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Re: TC Differential Equations



Tesla List wrote:

> There has been some discussion of 4th order differential equations and such
> that explain the operation of a TC.  I have been getting much more curious
> about this.  Could anyone refer me to some easily obtainable sources that
> explain this material?  I would really appreciated any help I can get on
this.

Actually, there is no need for differential equations. The solution for
this kind of problem is easily found by Laplace transforms.

(Possibly nontrivial math for some, but standard circuit theory,
follows.)

The model is:

    +--R1--+        +--R2--+
 +  |      |        |      |   +
Vc1 C1     L1 <-k-> L2     C2 Vc2
 -  |      |        |      |   -
    +--<---+        +-->---+
      Il1             Il2

The initial condition is a voltage Vc1(0) in C1.
Write two mesh equations in Laplace transform for the two loops:

(1/(sC1)+R1+sL1)*Il1(s) + sM*Il2(s) = Vc1(0)/s
sM*Il1(s) + (1/(sC2)+R2+sL2)*Il2(s) = 0

This system is solved for Il1(s) and Il2(s), and the voltages Vc1(s)
and Vc2(s) are computed as:

Vc1(s) = Vc1(0)/2-Il1(s)/(sC1)
Vc2(s) = -Il2(s)/(sC2)

These computations results in Laplace transforms in the form of
ratios of polynomials in "s", and the expressions in the time 
domain are obtained by inversion of these Laplace transforms
by standard methods. Best if numerically (essentially: factorize
the denominator, expand the polynomial ratios in partial fractions,
and identify the resulting terms in a table of inverse Laplace 
transforms).

The coefficients of the common Laplace transform denominator are:
d0=1/(L1*L2*C1*C2)/(1-k*k);
d1=(R2/(L2*L1*C1)+R1/(L1*L2*C2))/(1-k*k);
d2=(1/(L1*C1)+1/(L2*C2)+R1*R2/(L1*L2))/(1-k*k);
d3=(R1/L1+R2/L2)/(1-k*k);
d4=1;
For the primary voltage Vc1(s), the numerator coefficients are:
n0=R1/(L1*L2*C2)/(1-k*k);
n1=(1/(L2*C2)+R1*R2/(L1*L2))/(1-k*k);
n2=(R1/L1+R2/L2)/(1-k*k);
n3=1;
All multiplied by Vc1(0).
For the secondary voltage Vc2(s), the numerator coefficients are:
n0=0;
n1=1;
All multiplied by Vc1(0)*k/C2/sqrt(L1*L2)/(1-k*k).
For the primary current Il1(s), the numerator coefficients are:
n0=1/(L2*C2);
n1=R2/L2;
n2=1;
All multiplied by Vc1(0)/(L1*(1-k*k)).
For the secondary current Il2(s), the numerator coefficients are:
n0=0;
n1=0;
n2=1;
All multiplied by -Vc1(0)*k/sqrt(L1*L2)/(1-k*k).

The denominator is that 4th-order differential equation, in
Laplace transform. The roots of that polynomial are the natural 
frequencies of the circuit, that for a normal coil result in two pairs
of complex-conjugate roots close to the imaginary axis,
corresponding to two high-Q oscillatory modes. The interaction
between these two modes cause the familiar beats observed
in Tesla coil waveforms.

Antonio Carlos M. de Queiroz