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Question on resonance/transformers/

  1. May 11, 2007 #1
    1. The problem statement, all variables and given/known data
    The circuit below is an oscillator circuit driven bu a transformer coupled to the voltage source.

    a) replace the transformer with its Thevenin Equivalent ckt
    b) Find the resonant frequency of this circuit
    c) Find its 3db bandwidth
    d) Hence find the quality factor - Qfactor of resonance
    e) sketch the frequency response.
    http://img228.imageshack.us/img228/3462/20075107305163314379051gs0.jpg

    2. Relevant equations



    3. The attempt at a solution

    1) i simplified the transformer part by taking the r20 ohm resistor to the other side obtaining an equivalent resistance of

    (10^2 / 1) * 20 = 20,000 ohms

    then we're left with just a 20,000 ohm resistor + 100 ohms in parallell with the other 100 ohms in series with the inductor and the capacitor.


    1) thevenin equivalent - same voltage source - reduce the rest of the ckt into its equivalent impedence, Zth
    Code (Text):

    found
    Z(LC) = jwL + 1/jwc
    = j (wL - 1/wc)
    = j((w^2)LC-1/wc)

    Z(LC + R) = 100  + j((w^2)LC-1/wc)

    Z(LC+R) // R =
    {100 X [ 100  + j((w^2)LC-1/wc) ] }
    -------------------------------------
     {100 + [100  + j((w^2)LC-1/wc)]}

    Zt = 20,000 + Z(LC+R) // R

    = 20,000 + {[10,000wc + j100(w^2LC - 1)] / [200wc + j(w^2LC - 1 ) ] }

     using matlab:

    20,000 + [2.5E-15w^4  + 0.02 w^2 + 0.01 + j1E-6 -2w ]
                  ------------------------             -----------
                              D                                       D

    where D = -2.5E-13w^4 + 1E-8w^2 - 1

    using that I found the resonant freqency which is the imaginary part of the ckt set to zero - so its purely resistive.

    1E-6w^3  -   2w = 0

    w = sqrt(2 / 1E-6 ) = 1.4142E3 = 1.4142 K rad/s
     
     
    I think my omega is too large ... should i continue with this and find the power and then the 3db frenquncy - the frequency where the power is reduced to half.

    thanks in advance for any help.
     
  2. jcsd
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