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CHAPTER 10 PROPAGATION & REFLECTION OF PLANE WAVES 10.0 PROPAGATION & REFLECTION OF PLANE WAVES 10.1 ELECTRIC AND MAGNETIC FIELDS FOR PLANE WAVE 10.2 PLANE WAVE IN LOSSY DIELECTRICS – IMPERFECT DIELECTRICS 10.3 PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS 10.4 PLANE WAVE IN FREE SPACE 10.5 PLANE WAVE IN CONDUCTORS 10.6 POWER AND THE POYNTING VECTOR 1 10.0 PROPAGATION & REFLECTION OF PLANE WAVES Will discuss the effect of propagation of EM wave in four medium : Free space ; Lossy dielectric ; Lossless dielectric (perfect dielectric) and Conducting media. Also will be discussed the phenomena of reflections at interface between different media. Ex : EM wave is radio wave, TV signal, radar radiation and optical wave in optical fiber. Three basics characteristics of EM wave : - travel at high velocity - travel following EM wave characteristics - travel outward from the source These propagation phenomena for a type traveling wave called plane wave can be explained or derived by Maxwell’s equations. 10.1 ELECTRIC AND MAGNETIC FIELDS FOR PLANE WAVE From Maxwell’s equations : ∂ B ∂ H ∇ E - ∂ t ∂ t ∂ D ∂ E ∇ H J J ∂ t ∂ t ∇ D v ∇ B 0 Assume the medium is free of charge : ∂ H ∂ t ∂ E ∇ H ∂ t ∇ D 0 ∇ E - ∇ B 0 From vector identity and taking the curl of (1)and substituting (1) and (2) ∇ (∇ E ) ∇(∇ E ) -∇2 E where ∇(∇ E ) 0 v 0, J 0 (1) ( 2) (3) ( 4) ∂ H -∇2 E ∇ - ∂ t ∂ → ∇2 E (∇ H ) ∂ t ∂2 E 2 ∴ ∇ E ∂ t2 ie Helmholtz ' s equation for electric field ∂2 E 2 3 ∇ E Vm ∂ t2 In Cartesian coordinates : 2 E ∂2 E ∂2 E ∂2 E 2 2 2 2 x ∂ y ∂ z ∂ t Assume that : (i) Electric field only has x component (ii) Propagate in the z direction 2 2 ∂ Ex ∂ Ex 2 ∂ z ∂ t2 Similarly in the same way, from vector identity and taking the curl of (2)and substituting (1) and (2) ∂2 H 3 ∇ H A m ∂ t2 2 2 2 ∂ Ex ∂ Ex 2 ∂ z ∂ t2 The solution for this equation : E x E x cos(t - z ) E x- cos(t z ) Incidence wave propagate in +z direction To find H field : Reflected wave propagate in -z direction ∂ H ∇ E - ∂ t ∂ Ex ∂ Ex ∇ E yˆ zˆ ∂ z ∂ y E x sin( t - z ) - E x- sin( t z ) yˆ On the right side equation : ∂ Hy ∂ ∂ H Hx ∂ Hz - - xˆ yˆ ∂ t t ∂ t ∂ t ∂ zˆ Equating components on both side = y component - ∂ Hy ∂ t E x sin( t - z ) - E x- sin( t z ) E x E x- H y ∫ sin( t - z )dt - sin( t z ) dt E x cos(t - z ) E x cos(t z ) Hy E x cos(t - z ) E x - cos(t z ) H y cos(t - z ) - H y- cos(t z ) Hence : E x E x cos(t - z ) E x- cos(t z ) H y H y cos(t - z) - H y- cos(t z) These equations of EM wave are called PLANE WAVE. Main characteristics of EM wave : (i) Electric field and magnetic field always perpendicular. (ii) NO electric or magnetic fields component in the direction of propagation. (iii) E H will provides information on the direction of propagation. 10.2 PLANE WAVE IN LOSSY DIELECTRICS – IMPERFECT DIELECTRICS 0 ; 0 r ; 0 r Assume a media is charged free , ρv =0 ∂ D ∇ H J j E ∂ t ∂ B (2) ∇ E - jH ∂ t Taking the curl of (2) : (1) ∇∇ E - j ∇ H From vector identity : ∇∇ A ∇∇ A -∇2 A ∇∇ E -∇2 E - j j E ∇2 E - j j E 0 ∇2 E - 2 E 0 Where : ∇∇ E - j ∇ H Equating (4) and (5) for Re and Im parts : j j 2 - 2 - 2 (Re) (6) 2 - j propagatio n constant 2 Define : ∂ D j E (1) ∂ t ∂ B ∇ E - jH (2) ∂ t ∇ H J (4) j 2 2 - 2 2 j (5) 2 (Im) (7) Magnitude for (5) ; 2 2 2 Add (10) and (6) : (8) Hence : 2 2 2 2 - 2 Magnitude for (4) ; 2 - 2 2 2 2 2 (9) Equate (8) and (9) : 2 2 2 2 2 - 2 - 2 (Re) (6) (10) 2 2 1 2 2 - 2 2 2 2 2 1 2 2 - 1 2 2 1 2 2 - 1 Np / m (11) is known as attenuation constant as a measure of the wave is attenuated while traveling in a medium. Substract (10) and (6) : 2 2 2 2 2 2 1 2 2 1 2 rad / m (12) is phase constant If the electric field propagate in +z direction and has component x, the equation of the wave is given by : E ( z , t ) E0 e -z cost - z xˆ (13) And the magnetic field : H ( z, t ) H 0e-z cost - z - yˆ (14) H0 where ; E0 E ( z , t ) E0 e -z cost - z xˆ H ( z, t ) H 0e-z cost - z - yˆ (15) (15) Intrinsic impedance : j j ∠ e , () j where ; / 2 1/ 4 1 , tan 2 (16) , 0 ≤ ≤450 (17) Conclusions that can be made for the wave propagating in lossy dielectrics material : -z (i) E and H fields amplitude will be attenuated by e (ii) E leading H by (14) Wave velocity ; u / ; 2 / / 2 1 Jd E tan jE , tan 2 , 0 ≤ ≤450 (17) From (17) and (18) Loss tangent ; J 1/ 4 2 (18) Loss tangent values will determine types of media : tan θ small (σ / ωε < 0.1) – good dielectric – low loss tan θ large (σ /ωε > 10 ) - good conductor – high loss Another factor that determined the characteristic of the media is operating frequency. A medium can be regarded as a good conductor at low frequency might be a good dielectric at higher frequency. E ( z, t ) E0 e -z cost - z xˆ E0 e -z e jt - z xˆ H0 (14) H ( z , t ) H 0 e -z cost - z - yˆ E0 e -z e jt - z - (15) yˆ Graphical representation of E field in lossy dielectric E0 x E ( z , t ) E0 e -z cost - z xˆ e z z y E0 10.3 PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS Characteristics: 0, 0 r , 0 r Substitute in (11) and (12) : 0, u 1 2 , o 0 (20) (21) (19) 2 2 1 1 Np / m (11) 2 2 2 2 1 2 2 1 rad / m (12) j j ∠ e , () j (22) The zero angle means that E and H fields are in phase at each fixed location. 10.4 PLANE WAVE IN FREE SPACE Free space is nothing more than the perfect dielectric media : Characteristics: 0, 0 , 0 (23) 0, Substitute in (20) and (21) : 0 , 0 0 / c 1 2 u c , 0 0 where u (24) (25) u c 3 108 m / s 0 0 120 0 (20) 1 2 , (21) 0o (22) 0 4 10 7 H / m (26) 0 8.854 10 12 1 10 9 F / m 36 The field equations for E and H obtained : E E0 cos(t - z ) xˆ H E0 0 cos(t - z ) yˆ E ( z , t ) E0 e -z cost - z xˆ (27) H ( z, t ) H 0e-z cost - z - yˆ (28) E and H fields and the direction of propagation : x Ex+ Generally : ˆ zˆ k kos(-z) Eˆ Hˆ kˆ z (at t = 0) y Hy+ kos(-z) (14) (15) 10.5 PLANE WAVE IN CONDUCTORS In conductors : or →∞ With the characteristics : ~ ∞, 0 , 0 r (29) Substitute in (11 and (12) : 2 f 2 1 1 Np / m (11) 2 2 2 (30) 45o E leads H by 450 2 (31) The field equations for E and H obtained : E E0e-z cos(t - z ) xˆ H E0 0 (32) e-z cos(t - z - 45o ) yˆ (33) 2 1 2 2 1 rad / m (12) j j ∠ e , () j It is seen that in conductors E and H waves are attenuated by e -z From the diagram is referred to as the skin depth. It refers to the amplitude of the wave propagate to a conducting media is reduced to e-1 or 37% from its initial value. In a distance : E0 e - E0 e -1 ∴ 1 / 1 f It can be seen that at higher frequencies is decreasing. (34) x E0 0.368E0 z Ex.10.1 : A lossy dielectric has an intrinsic impedance of 200∠30 at the particular frequency. If at that particular frequency a plane wave that propagate in a medium has a magnetic field given by : o H 10e-x cos(t - x/2 ) yˆ A / m. Find E Solution : Eˆ Hˆ kˆ → Eˆ yˆ xˆ ∴E - zˆ and . From intrinsic impedance, the magnitude of E field : E0 200∠30o H0 → E0 2000∠30o It is seen that E field leads H field : 300 / 6 Hence : E -2000e-x cos(t - x / 2 / 6) zˆ (V / m) E -2000e-x cos(t - x / 2 / 6) zˆ (V / m) To find : 1 1 - 1 1 2 1/ 2 2 tan 2 tan 600 3 2 -1 1 ; and we know 1 / 2 ∴ 2 1 3 → 0.2887 Np / m 1/ 2 3 Hence: E -2000e-0.2887x cos(t - x / 2 / 6) zˆ (V / m) 10.6 POWER AND THE POYNTING VECTOR ∂ H ∇ E - ∂ t (35) ∂ E ∇ H E ∂ t Dot product (36) with (36) E: ∂ E E ∇ H E E ∂ t 2 (37) From vector identity: ∇ A B B ∇ A - A ∇ B Change A H, B E (38) in (37) and use (38) , equation (37) becomes : ∂ E H ∇ E ∇ H E E E ∂ t 2 (39) ∂ E H ∇ E ∇ H E E E ∂ t 2 (39) H ∂ ∇ E - t ∂ (35) And from (35): ∂ H ∂ H ∇ E H - H H ∂ t 2∂ t (40) Therefore (39) becomes: - ∂H 2 2 ∂ t -∇ E H E 2 ∂ E E ∂ t (41) where: ∇ H E -∇ E H Integration (41) throughout volume v : ∂ 1 2 1 2 2 ∇ E H dv E H dv E dv ∫ ∫ ∫ ∂ t v 2 2 v v (42) ∂ 1 2 1 2 2 ∇ E H dv E H dv E dv ∫ ∫ ∫ ∂ t v 2 2 v v (42) Using divergence theorem to (42): ∂ 1 2 1 2 2 E H d S E H dv E dv ∫ ∫ ∫ ∂ t v 2 2 s v Total energy flow leaving the volume The decrease of the energy densities of energy stored in the electric and magnetic fields Dissipated ohmic power Equation (43) shows Poynting Theorem and can be written as : E H W / m 2 (43) Poynting theorem states that the total power flow leaving the volume is equal to the decrease of the energy densities of energy stored in the electric and magnetic fields and the dissipated ohmic power. The theorem can be explained as shown in the diagram below : Output power σ Ohmic losses J E Stored electric field H Stored magnetic field Input power Given for lossless dielectric, the electric and magnetic fields are : E E0 cos(t - z ) xˆ H E0 cos(t - z ) yˆ The Poynting vector becomes: E H W / m2 E 20 cos 2 (t - z ) yˆ To find average power density : Integrate Poynting vector and divide with interval T = 1/f : Pave 1 T T E02 ∫ cos 2 (t z ) dt 0 2 T 0 1 E 2T 1 cos( 2t - 2 z )dt ∫ 0 T 1 E 1 sin( 2t - 2 z ) t 2T 2 0 1 E02 P ave W / m2 2 2 0 Average power through area S : 1 E02 P ave S (W ) 2 Given for lossy dielectric, the electric and magnetic fields are : E E0e-z cos(t - z ) xˆ H E0 0 e -z cos(t - z - ) yˆ The Poynting vector becomes: E 20 e 2z cos(t - z ) cos(t - z ) Average power : 1 E02 2z P ave e cos 2 Ex.10.2: A uniform plane wave propagate in a lossless dielectric in the +z direction. The electric field is given by : E ( z, t ) 377 cost 4 / 3z / 6xˆ (V / m) 2 The average power density measured was 377 W / m . Find: (i) Dielectric constant of the material if 0 (ii) Wave frequency (iii) Magnetic field equation Solution: (i) Average power : 1 E2 Pave 377 2 1 (377) 2 377 2 377 / 2 188.5 For lossless dielectric : 0 r 0 r 1 0 1.9986 0 r 4.0 (ii) Wave frequency : E ( z, t ) 377 cost 4 / 3z / 6xˆ (V / m) 4 / 3 0 4 3 0 2f 3.9946 1016 f 99.93 106 (100 MHz ) E ( z, t ) 377 cost 4 / 3z / 6xˆ (V / m) (iii) Magnetic field equation : H ( z, t ) 377 cos(t (4 / 3) z / 6) yˆ 2 cos(t (4 / 3) z / 6) yˆ ( A / m)