Finite spiral air-filled solenoid

[VictorQ]

Homework Statement

Hi guys.
Electrical engineering student here trying to get some real physics under his skin.
I'm trying to derive the field and ultimately the inductance of a finite solenoid based on the spiral shape of the windings (because I assume - for no good reason - that the exact geometry is somehow significant).

Parameters: N number of turns, l length of coil, R0 radius of coil, I current in coil.
Assumptions: Lossless coil, static field (no displacement current).

Homework Equations

Ampere's law : $$\nabla^2 A = -\mu J$$
Parametric description of J(z) in cylindrical coordinates:
$$J(r,\theta,z) = I \cdot \frac{\delta(r-R0) \cdot \delta(\theta-N/l \cdot 2 \pi \cdot z ) } {\sqrt{1+l^2/N^2}} \cdot \left[ {1 , N/l \cdot 2 \pi \cdot z + \pi/2, l/N}\right]$$
where I is the current, and the bracket on the right is a vector in cyl. coords., where the scalar factors obviously do not multiply the angle component...

The Attempt at a Solution

Direct numerical integration of Biot-Savart's law did draw me a pretty picture of the H-field, but it seems to be an inefficient approach to calculating the impedance. It simply takes longer than I want to wait.

More promising: I managed to transform the current distribution vector field onto a cylindrical Fourier-space inspired by http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.152.9411&rep=rep1&type=pdf". The transform I used is the polar transform described in the paper with a combined Fourier-series/BesselJ transform in the angular and radial directions respectively. I then tacked on a regular Fourier transform in the z-direction.

This diagonalized the Laplacian nicely and then gives me the transform of the vector potential. I can't manage the inverse transform, though, symbolically or numerically. Maple thought about it for about an hour before giving up completely...

The inductance can be written in terms of the vector potential, and via Stoke's theorem can be written as:
$$L=\frac{1}{I} \oint_C A \bullet d\vec{r}$$
where the curve C is up along the spiral and then directly down to the origin, nicely framing the flux linkage area of each turn.

I would like to calculate the inductance in the Fourier-domain directly, but how on Earth do I go about transforming the path integral?

Writing it in terms of anti-derivatives allows me to easily transform the functions, but what should the limits of integration (z=[0;l]) be in the transform domain (w=[0;1/l] obviously doesn't work as it does not evaluate to a numerical value)?
I feel like I'm SO close, but it just might not be possible at all (?).
Thanks so much!

 

[anathema]

Dear Electrical Engineering student,

First of all, it's great to see that you're delving deeper into the physics behind electrical engineering. It shows a true passion for the subject.

Your approach using Ampere's law and Biot-Savart's law is definitely on the right track. However, as you mentioned, it can be computationally intensive to directly integrate these equations. Have you considered using a finite element analysis software to simulate the field and calculate the inductance? This may be a more efficient approach, as it takes into account the exact geometry and material properties of the solenoid.

Alternatively, you could also try using a numerical integration technique such as the trapezoidal rule or Simpson's rule to approximate the path integral in the Fourier domain. This would allow you to transform the integral and then evaluate it numerically, potentially giving you a more accurate result.

I also want to point out that your assumption of a lossless coil may not be entirely accurate. In real-world applications, there will always be some amount of resistance and energy loss in the coil. This could potentially affect your calculations and should be taken into consideration.

Overall, your approach is solid and it seems like you're making good progress. Keep exploring different methods and don't be afraid to try out different software or numerical techniques. Good luck with your research!