5 Wave Propagation Mohammad Kolbehdari Intel Corporation Matthew N. O. Sadiku Prairie View A&M University Electromagnetic (EM) wave propagation deals with the transfer of energy or informa- tion from one point (a transmitter) to another (a receiver) through the media such as material space, transmission line, and waveguide. It can be described using both theoretical models and practical models based on empirical results. Here we describe the free-space propagation model, path loss models, and the empirical path loss formula. Before presenting these models, we first discuss the theoretical basis and characteristics of EM waves as they propagate through material media. 5.1. WAVE EQUATIONS AND CHARACTERISTICS The EM wave propagation theory can be described by Maxwell’s equations [1,2]. rÂEðr,tÞ¼À @ @t Bðr,tÞð5:1Þ rÂHðr,tÞ¼ @ @t Dðr,tÞþJðr,tÞð5:2Þ rÁDðr,tÞ¼rðr,tÞð5:3Þ rÁBðr,tÞ¼0 ð5:4Þ In the above equations, the field quantities E and H represent, respectively, the electric and magnetic fields, and D and B the electric and magnetic displacements. J and r represent the current and charge sources. This set of differential equations relates the time and space rates of change of various field quantities at a point in space and time. Furthermore, the position vector r defines a particular location in space (x,y,z) at which the field is being measured. Thus, for example, Eðx,y,z,tÞ¼Eðr,tÞð5:5Þ Hillsboro, Oregon Prairie View, Texas 163 © 2006 by Taylor & Francis Group, LLC An auxiliary relationship between the current and charge densities, J and r, called the continuity equation is given by rÁJðr,tÞ¼À @ @t rðr,tÞð5:6Þ The constitutive relationships between the field quantities and electric and magnetic displacements provide the additional constraints needed to solve Eqs. (5.1) and (5.2). These equations characterize a given isotropic material on a macroscopic level in terms of two scalar quantities as B ¼ H ¼  0  r H ð5:7Þ D ¼ "E ¼ " 0 " r E ð5:8Þ where  0 ¼ 4 Â10 À7 H/m (henrys per mete r) is the permeability of free space and " 0 ¼ 8:85 Â10 À12 F/m (farads per meter) is the permittivity of free space. Also, " r and  r , respectively, characterize the effects of the atomic and molecular dipoles in the material and the magnetic dipole moments of the atoms constituting the medium. Maxwell’s equations, given by Eqs. (5.1) to (5.4), can be simplified if one assumes time-harmonic fields, i.e., fields varying with a sinusoidal frequency !. For such fields, it is convenient to use the complex exponential e j!t . Applying the time-harmonic assumption to Eqs. (5.1) to (5.4), we obtain the time-harmonic wave propagation equations rÂEðrÞ¼Àj!BðrÞð5:9Þ rÂHðrÞ¼j!DðrÞþJðr Þð5:10Þ rÁDðrÞ¼rðrÞð5:11Þ rÁB ðrÞ¼0 ð5:12Þ The solution of Maxwell’s equations in a source free isotropic medium can be obtained by using Eqs. (5.9) and (5.10) and applying Eqs. (5.7) and (5.8) as follows: rÂEðrÞ¼Àj!HðrÞð5:13Þ rÂHðrÞ¼j!"EðrÞð5:14Þ Taking the curl of the Eq. (5.13) and using Eq. (5.14) we get rÂrÂEðrÞ¼Àj!rÂHðrÞ¼! 2 "EðrÞð5:15Þ Using a vector identity, and noting that r ¼0, we can write Eq. (15) as r 2 EðrÞþ! 2 "EðrÞ¼0 ð5:16Þ This relation is called the wave equation. For example, the x component of E(r)is @ 2 @x 2 þ @ 2 @y 2 þ @ 2 @z 2  E x ðrÞþ! 2 "E x ðrÞ¼0 ð5:17Þ 164 Ko lbe h dari an d S a di k u © 2006 by Taylor & Francis Group, LLC 5.1.1. Attenuat ion If we consider the general case of a lossy medium that is charge free (r ¼ 0), Eqs. (5.9) to (5.12) can be manipulated to yield Helmholz’ wave equations r 2 E À 2 E ¼ 0 ð5:18Þ r 2 H À 2 H ¼ 0 ð5:19Þ where  ¼  þj is the propagation constant,  is the attenuation constant in nepers per meter or decibels per meter, and  is the phase constant in radians per meters. Constants  and  are given by  ¼ ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ  !"  r 2 À 1 "# v u u t ð5:20Þ  ¼ ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ  !"  2 r þ 1 "# v u u t ð5:21Þ where ! ¼ 2 f is the angular frequency of the wave and  is the conductivity of the medium. Without loss of general ity, if we assume that the wave propagates in the z direction and the wave is polarized in the x direction, solving the wave equations in Eqs. (5.18) and (5.19), we obtain E x ¼ E 0 e Àz cos ð!t ÀzÞð5:22Þ H y ¼ E 0 jj e Àz cos ð!t Àz À  Þð5:23Þ where  ¼jjff  is the intrinsic impedance of the medium and is given by jj¼ ffiffiffiffiffiffiffiffi =" p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ =!"ðÞ 2 Âà 4 q tan 2  ¼  !" 0   45  ð5:24Þ Equations (5.22) and (5.23) show that as the EM wave propagates in the medium, its amplitude is attenuated to e Àz . 5.1.2 . Dispersion A plane electromagnetic wave can be described as E x ðrÞ¼E x0 e ÀjkÁr ¼ E x0 e Àjðk x xþk y yþk z zÞ ð5:25Þ where E x0 is an arbitrary constant, and k ¼ k x a x þk y a y þk z a z is the vector wave and r ¼ a x x þ a y y þ a z z is the vector observation point. The substitution of the assumed form Wave Propagation 165 © 2006 by Taylor & Francis Group, LLC of the plane wave in Eq. (5.17) yields k 2 x þ k 2 y þ k 2 z ¼ k 2 ¼ ! 2 " ð5:26Þ This equation is called the dispersion relation. It may also be written in terms of the velocity v defined by k ¼ ! v ð5:27Þ The other components of E(r) with the same wave equation also have the same dispersion equation. The characteristic impedance of plane wave in free space is given by  ¼ jEj jHj ¼ ffiffiffiffi  " r ¼ ffiffiffiffiffiffi  0 " 0 r ¼ 377  ð5:28Þ 5.1.3. Phase Velocity By assuming k ¼ k z ¼ ! ffiffiffiffiffiffi " p , the electric field can be described by Eðz,tÞ¼E 0 cos ð!t Àk z z þ ’Þð5:29Þ For an observer moving along with the same velocity as the wave, an arbitrary point on the wave will appear to be constant, which requires that the argument of the E(z,t)be constant as defined by !t Àk z z þ ’ ¼ constant ð5:30Þ Taking the derivative with respect to the z yields dz dt ¼ ! k z ¼ v p ð5:31Þ where v p is defined as the phase velocity; for free space it is ! k z ¼ 1 ffiffiffiffiffiffiffiffiffiffi  0 " 0 p ffi 3 Â10 8 m=s ð5:32Þ which is the velocity of light in free space. 5.1.4. Group Velocity A signal consisting of two equal-amplitude tones at frequencies ! 0 Æ Á ! can be represented by f ðtÞ¼2 cos ! 0 t cos Á!t ð5:33Þ 166 Ko lbe h dari an d S a di k u © 2006 by Taylor & Francis Group, LLC which corresponds to a signal carrier at frequency ! 0 being modulated by a slowly varying envelope having the frequency Á!. If we assume that each of the two signals travels along a propagation direction z with an associated propagation constant kð!Þ, then the propagation constant of each signal is kð! 0 Æ Á !Þ. An expansion in a first-order Taylor series yields kð! 0 Æ Á !Þffikð! 0 ÞÆÁ!k 1 ð! 0 Þð5:34Þ where k 1 ð! 0 Þ¼ dkð!Þ d! j !¼! 0 ð5:35Þ The substitution of Eq. (5.34) into Eq. (5.33) following some mathematical manipulation yields f ðt,zÞ¼2 cos ! 0 ðt À p Þ cos Á!ðt À g Þð5:36Þ where  p ¼ kð! 0 Þ ! 0 z ð5:37Þ and  g ¼ k 1 ð! 0 Þz ð5:38Þ The quantities  p and  g are defined as the phase and group delays, respectively. The corresponding propagation velocities are v p ¼ z  p ð5:39Þ v g ¼ z  g ð5:40Þ For a plane wave propagating in a uniform unbounded medium, the propagation constant is a linear function of frequency given in Eq. (5.2 6). Thus, for a plane wave, phase and group velocities are equal and given by v p ¼ v g ¼ 1 ffiffiffiffiffiffi " p ð5:41Þ It is worthwhile to mention that if the transmission medium is a waveguide, kð!Þ is no longer a linear function of frequency. It is very useful to use the !-k diagram shown in to the frequency ! 0 gives the phase velocity and the slope of the tangent to the curve at ! 0 yields the group velocity. Wave Propagation 167 © 2006 by Taylor & Francis Group, LLC Fig. 5.1, which plots ! versus k(!). In this graph, the slope of a line drawn from the origin 5.1.5. Polari za tion The electric field of a plane wave propagating in the z direction with no components in the direction of propagation can be written as EðzÞ¼ða x E x0 þ a y E y0 Þe Àjk z z ð5:42Þ By defining E x0 ¼ E x0 e j’ x ð5:43Þ E y0 ¼ E y0 e j’ y ð5:44Þ we obtain EðzÞ¼ða x E x0 e j’ x þ a y E j’ y y0 Þe Àjk z z ð5:45Þ Assuming A ¼ E y0 =E x0 and ’ ¼ ’ y À ’ x , and E x0 ¼ 1, we can write Eq. (5.45) as EðzÞ¼ða x þ a y Ae j’ Þe Àjk z z ð5:46Þ Case I: A ¼0. EðzÞ¼a x e Àjk z z and Eðz,tÞ¼a x cos ð!t À k z zÞ. The movement of the electric field vector in the z ¼0 plane is along the x axis. This is known as a linearly polarized wave along the x axis. Case II: A ¼1, ’ ¼ 0. EðzÞ¼ða x þ a y Þe Àjk z z ð5:47Þ and Eðz,tÞ¼ða x þ a y Þ cos ð!t Àk z zÞð5:48Þ Figure 5.1 !-k diagram. 168 Ko lbe h dari an d S a di k u © 2006 by Taylor & Francis Group, LLC This is again a linear polarized wave with the electric field vector at 45 degrees with respect to the x axis. Case III: A ¼2, ’ ¼ 0: Eðz,tÞ¼ða x þ 2 a y Þ cos ð!t À k z zÞð5:49Þ This is again a linear polarized wave with the electric field vector at 63 degrees with respect to the x axis. Case IV: A ¼1, ’ ¼ =2. EðzÞ¼ða x þ ja y Þe Àjk z z ð5:50Þ and Eðz,tÞ¼a x cos ð!t Àk z zÞÀa y sin ð!t Àk z zÞð5:51Þ In this case the electric field vector traces a circle and the wave is defined to be left-handed circularly polarized. Similarly, with ’ ¼À=2, it is a right-handed circularly polarized wave. Case VI: A ¼2 and ’ 6¼0. This is an example of an elliptically polarized wave. 5.1.6. Poynting’s Theorem The relationships between the electromagnetic fields can be described by Poynting’s theorem. For an isotropic medium, Maxwell’s curl equations can be written as rÂE ¼À @H @t ð5:52Þ rÂH ¼ " @E @t þ J ð5:53Þ where the current density J can be described as having two components: J ¼ J s þ J c ð5:54Þ where J c ¼ E represents conduction current density induced by the presence of the electric fields and J s is a source current density that induces electromagnetic fie lds. The quantity E ÁJ has the unit of power per unit volume (watts per unit cubic meter). From Eqs. (5.52) and (5.53) we can get E ÁJ ¼ E ÁrÂH À"E Á @E @t ð5:55Þ Applying the vector identity rÁðA ÂBÞ¼B ÁrÂA ÀA ÁrÂB ð5:56Þ Wave Propagation 169 © 2006 by Taylor & Francis Group, LLC gives E ÁJ ¼ H ÁrÂE ÀrÁðE ÂH ÞÀ"E Á @E @t ð5:57Þ Substituting Eq. (5.52) into Eq. (5.57) yields E ÁJ ¼ÀH Á @H @t ÀrÁðE  HÞÀ"E Á @E @t ð5:58Þ Integrating Eq. (5.58) over an arbitrary volume V that is bounded by surface S with an outward unit normal to the surface ^ nn shown in Fig. 5.2 gives ððð v E ÁJ dv ¼ @ @t ððð v 1=2 H jj 2 dv þ ððð v 1=2" E jj 2 dv  þ ðð  s ^ nn ÁðE  HÞds ð5:59Þ where the following identity has been used ððð v J ÁA dv ¼ ðð  s ^ nn Á A ds ð5:60Þ Equation (5.59) represents the Poynting theorem. The terms 1=2 H jj 2 and 1=2" E jj 2 are the energy densities stored in magnetic and electric fields, respectively. The term ÐÐ  s ^ nn ÁðE  HÞds describes the power flowing out of the volume V. The quantity P ¼ E ÂH is called the Poynting vector with the unit of power per unit area. For example, the Poynting theorem can be applied to the plane electromagnetic wave given in Eq. (5.29), where ’ ¼0. The wave equations are E x ðz,tÞ¼E 0 cos ð!t Àk z zÞð5:61Þ H y ðz,tÞ¼ ffiffiffiffi "  r E 0 cos ð!t Àk z zÞð5:62Þ The Poynting vector is in the z direction and is given by P z ¼ E x H y ¼ ffiffiffiffi "  r E 2 0 cos 2 ð!t À k z zÞð5:63Þ Figure 5.2 A volume V enclosed by surface S and unit vector n. 170 Ko l b e h d a r i a nd S a d i ku © 2006 by Taylor & Francis Group, LLC Applying the trigonometric identity yields P z ¼ ffiffiffiffi "  r E 2 0 1 2 þ 1 2 cos 2 ð!t Àk z zÞ ! ð5:64Þ It is worth noting that the constant term shows that the wave carries a time-averaged power density and there is a time-varying portion representing the stored energy in space as the maxima and the minima of the fields pass through the region. We apply the time-harmonic representation of the field components in terms of complex phasors and use the time average of the product of two time-harmonic quantities given by hAðtÞBðtÞi ¼ 1 2 ReðAB à Þð5:65Þ where B* is the complex conjugate of B. The time average Poynting power density is hPi¼ 1 2 ReðE ÂH à Þð5:66Þ where the quantity P ¼ E ÂH is defined as the complex Poynting vector. 5.1.7. Boundary Conditions The boundary conditions between two materials shown in Fig. 5.3 are E t1 ¼ E t2 ð5:67Þ H t1 ¼ H t2 ð5:68Þ In the vector form, these boundary conditions can be written as ^ nn ÂðE 1 À E 2 Þ¼0 ð5:69Þ ^ nn ÂðH 1 À H 2 Þ¼0 ð5:70Þ Figure 5.3 Boundary conditions between two materials. Wave Propagation 171 © 2006 by Taylor & Francis Group, LLC Thus, the tangential components of electric and magnetic field must be equal on the two sides of any boundary between the physical media. Also for a charge- and current- free boundary, the normal components of electric and magnetic flux density are continuous, i.e., D n1 ¼ D n2 ð5:71Þ B n1 ¼ B n2 ð5:72Þ For the perfect conductor (infinite conductivity), all the fields inside of the conductor are zero. Thus, the continuity of the tangen tial electric fields at the boundary yields E t ¼ 0 ð5:73Þ Since the magnetic fields are zero inside of the conductor, the continuity of the normal magnetic flux density yields B n ¼ 0 ð5:74Þ Furthermore, the normal electric flux density is D n ¼ r s ð5:75Þ where r s is a surface charge density on the boundary. The tangential magnetic field is discontinuous by the current enclosed by the path, i.e., H t ¼ J s ð5:76Þ where J s is the surface current density. 5.1.8. Wave Ref lection We now con sider the problem of a plane wave obliquely incident on a plane interface cases of the problem: the electric field is in the xz plane (parallel polarization) or normal to the xz plane (parallel polarization). Any arbitrary incident plane wave may be treated as a linear combination of the two cases. The two cases are solved in the same manner: obtaining expressions for the incident, reflection, and transmitted fields in each region and matching the boundary conditions to find the unknown amplitude coefficients and angles. For parallel polari zation, the electric field lies in the xz plane so that the incident fields can be written as E i ¼ E 0 ða x cos  i À a z sin  i Þe Àjk 1 ðx sin  i þz cos  i Þ ð5:77Þ H i ¼ E 0  1 a y e Àjk 1 ðx sin  i þz cos  i Þ ð5:78Þ 172 Ko l b e h d a r i a n d S a d i k u © 2006 by Taylor & Francis Group, LLC between two lossless dielectric media, as shown in Fig. 5.4. It is conventional to define two [...]... illustrated in Fig 5. 10 The values of A, B, C, and D are given in terms of the carrier frequency f (in MHz), the base station antenna height hb (in meters), and the mobile station antenna height hm (in meters) as A ¼ 69 :55 þ 26:16 log10 f À 13:82 log10 hb À aðhm Þ 5: 127aÞ B ¼ 44:9 À 6 :55 log10 hb   f 2 C ¼ 5: 4 þ 2 log10 28 5: 127bÞ 5: 127cÞ À Á2 D ¼ 40:94 À 19:33 log10 f þ 4:78 log10 f 5: 127dÞ where... Group, LLC !1=2 5: 108Þ ! 5: 109Þ Wave Propagation 179 Geometry of spherical earth reflection Figure 5. 8 and assume h1 G1 ¼ h2 and G1 G2 , using small angle approximation yields [5] G %þ þ p cos 2 3 5: 110Þ G2 ¼ G À G1 0i ¼ Gi , ae 5: 111Þ i ¼ 1,2  Ã1=2 Ri ¼ h2 þ 4ae ðae þ hi Þ sin2 ð0i =2Þ i 5: 112Þ i ¼ 1,2 5: 113Þ The grazing angle is given by   2ae h1 þ h2 À R2 1 1 ¼ sin 2ae R1 À1 5: 114Þ or ¼ sinÀ1... dt 5: 120Þ x and ! 1 1 a2 pffiffiffi e À a erfcðaÞ B¼ 4a % 5: 121Þ a¼ cot  2s 5: 122Þ s¼ 'h ¼ rms surface slope 'l 5: 123Þ In Eq (5. 123), 'h is the rms roughness height and 'l is the correlation length Alternative models for SðÞ are available in the literature Using Eqs (5. 103) to (5. 123), we can calculate the loss factor in Eq (5. 102) Thus 4%Rd l  à Lm ¼ À20 log 1 þ Àrs DSðÞeÀjÁ Lo ¼ 20 log 5. 4 5: 124Þ... i þ 1 cos t 5: 93Þ T? ¼ 22 cos i 2 cos i þ 1 cos t 5: 94Þ and 5. 2 FREE-SPACE PROPAGATION MODEL The free-space propagation model is used in predicting the received signal strength when the transmitter and receiver have a clear line -of- sight path between them If the receiving antenna is separated from the transmitting antenna in free space by a distance r, as shown in Fig 5. 5, the power received... 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., F¼ Em Eo 5: 97Þ The magnitude of F is always less than unity since Em is always less than Eo Thus, for a lossy medium, Eq (5. 95) becomes  Pr ¼ Gr Gt l 4%r 2 Pt jFj2 5: 98Þ For practical reasons, Eqs (5. 95) and (5. 98) are commonly expressed in logarithmic... 1 2 3 4 5 6 David M Pozar Microwave Engineering; Addison-Wesley Publishing Company: New York, NY, 1990 Kong, J.A Theory of Electromagnetic Waves; Wiley: New York, 19 75 Sadiku, M.N.O Elements of Electromagnetics, 3rd Ed.; Oxford University Press: New York, 2001; 621–623 Sadiku, M.N.O Wave propagation, In The Electrical Engineering Handbook; Dorf, R.C., Ed.; CRC Press: Boca Raton, FL, 1997; 9 25 937 Blake,... is random in nature The randomness arises out of the hills, structures, vegetation, and ocean waves It is found that the distribution of the heights of the earth’s surface is usually the gaussian or normal distribution of probability theory If 'h is the standard deviation of the normal distribution of heights, we define the roughness parameters g¼ 'h sin l 5: 117Þ If g < 1/8, specular reflection is dominant;... illuminate parts of the earth’s surface shadowed by higher parts In a geometric approach, where diffraction and multiple scattering effects are neglected, the reflecting surface will consist of well-defined zones of illumination and shadow As there will be no field on a shadowed portion of the surface, the analysis should include only the illuminated portions of the surface A pictorial representation of rough surfaces... Group, LLC 5: 86Þ 5: 87Þ 174 Kolbehdari and Sadiku while the reflected and transmitted fields are Er ¼ À? E0 ay eÀjk1 ðx sin r Àz cos r Þ Hr ¼ 5: 88Þ À? E 0 ðax cos r þ az sin r ÞeÀjk1 ðx sin r Àz cos r Þ 1 Et ¼ E0 T? ay eÀjk2 ðx sin t þz cos t Þ Ht ¼ 5: 89Þ 5: 90Þ E0 T? ðÀax cos t þ az sin t ÞeÀjk2 ðx sin t þz cos t Þ 2 5: 91Þ where k1 sin i ¼ k1 sin r ¼ k2 sin t ðSnell’s lawÞ 5: 92Þ À?... medium, the value of the reflection coefficient depends on the complex relative permittivity "c of the surface, the grazing angle , and the wave polarization It is given by À¼ sin sin Àz þz 5: 103Þ where z¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "c À cos2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "c À cos2 z¼ "c "c ¼ "r À j for horizontal polarization 5: 104Þ for vertical polarization 5: 1 05 ' ¼ "r À j60'l !"o 5: 106Þ "r and ' are, . ÁrÂE ÀrÁðE ÂH ÞÀ"E Á @E @t 5: 57Þ Substituting Eq. (5. 52) into Eq. (5. 57) yields E ÁJ ¼ÀH Á @H @t ÀrÁðE  HÞÀ"E Á @E @t 5: 58Þ Integrating Eq. (5. 58) over an arbitrary volume V that. per unit cubic meter). From Eqs. (5. 52) and (5. 53) we can get E ÁJ ¼ E ÁrÂH À"E Á @E @t 5: 55 Applying the vector identity rÁðA ÂBÞ¼B ÁrÂA ÀA ÁrÂB 5: 56Þ Wave Propagation 169 © 2006 by. the time-harmonic assumption to Eqs. (5. 1) to (5. 4), we obtain the time-harmonic wave propagation equations rÂEðrÞ¼Àj!BðrÞ 5: 9Þ rÂHðrÞ¼j!DðrÞþJðr Þ 5: 10Þ rÁDðrÞ¼rðrÞ 5: 11Þ rÁB ðrÞ¼0 5: 12Þ The