Re: NEC-LIST: calculating antenna bandwidth

At 02:24 PM 4/2/2002 -0500, John B. Wood wrote:
>Hello. At a given frequency the antenna can most certainly be viewed as
>an equivalent series R-L-C (or parallel G-L-C) circuit with appropriate
>values of resistance or reactance. The caveat is that the antenna is
>assumed to provide values of R, L, and C that are invariant with frequency
>over the frequency range that encompasses the 3-dB (half power) points.

I'm not sure that this is a reasonable assumption. Here's some sample data
from a NEC run on a multband dipole (where there are 3 dipoles in parallel,
1 for 7MHz, 1 for 14MHz, and 1 for 28 MHz. I applied the equations you
provided to calculate C and L, and while they do remain somewhat constant,
in a order of magnitude sense, they do change 10% or so over a fairly small
frequency range.

1.4175E+07F0
FRXCL
1.3900E+0718.1476-37.9456-1.1594E-11
1.3950E+0719.5161-31.109-1.1550E-11
1.4000E+0720.9445-24.2575-1.1500E-11
1.4050E+0722.4357-17.3858-1.1441E-11
1.4100E+0723.9925-10.4884-1.1358E-11
1.4150E+0725.6185-3.56-1.1135E-11
1.4200E+0727.31693.40491.0848E-05
1.4250E+0729.091810.41181.1076E-05
1.4300E+0730.947117.46631.1168E-05
1.4350E+0732.887524.5741.1243E-05
1.4400E+0734.917631.74061.1314E-05
1.4450E+0737.042838.97221.1386E-05

The actual X of the equivalent L and/or C is around 1000 ohms (at the
14.175 MHz frequency), so, if we fed from a 50 ohm resistive source, add
the 26 ohm resistive component (radiation resistance, mostly), the "Q"
would be 1000/76, or around 13. In an RLC this corresponds to a bandwidth
of 1.1 MHz, or thereabouts (i.e. Fc/BW = Q).

However, if you actually look at the feed point currents as a function of
frequency, or at the power, or at almost any measure, you can see that the
actual half power bandwidth is substantially narrower than that.. on the
order of a 100-200 kHz. 550 kHz away, the reactive impedance would be huge...

Currents (1 volt feed, 50 ohm source impedance, i.e. I= 1/(50+R+jX))
13.90 0.0112 + 0.0062i
13.95 0.0120 + 0.0054i
14.00 0.0126 + 0.0043i
14.05 0.0131 + 0.0031i
14.10 0.0132 + 0.0019i
14.15 0.0132 + 0.0006i
14.20 0.0129 - 0.0006i
14.25 0.0124 - 0.0016i
14.30 0.0118 - 0.0025i
14.35 0.0111 - 0.0033i
14.40 0.0103 - 0.0039i
14.45 0.0096 - 0.0043i

> If the driving point impedance characteristics of the antenna do not
> meet this criterion we will not arrive at the correct values. Also, if a
> matching network is required to tune out the antenna reactance at the
> center frequency of interest then the network is considered to be a part
> of the antenna.

Therefore, I think the summary is that you can't really approximate an
antenna as a RLC combination, at least as far as Q and Bandwidth is concerned.

The plain old ugly interpolation approach will be your best bet, because,
for simple antennas at least, the impedance or VSWR or actual power
radiated curve will be smooth, and as long as you bracket the "resonance"
(actually, more of a zero reactive impedance case) you'll do fine on
calculating 2:1 VSWR bandwidths, or whatever.

By the way, the condition where the current is distributed as a sinusoid on
the wire (i.e. an exact half wavelength long) is NOT where the impedance is
resistive. Since people, by and large, want their antennas to be a
resistive load (so they aren't pumping reactive power up and down the
feedline), the antenna winds up cut a bit shorter than the exact half wave,
so the current distribution has a little dip in the middle.

I don't think that this "cut the antenna a few percent short" can be
properly attributed (as in some literature) to "end effects" or "stray
capacitance", but is more a consequence of the finite propagation speed of
EM waves. Kraus has several ways to do the analysis, ranging from purely
analytical approaches after Schelkunoff and Hallen, to numerical moment
methods, and they all come up the same way, for idealized wires, etc.

> Jim Lux

Spacecraft Telecommunications Equipment Section
Jet Propulsion Laboratory
4800 Oak Grove Road, Mail Stop 161-213
Pasadena CA 91109

818/354-2075, fax 818/393-6875

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