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Math Help - system of 2║ order, Tesla coil characteristic equation

  1. #1
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    system of 2║ order, Tesla coil characteristic equation

    Hi,

    Tesla coil is represented by these two 2║ order equations if resistance is R1=R2=0

    (1/C1) q1 + L1 q1'' + M q2'' = 0 ("primary" net kirchoff equation)
    (1/C2) q2 + L2 q2'' + M q1'' = 0 ("secondary" net kirchoff equation)

    I know from two scientific articles that characteristic equation is:
    (1 - M^2/(L1 L2) ) D^4 + (w1^2 + w2^2) D^2 + (w1^2 w2^2) = 0

    with wi = 1/sqrt(Li Ci)

    I couldn┤t find any source where it is derived and would like to check it. The problem is that I have no clue about how to get to that 4th order characteristic equation!
    maybe a change of variable?

    If someone could help many thanks.
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  2. #2
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    Re: system of 2║ order, Tesla coil characteristic equation

    So you have two equations:

    \frac{q_{1}}{C_{1}}+L_{1}\ddot{q}_{1}+M\ddot{q}_{2  }=0\qquad (1), and

    \frac{q_{2}}{C_{2}}+L_{2}\ddot{q}_{2}+M\ddot{q}_{1  }=0\qquad (2).

    Take (2) and solve for \ddot{q}_{1} thus:

    \ddot{q}_{1}=-\frac{1}{M}\left[\frac{q_{2}}{C_{2}}+L_{2}\ddot{q}_{2}\right]\qquad (3).

    Differentiate this equation twice:

    q_{1}^{(4)}=-\frac{1}{M}\left[\frac{\ddot{q}_{2}}{C_{2}}+L_{2}q^{(4)}_{2}\right]\qquad (4).

    We also differentiate (1) twice thus:

    \frac{\ddot{q}_{1}}{C_{1}}+L_{1}q^{(4)}_{1}+Mq^{(4  )}_{2}=0\qquad (5).

    Now plug (3) and (4) into (5) thus:

    -\frac{1}{MC_{1}}\left[\frac{q_{2}}{C_{2}}+L_{2}\ddot{q}_{2}\right]+\frac{L_{1}}{M}\left[\frac{\ddot{q}_{2}}{C_{2}}+L_{2}q^{(4)}_{2}\right]+Mq^{(4)}_{2}=0\qquad (6).

    This equation can, most likely, be massaged to fit your characteristic equation. If not, then try solving (1) for \ddot{q}_{2}, differentiating twice, and substituting into the twice-differentiated version of (2). You'll have to see how your w_{1} and w_{2} fit into that scheme.

    Does that get you started?
    Last edited by Ackbeet; November 8th 2011 at 01:43 PM. Reason: Typo.
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  3. #3
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    Re: system of 2║ order, Tesla coil characteristic equation

    In (4) there is a missing minus, Massaging (6) corrected with this minus it can be derived the given characteristic equation, thanks
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  4. #4
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    Re: system of 2║ order, Tesla coil characteristic equation

    Quote Originally Posted by rulmismo View Post
    In (4) there is a missing minus, Massaging (6) corrected with this minus it can be derived the given characteristic equation, thanks
    Quite right. I will fix.
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  5. #5
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    Re: system of 2║ order, Tesla coil characteristic equation

    I did the detailed derivation (more or less...) of q1(t) and q2(t)
    if some folk has the time/motivation please check it to assure that I didn┤t do any dumb step....
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