# FeaturedI Magnetic flux is the same if we apply the Biot Savart?

1. Oct 6, 2017

### mertcan

Hi, initially I am aware that magnetic flux is conserved due to divergence of magnetic field is zero and for long solenoid, magnetic field inside is uniform. So magnetic field intensity (B) at point P,Q,S, T,R,U(in the attachment) must be same. But my question is : can we obtain the same magnetic field intensity at point S or R again at the point P or U using biot savart rule for constant current? I am asking because it seems to me that for instance if we apply biot savart at point S then we will obtain different magnetic field intensity from point U applying biot savart again( because point U is far away from current loops but point S is closer to current loops...)

Last edited by a moderator: Apr 18, 2018
2. Oct 6, 2017

### Staff: Mentor

Are you familiar yet with the difference between Magnetizing Inductance Lm and Leakage Inductance Lk?

3. Oct 7, 2017

### mertcan

Thank you for your return @berkeman
Yes, I am familiar with those notions( I am aware magnetic flux lines may not be conserved 100 percent, iron core is being used to maximize that flux). But I wonder that without using conservation of flux can we obtain the same similar result just using biot savart rule in order to find magnetic field intensity for point P and U????

Last edited: Oct 7, 2017
4. Oct 7, 2017

In applying Biot-Savart, you also need to apply it to the magnetic surface currents from the magnetized iron. This problem with the air-gap is somewhat unique and the equation $\oint H \cdot dl=N I$ is used. Let me give you a "link" to this problem which previously appeared on Physics Forums: (need to look for it=should have it momentarily=here it is: https://www.physicsforums.com/threads/absolute-value-of-magnetization.915111/ See also the "link" in post #2 of this "link" that has the Feynman lectures derivation of it. $\\$ For a problem that treats the magnetic surface currents, see https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/

Last edited: Oct 7, 2017
5. Oct 7, 2017

@mertcan Please see the edited post above: I found the "links" I was looking for.

6. Oct 7, 2017

### mertcan

7. Oct 7, 2017

Just an additional comment or two: The equation $\oint H \cdot \, dl =NI$ is quite exact, but the solution of this involves a couple of approximations based on the fact that because $\nabla \cdot B=0$ that the lines of flux of $B$ are conserved. Thereby, especially for a small gap, the assumption is made that $H$ takes on two values: $H_1$ in the magnetized material and $H_2$ in the gap. This assumption that $H_1$ is constant for the material, essentially assumes that $M$ is also constant in the material, because $B$ is assumed to be continuous and constant, and that $M$ takes on a value given by $M=\chi_m H_1$. The magnetic surface currents from this magnetization, along with the $B$ from the current in the windings, can be computed from Biot-Savart for a cylindrical geometry. For this rectangular geometry, any Biot-Savart calculations are much less precise, and the assumptions above are normally used to solve the problem.

8. Oct 8, 2017

### vanhees71

You can treat everything, i.e., magnetic fields from currents as well as those from magnetizations (ferromagnets) by using (in Heaviside Lorentz units):
$$\vec{j}_{\text{tot}}=\vec{j}_{\text{charges}}+\underbrace{c \vec{\nabla} \times \vec{M}}_{\vec{j}_\text{mag}}.$$
Note that at a surface of a ferromagnet this implies a surface-magnetization current.

9. Apr 14, 2018

### mertcan

@Charles Link While I was reviewing your answers above, I saw you mentioned the surface currents, and I deem that if surface currents exist then time varied magnetic field must exist. BUT my question is: when we have NO time varied magnetic field ( no time varied current), magnetic field intensity (B) at point P,Q,S, T,R,U still be same BUT now it is easy to apply biot-savart rule because no surface currents exist and if we apply biot-savart I do not deem that magnetic field intensity (B) at point P,Q,S, T,R,U be same because of distance difference between points???? What can you say about that situation??

10. Apr 14, 2018

The magnetic surface currents are present even in the static case, and occur also in a permanent magnet. The magnetic field for any magnet can be computed from Biot-Savart (or Ampere's law) if the magnetization $\vec{M}$ is known, so that the surface currents can be computed. (In complete detail, the magnetic current density $\vec{J}_m=\frac{\nabla \times \vec{M}}{\mu_o}$. The result is that there are surface currents at boundaries given by surface current per unit length $\vec{K}_m=\frac{\vec{M} \times \hat{n}}{\mu_o}$, but there can also be magnetic current densities in the bulk material in regions of non-uniform $\vec{M}$). $\\$ For a somewhat detailed example of the magnetic surface currents, see https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/

Last edited: Apr 14, 2018
11. Apr 14, 2018

And just one additional comment: If you do attempt a mathematical solution of this magnetic core with a gap problem using a model that assumes a linear response of the material so that you can write $M=\mu_o \chi_m H= \chi_m' B$, for some $\chi_m$ and a related $\chi_m'$, and computed the magnetic surface currents from the magnetization, and then used Biot-Savart to find the magnetic field, it would be an extremely complex calculation, in the sense that you would need to guess a magnetic field solution $B$ everywhere which would cause magnetization $M$ that would likely be somewhat non-uniform, and the $B$ that was computed from the magnetic currents that arose from the magnetization $M$ would need to be equal to the "guess" for $B$ that was proposed. If the original guess was a rather good one, the actual magnetization $M$ and the actual $B$ could be ultimately determined in an iterative and self-consistent manner. This approach is, at best, still a very difficult one. $\\$ The method that is normally used for this problem,[Edit: "magnetic circuit theory"], involves using a modified form of ampere's law $\oint H \cdot dl=NI$, and assuming $H$ is constant throughout each material, and is $H_1$ in the air gap, and $H_2$ in the iron. This is clearly using an approximation to arrive at the answer, but it gets an answer that has been verified repeatedly with experimentation that has shown it to be reasonably accurate, so it is considered the accepted way of solving this problem. $\\$ The alternative method of Biot-Savart and magnetic surface currents would likely get a similar answer with a very lengthy and cumbersome process of calculation, but in general, this problem of the magnetic transformer with an air gap simply is not done in this manner. $\\$ [Edit: There is also a magnetic "pole" method of doing magnetostatic calculations, that is equivalent to the surface current method, but sometimes gets to the answer a little quicker. In any case, I don't see any easy way of getting any quick results with the transformer with an air gap problem by using these alternative techniques].

Last edited: Apr 18, 2018
12. Apr 15, 2018

Last edited: Apr 15, 2018
13. Apr 15, 2018

### mertcan

thanks for return @Charles Link but according to faraday law change of magnetic field results in current or voltage but you say even we have no time varied current in transformer which also means no time varied magnetic field we again have surface currents. What is origin of those type of currents???

14. Apr 15, 2018

The Faraday effect of time variation is another problem. The Faraday EMF can create additional circulation of currents: Compute the direction=these are circular currents inside the material=these are in addition to magnetic surface currents but circulate in the opposite direction. They are often referred to as "eddy" currents, but therei s a simple solution for them: Transformers are often made in laminated layers to block these eddy currents. In these eddy currents, there is actual electrical charge transport, unlike the magnetic surface currents that are like the net result of a current circulating on the edges of eacgh square of a checkerboard, e.g. in the counterclockwise direction. The currents in adjacent squares will cancel, with the net result being a current along the outer edge of the board. With the magnetic surface currents, there is no charge transport=they are the result of "bound" quantum states in the same (e.g. counterclockwise) direction. Thereby, the laminated layers of a transformer do not block them, and the magnetic field from the surface currents occurs just as it should, and with very little eddy current in the material. The magnetic field $\vec{B}$ in a transformer is basically the result of the magnetic surface currents, computed from Biot-Savart and/or Ampere's law. You also need to add the contribution from the primary windings, (using Biot-Savart and/or Ampere's law), but if $\mu_r=500$ or more, the magnetic field is mostly from the magnetic surface currents.

15. Apr 15, 2018

@mertcan There is one more important practical detail that arises in these transformer equations, that probably isn't completely obvious from looking at the equations that have been considered so far in this thread. The current balance equation equation $N_p I_p=N_s I_s$ seems to be somewhat universal. $\\$ From Faraday's law $\mathcal{E}=-\frac{d \Phi}{dt}$, the result is the EMF in the secondary will be related to the EMF in the primary by the ratio of the number of turns: $\frac{\mathcal{E}_s}{\mathcal{E}_p}=\frac{N_s}{N_p}$. This is a voltage-based equation. [Editing: I made an error or two, so I revised this part]. $\\$ The law of current balance, $N_p I_p=N_s I_s$, a current-based equation, is also important in treating transformer circuits. Anyway, I think you might find the following thread of interest, where the current balance law is discussed, and particularly posts 25-29 : https://www.physicsforums.com/threa...ce-in-transformers.941936/page-2#post-5960705 $\\$ A little algebra with these two equations gives $\mathcal{E}_s I_s=\mathcal{E}_p I_p$, as expected from power considerations.

Last edited: Apr 15, 2018
16. Apr 15, 2018

And a follow-on: A slight puzzle arises on this one: I'd like to ask @jim hardy for his input to see if I have resolved it correctly: $\\$ Without any load=no secondary current, the transformer will develop a voltage that is a result of $\mathcal{E}=-\frac{d \Phi}{dt}$, and the magnetic flux $\Phi$ is in phase with the primary current. This means the voltage on the primary coil at zero load on the secondary will be 90 degrees out of phase with the (magnetizing) current from the primary. Now we are going to assume here that we are driving the primary coil with a voltage source that can deliver whatever current it needs to to maintain the voltage. $\\$ Let's assume we now attach a small resistor to the secondary,(a significant load). There will be current in the secondary that is in phase with the voltage that is generated in the primary and secondary from Faraday's law. This secondary current must be necessarily 90 degrees out of phase from the zero load (magnetizing) current of the primary. In addition, the equations that we have suggest there must necessarily be an increased current in the primary. This new current must satisfy $N_p I_p=N_s I_s$. It would appear that this primary current must be in phase with the secondary current, and thereby will be in phase with the primary voltage. That would be expected from power considerations=this primary current needs to be in phase with the primary voltage, and not 90 degrees out of phase. $\\$ A general comment here: The equations of magnetic coupling appear a little odd. Basically, because we are driving the circuit with a constant ac voltage, we are ensuring that $\frac{d \Phi}{dt}$ remains unchanged by any loading that occurs. It also appears that in putting a load on the secondary, the impedance of the transformer as seen by the voltage driving circuit has gone from inductive to resistive in its characteristics. In addition, putting a smaller resistor on the load will result in a smaller resistance of the transformer as seen by the voltage driving circuit. @jim hardy Might this be correct?

17. Apr 16, 2018

### jim hardy

Those statements are correct for the way we use most transformers, exciting them with sinewave voltage to move power.
I'm working on a basic explanation at nuts&bolts level, almost finished but my alleged brain is fried at the moment.

Will try to post it in the morning.

Transformers are beautifully self balancing and that's how the early experimenters described them in 1800's.
When they closed the magnetic circuit it was an AHA! moment.

old jim

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18. Apr 16, 2018

### jim hardy

Charles - i got past my confusion on dots and flux direction...

I am one of the less capable mathematicians here on PF
so kindly excuse the less-than-academic nature of this post.

I will describe the thought steps i use to get a mental picture of what's going on in a transformer. From that i can figure out a formula to approach most any problem i encounter.

To what degree of precision do we wish to describe the transformer ?

Your words above describe an ideal or near ideal transformer quite well. That would be one with zero resistance in its primary winding assuring 90 degrees between primary voltage and current.

A truly ideal transformer would have infinite inductance hence zero magnetizing current but we won't push our analysis to that extreme, just it's a point to be aware of.

Energy lost in the core looks like resistance in parallel , but for right now let's assume it's little enough energy we can ignore it. In a good transformer that's not far from the truth..

SO yes, with sinewave excitation constant voltage means constant flux. I use volts per turn as a measure of flux.

I think in terms of MMF's and let flux be MMF/Reluctance(of magnetic path). Remember units of MMF is amp-turns .

And being really a plodder i think in small simple steps.

Magnetizing current will be whatever amp-turns are required to push enough flux around the core to make a counter-emf equal to applied voltage.

A low reluctance core , ie good steel with no air gap will require very little MMF hence little magnetizing current.

A core with an air gap like you drew in first post will need a lot more amp turns to push the necessary flux through that gap. Magnetizing current will be higher.

So let's stick with a gapless core.

Now let secondary current flow. That current produces MMF that opposes primary MMF.

So primary current must change to restore flux and counter-emf to match the voltage applied to primary windings. But how does it change ?

Primary and secondary voltage must be in phase with one another because they're both the derivative of the same flux.

Secondary current has to be in phase with secondary voltage because the load is defined, for this thought experiment, as resistive.

So secondary current makes MMF that's out of phase with flux . Secondary current and MMF are both in phase with derivative of flux..

Hmm. Plodding along here,

Flux has to remain constant , defined by the applied voltage which we are holding constant for this thought experiment.

And since flux is ∑MMF's/ Reluctance and reluctance is constant

that means primary current must change to make a MMF that's equal and opposite to secondary MMF. Else sum of MMF's won't equal magnetizing MMF.

Well, MMF's are directly proportional to current, no derivative,

so primary current will pick up a component that's in phase with derivative of flux to balance the one from secondary current.

Now Faraday's Law acquires a minus sign when we incorporate Lenz's Law and you'll often see e = -NdΦ/dt and that can get confusing.

Add to that, transformers are usually drawn with flux going up one leg and down the other.

That's why we use the "Dot" convention. Dotted terminals will all move positive together, ie they're in phase.

But I resort to pictures. Snip from Hyperphysics, i don't think they mind so long as i credit them and don't cause lots of traffic there...

In top image

i added dots and arrows for flux and mmf.

Dot convention = dotted terminals will be positive at same instant. That's how you get phasing right.

Observe current ENTERS primary's dotted terminal but LEAVES secondary's dotted terminal. By right hand rule that gives upward MMF's in both windings .

Secondary MMF is Counterclockwise opposing Clockwise primary MMF . Lenz's law at work.

In Bottom image

I just moved secondary over to same side of core as primary . That makes one MMF point up and one down. Look carefully, i was rigorous to preserve winding directions relative to flux.
That helps some folks visualize right hand rule . It helps me remember that, in most training materials, the windings are drawn encircling the flux in opposite directions, Viewed from the top, primary is wound CCW and secondary is CW.

It might be clearer drawn this way with primary and secondary both wound same direction..…

That removes the confusion factor of winding direction. I need things simple.

....................................................................................................................................................................................................

All that having been said ,
now i want to point you toward Wikipedia's electrical model of a transformer . It's a pretty good thought tool. It's at https://en.wikipedia.org/wiki/Transformer

Rp and Xp repesent primary winding's leakage inductance and resistance.
Rc represents energy loss in the core due to eddycurrents, hysteresis , magnetorestriction ....
The transformer in the middle is ideal - zero magnetizing current , infinite inductance.
Xm represents the actual finite mutual inductance of the transformer.
Io is sum magnetizing current that makes flux and current that goes into core losses, ie heating the iron.
Rs and Xs are resistance and leakage inductance of the secondary winding.

Leakage inductance is present because of flux that doesn't couple both windings..
From the Wiki transformer page:

i just drew in a few loops of leakage flux..

I hope the above is of help to you.
As i say, when i get a mental picture that leads me to the right equation i can trust both.

old jim

Last edited: Apr 18, 2018
19. Apr 16, 2018

Thanks @jim hardy That was very helpful. It helped confirm a couple of my ideas in my post above. Right now, I like to work with a somewhat ideal transformer that may even have an air gap to give finite magnetizing currents, but computations with details such as core resistance and heating from eddy currents and hysteresis is more specialized than I presently need. $\\$ One additional item I mentioned above is laminations that are sometimes used to minimize eddy currents. Are these laminations a necessity, and how effective are they in minimizing eddy currents? And thanks again for your excellent inputs @jim hardy :) :)

20. Apr 16, 2018

### jim hardy

They are indeed complex. That was one of Steinmetz's contributions to the field, with his equation it became possible to design cores instead of trial and error them.
Search terms 'steinmetz equation magnetic losses' http://web.eecs.utk.edu/~dcostine/ECE482/Spring2015/materials/magnetics/CoreLossTechniques.pdf

They are indeed necessary.
Magnetization proceeds from the periphery toward the center of a piece of iron, because eddy currents oppose the change in flux per Lenz.
A solid unlaminated core, above some frequency, might as well not have any iron in the center.
Laminations impede eddy currents allowing your core to use its whole cross section to conduct flux. Analogous to skin effect in wires carrying current.

I once measured an inductor with a core made of unlaminated 400 series stainless steel bar stock. Above 3 hz it was really not a good inductor.

here is its response to triangle wave current at 3hz, 10 hz, and 60 hz.
Triangle wave has constant di/dt so should produce square wave voltage. Sinewave only gives phase shift, triangle wave makes the effect a lot more obvious to the eye
Observe voltage is di/dt only at low frequency

That system ran at 60 hz. It had terrible drift with temperature because resistivity of steel is a strong function of temperature, and eddy currents follow it.

old jim

21. Apr 16, 2018

Thank you @jim hardy Very helpful inputs. $\\$ One thing perhaps worth mentioning is that I think the physics books and physics curriculum of my generation, college days 1975-1980, really sidestepped the whole subject of magnetization/magnetostatics in treating transformers. They taught us even some of the finer details of Maxwell's equations, but they omitted the application of a variation of Ampere's law: $\oint H \cdot dl=NI$ which leads to the magnetic circuit equations. $\\$ It is so good to have the Physics Forums to fill in these "gaps", including the problem of the transformer with an air "gap", (no pun intended). :)

22. Apr 16, 2018

### jim hardy

I was in school perhaps ten years before you. We only studied Maxwell's equations in physics courses. Electric machinery course treated magnetics as a magnetic circuit, and that's how i still approach magnetics problems.

For a machine where virtually all the flux stays inside the iron it works well enough for slide rule accuracy. I've walked around huge machines with a search coil connected to a battery powered oscilloscope and indeed the field around them is nil. Where i found large fields was in the vicinity of air core inductors and high current busbars.

23. Apr 16, 2018

Perhaps the instruction has changed since then. In any case, the magnetic circuit equations, IMO, should not be the exclusive domain of the EE (electrical engineer). It has such a widespread application in what are countless transformers everywhere. For a physics person to be able to call himself an "expert" on Maxwell equations, he should also be able to perform at least simple computations with the magnetic circuit equation, that actually originates from a form of Maxwell's Ampere's Law equation.

24. Apr 17, 2018

### mertcan

@Charles Link here you say there is no charge transport in magnetic surface currents but also you say in previous posts in order to apply biot savart you should also take magnetic surface currents. In biot savart I deem that real currents(which there is a charge transports) are included but if biot savart includes charge transport current and magnetic surface currents are no real charge transport why you say that in biot savart you should also include magnetic surface currents??

25. Apr 17, 2018

Suggestion is to see this "link" https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/ This is an example of how magnetic surface currents are used in calculations. Griffiths E&M textbook derives the result that the magnetic surface current per unit length $K_m=\vec{M} \times \hat {n} /\mu_o$. Applying Biot-Savart to these magnetic surface currents does get the correct result for the magnetic field $B$ from them. $\\$ (This "link" was also given in post 4 of this same thread).