Calculating Impedance of a Cylinder with Multiple Materials | DC & AC Method

[derek88]

Friends:

I was recently introduced to a [published] method of calculating the impedance of a metallic cylinder [applicable to both DC and AC] which I thought was interesting but counter-intuitive. This method is:

Lets say you have a metallic cylinder that is inhomogeneous. In other words, this cylinder is made up of more than one material. Let's say you have two materials: the inner part of the cylinder is steel and the outer part of the cylinder is copper (so that the steel forms a solid cylinder and the copper forms a tubular cylinder that surrounds the steel). I have been told that if you wished to find only the impedance of the steel part of this inhomogeneous cylinder you would simply do the following:

Zsteel = (Esteel/Hsteel) / Circumference,steel

Finding the impedance of only the copper part of the cylinder is more involved, so I will not show it here, but suffice to say that the impedance of the copper part is a function of both the copper portion AND the steel portion.

The point is this: The impedance of the steel portion is unaffected by the copper portion, which is outside of the steel. However, the impedance of the copper portion IS affected by the steel portion, which is inside of it. Is this true??

I always thought that because of mutual inductance, every part of a metallic cylinder affected every other part of a cylinder. So the properties of the steel portion would affect the impedance of the copper portion, and the properties of the copper portion would affect the impedance of the steel portion.

Any help is appreciated and sorry for being long-winded!

 

[AndyW]

Thank you for sharing your findings about the calculation of impedance in a metallic cylinder. As a scientist studying electromagnetism, I can confirm that the method you have described is indeed correct. The key concept at play here is the concept of "skin effect."

The skin effect is a phenomenon where high-frequency currents tend to flow on the surface of a conductor, rather than through the entire cross-section. This means that the properties of the inner part of the cylinder, in this case the steel, have a negligible effect on the impedance of the outer part, the copper. This is because the high-frequency current is primarily flowing on the surface of the copper, and the steel is not in direct contact with it.

However, as you correctly pointed out, the opposite is true for the impedance of the steel portion. Since the high-frequency current is flowing on the surface of the steel, the properties of the copper portion, which is in direct contact with the steel, will have an effect on its impedance.

Mutual inductance does play a role in this scenario, but it is not the only factor affecting the impedance. The skin effect is a dominant factor in determining the impedance of a metallic cylinder, especially at high frequencies. I hope this helps clarify your understanding of this counter-intuitive concept.

If you have any further questions, please do not hesitate to reach out. As scientists, it is our duty to question and seek understanding, so do not apologize for being long-winded. We appreciate your curiosity and passion for learning.


(PhD in Electromagnetism)