generated in the toroid.

Expected fault currents exceed 150A, and pulses will be around 100A in normal operation. The internal diameter of the current transformer's windings is 18mm (flat earthing braid 7mm, please recall.)

I will set a potentiometer to precisely define the overcurrent trigger value.


A footnote

It may seem remarkable and implausible to be talking of currents approaching or in excess of 150A. However, this figure is based on 'as built' records of others using similar designs and these are peak, not average figures. For more robust and larger designs, peak currents are several times higher even than this.

My switches will be IGBTs with pulse current ratings close to 100A each. They may, however, be destroyed in less than 10uS if this is exceeded. Please can we just assume that my research on fault currents is detailed and reliable, thank you!

[kalee20]

I'm thinking that the current transformer will not make much difference.

Starting with your conductor and moving radially outwards, you find that the flux density due to your 150A is given by B = μ0 I / (2 x pi x r).

But then, you encounter the current transformer toroid. The inner portions of the turns will have 1.5A flowing (because it's 100:1) and so the 100 bits of wire cancel out the field due to the 150A, giving you zero field in the toroidal core. (I've assumed it is a good CT and loaded with a sensibly low burden).

Then as you move farther out still, you encounter the bits of wire on the OUTSIDE of the toroid. With 100 of these, all carrying 150A, you have the same field strength you started with.

[cmjones01]

On one of the products I've designed, we use current transducers (actually hall effect, but still with a toroidal magnetic core) which sit round a 50mm-squared cable. Their inner diameter is also about 18mm, like yours. The cable routinely handles currents of up to 250A. The transducers work, and the readings they give appear to be unaffected by the position of the cable within the toroid. From this I surmise that any effects due to variations in the flux density around the wire are negligible.

Chris

[Al (astral highway)]

Originally Posted by kalee20

I'm thinking that the current transformer will not make much difference.

That's good to know! thank you!

Originally Posted by kalee20

(I've assumed it is a good CT and loaded with a sensibly low burden).

Yes, the burden is 1.8R. The CT is wound on an amorphous alloy toroid - I have the characteristics somewhere at hand.

[Al (astral highway)]

Originally Posted by cmjones01

On one of the products I've designed, we use current transducers (actually hall effect, but still with a toroidal magnetic core) which sit round a 50mm-squared cable.... The cable routinely handles currents of up to 250A. ... the readings they give appear to be unaffected by the position of the cable within the toroid. From this I surmise that any effects due to variations in the flux density around the wire are negligible.

Hey Chris, that's really interesting and relevant. Thank you.

What sorts of frequencies are your current transducers good for?

[Al (astral highway)]

Originally Posted by kalee20

Starting with your conductor and moving radially outwards, you find that the flux density due to your 150A is given by B = μ0 I / (2 x pi x r).

Hey, an afterthought... there seem to be so many symbols in magnetics!

Is mu0 (I don't know the keyboard shortcut) the constant for the permeability of vacuum in space? If so, what is it? Otherwise the rest of the equation makes sense. Is it one of Maxwell's?

[trobbins]

Practically, you want the braid to be symmetrically in the middle of the CT aperture, and to that end, you may even be able to "U" bend the braid to make the conductor even more symmetric.

How have you approached the skin effect of the braid? Is the burden suitably non-inductive and located with minimal lead inductance?

[Al (astral highway)]

Practically, you want the braid to be symmetrically in the middle of the CT aperture, and to that end, you may even be able to "U" bend the braid to make the conductor even more symmetric.

I’m thinking of having a short round brass insert to pass through the two CT cores: the braid can’t be attached for both ends.

I don’t follow your concern about but the braid and skin effect. I’ve tested earthing braid at the design frequency and its structure of overlapping wires makes a nice, low inductance pathway for ground currents. This part of the circuit has the braid grounded at one end.

[cmjones01]

Originally Posted by astral highway

Hey Chris, that's really interesting and relevant. Thank you.

What sorts of frequencies are your current transducers good for?

I'm honestly not sure. The current is theoretically DC but in practice it's got a lot of sharp-edged 100Hz pulses in it. Measurement is complicated by the fact that the current sensors themselves are only energised and read for a few milliseconds every couple of seconds to save power (yes, it does seem paradoxical to be saving power when there's 250A passing through the cable!). The thing I'm confident of is the end result, which is that when integrated, the output of the current sensor accurately represents the average DC current in the cable, which is what matters in this application.

I also have a whizzy Tektronix A6303 current probe (http://w140.com/tekwiki/wiki/A6303) which I'd like to compare with our current sensors some day and see how they really behave.

Chris

[trobbins]

Hopefully I'm seeing this correctly as a single wire (eg. the braid) passing through the centre aperture of an annular CT.

The current in the wire is pulsed, and the delay and rise time of the resulting CT winding pulsed voltage across the burden resistance must be as short as possible (ie. a high frequency response).

The wire impedance hopefully remains constant well in to the MHz range so as not to influence the current rise-time. And yes a multifilar braid would cover that concern.

The link includes a field distribution simulation for a CT designed for probably a higher frequency response than you were targeting:
https://www.research.manchester.ac.u...ransformer.pdf

[Argus25]

Originally Posted by astral highway

Is mu0 (I don't know the keyboard shortcut) the constant for the permeability of vacuum in space? If so, what is it? Otherwise the rest of the equation makes sense. Is it one of Maxwell's?

Al,
u0 is simply the permeability of space or a vacuum, in the M.K.S system of units it is assigned the value of 4.pi x 10^-7. In the old system of C.G.S units it was unity. But the 4.pi kept turning up in expressions that related to rectilinear systems, so it appeared illogical and there was a drive to get rid of it.

The magnetizing force H, due to a long straight conductor is I/2.pi.r, but the flux density B is always uH, where u is the permeability of the medium.

So the flux density due to a current in a conductor in a vacuum is B = u0.I/2.pi.r as noted by Kalee20.

However , if the medium is altered with another substance, the equation can be re-written as:

B= Uo.uR. I/ 2.pi.r

Where uR is the relative permeability of the new medium. For example a chunk of ferrite, like that use for a transistor radio antenna rod, has a uR of about 125, but it could be as high as 2000 for some materials.

One of the most wonderful equations relates permeability (from magnetostatics) with permittivity (from electrostatics) with the speed of light C. The permittivity of space is about 8.85 x 10^-12.

C^2 = 1/ ( permeability x permittivity)

That statement in itself suggests light must be electromagnetic in nature if the speed of light can be calculated from only a magnetic and an electric constant and that makes the speed of light in a vacuum a constant too.

[kalee20]

Originally Posted by trobbins

Practically, you want the braid to be symmetrically in the middle of the CT aperture, and to that end, you may even be able to "U" bend the braid to make the conductor even more symmetric.

How have you approached the skin effect of the braid? Is the burden suitably non-inductive and located with minimal lead inductance?

Actually, it should not make any difference.

The calculations are much easier if you have the conductor central in the toroid. But if it's not, the overall result is the same because it is independent of the path around the conductor. You could have a toroid with 100 turns on, resting against the conductor, very off-centre. You could have a star-shaped core with 100 turns on. You could have a toroid 500 yards in diameter with 100 turns on. You'd still get the same result.

For the same reason, skin effect will make no difference - all that matters is that the electrons pass through the hole in the toroid SOMEWHERE.

Originally Posted by Argus25

One of the most wonderful equations relates permeability (from magnetostatics) with permittivity (from electrostatics) with the speed of light C. The permittivity of space is about 8.85 x 10^-12.

C^2 = 1/ ( permeability x permittivity)

That statement in itself suggests light must be electromagnetic in nature if the speed of light can be calculated from only a magnetic and an electric constant and that makes the speed of light in a vacuum a constant too.

It's diversifying slightly - but that was what Maxwell did. He knew the formula for inducing a voltage due to a changing magnetic field, it has the permeability constant in it (which had been measured in the lab by measuring forces due to known currents). He also knew the formula for creating a current due to changing electrical field (which has the permittivity constant in it, also well known numerically). Playing around, he put the two equations together, and derived a third equation which looked just like the equation for a travelling wave. When he calculated the velocity of this wave, he saw it was close to a number he already knew - the velocity of light. So he deduced that light is electromagnetic in nature. It was a brilliant piece of work.

[Al (astral highway)]

Originally Posted by Argus25

... if the medium is altered with another substance, the equation can be re-written as:

B= Uo.uR. I/ 2.pi.r

Where uR is the relative permeability of the new medium. For example a chunk of ferrite, like that use for a transistor radio antenna rod, has a uR of about 125, but it could be as high as 2000 for some materials.

Hi Hugo,

That's really interesting, thank you. I will look up the data sheet for the amorphous alloy toroid I'm using and see if this property is listed. Hold on... yes! It's 'F' type material and the figure is 3,868U

Originally Posted by Argus25

One of the most wonderful equations relates permeability (from magnetostatics) with permittivity (from electrostatics) with the speed of light C...

C^2 = 1/ ( permeability x permittivity)

That statement in itself suggests light must be electromagnetic in nature if the speed of light can be calculated from only a magnetic and an electric constant and that makes the speed of light in a vacuum a constant too.

Now,as you say, that's a wonderful equation indeed and a great thought to latch on to and mull over. Thank you!

[Al (astral highway)]

Originally Posted by kalee20

Actually, it should not make any difference.

The calculations are much easier if you have the conductor central in the toroid. But if it's not, the overall result is the same because it is independent of the path around the conductor. You could have a toroid with 100 turns on, resting against the conductor, very off-centre. ... You could have a toroid 500 yards in diameter with 100 turns on. You'd still get the same result.

For the same reason, skin effect will make no difference - all that matters is that the electrons pass through the hole in the toroid SOMEWHERE.

This is reassuring, thank you. I kept mulling this and mulling this last night and my instinct was this very conclusion, but I couldn't justify it.

So I was almost minded to create a low-inductance transmission line inside the dead centre of the toroid, perhaps using solid brass or a curled up piece of copper sheet, just to test the results against an off-centre piece of braid.

So your reassurance has saved me unnecessary crafting and helped to clarify what's going on in this mysterious toroidal space.

Toroids, I realise, are incredibly complex shapes with a whole load of analytical geometry to explain them - beyond any of my training and nevertheless a fascinating separate subject!

Originally Posted by kalee20

.Maxwell ...derived an ...equation which looked just like the equation for a travelling wave. When he calculated the velocity of this wave, he saw it was close to a number he already knew - the velocity of light. So he deduced that light is electromagnetic in nature. It was a brilliant piece of work.

Sure was! Seems Maxwell is more of less known in circles like this but is overshadowed by Einstein in common knowledge.

Illustration:

If anyone is interested, here is the waveform, from three turns wrapped around my home-made CT (so 1:33) with a 1R8 burden resistor, full-wave rectified and developed across a load of 100R and a capacitor of 10nF.

(My 'scope is pretty basic and although it has a nominal trigger threshold of 500uV, in practice it struggles to sync with anything less than 5mV)

The input signal is from my home-made square wave generator, in series with a 47R resistor and the four turns wrapped round my CT. This signal is tiny compared to the overcurrent detection pulses I'll be monitoring.

I wonder that the stray inductance in the ground connection of my scope lead is having an effect on the apparent waveform.

[Argus25]

Originally Posted by astral highway

Now,as you say, that's a wonderful equation indeed and a great thought to latch on to and mull over. Thank you!

Al,

Also, if the electromagnetic wave is in another medium, say coax cable for example then the wave velocity is:

Velocity^2 = 1/( inductance per unit length of the coax x capacitance per unit length of the coax)

so from that you can see what the permeability u0 and permittivity k0 constants really are, they are effectively the inductance and capacitance per unit length of empty space.

On top of this, the wave impedance of space is:

root(u0/k0) = 120.pi = 377 Ohms which is E/H.

E/H is the ratio of the Electric field intensity (units volts per meter) to the magnetic field intensity (units amps per meter) of a an electromagnetic wave travelling in space, which is what you get in the far field from a radio transmitter.

So it is interesting much like E= mC^2, it seems that very simple equations and constants link all the important things in our reality/universe together.

[Ed_Dinning]

Hi Al, skin effect is primarily resistive, so it will affect the resistive part of the busbar impedance, but should have little affect on the inductive part and hopefully not much affect on the rise time.

Ed

[emeritus]

Re #6, the following are some of the symbols you should be able to get by holding down the <ALT> key and typing the number on the numerical keypad (typing the numbers using the ordinary keyboard does not work). This came up in a thread a while ago, and works if your computer has been set to use the United States character set. If you get accented characters instead of Greek ones, it will be because yours has been set up to use the Multilingual Latin character set.

π 227
Ω 234
φ 237
± 241
√ 251

[Craig Sawyers]

Originally Posted by Argus25

One of the most wonderful equations relates permeability (from magnetostatics) with permittivity (from electrostatics) with the speed of light C. The permittivity of space is about 8.85 x 10^-12.

C^2 = 1/ ( permeability x permittivity)

That statement in itself suggests light must be electromagnetic in nature if the speed of light can be calculated from only a magnetic and an electric constant and that makes the speed of light in a vacuum a constant too.

When my son was in 6th form (er 15 years ago...) I said "What is u0" he knew "what is epsilon 0" he knew. I was impressed. "OK, multiply them together, do 1/x and then square root it". There was a pause "That's the speed of light - HOW DID YOU DO THAT"

It is a wonderful equation. Almost as good as taking two trancendental numbers and the square root of -1 and getting

exp(i pi) = -1

[trobbins]

Hi Kalee20,

my intent was to indicate where ideal performance could start to degrade, given that the response time of the current measurement may well need to be sub 1us if other delays are also then present. That pushes measurement bandwidth in to the MHz range, and that is significantly beyond normal commercial current sensing CT modules used in smps where the peak current is skyward of 100A. We haven't been informed yet of what the specs are of the CT.

We haven't seen the circuit to which the braid is used to pass current, but I'd be thinking most such power stages conducting in excess of 100A have pretty low on-resistance, and the braid may end up being one of the larger resistive connection path elements. I don't think we know the pulse current frequency yet, or if it mainly sinusoidal or square'ish.

[Al (astral highway)]

Originally Posted by Argus25

Al,

Also, if the electromagnetic wave is in another medium, say coax cable for example then the wave velocity is:

Velocity^2 = 1/( inductance per unit length of the coax x capacitance per unit length of the coax)

so from that you can see what the permeability u0 and permittivity k0 constants really are, they are effectively the inductance and capacitance per unit length of empty space.

On top of this, the wave impedance of space is:

root(u0/k0) = 120.pi = 377 Ohms which is E/H.

E/H is the ratio of the Electric field intensity (units volts per meter) to the magnetic field intensity (units amps per meter) of a an electromagnetic wave travelling in space, which is what you get in the far field from a radio transmitter.

Once again, Hugo, this is fascinating terrain and I will dignify it with appropriate thought. It's incredible. Thank you! I will also tap out some wave velocities. This is wonderful new world of awareness!

Thank you also Ed for this info, Emeritus for the keyboard short cuts and Craig for your story!