What Happens to Inductor Current When the Switch is Opened?

[cnh1995]

I was thinking about the discharging mechanism of an inductor and I have a question about it. It might sound stupid to the physics experts here, but I can't help thinking about it.
Before I ask my question, I want to mention a few points I already know about an inductor..
I know an inductor resists sudden changes in current through it. I am comfortable with the differential equations and their solutions for RL circuits with different types of excitation. So, I have enough understanding of the inductor as far as my EE domain is concerned.
When an inductor is charging, the energy is stored in the magnetic field and that stored energy is used to drive the current while discharging of the inductor. I understand this approach.
Now my question...
Consider the following circuit.

The switch was closed for a long time and the inductor was carrying a current of 1A. Now, when the switch is opened at t=0, current in the inductor is maintained momentarily at 1A and voltage across the 1 ohm resistor rises to 1V. So, at t=0⁺ (just after t=0), inductor current is 1A and induced emf=1V. But if the current is still 1A just after the switch is closed, how is emf induced in the inductor? There has to be some change in current for emf induction. My question is, how is this voltage of 1V developed across the 1 ohm resistor just after the switch is closed? If the resistor were 10 ohm, the induced emf at t=0 would be 10V. How does the inductor "know" that resistance value? I understand the behaviour mathematically, but I don't understand what maintains the current to 1A "just after" the switch is opened. I want to know the what happens at the electronic level just after opening the switch. I am having trouble understanding "what causes what" at t=0.
Thanks in advance.

 

[Dale]

But if the current is still 1A just after the switch is closed, how is emf induced in the inductor? There has to be some change in current for emf induction.

The current is 1 A, but it is not constant. It is changing.

 

[phinds]

The inductor doesn't know, or care, anything about the resistor, it just knows that it's got 1 amp flowing and it by damn is going to KEEP that 1 amp flowing. If there is no resistive path, then it will develop infinite volts. That of course, is a non-real-world idealized case but explains why inductors will spark-gap if suddenly removed from a circuit --- it develops however much voltage is needed to create an ionized air path for the current. If there is a 1 ohm path for the 1 amp, then only 1 volt needs to be produced and if 10 ohms then 10 volts. The inductor doesn't care about the voltage.

 

[cnh1995]

The current is 1 A, but it is not constant. It is changing.

I think I am having trouble understanding the concept of di/dt at t=0.
While charging, the initial value of di/dt is E/L, which is independent of the circuit resistance but initial value of di/dt depends on the resistance R. How does the inductor know the value of the connected resistance? For maintaining the current to 1A, emf induced should be 1V but for having an emf, the current has to change. How to visualize this situation?

 

[phinds]

I think I am having trouble understanding the concept of di/dt at t=0.
While charging, the initial value of di/dt is E/L, which is independent of the circuit resistance but initial value of di/dt depends on the resistance R. How does the inductor know the value of the connected resistance? For maintaining the current to 1A, emf induced should be 1V but for having an emf, the current has to change. How to visualize this situation?

Which part of my explanation is not clear?

 

[cnh1995]

The inductor doesn't care about the voltage.

How does the inductor know how many volts to generate at t=0? I mean if the resistance is 1 ohm, it will generate 1V and if it is 1k ohm, it will generate 1000V. How does it decide these values?

 

[phinds]

How does the inductor know how many volts to generate at t=0? I mean if the resistance is 1 ohm, it will generate 1V and if it is 1k ohm, it will generate 1000V. How does it decide these values?

AGAIN, it does NOT decide. It doesn't care. It just keeps on producing 1 amp. Ohm's Law does the rest.

 

[cnh1995]

It just keeps on producing 1 amp.

How does it "produce" this current? To "produce" a current, some voltage is needed. Now, if the resistance is 1 ohm, the inductor should generate 1V across the resistance first, then only 1A could flow through it. Am I missing something?

 

[Bystander]

You have a magnetic field collapsing.

 

[cnh1995]

You have a magnetic field collapsing.

Yes. But how does initial rate of collapsing of magnetic field depend on the external resistance? How does the field decide at what rate to collapse at t=0? Basically, what comes first at t=0? Induced emf or di/dt?

 

[Dale]

For maintaining the current to 1A, emf induced should be 1V

No, for maintaining the current to 1 A the voltage must be 0. Any other voltage will lead to a changing current. The resistor prevents the voltage from being 0, which forces the current to change. The bigger the voltage given by the resistor the faster the current changes.

As a fine point, in circuit theory you just have voltages at terminals. You don't speak of induced EMF since induced EMF is always contained within a circuit element which is treated as a black box with only terminal voltages and currents.

 

[Cutter Ketch]

I believe I can help. To explain, let's ignore the resistor for now. After you open the switch you have to think of the open circuit as a capacitor to ground (or depending on the design of the switch there may be some capacitance to the other side of the circuit too). This capacitance may be minuscule, but it is real, and it is critical for understanding what happens.

At the moment you open the switch there is no charge on this capacitor. It takes essentially no foward emf to push the first charge into the end of the open circuit. The continuing flow through the inductor charges up the capacitor creating a back emf opposing the continued current. The rate of change of the current is the rate of change of the magnetic field in the inductor. The faster the capacitors back emf stops the current, the larger the forward emf of the inductor. These two emfs will balance.

Q=CV so Vback = Q/C

I is defined as dQ/dT so

dVback/dt = I/C

The foward voltage is

Vforward = L dI/dt

So equating the two

L d^2 I / dt^2 = I/C

So I(t) = I0 * exp(-t/sqrt(LC))

and V(t) = I0 * sqrt(L/C) * exp(-t/sqrt(LC))

The resistor complicates things and I'll leave it to you to put in the capacitance to ground and solve that one in a similar way.

I hope that helps

 

[cnh1995]

The bigger the voltage given by the resistor the faster the current changes.

Thanks for the reply.
Say the switch is opened at t=0. How is this event "detected" by the inductor? What happens first? Can I visualize this in terms of "things happening one after another"?
Like just after the switch is opened, does the inductor send some EM signal around the circuit and detect a high resistance path, which makes the initial voltage across the terminals high? How does this interaction occur at t=0? I am asking this because I can visualize how current flows in a normal resistive network using the surface charge feedback mechanism but here, I am unable to visualize how the resistance decides initial di/dt. Is there any such initial mechanism or it's complete nonsense and I have to think differently?

 

[Cutter Ketch]

Or to put it more succinctly, the current will fall at a rate that develops a forward emf in the inductor equal to the back emf of the open circuit.

 

[Dale]

What happens first? Can I visualize this in terms of "things happening one after another"?

No, not in circuit theory. In circuit theory each circuit element is a black box which enforces the current-voltage relationship at its terminals at all times. You cannot say that one causes the other because they all happen at the same time.

Like just after the switch is opened, does the inductor send some EM signal around the circuit and detect a high resistance path, which makes the initial voltage across the terminals high? How does this interaction occur at t=0? I am asking this because I can visualize how current flows in a normal resistive network using the surface charge feedback mechanism but here, I am unable to visualize how the resistance decides initial di/dt. Is there any such initial mechanism or it's complete nonsense and I have to think differently?

You would indeed need to think differently. Specifically, you would need to use the full Maxwell's equations if you wish to consider time scales that are so short that you could distinguish which order things happen. One of the explicit assumptions of circuit theory is that you are only interested in longer time scales.

 

[Dale]

By the way, if you were to use a full EM solution then you would find that the EM wave propagates first from the switch, not the resistor or inductor. The order of the resistor and inductor changes would depend on the distances and conductor geometry. It all gets absurdly complicated.

 

[cnh1995]

By the way, if you were to use a full EM solution then you would find that the EM wave propagates first from the switch, not the resistor or inductor. The order of the resistor and inductor changes would depend on the distances and conductor geometry. It all gets absurdly complicated.

So can I say that using "some sort of EM feedback", the inductor "knows" the resistance of the discharging path and then generates a voltage accordingly at t=0 such that initial di/dt is equal to the voltage across the resistor?

 

[lychette]

Or to put it more succinctly, the current will fall at a rate that develops a forward emf in the inductor equal to the back emf of the open circuit.

This is the best response to a muddled thread !
Suppose that the current falls at a rate to produce an induced emf greater than the back emf
Suppose that the current falls at a rate to produce an induced emf smaller than the back emf...
There is only one logical answer... I think the logical technique is called 'reduction absurdium'...something like that
The circuit 'knows' exactly what to do.

 

[Dale]

So can I say that using "some sort of EM feedback", the inductor "knows" the resistance of the discharging path and then generates a voltage accordingly at t=0 such that initial di/dt is equal to the voltage across the resistor?

I wouldn't say that, I don't like anthropomorphizing if it can be avoided.

If I were using circuit theory then I would only say that the inductor enforces its current-voltage relationship at all times. That is all. Similarly for the resistor. Neither needs to receive any other information from the rest of the circuit in order to do its job. I would explicitly avoid any discussion about information in EM waves in the context of circuit theory.

If I were using Maxwell's equations then I would only say that the fields obey Maxwell's equations at each location and the materials similarly obey their constitutive equations at each location. No knowledge of the rest of the circuit is needed by any part.

 

[Cutter Ketch]

This is the best response to a muddled thread !

Thanks, lychette, even if apparently you are the only one who thinks so!

 

[lychette]

Thanks, lychette, even if apparently you are the only one who thinks so!

If you understand the physics there is only one logical answer.wanting to know how the circuit 'knows' what to do is 'absurd'
...I hold a stone in my hand...I let go...for a moment it is at rest...how does it know when to move...this is a similar 'absurd' question
Questions like this usually lead to 30 - 40 responses...all entertaining and educational

 

[cnh1995]

wanting to know how the circuit 'knows' what to do is 'absurd'

I don't think so. By that logic, everything in physics would be absurd. To answer 'how do things fall on the ground'?, "they just do or nature knows how to behave" are not acceptable explanations. There's an explanation in terms of gravity. I understand that the explanation of gravitation itself may get absurdly complicated but at least we can think of the motion in terms of gravitation and Newton's laws.
I was asking a similar question where I know how an inductor behaves and why, at the macroscopic level with a larger time scale (circuit theory approach), but I was more interested at the microscopic level, which is apparently very difficult to explain(read #15 and #16). But that should not make my question absurd because there is an explanation for it.

 

[Tom.G]

Let's try this approach. Hope it helps.

1) Switch closed, current = 1A, some amount of magnetic field around inductor, steady state.

2) Switch opens. magnetic field starts to collapse, inducing a voltage in inductor (the back EMF.)
2a) (note that without the 1 Ohm resistor, the field collapse would be essentially instantaneous, creating an infinite voltage)

3) With the 1 Ohm resistor in the circuit, the induced voltage causes a current to flow in the resistor.

4) This current, of course, also flows thru the inductor.

5) Since the current flows thru the inductor in the same direction as when the battery was connected, it (conceptually at least) 'replenishes' the collapsing magnetic field.

6) However, the resistor current is dissipating the energy that was stored in the magnetic field.

7) With less energy in the magnetic field there is less power transferred to the resistor, leading to the exponential decrease in voltage and current.

 

[cnh1995]

Let's try this approach. Hope it helps.

1) Switch closed, current = 1A, some amount of magnetic field around inductor, steady state.

2) Switch opens. magnetic field starts to collapse, inducing a voltage in inductor (the back EMF.)
2a) (note that without the 1 Ohm resistor, the field collapse would be essentially instantaneous, creating an infinite voltage)

3) With the 1 Ohm resistor in the circuit, the induced voltage causes a current to flow in the resistor.

4) This current, of course, also flows thru the inductor.

5) Since the current flows thru the inductor in the same direction as when the battery was connected, it (conceptually at least) 'replenishes' the collapsing magnetic field.

6) However, the resistor current is dissipating the energy that was stored in the magnetic field.

7) With less energy in the magnetic field there is less power transferred to the resistor, leading to the exponential decrease in voltage and current.

Thanks for the reply.
Actually that's the approach I already had in mind. I was asking about the interactions at the microscopic level which cause this behaviour, in terms of EM feedback or something similar. But I think I'll drop that approach for now since it is very complicated. I believe my current understanding of inductors is sufficient as far as my EE domain is concerned.

 

[Dale]

. I believe my current understanding of inductors is sufficient as far as my EE domain is concerned

I recommend that approach. There are three EM theories: circuit theory, Maxwell's equations, and quantum electrodynamics. I think that students (with the best of intentions) set themselves up for confusion by attempting to answer questions at one level by appeal to partially understood fragments of a deeper level.