CHAPTER12: PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIAIn pptx

CHAPTER 12 PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA In Chapter 11, we considered basic electromagnetic wave principles. We learned how to mathematically represent waves as functions of frequency, medium prop- erties, and electric field orientation. We also learned how to calculate the wave velocity, attenuation, and power. In this chapter, we consider wave reflection and transmission at planar boundaries between media having different properties. Our study will allow any orientation between the wave and boundary, and will also include the important cases of multiple boundaries. We will also study the practical case of waves that carry power over a finite band of frequencies, as would occur, for example, in a modulated carrier. We will consider such waves in dispersive media, in which some parameter that affects propagation (permittivity for example) varies with frequency. The effect of a dispersive medium on a signal is of great importance, since the signal envelope will change its shape as it propagates. This can occur to an extent that at the receiving end, detection and faithful representation of the original signal become problematic. Dispersion thus becomes an important limiting factor in allowable propagation distances, in a similar way that we found to be true for attenuation. 387 | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents 12.1 REFLECTION OF UNIFORM PLANE WAVES AT NORMAL INCIDENCE In this section we will consider the phenomenon of reflection which occurs when a uniform plane wave is incident on the boundary between regions composed of two different materials. The treatment is specialized to the case of normal inci- dence, in which the wave propagation direction is perpendicular to the boundary. In later sections we remove this restriction. We shall establish expressions for the wave that is reflected from the interface and for that which is transmitted from one region into the other. These results will be directly applicable to impedance- matching problems in ordinary transmission lines as well as to waveguides and other more exotic transmission systems. We again assume that we have only a single vector component of the electric field intensity. Referring to Fig. 12.1, we define region l  1 ; 1  as the half-space for which z < 0; region 2   2 ; 2  is the half-space for which z > 0. Initially we establish the wave traveling in the z direction in region l, E  x1 z; tE  x10 e À 1 z cos!t À  1 z or E  xs1  E  x10 e Àjk 1 z 1 where we take E  x10 as real. The subscript 1 identifies the region and the super- script + indicates a positively traveling wave. Associated with E  x1 z; t is a magnetic field H  ys1  1  1 E  x10 e Àjk 1 z 2 where k 1 and  1 are complex unless  HH 1 (or  1 ) is zero. This uniform plane wave in region l which is traveling toward the boundary surface at z  0 is called the incident wave. Since the direction of propagation of the incident wave is perpen- dicular to the boundary plane, we describe it as normal incidence. 388 ENGINEERING ELECTROMAGNETICS FIGURE 12.1 A wave E  1 incident on a plane boundary establishes a reflected wave E À 1 and a transmitted wave E  2 . | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents We now recognize that energy may be transmitted across the boundary surface at z  0 into region 2 by providing a wave moving in the z direction in that medium, E  xs2  E  x20 e Àjk 2 z 3 H  ys2  1  2 E  x20 e Àjk 2 z 4 This wave which moves away from the boundary surface into region 2 is called the transmitted wave; note the use of the different propagation constant k 2 and intrinsic impedance  2 . Now we must try to satisfy the boundary conditions at z  0 with these assumed fields. E x is a tangential field; therefore the E fields in regions l and 2 must be equal at z  0. Setting z  0 in (1) and (3) would require that E  x10  E  x20 . H y is also a tangential field, however, and must be continuous across the boundary (no current sheets are present in real media). When we let z  0 in (2) and (4), however, we find that we must have E  x10 = 1  E  x20 = 2 ; since E  x10  E  x20 , then  1   2 . But this is a very special condition that does not fit the facts in general, and we are therefore unable to satisfy the boundary conditions with only an incident and a transmitted wave. We require a wave traveling away from the boundary in region 1, as shown in Fig. 12.1; this is called a reflected wave, E À xs1  E À x10 e jk 1 z 5 H À ys1 À E À x10  1 e jk 1 z 6 where E À x10 may be a complex quantity. Since this field is traveling in the Àz direction, E À xs1 À 1 H À ys1 , for the Poynting vector shows that E À 1  H À 1 must be in the Àa z direction. The boundary conditions are now easily satisfied, and in the process the amplitudes of the transmitted and reflected waves may be found in terms of E  x10 . The total electric field intensity is continuous at z  0, E xs1  E xs2 z  0 or E  xs1  E À xs1  E  xs2 z  0 Therefore E  x10  E À x10  E  x20 7 Furthermore, H ys1  H ys2 z  0 PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA 389 | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents or H  ys1  H À ys1  H  ys2 z  0 and therefore E  x10  1 À E À x10  1  E  x20  2 8 Solving (8) for E  x20 and substituting into (7), we find E  x10  E À x10   2  1 E  x10 À  2  1 E À x10 or E À x10  E  x10  2 À  1  2   1 The ratio of the amplitudes of the reflected and incident electric fields is called the reflection coefficient and is designated by À (gamma), À  E À x10 E  x10   2 À  1  2   1 9 The reflection coefficient may be complex, in which case there is a phase shift in the reflected wave. The relative amplitude of the transmitted electric field intensity is found by combining (9) and (7), to yield the transmission coefficient, :   E  x20 E  x10  2 2  1   2  1 À 10 Let us see how these results may be applied to several special cases. We first let region 1 be a perfect dielectric and region 2 be a perfect conductor. Then, since  2 is infinite,  2   j! 2  2  j! H 2 s  0 and from (10), E  x20  0 No time-varying fields can exist in the perfect conductor. An alternate way of looking at this is to note that the skin depth is zero. 390 ENGINEERING ELECTROMAGNETICS | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents Since  2  0, then (9) shows that À À1 and E  x10 ÀE À x10 The incident and reflected fields are of equal amplitude, and so all the incident energy is reflected by the perfect conductor. The fact that the two fields are of opposite sign indicates that at the boundary (or at the moment of reflec- tion) the reflected field is shifted in phase by 180  relative to the incident field. The total E field in region 1 is E xs1  E  xs1  E À xs1  E  x10 e Àj 1 z À E  x10 e j 1 z where we have let jk 1  0 j 1 in the perfect dielectric. These terms may be combined and simplified, E xs1  e Àj 1 z À e j 1 z ÀÁ E  x10 Àj2sin 1 zE  x10 11 Multiplying (11) by e j!t and taking the real part, we may drop the s subscript and obtain the real instantaneous form: E x1 z; t2E  x10 sin 1 zsin!t12 This total field in region 1 is not a traveling wave, although it was obtained by combining two waves of equal amplitude traveling in opposite directions. Let us compare its form with that of the incident wave, E x1 z; tE  x10 cos!t À  1 z13 Here we see the term !t À  1 z or !t À z=v p1 , which characterizes a wave travel- ing in the z direction at a velocity v p1  != 1 . In (12), however, the factors involving time and distance are separate trigonometric terms. At all planes for which  1 z  m, E x1 is zero for all time. Furthermore, whenever !t  m, E x1 is zero everywhere. A field of the form of (12) is known as a standing wave. The planes on which E x1  0 are located where  1 z  m m  0; Æ1; Æ2; FFF Thus 2  1 z  m PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA 391 | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents and z  m  1 2 Thus E x1  0 at the boundary z  0 and at every half-wavelength from the boundary in region 1, z < 0, as illustrated in Fig. 12.2. Since E  xs1   1 H  ys1 and E À xs1 À 1 H À ys1 , the magnetic field is H ys1  E  x10  1 e Àj 1 z  e j 1 z ÀÁ or H y1 z; t2 E  x10  1 cos 1 zcos!t14 This is also a standing wave, but it shows a maximum amplitude at the positions where E x1  0. It is also 90  out of time phase with E x1 everywhere. Thus no average power is transmitted in either direction. Let us now consider perfect dielectrics in both regions 1 and 2;  1 and  2 are both real positive quantities and  1   2  0. Equation (9) enables us to calcu- 392 ENGINEERING ELECTROMAGNETICS FIGURE 12.2 The instantaneous values of the total field E x1 are shown at t  =2. E x1  0 for all time at multiples of one half-wavelength from the conducting surface. | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents late the reflection coefficient and find E À x1 in terms of the incident field E  x1 . Knowing E  x1 and E À x1 , we then find H  y1 and H À y1 . In region 2, E  x2 is found from (10), and this then determines H  y2 . h Example 12.1 As a numerical example let us select  1  100   2  300  E  x10  100 V=m and calculate values for the incident, reflected, and transmitted waves. Solution. The reflection coefficient is À  300 À100 300 100  0:5 and thus E À x10  50 V=m The magnetic field intensities are H  y10  100 100  1:00 A=m H À y10 À 50 100 À0:50 A=m The incident power density is   1;av  1 2 E  x10 H  y10  100 W=m 2 while  À 1;av À 1 2 E À x10 H À y10  25:0W=m 2 In region 2, using (10) E  x20  E  x10  150 V=m and H  y20  150 300  0:500 A=m Thus   2;av  1 2 E  x20 H  y20  75:0W=m 2 Note that energy is conserved:   1;av  À 1;av   2;av PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA 393 | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents We can formulate a general rule on the transfer of power through reflection and transmission by using Eq. (57) from Chapter 11:  av  1 2 Re E s  H à s ÈÉ We consider the same field vector and interface orientations as before, but we consider the general case of complex impedances. For the incident power density, we have   1;av  1 2 Re E  x10 H à y10 no  1 2 Re E  x10 1  à 1 E à x10 &'  1 2 Re 1  à 1 &' jE  x10 j 2 The reflected power density is then  À 1;av À 1 2 Re E À x10 H Àà y10 no  1 2 Re ÀE  x10 1  à 1 À à E à x10 &'  1 2 Re 1  à 1 &' jE  x10 j 2 jÀj 2 We thus find the general relation between the reflected and incident power:  À 1;av jÀj 2   1;av 15 In a similar way, we find the transmitted power:   2;av  1 2 Re E  x20 H à y20 no  1 2 Re E  x10 1  à 2  à E à x10 &'  1 2 Re 1  à 2 &' jE  x10 j 2 jj 2 and so we see that the incident and transmitted powers are related through   2;av  Re 1= à 2 ÈÉ Re 1= à 1 ÈÉ jj 2   1;av   1  2         2  2   à 2  1   à 1  jj 2   1;av 16 Eq. (16) is a relatively complicated way to calculate the transmitted power, unless the impedances are real. It is easier to take advantage of energy conservation by noting that whatever power is not reflected must be transmitted. Eq. (15) can thus be used to find   2;av  1 ÀjÀj 2 ÀÁ   1;av 17 As would be expected (and which must be true), Eq. (17) can also be derived from Eq. (16). \ D12.1. A 1 MHz uniform plane wave is normally incident onto a freshwater lake ( H R  78,  HH R  0,  R  1). Determine the fraction of the incident power that is (a) reflected and (b) transmitted; (c) determine the amplitude of the electric field that is transmitted into the lake. Ans. 0.63; 0.37; 0.20 V/m. 394 ENGINEERING ELECTROMAGNETICS | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents 12.2 STANDING WAVE RATIO One of the measurements that is easily made on transmission systems is the relative amplitude of the electric or magnetic field intensity through use of a probe. A small coupling loop will give an indication of the amplitude of the magnetic field, while a slightly extended center conductor of a coaxial cable will sample the electric field. Both devices are customarily tuned to the operating frequency to provide increased sensitivity. The output of the probe is rectified and connected directly to a microammeter, or it may be delivered to an electronic voltmeter or a special amplifier. The indication is proportional to the amplitude of the sinusoidal time-varying field in which the probe is immersed. When a uniform plane wave is traveling through a lossless region, and no reflected wave is present, the probe will indicate the same amplitude at every point. Of course, the instantaneous field which the probe samples will differ in phase by z 2 À z 1  rad as the probe is moved from z  z 1 to z  z 2 , but the system is insensitive to the phase of the field. The equal-amplitude voltages are characteristic of an unattenuated traveling wave. When a wave traveling in a lossless medium is reflected by a perfect con- ductor, the total field is a standing wave and, as shown by Eq. (12), the voltage probe provides no output when it is located an integral number of half-wave- lengths from the reflecting surface. As the probe position is changed, its output varies as jsin zj, where z is the distance from the conductor. This sinusoidal amplitude variation is shown in Fig. 12.3, and it characterizes a standing wave. A more complicated situation arises when the reflected field is neither 0 nor 100 percent of the incident field. Some energy is transmitted into the second region and some is reflected. Region 1 therefore supports a field that is composed of both a traveling wave and a standing wave. It is customary to describe this field as a standing wave even though a traveling wave is also present. We shall see that the field does not have zero amplitude at any point for all time, and the degree to which the field is divided between a traveling wave and a true standing wave is expressed by the ratio of the maximum amplitude found by the probe to the minimum amplitude. PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA 395 FIGURE 12.3 The standing voltage wave produced in a lossless medium by reflection from a perfect conductor varies as jsin zj. | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents Using the same fields investigated in the previous section, we combine the incident and reflected electric field intensities, E x1  E  x1  E À x1 The field E x1 is a sinusoidal function of t (generally with a nonzero phase angle), and it varies with z in a manner as yet unknown. We shall inspect all z to find the maximum and minimum amplitudes, and determine their ratio. We call this ratio the standing-wave ratio, and we shall symbolize it by s. Let us now go through the mechanics of this procedure for the case in which medium 1 is a perfect dielectric,  1  0, but region 2 may be any material. We have E  xs1  E  x10 e Àj 1 z E À xs1  ÀE  x10 e j 1 z where À   2 À  1  2   1 and  1 is real and positive but  2 may be complex. Thus À may be complex, and we allow for this possibility by letting À  À jj e j If region 2 is a perfect conductor,  is equal to ;if 2 is real and less than  1 ,  is also equal to ; and if  2 is real and greater than  1 ,  is zero. The total field in region 1 is E xs1  e Àj 1 z  À jj e j 1 z ÀÁ E  x10 18 We seek the maximum and minimum values of the magnitude of the complex quantity in the larger parentheses in (18). We certainly have a maximum when each term in the larger parentheses has the same phase angle; thus, for E  x10 positive and real, E xs1; max  1  À jj E  x10 19 and this occurs where À 1 z   1 z    2m m  0; Æ1; Æ2; FFF20 Thus z max À 1 2 1   2m 21 Note that a field maximum is located at the boundary plane z  0 if   0; moreover,   0 when À is real and positive. This occurs for real  1 and  2 when 396 ENGINEERING ELECTROMAGNETICS | | | | ▲ ▲ e-Text Main Menu Textbook Table of Contents [...]... for all time at multiples of one half-wavelength from the conducting surface | v v 392 | e-Text Main Menu | Textbook Table of Contents | PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA ‡ À late the reflection coefficient and find Ex1 in terms of the incident field Ex1 ‡ ‡ ‡ À À Knowing Ex1 and Ex1 , we then find Hy1 and Hy1 In region 2, Ex2 is found ‡ from (10), and this then determines Hy2 h Example... normally incident onto a freshwater lake (HR ˆ 78, HH ˆ 0, R ˆ 1) Determine the fraction of the incident power that is (a) R reflected and (b) transmitted; (c) determine the amplitude of the electric field that is transmitted into the lake Ans 0.63; 0.37; 0.20 V/m | v v 394 | e-Text Main Menu | Textbook Table of Contents | PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA 12.2 STANDING WAVE RATIO One... …11† ‡ ˆ Àj2 sin…1 z† Ex10 Multiplying (11) by e j!t and taking the real part, we may drop the s subscript and obtain the real instantaneous form: ‡ Ex1 …z; t† ˆ 2Ex10 sin…1 z† sin…!t† …12† This total field in region 1 is not a traveling wave, although it was obtained by combining two waves of equal amplitude traveling in opposite directions Let us compare its form with that of the incident wave,... standing wave is expressed by the ratio of the maximum amplitude found by the probe to the minimum amplitude | v v FIGURE 12.3 The standing voltage wave produced in a lossless medium by reflection from a perfect conductor varies as j sin zj | e-Text Main Menu | Textbook Table of Contents | 395 ENGINEERING ELECTROMAGNETICS Using the same fields investigated in the previous section, we combine the incident... previous section, we combine the incident and reflected electric field intensities, ‡ À Ex1 ˆ Ex1 ‡ Ex1 The field Ex1 is a sinusoidal function of t (generally with a nonzero phase angle), and it varies with z in a manner as yet unknown We shall inspect all z to find the maximum and minimum amplitudes, and determine their ratio We call this ratio the standing-wave ratio, and we shall symbolize it by s Let... into the second region and some is reflected Region 1 therefore supports a field that is composed of both a traveling wave and a standing wave It is customary to describe this field as a standing wave even though a traveling wave is also present We shall see that the field does not have zero amplitude at any point for all time, and the degree to which the field is divided between a traveling wave and. .. Main Menu | Textbook Table of Contents | 391 ENGINEERING ELECTROMAGNETICS and 1 2 Thus Ex1 ˆ 0 at the boundary z ˆ 0 and at every half-wavelength from the boundary in region 1, z < 0, as illustrated in Fig 12.2 ‡ ‡ À À Since Exs1 ˆ 1 Hys1 and Exs1 ˆ À1 Hys1 , the magnetic field is zˆm Hys1 ˆ ‡ Á Ex10 À Àj1 z e ‡ e j1 z 1 or Hy1 …z; t† ˆ 2 ‡ Ex10 cos…1 z† cos…!t† 1 …14† This is also a standing... which characterizes a wave traveling in the ‡z direction at a velocity vp1 ˆ !=1 In (12), however, the factors involving time and distance are separate trigonometric terms At all planes for which 1 z ˆ m, Ex1 is zero for all time Furthermore, whenever !t ˆ m, Ex1 is zero everywhere A field of the form of (12) is known as a standing wave The planes on which Ex1 ˆ 0 are located where 1 z ˆ m …m ˆ 0;... when it is located an integral number of half-wavelengths from the reflecting surface As the probe position is changed, its output varies as j sin zj, where z is the distance from the conductor This sinusoidal amplitude variation is shown in Fig 12.3, and it characterizes a standing wave A more complicated situation arises when the reflected field is neither 0 nor 100 percent of the incident field... rectified and connected directly to a microammeter, or it may be delivered to an electronic voltmeter or a special amplifier The indication is proportional to the amplitude of the sinusoidal time-varying field in which the probe is immersed When a uniform plane wave is traveling through a lossless region, and no reflected wave is present, the probe will indicate the same amplitude at every point Of course, . angle), and it varies with z in a manner as yet unknown. We shall inspect all z to find the maximum and minimum amplitudes, and determine their ratio. We call this ratio the standing-wave ratio, and. first interface and are now for- ward-propagating. The effect of combining many co-propagating waves in this way is to establish a single wave which has a definite amplitude and phase, determined. between a traveling wave and a true standing wave is expressed by the ratio of the maximum amplitude found by the probe to the minimum amplitude. PLANE WAVES AT BOUNDARIES AND IN DISPERSIVE MEDIA