# Re: formulas (fwd)

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---------- Forwarded message ----------
Date: Fri, 17 Jul 1998 23:23:35 -0700
From: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br>
To: Tesla List <tesla-at-pupman-dot-com>
Subject: Re: formulas (fwd)

Barry wrote:

>         Ever come across the derivation for the exact solution to the
> differential equations for tuned coupled circuits in mathcad?

Not in mathcad. The exact expressions for the general case are rather
complex, but analytical, and can be generated by algebra programs
(or by hand, in a few hours...).
I have below the exact expressions in Laplace transform, taken from
the documentation of my teslasim program
(ftp://coe.ufrj.br/pub/acmq/teslasim.zip):

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

The coefficients of the Laplace transform denominator (constant term first) 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)

If your program can invert Laplace transforms, you have the solution.
The teslasim program does exactly this, numerically.
I have also the 6th-order expressions for the magnifier, if someone is interested.
And in this case there is no exact closed-form analytical solution, but
a numerical solution is possible.

A note: My last post in this thread listed a formula for the wire
length of a conical or flat coil (r1=int. radius, r2=ext. radius, m=height, n=turns):
Wire length=sqrt((n*pi*(r1+r2))^2+(r2-r1)^2+m^2)
This formula is accurate, but -not- exact. The exact formula is the
expression in Jim Monte's program (we verified this offlist).

Antonio Carlos M. de Queiroz

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