system of 2º order, Tesla coil characteristic equation

Hi,

Tesla coil is represented by these two 2º 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.

Re: system of 2º order, Tesla coil characteristic equation

So you have two equations:

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

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

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

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

Differentiate this equation twice:

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

We also differentiate (1) twice thus:

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

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

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

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

Does that get you started?

Re: system of 2º 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

Re: system of 2º order, Tesla coil characteristic equation

Quote:

Originally Posted by

**rulmismo** 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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Re: system of 2º 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....(Surprised)