22 Phasor form of Maxwell s equations and damped waves in conducting media

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1 22 Phasor form of Maxwell s equations and damped waves in conducting media When the fields and the sources in Maxwell s equations are all monochromatic functions of time expressed in terms of their phasors, Maxwell s equations can be transformed into the phasor domain. In the phasor domain all t jω and all variables D, ρ, etc. are replaced by their phasors D, ρ, and so on. With those changes Maxwell s equations take the form shown in the margin. Also in these equations it is implied that D = ɛẽ B = µ H J = σẽ D = ρ B = 0 E = B t H = J + D t. D = ρ B = 0 Ẽ = jω B H = J + jω D where ɛ, µ, and σ could be a function of frequency ω (as, strictly speaking, they all are as seen in Lecture 11). We can derive from the phasor form Maxwell s equations shown in the margin the TEM wave properties obtained earlier on using the time-domain equations by assuming ρ = J =0. 1

2 We will do that, and and after that relax the requirement J =0with J = σẽ to examine how TEM waves propagate in conducting media. With ρ = J =0the phasor form Maxwell s equation take their simplified forms shown in the margin. Using [ Ẽ = jωµ H] 2 Ẽ = jωµ H which combines with the Ampere s law to produce Ẽ = 0 H = 0 Ẽ = jωµ H H = jωɛẽ 2 Ẽ + ω 2 µɛẽ =0. For x-polaried waves with phasors Ẽ =ˆxẼx(), the phasor wave equation above simplifies as Try solutions of the form where γ is to be determined. 2 2Ẽx + ω 2 µɛẽx =0. Ẽ x () =e γ or e γ 2

3 Upon substitution into wave equation both of these lead to which yields from which one possibility is (γ 2 + ω 2 µɛ)ẽx =0, γ 2 + ω 2 µɛ =0 γ 2 = ω 2 µɛ Thus viable phasor solutions are as we already knew. γ = jβ, with β ω ɛ. Ẽ x () =e jβ Furthermore, using the phasor form Faraday s law it is easy to show that H y = ± e jβ µ with η = η ω. Note that we have recovered above the familiar properties of plane TEM waves using phasor methods. Next, the phasor method carries us to a new domain that cannot be easily examined using time-domain methods. 3

4 With ρ =0but J = σẽ 0, implying non-ero conductivity σ, the pertinent phasor form equations are as shown in the margin. This is the same set as before, except that jωɛ has been replaced by σ + jωɛ. Thus, we can make use of phasor wave solutions above after applying the following modifications to γ and η: Ẽ = 0 H = 0 Ẽ = jωµ H H = σẽ + jωɛẽ = (σ + jωɛ)ẽ 1. γ 2 = ω 2 µɛ =(jωµ)(jωɛ) σ 0 γ = (jωµ)(σ + jωɛ) 2. η = ɛ = jωµ jωɛ σ 0 η = jωµ σ + jωɛ. Note that the modified γ and η satisfy γη = jωµ and γ η = σ + jωɛ leading to useful relations shown in the margin (assuming real valued σ and ɛ). µ = γη jω σ = Re{ γ η } ɛ = 1 ω Im{γ η } 4

5 In terms of γ and η above, we can express an x-polaried plane wave propagating in direction in terms of phasors (a) Damped wave snapshot at t=0 together with exponential envelope e α Ẽ =ˆxE o e γ and H = ±ŷ E o η e γ where E o is an arbitrary complex constant (complex wave amplitude). In expanded forms γ and η appear as: γ = (jωµ)(σ + jωɛ) α+jβ, so that α = Re{γ} and β = Im{γ}, e α cos(ωt β) t=0 (b) Snaphot at t>0, with t=0 waveform for comparison and η = jωµ jωµ σ + jωɛ η ejτ so that η = σ + jωɛ and τ = jωµ σ + jωɛ. 1. In the special case of a perfect dielectric with σ =0, we find e α cos(ωt β) β appears within cosine argument and determines the wavelength and, therefore, γ = jω ɛ jβ and η = ɛ, λ = 2π β and propagation speed v p = ω β. Ẽ =ˆxE o e jβ and H = ±ŷe oe jβ η as before. In this case α = τ =0. 5 α controls wave attenuation by e α factor in propagation direction.

6 2. Another case of imperfect dielectric (or lousy conductor) occurs when σ is not ero, but it is so small that are justified in using with p = 1 2 as follows: For σ ωɛ 1, (1 ± a) p 1 ± pa, if a 1, γ = (jωµ)(σ + jωɛ) =jω ɛ(1 j σ ωɛ )1/2 jω ɛ(1 j σ 2ωɛ )=σ 2 hence Ẽ ˆxE o e (α+jβ) with α = σ 2 also in the same case H ±ŷe oe (α+jβ) η with η = ɛ(1 j σ ωɛ ) ɛ and β = ωɛ; ɛ (1+j σ 2ωɛ ) ɛ +jωɛ; tan 1 σ ej ɛ such that η and τ = η σ ɛ 2ωɛ. Note: γ and η both are complex valued, the consequences of which will be examined later on. 3. A third case of good conductor corresponds to σ 1. In that case, ωɛ γ = jω µɛ(1 j σωɛ ) ω jµ σ ωµσ µ jωµ ω = (1+j) and η = 2 σ = 6 j σ ω 2ωɛ, ωµ σ ejπ/4.

7 Hence, α β ωµσ 2 = πfµσ while η = ωµ σ and τ = η = 45o. (a) Damped wave snapshot at t=0 together with exponential envelope e α 4. Finally, perfect conductor case corresponds to σ, in which case Ẽ x 0 as we will show later on. Wave fields cannot exist in perfect conductors. Summariing, in a homogeneous medium with arbitrary but constant µ, ɛ, and σ, time-harmonic plane TEM waves are in terms of E =ˆxRe{E o e (α+jβ) e jωt } =ˆx E o e α cos(ωt β + E o ) and accompanying magnetic fields H = ±ŷre{ E o η e (α+jβ) e jωt } = ±ŷ E o η e α cos(ωt β + E o η). Propagation velocity v p = ω β = ω Im{ (jωµ)(σ + jωɛ)}, now depends on frequency ω and it describes the speed of the nodes (ero-crossings, not modified by the attenuation factor) of the field waveform. Subscript p is introduced to distinguish v p also called phase velocity from group velocity v g discussed in ECE 450 (velocity of narrowband wave packets). 7 e α cos(ωt β) t=0 (b) Snaphot at t>0, with t=0 waveform for comparison e α cos(ωt β) β appears within cosine argument and determines the wavelength λ = 2π β and propagation speed v p = ω β. α controls wave attenuation by e α factor in propagation direction.

8 Wavelength λ = 2π β = v p f now depends on frequency f via both the numerator and the denominator, and measures twice the distance between successive nodes of the waveform. Penetration depth (also called skin depth if very small) δ 1 α = 1 Re{ (jωµ)(σ + jωɛ)} is the distance for the field strength to be reduced by e 1 factor in its direction of propagation. For a fixed σ, and a sufficiently large ω, the penetration depth δ 2 σ Imperfect dielectric formula ɛ which can be very small if σ is large with small δ the wave is severely attenuated as it propagates. For a fixed σ, and a sufficiently small ω, 2 δ µωσ = 1 Good conductor "skin depth" formula πfµσ which, although small with large σ, increases as ω decreases, making low frequencies to be preferable in applications requiring propagating through lossy media with large σ, such as in sea-water. 8 (a) Damped wave snapshot at t=0 together with exponential envelope e α e α cos(ωt β) t=0 (b) Snaphot at t>0, with t=0 waveform for comparison e α cos(ωt β) β appears within cosine argument and determines the wavelength λ = 2π β and propagation speed v p = ω β. α controls wave attenuation by e α factor in propagation direction.