Time dependent behavior of 'real' solenoids

Question

Consider a real(ish) solenoid composed of a finite number of coaxial conducting loops in series. The first loop will have a mutual inductance with all the other loops in the set ($M_{12}, M_{13}, ..., M_{1n}$), the second to have mutual inductance with the first loop and all the following ($M_{21}, M_{23}, ..., M_{2n}$), all the way up to the last loop. It might make sense to put this in a matrix: $$ M = \begin{matrix} 0 & M_{12} & M_{13} & ... & M_{1n} \\ M_{21} & 0 & M_{23} & & M_{2n} \\ M_{31} & M_{32} & 0 & & M_{3n} \\ ... & & & ... & \\ M_{n1} & M_{n2} & M_{n3} & & 0 \end{matrix} $$

Where M is symmetric due to reciprocity. If the current through the solenoid is some time varying function, then there is a time changing magnetic flux. The solenoid has some constant resistivity, so that any emfs produced result in knowable/finite currents. Several possibilities are available:

1. The mutual inductance doesn't apply, the simple DC equations are correct
2. The mutual inductance applies, and the current in each turn is modified by the emf contribution of every other turn, producing some more interesting, perturbed net current/B-field
3. The mutual inductance applies as described in 2, and the once-perturbed then couples through M to twice-perturb the net current, so on and so forth, until the net current has relaxed to some infinitely perturbed configuration.

My suspicion is currently the last option, in which case one would expect the inductance of this system to be more like a series expansion with smaller and smaller contributions due to different time derivatives of the signal, even before displacement current comes into the story. Does anyone else have any expertise on the topic?

Answer 1

If you're going to construct the inductance matrix $\mathbf M,$ where each entry $M_{ij}$ is the inductance between elements $i$ and $j,$ you may as well make it consistent and throw in the self inductances too. That is, each diagonal element $M_{ii}$ should each be the (self) inductance of loop $i.$ (What is the self inductance of loop $i$ except for the mutual inductance between loop $i$ and loop $i$?) That is, the diagonal elements aren't in principle zero.

Once you have the inductance matrix $\mathbf M$ right, you find it obeys the "same" law as just the scalar inductance. $$\mathbf V=\mathbf M\mathbf{\dot I},$$ where $\mathbf V=(V_1,\ldots,V_n)$ is the vector of EMFs across all the loops, and where $\mathbf I=(I_1,\ldots,I_n)$ is the vector of currents through all the loops. (Expand out the matrix multiplication in components: this equation says only that the EMF across each loop is the sum of all the EMFs due to the changes in currents through all of the loops via the respective inductances.)

You can use this equation to analyze your solenoid by setting $\mathbf I=(I,\ldots,I)$ (all currents the same) in order to reflect that all the elements have been placed in series and then considering the quantity $V=V_1+\cdots+V_n$ to reflect that you only care about the total voltage across the solenoid. Then you can derive the equation $$V=\begin{bmatrix}1&\cdots&1\end{bmatrix}\!\mathbf M\begin{bmatrix}1\\\vdots\\1\end{bmatrix}\dot I=\left(\sum_{i=1}^n\sum_{j=1}^nM_{ij}\right)\dot I.$$

So, no, the correct answer under your assumptions is really just 1. All the effects of mutual inductance can be summarized just by saying that the standard inductor law $V=\dot IL$ is valid for the whole solenoid, where $L$ is understood to contain contributions from all the self and mutual inductances between the coils: $L=\sum_{i=1}^n\sum_{j=1}^nM_{ij}.$ If you measured the inductance of the solenoid, the $L$ you would get is automatically this sum, and that's all you really need to describe the inductive character of the device.

Even if you now consider each loop to have both an inductance (and mutual inductance to all the other loops) and a finite conductivity, (which should be in series with the inductance, since you can't have current through a loop somehow not participating in the magnetic interaction), I don't think you change the equation except to $\mathbf V=\mathbf M\mathbf{\dot I}+\mathbf R^T\mathbf I$ and later $V=\dot IL+IR,$ where $L$ is the same total inductance from before and $R=R_1+\cdots+R_n$ is the total resistance. I.e. the solenoid can still just be summarized as a lump inductor in series with a lump resistor.

In order to get something more interesting than this, I believe you'd have to consider the capacitances between the coils, and/or leakage conductances (i.e. conduction through the insulation of the coils, which I guess is an even smaller effect). With a model of that sophistication, you could see the wavelike propagation of transients across the individual loops of the solenoid etc.. This takes you into the realm of transmission line theory.