Frequency dependent target response

[Aziz]

Hi all,


How far is infinity? The magnetic flux surface integral for the external inductance Le of the straight wire must go from its center position (if you place it at x=0) to infinity.
I have a 1 m lenght of wire of lets say 10 cm wire radius. Integration from 0.1 m to 10 m enough? Or 100 m far way? Let's say 1 km ok? Or should I take 1 million km? Upto Pluto, that should be really save I think.
You can't imagine what comes out. The magnetic field diminishes with 1/r², where r = distance. But the flux area gets very very large which compensates 1/r² behaviour for the external inductance.
55.5 km to 555.5 km integration delivers approx. 1 million part of 1 µH.
555.5 km to 500555.5 km (half million km) delivers 1.84e-007 µH part. You see, where I should take my integration limits for 0.1 % accuracy. 100 m - 500 m far away for a 1 m wire length. For short wires (1-5 mm), which is usual, 10 - 50 m.

I have to find a better formula for the real internal inductance Li. Something, which can be calculated quickly instead of making it via EMF simulations, which take longer time to calculate.
This is the missing part of my inductance calculation. The self internal inductance of a straight wire piece (usually very short). This is missing in my inductance calculations due to the integration method, which is subtracting it to zero.
Or I make an accurate lookup table and interpolate it. Or find the formula by myself.
Then I can say, whether Maxwell (the god father of EMF) and the other gurus made it correct or not.

Aziz

[Aziz]

Hi all,

I haven't found a formula for the internal partial self inductance Li of a straight wire yet. All the known formulas are not accurate und not for my conditions made. It's pure bullsh1t to calculate with 50 nH/m internal inductance Li.
But I have found a very accurate formula for the external inductance Le (I get same EMF simulations results). I could try to form the complex formulas for the best total self inductance (Lt=Li+Le) by subtracting out the external part (Le). I hope, Wolfram Alpha can handle such complex formula so I can start with a basic formula and tune it with MS Excel Resolver by trail and error method, which will be provided with my own calculated data. But when I get the damn 50 nH/m figure as a result, then it's time to go back the to black board.

The external inductance formula can be found in a book: Inductance - Loop and Partial by Clayton R. Paul, Wiley
Page 207, formula (5.18a or 5.18b)
That's a nice book, which explains very clearly the development of the formulas.

It is so much time consuming checking all the publications.
Cheers,
Aziz

[Aziz]

Hi all,

where is the wall for my head again? One more Give me more!

I can not extract a good formula for the partial internal inductance Li for a straight wire, if one of the formula is obviously flawed (Lt not exact).
But the partial external inductance Le for a straight wire in the formula 5.18a/5.18b is really exact! All my EMF simulations convergate very well to the formula result.
Li = Lt - Le still delivers inaccurate results. So it doesn't make to find a fancy formula for it.
The fkn exact Li formula is a function of wire length and wire radius. Only two parameters.

I show you my results (see attachment). My calculated values are only 4 digits accurate.
I probably have a bug in my software. I'll have to check it again.

Cheers,
Aziz

[Aziz]

ups... attachment

[Aziz]

Hi all,

BTW, by splitting the internal Li and external Le wire inductance is a real chance to calculate the frequency dependent inductance. External part is frequency independent, where the internal inductance Li approaches 0 with high frequency. A simple adaptation of the internal inductance and we can take the frequency dependent target response into account. This is my goal. There a some complex formulas available. But all assuming constant internal inductance of Li/l = 50 nH/m at DC (0 Hz frequency = constant current, evenly distributed over the cross section of the round straight wire).
Now here ist the wall and me , getting confused heavily about the constant nature of 50 nH/m internal inductance per meter at DC and my calculated results.
Most possible reasons:
1. I'm doing something wrong. Completely wrong. -> Maybe I have to check the Ampere's Law and magnetic field (MF) implementation once more. The external inductance calculation seams to be ok and so the external MF calculation. The internal MF (Ampere's Law) is a good candidate for such problems.
2. All the other gurus are doing it right. Believe there formula. You aren't understanding it at all. -> Maybe.
3. My computer doesn't calculate correnct. CPU-Error? -> Maybe. There was several times of Intel processors doing it wrong. But I don't think this is an issue for my computer. The external Le seems to be correct.
4. Rounding problem occurs during billions and trillions of calculations. -> Maybe. But the external Le dominates even at integrating very weak MF at large distances and this seems to be correct.
5. The weather isn't fine for such calculations. Wait for a good weather. -> Definitely true.

So I'm going back to 1. and look for some hidden bugs. Weak bugs. Severe bugs.
I'm so close to solve the frequency dependent inductance calculations. This is, what I need at the end.

Cheers,
Aziz

[Aziz]

I forgot the Monte Carlo method.

The pseudo random number generator for the Monte Carlo method isn't an issue. I have my own implementation and it has already been tested for evenly distribution. For 30 years ago.
I am solving the complex surface integrals for the magnetic flux calculation with a bunch of random points in the flux area (well, millions of random points of course).

The result distribution of magnetic flux or inductance calculation (so the accuracy of them) can be exactly calculated (simple stochastics math). So the integration method works fine. The external inductance calculation Le shows this and its approximation behaves according to stochastics math.
Thats all for now.

[Aziz]

Hi all,

I have been thinking about the mysterious results. I have still no idea or know the reason for. But let's look at the external inductance (Le) formulas for a straight wire of lenght l and circular cross section of radius r.

The book ("Inductance - Loop and Partial", Clayton, formula (5.18a) on page 207) for the external inductance (Le) states:
Le = (µ0/2pi) * l * ( ln( l/r + sqrt( (l/r)² + 1 ) ) - sqrt( (r/l)² + 1) + r/l )
or
Le = 4pi*10-7/2pi * ..(rest of the formula)
Le = 2e-7 * l * ( ln( l/r + sqrt( (l/r)² + 1 ) ) - sqrt( (r/l)² + 1) + r/l )

Rosa ("The self and mutual inductances of linear conductors", publication year 1908 states in formula (9) for the total inductance (L):
L = 2 * ( l * ln( (l + sqrt(l² + r²)) / r) - sqrt(l² + r²) + l/4 + r)
Lets form Rosas formula a bit and subtract the internal inductance part of the formula to get the external inductance Le. The internal inductance ist the summand l/4 in the brackets. We leave it. And we take the 10-7 from µ0 into account.
Rosa Le = 2e-7 * ( l * ln( (l + sqrt(l² + r²) ) / r) - sqrt(l² + r²) + r)
Book: Le = 2e-7 * l * ( ln( l/r + sqrt( (l/r)² + 1 ) ) - sqrt( (r/l)² + 1) + r/l )

They look quite different but they shoud be the same by forming it. Lets form the books formula:
Le = 2e-7 * ( l * ln( l/r + sqrt( (l/r)² + 1 ) ) - l * sqrt( (r/l)² + 1) + (l*r)/l )
Le = 2e-7 * ( l * ln( l/r + sqrt( (l/r)² + 1 ) ) - l * sqrt( (r/l)² + 1) + r )
Le = 2e-7 * ( a - b + r )
a = l * ln( c )
b = l * sqrt( (r/l)² + 1)
c = l/r + sqrt( (l/r)² + 1 )
I have taken out the summands and replaced them by a, b and c for simplicity. Let's form them individually.

c = l/r + sqrt( (l/r)² + 1 ) = l/r + r*sqrt( (l/r)² + 1 )/r = (l + d ) / r, where d = r* sqrt( (l/r)² + 1 )
lets form d further
d = r* sqrt( (l/r)² + 1 ) = sqrt( r² * ( (l/r)² + 1 ) ) = sqrt (e), where e= r² * ( (l/r)² + 1 )
e = r² * ( l²/r² + 1 ) = l² + r²
put back e and d to get the c again:
c = (l+d) / r = (l + sqrt(e) ) / r = (l + sqrt( l² + r² ) ) / r
c = (l + sqrt( l² + r² ) ) / r

Let's form b now:
b = l * sqrt( (r/l)² + 1)
b = l * sqrt( (r/l)² + 1) = sqrt( l² * (r²/l² + 1 ) )
b = sqrt ( r² + l²)

Let's put b and c back into the formula above:
Le = 2e-7 * ( a - b + r )
Le = 2e-7 * ( l * ln( (l + sqrt( l² + r² ) ) / r) - sqrt ( r² + l² ) + r )

Lets look at Rosa's formula:
Le = 2e-7 * ( l * ln( (l + sqrt( l² + r² ) ) / r) - sqrt ( l² + r² ) + r )

Q.E.D. Ok, both formula are the same.

Rosa's internal inductance formula Li is (remember, we have left the (l/4) summand for the external inductance calculation):
Li = 2e-7*(l / 4) = 1e-7 * l / 2, for l=1 m for the unity wire length:
Li = 1e-7/2 = 0,050 µH or
Li = 50 nH

There we have it. My personal headache.

My own calculations approximate very well to the external inductance Le so it must be correct. Lets assume, I'm calculating the internal inductace Li totally wrong.
But if I take a look into the publication ("Improved Formulae for the Inductance of Straight Wires") and implement even the correction with Taylor Series method (formulas 49, 50, 51, and 52) to get a better inductance calculation, and subtract the calculated external inductance Le from it (formula above), I get nearly the same shape of the behaviour.


WTF? Let's assume, they calculate it right. But I get the same shape with a scaling factor to my calculations (see attachment on the graph above on the bottom left side).
Let's assume, there reference for the error calculation is not correct and they calculate the wrong way. But I get nearly same results behaviour.
Let's assume, there reference is wrong, they and myself are doing it wrong way. -> Rosa's internal inductance Li must be right.


So who is correct? I have headache.

I will get rid of the Monte Carlo method and will implement a much faster approximation algorithm so I can calculate with more accuracy and reduced time. We will see it.
Cheers,
Aziz

PS: I have tried with Latex formulas but some formulas won't work. So I have chosen the classic formulation.

[Aziz]

Hi all,

I have decided to implement the whole core part of the coil calculations. This will deliver much more accurate results and it will be prepared to take the frequency dependent inductance, magnetic flux and resistance of coils into account. And the missing part of the calculation (the self partial internal inductance) will be included.
This will take a while of course. My headache will arise once again for single turn circular loop coil of course. There isn't an exact closed form formula for it. Only approximation formulas or an exact formula only for the external inductance with elliptical integrals.

Cheers,
Aziz

[Aziz]

Hi all,

EM physics as known for 150 years or so must be redefined.
The mystery of my internal inductance calculation for a straight wire isnt a bug or mystery at all, it is a fact now.
The 50 nH/m inductance saga istn anymore valid. At least not in practical use.

More later

[Carl-NC]

I've lost track of what you are trying to do. I thought it was to model the frequency dependency of a target tau.

[Aziz]

Hi Carl,

I've lost track of what you are trying to do. I thought it was to model the frequency dependency of a target tau.

yep, I'm still working on it.

I'm following the strategy to split the internal and external inductance (Li, Le) of a coil arrangement. Applying at the end the Bessel/Kelvin functions to the internal inductance (Li) to model the frequency dependent inductance and resistance.

Now here is the crucial point. Former scientists made a simplification to derive the formula for the internal inductance calculation for a straight wire and we end up with the famous 50 nH/m internal inductance. Instead of calculating the magnetic field inside the conductor of finite length, they took the magnetic field calculation for infinite length straight wire. But the magnetic field for a finite length straight wire inside the wire diminishes to the end points, so it is not constant along the straight wire as it would be for an infinite lenght straight wire.

So the 50 nH/m internal inductance is only valid for long, very long (infinite long) straight wires with low wire diameter. The relation length to wire radis (l/r) must be very very large. In my magnetic field simulations it isn't the case at all. I have a wire radius which is much much larger than its length (a coil consists of small pieces of straight wires). Or at least l/r = 1.

Another crucial point. The frequency dependent inductance calculations are all based on the 50 nH/m internal inductance. So all the nice formulas for are being screwed too now.

The attached image below shows the cross section of magnetic field density for a straight wire of some lenght and thickness (relative thick to show the MF inside the wire):


At the center axis of the wire, there isn't a MF (B=0). To the ends, it diminishes. Now make the wire length smaller and it will end up with wrong calculations using the well-known classic formulas.

I'm surprized, that this issue isn't discussed in the literature.
I declare the 50 nH/m internal inductance for obsolete.

And all the nice other formulas too.

Cheers,
Aziz

[Aziz]

Hi again,

I've to clarify the discrepancy of my calculated figures to the published figures of the internal inductance. Someone is making it wrong. Either myself or the others.
I want to try a different way of MF calculation for a straight wire. BTW, I also must add a straight wire model into my coil software. The straight wire shown recently is only a piece of "round coil". So it is a work around.

The most critical part is the Ampere's Law inside the wire and at the surrounding ends of the straight wire.
I have found some sources for the different solution (formula), but some seems to be incorrect. Which one is the correct one? This is what makes me angry. Even Wikipedia has blatant misinformations. It's full of bugs.

Aziz

[Aziz]

Hi all,

the last EMF picture of a straight wire was buggy. I had the old calculation routines activated and they aren't accurate. I forgot to deactivate them.
The following EMF cross section of a straight wire is correct. You see the magnetic field density is much more critical at the ends of the wire. The outline of the straight wire is shown (black).

[Aziz]

Hi all,

Good news.
I have implemented and tested 3 different magnetic field (MF) calculations now. Two of them are the fast implementations. And one is the standard Biot-Savart implementation by dividing the conductor into very small pieces (at least 1000 pieces). They all deliver the same results now. I am sure now, that the MF calculation is corrent and the Ampere's Law inside the conductors works fine. And if I find a discrepancy, all the others do it simply the wrong way!

Here is the Biot-Savart with 1000 pieces divisions:


Here is the Formula 9 implementation:


And here is the new implementation:


Everything looks same and ok.

I'll continue with the implementation of different surface integral part methods, which convergate faster than the Monte Carlo method. And will do the calculations once again.
I'm sure, I can make a lot of well-known inductance calculation formulas obsolete. Even Maxwell failed! He simply failed! He and others failed! Totally failed!

Cheers
Aziz

[Aziz]

Hi all,

I'm tired to search for appropriate magnetic field cross section views of single straight finite length wires. The only one I have found is here:

Source: https://www.comsol.com/blogs/computi...ce-with-comsol

This is my take:

Oh well, I have tried to make the same color palette. It is approximately the same however.
There is some artifact and the ends of the thick wire. I haven't found how to solve it as it is nowhere specified how to handle or calculate it. I'll leave it for now until I have an idea.

Now, I'm really thick of bad physics, bad books, bad publications, bad formulas, bad wikipedia...it's is full of bugs, inconsistency, lack of further information, .... f!_!ck them all.

It is time for me to see more. I'll implement the current density model (for skin depth), surface current model (max. skin effect), etc..
Cheers,
Aziz