Question about magnetic circuit

[CheyenneXia]

recently I am working on magnetic solenoid and got a question.

The attachment shows the simple cross-section. Blue green represents the steel providing the magnetic flux path and red is the copper wire with current flowing into the screen. Also, the whole structure is in cylindrical shape. It is like rotating the cross-section in the attachment with the axis of the left line 360 degree. A1=2A2.

According to magnetic circuit theory, magnetic flux is on the left part of the steel should be approximately two times of the one on the other parts of the steel. Since B=phi/A, H=B/mu, and A1=2A2, magnetic field intensity is the same everywhere. Am I right?
What I confused is I believe the root cause of the magnetic field is the current. If we calculate H using the Biot-Savart law, How can H be the same at the whole path of the steel?

Also, does saturation in the steel generate more loss compared to the situation without saturation? I read it from my colleague's report and I couldn't figure it out. I only read about hysteresis loss.

Thank you for your help.

 

[Enthalpy]

Because of the rotation symmetry, the cross area at left is not twice that at right. Good design would put the same cross area everywhere, so iron gets a uniform induction; it could well be your case.

Biot-Savart works without iron only - and for simple shapes...

In real life, forget everything differential. Compute the reluctance from the magnetic path's length, permeability... and deduce the induction and flux. Stick to these circuit global figures.

Saturation itself doesn't create iron losses. Iron losses increases with the induction nearly squared. Saturation can prevent your part work properly (but is sometimes desired: fluxgate magnetometers, magnetic amplifiers, pulse generators...). Say, a transformer can't resist the primary voltage any more when its core saturates, and then the current increases exaggerately.

 

[CheyenneXia]

You are right. The cross area at left is not twice of the right after rotating. It is not the real design. I just use this drawing for my questions.

I went back to the book and the internet. I only got the info that Biot-Savart works for magnetostatic case. Other than that, not much. Do not take it wrong. It is not that I do not believe you that it doesn't work with iron. I just want to learn more about it.

When you say "Iron losses increases with the induction nearly squared", do you mean the eddy loss?

Coincidently I am reading a book related to electrical machines and i am on the transformer charpter. Excitation current and primary current really confuse me. I posted my question here
https://www.physicsforums.com/showthread.php?p=4207887#post4207887

If you know the answer. Please help me.

Also, I didnt get why saturated core generates exaggerately larger current. When the core is saturated, should emf which is related to dphi/dt be reduced?

 

[Enthalpy]

Biot&Savart won't work, because at some places you have a magnetic material and at some others none.

Hysteresis losses increase more or less as the induction squared, too.

When the core is saturated, dphi/dt is limited by phi, so the primary can't respond to the external voltage by the emf, and all the excess external voltage creates a current limited only by ohmic loss. Since coil and transformer designs want small ohmic losses, the current can increase a lot when the core saturates.

 

[alphysicist]

Hello,

Thank you for your question about magnetic circuits and solenoids. Based on the information and diagram provided, it appears that you are working on a cylindrical solenoid with a steel core and copper wire winding. You are correct in saying that the magnetic flux on the left part of the steel should be approximately two times that of the other parts, as the cross-sectional area is doubled on the left side.

In terms of magnetic field intensity (H), it is important to note that H is dependent on the material's permeability (mu) and the current density (J). In your case, the steel core has a higher permeability compared to the air or other materials on the right side, but the current density is the same throughout the solenoid. Therefore, the magnetic field intensity will be higher on the left side due to the higher permeability, but the current density will remain the same.

As for your question about the Biot-Savart law, it is a valid method for calculating magnetic field intensity, but it is not the only method. In this case, the magnetic field intensity is also affected by the permeability of the material, which is why it may seem counterintuitive that H is the same throughout the steel core. However, the Biot-Savart law can still be used to calculate the magnetic field intensity at specific points along the steel core.

Regarding saturation in the steel, it is true that saturation can lead to more losses compared to a situation without saturation. This is because when a material reaches its saturation point, it can no longer increase its magnetic flux density, which can result in eddy current and hysteresis losses. However, the extent of these losses will depend on various factors such as the material's properties, the strength of the magnetic field, and the frequency of the current. It is important to consider these factors in your calculations and experiments.

I hope this helps clarify your questions. If you have any further inquiries, please do not hesitate to ask. Best of luck with your research.