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 A C,
and that the driving voltage be V zero cosine omega T.
We have to set up the differential equation for this,
and I want to remind you that Kirchoff's Loop Rule does not
hold. So the closed loop
integral of E dot D L, in spite of what the author of
your book wants you to believe, that is not zero.
So how do we set it up? There are various ways that you
can do that, I have my own discipline.
I, in my [unintelligible], 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 zero, 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 minus L D I D T. 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 D L there is zero.
Then I go through the resistor, so I get I R,
and then I have here my power supply, so I get minus V zero
cosine omega T, and that,
now, according to Faraday's Law, equals L minus L D I D T.
The current equals D Q D T. 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 D L D I D T 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, D I D T now becomes D two Q D T
squared -- my goal is to get everything in terms of Q -- then
my I R become R times D Q D T, and my V of C becomes Q divided
by C -- notice that I ranked them in order,
D two Q D T squared, D Q D T, and then Q,
you don't have to do that, but there is nothing wrong with
doing that -- And then we get here, equals V zero 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
eighteen oh three, 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 V zero divided by [whistles] R squared plus omega
L minus 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 xi, 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, as resonance.
It is at resonance when X equals
zero, 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 zero to remind you that you're dealing with the
resonance frequency. And Z is then just R,
because when X is zero, 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,
V zero 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 zero, 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 two
pi divided by 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 vale -- and, of course, the maximum value is
also here, except that the cosine is minus one here,
and the cosine is plus one 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 V zero 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
zero, then notice that Z goes to infinity, and so the maximum
current becomes zero. And the person responsible for
that is the capacitor, because if omega goes to zero,
this goes to infinity. And that's intuitively
pleasing, because omega zero really means you have no A C any
more, you have D C. And with D C,
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 zero. 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 zero 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 zero. 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 zero,
then I Max is zero. But when omega is very high,
it's also zero. But when omega is at resonance,
omega zero, 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 side,
it's the X that you want to make zero, and X is only a
function of L and C -- at this frequency, we have a value here
which is V zero 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 zero.
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 zero -- here it is the inductor that plays the key
role, and here phi equals larger than zero, and right here,
phi, and only there, phi is zero,
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, five times ten to the minus two
Henry, and three times ten to the minus seven farads.
The resonant frequency is a little over eight thousand
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 ten percent
below the resonance frequency. And I calculate for you,
here omega L, which is three hundred sixty
seven ohms, and one over omega C, which is four fifty three
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 eighty six,
and Z is only a hair larger than eighty
six, because this R almost doesn't add to Z,
because you get here the square root of ten squared plus
eighty-six squared, that is almost eighty six.
It becomes eighty seven. And then you see that the
current, the maximum current, which is this value for V zero
divided by, uh, the Z, by eighty seven,
becomes oh point one one 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 zero,
so the impedance becomes ohm -- ten ohms, which is the
resistance, and so the maximum current is now V zero divided by
R, which is one amperes. And when you're ten percent
over resonance, then the self-inductor becomes
to be more powerful than the capacitor, and again,
your current is substantially down,
in this, case, eight times lower than at
resonance. We define, at a height of oh
point seven 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 ten ohms,
divided by five times ten to the minus two,
and that is about two hundred 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
zero divided by delta omega. Now, omega zero itself is one
over the square root of L C, and delta omega as R divided by
L. And so that makes the quality
one over R times the square root of L over C.
And the quality is the measure for omega zero,
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 seventy percent 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 point five.
And point five means, then, that this is really the
width at half-power. And so that's the reason why we
chose the oh point seven 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 sixty ohms, and the self-inductance is
fifty milliHenry, and the capacitance is point
three microfarads. So that's a given there.
And I give you here the resonance frequency,
eight thousand, 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 zero to
sixteen thousand radians per second.
So my omega can go from zero to sixteen thousand.
And I leave the values as they are, here.
So I'm going to sweep, sweep over this eight thousand.
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.
Now omega is zero here, omega goes up,
I go through resonance, and omega is here,
the value would be sixteen thousand radians per second that
we have here. And it sweeps back and forth
between zero and sixteen thousand.
So you see a dramatic increase at resonance.
So now what I'm going to do, I'm going to double the
self-inductance. And when you double the
self-inductance, this one goes up,
omega zero goes down. So all I want you to see,
that the resonance frequency, which is here -- that's the
maximum -- that the resonance frequency will shift.
Because if L goes up by a factor of two,
the resonant frequency will come down by the square root of
two. And so I am going to increase L
by a factor of two -- let me make sure that I have the right
know here -- and this is my L. Notice that the
f- the resonant frequency is now at a lower value,
it is here. Also notice that the peak value
at resonance has not changed. Because the peak value at
resonance, you see on the blackboard here,
is V zero over R. So as long as I don't change R,
that doesn't change. It's only the frequency that
changes. Omega zero is one over the
square root of L C. That changes.
It is now here. So I have increased L by a
factor of two, I can bring it back to my
original resonance frequency by now changing C.
If you increase by a factor of two, all you have to do is
decrease C by a factor of two, and you're back at the same
resonance. So I'm going to make C down --
C lower by a factor of two, which I'm doing now,
and if you look, now, here, and you have a good
memory, you will see that the resonant frequency is back where
it was. Again, V zero over R is not
changed, and the resonant frequency is
back here, even though L is now twice is higher,
and C is twice lower. To show you the effect of R,
I will double R now, and I will leave everything
else alone, so the resonance frequency will stay here,
but of course the maximum current -- this high value will
now come down, because, you see,
at resonance, this value is v zero divided by
R, and since R goes up by a factor of two,
you will see that the maximum current will go
down by a factor of two. And so, I go now from sixty --
from fifty -- oh, no, I go from sixty to a
hundred, I don't double it -- I can't go any further than a
hundred. But you'll see a substantial
reduction. So if you're ready for this --
remember this height -- and now you see hundred ohms,
and it is much lower. It was this high before,
and now it's only here. But notice the resonance
frequency has not changed. So this is an extremely
interesting behavior, and every time,
you have to think through what is happening,
you have not much intuition for it, you're not alone,
I don't have much intuition for that either.
But with these rather simple equations -- they're really not
that difficult -- here, this is really the heart of the
equation, and then, of course, your tangent phi,
in case you're interested in the phase lag -- that they're
really not that difficult. I will not hold you responsible
for being able to derive the r- solution.
I give you the solution. But with that solution,
you can do a lot, and you can understand the
behavior of these circuits quite well.
Now I'm going to do a demonstration,
and I warn you, you have to very closely
follow, step-by-step, what I'm going to do.
Because if you miss one small step, you're lost for the next
twelve minutes. Completely lost.
You will just see nice things, but you don't know what you're
looking at. So follow me closely.
I have an LRC circuit. And the LRC circuit is right
here. This is my L,
oh point one Henry, these are my Cs,
and this is my R, it's a two-hundred watt light
bulb. I have an LRC circuit.
And I will give you the values fro L, for R,
and for C. I'm going to drive it at sixty
Hertz. So omega equals three hundred
and seventy seven radians per second.
And whatever you're going to see, for the next twelve
minutes, that is not going to change.
That frequency is a given. I simply plug it into the wall,
and so my driving voltage is hundred and ten times the square
root of two, times the cosine of omega T, and it is this omega.
So it's the hundred ten volts that comes out of
the wall, so to speak. My light bulb R is sixty ohms
when it is hot, and it is a two hundred watt
bulb. So if it's -- bright
two-hundred watt bulb. The self-inductance L is oh
point one Henry, and the capacitance is eight
microfarads. So this is God-given.
With these three values, I can calculate what Z is,
and Z is three hundred ohms. In case you're interested,
I can also give you omega L, that is thirty-eight ohms,
and I can give you one over omega C, that is three hundred
and thirty two ohms. So notice that this is,
by far, the dominant player in the game, compared to omega L.
In case you're interested in the resonance frequency,
omega zero is one over the square root of L C,
and that is about eleven hundred and twenty radians per
second. In other words,
my driver is nowhere near resonance.
I am way below resonance. When you're way below
resonance, it is the C that dominates.
Look at the three hundred and thirty two ohms,
and compare that with the thirty-eight.
I can calculate, now, what the maximum current
is, that we get, I Max.
That is V zero, divided by Z,
that impedance, so that is hundred and ten
square root two, divided by three hundred ohms,
and that is oh point five amperes.
And now I can tell you how much power is dissipated in the light
bulb, which is I square R. So in the light bulb,
which is the only component that has a resistance -- well,
maybe the L has, also, a little bit of
resistance, but we will ignore that for now -- so in the light
bulb, the average power over one full cycle of my oscillations
equals the mean value between I squared R, time averaged,
the time average value of cosine omega T squared is always
one-half, so this one-half that comes here is the result of the
fact that the I there has a cosine omega T,
and you square that, you get one half,
if you average it over oscillations,
so now I get I Max squared times R.
So this is one-half times my oh point five squared,
and the R of the light bulb is sixty, and this turns out to be
seven point five watts. Nothing.
Why is it nothing? Well, that's the way we
designed this system.
We're way below resonance. So if I plug the system in,
which I will do for you, this two-hundred watt light
bulb will only dissipate seven and a half watts,
and you won't see anything. You ready for this?
Three, two, one, zero.
Physics works. You see nothing.
Light bulb, not working. Great.
So now comes the question, what could we do
to get the system back on resonance?
Well, we know that our driving frequency is three hundred and
seventy seven radians per second.
And so what we could do, we could change the C in the
circuit, and we could change L in the circuit.
If we increase C, or we increase L,
then the resonance frequency will shift from eleven twenty
down, make L larger, or make C larger,
then, obviously, you shift your resonance
frequency down. And my goal is to shift it down
to three hundred and seventy seven radians per second.
I first want to make a curve for you here,
very roughly, of I Max as a function of
frequency, omega, and that's the de-
demonstration I just did. When this was eleven hundred
and twenty, we had about oh point five amperes at R value of
three hundred and seventy seven. So this is one point over I Max
versus omega curve. I can now ask the question,
if we drive the system at resonance, what then,
would have been the maximum current?
We're not driving it at resonance.
But suppose we had driven it at resonance.
Well, at resonance -- so at resonance, I Max is V zero
divided by R. There is no resi- there is no
self-inductor, and there is no capacitance at
resonance. And so that would then be
hundred and ten, square root of two,
divided by sixty. But that is two point six
amperes. That is substantially larger
than one-half. So this value here is two point
six amperes. It is too bad that I can't go
there, because I am stuck to me three hundred and seventy seven,
and I'm not going to change that.
Throughout the demonstration, I stick to my driver here.
So keep in mind that the driver is always here,
I don't change that. So the resonance curve -- we
call this the resonance curve -- is very broad,
something like this. And I believe the Q is around
2, you can check that for yourself.
I can calculate what the power is, that the bulb is
dissipating at resonance. Well, that is my one-half,
which is the time-average cosine squared --
my I Max is now point -- two point six amperes,
so now I get two point six squared -- and my resistance is
still sixty ohms -- and this is two hundred watts.
What a coincidence, that's the way we designed the
experiment. So if we could be at resonance,
the light bulb would be happy like a clam at high tide,
it's exactly two hundred watts. It wants two point six amperes,
that's what it wants. Then it is a
two hundred watt light bulb. And so we want to make the
light bulb happy. And how can we do that?
Well, we can either increase L, or we can increase C.
And I want to catch two birds with one stone during this
demonstration, I'm going to teach you a way
that you can increase L, something that we have never
discussed before. I have here,
ferromagnetic material, which has a value for kappa
somewhere around ten, twelve -- let's say it is ten
for now -- so that means that if I bring that iron core inside,
that I could make this go up to one Henry.
Because remember, the meaning of self-inductance
is the flux divided by the current in the solenoid.
And so if you bring in ferromagnetic material with a
kappa M of ten, then the flux goes up by a
factor of ten. The current remains the same,
and therefore your self-inductance goes up by a
factor of ten. So you see that you can make a
variable self-inductance by bringing in ferromagnetic
material, and by removing it again.
And so the question, now, is, how high should L be
to get on resonance? Well, that's very easy,
because we know that one over the square root of L C,
we want to make that three hundred and seventy [break],
that's our driver. But we know that
C is eight microfarads, that's a God-given,
we're not going to change that. And so you can easily show now
that if we could only make L oh point eight eight Henry,
we should be back on resonance. Then we have three hundred and
seventy seven radians per second as our resonance.
And you shouldn't be surprised that omega L is then three
hundred and thirty two ohms. Of course it is three hundred
and thirty two ohms, because it's going to
eat up that one over omega C, they're going to cancel each
other. That's another way you could
have calculated the value of L, by simply saying omega L has to
be three hundred and thirty two. So there are various ways to
get your resonance frequency. And so what I have done,
if I make L oh point eight eight Henry, I shift the
resonance frequency to this value, and my resonance curve
now will look like this. But this height is going to be
the same, because the height is only
determined by V zero and R. And so what I'm going to do now
is I'm going to turn on the instrument where it was before,
[unintelligible] this was the resonance curve,
you won't see any light. And I'm going to move the bar
in, I probably don't have to move it in all they way,
and then the light bulb will go through resonance,
the system will go through resonance, you will get your two
point six amperes, and the light bulb will be
happy. And so let's do that now.
All right? So here, system is now --
current is going. Sixty Hertz,
hundred and ten volts. [unintelligible] square as we
call that, and it's way off resonance, seven and a half
watts, the light bulb doesn't even feel warm.
And now I bring the -- this enormous piece of iron in there,
and when I shove it in, the self-inductance will slowly
go up, the resonance frequency will shirt, and the current
through the system will increase. And there we go.
There it is. I'm on resonance now.
So the light bulb is quite happy.
I want to do one more thing. I'm going to double the
capacitance. And what happens when I double
the capacitance? The frequency,
the resonance frequency goes further down that it was
already, because remember, when you increase L or C,
the resonance frequency goes down.
So I'm going to double the capacitance.
So that curve that you see on the
left there, has shifted, now, even further to the left.
And the light bulb is not very happy.
The light bulb is -- the system is off resonance again.
But if I increase C -- I doubled it -- I can decrease L
by a factor of two, all I have to do is pull the
iron slowly out. And then the resonance
frequency comes back at three hundred and seventy seven
radians per second. Watch it.
Back on resonance. Amazing, isn't it?
Physics works. So I showed you this
demonstration, and I spent so much time on it
because I wanted you to be able to appreciate -- I want you to
be able to see through these equations.
The answer, in physics, in general, lies in the
equations, but it only works if you build up a certain amount of
understanding. And that is not always easy,
and the demonstrations help that.
Seeing is believing. And you may have to go over
this at home in order to digest it a little bit.
I don't expect you to get all this just in one lecture,
of course not. We have a new impedance here,
we have the reactance here, we have the idea that omega L
eats up omega C, then we get into resonance,
all these phenomenon take time to digest.
There are many key practical applications in LRC circuits,
more than you may think. Your radio and your TV are
systems that you, without realizing it,
you tune them to resonance. There's an antenna,
and that receives many, many stations at all different
frequencies, and you are changing -- in general,
you change the capacitor. And when you change the
capacitor in the LRC circuit, you are changing the resonance
frequency, and at that resonance frequency, the system is very
sensitive, drives a very high current, and it selects,
then, a particular station at that frequency.
That's what you're doing with your TV, and that's what you're
doing with your radio. You're changing a variable
capacitor. Another application which is
quite common are metal detectors.
Metal detectors are resonant circuits, they are set at
resonance, and then, when you bring metal nearby,
you bring them off resonance, and then the alarm goes.
In general, they have two coils -- this is one coil,
and this is another coil. Inductance L one,
resistor R one, capacitance C one,
L two, R two, C
two. And so there's a current,
I one, running through this one, a current I two running
through that one. To set up the differential
equations for this system is not so easy.
You get two differential equations, of course,
one for this system, and one for this system.
The problem -- but at the same time the trick -- is that there
is a mutual inductance between the two, because when this one
produces a current I two, there is a magnetic flux going
through this one. That's why we call it mutual
inductance, which we have never really dealt with in any detail
in eight oh two, and we won't.
But I want to mention that this differential equation here is an
equation in I one, L one, R one,
C one, but also contains a term N times D I two D T.
And this differential equation is an equation in L two,
R two, C two, I two, but also contains a
term, N times D I one D T. And so now you get two coupled
differential equations. It's not just one in I one,
and one in I two. It is one in I one and I two,
and it is another one in I one and I two.
And that is a pain in the neck, and it's not easy to solve.
But there are mathematicians who can do all that,
and they solved that for us physicists.
In any case, what is important that clearly,
since you have two coils, you get two resonance
frequencies, not just one, but you have two resonance
frequencies. And at least one of those
resonance frequencies depends very
strongly on the mutual inductance.
And so the system is tuned to set at one of these two
resonance frequencies. And now someone brings in
metal. You bring in a chu- chunk of
metal. And in this metal the metal
will experience the magnetic field, flux changes,
A C of course, always, in the metal,
it will start building up eddy currents.
And so the eddy currents will change the magnetic coupling
between these two loops. And so M changes.
And when M changes, you go off resonance,
and bingo, your alarm goes off. Clearly, and very
unfortunately, this never works for plastic
bombs. You must have conducting
material that you bring in, so it must be a metal.
These systems have very high Q, very high Q means they are
extremely sensitive. If you have very high Q,
that curve is extremely narrow, and so the slightest change in
M, you go off resonance, and you get
your alarm. And so they come in various
forms, and in various applications.
At the airport, it is very simple.
You walk through these two coils.
You may not have realized that, but you simply walk through.
You walk through, like, there's one coil here,
one coil there, and that's how they detect
whether you have any, uh, metal on you.
The one -- the metal detectors that are used to search your
body, including the ones that are used to
search the ground for coins -- uh, when I was a kid,
I went often to the beach in the Netherlands,
and there were often people who were having this metal
detectors, they were looking for coins.
And those metal detectors, basically the same idea,
always two coils -- this is this coil, and this is this
coil, and when metal comes nearby, you have a system that
goes off resonance, you get a difference --
magnetic coupling between the two
coils, and the system goes into alarm.
And I'd like to demonstrate that, for wh- for which I need a
student. Any student who wants to
volunteer, I have here a metal detector, and,
thank goodness, we don't have to go through
metal detectors yet, at MIT -- you want to
volunteer? Boy, you're brave,
you're all the way at the end of the -- can't wait what you
have in store for us. Oh, you look like you're made
of steel. OK, you -- I hope you left all
your secret weapons up there,
did you? [unintelligible].
So this is a -- this is a metal detector, and if you opened this
up here, you would see two coils, one in the center,
and one near the edge. [beep].
And we have to -- I have to calibrate it first -- OK.
Oh, you left your cigarettes out there?
They -- they won't record it. Student: Definitely not
working. No, that's probably not
working. Well, just a second, may have to recalibrate this.
Pretty quiet. Doesn't look like you have any
metal there. I'm pr- fairly sure I turned it
on. They don't always work,
do they? Marcos, you have any
[unintelligible]? [beep] Uh-uh.
It's working. No, it's not working very well.
It's working well here. [laughter].
Wonder what you got there. Are you metal -- is this not
made of metal, all these things in here?
Student: I have lots of metal on me, and [unintelligible] I'm
surprised it's only going off at the belt buckle,
it's weird. Oh, you may be a weird person,
that may be [laughter]. [beeping].
Can you turn around? Oh, that's your watch.
[beeping]. [laughter].
You'd better turn around again. Ah, there are the keys.
You're not made of steel, but you're a great guy,
thank you very much. [applause].
Today is Wednesday, right?
See you Friday.