Re: NEC-LIST: Q of small antennas.

Doug:

The conventional definition of Q uses the frequency
derivative of the tuned antenna's reactance to get

Q = (w/2R) (dX/dw + |X|/w)

The |X|/w term is included to account for the fact
that the definition of Q is for the tuned antenna
and the frequency derivative of X is taken after the
antenna has been tuned with a lossless series reactance.

The antenna must be tuned at each frequency w in order
to define Q at that frequency.

The same is true for the formula I posted yesterday. The
frequency derivative of Z is taken after the antenna has
been tuned with a lossless series reactance at each
frequency w. |dZ/dw| for the tuned antenna is given by

|dZ/dw| = SQRT [ (dR/dw)^2 + (dX/dw + |X|/w)^2 ]

Q for the tuned antenna is given by (w/2R) |dZ/dw|

Another point that should be noted is that there is only
one value of Q for an antenna. Q is exactly defined by
wW/P, where W is reactive energy (not stored energy) and P
is accepted power.

Terms like "loaded Q" and "unloaded Q" are generally used
when one "defines" Q using inverse bandwidth. The ambiguity
in this definition of Q (loaded vs unloaded) arises depending
on how one choses to define bandwidth. i.e. conductance
bandwidth or matched VSWR bandwidth. It is again very
important to note that in defining bandwidth (for the purpose
of relating bandwidth and Q) the antenna must still be tuned
at each frequency w before the bandwidth is determined at that
frequency.

The most appropriate bandwidth to use when relating Q and
bandwidth is matched VSWR bandwdith. Conductance bandwidth
does not exist in frequency ranges of anti-resonance.

For all ranges of frequency, Q and matched VSWR bandwidth
(MVBW) for the tuned antenna are related as follows:

Q = wW/P ~ (w/2R) |dZ/dw| ~ 2 SQRT[B]/MVBW

SQRT[B] = (S-1)/(2 SQRT[S]) < 1, where S is an arbitrary
choice of VSWR. For half-power matched VSWR bandwidth,
S = 5.828 and B = 1.

Steve Best

-----Original Message-----
From: D. B. Miron
To: nec-list_at_gweep.ca
Sent: 12/11/2003 9:24 PM
Subject: Re: NEC-LIST: Q of small antennas.

Hello Steven,

Now that you mention it, I have seen data sets for antennas
at anti-resonance which display the negative reactance
slope. I expect Geyi's assumptions don't fit this case.
But where does the |X|/w term in your expression come from?

Z=R+jX, dZ/dw=dR/dw+jdX/dw, so

|dZ/dw|=sqrt[(dR/dw)^2+(dX/dw)^2]

No?

Regards,
Doug

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