# Re: Coupling coeff. vs Voltage gain (was Re: Who needsaquenching gap ?)

```Original poster: "Bert Hickman by way of Terry Fritz <twftesla-at-uswest-dot-net>" <bert.hickman-at-aquila-dot-net>

Ed and all,

The equations for estimating sideband frequencies in my earlier post
were indeed approximations. They assumed that the primary and secondary
Q's were not too different in size. If Qp and Qs are significantly
different, and both circuits are still tuned to Fo, the exact form of
the equation gets a bit nastier, with another "adjustment" term in it
(shown as "X" below):

Flower =  Fo/Sqrt(1+K*X)
Fupper =  Fo/Sqrt(1-K*X)

where:
X = [1-(Kc^2/2*K^2)*(Qp/Qs + Qs/Qp)]
and
Kc = 1/sqrt(Qp*Qs)  (Critical coupling)

However, from a practical standpoint, X is usually close to 1 for
typical values of Qp, Qs, and K for Tesla Coils, even when Qp and Qs are
fairly different. For example, let's assume that:
Qp = 10
Qs = 200
K = 0.22.

resulting in a maximum change of about 1.5% in the sideband frequencies
from the approximate formulas. And with the exact form as well, Flower
approximately equals Fo/Sqrt(2) as K approaches 1.

BTW, my hat's off to you - I've never actually tried to derive these
formulas! Thankfully, Prof. Frederick Terman figured them out in 1943...
:^)

Best regards,

-- Bert --
--
Bert Hickman
Stoneridge Engineering
Email:    bert.hickman-at-aquila-dot-net
Web Site: http://www.teslamania-dot-com

Tesla list wrote:
>
> Original poster: "Ed Phillips by way of Terry Fritz
<twftesla-at-uswest-dot-net>" <evp-at-pacbell-dot-net>
>
> Tesla list wrote:
> >
> > Original poster: "Bert Hickman by way of Terry Fritz
> <twftesla-at-uswest-dot-net>" <bert.hickman-at-aquila-dot-net>
> >
> > Ed,
> >
> > A minor quibbling point - when k=1, the upper sideband goes to
> > infinity, but the lower sideband only drops down to Fo/Sqrt(2), not
> > zero.
> >
> > This is because (for large coupling coefficients and Q's of primary and
> > secondary not too unequal):
> >    F upper = Fo/Sqrt(1-k)
> > and
> >    F lower = Fo/Sqrt(1+k)
> >
> > I fully agree that it's amazing that such a simple circuit leads to such
> > interesting and complex behavior...
> >
> > -- Bert --
>
> Bert:
>
>         Are you sure those are the EXACT expressions for such high
coupling?  I
> don't have the derivation here and am too lazy to work it out again, so
> can't check it.
>
> Ed

```