With this post I intend to show mathematically the nuances of this relationship between number of antennas in a system, exposure to data carrying light rays, and energy usage. My model will make some assumption for the sake of keeping the model relatively simple to explain. Realize that it is certainly not intended to be a model for actual systems, only to demonstrate these relationships in an elementary way without falling prey to odious complexity.
Here is a list of those assumptions
Again, realize that these are idealizations (which is to say not necessarily attempts to model reality) and that this list is not comprehensive. These are just the assumptions I think would be best to state.
Keep in mind that I do some rounding, so these often are not exact. I try to state when I'm estimating.
I also make use of weak not to scale ASCII art to create diagrams since I'm too lazy to do image hosting, so steel yourself.
Say we have antennas that can detect a signal with 1 unit of intensity, and send a signal with any (positive) power.
We want to send a detectable signal from point a to point b. No more, no less. How much power do the antennas need to output? In other words, how brightly must they shine to be seen by each other?
The power is being spread out over a larger and larger surface the farther it gets from the antenna to give the intensity as I=P/A where P and A are the power output and the area it's spread over respectively. The power is spread in the shape of a sphere, whose surface area is given by A=4*π*r^2. So the equation relating power (P), intensity (I) and the distance from the source (r), is I=P/(4*π*r^2).
Let's first consider the case where a is 1 distance unit away from b and we have 2 antennas, one at a and one at b, like so.
a b
| |
We know the intensity of the signal must be 1 when it reaches the second antenna. With a scrap of algebra, we get P=I*(4*π*r^2) and filling in the blanks, P=1*(4*π*1^2). Simplifying that yields P=4*π which is approximately 12.5.
So to get a signal between these two antennas, the sender must send a signal with power a little over 12.5.
Lets increase the distance between a and b by 10 times. Many of these types of technologies do their work over great distances after all, like radio.
a b
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Using our power equation we get P=1*(4*π*10^2) which makes P=400*π by simplification and that's about 1,256. So the antenna must now put out a bit over 1256 power to be detectable at distance 10. That is, the first antenna must shine 100 times more powerfully for the signal to be detectable to the second this time.
Let's say we put 4 people between the antennas, a person every 2 units starting at 1.5 units in our 10 unit distance from a to b.
a b
| p p p p |
Now we are interested in how much each person is exposed to the power output of the antennas. So the person 1.5 units away from the sending antenna receives an intensity of around 44, or exactly 1256/(4*π*1.5^2). The person at 3.5 units away gets intensity about 8.16. 5.5 units away is about 3.3 intensity, and the person at 7.5 is hit by a signal of intensity near 1.7 units.
Let's change our antenna array up a bit. Instead of sending one signal pulse from the antenna at a to the one at b, we will instead have 11 antennas. They will be placed one every distance unit from a to b inclusive, and will each send a signal only powerful enough to be detectable to the next one along the line.We already know that to send a signal distance 1 the antenna must shine with (approximately) intensity 12.5. So each of the first 10 antennas will send a signal with power 4*(pi). We send the signal from the first one, and have the rest set to send their signal the first time they detect a signal. In this way the signal will be passed between them from a to b. We only use roughly 12.5 * 10 = 125 power to send the signal, 10 times less.
a b
| |p| |p| |p| |p| | |
Let's examine how much each person was exposed to this time. Person 1.5, which is to say the person 1.5 distance units away from point a, is first hit with a signal of intensity a bit over 0.44 from the first antenna's signal that we sent. This wave keeps going all the way through each person with diminishing intensity along with each of the nine other waves propagated. Next, the second signal flash hits person 1.5 with intensity 4, and continues along. I'm going to need to calculate and sum 10 different values for each person, and for the sake of concision I'm going to just do the math and show the results instead of going step by step at this point.
Person 1.5 gets total intensity 9.2536
Person 3.5 gets total intensity 9.4537
Person 5.5 gets total intensity 9.4537
Person 7.5 gets total intensity 9.2536
sum = 37.4146
simple mean = 9.3537
geometric mean = 9.3531
harmonic mean = 9.3525
quadratic mean = 9.3541
For comparison here are the numbers from the test with only 2 antennas over 10 units distance.
Person 1.5 gets total intensity 44.4219
Person 3.5 gets total intensity 8.1591
Person 5.5 gets total intensity 3.3041
Person 7.5 gets total intensity 1.7777
sum = 57.6912
simple mean = 14.4228
geometric mean = 6.7952
harmonic mean = 3.9602
quadratic mean = 22.6717
I wasn't sure which mean would be most appropriate, so I included a few different ones for your consideration. Some rounding did happen with these values too, but they should be pretty close.
I'll confess that I didn't know exactly how this would turn out before doing the math, but that's what made it exciting!
One thing that really jumped out at me are how much less power the many antenna system used. Maybe this kind of effect is why some sources say that 5G networks will use less energy.
With the 11 antenna scenario even though all the people were exposed to more radiation than 3/4 of the people from the 2 antenna scenario, the total exposure over all people was about %30 less. My model fails to capture some things I think are noteworthy. For example, in the 2 antenna scenario a person at -5 units from point a would be exposed to just as much as a person at 5 units from point a. This is in great contrast to the 11 antenna experiment, where the person at -5 will be exposed to far less. Exposure also goes down much more quickly in the 11 antenna scenario as the people move perpendicularly away from the line segment between point a and point b. In effect, where the people were placed was the area where their exposure would be maximized.
Generally it seems to me that there is less risk of high exposure with the 11 antenna scenario vs the 2 antenna scenario, but this is fertile ground for confirmation bias so, please dear reader, set me straight if I need it.
In closing, I'm going to take one more opportunity to emphasize that I don't think this is exactly how real systems work. There are technologies like beamforming that make signals propagate less uniformly (more directionally) for instance. I suppose I could've made a more realistic and more compelling model by including details like this, or generalizing into three dimensions. I think my model is fairly compelling however, and not entirely unsophisticated. To do so would have been much less accessible, and I'm already worried that this might not be very accessible. Not to mention it would have taken me much more time and effort and I probably put in an inordinate amount of both already anyway. I don't expect to make even half a Steem from this if previous author rewards are anything to go by, but low expectations immunize against disappointment after all.
Thanks for reading. Kudos for making it this far! I know it wasn't exactly action-packed. Hopefully you got something out of this and it's not just blustery numbers beating a dead horse.
