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). Thus, the characteristics of wave propagation through other types of media can be obtained as special cases of wave propagation in a lossy medium as follows. 1. Good conductors:a>>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). Thus, the characteristics of wave propagation through other types of media can be obtained as special cases of wave propagation in a lossy medium as follows: 1. Good conductors: 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