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Electricity circuit question, amplitude resistor?

  1. Nov 1, 2013 #1
    1. The problem statement, all variables and given/known data

    The ac circuit has 3 components in series: a Capacitor C=120μF, an Inductor L=100mH, and a resistor R=7Ω.

    A) Find the amplitude of the voltage across the resistor R in the LCR circuit below, when
    operated at a frequency of 45 Hz?

    B) What is the phase difference between this voltage and
    the supply voltage, Vac (= (15 V)cosωt)?

    C) At what frequency is this phase difference zero?

    D)What is the Q value of this resonant circuit?


    2. Relevant equations

    V(t)=Vocos(ωt + ∅)

    Z=Vo/Io=R (Z=impedance)

    ω=2πf

    Q=(ωo)(L)/R

    There the only relatable I can find


    3. The attempt at a solution

    Attempt for part (A) Using V(t)=Vocos(ωt + ∅) I tried searching for Vo as that is the amplitude but we are not given any Voltage? Then I tried using Z=Vo/Io and I found the impedance but then I realised I don't have Io (current). So I was stuck plugging in numbers and trying to manipulate formulae to no avail.

    Attempt for part (B) Well I think the voltage amplitude here is 15V with no phase change as ∅ is not present. So I said there is no difference as they must be in phase if there doesn't exist a ∅. Not sure if that is right.

    Attempt for part (C) I used Q=(ωo)(L)/R, then I used the fact that ω=2πf which yielded an equation of the form Q=(2π)(f)(L)/R . I plugged in the figures and got a value of 4.04.


    Please help! I am very stuck.
     
  2. jcsd
  3. Nov 1, 2013 #2

    gneill

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    Staff: Mentor

    Hi Michael.

    You've used the term "impedance". Have you learned about complex impedance yet, or just reactance, for reactive components (L, C)?
     
  4. Nov 1, 2013 #3
    Hi gneil,

    Yes I know how to calculate complex impedance. I think I am pretty comfortable doing so. Is that needed for this particular question? I was thinking that Z should be the complex impedance but that still leaves me with two unknowns whatever equation I use? I wouldn't be too familiar with the reactance componants if I am honest.
     
  5. Nov 1, 2013 #4

    gneill

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    Okay, you're given the frequency of operation. Can you calculate the individual impedances of the capacitor and inductor?
     
  6. Nov 1, 2013 #5
    Yes, the complex impedance of a capacitor is (1/(iwC)) and the Complex impedance of an Inductor is iwL. So I could calculate the individula complex impedances of both comfortably.
     
  7. Nov 1, 2013 #6

    gneill

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    Staff: Mentor

    So then, what's the total impedance of the series circuit (numerically)?
     
  8. Nov 1, 2013 #7
    This is frustrating. When I use the complex impedance formula for a capacitor to be (1)/(iwC) I get a total circuit complex impedance to be = ((0.042)/(i0.0339)) + 7. The when I use the other capacitor formula for complex impedance -(i)/(wC) I get the total complex impedance of the circuit to = -i0.042 + 7. So lost with the whole question :(
     
  9. Nov 1, 2013 #8

    gneill

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    Staff: Mentor

    Let's do it one step at a time. What are the individual impedances for the components?
     
  10. Nov 2, 2013 #9
    1) Complex Impedance (C.I) of capacitor = 1/(i)(w)(C) where w=2pif.

    Therefore plugging it ---> 1/(i)(2pi)(45)(120E-6) = 1/i(29.5)

    2) C.I of Inductor = iwL therefore plugging in yields ---> i(2pi)(f)L = (i)(2pi)(45)(100E-3) = i9pi

    3) C.I of resistor = R therfore = 7


    They are all the complex impedances arn't they? What now? Thank you!!
     
  11. Nov 2, 2013 #10

    gneill

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    Presumably that's meant to be (1/i)(29.5) Ω. Which is -29.5i Ω, after multiplying top and bottom by i. That's a good value. A better one would keep a couple more decimal places as guard digits for future calculations -- Don't round intermediate values.

    Something went awry there. ##2 \cdot \pi \cdot 45## is about 283, and multiplying by 100 E-3 can't change the digits...

    Correct.

    Once you've sorted out the inductive impedance value you can write the total impedance for the circuit. Z = ?.

    With that you can use Ohm's law to find the current, potential across R, and so on.

    And while we're at it, in your first post you calculated the Q using the driving frequency of the source (45 Hz). It should be ωo as you wrote in the expression for Q. ωo is the resonant frequency of the RLC circuit.
     
  12. Nov 3, 2013 #11
    Hi.
    I fixed the impedances and found the total complex impedance to hold a value of:

    -29.5i + 28.3 + 7 = 35.3 - 29.5i = Z total.

    but if I were to use ohms law to find the voltage amplitude like part A wants me too would the following be correct: Ohms Law: Zr = Vo/Io = R. Therfore Io = Z/R = 7/7 = 1. Then rearranging Vo = RIo = (7)(1) = 7.

    Is that the Voltage amplitude across the resistor? Surely not. And I don't quite understand what you are saying about the Q expression. Thanks
     
  13. Nov 3, 2013 #12

    gneill

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    Careful, the impedance of the inductor is also imaginary, like the capacitor's. jωL.

    Ohm's law: E = I*Z, so that I = E/Z.

    Find the total current I first. Then for individual components with impedance z, Ez = I*z.

    The Q of a circuit is related to the resonant frequency ωo of the circuit by the given formula. The resonant frequency is inherent to the circuit, not determined by the source driving it. You need to find the resonant frequency of the circuit.
     
  14. Nov 3, 2013 #13
    Oh yes my mistake. Ztotal = -1.2i + 7.

    Maybe this is me just being an idiot but E is Electric potential or voltage, which we do not have. This is where I keep getting lost. I am obviously wrong but to get the flowing current, we need to know the voltage. We don't have that info yet. All we have is the complex impedance of each component and the frequency....
     
  15. Nov 3, 2013 #14

    gneill

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    Much better. You should get into the habit of keeping a few more decimal places in values that will be used in additional computations. These "guard digits" will prevent rounding errors from creeping into your results.

    E is the potential (or voltage). I use E rather than V since V is also the unit (Volt). Avoids confusion that way. Since you're not given a specific value for E at this point in the problem, express the result in terms of E.
     
  16. Nov 3, 2013 #15
    Done! :)

    Total current I = E/Ztotal = E/(-1.2i + 7) then multiplying by it's conjugate yields:
    I total = ( (E)(7 + 1.2i) ) / 50.44

    I for the individual componants: For capacitor I = -29.5iE
    I for Inductor I= 28.3iE
    I for Resistor I= 7E.

    What I then did was sum the individual I values and let it equal the total value. But I didn't get anywhere with it. Am I getting somewhere?
     
  17. Nov 3, 2013 #16

    gneill

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    Since it's a series circuit the current must be the same for all components. That'll be your E/Z value, where Z is the total impedance. That same current flows through each component and produces potentials across them.

    Your value for I total is okay, but you might as well normalize the complex value (divide both terms by the 50.44 so you have a canonical form of complex number: a + ib). As it turns out, that complex number that you determined is also known as the admittance of the circuit. Admittance is the complex analog of conductance. Where G = 1/R for resistances, Y = 1/Z for impedances.

    So if you calculate Y = 1/Z, then the current in the circuit is E*Y. That's Ohm's law in terms of admittance (conductance). And the potential across a component with impedance z is then E*Y*z. You can pick out the expression that represents the "multiplier" of E for the component's voltage.
     
  18. Nov 3, 2013 #17
    I just realised that my individual I values were wrong. I multiplied E by the impedance instead of dividing by it.

    To be perfectly honest I have no idea about the above message but I will try to follow what you are saying and see where I get. Is there not a formula that gives me voltage amplitude across a component or is it not that simple? Thanks again.
     
  19. Nov 3, 2013 #18

    gneill

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    I*z is the give you the complex voltage across z. |I*z| is its amplitude.
     
  20. Nov 3, 2013 #19
    Y=1/Z so Y = 1/7-1.2i. You then said potential is E*Y*z so my potential across my resistor is then :
    = ((E)(7))/(7-2.1i)??? I am struggling to find how I am going to get the amplitude across the resistor. Sorry for being slow at this but I can't see where we are going with this.

    Thanks
     
  21. Nov 3, 2013 #20
    Due to my severe lack of understanding of circuits as you have probably identified, do you think we should maybe move on to the other parts of the question and hopefully I will be capable of understanding how they work?
     
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