in this video we briefly introduced the series rlc circuit capacitive reactants inductive reactants
impedance and resonance so this is the series rlc circuit with a sinusoidal voltage of sine omega t
applied across it and i see my resistor inductor and capacitor in series now just a reminder about
the way i've written this with an omega in front of t you could express this in terms of period
by remembering that period is 2 pi divided by the coefficient of little t in the sinusoidal function
so 2 pi over omega and that means that frequency which is the reciprocal of period can be written
as omega over 2 pi and then we could write that omega is equal to 2 pi times the frequency in
hertz now the idea of the reactants of a capacitor or an inductor is that inductors and capacitors
have some effective resistance in a circuit in the sense that they resist the flow of current
but this isn't the same as the resistance of a resistor that dissipates power
so it has to have a different name so you can think of reactants as effective resistance
for one of these circuit elements then to motivate the formula for capacitive reactants
in other words the effective resistance of a capacitor we think about how a capacitor
behaves with really high frequency versus really low frequency signals
if you imagine a really low frequency sinusoidal signal across this circuit for example like
one hertz one oscillation per second then for such long time periods it's very easy for the
capacitor to get completely charged up and block the flow of current in the circuit completely
in other words in the low frequency limit the effective resistance of a capacitor
is infinite for very high frequencies the capacitor would barely charge at all and
you could barely tell it's there in the circuit in other words capacitors are sort of transparent to
the highest frequency signals so we can quantify this with a formula for capacitive reactants
and what i want is a very high reactance if the frequency is low and a very low reactance if the
frequency is high and the way the formula turns out is like this we use the greek letter chi
for this the capacitive reactance is just 1 over omega c in terms of frequency i could write it as
1 over 2 pi f c now with very high frequencies i get very low reactants and vice versa so it's
doing what it's supposed to do i do want to look at the units of capacitive reactants real quick
frequency has units of hertz or one over seconds so i end up with seconds in the numerator and
then capacitance is charged per volt coulombs per volt so i end up with coulombs in the denominator
and volts in the numerator then i see seconds per coulomb well that's the reciprocal of coulombs per
second which is current so this must be equal to volts over amps and then using ohm's law i
can express volts with v equals ir as units of amps times ohms and the amps cancel out
and i get ohms for my units of reactants so again this is a measure of effective resistance in the
circuit it's just that it's frequency dependent unlike a resistor what about inductive reactants
well using the same sort of physical reasoning about high and low frequencies we know that
inductors resist changes in current so when you change the current through an inductor
that creates flux through the coil and there's an induced emf opposing that flux change the
bigger the inductance the more intense the induced emf so if we were to run a really low frequency
signal over an inductor it's going to offer very little effective resistance because the flux just
isn't changing that fast so inductors give you really small resistance when the frequency is low
if the frequency was really high then you're just constantly reversing the direction of the current
very rapidly and you get huge induced emfs opposing those changes in the inductor
and the effective resistance of an inductor can become infinite in the high frequency
limit as you're trying to shift the current back and forth infinitely fast
so the simplest way to get this done is with the formula chi l equals omega l replacing
omega with 2 pi f i get 2 pi f l again it's instructed to look at the units here so the units
of chi l for frequency that's one over seconds for inductance that's henry's so henries per second
and then i have to think about where the self-inductance was first defined
in order to figure out a more useful way of expressing the henrys
and i'll post a link to the video where this was done but just as a quick reminder
we have that the induced voltage as a magnitude anyway would be l times the rate of change in
the current that was measured in henry's the induced emf measured in volts and the current
of course is amps and so i'm going to solve for l in terms of all these other quantities and i
would get volts times seconds divided by amps for a henry so henry is a volt second per amp
our factors of seconds are going to cancel and again i have volts per amp which i already showed
gives me units of ohms so again my inductive reactance has units of resistance on it this
is an effective resistance in the circuit now the total effective resistance in an rlc circuit
is called the impedance of the circuit and ohm's law will still apply so our impedance that
gets the symbol z instead of r because it's not exactly a resistance and i can say that the peak
current in the circuit will be equal to the peak voltage over z the total impedance
or just depending on the context maybe i want an rms current instead and it would just be v
rms over z and the point of this video is not to get into the weeds of how we derive these things
but our total impedance is given by this formula where r is the resistance of the resistor chi l
is the inductive reactants chi c is the capacitive reactants and they combine in this very non-simple
way in order to create a total effective resistance or impedance for the circuit
now something really interesting happens if you look at this formula and think about well
when is the total impedance minimized and it's minimized when this chi l minus chi c term is zero
and those are both frequency dependent quantities so what we can do is look for a frequency that
minimizes the impedance remember the effective resistance of capacitors and inductors depends
on the frequency so what we're going to do is try to find the special frequency giving us
the lowest resistance in the circuit that will give us the highest peak current or rms current
if we want to evaluate it that way and this is called the resonant frequency for the circuit
so we're searching for resonant frequency and we just say okay let's let the inductive
reactants equal the capacitive reactants express those in terms of frequency
and then solve for the frequency so i'm going to multiply both sides by 2 pi f
which gives me a 2 pi f quantity squared on the left 2 pi f quantity squared at the same time
i'm going to divide by l and i get 1 over lc i take the square root and i get 2 pi f
is 1 over the square root of lc and finally we divide by 2 pi
and we get an expression for the resonant frequency of our circuit
1 over 2 pi root lc so for an rlc circuit we can imagine sweeping through frequencies from low to
high for our signal generator and that's what we have a picture of in the lower right hand corner
i'm looking at the peak current as a function of applied frequency and you'll see that peak current
grow and grow and grow to some maximum value that's right when you hit the resonant frequency
you get the maximum peak current and then it starts to drop as you go to a frequency that's
too high so there's this one frequency that sort of perfectly balances the effective resistance
of the capacitor in the inductor to minimize the impedance and give you maximum current
the farther you get away from that special frequency the lower the peak current becomes
so let's wrap things up with an rlc example i'm given an rlc circuit with a resistance of 3.5
ohms a capacitance of two microfarads and an inductance of 0.1 millihenries
and i'm told the circuit is driven by a sinusoidal power source with an rms voltage of 2.0 volts
part a gives me the resonant frequency for the circuit so we're just plugging into that little
formula resonant frequency is one over two pi root lc changing my millihenries to henry's i
have point zero zero zero one henries for the inductance and then my capacitance was
two times ten to the negative 6 fair ads and i get 11 254 out of this so i should round it somewhere
and i'll go ahead and express this in kilohertz so i'm going to call it 11.3
kilohertz so that's the frequency where you will see the maximum current response in the circuit
in part b i want to get the peak current corresponding to this resonant frequency
so the way we're going to do that again is by using ohm's law except using the impedance
for the total effective resistance and the peak current would be the peak voltage over
my impedance which again is a frequency dependent quantity
so to begin with i think i need to get the peak voltage i was given the rms value
and the peak value is just square root 2 times that and this gives me 2.83
volts then i write down the formula for the total impedance and then it's not as much work as it
seems here because the whole premise of how we get the resonant frequency is we found the special
frequency for which for which the inductive reactants and capacitive reactants were equal
in other words that piece of the impedance vanishes and all we have is the square root
of r squared or just the ordinary resistance in the circuit so my total impedance has to come out
to three and a half ohms so then i just apply ohm's law i is v over r and i have 2.83 volts
for the peak voltage over three and a half ohms for the total impedance
and i've got .809 amps for the peak current in the circuit
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