We have a circuit shown in the picture. Known values are: R= 10 Ω, L=10mH and the current of the generator Ig is not dependent on time and it's Ig = 100mA. Switch P is opened until the moment =0, then it closes. Find the current of inductor L in function of time when t>0.

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First, current of generator Ig is not dependent on time, does that mean that Ig is DC source? If it is, how that will affect the circuit, and if if it isn't how that is going to affect the circuit?

Now, after the switch is closed, there's no Ig in the circuit, and, since there's no generator, there shouldn't be any current or voltage through it. Is this correct?

 

There is current through the inductor until time = 0

How does an inductor react to a change in current flow?

 

Yes, Ig is a DC current source and that current is flowing through the inductor at t < 0 since an ideal inductor has no resistance.
At t=0 the inductor and source current both start flowing through P.

That looks to be sort of a trick question as it can be answered by inspection.

 

At t=0- what is the voltage across switch P? Will closing the switch at t=0 change anything?

At t=0+ can the voltage across switch P ever change? What does that say about changing current in the inductor?

How does closing switch P remove the current source from the circuit? Does the same current Ig still flow through the current source?

Since I thought this might be the sort of situation that could be encountered when working with superconducting coils, I googled the phrase "switching transients in superconducting coils" and found a technical paper by that title at Lawrence Livermore National Laboratory published in 1983.

 

There is current through the inductor until time = 0

How does an inductor react to a change in current flow?

Well, u=Ldi/dt, but current flowing through inductor before the switch is closed is constant, does that means that current and voltage on inductor are zero after closing the switch?

 

At t=0- what is the voltage across switch P? Will closing the switch at t=0 change anything?

At t=0+ can the voltage across switch P ever change? What does that say about changing current in the inductor?

How does closing switch P remove the current source from the circuit? Does the same current Ig still flow through the current source?

Since I thought this might be the sort of situation that could be encountered when working with superconducting coils, I googled the phrase "switching transients in superconducting coils" and found a technical paper by that title at Lawrence Livermore National Laboratory published in 1983.

There's no voltage across switch P before closing at t=0- , yet, when it closes, voltage will still be zero, since there's no resistance in the branch with switch P. If voltage is same after closing switch in the branch with switch then voltage on the inductor remains the same too. But, if that is correct, that would mean that current through the inductor should remain the same as it was before closing the switch. About Ig, i an not sure what actually happens, if passive element was there instead of generator, it could be removed from the circuit. Correct me if i made any mistakes.

 

This is not a physical problem but rather a mathematical abstraction. How does current divide between two paths of zero resistance? All you can do is use the mathematical relations that apply, and decide which ones should take precedence.

 

This is not a physical problem but rather a mathematical abstraction. How does current divide between two paths of zero resistance? All you can do is use the mathematical relations that apply, and decide which ones should take precedence.

How can i decide which ones take precedence? Which mathematical relations i should apply? I don't know how current divides between two paths of zero resistance. If in the one branch resistance is zero, it would be easy.

 

Energy is stored in the magnetic field of an inductor.

What happens when the current generating the magnetic field stops at t=0? What occurs in terms of voltage across the inductor as the magnetic field collapses during t>0?

Go back to Google and learn more about back EMF and the behavior of inductors. You almost have this .... don't stop now.

 

How can i decide which ones take precedence?

I would think that basic definitions would take precedence. For instance, a current source is defined by the current which flows through it. The same current always flows through the current source regardless what may happen in the circuit. The behavior of an inductor is defined by the relation \(V_L = L \frac {di} {dt}\) so in the case that \(V_L = 0\) the rate at which the inductor current will change must also be zero.

 

I would think that basic definitions would take precedence. For instance, a current source is defined by the current which flows through it. The same current always flows through the current source regardless what may happen in the circuit. The behavior of an inductor is defined by the relation \(V_L = L \frac {di} {dt}\) so in the case that \(V_L = 0\) the rate at which the inductor current will change must also be zero.

Ok, so at t=0 current that generates magnetic field stops, in that moment, switch is on, and we have no voltage on inductor because both ends are at the same potential, as you stated VL= 0 which means that di/dt = 0 since L isn't zero, that means that current in branch with inductor is constant, but, can i find exact value?

 

But you know the value of Ig? Is this not enough for you?

 

Ok, so at t=0 current that generates magnetic field stops,

Why do you think the current flowing through the inductor should suddenly stop? What causes a spike in the value of L di/dt?

 

This is not a physical problem but rather a mathematical abstraction. How does current divide between two paths of zero resistance? All you can do is use the mathematical relations that apply, and decide which ones should take precedence.

Actually, this is the model for a basic persistent-mode superconducting magnet system.

 

How can i decide which ones take precedence? Which mathematical relations i should apply? I don't know how current divides between two paths of zero resistance. If in the one branch resistance is zero, it would be easy.

What is required in order for the current in the inductor to change?

What does closing the switch tell you about whether the conditions required to get the current in the inductor to change can every happen?

 

Energy is stored in the magnetic field of an inductor.

What happens when the current generating the magnetic field stops at t=0? What occurs in terms of voltage across the inductor as the magnetic field collapses during t>0?

Go back to Google and learn more about back EMF and the behavior of inductors. You almost have this .... don't stop now.

Why and how does the current generating the magnetic field stop at t=0?

What is the basis for claiming that the magnetic field collapses "during t>0"? (Not sure what "during t>0" quite means.)

 

Ok, so at t=0 current that generates magnetic field stops, in that moment, switch is on, and we have no voltage on inductor because both ends are at the same potential, as you stated VL= 0 which means that di/dt = 0 since L isn't zero, that means that current in branch with inductor is constant, but, can i find exact value?

What is the value of the current in the inductor for t = 0- ?

What is the value of the current in the inductor for t = 0+ ?

If di/dt = 0 for t > 0, then what is the current in the inductor for t > 0?

 

What is the value of the current in the inductor for t = 0- ?

What is the value of the current in the inductor for t = 0+ ?

If di/dt = 0 for t > 0, then what is the current in the inductor for t > 0?

If u= Ldi/dt, then i=(1/L)∫udt which means that i should find u(t) in order to find i(t) i am just not quite sure how, when the switch is closed, then u=0 which means that di/dt=0 so i should be some constant (it shouldn't depend on time) since first derivative of constant is zero.

 

If u= Ldi/dt, then i=(1/L)∫udt which means that i should find u(t) in order to find i(t) i am just not quite sure how, when the switch is closed, then u=0 which means that di/dt=0 so i should be some constant (it shouldn't depend on time) since first derivative of constant is zero.

You say that you don't know how to find u(t) yet then immediately (and correctly) say that u(t) = 0 for t >=0. Sounds like you found u(t), doesn't it?

I'll ask again.

What is i(t=0-)?

What is i(t=0+)?

If you know what the current is just after the switch closes, and if you know that the current can't change in the inductor once the switch closes, then what is the current in the inductor for all time after the switch closes?

Don't make it harder than it is -- this is a trivially simply problem!

 

You say that you don't know how to find u(t) yet then immediately (and correctly) say that u(t) = 0 for t >=0. Sounds like you found u(t), doesn't it?

I'll ask again.

What is i(t=0-)?

What is i(t=0+)?

If you know what the current is just after the switch closes, and if you know that the current can't change in the inductor once the switch closes, then what is the current in the inductor for all time after the switch closes?

Don't make it harder than it is -- this is a trivially simply problem!

Let's see, i'll give it a shot

for i(t=0-) switch is still off so current through the inductor is:

IgR=Ldi/dt

di=IgRdt/L

i=IgR/L∫dt = IgRt/L is this correct?

for i(t=0+) switch is on and since there's no voltage on inductor, there no current either.