Ampere's law and magnetic circuit

[kelvin490]

For magnetic circuit there is an equation NI=HL. This equation is obtained by applying Ampere's law. H is magnetic field intensity and L is the length of the circuit. In following picture the equation reduced to I=HL as there is only one turn:

https://www.dropbox.com/s/22rgm63vuduhyr1/Untitled.png?dl=0
https://www.dropbox.com/s/22rgm63vuduhyr1/Untitled.png?dl=0

However, if take a shorter path and apply the Ampere's law, HL must be smaller than before due to shorter path but there is still same current I enclosed (see below). Why?

https://www.dropbox.com/s/9mbbi49q386jq8o/Untitled.jpg?dl=0

https://www.dropbox.com/s/9mbbi49q386jq8o/Untitled.jpg?dl=0btw, sorry for the ugly picture.

 

[Windadct]

Why do you say H is the same?

 

[kelvin490]

Why do you say H is the same?

It is common assumption in magnetic circuit.

 

[Windadct]

But you changed the circuit. -- the clue is what are the units of H

Meant to say "magnetic circuit"..

 

[kelvin490]

H is magnetic field intensity, unit is A/m.

The circuit in the two cases are the same. In my picture only a small section of the iron core is wrapped with electric wire (one single turn in this case). The enclosed current is the same for both the large and the small loops in the pictures. Only the loops taken to calculate the integral to apply Ampere's law are different. My concern is, the integral are supposed to be different for different path (one is longer and the other is shorter), but the current enclosed by both loops are the same. That is the dilemma.

 

[Windadct]

Maybe I mis-understood your OP - you shortened the "magnetic circuit" (I should have said that above) correct? H is not flux... in your case I ( one turn) = Flux x Reluctance = H x L = when you change the magnetic circuit ( by shortening it) everything changes.

Analogus to reducing the length of a loop of wire, where the circuit resistance is Resistance per unit length(defined by the wire) x length.

 

[kelvin490]

No no no... the circuit is not shortened. Those loops are just for integration in applying Ampere's law. H is magnetic field strength B divided by permeability.

 

[Windadct]

Magnetic Path? That is the magnetic circuit- not the electrical circuit represented by the wire. That is what you mean - the path that the magnetic flux follows in the core material? - If not then I completely am missing the point of your 2nd image.

 

[kelvin490]

Actually there are magnetic flux in the magnetic circuit. The iron core is used to keep all the flux in the core. You can assume the leakage is zero. Now I apply Ampere's law to two different paths - one longer and the second one is shorter. The integration along these two paths are supposed to be different, but they are enclosing the same electric current ! That's the problem.

 

[Windadct]

Path of integration I think I now I understand - the basic assumption is the field strength is uniform, and lm is to be the path of the mean path of flux. When your path of integration "exists" the core, this assumption is gone - and you can not use the NI=HL simplification... i.e. the "H" is not the same around your path.

 

[kelvin490]

Just see the pictures and compare, H doesn't seem to be different, right?

 

[kelvin490]

For those who are interested. I would like to explain more about my question.

Yes, H is supposed to be different. But in the iron core it is uniform or approximately uniform. Now try to apply Ampere's law. If you do the integration in the path of picture 1 you get HL where L is length of core. Then do the integration for the second path in picture 2, you will find that part of it have the same H because it is inside the iron core, but part of it is outside the core so the H is essentially zero. Therefore, the second integration is smaller than the first one. The problem is, Ampere's law says that the integration is always equals to I (only one single turn of electric wire). That is the dilemma.

 

[Windadct]

Again - the integration for the two cases is different - the total field you are enclosing is different for the two cases. The law says that the field is proportional to I - for a specific path of integration. If you change the path you change the ratio of field strength and current, that is the function of the H in the simplified version of NI-HL. It seems you are still oversimplifying the Law --

 

[kelvin490]

In ideal magnetic circuit with iron core, H is uniform inside, no flux leakage so no flux in air.

Consider the rectangular second path which shown in picture 2, H_iron is the same as in path 1 but the path length is shorter.

For the horizontal sections of the rectangular path, the path is perpendicular to the flux so the dot product is zero.

For all other portion of the path that is outside the iron core, the dot product is zero because no flux in air.

Therefore the integration results is smaller than integration along the path in picture 1.

 

[Windadct]

I do not know how else to describe the issue - the total flux in the area - which you get from the integration, is not the same in these two cases. The NI = HL case is for a complete Magnetic Circuit - in your smaller area of integration you do not have that case, so the HL equivalency is not valid. Basically you are asking now to calculate the flux in a portion of the core, and some air - it is not a trivial question. Yes it is a smaller amount, not all of the flux created by I is included in your integration path. However -- if you keep that seccond integration path constant - and vary the current - the Flux in that area will vary proportionally to the current - that is Ampere's

 

[kelvin490]

Maybe I quote from a textbook:

The analysis of a transformer is based on three stated assumptions:
1. The magnetic fluxes of all turns of the coil are linked.
2. The magnetic flux is contained exclusively within the magnetic core.
3. The flux density is uniform across the cross-sectional area of the core.Although they are only assumptions they supposed to be closed to real situation.

I have also considered the flux in air is not zero before I posted the question. However, we may still get a smaller result if the field is close to zero. That's why I asked.