Chapter 2 Fundamentals of VHF and UHF Propagation 2.1 INTRODUCTION Having established the suitability of the VHF and UHF bands for mobile communications and the need to characterise the radio channel, we can now develop some fundamental relationships between the transmitted and received power, distance (range) and carrier frequency. We begin with a few relevant de®nitions. At frequencies below 1 GHz, antennas normally consist of a wire or wires of a suitable length coupled to the transmitter via a transmission line. At these frequencies it is relatively easy to design an assembly of wire radiators which form an array, in order to beam the radiation in a particular direction. For distances large in comparison with the wavelength and the dimensions of the array, the ®eld strength in free space decreases with an increase in distance, and a plot of the ®eld strength as a function of spatial angle is known as the radiation pattern of the antenna. Antennas can be designed to have radiation patterns which are not omnidirec- tional, and it is convenient to have a ®gure of merit to quantify the ability of the antenna to concentrate the radiated energy in a particular direction. The directivity D of an antenna is de®ned as D  power density at a distance d in the direction of maximum radiation mean power density at a distance d This is a measure of the extent to which the power density in the direction of maximum radiation exceeds the average power density at the same distance. The directivity involves knowing the power actually transmitted by the antenna and this diers from the power supplied at the terminals by the losses in the antenna itself. From the system designer's viewpoint, it is more convenient to work in terms of terminal power, and a power gain G can be de®ned as G  power density at a distance d in the direction of maximum radiation P T =4pd 2 where P T is the power supplied to the antenna. The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright & 2000 John Wiley & Sons Ltd Print ISBN 0-471-98857-X Online ISBN 0-470-84152-4 So, given P T and G it is possible to calculate the power density at any point in the far ®eld that lies in the direction of maximum radiation. A knowledge of the radiation pattern is necessary to determine the power density at other points. The power gain is unity for an isotropic antenna, i.e. one which radiates uniformly in all directions, and an alternative de®nition of power gain is therefore the ratio of power density, from the speci®ed antenna, at a given distance in the direction of maximum radiation, to the power density at the same point, from an isotropic antenna which radiates the same power. As an example, the power gain of a half- wave dipole is 1.64 (2.15 dB) in a direction normal to the dipole and is the same whether the antenna is used for transmission or reception. There is a concept known as eective area which is useful when dealing with antennas in the receiving mode. If an antenna is irradiated by an electromagnetic wave, the received power available at its terminals is the power per unit area carried by the wave6the eective area, i.e. P WA. It can be shown [1, Ch. 11] that the eective area of an antenna and its power gain are related by A  l 2 G 4p 2:1 2.2 PROPAGATION IN FREE SPACE Radio propagation is a subject where deterministic analysis can only be applied in a few rather simple cases. The extent to which these cases represent practical conditions is a matter for individual interpretation, but they do give an insight into the basic propagation mechanisms and establish bounds. If a transmitting antenna is located in free space, i.e. remote from the Earth or any obstructions, then if it has a gain G T in the direction to a receiving antenna, the power density (i.e. power per unit area) at a distance (range) d in the chosen direction is W  P T G T 4pd 2 2:2 The available power at the receiving antenna, which has an eective area A is therefore P R  P T G T 4pd 2 A  P T G T 4pd 2 l 2 G R 4p  where G R is the gain of the receiving antenna. Thus, we obtain P R P T  G T G R l 4pd  2 2:3 16 The Mobile Radio Propagation Channel which is a fundamental relationship known as the free space or Friis equation [2]. The well-known relationship between wavelength l, frequency f and velocity of propagation c (c f l) can be used to write this equation in the alternative form P R P T  G T G R c 4pfd  2 2:4 The propagation loss (or path loss) is conveniently expressed as a positive quantity and from eqn. (2.4) we can write L F dB10 log 10 P T =P R  À10 log 10 G T À 10 log 10 G R  20 log 10 f  20 log 10 d  k 2:5 where k  20 log 10 4p 3  10 8  À147:56 It is often useful to compare path loss with the basic path loss L B between isotropic antennas, which is L B dB32:44  20 log 10 f MHz  20 log 10 d km 2:6 If the receiving antenna is connected to a matched receiver, then the available signal power at the receiver input is P R . It is well known that the available noise power is kTB, so the input signal-to-noise ratio is SNR i  P R kTB  P T G T G R kTB c 4p fd  2 If the noise ®gure of the matched receiver is F, then the output signal-to-noise ratio is given by SNR o  SNR i =F or, more usefully, SNR o  dB SNR i  dB À F dB Equation (2.4) shows that free space propagation obeys an inverse square law with range d, so the received power falls by 6 dB when the range is doubled (or reduces by 20 dB per decade). Similarly, the path loss increases with the square of the transmission frequency, so losses also increase by 6 dB if the frequency is doubled. High-gain antennas can be used to make up for this loss, and fortunately they are relatively easily designed at frequencies in and above the VHF band. This provides a solution for ®xed (point-to-point) links, but not for VHF and UHF mobile links where omnidirectional coverage is required. Sometimes it is convenient to write an expression for the electric ®eld strength at a known distance from a transmitting antenna rather than the power density. This can be done by noting that the relationship between ®eld strength and power density is Fundamentals of VHF and UHF Propagation 17 W  E 2 Z where Z is the characteristic wave impedance of free space. Its value is 120p ($377 O) and so eqn. (2.2) can be written E 2 120p  P T G T 4pd 2 giving E   30P T G T p d 2:7 Finally, we note that the maximum useful power that can be delivered to the terminals of a matched receiver is P  E 2 A Z  E 2 120p  l 2 G R 4p  El 2p  2 G R 120 2:8 2.3 PROPAGATION OVER A REFLECTING SURFACE The free space propagation equation applies only under very restricted conditions; in practical situations there are almost always obstructions in or near the propagation path or surfaces from which the radio waves can be re¯ected. A very simple case, but one of practical interest, is the propagation between two elevated antennas within line-of-sight of each other, above the surface of the Earth. We will consider two cases, ®rstly propagation over a spherical re¯ecting surface and secondly when the distance between the antennas is small enough for us to neglect curvature and assume the re¯ecting surface to be ¯at. In these cases, illustrated in Figures 2.1 and 2.4 the received signal is a combination of direct and ground-re¯ected waves. To determine the resultant, we need to know the re¯ection coecient. 2.3.1 The re¯ection coecient of the Earth The amplitude and phase of the ground-re¯ected wave depends on the re¯ection coecient of the Earth at the point of re¯ection and diers for horizontal and vertical polarisation. In practice the Earth is neither a perfect conductor nor a perfect dielectric, so the re¯ection coecient depends on the ground constants, in particular the dielectric constant e and the conductivity s. For a horizontally polarised wave incident on the surface of the Earth (assumed to be perfectly smooth), the re¯ection coecient is given by [1, Ch. 16]: r h  sin c À  e=e 0 À js=oe 0 Àcos 2 c p sin c   e=e 0 À js=oe 0 Àcos 2 c p where o is the angular frequency of the transmission and e 0 is the dielectric constant of free space. Writing e r as the relative dielectric constant of the Earth yields 18 The Mobile Radio Propagation Channel r h  sin c À  e r À jxÀcos 2 c p sin c   e r À jxÀcos 2 c p 2:9 where x  s oe 0  18  10 9 s f For vertical polarisation the corresponding expression is r v  e r À j xsin c À  e r À jxÀcos 2 c p e r À jxsin c   e r À jxÀcos 2 c p 2:10 The re¯ection coecients r h and r v are complex, so the re¯ected wave will dier from the incident wave in both magnitude and phase. Examination of eqns (2.9) and (2.10) reveals some quite interesting dierences. For horizontal polarisation the relative phase of the incident and re¯ected waves is nearly 1808 for all angles of incidence. For very small values of c (near-grazing incidence), eqn. (2.9) shows that the re¯ected wave is equal in magnitude and 1808 out of phase with the incident wave for all frequencies and all ground conductivities. In other words, for grazing incidence r h jr h je jy  1e j p À1 2:11 As the angle of incidence is increased then jr h j and y change, but only by relatively small amounts. The change is greatest at higher frequencies and when the ground conductivity is poor. Fundamentals of VHF and UHF Propagation 19 Figure 2.1 Two mutually visible antennas located above a smooth, spherical Earth of eective radius r e . For vertical polarisation the results are quite dierent. At grazing incidence there is no dierence between horizontal and vertical polarisation and eqn. (2.11) still applies. As c is increased, however, substantial dierences appear. The magnitude and relative phase of the re¯ected wave decrease rapidly as c increases, and at an angle known as the pseudo-Brewster angle the magnitude becomes a minimum and the phase reaches 7908. At values of c greater than the Brewster angle, jr v j increases again and the phase tends towards zero. The very sharp changes that occur in these circumstances are illustrated by Figure 2.2, which shows the values of jr v j and y as functions of the angle of incidence c. The pseudo-Brewster angle is about 158 at frequencies of interest for mobile communications (x ( e r ), although at lower frequencies and higher conductivities it becomes smaller, approaching zero if x ) e r . Table 2.1 shows typical values for the ground constants that aect the value of r. The conductivity of ¯at, good ground is much higher than the conductivity of poorer ground found in mountainous areas, whereas the dielectric constant, typically 15, can be as low as 4 or as high as 30. Over lakes or seas the re¯ection properties are quite dierent because of the high values of s and e r . Equation (2.11) applies for horizontal polarisation, particularly over sea water, but r may be signi®cantly dierent from 71 for vertical polarisation. 20 The Mobile Radio Propagation Channel Figure 2.2 Magnitude and phase of the plane wave re¯ection coecient for vertical polarisation. Curves drawn for s  12 Â10 À3 , e r  15. Approximate results for other frequencies and conductivities can be obtained by calculating the value of x as 18  10 3 s=f MHz . 2.3.2 Propagation over a curved re¯ecting surface The situation of two mutually visible antennas sited on a smooth Earth of eective radius r e is shown in Figure 2.1. The heights of the antennas above the Earth's surface are h T and h R , and above the tangent plane through the point of re¯ection the heights are h H T and h H R . Simple geometry gives d 2 1 r e h T À h H T  2 À r 2 e h T À h H T  2  2r e h T À h H T 92r e h T À h H T  2:12 and similarly d 2 2 9 2r e h R À h H R 2:13 Using eqns. (2.12) and (2.13) we obtain h H T  h T À d 2 1 2r e and h H R  h R À d 2 2 2r e 2:14 The re¯ecting point, where the two angles marked c are equal, can be determined by noting that, providing d 1 , d 2 44h T , h R , the angle c (radians) is given by c  h H T d 1  h H R d 2 Hence h H T h H R 9 d 1 d 2 2:15 Using the obvious relationship d d 1 +d 2 together with equations (2.14) and (2.15) allows us to formulate a cubic equation in d 1 : 2d 3 1 À 3dd 2 1 d 2 À 2r e h T  h R d 1  2r e h T d  0 2:16 The appropriate root of this equation can be found by standard methods starting from the rough approximation d 1 9 d 1  h T =h R To calculate the ®eld strength at a receiving point, it is normally assumed that the dierence in path length between the direct wave and the ground-re¯ected wave is negligible in so far as it aects attenuation, but it cannot be neglected with regard to the phase dierence along the two paths. The length of the direct path is Fundamentals of VHF and UHF Propagation 21 Table 2.1 Typical values of ground constants Surface Conductivity s (S) Dielectric constant e r Poor ground (dry) 1Â10 73 4±7 Average ground 5Â10 73 15 Good ground (wet) 2Â10 72 25±30 Sea water 5Â10 0 81 Fresh water 1Â10 72 81 R 1  d 1  h H T À h H R  2 d 2  1=2 and the length of the re¯ected path is R 2  d 1  h H T  h H R  2 d 2  1=2 The dierence DR  R 2 À R 1 is DR  d 1  h H T  h H R  2 d 2  1=2 À 1  h H T À h H R  2 d 2  1=2 () and if d ) h H T , h H R this reduces to DR  2h H T h H R d 2:17 The corresponding phase dierence is Df  2p l DR  4ph H T h H R ld 2:18 If the ®eld strength at the receiving antenna due to the direct wave is E d , then the total received ®eld E is E  E d 1  r expÀj Df where r is the re¯ection coecient of the Earth and r jrjexp jy. Thus, E  E d f1 jrjexpÀjDf À y g 2:19 This equation can be used to calculate the received ®eld strength at any location, but note that the curvature of the spherical Earth produces a certain amount of divergence of the ground-re¯ected wave as Figure 2.3 shows. This eect can be taken into account by using, in eqn. (2.19), a value of r which is dierent from that derived in Section 2.3.1 for re¯ection from a plane surface. The appropriate modi®cation consists of multiplying the value of r for a plane surface by a divergence factor D given by [3]: D 9 1  2d 1 d 2 r e h H T  h H R   À1=2 2:20 The value of D can be of the order of 0.5, so the eect of the ground-re¯ected wave is considerably reduced. 2.3.3 Propagation over a plane re¯ecting surface For distances less than a few tens of kilometres, it is often permissible to neglect Earth curvature and assume the surface to be smooth and ¯at as shown in Figure 2.4. If we also assume grazing incidence so that r À1, then eqn. (2.19) becomes 22 The Mobile Radio Propagation Channel E  E d 1 À exp ÀjDf  E d 1 À cos D f j sin D f Thus, jEjjE d j1  cos 2 D f À2cosDf  sin 2 Df 1=2  2jE d jsin Df 2 and using eqn. (2.18), with h H T  h T and h H R  h R , jEj2jE d jsin 2ph T h R ld  The received power P R is proportional to E 2 so P R G 4jE d j 2 sin 2 2ph T h R ld   4P T l 4pd  2 G T G R sin 2 2ph T h R ld  2:21 If d44h T , h R this becomes P R P T  G T G R h T h R d 2  2 2:22 Fundamentals of VHF and UHF Propagation 23 Figure 2.3 Divergence of re¯ected rays from a spherical surface. Figure 2.4 Propagation over a plane earth. Equation (2.22) is known as the plane earth propagation equation. It diers from the free space relationship (2.3) in two important ways. First, since we assumed that d44h T , h R , the angle Df is small and l cancels out of eqn. (2.22), leaving it frequency independent. Secondly, it shows an inverse fourth-power law with range rather than the inverse square law of eqn. (2.3). This means a far more rapid decrease in received power with range, 12 dB for each doubling of distance in this case. Note that eqn. (2.22) only applies at ranges where the assumption d44h T , h R is valid. Close to the transmitter, eqn. (2.21) must be used and this gives alternate maxima and minima in the signal strength as shown in Figure 2.5. In convenient logarithmic form, eqn. (2.22) can be written L P dB10 log 10 P T =P R  À10 log 10 G T À 10 log 10 G R À 20 log 10 h T À 20 log 10 h R  40 log 10 d 2:23 and by comparison with eqn (2.6) we can write a `basic loss' L B between isotropic antennas as L B dB40 log 10 d À 20 log 10 h T À 20 log 10 h R 2:24 2.4 GROUND ROUGHNESS The previous section presupposed a smooth re¯ecting surface and the analysis was therefore based on the assumption that a specular re¯ection takes place at the point where the transmitted wave is incident on the Earth's surface. When the surface is 24 The Mobile Radio Propagation Channel Figure 2.5 Variation of signal strength with distance in the presence of specular re¯ection. [...]... complexity by increasing the true value of the Earth's radius until ray paths, modi®ed by the refractive index gradient, become straight again The modi®ed radius can be found from the relationship where dn/dh is the rate of change of refractive index with height The ratio p/r is thep re e€ective Earth radius factor k, so the distance to the radio horizon is 2krh …ˆ 2re h † The average value... Variations in the climatic conditions within the troposphere, i.e changes of temperature, pressure and humidity, cause changes in the refractive index of the air Large-scale changes of refractive index with height cause radio waves to be refracted, and at low elevation angles the e€ect can be quite signi®cant at all frequencies, especially in extending the radio horizon distance beyond the optical horizon... on radio signals, refraction is the one that has the greatest e€ect on VHF and UHF point-to-point systems; it is therefore worthy of further discussion We start by considering an idealised model of the atmosphere and then discuss the e€ects of departures from that ideal An ideal atmosphere is one in which the dielectric constant is unity and there is zero absorption In practice, however, the dielectric... interference between di erent radio communication systems Forward scattering of radio energy is suciently dependable that it may be used as a mechanism for long-distance communications, especially at frequencies between about 300 MHz and 10 GHz Nevertheless, this troposcatter is not used for mobile radio communications and we will not consider it any further Re¯ection and ducting are much less predictable Variations... and use of this four-thirds Earth radius is very widespread in the calculation of radio paths It leads to a very simple relationship for the horizon p distance: d ˆ 2h where d is in miles and h is in feet In practice the atmosphere does not always behave according to this idealised model, hence the radio wave propagation paths are perturbed 2.5.1 Atmospheric ducting and non-standard refraction... must propagate in a nearly horizontal direction, so to satisfy 30 Figure 2.8 The Mobile Radio Propagation Channel The phenomenon of ducting conditions for guiding within the duct the wavelength has to be relatively small The maximum wavelength that can be trapped in a duct of 100 m thickness is about 1 m, (i.e a frequency of about 300 MHz), so the most favourable conditions for ducting are in the VHF... ducting can be expressed in terms of a modi®ed index of refraction that is the di erence between the actual refractive index and the value of 7157 N-units per kilometre that causes rays to remain at a constant height above the curved surface of the Earth [4, Ch 6] Under non-standard conditions the refractive index may change either more rapidly or less rapidly than 7157 N-units per kilometre When the decrease... trapping or ducting The radio waves will then propagate over quite long distances with much less attenuation than for free space propagation; the guiding action is in some ways similar to the Earth±ionosphere waveguide at lower frequencies Ducts can form near the surface of the Earth (surface ducts) or at heights up to about 1500 m above the surface (elevated ducts) To obtain long-distance propagation,... Relative to the wavefront AAH shown, the di erence in path length of the two rays when they reach the points C and C H after re¯ection is Dl ˆ …AB ‡ BC† À …AH BH ‡ BH CH † d …1 À cos 2c† ˆ sin c ˆ 2d sin c Figure 2.6 model …2:25† Re¯ections from a semi-rough surface: (a) practical terrain situation, (b) idealised 26 The Mobile Radio Propagation Channel The phase di erence between C and CH is therefore... surface: 28 The Mobile Radio Propagation Channel Figure 2.7 An e€ective Earth radius of 8490 km (6730Â4/3) permits the use of straight-line propagation paths N…h† ˆ Ns exp…Àh=H† …2:31† where H is a scale height (often taken as 7 km) Over the ®rst kilometre or so, the exponential curve can be approximated by a straight line and in this region the refractivity falls by about 39 N-units Although this may . Ltd Print ISBN 0-4 7 1-9 8857-X Online ISBN 0-4 7 0-8 415 2-4 So, given P T and G it is possible to calculate the power density at any point in the far ®eld that lies in the direction of maximum radiation  power density at a distance d in the direction of maximum radiation P T =4pd 2 where P T is the power supplied to the antenna. The Mobile Radio Propagation Channel. Second Edition. J. D. Parsons Copyright. density at a distance d in the direction of maximum radiation mean power density at a distance d This is a measure of the extent to which the power density in the direction of maximum radiation exceeds