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Various RLC Circuit Problems (Resonance Frequency, Phase, Current)

  1. Jun 10, 2012 #1
    1. The problem statement, all variables and given/known data
    An RLC circuit has L = 20mH, R = 20kΩ, C = 250μF.

    1) Calculate the resonance frequency for this circuit.

    2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.

    3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

    4) Draw the phasor diagram for t=0 seconds.


    2. Relevant equations
    ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]

    ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

    [itex]X_{L}[/itex] = ωL

    [itex]X_{C} = \frac{1}{ωC}[/itex]

    Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}}[/itex]

    I(t) = [itex]\frac{ε}{Z}[/itex]

    3. The attempt at a solution
    So I wasn't sure if I was doing any of this correctly; with all of the formulas/equations, it looked like the questions were mostly some simple plug-in questions, but I feel like my numbers just weren't coming out quite right.

    1) Calculate the resonance frequency for this circuit.

    ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]
    L = 20mH = 0.02H; C = 250μF
    ω[itex]_{o} = \frac{1}{\sqrt{(0.02H)(250μF)}} = \frac{1}{2.24 sec}[/itex] = 0.447 Hz

    2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.
    Phase Angle ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

    [itex]X_{L} = ωL = (400)(0.02) =[/itex] 8
    [itex]X_{C} = \frac{1}{(400)(250)} = \frac{1}{100,000} =[/itex] 0.00001
    R = 20kΩ

    ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right] = tan^{-1} \left[\frac{8-0.00001}{20,000}\right] = tan^{-1} \left[\frac{7.99999}{20,000}\right] = tan^{-1} \left[3.99x10^{-4}\right] =[/itex] 0.0229°

    Impedance Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}} = \sqrt{20,000^{2} + (8 - 0.00001)^{2}} = \sqrt{4.0x10^{8} + 63.99} =[/itex] 20000.0016Ω

    3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

    I(t) = [itex]\frac{ε}{Z} = \frac{30[V]sin(400[rad/s]t)}{20000.0016[Ω]} = \frac{30[V]sin(400[rad/s]t)}{20000.0016[Ω]} = 0.00149sin(400t)[/itex]

    So I(t) = 0.00149sin(400t)

    4) Draw the phasor diagram for t=0 seconds.

    I wasn't sure at all how to go about doing this one. As far as I can tell, this one ends up being a graph with voltage and current functions (current found in #3) drawn at t=0, showing how out of phase they are at that time, although I'm not sure how to find the voltage/what to use to find it. Of course, I don't want anyone to draw the graph for me, but some advice on phasor diagrams in general/a push in the right direction would be very nice.

    Any help would be greatly appreciated. Thank you in advance!
     
    Last edited: Jun 10, 2012
  2. jcsd
  3. Jun 10, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    The capacitance is given in microfarads: C=250 *10-6 F. Repeat the calculations.

    ehild
     
  4. Jun 10, 2012 #3
    Alrighty. I guess my mind totally drew a blank around that capacitance. :blushing:

    I feel a lot better about the frequency now; I'm hoping the others are okay, too.

    1) Calculate the resonance frequency for this circuit.
    ω[itex]_{o} = \frac{1}{\sqrt{LC}}[/itex]
    L = 20mH = 0.02H; C = 250x10^{-6}F
    ω[itex]_{o} = \frac{1}{\sqrt{(0.02H)(250x10^{-6}F)}} = \frac{1}{.00223 sec}[/itex] = 447.21 Hz

    2) For a frequency ω=400 rad/s, determine the phase angle and the circuit impedance.
    Phase Angle ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right][/itex]

    [itex]X_{L} = ωL = (400)(0.02) =[/itex] 8
    [itex]X_{C} = \frac{1}{(400)(250x10^{-6})} = \frac{1}{.1} =[/itex] 10
    R = 20kΩ

    ϕ = tan[itex]^{-1} \left[\frac{X_{L}-X_{C}}{R}\right] = tan^{-1} \left[\frac{8-10}{20,000}\right] = tan^{-1} \left[\frac{-2}{20,000}\right] = tan^{-1} \left[-1.0x10^{-4}\right] =[/itex] -0.0057°

    Impedance Z = [itex]\sqrt{R^{2} + (X_{L} - X_{C})^{2}} = \sqrt{20,000^{2} + (8 - 10)^{2}} = \sqrt{4.0x10^{8} + 4} =[/itex] 20000.0001Ω

    3) Using ε = 30[V]sin(ωt), calculate the current in the circuit. This will be a function of time.

    I(t) = [itex]\frac{ε}{Z} = \frac{30[V]sin(400[rad/s]t)}{20000.0001[Ω]} = \frac{30[V]sin(400[rad/s]t)}{20000.0001[Ω]} = 0.0015sin(400t)[/itex]

    So I(t) = 0.0015sin(400t)

    4) Draw the phasor diagram for t=0 seconds.
    I'm still a little confused about this phasor diagram, but I think I can get by. Again, if anyone could offer a little advice on this, it would be great.

    And thank you for catching my mistake, ehild.
     
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