Switching DC battery on ideal choke?

(Google translation). A switch is activated at the instant t = 0 in a series circuit containing a battery of voltage U (ideal) and an inductor L (ideal). I am in doubt with the answers I have found on certain sites (not very clear).

<< NB : this question is more general and can be also applied to a resistance or a capacitor or anything else. In fact, the real question is : "how is it really going?" >>

What are, in the end, the "theoretical" hypotheses which make it possible to validate and use the equation: L * di / dt = U, as well as the final equation i = U / L * t valid even for t = 0 up to 'infinity' ?

I think the question asked requires an answer over 3 (?) differents levels.

1 - Mathematical background (I am not mathematician, but I used maths as an electrician like many as me . Some times, we forget our bases. https://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Lebesgue

What I retain from this, it is the fact that we can write differential equation, we can "derivate", "integrate" ... Function is well defined even in a singular point. Ok, it is the traditional way we follow to find all behavior of our circuits, even in singular point.

2 - Physical background - some times, we used analogy as pump and water circulating in pipes for electrons circulating in wire ... not true in this case really, here, because electrons moves very very slowly ... but they are part of the answer.

3 - Electromagnetical point of view - it is one more serious explanation for the transient phenomena we used every day in electronics. It can also explain, however, the steady state we used on the power grid and other electronics systems we use everyday.

The reasoning which follows is simple but important and it applies to all (?) cases. I'm going to suppose that closing the switch generates a "wave" (point 2 - at 300,000 km / s ?) Which will propagate in the wire connecting the battery to the choke. I start from the battery.

I am the "electromagnetic" wave (I know, a "wave", it doesn't think so, but are you sure of this ? ...). What do I see in my neighborhood? A wire. (what I see is that this wire has an operational Laplace impedance (that I know - see point 1- and I know what to do ...) and I know, at this moment, how I should behave ... I commit to it ... I don't yet know what is on the end of the line, especially if it is long. Finally, I get to the end of the line and I encounter a "self" of which I know the operational impedance of Laplace ... I react accordingly ... which means for me that I have to turn around ... The choke, it follows its little path with the "slap" that I gave it. Etc ... If the wire is short, it will go very, very quickly. Otherwise, it will take a while. So I go back to the battery. I also know its operational impedance. again in the other direction ... He yes, it is likely to last ... Is this reasoning valid?

4 - Others ?

What are, in the end, the "theoretical" hypotheses which make it possible to validate and use the equation: L * di / dt = U, as well as the final equation i = U / L * t valid even for t = 0 up to 'infinity' ?

These equations are the definitions of ideal inductor behaviour.

As such, they don't need a justification, but we should find out whether they are useful.

When we test some real inductors, we find that the ratio of U and di/dt is constant within measurement uncertainties over a range of values for which they are still operating ideally. For instance, $$\I=\frac{Ut}{L}\$$ predicts an infinite current at infinite time. We haven't got that long to do the experiment, neither have we any inductors that would handle an infinite current without melting or tearing themselves apart.

We can create all sorts of equations. Some match real life behaviour and so are useful, some don't and so are ignored.

$$\V=IR\$$ is the definition of an ideal resistance, the constant R. Many components follow this equation quite closely, so it gets used a lot.

$$\V=k\int_{0}^t{I^3}dt\$$ is attempting to be the definition of some constant k, but it doesn't match any physical situations, so it's not useful, and I doubt you'll ever see it again.

• Here's that non-useful formula Neil mentioned just in case you thought you might not see it again $V = k\int_0^t I^3 dt$ LOL. Jun 29 '21 at 14:21
• Ok for all this. In fact, the problem occurs at t=0. The function is "particular" not defined for t<0, and for t=0, di/dt is indeterminate = 0/0, but has a true value of U/L. How the theory can calculate a derivative in a point where the function "stops" or "begin" ? Jun 29 '21 at 16:31
• @Antonio51 and there was me thinking you wanted electronics, when you're quibbling over the maths. Take it to the maths stack if you want derivatives defined to the left, or right, or at a discontinuity. Anyhow, t<0 is not defined. Who says U isn't continuous? Perhaps current has been ramping from minus inifnity with constant U, and we define t=0 to occur when I=0? How's about that then! We're doing that sort of thing all the time with transformer inductance energised by a mains U(t), wondering where the big inrush current comes from (L too small, so t*U/L too big!) Jun 30 '21 at 4:53
• @Neil_UK ok with you, it is first a mathematical point of you (the answer will came later). In fact, this question if the first part (on 3) of the answer. The second part is the physical point of view ... and the third is ... the electromagnetical point of view. The electrician or electronician forget easily his "origin" and some time, a recall is necessary ... for the beginners, especially. :) Jun 30 '21 at 6:34
• I edit my question to be complete. Jun 30 '21 at 6:41

To attempt to answer the second part of the question, the (meta)physical: if you consider a model where a force (voltage) accelerates an object (electrons) by pushing against their inertia (inductance) then the analogy seems to work well; indeed many of the real-world limitations apply. In the real world you can’t instantly apply a force at t=0, and most things that try to apply a constant force can only do so up to a certain speed, after which they struggle. Nevertheless, the definitions are what they are and when we apply them we need to be mindful of the real-world limitations.

• @ Frog : ... yes ... I will add a picture (link to a physical site ... ) which explain a little what can happen ... and be interpreted for electrons ... Jun 30 '21 at 7:39