Hi Mike,
Yes, the math can get tricky and cause you to get lost in the forest. I'm not
afraid of doing math (I did a mathematics minor in graduate school) but as an
engineer, I prefer to seek the simplest possible solutions to engineering
problems. This is a case where you can avoid messy math by falling back on
basic principles.
First, I have invoked a passive, lossless phase combiner circuit for the array.
There is no loss in generality in doing so. Gain or loss are just scaling
factors that affect everything equally. This is why I said the total amplifier
circuit noise power coming out of the combiner is exactly equal to the sum of
the individual amplifier noise powers entering the combiner. There is no
combining loss and energy is conserved. For circuit noise it doesn't matter
what phases are used in the combiner because random noise is incoherent. Phase
only matters for coherent signals.
Next I fix the gain of a single vertical at 0 dB. Antenna gain is just another
scaling factor and here were are only interested in relative gain (a single
vertical vs. a phased array) when determining S/N ratios coming out of the
combiner. When I say the received atmospheric power is one "atmospheric noise
power unit", the gain is imbedded in that quantity but there is no need to
actually calculate the absolute gain for our purposes.
Next we invoke the principle that the spatially-averaged gain of a single
vertical and a phased vertical array are exactly the same. This allows us to
determine the atmospheric noise power coming out of the combiner without
actually doing any phasing calculations. For our combiner the total combined
atmospheric noise for the array is exactly the same as the total atmospheric
noise received by a single vertical.
For signals of interest, again we don't need to do any brute-force mathematical
phasing calculations. We can fall back on antenna modeling programs. Of
course you can do the math, but modeling programs will spare you the effort.
Because we have already fixed the gain of a single vertical (at 0 dB), we can
use EZNEC (or your favorite modeling program) to determine how much additional
gain the array provides on the signal of interest, based on its direction of
arrival. We have already established the array gain for atmospheric noise is
the same for the array and the single vertical. Therefore the S/N improvement
of the array vs. the single vertical (assuming atmospheric noise is the
dominant noise) is just the difference between the signal gain of the array and
the single vertical. Antenna modeling gives you that number. No ugly math is
needed.
Getting back to my earlier e-mail about circuit noise vs. atmospheric noise, we
do need to keep track of the actual amplifier circuit noise relative to the
atmospheric noise to insure that the circuit noise doesn't degrade the overall
noise performance of the system. That will guide us in the design of a
low-noise amplifier. To put numbers on these noise quantities, we need to
either calculate circuit noise power, as others have already started to do, or
to measure it (as I have done for different amplifiers). Similarly for
atmospheric noise, we need to calculate it or measure it. I leave that for
others to do. For an array of N amplified elements, we need to insure that N
times the circuit noise power of a single amplifier remains well below the
atmospheric received by the array.
73, John W1FV
-----Original Message-----
From: Topband [mailto:topband-bounces+john.kaufmann=verizon.net@contesting.com]
On Behalf Of Michael Tope
Sent: Friday, March 13, 2020 7:34 AM
To: topband@contesting.com
Subject: Re: Topband: Hi Z amplifiers for 160m (LONG)
I agree with your conclusions regarding the case of isotropic
atmospheric noise. This is the same reason that cold space looks like 3
Kelvin regardless of how high the antenna gain. As the antenna gain goes
up you reinforce to a greater degree a lesser slice of the overall pie.
This ends up being a wash.
Where I think you may be mistaken, is the relationship between the
number of amplified elements (N), the gain of the antenna, and properly
book keeping combining losses. If I have N amplified elements and I
mathematically sum the amplifiers outputs with zero combining loss (this
would be equivalent to digitizing the output of each amplifier and then
summing the results in digital processing), then the uncorrelated noise
from the amplifiers (as you correctly point out) sum to 10*log(N).
Double the number of amplified elements and you double the noise power
due to the amplifiers (i.e. 3dB amplifier noise increase). So far we agree.
Where it gets tricky is when you consider the mathematical addition of
the over-the-air contributions. If I have a linear broadside array and I
double the number of elements from N to 2*N, the mathematical sum of the
components of the signal-of-interest in the bore site of the main lobe
goes up by 20*log(2) = 6dB. That would imply that the gain of the array
has gone up by 6dB and that the azimuth beamwidth of the main lobe has
gone down by a factor of 4. However, if you look at gain of a linear
broadside array when you double the number of elements (assuming
constant element spacing), the gain goes up by at most 3dB. If the
average gain for the isotropic atmospheric noise is a constant 0dB, then
signal-of-interest in the antenna bore site can only go up by at most
3dB relative to the atmospheric noise. But the mathematical sum of the
components of the signal-of-interest have gone up 6dB, so the
mathematical sum of the components of the isotropic atmospheric noise
have to go up by at least 3dB.
I think I have this right, John, but feel free to shoot holes in it if I
don't. I know thinking about it made my head hurt.
73, Mike W4EF........
On 3/12/2020 4:37 PM, John Kaufmann via Topband wrote:
> To assess the impact of amplifier circuit noise in "active" receive arrays,
> we only need to be concerned with the contribution of amplifier circuit
> noise relative to atmospheric noise. The details of how signals are phased
> in any particular array do not matter. The objective is to keep the total
> contribution of amplifier noise far below the atmospheric noise so as not to
> degrade the overall system noise floor in any significant way. However, we
> need to understand that the combiner circuit that phases up the signals in a
> receive phased array responds very differently to amplifier noise and
> atmospheric noise. This makes it less obvious how to determine whether the
> circuit noise of a particular amplifier is "low enough". Fortunately, there
> is a simple way to determine that using basic principles.
>
> Let's start with a single amplified vertical antenna. To simplify the
> analysis, we just set the gain of the vertical to 0 dB. In practice we can
> do a NEC analysis to calculate absolute gain in dBi, factoring in real
> losses but that is not necessary and does not change the conclusions. The
> antenna feedpoint amplifier adds its own noise to whatever signal plus
> atmospheric noise is received by the vertical. Let's set the circuit noise
> power equal to one "circuit noise unit" and the atmospheric noise power to
> one "atmospheric noise unit". Of course we can put voltage (or power)
> numbers on those units, based on properties of the amplifier, the
> atmospheric noise, the actual antenna gain, and the measurement bandwidth.
> However, that makes things unnecessarily complicated, so we won't do that.
>
> Next we create an array of N amplified vertical antennas, each one identical
> to the single vertical we started out with. We feed the signals from all
> the antenna amplifiers into an ideal combiner circuit that does not add its
> own noise. The combiner circuit phases up signals to create a directive
> beam pattern. Now we ask how much atmospheric noise appears in the phased
> up sum compared to the amount of total amplifier circuit noise.
>
> The atmospheric noises received at the various verticals are all correlated.
> The correlation comes about because the atmospheric noise is the same at
> each vertical except for time delay differences caused by geometric path
> length differences to each antenna element. However, as I described in an
> earlier e-mail, the amplifier circuit noises coming from each of the antenna
> amplifiers are all uncorrelated.
>
> For uncorrelated noises, the combiner simply adds the circuit noise powers
> of the individual amplifiers as I described previously. For N elements with
> N amplifiers, the total circuit noise power out of the combiner is then N
> times one "circuit noise unit" (ignoring any additional gain or throughput
> loss imparted by the combiner circuit).
>
> To determine the total atmospheric noise coming out of the combiner circuit,
> let's assume the atmospheric noise has a completely uniform distribution in
> 3-dimensional space. That is, the strength of the atmospheric noise is the
> same in every direction. This is an idealized assumption, but is often a
> reasonable approximation to reality. Under these assumptions, the total
> atmospheric noise out of the combiner turns out to be just one "atmospheric
> noise unit"! In other words, it is exactly the same as the atmospheric
> noise coming out of a single vertical. This is because the total
> atmospheric noise power picked up by the array is just the gain of the array
> (relative to a single vertical) averaged over all of 3-dimensional space
> times one "atmospheric noise unit" (the noise picked up by a single
> vertical). That average gain is exactly 0 dB, so the total atmospheric
> noise doesn't change in our idealized system. It doesn't matter what the
> antenna pattern is; the average gain is always 0 dB, which is why we did not
> need to be concerned with details of how signals are phased up to form a
> beam pattern. Of course, a different gain applies to actual signals that
> are coming from a specific direction and are not uniformly distributed like
> atmospheric noise, which is why we see a S/N improvement when the array is
> aimed at a signal of interest.
>
> So, we have demonstrated that in relative terms, the amplifier circuit noise
> power in an array of N amplified antennas goes up by a factor N whereas the
> atmospheric noise does not change. That increase in the amplifier noise
> contribution relative to atmospheric noise degrades the overall noise figure
> of the system. However, as long as we keep the amplifier noise contribution
> small enough, the noise figure degradation can also be kept to a minimum.
> That is why having more amplified elements makes it more important to design
> the antenna amplifiers for low circuit noise.
>
> 73, John W1FV
>
>
>
>
>
>
> -----Original Message-----
> From: Topband
> [mailto:topband-bounces+john.kaufmann=verizon.net@contesting.com] On Behalf
> Of Michael Tope
> Sent: Thursday, March 12, 2020 4:37 PM
> To: topband@contesting.com
> Subject: Re: Topband: Hi Z amplifiers for 160m
>
> Hi Lee,
>
> Yes, if you are combining coherent signals that are not in phase, then
> the each of the voltage vectors is weighted by cos(phi-i) where phi-i is
> the angle between the i-th voltage vector and the 1st vector. If phi=0,
> then you have the case I described previously. I can see how this can
> get tricky, however, with an electrically short baseline where you are
> striving for cancellation in the rearward looking direction. It's like
> you cancel in the rearward direction and almost cancel in the preferred
> direction :-). This degrades the SNR not because the noise is adding up,
> but because the signals are subtracting down.
>
> 73, Mike W4EF.............
>
>
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