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Old 20th Mar 2017, 9:12 pm   #28
Skywave
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Join Date: Jun 2006
Location: Chard, South Somerset, UK.
Posts: 7,457
Arrow Re: Permeability Tuning

Quote:
Originally Posted by kalee20 View Post
If the loss resistance is independent of frequency - ie mostly the DC resistance of the coil, then:

Q = (1/R)[√(L/C)]

We also have:

Q = fres / BW (bandwidth)

and fres = 1/[2π√(LC)].

Putting these together, we have:

BW = R/(2πL)

Thus for constant bandwidth, you should vary C because bandwidth is independent of C!

This analysis is revealing, nonetheless!
"Revealing": indeed it is - and in all my years of fooling around with circuits (and studying RF theory), I've never seen that analysis before. Now not wishing to suggest for one moment that I had any doubts in your algebra, but out of sheer curiosity, I did the sums myself. For those who are interested, they go like this . . .

Q = [1/R].[√(L/C)], from which we get: C = L/(Q².R²) . . . . . eqn. 1.
fo = 1/[2π√(L.C)], from which we get: C = 1/(4π.L.fo²) . . . . . eqn. 2.

Equate the values of C from eqns. 1 & 2:
i.e.: L/(Q².R²) = 1/(4π.L.fo²), and simplify.

We get:
fo² = (Q².R²)/(4π.L²).
So: (fo/Q)² = R²/(4π.L²).

But: (fo/Q)² = (BW)², where BW = bandwidth, by definition.

Hence: BW = R/2π.L . . . . as derived and shown by kalee20.
And, moreover, it is independent of C.

My further thoughts . . . .
The 'problem' there is that that result implies that in an LC cct., L should be kept constant and C made the variable. (So much for the benefits of permeability tuning ). However, the fly in the ointment with a coupled LC cct. used for tuning is the loading effect of the primary on the secondary and that loading will affect the BW of that LC cct. (Usually to reduce the BW). In other words, the 'R' in Q = [1/R].[√(L/C)] needs to be taken fully into account and that R - as a consequence of the effect of the primary - will be frequency dependent, since the effective degree of the coupling will be also. And that tells us why in most radios where the aerial input coil is quite loosely coupled to the first tuned cct., why that coupling is usually quite low.

Al.

Last edited by Skywave; 20th Mar 2017 at 9:17 pm.
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