Time-harmonic form Phasor form. =0 (1.11d)

Transcription

1 Chapter 2 Wave in an Unbounded Medium Maxwell s Equations Time-harmonic form Phasor form (Real quantity) (complex quantity) B E = Eˆ = jωbˆ (1.11 a) t D= ρ Dˆ = ρ (1.11 b) D H = J + Hˆ = Jˆ+ jωdˆ ( 1.11 c) t B = 0 Bˆ =0 (1.11d) with E(x,y,z,t)=Re { j t Eˆ (x,y,z)e ω }

2 Chapter 2.1 Plane Waves in a Simple, Source-Free, and Lossless Medium Maxwell s Equations B E = (2.1 a) t D= ρ (2.1 b) D H = J + (2.1 c) t B = 0 (2.1 d)

3 2.1 Plane Wave in a Simple, Source-free, and Lossless Medium The general equations for electric and magnetic fields: E 0 (2.2) t H 0 (2.3) t 2 2 E με = H με = 2

4 If we restric the electric field as the function z, and the electric field to one component E only, equation (2.2) becomes: E E = t 2 2 x x με 2 2 z x 0 (2.5) the general solution of (2.5) is in the form of: E x =p (z-v t) + p (z+v t) (2.6) 1 p 2 p

5 2.1.1 The Relation between E and H Assume we have a electric field of uniform plane wave: E x =p (z-v t) 1 p The corresponding H(z) is not independent, and must be resolved from Maxwell equations (2.1a) and /or (2.1c): ε H y =( )p (z-v t)= μ 1 ( )p (z-v η 1 p 1 p t) where η = μ/ ε is called " intrinsic impedance" of the medium (in unit ohms).

6 2.2 Time-Harmonic Uniform Plane Waves in a Lossless Medium The electric field of a uniform plane wave in a lossless medium is { jβt + jβt jωt 1 1 } E (,) z t = Re( Ce + Ce ) e ) (2.11) x E ( z, t) = C cos( ωt βz) + C cos( ωt+ βz) (2.12) x or 1 2

7 The electric and magnetic fields of a time-harmonic wave propagating in the + Z direction are E(,) zt = Ccos( wt βt) x 1 1 Hy (,) z t = C1 cos( wt βt) (2.13) η where the " intrinsic impedance" (in unit ohms) η = μ/ ε = ωμ/ β (note: β= ω με ).

8 Example 2-1: AM broadcast Singnal The instantaneous expression for the electric field component of an AM broadcast signal propagating is given by 6 E(x,t in air )=z10c os(1. 5π 10 t+ βx) V m 1 (a) Determine the direction of propagation and frequency f. (b) Determine the phase constant β and wavelength. (c) Find the instantaneous expression for the corresponding H(x,t).

9 Example 2-2: FM broadcast Singnal An FM broadcast signal traveling in y direction in air has a magnetic field given by the phasor: -3 -j0.68π y 1 Ĥ(y)= e (-x+zj) A m (a) Determine the frequency. (b) Find the corresponding E(y). ˆ (c) Find the instantaneous expression for the E(x,t) and H(x,t).

10 2.3 Plane Waves in Lossy Media The electric field of a uniform plane wave in a lossy medium is { αz jβt αz + jβt jωt 1 2 } E (,) z t = Re( Ce e + C e e ) e ) x αz E ( z, t) = Ce cos( ωt βz) + C e cos( ωt+ βz) (2.18) x or αz 1 2 The propagation constant γ = α+i β : με σ α = ω [ 1 + ( ) 1] np m 2 ωε 2 1/2 1 με σ β = ω ωε 2 1/2 1 [ 1 ( ) 1] rad m (2 19)

11 The waves propagate in the +Z directions are E (,) z t = Ce cos( ωt βz) (2.21) + α z x 1 1 H (,) z t = Ce cos( ωt βz+ φ ) (2.23) + α z x 1 ηc where intrinsic impedance η is: c n η c η jφn = e = = c μ ε 1 (2 22) j(1/ 2) tan [ σ /( ωε)] = e σ 2 1/4 [1 + ( ) ] ωε μ ε eff μ σ ε j ω

12 Example 2-5 Find the complex propagation constant γ and the intrisic impedance η of a microwave signal in muscle tissue at 915 MHz ( =1.6 c -1 σ s-m ε r = 5) 1

13 Example 2-6 Consier distilled water at 25 GHz ( ε = 34 j9.01). Calculate the attenuation constant α, phase constant β, penetration depth d, and the wavelength λ. cr

14 2.4 Electromagnetic Energy Flow and the Poynting Vector Poynting Theorem: E Jdv = t ( μ H 2 + ε E 2 ) dv ( E H ) ds (2.31) V V S Poynting Vecor: S = E H

15 Example 2-12: Wire carrying direct current. Consider a long cylindrical conductor of conductivity σ and radius a, carrying a direct current I as shown in the figure. Find the power dissipated in a portion of the wire of length l, using (a) the left-hand side of (2.31); (b) he right-hand side of (2.31)

16 Example 2-13 Power flow in a coaxial line Consider a coaxial line delivering power to a resistor as shown in the figure. Assume the wire to be perfect conductor, so that there is no power dissipation in the wires, and the electric field inside them is zero. The configurations of the electric and magnetic field lines in the coaxial line are shown. Find the power delivered to the resistor R.

17 Example 2-18 VHF/UHF broadcast radiation A survey conducted in the United State indicated that ~50% of the population is exposed to average power densities of μ -2 W-(cm) due to VHF and UH F broadast radiaction. Find the correcponding amplitudes of the electric and magnetic fields.

18 Example 2-19 VLF waves in the ocen. a Consider VLF wave progation in the ocean. Find the time-average Poynting flux at the sea surface (z=0) and at the depth of z=100m. Assume: z e e j 18 z k V-m -1 Ê(z)=x2. Ĥ(z)=x2.84e j e z j e π k A-m z / 4-1