in this video we're given an rlc circuit and we know the value of the inductance
is 33 millihenries and we're told that inductor has negligible resistance our resistor has four
and a half ohms resistance our capacitor is 40 microfarads and we have this sinusoidal voltage
applied to the circuit with an rms value of 3.0 volts and we're asked five different questions in
this circuit we want the resonant frequency the rms current at the resonant frequency
the inductive and capacitive reactances and rms current when i'm away from the resonant frequency
at 200 hertz and i'll start by summarizing some of the formulas that we're going to use
as we answer these questions so first off when we're dealing with alternating current
we use the word impedance instead of resistance and the impedance is the effective resistance of
a circuit with an alternating current running through it and so there's the formula for
impedance and it has the ordinary resistance in there the square root of r squared plus
the term that accounts for the effective resistance of the inductor in the capacitor
and this is the way the impedances add up you have to take the difference
between the contribution from the inductor which is called the inductive reactance
and the contribution from the capacitor that's called the capacitive reactants and you square
that inside the square root before you add to the square of the resistance and those effective
resistances or reactances for the inductor and the capacitor those are frequency dependent quantities
an inductor will give you more resistance if you hit it with a higher frequency because inductors
resist changes in current and that one is given by the formula two pi f times the inductance
are capacitive reactants capacitors have lower resistance if you hit them with a high frequency
because if you hit them with a high frequency they can't build up very much charge to resist the flow
of current and i'll go ahead and post a link to the video where resonance in an rlc circuit was
first discussed but the idea behind the resonant frequency for an rlc circuit is that there's
one special frequency for which the capacitive and inductive reactances are equal that makes
this part of the impedance term vanish it makes the impedance its minimum possible value and you
get the maximum possible current if you set the inductive and capacitive reactance equal and solve
for the frequency that's the resonant frequency that maximizes the current in the circuit
and i'm just going to quote the formula for it here all right so there's our summary of all
the tools we're going to use let's get into the parts of the problem so first i want the resonant
frequency for the circuit and that's just a matter of quoting the formula i just wrote down so 1 over
2 pi square root of lc l is 33 millihenry's i'm going to change that into .033 henry's
c was 40 microfarads so 40 times 10 to the negative 6. and when i run the numbers on
this i get 139 hertz keeping three sig figs as a matter of convention in part b i want the
root mean squared or rms current at the resonant frequency and so the current follows a version
of ohm's law i equals v over r and the rms current is given by the rms voltage divided by the total
resistance but in an alternating current circuit we use the total impedance but things get simpler
here because i said at the resonant frequency of 139 hertz there's no calculation necessary to get
the impedance because i know at the resonant frequency the two reactances are equal to each
other and that term in the impedance vanishes so as long as we're at the resonant frequency
the total impedance is equal to the square root of r squared which is just the resistance in the
circuit so i have my 3.0 volts rms divided by my impedance which turns out to just be
equal to the resistance which was 4.7 ohms and i've got my rms current this turns out to be
0.638 amps next we're going to hit this circuit with a 200 hertz signal so that's quite a bit
above the resonant frequency we should expect to see a smaller current resulting from that
and this time we do need to carefully compute the total impedance so we start by getting the
inductive reactants again that's 2 pi f l 2 pi times 200 times our inductance which was 0.033
this gives me an inductive reactance of 41.5 ohms remember the reactance of a circuit element is its
effective resistance then we want the capacitive reactants and we plug into the formula for that
so we have 1 over 2 pi times the frequency we're driving this thing at as 200 hertz
times our capacitance which was 40 microfarads or 40 times 10 to the negative 6. we run the numbers
on this and get 19.9 ohms for the reactants and then finally we want the rms current when we
drive this circuit at 200 hertz with a 3 volt signal so we need to get the total impedance
that's our effective resistance for the circuit and that starts with r squared that's 4.7 squared
and then we have this term with the difference of the inductive and capacitive reactances it doesn't
matter what order you put them in because we're going to square it and when i run the numbers
on this i get 22.1 ohms now the rms current is simply found from ohm's law so i rms is v over r
but again instead of a simple r we use a z for impedance and i have this 3 volt signal that we're
applying to the circuit and i'm dividing by this total impedance of 22.1 ohms and when i run the
numbers i get 0.136 amps and as we expected this is a lower amount of current than we got when we
were driving the circuit at its resonant frequency at the resonant frequency we're maximizing the
current for the particular voltage that we're applying to the circuit if you enjoyed this video
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