in this video we're given the specs on an rlc circuit and then we're asked a bunch of standard
questions about it so here's our rlc circuit i see an inductor a capacitor and a resistor and
i have an oscilloscope across the resistor and this is being used as a probe for measuring the
current in the circuit very similar to a lab that we've done and i'm asked first for the
resonant frequency so this is just a matter of quoting the formula that's 1 over 2 pi root lc
8.2 micro henrys that's 8.2 times 10 to the negative 6 for the inductance 10 microfarads so 10
times 10 to the negative 6 which is 10 to the negative fifth
and when i run the numbers on this i get 17576 hertz i'm going to write this as 17.6 kilohertz
in part b i'm asked for the total impedance if we apply a signal at the resonant frequency
and i'll post a link back to where this was first introduced but the impedance
is given by this formula square root of r squared plus chi l minus chi c quantity squared and that
chi l is called the inductive reactance and the chi c is called the capacitive reactants well
the resonant frequency occurs at the point where this term is equal to zero so we have
a special frequency that makes the inductive and capacitive reactances equal to each other
that term cancels out and the impedance reduces to the simple resistance for the circuit
so this is kind of just a concept question because you can just read it right off the
diagram the impedance is going to be 10 ohms right when we drive this thing at 17.6 kilohertz
that is the minimum possible impedance it will maximize the current in the circuit
to look at what happens at other frequencies i picked a frequency of 10 kilohertz for questions
c d and e and we're going to look at the inductive reactants capacitive reactants and total impedance
when we're rather far from the resonant frequency so just to remind you of these
formulas the inductive reactants that's the effective resistance of an inductor and that
gets bigger when the frequency gets bigger because the induced back emf in the inductor gets larger
if we start changing the current faster so at ten thousand hertz
that's two pi times ten thousand
times our inductance which was eight point two times ten to the negative six henrys
and this gives me 0.515 ohms then to get capacitive reactants that's 1 over 2 pi f
times the capacitance and the lower the frequency is the bigger the capacitive reactants because the
capacitor can start to have a rather large voltage induced on it by charge building up
so f is in the denominator and this gets bigger when f is smaller and i get 1 over 2 pi
times the frequency 10 000 hertz times the capacitance 10 microfarads 10 times 10 to
the negative 6 that was just 10 to the negative 5th farads and i get 1.59 ohms for the capacitive
reactants now i get my total impedance using the impedance formula and my resistance was 10 ohms
and my inductive reactance was 0.515 ohms minus my capacitive reactants
so these are out of balance now because we're not at the resonant frequency
and rounding to three sig figs i get a total impedance of 10.1 ohms so just a little bit
bigger than it was at the resonant frequency i didn't ask these questions here but that larger
total impedance means the total current will be a little bit less when you apply ohm's law
like the peak current will be my signal voltage over the impedance and the bigger the impedance
gets the smaller the current gets and that current will be maximized when we run this
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