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Phasor circuit problem: 1 resistor, 1 capacitor, 1 inductor, 1 Voltage source

  1. Apr 3, 2007 #1
    1. The problem statement, all variables and given/known data

    What value of [itex]\omega[/itex] will cause the forced response [itex]v_0[/itex] in the circuit below to be zero?

    [​IMG]



    2. Relevant equations

    Phasor eqs.



    3. The attempt at a solution

    [tex]i\,=\,\frac{50\,-\,V_1}{2\Omega}\,=\,\frac{V_1\,-\,V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}[/tex]

    If I substitute [itex]V_1\,=\,0[/itex], then I get:

    [tex]25\,=\,-\frac{V_2}{5\,j\,\Omega}\,=\,\frac{V_2}{20\,j\,\Omega}[/tex]

    How do I use this to find [itex]\omega[/itex]?
     
    Last edited: Apr 3, 2007
  2. jcsd
  3. Apr 3, 2007 #2

    SGT

    User Avatar

    If you want Vo to be zero, the impedance of the series connection of the capacitor with the inductor must be zero.
    You got the impedance of the capacitor wrong.
    [tex]Z_C = \frac{-j}{\omega C}[/tex]
    [tex]Z_L = j \omega L[/tex]
     
  4. Apr 3, 2007 #3
    [​IMG]

    [tex]Z\,=\,\frac{1}{5j\omega}\,+\,20j\omega[/tex]

    [tex]V_{OUT}\,=\,\frac{Z}{Z\,+\,2\Omega}[/tex]

    We want [itex]V_{OUT}[/itex] to be zero:

    [tex]0\,=\,\frac{\frac{1}{5j\omega}\,+\,20j\omega}{\frac{1}{5j\omega}\,+\,20j\omega\,+\,2\Omega}\,V_{IN}[/tex]

    [tex]0\,=\,\frac{1}{5j\omega}\,+\,20j\omega}[/tex]

    [tex]-\frac{1}{5j\omega}\,=\,20j\omega[/tex]

    [tex]\omega\,=\,\frac{1}{10}[/tex]

    Is that right?
     
    Last edited: Apr 3, 2007
  5. Apr 3, 2007 #4

    SGT

    User Avatar

    Yes, the impedance is zero at the resonant frequency [tex]\omega_0 = \frac{1}{(LC)^\frac{1}{2}} = \frac{1}{10}[/tex]
     
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