Why does an LC cct (tank cct) oscillate

[curiouschris]

I was reading a very old article this morning about electrical theory, it was talking about the flow of current in a tank circuit otherwise known as a tuned cct. (parallel LC cct).

A strange and disturbing thought struck me. I know *how* a tank cct oscillates. but I don't have the foggiest *why* a tank cct oscillates.

Let me explain further.

The How...

If you charge a capacitor and then attach it to an inductor such that the cct is in parallel, the charge in the capacitor will rush through the inductor creating a magnetic field around the inductor until such time as the capacitor is discharged. when the current stops flowing due to the discharged capacitor, the flux collapses creating a current which then charges the capacitor in the opposite polarity, once the flux has collapsed the capacitor starts to discharge and the cycle repeats.

In a cct with a high Q factor this can continue for quite a while. We will ignore losses due to resistance for now please.

My electronics is a bit rusty but I am pretty sure at the basic level the above describes the process or the HOW a LC circuit resonates.

Now the question is *Why* does it resonate?

Even in a high Q cct, shouldn't the capacitor only discharge to 50% of its charge, where upon the magnetic flux which now contains the equivalent amount of energy as the remaining charge in the capacitor prevent further current flow?

It sort of indicates that electrons are massive and once in motion don't want to stop even when an equal and opposite force is applied.

Perhaps the opposite.

The magnetic field is extraordinarily easy to create, even a small current can support all the potential energy it contains.

But in this case as soon as the flux starts to collapse and the capacitor accepts even a tiny amount of charge it should then prevent the flux from collapsing any further.

I hope I have made myself and my quandary clear.

Can anyone tell me *Why* such a cct can resonate. in fact why does anything resonate, a tuning fork for example?


CC

 

[jambaugh]

It sort of indicates that electrons are massive and once in motion don't want to stop even when an equal and opposite force is applied.

That is what the L part of the LC circuit does. The self inductance of the circuit is the inertial analogue to a mass spring system while the capacitance (reciprocal C) is the elastic component.

Recall that an inductor will produce a voltage in proportion to the change in current and in the direction opposing that change. It is the energy stored in the B field of the inductor that does this (it is the mechanical analogue of kinetic energy). To increase the current requires you apply voltage across the inductor to do work (work = VA) on the inductor increasing its B field while to decrease the current the L circuit must do work using up the energy stored in its B field.

If you imagine the starting point being a charged capacitor and 0 current just as you close a switch shorting the capacitor through the inductor. All the energy is potential energy stored in the E field of the capicator. Then the current builds as the cap discharges until just as the capacitor's charge reaches zero the current is at a maximum and so is the B field. The current then decreases as the waning B field's energy is used to recharge the capacitor in reverse. Then the whole thing swings back completing one cycle.

 

[Antiphon]

Footnote: the electron mass is negligible. All of the inertia-like effects are due to the magnetic field.

 

[curiouschris]

Footnote: the electron mass is negligible. All of the inertia-like effects are due to the magnetic field.

Can you please explain this further?

I guess this is my problem. a magnetic field can't have inertia. nor can an electric current (well a minimal amount) the former has no mass an the latter has negligible mass. So nothing to store kinetic energy in.

What is it then that allows the capacitor to continue discharging into the inductor until the capacitor reaches (near) zero voltage. Surely as soon as the energy in the inductor matches the remaining charge across the capacitor (50%) the magnetic field will start to collapse and thus cancel the current supplied by the voltage across the capacitor.

Why does the inductor 'politely wait' until the capacitor has discharged before it starts to exert its own influence on the cct? and why then does the capacitor 'politely wait' until the energy in the inductor has returned to it before it then starts to discharge back into the inductor. What is the driver behind this politeness?

CC

 

[jambaugh]

Can you please explain this further?

I guess this is my problem. a magnetic field can't have inertia. nor can an electric current (well a minimal amount) the former has no mass an the latter has negligible mass. So nothing to store kinetic energy in.

Right but we aren't talking inertia in the usual sense since we are talking current and not velocity. There is an inertia analogue in the inductance of the coil. So don't think about inertia as a substance but think in terms of how it behaves.

The kinetic energy of a moving mass is $\frac{1}{2}m v^2$. The mass (inertia) is the proportion of energy one gets from the motion.

The energy stored in the magnetic field of an inductor coil is given by $\frac{1}{2}L I^2$ where L is the inductance and I is the current.

This magnetic energy affects the current and charge just as kinetic energy affects velocity and position.

If you apply a voltage to an inductor (with zero resistance) its current can't just instantly jump to infinity because the current generates a magnetic field with field energy given above.
What happens is that the current will increase at a rate such that the increasing B field induces a matching reverse voltage opposing the voltage trying to push the charge through.
This is just like the reaction force when you push on a mass... it pushes back.

Similarly once a current is established if you remove the voltage the current can't just drop to zero instantaneously. The energy in the B field must go somewhere. So what happens is that the current will decrease at a rate producing a forward voltage which together does work on whatever the inductor is connected to...in this case the discharged capacitor.

By the time the inductor's current drops to zero the capacitor is recharged in the opposite polarity.

If in fact the inductor is super-conducting and you short it out then the current will flow forever (in analogue to a mass moving at constant speed).

In short the inductance is not inertia per se but acts just like inertia does on a moving object with respect to moving charge.

Here is what I suggest. Forget your question as stated for now. Don't worry about it because it throws you off. Instead do this...

Picture the tank circuit (LC circuit) as it is described to behave (oscillation) and consider both the energy stored in (the E field of) a capacitor ($E_c = \frac{1}{2C}V^2$)
and the energy stored in (the B field of) an inductor ($E_L= \frac{L}{2} I^2$).

If you have no resistance and no radio waves radiating from the circuit then the total energy must be conserved, right?

As the circuit oscillates the energy is moving back and forth between the inductor and the capacitor. Now work out the text-book voltage and current of an oscillating LC circuit and pick an arbitrary time in the cycle.

Say $V = V_{max} \cos(\omega t)$ and $I = I_{max} \sin(\omega t)$. Let the inductance be 1 henry, the capacitance be 1 farad, and the capacitor initially charged to 1 volt (so it has a charge of 1 coulomb). Then the angular frequency will be 1 radian per second and the period will be 2pi seconds. (I think... double check my memory). Now...

a.) Calculate the energy in each part $E_c$ and $E_L$.
b.) Determine which way the energy is flowing, i.e. which of the two is increasing and which is decreasing.
c.) Determine the power which is the rate at which work is being done on/by each component in terms of voltage times current.

Now work this out for a couple of different points in time and include both the starting time when the capacitor is fully charged and the point in time when the capacitor is fully discharged but the current is at a maximum (and thus the inductor is fully "charged" with magnetic energy).

I think working through the details of a concrete example is the best way to gain full insight into the behavior.

 

[NascentOxygen]

If you push a child on a swing, they don't return to the lowest point and suddenly stop. They overshoot by an equal amount, due to the interplay of gravity and inertia. The same with current. The inductor has current flowing through it, wants it to stay that way, so continues to draw charge from the capacitor until the capacitor is reverse charged to the same potential.

 

[curiouschris]

Thanks jambaugh. I will work through some examples as you have described.

I understand NascentOxygen. An LC cct mimics a swing.

What I am looking for is the reason *why* it mimics a swing. Without momentum how does the energy continue to move into the inductor. The statement "wants it to stay that way, so continues to draw charge from the capacitor" is patently wrong, an electron doesn't want for anything. There is negligible inertia to give the current momentum (as electrons have negligible mass), without which the swing would in fact stop at the bottom, the reason being no mass = no inertia = no momentum.

CC

 

[jambaugh]

Chris,
I don't know what else to suggest other than work through details and reread the posts here... well I though of a few other points to consider and integrate into you understanding. Take them as useful and ignore them if they confuse things...

Here's one thing that might spark some insight. If you have some grasp of how a transformer works, changing current in one coil induces changing voltage in the other, then think of an inductor as a transformer with the primary and secondary coils being one and the same.

The inductor acting as a "self transformer" recharges the capacitor.

Here's another point, though kind of advanced physics. If you've ever heard anyone talking about renormalized masses of particles, this is what is going on. If you treat an electron as a charged sphere of a certain radius, then the electromagnetic field around that electron carries a certain amount of energy and via E=mc^2 that contributes to the effective mass of the electron. The mass we measure is the "bare mass" plus the energy stored in the electron's field. BTW the smaller the radius the more e-m energy since the field energy density grows faster than the scale shrinks.

Now your intuition about the mass of the electrons being too small to account for the "inertia" in the circuit applies to this extra mass if you're talking about electrons traveling individually, but when they travel in concert, and especially when you coil the conductor so the B fields of the many moving electrons both reinforce each other and affect each other, you are increasing this "effective mass" of each electron. In this way we can say "yes it is the 'inertia' of the electrons".

Here's another thought. Look at Maxwell's mechanical model for EM (remembering it is only a model and space really isn't packed full of spinning spheres.) In his model though you can treat electrical current as a current of these spheres, the electric field as the pressure gradient, and the magnetic field as the vorticity. The magnetic permeability and electric permittivity constants are functions of the spheres' moments of inertia and masses. Thus currents induce "real" inertia in this fluid of spheres which is reinforced by coiling the current.

Finally I would suggest you look at the time reversal symmetry for the electrical dynamics of the tank circuit. If you understand the process from the point of maximum capacitor charge up to 0 charge and maximum inductor current. Then you just fold this over and look at the process in reverse. The same force which prevents instantaneous discharge of the capacitor, will in time's reversal recharge the capacitor.

 

[curiouschris]

I am feeling very sheepish this morning. After a few hours extra sleep and a nice warm shower, I realized what my problem was.

I was thinking that the magnetic field that was building around the inductor would at some point reach a strength that would overcome the energy that was creating it. ie the emf in the capacitor, thus prevent the capacitor from ever discharging fully.

What I failed to account for was, yes the magnetic field around the inductor was trying to collapse, but that the voltage was reversed. duh!

To explain further, if you have two connectors A and B the fully charged capacitor is connected to A and B such that the (currently) positive terminal on the capacitor is connected to A and the negative is connected to B.

The inductor is then connected to A and B. Therefore the A side of the inductor is positive and the B side is negative. Current flows through the inductor resulting in a B field building around the inductor.

The B field does try to collapse (which was my point) but rather than opposing the current (which was my folly) it tries to drive the A side to a negative potential. It cannot do this because of the voltage supplied by the capacitor (assuming no resistance in any part of the cct). Thus it *does not* resist the flow of current.

Of course when the capacitor is fully discharged. There is no more capacitor supplied current flowing through the inductor. but the B field is now able to collapse thus current continues to flow, pushing the once positive side of the capacitor negative and the once negative side of the capacitor positive. It is also important to remember that, starting from a fully charged capacitor to the point it is fully discharged is only one quarter of the cycle.


CC

 

[sophiecentaur]

The Swing analogy is fine as long as no one tries to 'make' the electrical quantities exactly the same as the mechanical quantities in the analogy. It is only the relationship that is the same - which all comes down to the common Maths involved in both. (Maths not being magic, of course, just a language for description and prediction)

 

[jambaugh]

After a few hours extra sleep and a nice warm shower, I realized what my problem was. [...]

Great! Yea sometimes an hour's sleep is worth a day's calculations!

 

[NascentOxygen]

The statement "wants it to stay that way, so continues to draw charge from the capacitor" is patently wrong, an electron doesn't want for anything.

Patently wrong, but perfectly correct as an illustrative way for interpreting the mathematics.

I understand NascentOxygen. An LC cct mimics a swing.

Your statement "an LC cct mimics a swing" is patently wrong, an LC cct can't modify its natural behavior to mimic anything.