Pseudo free space communication ranges
There's a rule of thumb that says free space propagation occurs between a base station antenna on a hilltop or tower and a mobile unit at that base station's geographic horizon. This type of radio path is extremely reliable; it's more likely to be interrupted by artificial interference than by any natural phenomenon.
For spherical earth, the range calculation is a straightforward exercise in trigonometry. At ordinary hilltop heights, the horizon distance can be approximated accurately enough by a simplified equation in the REA handbook:
Dfree=sqrt(2H1)+sqrt(2H2)
where
Dfree=free space path distance in miles
H1=height of base station antenna above ground level in feet
H2=height of mobile station antenna above ground level in feet
In kilometers and meters, that would be
Dfree=4.13[sqrt(H1)+sqrt(H2)]
There's also a rule of thumb that the distance to the radio horizon is about 4/3 the distance to the geographic horizon. That's because of refraction due to the density gradient in the lower atmosphere. However, that effect can vary a lot with weather conditions, especially at microwave frequencies. Relying on over-the-horizon tropospheric bending can result in random outages.
With the free-space geographic range calculated, the curves in the REA handbook can then supply the path loss. The curves assume an antenna height of 6 feet at the mobile end. That's a reasonable assumption for our wagon-mobile units if they don't find a convenient hill or a tall tree. We'll do the path loss and power calculations for 15 MHz only, extrapolated from the 40 MHz loss curves. That's less of an extrapolation than doing it for 5 MHz as well; also, the noise is much lower at 15 MHz, so that band is a good choice for land mobile use anyway.
The free space distance equation obviously includes the case in which both station antennas are elevated, as in hilltop-to-hilltop operation between relay stations. The range between them is the sum of their horizon distances. Of course, this assumes no terrain obstacles tall enough to obstruct the direct path between the stations, and no interfering bounces off highly reflective surfaces. For that case, where the wavefront continues to expand and the power density declines after the wave passes the transmitting station's horizon, the path loss is not the sum of the losses of the two separate paths from each station to a 6-ft high station at their mutual horizon. Instead, the inverse square law applies to the second part of the path. For the case of two stations at equal heights, the additional path loss would be -6 dB due to the quadrupling of the wave front's area in the second half of the path. Quadrupling the transmitter power would compensate for that. For example, taking the case in the table below where a 250-mW transmitter on a 1600-ft hill could reach a 6-ft high antenna 60 miles away, 1 W would be sufficient to reach a station on another 1600-ft hill 120 miles away.
The REA curves show path loss at distances continuing well beyond the horizon (for a mobile station on flat ground), at which point the free space wave transitions into ground wave, and the losses increase. In other words, with increased power it's possible to communicate reliably for some distance beyond the geographic horizon. We'll explore the distances where the received power falls to the desired +16 dB S/N, for 25 W, 5 W, and ¼ W.
The curves assume a half-wave vertical at the base station and a quarter-wave vertical at the mobile. That's a reasonable setup for a fixed base station communicating with a mobile unit. A quarter-wave antenna at a mobile set-up on a hilltop might send 3 dB less power toward the lower elevations.
For this calculation, we'll use the assumed antenna combination, and add a correction for the 15 MHz receiving antenna's larger capture area relative to the same antenna design at 40 MHz. That comes out to G=+8.5 dB.
As with the ground wave case at 15 MHz:
Pnoise=-116 dBm
Receiver Preqd=-100 dBm
In the following table
Lfree is the path loss at Dfree
G=+8.5 dB
Pfree is the transmitter power required to achieve +16 dB S/N at Dfree
The published curves are drawn for eight base station heights given in feet, with distances in miles, so we'll use those as the primary units and calculate the metric equivalents.
In short, at any distance up to 60 miles, a quarter-watt tuna can transmitter is powerful enough to communicate by Morse code, as long as the station at the far end is on a high enough hill and has a receiving filter just wide enough to pass a CW signal.
With relatively simple first-generation tube equipment, five watts would be adequate to send messages over a 100-kilometer path, from even a fairly modest hilltop. Even a kilowatt wouldn't be sufficient to do that by ground wave.
(Operators generally prefer not to select the narrowest possible filter that will pass the signal, unless they need it either to suppress noise or to separate closely-spaced signals in a crowded band. Given that mobile tube transmitters would usually fall into the 5- to 25-watt range, this propagation mode offers the less fatiguing sound of a somewhat wider filter.)
These examples demonstrate the great value of high locations for communication across land. With the coming of radio, mountaintops have become strategic terrain. That's why the USE government maintains a major relay station on top of the Brocken in the Harz Mountains. Though canon doesn't mention a permanent station atop the Großer Beerberg in the Thüringerwald, it's reasonable to expect one there as well. Let's look briefly at a few terrain altitudes of interest in the USE.
Diffraction losses
Calculating the additional path loss due to diffraction over an obstacle can be very complex. The Radio Propagation Handbook devotes an entire chapter to it, and we can't do full justice to the topic here. What we can do is examine a few simple cases that a mobile communication crew is likely to encounter.
If a radio path that's otherwise free-space or close to it is obstructed by higher terrain near one end, it can be modeled mathematically to a good approximation as an additional loss term added to the free space path loss. The diffraction loss can be compensated by increasing the transmitter power. That loss is a nonlinear function of the frequency and the angle through which the path must bend to reach the receiver. The handbook provides a nomograph for the purpose.
If both ends of the path are obstructed, then two diffraction loss terms must be added to the free space path loss.
The handbook says that diffraction is likely to dominate the path over the obstacle if the bend angle is less than 0.02 radian. That's equivalent to a 2% grade, if the wave arrives at the obstacle parallel to the horizon. For example, that would be the case if there were a 50-meter-high ridge line 1 km from the receiver. If the angle is greater than that, the handbook indicates that other less lossy propagation mechanisms such as forward scatter might deliver a stronger signal to the receiver. However, the chapter is written with UHF and microwave signals in mind. At the wavelength of a 15 MHz signal (20 meters) other modes are less likely to be of much help, so we'll make the pessimistic assumption that an obstructed station depends on diffraction.
For example, a mobile unit communicating with a base station on a 30-meter hill 28 km away over an unobstructed path would require about 90 mW. But if the mobile's path is blocked by a 50-meter ridge 1 km away, the route would require 12.6 times as much power, or 1.13 W. If both ends are similarly obstructed, 14.2 W would be needed.
50 watts should be sufficient to communicate over the majority of pseudo-free space paths where there's a single obstruction at one end. Mobile-to-mobile paths where both stations have limited power and nearby obstructions tend to have significantly reduced range. That situation is common in hilly terrain. For that reason, there would be a tactical advantage to setting up a relay station or a command post on a high location, if one can be secured in the operation area.
Reflections
Radio waves reflect from conductive surfaces and can bounce into shadowed areas. In up-time cites reflections off the metal structu
res of buildings are very common. In the seventeenth-century world it's possible that 15 MHz signals might reflect from metal-roofed church spires or off steep cliffs. Otherwise, they're not likely to be a frequent source of help to radio operators. The handbooks give little information on estimating their magnitude, except for billboard-sized metal mirrors used on microwave fixed routes and oriented with great care.
Sky wave
To recap very briefly the discussion in "1636: Marine Radio in the Mediterranean," sky wave below the AM broadcast band is likely to be reliable while the ionospheric path is in full darkness. That lasts from about an hour after sunset at the west end of the path to an hour before sunrise at the east end. The range estimated in that article, using a standard full-size antenna on a single-hop path, came out to about 950 km with 100 watts, and 2000 km with 1 kilowatt.
Shorter ranges may be iffy. The published loss curves for 100 and 200 kHz show reasonable path losses down to 200 km or so, but this may not be reliable in the weak ionization conditions of the seventeenth century. A signal striking the ionosphere at a steep angle may not be bent enough to return to earth. The shallower incidence angles at longer ranges are more likely to be reflected back to the ground.
Thus, we could encounter a skip-over zone somewhere between 350 km where 24/7 ground wave becomes too weak to copy, and 500 km or so, where night-time sky wave first reaches the ground. Message traffic for that dead zone might have to be relayed by a station 1000 km away.
Interestingly, the "gray line" mode canonized in a number of places, beginning in 1632 itself, is a manifestation of sky wave, but at higher frequencies. The canon contacts made with this mode used unmodified up-time ham gear. Ham transmitters aren't built to operate at 500 kHz and lower. Their lowest band starts at 1.8 MHz. At that frequency, the weak ionization of the seventeenth-century Maunder Minimum is barely enough to offer skip for a short period around twilight. Sky wave opens when the ionosphere's high-loss lower layer fades shortly after sunset, allowing the signals to reach the higher layers where skip happens. But the higher-layer ionization degrades with time, too. The higher frequencies need stronger ionization to be bent back to earth, so the ham band opening fades quickly. The short duration of that opening limits communication to stations near the same longitude, hence the term "gray line." But based on long historical experience, medium and low frequencies should be able to reflect off much weaker ionization than the ham bands need, and persist for many hours during the night.
Spark communication capabilities
Having covered with reasonable confidence what CW could do using the major propagation modes, it's time to take a look at spark. Here, we're on much shakier ground. In fact, any estimate of what could be done with spark verges on outright speculation. The problem is that most of the published material dealing with spark that can be easily found nowadays is more historical than technical. There are a great many unknowns.
We do have Rick Boatright's spark article "Radio FAQ Part 1: Spark and Crystal Radios" posted at http://www.1632.org/1632tech/faqs/radio-rfe.html. We also have "The History of Amateur Radio, Part IV" at http://www.astrosurf.com/luxorion/qsl-ham-history4.htm. They're in good agreement that after the U.S. 1912 Radio Act pushed hams above 1.5 MHz (200 meters), the working ranges were typically between 25 and 75 miles. That's with 600 watts input, on ground wave across North American "good earth," with the equipment and antenna a ham could set up at home. So, what should we expect in our early NTL years in Europe?
It's a fair guess that anyone who doesn't have access to tube gear probably won't be getting power from a commercial electric utility, either. That's doubly true for a mobile station. So, 20 or 30 watts from storage batteries or pedal generators is a lot more likely than 600 watts. Not to mention, components for that power level would be a lot easier to fabricate from materials generally available in early modern times, and a lot more reliable as well. And, as noted above, European soil is typically considered to be "poor earth" in the absence of specific data.
Going from 600 watts down to 30 watts is a change of -13 dB. We can look up the loss for ground wave propagation across 75 miles on good earth at 1.5 MHz, and then look up the distance on the poor-earth chart for 13 dB less loss at the same frequency.
All other things being equal, we get 13 miles (21 km).
But all other things may very well not be equal.
For a typical early twentieth-century ham living in a suburban lot with trees for antenna supports, a full-size antenna for 1.5 MHz was impossible, let alone an optimally constructed full-size antenna. That would be 50 meters high on a ground plane 100 meters in diameter. The obvious solution would be a "T" antenna. But there usually wasn't the space or budget for that either.
What a ham in that period could usually put up would be a single wire going up at a roughly vertical angle to somewhere in a tree's branches. A single horizontal wire from the top to another tree would provide some amount of capacitive top-loading, but the more-or-less vertical wire would connect to one end, not to the precise center. Then, instead of multiple radial wires at ground level for the return current to flow into with low resistive loss, there would usually be a clamp on a single water pipe running out to the water main in the street. Lacking city water, there would often be nothing more than an eight-foot metal rod driven into the ground outside the window, which might or might not reach the water table.
Everything is wrong with this. It's a random shortened vertical, which has a broadened vertical radiation pattern to begin with. The antenna current is less than optimally coupled to the electromagnetic field; that requires the electrical resistances elsewhere in the RF circuit to be very low if the antenna is to be at all efficient. But the ground resistance is high, so power is wasted. Because the horizontal wire isn't centered on the top of the vertical wire, horizontally polarized emission isn't cancelled out, so some of the power is wasted in a horizontally polarized signal that can't couple into the ground wave. The random-wire top loading is generally insufficient to resonate the antenna, so inductance must be added, but an imperfect inductor adds more resistance to the circuit and wastes more power. The tilt of the vertical wire further de-optimizes coupling into the ground wave.
It does radiate. Any conductor that carries RF current will radiate something. It just doesn't do an efficient job of converting RF power into a ground wave signal.
A military communications detachment in the field, or a commercial enterprise with money to spend and the connections to obtain an unobstructed site, need not accept these limitations. They could locate where there's room to put up a proper antenna for the band they're using. (This becomes easier if they select a higher frequency, where antennas aren't so large.)
Another limitation of the early ham station was the crystal set. A crystal set's only source of power to drive the headset is the radio wave. The signal-to-noise ratio was not the only determinant of whether the signal could be heard; it was the absolute strength of the signal itself. The capture area and efficiency of the receiving antenna had as much to do with that as the transmitting antenna. (Not only that, the passive crystal set imposes an unfortunate trade-off between selectivity and sensitivity. The more tightly the resonant tank circuit is coupled to the antenna, the more signal can be passed through to the headset, but the broader its bandwidth becomes, and less effective it is in separating signals on nearby frequencies.)
Those limitations would be the case for most of the crystal sets in our fictional universe, but it need not be true for all of them. One of the pieces of bypassed technology, which is certainly known to Grantville's radio scholars, is the electromechanical audio amplifier. In principle, it's an earphone mechanically coupled to a carbon microphone. Drive one or two stages of audio amplification from a crystal set, and a weak signal could be brought up to audibility. Noise would again become the limitation.
Combine optimally designed and installed antennas with crude audio amplification, and perhaps that -13 dB could be made up. That does
n't take deep knowledge and years of experience, it just takes money, materials, and manpower. Any army that's had a spy in the libraries could at least optimize its antennas.
Which bands would likely be used for spark radio is another major area of uncertainty. 1.5 MHz is inside the upper end of the AM broadcast band. All the thousands of legacy up-time broadcast band receivers cover 530 kHz to 1.710 MHz with 10 kHz channel spacing. That's a large enough installed base of equipment to permanently nail down that band for broadcasting. Spark would be most unwelcome there. Besides, an optimum antenna for such a low frequency is inconveniently large for most users, especially mobile stations. Broadcasting stations are few in number and commercially funded, hence can afford good antennas and enough power to reach crystal sets.
The next band up in the spectrum is the 160-meter ham band at 1.8 to 2.0 MHz, which has been in use for vital government and military communication almost from the time the up-timers arrived. The spectrum plot of a spark transmitter in Rick's article shows most of the power concentrated in a 10 kHz bandwidth, but with splatter spreading out for 100 kHz on each side. That kind of interference would be even more unwelcome in this busy piece of spectrum.
Up-time band allocations place a marine band at 2.0 to 2.5 MHz, which appears in "Storm Signals" (Grantville Gazette 31). That's a broad enough chunk of spectrum to accommodate at least one spark channel without seriously inconveniencing all the CW stations. The experimental spark transmitter Rick cites was tested at 2 MHz, so we know it's feasible. It's reasonable to expect some spark stations at somewhat higher frequencies as well, to take advantage of the smaller and less expensive antennas, the more modest demands on real estate, and the less demanding logistics.
Grantville Gazette, Volume 73 Page 17
