Symmetrical parallel inductors, different currents? State Preparation

[yucheng]

Homework Statement

JEE Pathfinder
https://youtu.be/a7wi7F_amJw
See below

Relevant Equations

N/A

We have there parallel inductors (i.e. having the same voltage) with the same inductance, having different currents at a particular time.

It appears that this is only possible if the current phasors have different phases
See bottom left of video below, $I_1, I_2, I_3$ (time already set to 10:18)


However, how is this 'state' 'prepared'? By symmetry, if we start by connecting a charged capacitor to the open-circuited parallel inductors, shouldn't the current phasors be the same?

Thanks in advance!

 

[berkeman]

We have there [sic] parallel inductors (i.e. having the same voltage) with the same inductance, having different currents at a particular time.

I didn't watch the video, but it would seem to me that you can only have different currents in the 3 inductors if they are connected one at a time to the capacitor. The 2nd and 3rd inductors connected will have smaller currents in them compared to the first one connected.

 

[DaveE]

I didn't watch the video

Just as well it's not in english. This is a mini-study in how to make people not reply to your post.

 

[berkeman]

Just as well it's not in english.

Trickery!!

 

[DaveE]

Nope, I didn't watch your video either (OK about 15 seconds of it).

Note to others: this is a transient response problem where the inductors start with different ICs. Which would have been nice to know from the beginning.

I don't have time to answer yet. Except to say that the state of each inductor is fully determined by its current. So for a rigorous treatment of transient response this is a 4th order system. But an intelligent approach using superposition can simplify the dynamic response to a 2nd order system. The inductors can still be modelled as a single impedance if you don't care about how they share the DC current. Then you can separately solve for the different inductors, which should be equal dynamics with DC offsets. This is because the state change (current) of each inductor is fully determined by the applied voltage, which is equally shared amongst them.

 

[DaveE]

BTW, I'm not a fan of "phasors" for transient response problems. It's not wrong, but when you get to this level of complexity, you will do better to think in real math terms. For example, "phasors" contain no information about the DC offsets in a problem like this. Their existence depends on the steady state assumption that there is a single oscillation frequency and nothing else matters.

Anyway, yes, the "phasors" are the same, but the currents aren't.

 

[yucheng]

Note to others: this is a transient response problem where the inductors start with different ICs. Which would have been nice to know from the beginning.

Just as well it's not in english.

I don't know either, to be honest!

I should have mentioned that I only extracted the problem statement (in English!) and the formulas from the video.

 

[yucheng]

@DaveE Do you mean DC offsets in the context of analog circuits?

I guess the right way to approach this problem is by setting up the differential equations and with the stated initial conditions right?

By the way, why is the system of fourth order though? Is there a way to see it immediately? Let me write down the differential equation first...