# Induction and flux linkage clarification

So if you have a single inductor then I assume that there is no flux linkage and that L = flux*N / i = L11
and so that flux is total leakage flux. If you have two inductors then the flux leakage is any flux that isn't flux linkage (the self flux is flux leakage).

(flux21 is the flux in 2 from 1, flux11 is the flux in 1 from 1 etc.)
According to my book the coupling coefficients are:
k1 = flux21/flux11
k2 = flux12/flux22
k = sqrt(k1*k2)
And mutual inductance = L12 = L21 = k*sqrt(L11*L22)

What I'm wondering is that how do you calculate flux12 or flux 21? And that value for inductance of:
N2/reluctance
is that the total inductance of L11 + L12 ?

And if you have two inductors how do you calculate flux11 and flux22 and their respective linkages?

If you have a transformer can you say inductance of side one is: L = NBA/i
(where N is turns of that side, i is current of that side and BA is the flux going through that side)
Is that the total inductance given to you by that formula, L11 + L12 and how do you calculate wich is wich flux, like what the leakage flux is?

Dammit I was hoping to get some interesting feed back on this.

Baluncore
2021 Award
And if you have two inductors how do you calculate flux11 and flux22 and their respective linkages?
Where you have two remote coils the mutual inductance will be low. Where you have a transformer the coupling will be close to one. The closest coupling between two inductors is when they share a common magnetic core.

The inductance of L1 is the sum of the coupling of all it's geometrical parts. The same is true for L2. The mutual inductance between L1 and L2 can be solved algebraically for most simple geometries. You have not identified the geometry of your inductors or their relative positions. For that reason there is no simple answer to your question. Can you specify the geometric configuration of your two inductors?

I would suggest you find a copy of “Inductance Calculations” by G.W.Grover. Use http://www.bookfinder.com/ to locate a low cost copy, or maybe a new Dover Publications reprint.

The inductance of L1 is the sum of the coupling of all it's geometrical parts. The same is true for L2. The mutual inductance between L1 and L2 can be solved algebraically for most simple geometries. You have not identified the geometry of your inductors or their relative positions. For that reason there is no simple answer to your question. Can you specify the geometric configuration of your two inductors?
.

Ok I am concidering 3 different geometries, two are torroids, wound identically, from the top view and the third one is just bifilar coils, I suppose you could imagine a small magnetic core in the centre of the bifilar coils if you wanted:  Say geometry 1 is on an angle of zero degrees and geometry 2 is offset by an angle of 180 degrees. I imagine that if they were 90 degrees offset there would be no mutual flux linkage as they would be perpendicular. I don't know how to calculate the coupling coefficients from the geometries but I imagine that geometry 1 would have more flux linkage than geometry 2 (as are functions of 'd') I woul guess it would be double the flux linkage of geometry 2.

As for geometry 3, from what I have read (but don't really fully understand) that if you just took L to be the inductance of winding A or B, equal to:
'N' (of A or B) squared / Reluctance
Then this series layout adds to yeild a total inductance of Ltotal = (LA + LB)*M = (L + L)*2 = 4*L, because the mutual coupling coefficient is 2 (how? I don't know), which kind of makes sense, I suppose there would be no stray inductance, but still sounds a bit high, too good to be true, unless the reluctance was something really small, not sure what, maybe the centre diameter of the wire length with a permittivity of air or if there was a core in there.
Thanks

Baluncore
2021 Award
I assume your toroids have magnetic cores. The thing about a toroid is that the magnetic field is contained within the magnetic core. Geometry 1. If you stack two toroidal cores, each with it's own winding, then there will be very little coupling between the two windings. Geometry 2. The same containment applies, so there is little coupling.
http://en.wikipedia.org/wiki/Toroid...tal_B_field_confinement_by_toroidal_inductors

The third is a bifilar pancake coil. Because the windings are spread the coupling is not as good as you might expect. These can be complex to calculate, (see chapter 17 of Grover). There is also a correction needed because of the thickish insulation between the windings. When you connect the two filaments in series you simply double the number of turns which gives you four times the inductance of either filament by itself with the other open circuit.

I assume your toroids have magnetic cores. The thing about a toroid is that the magnetic field is contained within the magnetic core. Geometry 1. If you stack two toroidal cores, each with it's own winding, then there will be very little coupling between the two windings. Geometry 2. The same containment applies, so there is little coupling.
http://en.wikipedia.org/wiki/Toroid...tal_B_field_confinement_by_toroidal_inductors

Hmm, say that the toroids were more like an almost closed 'U' rather than a totally confined B-field, or a rectangle, there'd be more flux linkage then?

The third is a bifilar pancake coil. Because the windings are spread the coupling is not as good as you might expect. These can be complex to calculate, (see chapter 17 of Grover). There is also a correction needed because of the thickish insulation between the windings. When you connect the two filaments in series you simply double the number of turns which gives you four times the inductance of either filament by itself with the other open circuit.

I haven't been able to get Grover yet, but I did find the attached .pdf of 'inductance calculation' excert from it (haven't had a good read yet even though its small).
Yes, that did cross my mind, that it is just like N = 2*Na = 2*Nb
so even if you connected the pancake like this: The inductance would be the same? because the number of turns is the same? and its like [2^2 * N^2 ] / Reluctance
If so, what's the point of putting them in series rather than parallel?
Thanks

#### Attachments

• Grover_inductance calculation.pdf
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Baluncore
2021 Award
If two wires, joined at the ends, are wound together to form a spiral inductor, the inductance will be close to that of one wire only, while the resistance will be half of one wire only.
In parallel you are making Litzendraht wire. http://en.wikipedia.org/wiki/Litz_wire

If two wires, joined at the ends, are wound together to form a spiral inductor, the inductance will be close to that of one wire only, while the resistance will be half of one wire only.

Really good point! So what was the advantage of winding it out of two wires as opposed to more turns of one wire?

P.S did you think much of that .pdf?

Baluncore
2021 Award
P.S did you think much of that .pdf?
It is the errata for Grover's book. I made the corrections to my copy a couple of years ago.

I don't know where you found the bifilar pancake picture, or who drew it and why. You can wind a bifilar coil twice as fast as two mono-filar coils. Maybe they wanted a centre tapped coil to couple with another nearby coil.

Baluncore
2021 Award
Take a look at this archive. https://archive.org/browse.php?field=subject&mediatype=texts&collection=NISTJournalofResearch

Some of the papers written by or used by Grover in writing his book are listed below. I have renamed them but original filename is at end of new filename.
They are available here; https://archive.org/details/NBSBulletin

BBS Vol 2 no 2. Self—Inductance of Single—Layer Coils. Rosa. calcul216118719063131unse.pdf
BBS Vol 4 no 1 Inﬂuence of Frequency on Coib inﬂue416117819077676cohe.pdf
BBS Vol 4 no 1 Self—inductance of Circles. Rosa,Cohen. onself414915919077575unse.pdf
BBS Vol 4 no 2 B&W Inductance of ﬁnear conductors. Bureau of Standards, E.B.Rosa 1907.pdf
BBS Vol 4 No 2 Inductance of Linear Conductors. Rosa. selfmu430134419088080unse.pdf
BBS Vol 4 no 2. The Self and Mutual Inductances of Linear Conductors. Rosa selfmu430134419088080unse.pdf
BBS Vol 5 no 1. Formulae and Tables for the Calculation of Mutual and Self—Inductance formulae5113219089393rosa.pdf
BBS Vol 8 no 1 Formulas for Mutual and Self—Inductance formul812871912169169unse.pdf
BBS Vol 8, No 1. Formulas for Self and Mutual Inductance, Sci169noerr.pdf
BBS Vol 12 Effective Resistance and Inductance of Iron and Bimetalic Vlnres. Miller. eff122072671915252252mﬂl.pdf
BBS Vol 17, part 2 Capacity Between Inductance Coils and the Ground. Breit. effectsofdistrib17521unse.pdf
BBS Vol 18 Inductance of Polygonal Coib. Grover formulaetablesfo18737unse.pdf
BBS Vol 19. Skin effect in solenoids, Hickman_Sci472.pdf
BBS Vol 21, Inductance of a helix, Sci537.pdf

Take a look at this archive. https://archive.org/browse.php?field=subject&mediatype=texts&collection=NISTJournalofResearch

Some of the papers written by or used by Grover in writing his book are listed below. I have renamed them but original filename is at end of new filename.
They are available here; https://archive.org/details/NBSBulletin

BBS Vol 2 no 2. Self—Inductance of Single—Layer Coils. Rosa. calcul216118719063131unse.pdf
BBS Vol 4 no 1 Inﬂuence of Frequency on Coib inﬂue416117819077676cohe.pdf
BBS Vol 4 no 1 Self—inductance of Circles. Rosa,Cohen. onself414915919077575unse.pdf
BBS Vol 4 no 2 B&W Inductance of ﬁnear conductors. Bureau of Standards, E.B.Rosa 1907.pdf
BBS Vol 4 No 2 Inductance of Linear Conductors. Rosa. selfmu430134419088080unse.pdf
BBS Vol 4 no 2. The Self and Mutual Inductances of Linear Conductors. Rosa selfmu430134419088080unse.pdf
BBS Vol 5 no 1. Formulae and Tables for the Calculation of Mutual and Self—Inductance formulae5113219089393rosa.pdf
BBS Vol 8 no 1 Formulas for Mutual and Self—Inductance formul812871912169169unse.pdf
BBS Vol 8, No 1. Formulas for Self and Mutual Inductance, Sci169noerr.pdf
BBS Vol 12 Effective Resistance and Inductance of Iron and Bimetalic Vlnres. Miller. eff122072671915252252mﬂl.pdf
BBS Vol 17, part 2 Capacity Between Inductance Coils and the Ground. Breit. effectsofdistrib17521unse.pdf
BBS Vol 18 Inductance of Polygonal Coib. Grover formulaetablesfo18737unse.pdf
BBS Vol 19. Skin effect in solenoids, Hickman_Sci472.pdf
BBS Vol 21, Inductance of a helix, Sci537.pdf
Thank you very much for the papers, I'll follow them up tomorrow.

As for were I got that picture, it was just from the wiki page (as you might have guessed, sorry, I remember your past advice) :
http://en.wikipedia.org/wiki/Bifilar_coil
It states "Some bifilars have adjacent coils in which the convolutions are arranged so that the potential difference is magnified (i.e., the current flows in same parallel direction). Others are wound so that the current flows in opposite directions. The magnetic field created by one winding is therefore equal and opposite to that created by the other, resulting in a net magnetic field of zero (i.e., neutralizing any negative effects in the coil). In electrical terms, this means that the self-inductance of the coil is zero." And I think the picture is from Tesla's patent.
Which to mean indicates the design is for more than just the convenience of centre-tapping.

Thanks!

Baluncore
2021 Award
Back in Tesla's day, low value resistors were wire-wound. That usually meant they were also inductive. Inductance was not a problem with DC circuits, but it was with AC. By connecting the two filaments in series so the current flowed in both directions, most of the inductance could be cancelled.

The pancake construction allows the W = I2R heat dissipated in the resistance to escape quickly.

Back in Tesla's day, low value resistors were wire-wound. That usually meant they were also inductive. Inductance was not a problem with DC circuits, but it was with AC. By connecting the two filaments in series so the current flowed in both directions, most of the inductance could be cancelled.

The pancake construction allows the W = I2R heat dissipated in the resistance to escape quickly.

Interesting, ok well my main question was whether there was any special coupling interaction between: parallel-wound serise connected coils compared to a standard single coil with the same amount of turns. But from what you're saying there isn't and there is other reasons as to why the layout was is used.
I do have two side questions though about what you raised about the number of turns: comparing a parallel-wound, series connected set of coils to a parallel-wound, parallel connected set of coils, what is it about the potential difference between turns that distinguishes each turn as being another turn? As opposed as to when they're parrallel and just appear to be different strands of the same turn?
Furthermore, if you connect a bifilar set of coils from one connection to the other, does the value of 'N*I' remain the same with N and I changing in proportion, or would one set have a larger N*I?

Thanks

Baluncore
2021 Award
Where multiple filaments take the same path through a changing magnetic field, they will have identical voltages induced in them by the changing field. Consider two wires, each has a voltage V induced in them by the changing magnetic field. The magnetic field is proportional to the ampere * turns. Inductance obeys the relationship V = L * di/dt, therefore L = V * dt/di

In parallel the voltage is once, but the current is twice. Lp = 1 * dt/2 ∝ 1 / 2 = 0.5
In series the voltage is twice, but the current is once. Ls = 2 * dt/1 ∝ 2 / 1 = 2.0
So series has 4 times the inductance of parallel, as is expected from N2 relationship.

Where multiple filaments take the same path through a changing magnetic field, they will have identical voltages induced in them by the changing field. Consider two wires, each has a voltage V induced in them by the changing magnetic field. The magnetic field is proportional to the ampere * turns. Inductance obeys the relationship V = L * di/dt, therefore L = V * dt/di

In parallel the voltage is once, but the current is twice. Lp = 1 * dt/2 ∝ 1 / 2 = 0.5
In series the voltage is twice, but the current is once. Ls = 2 * dt/1 ∝ 2 / 1 = 2.0
So series has 4 times the inductance of parallel, as is expected from N2 relationship.
Ok, so NI will remain the same but the inductance will change. But what I meant when I asked:
comparing a parallel-wound, series connected set of coils to a parallel-wound, parallel connected set of coils, what is it about the potential difference between turns that distinguishes each turn as being another turn?
Irrespective of why there is moving charges flowing around the coils (magnetic induction or battery) there seems to need to be to be a potential difference between coils for them to count as different turns. So when you have them in parallel, they just look like a copy of eachother and don't count as extra turns (although as we discussed there is more current). I'm just wondering what it is about this potential difference that distinguishes between turns.

Baluncore
2021 Award
For the same di/dt, terminal voltage is proportional to inductance. V = L * di/dt.
If N turns are in series then the N voltages add, so the inductance increases by N times due to series connection.
When in parallel, there is no change in voltage, but for the same ampere turns the inductor terminal current is N times greater.

For the same di/dt, terminal voltage is proportional to inductance. V = L * di/dt.
If N turns are in series then the N voltages add, so the inductance increases by N times due to series connection.
When in parallel, there is no change in voltage, but for the same ampere turns the inductor terminal current is N times greater.
That's something to think about, are you saying that if there is a coil of N turns, throughout each N1 + N2 ...+ NN, there is a slightly different di/dt for each turn at that point as the current flows down the wire?
So V = L1*di/dt +...+ LN*di/dt
Assuming all the points of inductance for each turn are the same.

Baluncore
2021 Award
are you saying that if there is a coil of N turns, throughout each N1 + N2 ...+ NN, there is a slightly different di/dt for each turn at that point as the current flows down the wire?
No, current is the same in all turns and so is di/dt, if that was not the case it would be a transmission line. The inductor must be treated as an N x N matrix of mutual inductance between turns. The self inductance of the winding is the total.
One thing about transformers and inductors is that they are optimised to operate close to a particular voltage per turn.

No, current is the same in all turns and so is di/dt, if that was not the case it would be a transmission line. The inductor must be treated as an N x N matrix of mutual inductance between turns. The self inductance of the winding is the total.
One thing about transformers and inductors is that they are optimised to operate close to a particular voltage per turn.
Yes, I think I see what you're saying. And increasing the diameter of the wire, with another strand (in parallel) doesn't increase inductance, so in a way 'N' is really more of an arbitrary thing.

Baluncore
2021 Award
And increasing the diameter of the wire, with another strand (in parallel) doesn't increase inductance,
It will actually lower the total inductance very slightly because the self inductance of the wire will be less. That is because the bigger wire will have parallel current filaments that are further apart, so there is less "self coupling". A single filament has greater self inductance than a tape or sheet.

It will actually lower the total inductance very slightly because the self inductance of the wire will be less. That is because the bigger wire will have parallel current filaments that are further apart, so there is less "self coupling". A single filament has greater self inductance than a tape or sheet.
I've been reflecting on your last statement because I find that detail a revelation and extremely fascinating, but do you think you could please indulge me and elaborate a little bit more, I'm just not seeing exactly why. Pretend I'm a complete idiot...but not too much.

Thanks

Baluncore
2021 Award
The closer that parallel currents flow to each other, the greater is the coupling of their fields, hence the greater is their mutual inductance.
The inductance of a short wire can be modelled as many short segments. The sum of the mutual inductance of all the short segments plus the self inductance of each of those short segments gives the total inductance of the wire.

At low frequencies the current flows throughout the cross sectional area of a round wire. At high frequencies, because of skin effect, the current can be treated as flowing only in the surface skin of a round wire. The inductance of a wire is therefore frequency dependent because the Geometric Mean Distance, GMD, of a circular area from itself is different to the GMD of the circumference of a circle from itself.

When a conductor has a shape that is not a single filamentary conductor, the self inductance and mutual inductance between conductor segments must be adjusted for the change in self and mutual GMDs. Litz wire and flat tape are used to wind real inductors for high frequencies because the resistance can be lower due to greater surface skin area to volume ratio. A Litz wire bundle can be treated as being a round area of current at high frequencies. But tape has not only lower self inductance but also lower mutual inductance between turns due to the significantly different GMD between turns. That makes the computation of inductance more difficult for coils that deviate from a simple geometry.

See; F. W. Grover, “Inductance Calculations”, Chapter 3 covers the computation of GMDs for different sections.
Grover is essential reading for anyone interested in real inductors.

See the paper by E. B. Rosa, “On the Geometrical Mean Distances of Rectangular Areas and the Calculation of Self-Inductance”, in Bulletin of the Bureau of Standards, BBS Vol 3, 1907.
https://archive.org/search.php?query=geometrical mean distance AND mediatype:texts AND collection:americana

The Bulletin of the Bureau of Standards, now called NIST, is a gold mine of inductance computation.
https://archive.org/search.php?query=Inductance AND collection:americana

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When you said: "the bigger wire will have parallel current filaments that are further apart, so there is less "self coupling"."
What I want to clarify about that was 'so if you have two single strand conductors, the further appart they are (GMD) the less inductance they have because of the distance, I understand that. But say
you have conductor A, it has a self inductance, then you add another one, so conductors A and B are running parallel toughing eachother, then they have a self and coupling inductance. Does this second sinario always lower the total inductance from just having conductor A on it's own? Or is it at a surtain GMD separation of the two that results in a lower self inductance in A and B rather than just having A?

The closer that parallel currents flow to each other, the greater is the coupling of their fields, hence the greater is their mutual inductance.
The inductance of a short wire can be modelled as many short segments. The sum of the mutual inductance of all the short segments plus the self inductance of each of those short segments gives the total inductance of the wire.
How could a wire have a mutual inductance with itself? Because the magnetic field is perpendicular to the way the wires running (the next small segment)

When a conductor has a shape that is not a single filamentary conductor, the self inductance and mutual inductance between conductor segments must be adjusted for the change in self and mutual GMDs. ... But tape has not only lower self inductance but also lower mutual inductance between turns due to the significantly different GMD between turns. That makes the computation of inductance more difficult for coils that deviate from a simple geometry.
I am interested in real inductors, so I will be looking into those links, thanks. I sort of see how a flat tape could have a lower mutual inductance, through some sort of strange GMD, however I've never seen an example of tape for GMD. But why would it have a lower self inductance??
Thanks

Baluncore
2021 Award
Does this second sinario always lower the total inductance from just having conductor A on it's own? Or is it at a surtain GMD separation of the two that results in a lower self inductance in A and B rather than just having A?
If the parallel connected round wires carry half the current each then they have less inductance together than one round wire of the same diameter would have. If you increased the diameter of the one wire so it's GMD with itself was the same as the GMD of the two parallel wires, then the inductance would be the same. That is the elegance of GMD. The GMD applies both between the close turns of wire in an inductor and between the virtual filaments of current in a single conductor.

How could a wire have a mutual inductance with itself? Because the magnetic field is perpendicular to the way the wires running (the next small segment)
Think of a wire as being filled with many filaments of current, the self inductance is the mutual inductance between all those filaments. A thin wire has higher inductance than a thick wire or metal tape.

I've never seen an example of tape for GMD. But why would it have a lower self inductance??
As above, the cross section of a very thin tape is a straight line. Investigate the GMD of a line with itself to understand why a wide surface has lower inductance than a single thin filamentary conductor.

Don't expect inductance computation to be simple. It is not. Read the references, or abandon the field.

The Electrician
Gold Member
After reading all those NBS papers from the early 1900s, one has to be impressed. Those people did all their computations with log tables.

At the bottom of page 18 of Grover's book, he says concerning the GMD of equal rectangles with sides parallel:

"The general formula for LOGe R is known, but involves many terms and is ill suited for computation." The formula he refers to is formula 8 in the paper referenced:

https://archive.org/search.php?quer... AND mediatype:texts AND collection:americana

Grover managed to do all the calculations to produce table 1 and table 2 on pages 19 and 20 of his book, with only a single error! Most of the other important tables (Table 13, 21, 36, etc.) are essentially error free, with only a few rounding errors. If only he had something like Mathcad, or even a scientific calculator, imagine the savings of his time.

• Baluncore
Baluncore