# resonance

#### PG1995

##### Active Member
Thank you.

For a capacitor i(t)=C*dv/dt and the derivative is maximum is at zero crossing for a sinusoid. i(t)=V/X=10/1000=10 mA.

So, I was right in saying that energy is exchanged twice between inductor and capacitor every half cycle. Thanks.

#### The Electrician

##### Active Member
A lossless parallel resonant circuit has only two nodes, in my example node p and ground (gnd symbol). You can only talk about the voltage at V(p) because by definition V(gnd) = 0.

There is only one voltage to display. The energy exchange is shown by the quadrature relationship between the voltage V(p) and the current I(L1). Note that this is a series circuit, so I(L1) = - I(C1).
How paradoxical! A parallel circuit and a series circuit at the same time!

#### MikeMl

##### Well-Known Member
How paradoxical! A parallel circuit and a series circuit at the same time!
Yes, the capacitor is in "parallel" with the inductor, hence only two electrical nodes in the ideal circuit.
Yes, the capacitor current flows through the inductor and vice-versa, hence the capacitor forms a "series" circuit (one loop) with the inductor...

The schematic representation of lossy "series-resonant" and "parallel-resonant" will differ only after sources and loss elements are added to the circuit model.

#### MrAl

##### Well-Known Member
Thank you, MrAl

It's amazing that you still have your notes.

How is it so? I was thinking more in terms of this picture where capacitor in connected in the middle rather than an inductor. Thank you.
Hi,

Sadly i have notes even older than that one

Maybe you are trying to show the MEASURED polarity?
It's hard to show the measured polarity because it may change over time from plus left minus right to plus right minus left (graphically).

My point is like this...
If i tell you i have a circuit where we have just one 5v DC voltage source and once capacitor and the capacitor voltage we will call vL, what is the polarity of the capacitor voltage? You can not tell me because i did not yet assign any reference polarity to the capacitor voltage. Note it is a DC source yet you still cant tell me because we did not show how we wish to interpret that voltage yet.

If we draw the circuit and show everything, then you can tell me what it is. But then if we reverse the DC source polarity then we have to reverse the cap polarity so it becomes depending on the direction of the source.

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#### MikeMl

##### Well-Known Member
...

So, I was right in saying that energy is exchanged twice between inductor and capacitor every half cycle. Thanks.
Half, or Full?

#### MikeMl

##### Well-Known Member
...
If i tell you i have a circuit where we have just one 5v DC voltage source and once capacitor and the capacitor voltage we will call vL, what is the polarity of the capacitor voltage? You can not tell me because i did not yet assign any reference polarity to the capacitor voltage. Note it is a DC source yet you still cant tell me because we did not show how we wish to interpret that voltage yet.....
There is never any ambiguity if you mark one of the nodes (can be any of the circuit nodes) with a ground symbol, and follow the convention in post #16!

#### MrAl

##### Well-Known Member
Hello again,

About the "parallel" circuit with just L and C....

The capacitor voltage is easiest to analys
There is never any ambiguity if you mark one of the nodes (can be any of the circuit nodes) with a ground symbol, and follow the convention in post #16!
Hello,

Well a convention of any kind is what clears it up, and that was my point.

#### MrAl

##### Well-Known Member
Hello again,

For the "parallel" LC circuit...

First, the cap voltage is easier to analyze and with initial energy in the cap only that response is of the form:
vC(t)=A*cos(t/sqrt(LC))

where we can see it is just a cosine wave where w=1/sqrt(LC) as expected.

The energy in the cap with initial voltage E in the cap only is:
W=E^2*C(cos((2*t)/(sqrt(C*L))+1)/4

and comparing with above we see that this has frequency that is twice that of above because of the "2*t" rather than just "t" so the energy transfers at a 2x rate.

It's the same if there is initial current in the inductor or both elements have initial energy but the expression is more complicated.

Note that if we evaluate that at t=0 we get:
W=(C*E^2)/2

which we usually see as (1/2)* C*V^2 which is the maximum energy in the cap. So the total energy stays constant and is the same as the initial energy injected into the system via the initial cap voltage. The total energy would be different if we had initial current in the inductor also.

Again a little circuit analysis shows exactly what is happening. This was a simpler circuit though so it was not too hard to do and extract the most important ideas from. As the circuits get more complicated this gets harder to do of course, but it provides a really good overall view of what is happening most of the time.

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#### PG1995

##### Active Member
Thank you, MikeMl, MrAl.

Half, or Full?
Please have a look on the attachment.

Point A: The capacitor has fully discharged. The inductor takes over and keep pushing the current in the same direction.
Point B: The magnetic field of indcutor has fully vanished and hence no more stored energy. The capacitor starts pushing the current in the other direction.
Point C: The capacitor has fully discharged and the inductor takes over and keeps the current flowing in the same direction.
Point D: The inductor has fully discharged and at this point the capacitor takes over.

The energy exchange takes place four time in a single cycle; therefore it was really "half cycle". Please let me know if I'm wrong. Thank you.

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#### MikeMl

##### Well-Known Member
I would just say that the capacitor and inductor exchange their energy twice per cycle. Added a plot of 1/2CV^2 and 1/2LI^2 to the previous plot...
peaks at 7.95uJ

#### PG1995

##### Active Member
Thank you but sorry I'm having difficulty understanding it. I still think that energy is exchanged four time each cycle between inductor and capacitor.

Between point T and A: The capacitor delivers its energy to inductor.
Between point A and B: The inductor delivers its energy to the capacitor.
Between point B and C: The capacitor delivers its energy to the inductor.
Between point C and D : The inductor delivers its energy back to the capacitor.

Could you please guide me where I'm going wrong with it? Is it just a difference of viewpoint? Thank you.

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#### MrAl

##### Well-Known Member
Thank you, MikeMl, MrAl.

Please have a look on the attachment.

Point A: The capacitor has fully discharged. The inductor takes over and keep pushing the current in the same direction.
Point B: The magnetic field of indcutor has fully vanished and hence no more stored energy. The capacitor starts pushing the current in the other direction.
Point C: The capacitor has fully discharged and the inductor takes over and keeps the current flowing in the same direction.
Point D: The inductor has fully discharged and at this point the capacitor takes over.

The energy exchange takes place four time in a single cycle; therefore it was really "half cycle". Please let me know if I'm wrong. Thank you.
Hi,

That's not what appears to be happening. If you look at Mike's plot you see the EXCHANGE happen just two times for one cycle, and if you look at the mathematical derivation you see the "2*t" instead of just "t" in the cos() expression which says that there is something happening twice as often not four times as often.

It's not a matter of following the peaks, it's a matter of following the entire exchanges. For one cycle, the cap has max energy just twice so the exchange can only be twice per cycle. The total energy is handed to the inductor twice per cycle, and is handed to the cap twice per cycle. That does not add up to four times.

If at t=1 second i give you an apple, then at t=2 seconds you give it back to me, that's ONE exchange of the apple not two.

#### PG1995

##### Active Member
Thank you.

I think that it's a matter of viewpoint.

The energy exchange between electric field of capacitor and magnetic field of inductor takes place twice each cycle. As an analogy we can look at a pendulum example where energy exchange between potential energy and kinetic energy takes place twice. You can have a look here: https://www.allaboutcircuits.com/textbook/alternating-current/chpt-6/electric-pendulum/

From my viewpoint, energy exchange between the capacitor and inductor takes place four times each cycle but the exchange between 'electrically' stored energy and 'magnetically' stored energy takes place twice each cycle. Thanks a lot.

#### MrAl

##### Well-Known Member
Thank you.

I think that it's a matter of viewpoint.

The energy exchange between electric field of capacitor and magnetic field of inductor takes place twice each cycle. As an analogy we can look at a pendulum example where energy exchange between potential energy and kinetic energy takes place twice. You can have a look here: https://www.allaboutcircuits.com/textbook/alternating-current/chpt-6/electric-pendulum/

From my viewpoint, energy exchange between the capacitor and inductor takes place four times each cycle but the exchange between 'electrically' stored energy and 'magnetically' stored energy takes place twice each cycle. Thanks a lot.
Hi,

Well if we wanted to we could say that the cap and inductor change energy an infinite number of times because at say t=1us the C had some energy and the L has some other energy level, then at t=2us the cap gives a little more to the inductor, then at t=3us the cap gives still more to the inductor etc., etc., but that's not a total exchange. A total exchange means that at some point the cap has all the energy and at another point in time the inductor has all the energy. That's a total exchange, and that happens exactly two times per electrical cycle.

If i wanted to make up my own definitions i could say that the sun rises twice per day and sets twice per day
Now that might fly, but i'd have to have some proof and show how we could get away with that viewpoint, otherwise i would be the only person who thinks that is true.

You might show an example where your view makes sense if you like.

#### PG1995

##### Active Member
Thank you, MrAl.

A total exchange means that at some point the cap has all the energy and at another point in time the inductor has all the energy.
I'd refer back to my post #31.

How much energy does the capacitor have at point "A"? And how much energy does the inductor have at point "B"? Thanks a lot.

#### MikeMl

##### Well-Known Member
Your point A is at 0.25ms; B is at 0.5ms.

The energy in the inductor is (LI^2)/2, the energy in the capacitor is (CV^2)/2

In the plot in post #30, the blue trace is the inductor energy, and it peaks at 0.25ms(A) and 0.75ms. It peaks at 7.95uA^2= 7.95uJ.
In the plot in post #30, the red trace is the capacitor energy, and it peaks at 0ms, 0.5ms(B) and 1ms. It peaks at 7.95uV^2=7.95uJ.

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#### MrAl

##### Well-Known Member
Thank you, MrAl.

I'd refer back to my post #31.

How much energy does the capacitor have at point "A"? And how much energy does the inductor have at point "B"? Thanks a lot.

Hi,

Yes i saw that. That shows voltages and currents not the energy itself. These next two diagrams show the energy itself over one cycle between the violet 'bars'. You need to look at the energy itself to know how many times it is completely exchanged in one cycle. The violet bars enclose one cycle only.

Note in the first diagram _1 the cap has maximum energy exactly two times in that one enclosed cycle. That's not four times.

vC is the capacitor voltage to use as a reference as that shows the oscillation cycle.

Note that a complete "exchange" is not the same as simply counting the peaks of each energy cycle. Just because there are four peaks total that does not mean that they exchanged energy four times. An exchange is a view of two things at once not just one thing at a time. This is why i show the cap energy ONLY in the first pic, so you can see that it only has max energy twice per cycle more clearly. The cap has all the energy twice and the inductor has all the energy twice, and that is an exchange of two times not four times.

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#### PG1995

##### Active Member
Thank you, MikeMl , MrAl.

I'm sorry about the confusion but I was completely wrong. I don't know what I was thinking at that moment.

Best wishes,
PG

#### PG1995

##### Active Member
Hi,

Please have a look on the attachment. I'm not sure how they are getting to this final expression. I had thought that perhaps they started with the 'initial expression' and after simplification got this final expression but it looks like I was wrong. You can see below that I tried to simplifyy the initial expression but it ended up to something different from final expression. Thank you!

Note to self:
Q (the “quality factor”), is a measure of how much energy is not lost in a reactive element. The higher the Q, the less energy is lost. Quality factor exist for inductor as well as capacitor. Q_C=Xc/Rc=1/(2*pi*f*C*Rc) and Q_L=X_L/R_w=2*pi*f*L/Rw where Xc=1/(2*pi*f*C) is capacitive reactance, Rc is equivalent series capacitor resistance, X_L=2*pi*f*L and Rw is equivalent series winding resistance. When the resistance is just the winding resistance of the coil, the circuit Q and the coil Q are the same.

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#### Ratchit

##### Well-Known Member
Hi,

Please have a look on the attachment. I'm not sure how they are getting to this final expression. I had thought that perhaps they started with the 'initial expression' and after simplification got this final expression but it looks like I was wrong. You can see below that I tried to simplifyy the initial expression but it ended up to something different from final expression. Thank you!

Note to self:
Q (the “quality factor”), is a measure of how much energy is not lost in a reactive element. The higher the Q, the less energy is lost. Quality factor exist for inductor as well as capacitor. Q_C=Xc/Rc=1/(2*pi*f*C*Rc) and Q_L=X_L/R_w=2*pi*f*L/Rw where Xc=1/(2*pi*f*C) is capacitive reactance, Rc is equivalent series capacitor resistance, X_L=2*pi*f*L and Rw is equivalent series winding resistance. When the resistance is just the winding resistance of the coil, the circuit Q and the coil Q are the same.
Your attempt to derive the resonant frequency with resistance present in the coil tells me that you do not quite understand the principle of parallel resonance (PR). Unlike series resonance, PR depends on the different amplitudes and phases of the currents in the branches. Lets start out with the admittance of the coil with resistance and capacitor without resistance in parallel.

To achieve resonance, the orthogonal term designated by "j" must be zero.

Substituting into the above equation the formula for reactance.

And, we get the formula you are seeking.

If resistance is allowed in the capacitor branch, then it is possible for the circuit to be resonant at all frequencies by adjusting the two resistances.

Ratch