Electromagnetic Waves, Antennas, and RF: Wave Propagation in Materials

(Topic)

Electromagnetic Waves, Antennas, and RF: Wave Propagation in Materials

EM16 derived the electromagnetic wave equation in vacuum. EM17 and EM18 showed that the wave transports energy and momentum, and EM19 described the geometry of its transverse polarization. We now ask what changes when the wave travels through matter rather than empty space.

The answer is controlled by three material parameters:

|𝜖,----μ,-----σ,-| -----------------|

(1)

where 𝜖 is permittivity, μ is permeability, and σ is electrical conductivity. These quantities determine how rapidly phase advances, how strongly amplitude decays, and how the electric and magnetic fields are related [1, 2, 3, 4].

The central result is that a time-harmonic plane wave propagating in the +z direction through a homogeneous linear medium has the form

---------------------------- | { −γz iωt} | -E(z,t)-=-Re--E0e----e----,-

(2)

with the complex propagation constant

|------------| -γ-=-α-+-iβ.-|

(3)

The real part α controls attenuation. The imaginary part β controls phase propagation.

1 Constitutive relations connect fields to material response

In a homogeneous, linear, isotropic material,

|--------| |---------| |--------| D--=-𝜖E,-- B--=-μH,--- -J-=-σE.--

(4)

The first two equations describe electric and magnetic material response. The third is Ohm’s law in local field form.

It is often useful to write

𝜖 = 𝜖r𝜖0, μ = μrμ0.

(5)

2 Maxwell’s curl equations inside the medium

In a source-free homogeneous propagation region,

|-----------------| | ∂H-- | ∇ × E = − μ ∂t ,| -------------------

(6)

and

|--------------------| ∇ × H = σE + 𝜖∂E-.| -----------------∂t---

(7)

The conduction-current density and displacement-current density are

J = σE, J = 𝜖∂E-. c d ∂t

(8)

For a sinusoidal field, the ratio of their characteristic magnitudes is

|--------| | -σ- | |p ≡ ω 𝜖.| ---------

(9)

Thus

p ≪ 1 : dielectric-like, p ≫ 1 : conductor -like.

(10)

3 Derivation of the wave equation in matter

Take the curl of Faraday’s law:

∇ × (∇ × E) = − μ-∂-(∇ × H ). ∂t

(11)

Insert the Ampere-Maxwell law:

( ) ∇ × (∇ × E) = − μ ∂-- σE + 𝜖∂E- . ∂t ∂t

(12)

For constant material parameters,

∂E ∂2E ∇ × (∇ × E ) = − μ σ-- − μ 𝜖--2-. ∂t ∂t

(13)

Use

2 ∇ × (∇ × E) = ∇ (∇ ⋅ E ) − ∇ E.

(14)

In a homogeneous source-free region,

∇ ⋅ D = 0 =⇒ ∇ ⋅ E = 0.

(15)

Therefore

|------------------------| | 2 ∂E ∂2E | |∇ E = μ σ--- + μ 𝜖--2-.| ------------∂t------∂t---

(16)

Similarly,

|------------------------| | 2 ∂H ∂2H | ∇ H = μσ ----+ μ 𝜖---2 . ------------∂t------∂t----

(17)

The term proportional to σ is the key addition relative to the vacuum wave equation. It produces attenuation because electromagnetic energy is transferred to the material.

4 The lossless dielectric limit

Set σ = 0. Then

2 ∇2E = μ𝜖∂-E-. ∂t2

(18)

Comparison with the standard wave equation gives

|--------------------| | --1-- --c--- | |vp = √ μ𝜖-= √ μ-𝜖-.| -----------------r-r-

(19)

For a nondispersive lossless medium, the refractive index is

|-----c---√------| n = -- = μr𝜖r.| -----vp-----------

(20)

For many ordinary RF dielectrics, μ_r ≈ 1, so

-- n ≈ √ 𝜖r.

(21)

Figure 1. A lossless plane wave propagates without amplitude decay. The phase advances by 2π over one wavelength λ.

5 Frequency, wavelength, and phase constant

A sinusoidal plane wave may be written

E (z,t) = E0 cos(ωt − βz ).

(22)

One wavelength corresponds to a phase change of 2π, so

|--------| | 2π | |λ = ---.| ------β---

(23)

The phase velocity is

|--------------| |vp = fλ = ω-.| ------------β--|

(24)

In a lossless material,

|-----------| β = ω√ μ𝜖. | ------------

(25)

6 Example: wavelength in a lossless dielectric

Let

𝜖r = 4, μr = 1.

(26)

Then

vp = c-. 2

(27)

f = 1.0 GHz,

(28)

the vacuum wavelength is approximately

λ0 = c-= 0.300 m. f

(29)

Inside the dielectric,

|------------------| |λ = λ0-= 0.150 m. | ------2-------------

(30)

The frequency is fixed by the source; the material changes propagation speed and therefore wavelength.

7 Time-harmonic fields and phasors

Adopt the time convention

eiωt.

(31)

Write

{ } E(r,t) = Re E^(r)eiωt .

(32)

Then

∂-- ∂2-- 2 ∂t −→ iω, ∂t2 −→ − ω .

(33)

The field equation becomes

( ) ∇2 ^E = iωμσ − ω2μ𝜖 ^E.

(34)

Define

|------------------| γ2 = iω μ(σ + iω𝜖).| --------------------

(35)

Then

∇2E^ = γ2E^.

(36)

For one-dimensional propagation in +z,

|----------------| |^E (z) = E e− γz. | ----------0------

(37)

8 The propagation constant separates attenuation and phase

Write

|------------| |γ = α + iβ. | -------------

(38)

Then

e− γz = e− αze−iβz.

(39)

The physical field is

|-----------------------------------| E (z,t) = E0e− αz cos(ωt − βz + ϕ0).| ------------------------------------

(40)

Thus

|-−αz--------------------------------------------------------| -e----:-amplitude-attenuation,-----βz-:-phase--accumulation.-|

(41)

The units are

[α ] = Np/m, [β] = rad/m.

(42)

Figure 2. In a lossy medium, the sinusoid propagates while its amplitude envelope decays as e^−αz.

9 Exact attenuation and phase constants

Let

σ p = ---. ω 𝜖

(43)

From

2 2 (α + iβ ) = − ω μ 𝜖 + iω μσ,

(44)

the real and imaginary parts give

2 2 2 α − β = − ω μ 𝜖,

(45)

and

2αβ = ωμ σ.

(46)

Solving gives

|------------------------------| | ∘ -μ𝜖[∘ ------ ]1∕2 | |α = ω --- 1 + p2 − 1 ,| ----------2--------------------

(47)

and

|------∘-----------------------| | μ𝜖[∘ ------ ]1∕2 | |β = ω --- 1 + p2 + 1 . | ----------2--------------------

(48)

10 A material-regime map

The ratio

-σ- p = ω 𝜖

(49)

compares conduction current with displacement current.

Figure 3. Material propagation regimes are organized by the ratio σ∕(ω𝜖). The same material can move between regimes as frequency changes.

Because p contains ω in the denominator, increasing frequency tends to make a fixed material appear more dielectric-like.

11 Low-loss dielectric approximation

For

p ≪ 1,

(50)

use

∘ ------ p2 1 + p2 ≈ 1 + --. 2

(51)

Then

|------∘-----| | σ μ | |α ≈ -- -, | ------2---𝜖--

(52)

and

|-----√-----| β-≈--ω--μ𝜖.-|

(53)

Thus a weakly conducting dielectric has almost the same phase behavior as a lossless dielectric but slowly loses amplitude.

The ratio

|------------| | σ--| |tan δc ≡ ω𝜖 | -------------

(54)

may be interpreted as a conduction-loss tangent. Real dielectric loss can also be represented by a complex, frequency-dependent permittivity, so the full loss tangent can contain more than this ohmic contribution [3, 4].

12 Example: a slightly lossy dielectric

Take

−3 f = 100 MHz, 𝜖r = 4, μr = 1, σ = 1.0 × 10 S/m.

(55)

Then

p = σ--≈ 0.0449, ω𝜖

(56)

so the low-loss approximation is appropriate.

The exact formulas give

|----------------------| α ≈ 9.42 × 10−2Np/m, | ------------------------

(57)

and

|-----------------| β ≈ 4.193 rad/m. | -------------------

(58)

Therefore

λ = 2π-≈ 1.499m, β

(59)

and

v = ω-≈ 1.499 × 108m/s. p β

(60)

After 10 m the field amplitude is reduced by

e−αz = e− (0.0942)(10) ≈ 0.390.

(61)

13 Attenuation in nepers and decibels

− αz E (z) = E(0)e ,

(62)

then

20 log ₁₀	= 20 log ₁₀(e^−αz)	(63)
	= −8.686 αz.	(64)

Hence

|----------------------| -Lamp-=-−-8.686-αz-dB.--

(65)

Since power is proportional to field amplitude squared,

P-(z)= e−2αz, P (0)

(66)

and therefore

P(z)- 10log10 P(0) = − 8.686 αz dB.

(67)

The same numerical dB attenuation results whether one starts from field amplitude with 20 log ₁₀ or power with 10 log ₁₀.

14 Intrinsic impedance of a material

For a uniform plane wave define

|--------| | E | |η ≡ H--.| ---------

(68)

With the e^iωt convention,

|---∘-----------| | --iωμ--- | η = σ + iω𝜖. | -----------------

(69)

For a lossless dielectric,

|---∘-----| | μ | η = -. | -------𝜖---

(70)

In vacuum,

∘ --- η0 = μ0-≈ 376.73Ω. 𝜖0

(71)

For 𝜖_r = 4 and μ_r = 1,

η = η0-≈ 188.4Ω. 2

(72)

This impedance is a field ratio, not a lumped resistor inserted in the path of the wave.

15 The good-conductor limit

For

σ ≫ ω𝜖,

(73)

the exact expressions simplify to

|---------∘---------∘--------| |α ≈ β ≈ ω-μσ- = πf μσ. | --------------2--------------|

(74)

The intrinsic impedance becomes

|-----------------| | ∘ ω-μ- | η ≈ (1 + i) ---. | --------------2σ---

(75)

The wave both accumulates phase and loses amplitude rapidly.

16 Skin depth

Define

|-------| | 1- | δ ≡ α. | --------

(76)

For a good Conductor,

|----∘-------| |δ ≈ --2--.| | ω μσ | -------------

(77)

At one skin depth,

E-(δ-) −1 E (0 ) = e ≈ 0.368.

(78)

At two and three skin depths,

E(2δ ) −2 E(3δ ) −3 E-(0)-= e ≈ 0.135, E-(0)-= e ≈ 0.0498.

(79)

Figure 4. Skin depth is the distance over which field amplitude falls by the factor 1∕e.

17 Example: copper at 1 MHz

Take

σ = 5.8 × 107 S/m, μ ≈ μ , f = 1.0MHz. 0

(80)

The good-conductor approximation gives

∘ ------ 2 −5 δ = ωμ-σ--≈ 6.61 × 10 m, 0

(81)

|------------| δ ≈ 66.1 μm. | --------------

(82)

The field is strongly attenuated within a fraction of a millimeter. This is the electromagnetic basis of the RF skin effect in good conductors.

18 Power flow also decays

EM17 introduced the time-averaged Poynting vector. In a lossy medium,

E ∝ e−αz,

(83)

so power density behaves as

|---------------| ⟨S (z )⟩ ∝ e− 2αz. | -----------------

(84)

The lost electromagnetic power is transferred to matter. For an ohmic conductor,

J ⋅ E = σE2 ≥ 0,

(85)

which is the local electromagnetic-to-thermal energy transfer appearing in Poynting’s theorem.

19 Complex permittivity viewpoint

The conduction term can be absorbed algebraically into a complex permittivity:

( ) σ + iω𝜖 = iω 𝜖 − iσ- . ω

(86)

Define

|------------| |𝜖 = 𝜖 − iσ-.| --c-------ω---

(87)

Then

γ2 = − ω2μ 𝜖 . c

(88)

This form is useful because real material polarization can itself introduce frequency-dependent complex permittivity.

20 Dispersion: when phase speed depends on frequency

Real material parameters can depend on frequency:

𝜖 = 𝜖(ω), μ = μ(ω), σ = σ (ω ).

(89)

Then

β = β (ω ),

(90)

and

|--------------| | --ω-- | |vp(ω) = β(ω ).| ----------------

(91)

A pulse contains a range of frequencies. The speed of a narrowband envelope is described by the group velocity

|---------| v = d-ω. | |g dβ | -----------

(92)

In a nondispersive medium, β ∝ ω and v_g = v_p. In a dispersive medium they need not be equal.

21 What remains unchanged from the vacuum plane wave

A uniform plane wave in a homogeneous isotropic medium remains transverse:

E ⊥ ^k, H ⊥ ^k.

(93)

The fields are related by

|--------------| | 1 | |^H = --^k × ^E. | ------η--------

(94)

In a lossless medium, η is real and E and H are in phase. In a lossy medium, η is complex, so they generally have a phase offset.

Polarization from EM19 remains a transverse-vector property. An isotropic medium can change phase velocity and amplitude without selecting a preferred transverse direction. Anisotropic materials can do more and can alter the polarization state itself; that lies beyond the scope of this introductory propagation article.

22 A compact hierarchy of propagation models

|--------------------------------------------------| | vacuum : 𝜖 = 𝜖0, μ = μ0, σ = 0, | | | | lossless dielectric : σ = 0, | | low -loss dielectric : σ ≪ ω 𝜖, | | | | general lossy medium : σ ∼ ω𝜖, | | good conductor : σ ≫ ω 𝜖. | ---------------------------------------------------

(95)

The same Maxwell equations underlie every regime. What changes is the relative importance of displacement current, conduction current, and material response.

23 Connection to RF propagation

The propagation constant

γ = α + iβ

(96)

is the compact bridge from Maxwell’s Equations to engineering propagation calculations. The attenuation constant α determines loss through matter. The phase constant β determines wavelength and phase delay. The intrinsic impedance η determines the electric-to-magnetic field ratio.

The next natural question is what happens when a wave reaches a boundary between two different media. That problem introduces impedance mismatch, reflection, transmission, refraction, and the boundary conditions that connect the fields on the two sides.

24 Summary

For a homogeneous linear isotropic medium,

|------------------------| | 2 ∂E ∂2E | |∇ E = μ σ--- + μ 𝜖--2-.| ------------∂t------∂t---

(97)

For time-harmonic propagation,

|--------------∘---------------| |γ = α + iβ = iωμ (σ + iω 𝜖).| -------------------------------

(98)

The corresponding plane wave is

|-----------------------------------| E (z,t) = E0e− αz cos(ωt − βz + ϕ0).| ------------------------------------

(99)

The principal propagation quantities are

|-----------------------------∘----------| | 2π ω iω μ | |λ = ---, vp = -, η = --------.| ------β------------β------------σ-+--iω-𝜖--

(100)

For a lossless dielectric,

|-------------------∘----------------| | --1-- μ- | |vp = √ μ-𝜖, η = 𝜖, α = 0. | -------------------------------------|

(101)

For a good conductor,

|----------------------------------| | ∘ ωμσ ∘ 2 | |α ≈ β ≈ ----, δ ≈ ----. | --------------2--------------ω-μσ--

(102)

These relations explain how the same electromagnetic wave can propagate nearly without loss through one material, slow down and shorten its wavelength in another, or decay within micrometers of the surface of a good conductor.

References

[1] D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017.

[2] J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.

[3] D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.

[4] F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 7th ed., Pearson, 2015.

[5] C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed., Wiley, 2012.

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Electromagnetic Waves, Antennas, and RF: Wave Propagation in Materials