By Bruce A. Fette

It is fitting in a book about wireless communication to look at the phenomena that lets us transfer information from one point to another without any physical medium—the propagation of radio waves. If you want to design an efficient radio communication system, even for operation over relatively short distances, you should understand the behavior of the wireless channel in the various surroundings where this communication is to take place. While the use of "brute force" —increasing transmission power—could overcome inordinate path losses, limitations imposed on design by required battery life, or by regulatory authorities, make it imperative to develop and deploy short-range radio systems using solutions that a knowledge of radio propagation can give.

The overall behavior of radio waves is described by Maxwell’s equations. In 1873, the British physicist James Clerk Maxwell published his *Treatise on Electricity and Magnetism* in which he presented a set of equations that describe the nature of electromagnetic fields in terms of space and time. Heinrich Rudolph Hertz performed experiments to confirm Maxwell’s theory, which led to the development of wireless telegraph and radio. Maxwell’s equations form the basis for describing the propagation of radio waves in space, as well as the nature of varying electric and magnetic fields in conducting and insulating materials, and the flow of waves in waveguides.

From them, you can derive the skin effect equation and the electric and magnetic field relationships very close to antennas of all kinds. A number of computer programs on the market, based on the solution of Maxwell’s equations, help in the design of antennas, anticipate electromagnetic radiation problems from circuit board layouts, calculate the effectiveness of shielding, and perform accurate simulation of ultra-high-frequency and microwave circuits. While you don’t have to be an expert in Maxwell’s equations to use these programs (you do in order to write them!), having some familiarity with the equations may take the mystery out of the operation of the software and give an appreciation for its range of application and limitations.

**Mechanisms of Radio Wave Propagation**

Radio waves can propagate from transmitter to receiver in four ways: through ground waves, sky waves, free space waves, and open field waves. Ground waves exist only for vertical polarization, produced by vertical antennas, when the transmitting and receiving antennas are close to the surface of the earth. The transmitted radiation induces currents in the earth, and the waves travel over the earth’s surface, being attenuated according to the energy absorbed by the conducting earth. The reason that horizontal antennas are not effective for ground wave propagation is that the horizontal electric field that they create is short circuited by the earth.

Ground wave propagation is dominant only at relatively low frequencies, up to a few MHz, so it needn’t concern us here. Sky wave propagation is dependent on reflection from the ionosphere, a region of rarified air high above the earth’s surface that is ionized by sunlight (primarily ultraviolet radiation).

The ionosphere is responsible for long-distance communication in the high-frequency bands between 3 and 30 MHz. It is very dependent on time of day, season, longitude on the earth, and the multiyear cyclic production of sunspots on the sun. It makes possible long-range communication using very low power transmitters. Most short-range communication applications that we deal with in this chapter use VHF, UHF, and microwave bands, generally above 40 MHz. There are times when ionospheric reflection occurs at the low end of this range, and then sky wave propagation can be responsible for interference from signals originating hundreds of kilometers away. However, in general, sky wave propagation does not affect the short-range radio applications that we are interested in.

The most important propagation mechanism for short-range communication on the VHF and UHF bands is that which occurs in an open field, where the received signal is a vector sum of a direct line-of-sight signal and a signal from the same source that is reflected off the earth. Later we discuss the relationship between signal strength and range in line-of-sight and open field topographies.

The range of line-of-sight signals, when there are no reflections from the earth or ionosphere, is a function of the dispersion of the waves from the transmitter antenna. In this free-space case the signal strength decreases in inverse proportion to the distance away from the transmitter antenna. When the radiated power is known, the field strength is given by equation (5.1):

where P_{t} is the transmitted power, G_{t} is the antenna gain, and d is the distance. When P_{t} is in watts and *d* is in meters, *E* is volts/meter. To find the power at the receiver (P_{r}) when the power into the transmitter antenna is known, use (5.2):

G_{t} and G_{r} are the transmitter and receiver antenna gains, and λ is the wavelength.

Range can be calculated on this basis at high UHF and microwave frequencies when high-gain antennas are used, located many wavelengths above the ground. Signal strength between the earth and a satellite, and between satellites, also follows the inverse distance law, but this case isn’t in the category of short-range communication! At microwave frequencies, signal strength is also reduced by atmospheric absorption caused by water vapor and other gases that constitute the air.

**Open Field Propagation**

Although the formulas in the previous section are useful in some circumstances, the actual range of a VHF or UHF signal is affected by reflections from the ground and surrounding objects. The path lengths of the reflected signals differ from that of the line-of-sight signal, so the receiver sees a combined signal with components having different amplitudes and phases.

The reflection causes a phase reversal. A reflected signal having a path length exceeding the line-of-sight distance by exactly the signal wavelength or a multiple of it will almost cancel completely the desired signal ("almost" because its amplitude will be slightly less than the direct signal amplitude). On the other hand, if the path length of the reflected signal differs exactly by an odd multiple of half the wavelength, the total signal will be strengthened by "almost" two times the free space direct signal.

In an open field with fl at terrain there will be no reflections except the unavoidable one from the ground. It is instructive and useful to examine in depth the field strength versus distance in this case. The mathematical details are given in the Mathcad worksheet "Open Field Range."

In **Figure 5.1** we see transmitter and receiver antennas separated by distance *d* and situated at heights h_{1} and h_{2}. Using trigonometry, we can find the line of sight and reflected signal path lengths d_{1} and d_{2}. Just as in optics, the angle of incidence equals the angle of reflection θ. We get the relative strength of the direct signal and reflected signal using the inverse path length relationship. If the ground were a perfect mirror, the relative reflected signal strength would exactly equal the inverse of d_{2}. In this case, the reflected signal phase would shift 180 degrees at the point of reflection. However, the ground is not a perfect reflector. Its characteristics as a reflector depend on its conductivity, permittivity, the polarization of the signal, and its angle of incidence.

*5.1. Open field signal paths*

In the Mathcad worksheet we have accounted for polarization, angle of incidence, and permittivity to find the reflection coefficient, which approaches -1 as the distance from the transmitter increases. The signals reaching the receiver are represented as complex numbers, since they have both phase and amplitude. The phase is found by subtracting the largest interval of whole wavelength multiples from the total path length and multiplying the remaining fraction of a wavelength by 2π radians, or 360 degrees.

**Figure 5.2** gives a plot of relative open field signal strength versus distance using the following parameters :

Polarity: horizontal

Frequency: 300 MHz

Antenna heights: both 3 meters

Relative ground permittivity: 15

*5.2. Field strength versus range at 300 MHz*

Also shown is a plot of free space field strength versus distance (dotted line). In both plots, signal strength is referenced to the free space field strength at a range of 3 meters. Notice in Figure 5.2 that, up to a range of around 50 meters, there are several sharp depressions of field strength, but the signal strength is mostly higher than it would be in free space. Beyond 100 meters, signal strength decreases more rapidly than for the free space model. Whereas there is an inverse distance law for free space, in the open field beyond 100 meters (for these parameters) the signal strength follows an inverse square law. Increasing the antenna heights extends the distance at which the inverse square law starts to take effect. This distance, dm, can be approximated by

where h_{1} and h_{2} are the transmitting and receiving antenna heights above ground and λ is the wavelength, all in the same units as the distance d_{m}. In plotting Figure 5.2, we assumed horizontal polarization. Both antenna heights, h_{1} and h_{2}, are 3 meters. When vertical polarization is used, the extreme local variations of signal strengths up to around 50 meters are reduced because the ground reflection coefficient is less at larger reflection angles. However, for both polarizations, the inverse square law comes into effect at approximately the same distance. This distance in Figure 5.2 where λ is 1 meter is, from equation (5.3): d_{m} _ (12 _ 3 _ 3)/ λ _ 108 meters. In Figure 5.2 we see that this is approximately the distance where the open-field field strength falls below the free-space field strength.

Part 2 will cover Multipath Phenomena

Part 3 will cover Diversity Techniques

**References**

Gibson, J. D. (ed.), The Mobile Communications Handbook, CRC Press, Inc., 1996.

Rappaport, T. S., Wireless Communications, Principles and Practice, Prentice Hall, Upper Saddle River, NJ, 1996.

Spix, G. J., "Maxwell’s Electromagnetic Field Equations," unpublished tutorial, copyright 1995 (http://www.connectos.com/spix/rd/gj/nme/maxwell.htm).

## Speak Your Mind