Re: NEC-LIST: Q of small antennas.

The expression for antenna Q, valid for all ranges of
frequency, that is through ranges of both resonance and
anti-resonance is

Q = (w/2R) |dZ/dw|,

where dZ/dw (the frequency derivative of impedance) is
calculated for the antenna after it has been tuned at each
frequency, w, with a lossless series reactance. Q is only
meaningful for the tuned antenna. (The expression posted
by Doug makes this same assumption.) |dZ/dw| for the tuned
antenna can be written in terms of dR/dw and dX/dw so that
the formula for Q can be written as follows:

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

This is a new modification of the formula posted by Doug to
include the dR/dw term. This expression is more accurate in
frequency ranges of resonance (dX/dw > 0) and it corrects the
inherent error in the other formula in regions of
anti-resonance (dX/dw < 0). Using the other formula in
regions of anti-resonance would give negative values for Q.

[Note also that the Foster reactance theorem is not valid
for antennas. In regions of anti-resonance, dX/dw < 0 for
the lossless and lossy antenna.]

Steve Best

-----Original Message-----
From: D. B. Miron [mailto:dbmiron_at_paulbunyan.net]
Sent: Wednesday, December 10, 2003 3:27 PM
To: nec-list_at_gweep.ca
Subject: NEC-LIST: Q of small antennas.

Good day,

Several times over the last few years, antenna Q has been a
hot topic on NEC-LIST. Recently, in an effort to be less
behind on my reading, I found several articles on the
subject in this year's IEEE Trans. A&P, and went back to a
referenced paper from 2000, all listed below. [1] and [3]
give different methods of calculating Q from the numerical
solution of the antenna's current and radiation. [1] argues
that the traditional expressions for minimum Q give values
that are too small, and discusses some factors that affect Q
for small antennas. [2] and [4] are re-examina;tions of
minimum Q and maximum gain/Q. For NEC users, [4] gives a
circuit-based expression for Q that I like (eq. (55))

Q=0.5[w dX/dw + |X|)/R, w=2 pi frequency.

I've left off the sub and super scripts. Be wary of typos
in [2-4]. I think, but I'm not certain, they are all
obvious.

Doug Miron

[1] Thiele, Detweiler, and Penno, "On the Lower Bound of the
Radiation Q for Electrically Small Antennas", IEEE Trans.
A&P, vol. 54, no. 6, pp1263-1269, June, 2003.
[2] W. Geyi, "Physical Limitations of Antennas", IEEE Trans.
A&P, vol. 54, no. 8, pp2116-2123, August, 2003.
[3] W. Geyi, "A Method for the Evaluation of Small Antenna
Q", IEEE Trans. A&P, vol. 54, no. 8, pp2124-2129, August,
2003.
[4] Geyi, Jarmuszewski, and Qi, "The Foster Reactance
Theorem for Antennas and Radiation Q", IEEE Trans. A&P, vol.
48, no. 3, pp401-408, March, 2000.

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