Mutual inductance in concentric cylinders

[derek88]

Friends:

I have been thinking about this question for a while, so I hope you can help! Here it is:

Lets say you have two concentric, conducting cylinders.

The outer cylinder ("Cylinder 2") has the following properties: radius R2, carries a current of I2, relative magnetic permeability mu2, impedance of Z2 = R2 + jwL2.

The inner cylinder ("Cylinder 1") has the following properties: radius R1, carries a current of I1 (in the same direction as I2), relative magnetic permeability mu1, impedance of Z1 = R1 + jwL1.

My question is: how does the mutual inductance between these cylinders increase their impedance?

My guess is the following:

"Cylinder 1" creates a magnetic field B = (mu0)(mu1)(I1)/(2*pi*R1)
This creates a mutual inductance on "Cylinder 2". The mutual inductance created on "Cylinder 2" is M21 = [(mu0)(mu1)(I1)/(2*pi*R1)]*LN(R2/R1).
Thus the impedance of "Cylinder 2" increases to Z2 = R2 + jwL2 + jwM21.

"Cylinder 2" creates a magnetic field B = (mu0)(mu2)(I2)/(2*pi*R2). However, this field exists only outside the cylinder. Inside the cylinder, the magnetic field is 0. Therefore, the mutual inductance created on "Cylinder 1" by "Cylinder 2" is M12 = 0.
Thus the impedance of "Cylinder 1" remains unchanged at Z1 = R1 + jwL1.

Please let me know if this is correct or if I'm terribly mistaken! Thank you kindly!

 

[eljose79]

Thank you for your question about the mutual inductance between two concentric conducting cylinders. Your understanding of the concept is mostly correct, but I would like to clarify a few points for you.

First, you are correct in saying that "Cylinder 1" creates a magnetic field that induces a mutual inductance on "Cylinder 2". This mutual inductance is determined by the equation M21 = (mu0)(mu1)(I1)(LN(R2/R1))/2pi. However, this mutual inductance also contributes to the impedance of "Cylinder 1" because it is affected by the magnetic field created by "Cylinder 2". Therefore, the impedance of "Cylinder 1" will also increase to Z1 = R1 + jwL1 + jwM21.

Additionally, you mentioned that the magnetic field created by "Cylinder 2" is only present outside the cylinder. This is true, but it still contributes to the mutual inductance on "Cylinder 1". The equation for this mutual inductance is M12 = (mu0)(mu2)(I2)(LN(R2/R1))/2pi, and it does affect the impedance of "Cylinder 1" as well.

In summary, the mutual inductance between the two cylinders will contribute to the impedance of both cylinders, and the equations for mutual inductance in both cases should include the term LN(R2/R1). I hope this helps clarify your understanding. Let me know if you have any further questions.