Re: Phase array question

Richard Clark wrote:
On Tue, 19 Aug 2008 09:17:01 -0700, Jim Lux <james.p.lux@xxxxxxxxxxxx>
wrote:

But your search for small, broadband antennas puts you bump-up against the principle "small - broadband - efficient, choose any two".
But actually, that's not the principle.. The actual limits have to do with the ratio of stored energy vs radiated power in the antenna (Q, in the energy storage sense, not in the "resonant circuit" sense).

Hi Jim,

Different meanings of Q? The measure of Q may vary according to
arbitrary usage: the choice of SWR limits to define bandwidth which
infers Q. Some choose 2:1, classic Q would go further. Either way,
and for either quantitative result, the meaning of Q remains
essentially the same.


Actually, the Q you use is not the actual definition, which is the ratio of the energy stored in the system vs the energy lost per cycle (or possibly per radian, so there's factor of 2pi in there)

The articles that talk about size, efficiency, and Q, use this definition (Energy stored in near field vs energy radiated away).
(e.g. the papers by Chu, Harrington, etc.)


For a single LCR tuned circuit with reasonably high Q, it just happens that the BW/CF works out to the same thing (because it's a quadratic equation that determines both..)


Chu, 1948, defines Q = 2*omega*mean electric energy stored beyond input terminals/(power dissipated in radiation) (page 1170). he goes on to say,"We have computed the Q of an antenna from the energy stored in the equivalent circuit and the power radiated, and *interpreted it freely* as the reciprocal of the fractional bandwidth." (my emphasis added) " To be more accurate, one must define the bandwidth in terms of allowable impedance variation or the tolerable reflection coefficient over the band. For a given antenna, the bandwidth can be increased by choosing a proper matching network. The theoretical aspect of this problem has been dealt with by R.M. Fano."

Harrington, 1965, considering directional antennas (Chu dealt with omnis) defines Q as = 2*omega*W/Pin [eq 54], which is slightly different than Chu. W is either We or Wm (the energy stored in the E or H field respectively), and Pin is the input power to the array. He goes on to say,"If the Q is large, it is related to the frequency bandwidth of the array as follows. Consider the array to be resonated by a suitable reactance network at the frequency of interest, omega/sub/r. Define the ferquency bandwidth of the array in the usual manner to be the fractional frequency increment between the 0.707 points on the normalized input |Z|,
beta = deltaOmega/Omega/sub/r. [eq 56]

If the Q is high (say Q>10) then we have the relationship:
Q approx= 1/beta [eq 57]"


Note well the "approximately equal" and the "resonated by a suitable reactance network"


SWR bandwidth is something totally different of course..

But again, if you pick the right SWR value for your bandwidth measurement, then a dipole antenna is modeled pretty well by a single pole LCR resonant, so the math works out conveniently the same.

Once you start straying away from antennas that can be modeled as a single LCR, the "bandwidth" vs "Q" relationship goes away.

A good example would be some forms of phased arrays with non-reciprocal devices. Antennas with multiple resonances would be another case (except if you are only working over a restricted range, where the single resonance in your range is approximated well by a single LCR)



.

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