Sadik, M.N.O., Demarest, K " Wave Propagation The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton CRC Press llc. 2000
Sadiku, M.N.O., Demarest, K. “Wave Propagation” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
37 Wave Propagation 37.1 Space Propagation Matthew n.o. Sadiku Propagation in Simple Media.Propagation in the Atmosphere Waveguide Modes. Rectangular Waveguides. Circula Kenneth Demarest Waveguides. Commercially Available Waveguides University of Ka osses. Mode launching 37.1 Space Propagation Matthew N.o. Sadiku This section summarizes the basic principles of electromagnetic(EM)wave propagation in space. The principles essentially state how the characteristics of the earth and the atmosphere affect the propagation of EM waves. Understanding such principles is of practical interest to communication system engineers. Engineers cannot competently apply formulas or models for communication system design without an adequate knowledge of Propagation of an EM wave may be regarded as a means of transferring energy or information from one point(a transmitter)to another (a receiver). EM wave propagation is achieved through guided structures such as transmission lines and waveguides or through space. Wave propagation through waveguides and microstrip lines will be treated in Section 37. 2. In this section, our major focus is on EM wave propagation in space and the power resident in the wave For a clear understanding of the phenomenon of EM wave propagation, it is expedient to break the discussion of propagation effects into categories represented by four broad frequency intervals [Collin, 1985 Very low frequencies(VLF), 3-30 kHz LF)band, 30-300 kH High-frequency(HF)band, 3-30 MHz Above 50 MHz In the first range, wave propagates as in a waveguide, using the earths surface and the ionosphere as boundaries Attenuation is comparatively low, and hence vlF propagation is useful for long-distance worldwide telegraphy and submarine communication. In the second frequency range, the availability of increased bandwidth makes standard AM broadcasting possible. Propagation in this band is by means of surface wave due to the presence of the ground. The third range is useful for long-range broadcasting services wave reflection and refraction by the ionosphere. Basic problems in this band include fluctuations in the ionosphere and a limited usable frequency range. Frequencies above 50 MHz allow for line-of-sight space wave propagation, FM radio and TV channels, radar and navigation systems, and so on. In this band, due consideration must be given to reflection from the ground, refraction by the troposphere, scattering by atmospheric hydrometeors, and mul- tipath effects of buildings, hills, trees, etc c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 37 Wave Propagation 37.1 Space Propagation Propagation in Simple Media • Propagation in the Atmosphere 37.2 Waveguides Waveguide Modes • Rectangular Waveguides • Circular Waveguides • Commercially Available Waveguides • Waveguide Losses • Mode Launching 37.1 Space Propagation Matthew N. O. Sadiku This section summarizes the basic principles of electromagnetic (EM) wave propagation in space. The principles essentially state how the characteristics of the earth and the atmosphere affect the propagation of EM waves. Understanding such principles is of practical interest to communication system engineers. Engineers cannot competently apply formulas or models for communication system design without an adequate knowledge of the propagation issue. Propagation of an EM wave may be regarded as a means of transferring energy or information from one point (a transmitter) to another (a receiver). EM wave propagation is achieved through guided structures such as transmission lines and waveguides or through space. Wave propagation through waveguides and microstrip lines will be treated in Section 37.2. In this section, our major focus is on EM wave propagation in space and the power resident in the wave. For a clear understanding of the phenomenon of EM wave propagation, it is expedient to break the discussion of propagation effects into categories represented by four broad frequency intervals [Collin, 1985]: • Very low frequencies (VLF), 3–30 kHz • Low-frequency (LF) band, 30–300 kHz • High-frequency (HF) band, 3–30 MHz • Above 50 MHz In the first range, wave propagates as in a waveguide, using the earth’s surface and the ionosphere as boundaries. Attenuation is comparatively low, and hence VLF propagation is useful for long-distance worldwide telegraphy and submarine communication. In the second frequency range, the availability of increased bandwidth makes standard AM broadcasting possible. Propagation in this band is by means of surface wave due to the presence of the ground. The third range is useful for long-range broadcasting services via sky wave reflection and refraction by the ionosphere. Basic problems in this band include fluctuations in the ionosphere and a limited usable frequency range. Frequencies above 50 MHz allow for line-of-sight space wave propagation, FM radio and TV channels, radar and navigation systems, and so on. In this band, due consideration must be given to reflection from the ground, refraction by the troposphere, scattering by atmospheric hydrometeors, and multipath effects of buildings, hills, trees, etc. Matthew N.O. Sadiku Temple University Kenneth Demarest University of Kansas
M wave propagation can be described by two complementary models. The physicist attempts a theoretical model based on universal laws, which extends the field of application more widely than currently known. The ngineer prefers an empirical model based on measurements, which can be used immediately. This section presents complementary standpoints by discussing theoretical factors affecting wave propagation and the semiempirical rules allowing handy engineering calculations. First, we consider wave propagation in idealistic mple media, with no obstacles. We later consider the more realistic case of wave propagation around the earth, as influenced by its curvature and by atmospheric conditions Propagation in Simple Media The conventional propagation models, on which the basic calculation of radio links is based, result directly from Maxwells equations v·D=p (37.1) V×E dB at V×H D at In these equations, E is electric field strength in volts per meter, H is magnetic field strength in amperes per meter, D is electric flux density in coulombs per square meter, B is magnetic flux density in webers per square meter, J is conduction current density in amperes per square meter, and p, is electric charge density in coulombs per cubic meter. These equations go hand in hand with the constitutive equations for the medium E (37.5) LH J=OE (377) where e=EE,u=pu, and o are the permittivity, the permeability, and the conductivity of the medium, respectively Consider the general case of a lossy medium which is charge-free(p,=0). Assuming time-harmonic fields and suppressing the time factor e of, Eqs.(37. 1)to(37.7)can be manipulated to yield Helmholtz's wave V2E-Y2E=0 (378) V2H-YH=0 (379) where y=a+ p is the propagation constant, a is the attenuation constant in nepers per meter or decibels per meter, and B is the phase constant in radians per meter. Constants a and B are given by (37.10) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC EM wave propagation can be described by two complementary models. The physicist attempts a theoretical model based on universal laws, which extends the field of application more widely than currently known. The engineer prefers an empirical model based on measurements, which can be used immediately. This section presents complementary standpoints by discussing theoretical factors affecting wave propagation and the semiempirical rules allowing handy engineering calculations. First, we consider wave propagation in idealistic simple media, with no obstacles. We later consider the more realistic case of wave propagation around the earth, as influenced by its curvature and by atmospheric conditions. Propagation in Simple Media The conventional propagation models, on which the basic calculation of radio links is based, result directly from Maxwell’s equations: — × D = rv (37.1) — × B = 0 (37.2) (37.3) (37.4) In these equations, E is electric field strength in volts per meter, H is magnetic field strength in amperes per meter, D is electric flux density in coulombs per square meter, B is magnetic flux density in webers per square meter, J is conduction current density in amperes per square meter, and rv is electric charge density in coulombs per cubic meter. These equations go hand in hand with the constitutive equations for the medium: D = eE (37.5) B = mH (37.6) J = sE (37.7) where e = eoer, m = momr, and s are the permittivity, the permeability, and the conductivity of the medium, respectively. Consider the general case of a lossy medium which is charge-free (rv = 0). Assuming time-harmonic fields and suppressing the time factor ejwt , Eqs. (37.1) to (37.7) can be manipulated to yield Helmholtz’s wave equations —2 E – g 2E = 0 (37.8) —2 H– g 2 H = 0 (37.9) where g = a + jb is the propagation constant, a is the attenuation constant in nepers per meter or decibels per meter, and b is the phase constant in radians per meter. Constants a and b are given by (37.10) — ¥ E = - ¶B ¶t — ¥ H = + D J ¶ ¶t a w m s w = + Ê Ë Á ˆ ¯ ˜ - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e 2 e 1 1 2
+1 where a= 2T is the frequency of the wave. The wavelength A and wave velocity u are given in terms of p as 0=f (37.13) without loss of generality, if we assume that wave propagates in the z-direction and the wave is polarized in the x-direction, solving the wave equations(37. 8)and(37.9)results in E(zt)=E H(z, t) /n/ cos(ot-Bz-0,)a (37.15) where n=Inle, is the intrinsic impedance of the medium and is given by Equations(37. 14)and(37.15)show that as the EM wave travels in the medium, its amplitude is attenuated according to e-a, as illustrated in Fig. 37. 1. The distance 8 through which the wave amplitude is reduced by a factor of e-(about 37%)is called the skin depth or penetration depth of the medium, (37.17) FIGURE 37.1 The magnetic and electric field components of a plane wave in a lossy medium. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC (37.11) where w = 2pf is the frequency of the wave. The wavelength l and wave velocity u are given in terms of b as (37.12) (37.13) Without loss of generality, if we assume that wave propagates in the z-direction and the wave is polarized in the x-direction, solving the wave equations (37.8) and (37.9) results in E(z,t) = Eoe – az cos(wt – bz)ax (37.14) (37.15) where h = ˜h˜ –qh is the intrinsic impedance of the medium and is given by (37.16) Equations (37.14) and (37.15) show that as the EM wave travels in the medium, its amplitude is attenuated according to e –az , as illustrated in Fig. 37.1. The distance d through which the wave amplitude is reduced by a factor of e –1 (about 37%) is called the skin depth or penetration depth of the medium, i.e., (37.17) FIGURE 37.1 The magnetic and electric field components of a plane wave in a lossy medium. b w m s w = + Ê Ë Á ˆ ¯ ˜ + È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e 2 e 1 1 2 l p b = 2 u = = f w b l H(z t, ) cos ( )a E e t z o z = - - y - *h* w b q a h *h* m s w q s w q h h = + Ê Ë Á ˆ ¯ ˜ È Î Í ˘ ˚ ˙ = £ £ ° / tan 2 , 0 45 e e e 4 1 1 4 , d a = 1
The power density of the EM wave is obtained from the Poynting vector P=E×H (37.18) with the time-average value of Re(E×H) -201Z cos B. a n It should be noted from Eqs.(37. 14)and(37.15)that E and H are everywhere perpendicular to each other and also to the direction of wave propagation. Thus, the wave described by Eqs.(37. 14)and(37. 15)is said to be plane-polarized, implying that the electric field is always parallel to the same plane(the xz-plane in this case) and is perpendicular to the direction of propagation. Also, as mentioned earlier, the wave decays as it travels in the z-direction because of loss. This loss is expressed in the complex relative permittivity of the medium d measured by the loss tangent, defined by 16 (37.21) The imaginary part E,=o/oE, corresponds to the losses in the medium. The refractive index of the medium n is given by Having considered the general case of wave propagation through a lossy medium, we now consider wave propagation in other types of media. A medium is said to be a good conductor if the loss tangent is large(o>> oe) or a lossless or good dielectric if the loss tangent is very small (o >oe,∈=∈μ=H2 2. Good dielectric:a<<0∈,∈=∈∈=μ 3. Free space:o=0,∈=∈oμ=μ where,=8.854X10-12F/m is the free-space permittivity, and H =4T X 10-7H/m is the free-space permeability. The conditions for each medium type are merely substituted in Eqs.(37. 10)to(37. 21)to obtain the wave properties for that medium. The formulas for calculating attenuation constant, phase constant, and intrinsic impedance for different media are summarized in Table 37.1 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The power density of the EM wave is obtained from the Poynting vector P = E 2 H (37.18) with the time-average value of (37.19) It should be noted from Eqs. (37.14) and (37.15) that E and H are everywhere perpendicular to each other and also to the direction of wave propagation. Thus, the wave described by Eqs. (37.14) and (37.15) is said to be plane-polarized, implying that the electric field is always parallel to the same plane (the xz-plane in this case) and is perpendicular to the direction of propagation. Also, as mentioned earlier, the wave decays as it travels in the z-direction because of loss. This loss is expressed in the complex relative permittivity of the medium (37.20) and measured by the loss tangent, defined by (37.21) The imaginary part er¢¢ = s/weo corresponds to the losses in the medium. The refractive index of the medium n is given by (37.22) Having considered the general case of wave propagation through a lossy medium, we now consider wave propagation in other types of media. A medium is said to be a good conductor if the loss tangent is large (s >> we) or a lossless or good dielectric if the loss tangent is very small (s > we, e = eo, m = momr 2. Good dielectric: s << we, e = eoer , m = momr 3. Free space: s = 0, e = eo, m = mo where eo = 8.854210–12 F/m is the free-space permittivity, and mo = 4p210– 7 H/m is the free-space permeability. The conditions for each medium type are merely substituted in Eqs. (37.10) to (37.21) to obtain the wave properties for that medium. The formulas for calculating attenuation constant, phase constant, and intrinsic impedance for different media are summarized in Table 37.1. P E e o z z ave = ¥ = * - 1 2 2 2 2 Re( ) cos E H a * h* q a h e e e e e c r r r = ¢ - j ¢¢ = - j Ê Ë Á ˆ ¯ ˜ 1 s w tan d s w = ¢¢ ¢ = e e e r r n = c e
TABLE 37.1 Attenuation Constant, Phase Constant and Int trinsic Impedance fo Different Media Good Good Conductor electric Lossy Medium o/oE >>1 o/oE<<1 Space Attenuation constant a o Phase constant B Oyp。。 Intrinsic impedance n Vo+joe 2(+n The classical model of a wave propagation presented in this subsection helps us understand some basic concepts of EM wave propagation and the various parameters that play a part in determining the motion of a wave from the transmitter to the receiver. We now apply the ideas to the particular case of wave propagation in the atmosphere Propagation in the Atmosphere Wave propagation hardly occurs under the idealized conditions assumed in the previous subsection. For most communication links, the analysis must be modified to account for the presence of the earth, the ionosphere. and atmospheric precipitates such as fog, raindrops, snow, and hail. This will be done in this subsection. The major regions of the earths atmosphere that are of importance in radio wave propagation are the troposphere and the ionosphere. At radar frequencies(approximately 100 MHz to 300 GHz), the troposphe is by far the most important. It is the lower atmosphere consisting of a nonionized region extending from the earths surface up to about 15 km. The ionosphere is the earths upper atmosphere in the altitude region from 50 km to one earth radius(6370 km). Sufficient ionization exists in this region to influence wave propagation Wave propagation over the surface of the earth may assume one of the following three principal modes: Surface wave propagation along the surface of the earth Space wave propagation through the lower atmosphere ation by ref These modes are portrayed in Fig. 37. 2. The sky wave is directed toward the ionosphere, which bends the propagation path back toward the earth under certain conditions in a limited frequency range(0-50 MHz approximately). The surface wave is directed along the surface over which the wave is propagated. The space wave consists of the direct wave and the reflected wave. the direct wave travels from the transmitter to the receiver in nearly a straight path, while the reflected wave is due to ground reflection. The space wave obo the optical laws in that direct and reflected wave components contribute to the total wave. Although the sk and surface waves are important in many applications, we will only consider space waves in this section. Figure 37.3 depicts the electromagnetic energy transmission between two antennas in space. As the wave radiates from the transmitting antenna and propagates in space, its power density decreases, as expressed ideally in Eq.(37. 19). Assuming that nas are in free space, the power received by the receiving antenna given by the Friis transmission equation [Liu and Fang, 1988] P=Gc/a P 4πr e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The classical model of a wave propagation presented in this subsection helps us understand some basic concepts of EM wave propagation and the various parameters that play a part in determining the motion of a wave from the transmitter to the receiver. We now apply the ideas to the particular case of wave propagation in the atmosphere. Propagation in the Atmosphere Wave propagation hardly occurs under the idealized conditions assumed in the previous subsection. For most communication links, the analysis must be modified to account for the presence of the earth, the ionosphere, and atmospheric precipitates such as fog, raindrops, snow, and hail. This will be done in this subsection. The major regions of the earth’s atmosphere that are of importance in radio wave propagation are the troposphere and the ionosphere. At radar frequencies (approximately 100 MHz to 300 GHz), the troposphere is by far the most important. It is the lower atmosphere consisting of a nonionized region extending from the earth’s surface up to about 15 km. The ionosphere is the earth’s upper atmosphere in the altitude region from 50 km to one earth radius (6370 km). Sufficient ionization exists in this region to influence wave propagation. Wave propagation over the surface of the earth may assume one of the following three principal modes: • Surface wave propagation along the surface of the earth • Space wave propagation through the lower atmosphere • Sky wave propagation by reflection from the upper atmosphere These modes are portrayed in Fig. 37.2. The sky wave is directed toward the ionosphere, which bends the propagation path back toward the earth under certain conditions in a limited frequency range (0–50 MHz approximately). The surface wave is directed along the surface over which the wave is propagated. The space wave consists of the direct wave and the reflected wave. The direct wave travels from the transmitter to the receiver in nearly a straight path, while the reflected wave is due to ground reflection. The space wave obeys the optical laws in that direct and reflected wave components contribute to the total wave. Although the sky and surface waves are important in many applications, we will only consider space waves in this section. Figure 37.3 depicts the electromagnetic energy transmission between two antennas in space. As the wave radiates from the transmitting antenna and propagates in space, its power density decreases, as expressed ideally in Eq. (37.19). Assuming that the antennas are in free space, the power received by the receiving antenna is given by the Friis transmission equation [Liu and Fang, 1988]: (37.23) TABLE 37.1 Attenuation Constant, Phase Constant, and Intrinsic Impedance for Different Media Good Good Conductor Dielectric Free Lossy Medium s/we >> 1 s/we << 1 Space Attenuation constant a . 0 0 Phase constant b Intrinsic impedance h 377 w m s w e 2 e 1 1 2 + Ê Ë Á ˆ ¯ ˜ - È Î Í Í ˘ ˚ ˙ ˙ wms 2 w m s w e 2 e 1 1 2 + Ê Ë Á ˆ ¯ ˜ + È Î Í Í ˘ ˚ ˙ ˙ wms 2 w me w mo o e j j wm s + we wm 2s (1+ j) m e P G G r r = r t Pt Ê Ë Á ˆ ¯ ˜ l 4p 2
ei nosp direct wave reflected wave surface wave FIGURE 37.2 Modes of wave propagation. FIGURE 37. 3 Transmitting and receiving antennas in free space. where the subscripts t and r respectively, refer to transmitting and receiving antennas In Eq (37. 23), P is the power in watts, G is the antenna gain(dimensionless), r is the distance between the antennas in meters, and n is the wavelength in meters. The Friis equation relates the power received by one antenna to the power transmitted by the other provided that the two antennas are separated by r>2d2/m, where d is the largest dimension of either antenna. Thus, the Friis equation applies only when the two antennas are in the far-field of each other. In case the propagation path is not in free space, a correction factor F is included to account for the effect of the medium. This factor, known as the propagation factor, is simply the ratio of the electric field intensity Em in the medium to the electric field intensity E, in free space, i.e The magnitude of Fis always less than unity since Em is always less than Eo. Thus, for a lossy medium, Eq. (37 23 P=GG P (37.25) For practical reasons, Eqs. (37. 23)and(37. 25) are commonly expressed in the logarithmic form. If all terms are expressed in decibels(dB), Eq.(37. 25) can be written in the logarithmic form as e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where the subscripts t and r, respectively, refer to transmitting and receiving antennas. In Eq. (37.23), P is the power in watts, G is the antenna gain (dimensionless), r is the distance between the antennas in meters, and l is the wavelength in meters. The Friis equation relates the power received by one antenna to the power transmitted by the other provided that the two antennas are separated by r > 2d2 /l, where d is the largest dimension of either antenna. Thus, the Friis equation applies only when the two antennas are in the far-field of each other. In case the propagation path is not in free space, a correction factor F is included to account for the effect of the medium. This factor, known as the propagation factor, is simply the ratio of the electric field intensity Em in the medium to the electric field intensity Eo in free space, i.e., (37.24) The magnitude of F is always less than unity since Em is always less than Eo . Thus, for a lossy medium, Eq. (37.23) becomes (37.25) For practical reasons, Eqs. (37.23) and (37.25) are commonly expressed in the logarithmic form. If all terms are expressed in decibels (dB), Eq. (37.25) can be written in the logarithmic form as FIGURE 37.2 Modes of wave propagation. FIGURE 37.3 Transmitting and receiving antennas in free space. F E E m o = P G G r r = r t P F t Ê Ë Á ˆ ¯ ˜ l 4p 2 2 * *
Pt+G+ g,-lo-lm (3726) where P is power in decibels referred to 1 w(or simply dBW), G is gain in decibels, L, is free-space loss in decibels, and L is loss in decibels due to the medium The free-space loss is obtained from standard monograph or directly from while the loss due to the medium is given by Our major concern in the rest of the section is to determine Lo and Lm for two important cases of space propagation that differ considerably from the free-space conditions Effect of the earth The phenomenon of multipath propagation causes significant departures from free-space conditions. The term multipath denotes the possibility of EM wave propagation along various paths from the transmitter to the receiver In multipath propagation of an EM wave over the earths surface, two such paths exist: a direct path and a path via reflection and diffractions from the interface between the atmosphere and the earth. a simplified geometry of the multipath situation is shown in Fig. 37. 4. The reflected and diffracted component is commonly separated into two parts, one specular(or coherent)and the other diffuse(or incoherent), that can be separately analyzed. The specular component is well defined in terms of its amplitude, phase, and incident direction. It main characteristic is its conformance to Snells law for reflection, which requires that the angles of incidence and reflection be equal and coplanar. It is a plane wave and, as such, is uniquely specified by its direction. The diffuse component, however, arises out of the random nature of the scattering surface and, as such, is nonde terministic. It is not a plane wave and does not obey Snell's law for reflection. It does not come from a given direction but from a con Transmitter Curved earth FIGURE 37.4 Multipath e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Pr = Pt + Gr + Gt – Lo – Lm (37.26) where P is power in decibels referred to 1 W (or simply dBW), G is gain in decibels, Lo is free-space loss in decibels, and Lm is loss in decibels due to the medium. The free-space loss is obtained from standard monograph or directly from (37.27) while the loss due to the medium is given by Lm = –20 log *F* (37.28) Our major concern in the rest of the section is to determine Lo and Lm for two important cases of space propagation that differ considerably from the free-space conditions. Effect of the Earth The phenomenon of multipath propagation causes significant departures from free-space conditions. The term multipath denotes the possibility of EM wave propagation along various paths from the transmitter to the receiver. In multipath propagation of an EM wave over the earth’s surface, two such paths exist: a direct path and a path via reflection and diffractions from the interface between the atmosphere and the earth. A simplified geometry of the multipath situation is shown in Fig. 37.4. The reflected and diffracted component is commonly separated into two parts, one specular (or coherent) and the other diffuse (or incoherent), that can be separately analyzed. The specular component is well defined in terms of its amplitude, phase, and incident direction. Its main characteristic is its conformance to Snell’s law for reflection, which requires that the angles of incidence and reflection be equal and coplanar. It is a plane wave and, as such, is uniquely specified by its direction. The diffuse component, however, arises out of the random nature of the scattering surface and, as such, is nondeterministic. It is not a plane wave and does not obey Snell’s law for reflection. It does not come from a given direction but from a continuum. FIGURE 37.4 Multipath geometry. L r o = Ê Ë Á ˆ ¯ ˜ 20 4 log p l
The loss factor F that accounts for the departures from free-space conditions is given by F=l+TpDs(e where r is the Fresnel reflection coefficient, P, is the roughness coefficient, D is the divergence factor, S(e)is the shadowing function, and A is the phase angle corresponding to the path difference. We now account for each of these terms The Fresnel reflection coefficient I accounts for the electrical properties of the earths surface. Because the earth is a lossy medium, the value of the reflection coefficient depends on the complex relative permittivity ec of the surface, the grazing angle v, and the wave polarization. It is given by (37.30) where z=VE.-cos'y for horizontal polarization (37.31) for vertical polarization (37.32) (37.33) e, and o are the dielectric constant and conductivity of the surface; o and n are the frequency and wavelength of the incident wave; and v is the grazing angle. It is apparent that 0< Irk<1 To account for the spreading (or divergence)of the reflected rays because of the earths curvature,we introduce the divergence factor D. The curvature has a tendency to spread out the reflected energy more than a corresponding flat surface. The divergence factor is defined as the ratio of the reflected field from curved surface to the reflected field from flat surface[Kerr, 1951]. Using the geometry of Fig 37.5, D is given by G,G, where G= G+ G is the total ground range and a=6370 km is the effective earth radius. Given the transmitter height h,, the receiver height h2, and the total ground range G, we can determine Gr, G2, and w. If we define 2 l(h1+h2) (37.35) e(h,,)G e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The loss factor F that accounts for the departures from free-space conditions is given by F = 1 + G r s D S(q)e –jD (37.29) where G is the Fresnel reflection coefficient, rs is the roughness coefficient, D is the divergence factor, S(q) is the shadowing function, and D is the phase angle corresponding to the path difference. We now account for each of these terms. The Fresnel reflection coefficient G accounts for the electrical properties of the earth’s surface. Because the earth is a lossy medium, the value of the reflection coefficient depends on the complex relative permittivity ec of the surface, the grazing angle y, and the wave polarization. It is given by (37.30) where (37.31) (37.32) (37.33) er and s are the dielectric constant and conductivity of the surface; w and l are the frequency and wavelength of the incident wave; and y is the grazing angle. It is apparent that 0 < ˜G˜ < 1. To account for the spreading (or divergence) of the reflected rays because of the earth’s curvature, we introduce the divergence factor D. The curvature has a tendency to spread out the reflected energy more than a corresponding flat surface. The divergence factor is defined as the ratio of the reflected field from curved surface to the reflected field from flat surface [Kerr, 1951]. Using the geometry of Fig. 37.5, D is given by (37.34) where G = G1 + G2 is the total ground range and ae = 6370 km is the effective earth radius. Given the transmitter height h1, the receiver height h2, and the total ground range G, we can determine G1, G2, and y. If we define (37.35) (37.36) G = - + sin sin y y z z z = ec - cos2 y for horizontal polarization z c c = e - e cos2 y for vertical polarization e e e e c r o r = - j = - j s w 60sl D G G a Ge . 1 2 1 2 1 2 + Ê Ë Á ˆ ¯ ˜ - sin y / p a h h G = e + + È Î Í Í ˘ ˚ ˙ ˙ 2 3 4 1 2 2 1 2 ( ) / a = È - Î Í Í ˘ ˚ ˙ ˙ - cos 1 ( ) 1 2 3 2a h h G p e
Transmitte Earth Center FIGURE 37.5 Geometry of spherical earth reflection. and assume h, s h2, G, s G, using small angle approximation yields [Blake, 1986] G 1,2 R=[h2+4a(a2+h1)sin2(/2)12,i=1,2 The grazing angle is given by a h thi-r (37.41) R Y =sIn/22 h+hi+ ri (37.42) 2(a+h1)R1 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC and assume h1 £ h2, G1 £ G2, using small angle approximation yields [Blake, 1986] (37.37) G2 = G – G1 (37.38) (37.39) (37.40) The grazing angle is given by (37.41) or (37.42) FIGURE 37.5 Geometry of spherical earth reflection. G G p 1 2 3 = + Ê + Ë Á ˆ ¯ ˜ cos p a fi i e G a = = , , i 1 2 R h aa h i i i ee i i =+ + = [ ( ) sin ( )] , 2 2 12/ 4 2 12 f / , y = È + - Î Í Í ˘ ˚ ˙ ˙ - sin 1 1 1 2 1 2 1 2 2 ah h R a R e e y f = + + + È Î Í Í ˘ ˚ ˙ ˙ - - sin ( ) 1 1 1 2 1 2 1 1 1 2 2 ah h R a hR e e