Re: Effective Coupling Coefficient of Directly Coupled Resonant Networks

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:
 >
 > Original poster: "Mccauley, Daniel H" <daniel.h.mccauley-at-lmco-dot-com>
 >
 > In a directly coupled double-resonant circuit, Antonio stated there is
 > an equivalent coupling coefficient which is
 > similar to that of the magnetic coupling in a transformer based
 > double-resonant circuit.
 >
 > In this version, k = SQRT (L1/(L1+L2))
 >
 > I was wondering if someone (Antonio?) could expand in more detail the
 > role and its derivation within a directly
 > coupled system.

This equivalence comes from the equivalence between a transformer
with two coupled coils and an ideal transformer and two inductors:
(See with a fixed width font, as always.)

        k
o-----+  +-----o
       |  |
       L1 L2
       |  |
o-----+  +-----o

        1:n
o-----+  +--+--Lb--o
       )  (  |
       )  (  La
       )  (  |
o-----+  +--+------o

The circuits are equivalent if:

n=k*sqrt(L2/L1)
La=L1*n^2
Lb=L2*(1-k^2)

or, inverting the relations:

L2=La+Lb
L1=La/n^2
k=sqrt(La/(La+Lb))

In a directly coupled system, the second circuit appears, with n=1.
Lb is the secondary inductor and La is the primary.
The equivalent transformer has that value for k, sqrt(La/(La+Lb)).

 > For example, how could I at least mathematically derive
 > this coefficient when doing so for 6th order and higher directly coupled
 > resonant systems.

There is an equivalence between a 6th-order directly coupled system and
a magnifier, if you replace the first two inductors by a transformer
according to the equivalence above. Note that the role of k in a
6th-order
system is different from in a 4th-order system (a conventional Tesla
coil).
it's possible to use n different from 1 if the primary capacitor is
divided by n^2, in any case. This is where the advantage of the
transformer
appears (in a 6th, 4th, or mth-order system): To change n doesn't change
the waveforms if the primary capacitor is divided by n^2. The energy
transfer time is not changed, but the voltage gain is multiplied by n.

For how to design these systems, see my papers about multiple resonance
networks (http://www.coe.ufrj.br/~acmq/papers/papers.html).

Antonio Carlos M. de Queiroz