Topband: Flag theory by Dr Dallas

[Jose_Carlos]

Hi guys


I would like to share some papers from my good friend (SK) Dr Dallas was a great experimental guy that loved Mathematic and AM DXing,. Dr Dallas with a strong personality and seniority some times was hard to follow and some time the result of tests were not as planned, like the multi turn flags.

Dr Dallas interest was on MF 500KHz to 1600 MHz  and some results are not the same on 160 or 80m. I can send the original paper attached if requested.


 Flag Theory
Dallas Lankford, 1/31/09, rev. 9/9/09
The derivation which follows is a variation of Belrose's classical derivation for ferrite rod loop antennas,
“Ferromagnetic Loop Aerials,” Wireless Engineer, February 1955, 41– 46.
Some people who have not actually compared the signal output of a flag antenna to other small antennas have
expressed their opinions to me that the signal output of a flag antenna has great attenuation compared to those
other small antennas, such as loops and passive verticals. Their opinions are wrong. One should never express
opinions which are based, say, on computer simulations alone, without actual measurements. The development
below is based on physics (including Maxwell's equations), mathematics, and measurements.
Measurements have confirmed that the flag signal to noise formula derived below is approximately correct
despite EZNEC simulations to the contrary. For example, EZNEC simulation of a 15' square loop at 1 MHz
predicts its gain is about +4 dbi, while on the other hand EZNEC simulation of a 15' square flag at 1 MHz
predicts its gain is about –46 dBi. But if you construct such a loop and such a flag and observe the signal
strengths produced by them for daytime groundwave MW signals, you will find that the maximum loop and flag
signal outputs are about equal. Although somewhat more difficult to judge, the nighttime skywave MW signals
are also about equal.
Also, the signal to noise ratio formula below for flag arrays has been verified by man made noise measurements
in the 160 meter band using a smaller flag array than the MW flag array discussed below. Several years ago a
similar signal to noise ratio formula for small untuned (broadband) loop antennas was verified at the low end of
the NDB band.
The signal voltage es in volts for a one turn loop of area A in meters and a signal of wavelength λ for a given
radio wave is
es = [2πA Es /λ] COS(θ)
where Es is the signal strength in volts per meter and θ is the angle between the plane of the loop and the radio
wave. It is well known that if an omnidirectional antenna, say a short whip, is attached to one of the output
terminals of the loop and the phase difference between the loop and vertical and the amplitude of the whip are
adjusted to produce a cardioid patten, then this occurs for a phase difference of 90 degrees and a whip amplitude
equal to the amplitude of the loop, and the signal voltage in this case is
es = [2πA Es /λ] [1 + COS(θ)] .
Notice that the maximum signal voltage of the cardioid antenna is twice the maximum signal voltage of the loop
(or vertical) alone. A flag antenna is a one turn loop antenna with a resistance of several hundred ohms inserted
at some point into the one turn. With a rectangular turn, with the resistor appropriately placed and adjusted for
the appropriate value, the flag antenna will generate a cardioid pattern. The exact mechanism by which this
occurs is not given here. Nevertheless, based on measurements, the flag antenna signal voltage is approximately
the same as the cardioid pattern given above. The difference between an actual flag and the cardioid pattern
above is that an actual flag pattern is not a perfect cardioid for some cardioid geometries and resistors. In
general a flag pattern will be
es = [2πA Es /λ] [1 + kCOS(θ)]
where k is a constant less than or equal to 1, say 0.90 for a “poor” flag, to 0.99 or more for a “good” flag. This
has virtually no effect of the maximum signal pickup, but can have a significant effect on the null depth.
1
The thermal output noise voltage en for a loop is
en = √(4kTRB)
where k (1.37 x 10^–23) is Boltzman's constant, T is the absolute temperature (taken as 290), (Belrose said:) R is
the resistive component of the input impedance, (but also according to Belrose:) R = 2πfL where L is the loop
inductance in Henrys, and B is the receiver bandwidth in Hertz. When the loop is rotated so that the signal is
maximum, the signal to noise ratio is
SNR = es/en = [2πA Es /λ]/√(4kTRB) = [66Af/√(LB)]Es .
The point of this formula is that the sensitivity of small loop antennas can be limited by internally generated
thermal noise which is a characteristic of the loop itself. Even amplifying the loop output with the lowest noise
figure preamp available may not improve the loop sensitivity if man made noise drops low enough.
Notice that on the one hand Belrose said R is the resistive component of the input impedance, but on the other
hand R = 2πL. Well never mind. Based on personal on hands experience building small loops, I believe R =
2πL is approximately correct. What I believe Belrose meant is that R is the magnitude of the output impedance.
For a flag antenna rotated so the the signal is maximum, the signal to noise ratio is
SNR = es/en = 2[2πA Es/λ]/√(4kT√((2πfL)^2 + (Rflag)^2)B) = [322Af/√(√((2πfL)^2 + (Rflag)^2)B)]Es .
Now let us calculate a SNR. Consider a flag 15' by 15' with inductance 24 μH at 1.0 MHz with 910 ohm flag
resistor, and a bandwidth of B = 6000 Hz. Then A = 20.9 square meters and SNR = 2.86x10^6 Es . If Es is in
microvolts, the the SNR formula becomes
SNR = 2.9 Es .
Any phased array has loss (or in some cases gain) due to the phase difference of the signals from the two
antennas which are combined to produce the nulls. This loss (or gain) depends on (1) the separation of the two
antennas, (2) the arrival angle of the signal, and (3) the method used to phase the two flags. Let φ be the phase
difference for a signal arriving at the two antennas. It can be shown by integrating the difference of the squares
of the respective cosine functions that the amplitude A of the RMS voltage output of the combiner given RMS
inputs with amplitudes e is equal to to e√(1 – COS( φ)) where e is the amplitude of the RMS signal, in other
words,
A= 1
2π∫
0
2π
2 e2cost−costφ2dt=e21−cosφ
The gain or loss for a signal passing through the combiner due to their phase difference is thus √(1 – COS( φ)).
Let us consider the best case, when the signal arrives from the maximum direction. For a spacing s between the
centers of the flags, if the arrival angle is α, then the distance d which determines the phase difference between
the two signals is d = s COS(α). If s is given in feet, then the conversion of d to meters is d = s COS(α)/3.28.
The reciprocal of the velocity of light 1/2.99x10^8 = 3.34 nS/meter is the time delay per meter of light (or radio
waves) in air. So the phase difference of the two signals above in terms of time is T = 3.34 s COS(α)/3.28 nS
when s is in meters. The phase difference in degrees is thus φ = 0.36Tf = 0.36 f x 3.34 s COS(α)/3.28 where f is
the frequency of the signals in MHz. If additional delay T' is added (phase shift to generate nulls or to adjust the
reception pattern), then the phase difference is φ = 0.36(T + T')f = 0.36f(T' + 3.34 s COS(α)/3.28) . If the
additional delay is implemented with a length of coax L feet long with velocity factor VF, then the phase delay is
φ = 0.37f(L/VF + s COS(α))
where f is the frequency of the signal in MHz, s is in feet, L is in feet, and α is the arrival angle.
2
In the case of the flag array above in the maximum direction there are two sources of delay, namely 60.6 feet of
coax with velocity factor 0.70, and 100 feet of spacing between the two flag antennas. The phase delay at 1.0
MHz for a 30 degree arrival angle is thus
φ = 0.37 x 1.0 x (60.6/0.70 + 100 COS(30)) = 64.1 degrees.
Thus the signal loss in the maximum direction at 30 degree arrival angle due to spacing and the phaser is
√(1 – COS( 64.1)) = 0.75 or 20 log(0.75/2) = –8.5 dB.
Now comes the interesting part. What happens when we phase the WF array with dimensions and spacing given
above? The flag thermal noise output doubles (two flags), and the flag signal output decreases (due to spacing
and phaser loss), so the SNR is degraded by 14.5 dB to
SNR = 0.55 Es .
So a signal of 1.8 microvolts per meter is equivalent to the thermal noise floor of the flag array.
On some occasions, when man made noise drops to very low levels at my location, it appeared to fall below the
thermal noise floor of the WF array. By that I mean that the characteristic “sharp” man made noise changed
character to a “smooth” hiss. To determine whether this was the case, I measured the man made noise at my
location for one of these low noise events at 1.83 MHz.
To measure man made noise at my location I converted one of the flags of my MW flag array to a loop. The
loop was 15' by 15', or 20.9 square meters. I used my R-390A whose carrier (S) meter indicates signals as low
as –127 dBm. The meter indication was 4 dB. Then I used an HP-8540B signal generator to determine the dBm
value for 4 dB on the R-390A meter. It was –122 dBm. Now the fun begins. The RDF of a loop for an arrival
angle of 20 degrees (the estimated wave tilt of
man made noise at 1.83 MHz) was 4 dB. So now
man made noise after factoring out the loop
directionality was estimated as –118 dBm. Field
strength is open circuit voltage equivalent, which
gives us –112 dBm. I measured MM noise on the
R-390A with a 6 kHz BW. The conversion to
500 Hz is –10 log(6000/500) = –10.8, which
gives us –122.8 or –123 dBm. The conversion to
500 Hz was necessary in order to be consistent
with the SNR above which was calculated for a
500 Hz BW. The loop equation is es = 2πAEs
/lambda = 0.41 Es, and 20 log(0.41) = –7.7,
rounded off to - 8, so we have -115 dBm, or 0.40
microvolts per meter for my lowest levels of man
made noise at 1.83 MHz in a 500 Hz bandwidth.
This seemed impossibly low to me until I came
across the the ITU graph at right. Man made
noise at quiet rural locations may be even lower
than 0.40 microvolts per meter at 1.83 MHz. But what about the MW band? From the CCIR Report 322 we
find that the man made noise field strength on the average is about 10 dB higher at 1.0 MHz than 1.83 MHz,
which would make it 1.26 microvolts per meter at 1.0 MHz. Another 4 dB is added because of impedance
mismatch between the R-390A and the loop, which brings man made noise up to 2.0 microvolts per meter at 1.0
MHz. The RDF of one of these flags is about 7 dB, which lowers the man made noise to 0.89 microvolts per
meter. Observations in the 160 meter band do not seem to agree exactly with this analysis because flag thermal
noise has never been heard on the MW flag array. But it would not surprise me at all if the flag array thermal
noise floor were only a few dB below received minimum daytime man made noise and that measurement error
(for example, calibration of my HP 8640B) accounts for the difference between measurement and theory. Also,
observations with a flag array having flag areas half the size of the MW flag elements in the 160 meter band do
confirm the signal to noise ratio formula; in this case, flag thermal noise does dominate minimum daytime man
made noise at my location (0.40 microvolts per meter field strength measured as described above.




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