Kelistrikan dan Kemagnetan

**Topics covered:**

Driven LRC Circuits

Resonance

Metal Detectors (Beach/Airport)

**Instructor/speaker:** Prof. Walter Lewin

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So we have covered RC circuits and RL circuits, and today, we will spend the entire lecture on LRC circuits.

We will only discuss them in series so that you get the basic idea.

I have here a driving power supply, alternating, and here I have a capacitor C, self-inductor L, and a resistance R, this is AC, and let the driving voltage be V0 cosine omega T.

We have to set up the differential equation for this, and I want to remind you that Kirchhoff's Loop Rule does not hold.

So the closed loop integral of E dot dL, in spite of what the author of your book wants you to believe, that is not 0.

So how do we set it up?

There are various ways that you can do that, I have my own discipline.

I, in my mind, I think of this first being a, a battery -- by this is the plus side, and this is the minus side -- a current is going to flow, capacitor is going to charge up, electric field inside the capacitor is in this direction, the electric field in the self-inductor is always 0, because the self-inductor has no resistance.

There's no electric field inside the self-inductor, no matter what some of your books want you to believe.

Then, the electric field in the resistor is in this direction, and the electric field inside the power supply goes from plus to minus, would be in this direction.

So if I set up the differential equation, I start here, I always go in the same direction as I, because only then is the closed loop integral -L dI/dt.

So I go over this capacitor, that is V of C, then I go through the wire of the self-inductor.

There is no electric field, so the integral E dot dL there is 0.

Then I go through the resistor, so I get IR, and then I have here my power supply, so I get -V0 cosine omega T, and that, now, according to Faraday's Law, equals L, minus L dI/dt.

The current equals dQ/dt.

If the current is positive -- this is my positive direction -- then the charge of the capacitor will increase.

And I also know that V of C, the potential difference over the capacitor is the charge on each one of the capacitor plates, divided by C.

And so I substitute that in this equation, and I bring the L dI/dt to the left side.

That is conventionally done.

You don't have to do that, but that's often done.

So I get a plus L, dI/dt now becomes d2Q/dt squared -- my goal is to get everything in terms of Q -- then my IR become R dQ/dt, and my V of C becomes Q / C -- notice that I ranked them in order, D2Q/dt squared, dQ/dt, and then Q, you don't have to do that, but there is nothing wrong with doing that -- and then we get here, equals V0 cosine omega T.

And this is the form in which most books would present this differential equation.

And they arrive that in various ways, most books arrive at this equation in a completely wrong way, but they get -- anyhow, they end up with this equation.

And so, you have to solve this equation, which is really beyond your present abilities, it's second-order differential equation, it's really part of 18.03, so I will give you the solution.

The basic idea being that you find a solution for Q as a function of time, and once you know Q as a function of time, you have, of course, the current, because then you take the derivative of your solution, and you get the current.

I will give you the current as a function of time.

So I, that satisfies that differential equation, is the V0 divided by [whistles] R squared + omega L - one over omega C squared, and the whole thing times cosine omega T minus phi.

And the tangent of phi equals omega L minus one over omega C divided by R.

We give this upstairs here a name, we call that the reactance.

The reactance, and that X, or sometimes it's called chi, is omega L minus one over omega C.

And the units are also ohms.

We call the entire square root that you see here, we call that capital Z, which is called the impedance, so the square root of R squared plus that X squared equals Z, that also has units of ohm, and that is called the impedance.

And so Z is an effective resistance, because this whole thing behaves like a resistance.

But the resistance depends not only on R, L and C, but also on the values of omega.

This solution is what we call a steady-state solution, it is the solution that you get if you wait a certain amount of time.

If you turn the instrument on, so you all of a sudden start this experiment, then in the beginning, you get a different solution, which is more complicated, you get transient phenomenon, but these transient phenomenon die out, and you end up with this solution.

Now, there are several interesting things that you can see in this solution.

We have to start digesting, this whole hour, this solution.

It has very interesting aspects.

For one thing, you can see that the current can be delayed over the driving voltage when phi is positive.

Then the current comes later than the voltage.

And that's the result of the inductor, we've discussed that before.

But now, that's also possible that the current is leading the voltage, which is very hard to understand intuitively.

That is the case when this term dominates over this one, then phi becomes negative, and so minus phi becomes positive.

If minus phi is positive, the current is leading the voltage.

Now you may say, "How can it possibly be?

Does that meant that before I switch the instrument on, that I already have a current?" Of course it doesn't mean that.

But that's the transient solution, remember?

When you turn something on, when you switch it on, this solution doesn't hold yet.

This is the steady-state solution.

So the value for I Max, we have always called what is front of the cosine term, we've always called that I Max, that value for I Max is a function of omega itself -- as we will analyze in detail today -- and of course, also, of R, L and C.

And there is one particular value for Z, and therefore for omega, whereby this value reaches a maximum, and that's what we call resonance.

There is no value for omega for which the current is any higher.

And so I will call here, the situation, at resonance.

It is at resonance when X equals 0, so when omega L is one over omega C, so when omega is one over the square root of L C.

And we call that the resonance frequency, and we often give a little subscript 0 there to remind you that you're dealing with the resonance frequency.

And Z is then just R, because when X is 0, the omega L and the one over omega C eat each other up.

They are not there any more, it's gone.

And so the system behaves as if there were only a resistor.

And so you also see that the maximum current that you get is then, simply, V0 divided by that value for R, because Z, the impedance, is now R.

And in addition, if you're interested in phi, phi then becomes 0, so the driving voltage is then in phase with the current that follows.

And so the signal that you will see is a cosinusoidal variation in the current, so if I have here the current as a function of time, and you get a signal like so, and this here, this period T equals your 2 pi / omega.

So that is the -- directly connected to your driving frequency.

And if the impedance Z is very low, then this maximum value of the current, this is what we call the maximum value -- and, of course, the maximum value is also here, except that the cosine is -1 here, and the cosine is +1 here -- so if Z is very low, then this will be high.

If Z is very high, this will be low.

And there is only one and one value of Z for which the system is at resonance, and that is when the self-inductance and the capacitor eat each other up, and then you get the maximum possible value for the current at maximum, which is V0 over R.

And that's the highest value that you could ever get them.

Imagine that we have an LRC circuit, and we have L and R and C fixed, but we change the driving frequency.

So we move over various values of Z by changing omega from a very low value to a very high value.

If you start at a very low value for omega, let's say it approaches 0, then notice that Z goes to infinity, and so the maximum current becomes 0.

And the person responsible for that is the capacitor, because if omega goes to 0, this goes to infinity.

And that's intuitively pleasing, because omega 0 really means you have no AC any more, you have DC.

And with DC, what you're doing is, you charge up the capacitor when it's fully charged, no current can flow any more.

So that's intuitively pleasing.

When omega becomes very high, let's call it infinity, then Z, again, goes to infinity.

So again, the maximum current, again goes to 0.

And the person responsible for that is the self-inductor, because when omega goes to infinity, again, Z goes to infinity.

So again, you get 0 here.

And that's also intuitively pleasing, because if you have an infinitely high frequency, that means the self-inductance puts up an enormous fight.

It's ideal for a self-inductor to fight currents if the time over which the changes occur go to 0.

And so, then, again, it says, "Sorry, you can't have any current." So that's also intuitively pleasing, that the self-inductance, then, becomes the dominant factor.

And so what I can do now, I can plot the I Max as a function of omega.

So here is omega, and here is I Max, and we already agreed that when omega is 0, then I Max is 0.

But when omega is very high, it's also 0.

But when omega is at resonance, omega 0, which is one over the square root of L C -- notice that R has nothing to do with the resonant frequency, it's really determined by L and C, because it's the chi, it's the X that you want to make 0, and X is only a function of L and C -- at this frequency, we have a value here which is V0 divided by R.

And so the curve that you're going to see, which we call the resonance curve, is something like this.

You start out with an extremely small current, you go through resonance, we have a high current, and then at high frequencies, again, you go down to 0.

And so the left part, when you are below resonance, it's really the capacitance which is the dominant guy in the whole game -- and phi, by the way, is here, less than 0 -- here it is the inductor that plays the key role, and here phi equals larger than 0, and right here, phi, and only there, phi is 0, only when you're exactly at resonance.

I'd like to show you some numerical results, and for that I have a transparency -- it's also on the web, so you don't have to copy the numbers, uh, you can download them -- these are just some numerical numbers which I want to digest with you, so that you get a feeling for the effect, that you see it in front of your own eyes, what is happening, how this curve evolves.

We have here a given R, L, and C: ten, 5 times 10 to the -2 henry, and 3 times 10 to the -7 farads.

The resonant frequency is a little over 8000 radians per second, you see it here in kilohertz, and you see here the impedance -- and what I do here, I have a driving frequency which is 10% below the resonance frequency.

And I calculate for you, the omega L, which is 367 ohms, and one over omega C, which is 453 ohms.

You are a little bit below resonance, and so C dominates.

And you can see, indeed, that this ohm value is larger than this one.

And so out of that pops a value for X, out of that pops a value for Z.

Notice that X is 86, and Z is only a hair larger than 86, because this R almost doesn't add to Z, because you get here the square root of 10 squared plus 86 squared, that is almost 86.

It becomes 87.

And then you see that the current, the maximum current, which is this value for V0 divided by, uh, the Z, by 87, becomes 0.11 amperes.

And now, the system is driven at resonance, and notice that it's exactly characteristic for resonance that omega L and one over omega C have the same value.

They are not there any more, they're gone.

And so X becomes 0, so the impedance becomes ohm -- 10 ohms, which is the resistance, and so the maximum current is now V0 divided by R, which is 1 amperes.

And when you're 10% over resonance, then the self-inductor becomes to be more powerful than the capacitor, and again, your current is substantially down, in this, case, 8 times lower than at resonance.

We define, at a height of 0.7 times the value at resonance, we define a width of this curve.

And this width is given in terms of delta omega.

And that width -- and I will give you the answer without mathematical proof, it's not so difficult, but it's a little bit of a headache -- that value is R divided by L.

So the larger R is, the broader it becomes.

So if we look at delta omega, for the numbers that we have there, the numbers of the transparency -- so this is for, for the numbers that we have there, we have delta omega, would be R, which is 10 ohms, divided by 5 times 10 to the -2, and that is about 200 radians per second.

We define Q not as charge -- don't never confuse that with charge -- we call that the quality of the resonance, and the quality is defined as omega 0 divided by delta omega.

Now, omega 0 itself is one over the square root of L C, and delta omega is R divided by L.

And so that makes the quality 1/R times the square root of L/C.

And the quality is the measure for omega 0, which is this, what I'm pointing at now, divided by delta omega, which is this.

So if the quality is high, this peak is relatively narrow, and if the quality is low, it's relatively wide.

You may ask yourself the question, why do we define delta omega at 70% of the maximum current at resonance?

Why not at half?

There's a good reason for that, because, in practice, we are more interested in power than that we are in currents.

And power is proportional with I squared.

And so when you square this, you get 0.5.

And 0.5 means, then, that this is really the width at half-power.

And so that's the reason why we chose the 0.7 times the maximum current at resonance.

It's really the half-power width.

Resonance can be destructive.

Uh, imagine, if you have a very high-Q system, if you're slightly off-resonance, there's almost no current, no power dissipated in your resistor, and now, you come, all of a sudden, on the resonance, you can an enormous current, and that means there's an enormous power dissipation in your resistor, and you can burn out your resistor.

You can destroy your circuits, if you're not careful.

And next lecture and Monday, I will also discuss with you some med- mechanical resonances.

Mechanical systems can also go into [unintelligible] can also be destructive.

At certain frequencies, the systems behave -- call it k- violently, they respond extremely strongly to their input frequency, and things can break.

Humans also have resonance frequencies, you can call them, if you want, emotional resonances.

All have sensitive nerves.

Someone makes a particular remark, go through the roof.

Also, falling in love, when you think about it, is a resonance phenomenon, and that, too, can be rather destructive.

As many of us know.

But now I would like to demonstrate to you the resonance curves -- I'm going to choose particular values of, um, R, L, and C, which I can change, and then I will show you the current as a function of frequency.

And these are the values that I have chosen.

Again, this is on the Web, you can download it, so you don't have to copy it now.

And I will change the -- the light setting so that we can also enjoy the demonstration.

The idea being that, for these values that I have there, in the first line you see R, 60 ohms, and the self-inductance is 50 millihenry, and the capacitance is 0.3 microfarads.

So that's a given there.

And I give you here the resonance frequency, 8000, in terms of omega radians per second, this is the resonance frequency in Hertz -- and just in case you're interested, I gave you the Q value there as well.

And what I'm going to do now for you, is I'm going to sweep the input frequency from 0 to 16000 radians per second.

So my omega can go from 0 to 16000.

And I leave the values as they are, here.

So I'm going to sweep, sweep over this 8000.

And so you're going to see that curve.

Except that I'm show -- I'm going to show you I as a function of frequency, not I Max.

And I is oscillating, because there's a cosine term.

And so, for instance, if I were here, with this value for omega, you would see then that it goes up, it goes down, it goes up, it goes down, it goes up, and it goes down.

And when I'm here, you will see this.

And keep that in mind when you look at the curve that you're going to see there -- and so it's only the envelope, then, that is the I Max.

But you actually see the entire current as a function of frequency.

And I am going to do that, then, for all these four values that you see there.

So, let's first change the light so that we get an optimum situation for you.

And now, I will show you.

Already, the results of the first line -- so these are the values that you see there.

And I go -- I sl- I go very slowly.

Pengembangan Perkuliahan

1. Buatlah sebuah Esai mengenai materi perkuliahan ini

2. Buatlah sebuah kelompok berjumlah 5 orang untuk menganalisis materi perkuliahan ini

3. Lakukan Penelitian Sederhana dengan kelompok tersebut

4. Hasilkan sebuah produk yang dapat digunakan oleh masyarakat

5. Kembangkan produk tersebut dengan senantiasa meningkatkan kualitasnya#### Staff

Visualizations:

Prof. John Belcher

Instructors:

Dr. Peter Dourmashkin

Prof. Bruce Knuteson

Prof. Gunther Roland

Prof. Bolek Wyslouch

Dr. Brian Wecht

Prof. Eric Katsavounidis

Prof. Robert Simcoe

Prof. Joseph Formaggio

Course Co-Administrators:

Dr. Peter Dourmashkin

Prof. Robert Redwine

Technical Instructors:

Andy Neely

Matthew Strafuss

Course Material:

Dr. Peter Dourmashkin

Prof. Eric Hudson

Dr. Sen-Ben Liao

#### Acknowledgements

The TEAL project is supported by The Alex and Brit d'Arbeloff Fund for Excellence in MIT Education, MIT iCampus, the Davis Educational Foundation, the National Science Foundation, the Class of 1960 Endowment for Innovation in Education, the Class of 1951 Fund for Excellence in Education, the Class of 1955 Fund for Excellence in Teaching, and the Helena Foundation. Many people have contributed to the development of the course materials. (PDF)

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