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:

and

Take (2) and solve for thus:

Differentiate this equation twice:

We also differentiate (1) twice thus:

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

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