Plane Waves in Lossy Dielectric

Lecture #9

 Topics to be covered: (i) (i) Wave Wave Prop Propag agat atio ion n in in Los Lossy sy Diel Dielec ectr tric ics s (ii)W (ii)Wav ave e Powe Powerr – Poyn Poynti ting ng Vect Vector or

Reference Reference:: Hayt Hayt & Buck (page (page 404-416) 404-416)

Objectives

1. To study the wave propagation behaviour in lossy  dielectrics in terms of intrinsic impedance, wave velocity, wavenumber (propagation constant) and relation between propagating electric field and magnetic field. 2. To calculate the wave power by utilizing the Poynting  Theorem

Recall: Lossy Dielectrics  The concept has been discussed in the topic of displacement current.  The lossy dielectric  permittivity is complex

ε  = ε ' − j

σ  ω 

= ε r ' ε o −  j

σ  ω 

Where;

ε '  = dielectric permittivity  σ  = dielectric conductivity  ω  = frequency in rad/s

(F  /  m)

(1)

Plane Wave in Lossy Dielectrics Recall: For wave propagation in lossless dielectric; Propagation constant,

k  = ω  µε 

(rad/m)

(2)

However, in lossy dielectric; ε   complex

∴ k   complex  Thus, from (2):

σ  ⎞ ⎛  k  = ω  µ ⎜ ε '− j ⎟ ω  ⎠ ⎝  k  = ω  µε  ' 1 − j

σ  ωε  '

(3)

Plane Wave in Lossy Dielectrics Let:

 jk = α + j

⇒ −k 2 = (α + j β )

2

⇒ k 2 = − α 2 − β 2 ) −  j 2αβ 

(4)

From (3):

k 2 = ω 2 µε  '− j ωµσ 

(5)

From (4) and (5):

α 2 −  β  2 ) = −ω 2 µε ' 2αβ  = ωµσ 

(real)

(6)

(imag)

(7)

Plane Wave in Lossy Dielectrics (6) and (7) are used to determine the  and  β :

α  = ω 

µε  ' ⎡

⎤ ⎢ 1 + 2 2 - 1⎥  Np  / m (8 ) 2 ⎢ ω  ε ' ⎥⎦ ⎣ σ  2

 is an attenuation constant: A measure of wave attenuation while travelling in a medium.

 β  = ω 

µε  ' ⎡

⎤ ⎢ 1 + 2 2 + 1⎥ 2 ⎢ ω  ε ' ⎥⎦ ⎣ σ  2

rad  / m (9 )

 β  is a phase constant. A measure of phase change while travelling in a medium.

Please prove (8) and (9) on your own !!!

Plane Wave in Lossy Dielectrics Recall: From the Lecture #22; the wave propagation equation can be written as: + −  jk o z

+  E  x− e +  jk  z

 E  x ( z ) =  E  x e

o

(10)

Where k o  is a wave propagation constant in free space. In lossy dielectrics, replace jk o  with α  + j β  . (10) becomes: + − (α +  j β  ) z

 E  x ( z ) =  E  x e

+  E  x− e + (α +  jβ  ) z

(11)

Consider a wave propagation in +z direction: + −α  z −  jβ z

 E  x ( z ) =  E  x e In real form:

+ −α z

 E  x ( z ) =  E  x e

e

cos(ω t  − β  z )

(12)

(13)

Plane Wave in Lossy Dielectrics From (13); wave will be attenuated by e - z  when propagate in the lossy dielectrics.

 E  x ( z ) =  E  x+ cos (ω t  − k o z ) (free space)

 E  x ( z ) =  E  x+ cos (ω t  − kz ) (lossless dielectrics)

 x

z

+ −α z

 E  x ( z ) =  E  x e

cos(ω t  − β  z )

(lossy dielectrics)

 x

e - z 

z

Plane Wave in Lossy Dielectrics In lossless dielectric; the intrinsic impedance: η  =

µ 

(Ω)

ε 

In lossy dielectric; the intrinsic impedance:

η  =

µ  σ   ⎞ ⎛  ε ' ⎜ 1 −  j ⎟ ' ωε  ⎝   ⎠

=

µ  ε ' j θ  = η  ∠ θ η  = η  e η  σ   ⎞ ⎛  1 −  j ⎜ ⎟ ' ωε  ⎝   ⎠

(Ω)

Where:

η  =

µ   /  ε  ' 

⎡ ⎛  σ   ⎞ ⎤ ⎟ ⎥ ⎢1 + ⎜ ωε  '  ⎝   ⎠ ⎥⎦ ⎢⎣ 2

1  /  4

tan 2θ η  =

σ  ωε ' 

Plane Wave in Lossy Dielectrics +

From:

η  =

−

 E  x

η  = −

+

 H  y

 E  x

−

 H  y

+

+

⇒  H  y =

−

 E  x

⇒  H  y− = −

η  ∠θ η 

 E  x

η ∠θ η 

 Thus, consider only the +z propagation of the magnetic field wave: +

 H  y ( z ) =

x 

 E  x

η 

e

−α z

(

cos ω t  − β  z − θ η 

E x 

z

 y 

H y 

)

Plane Wave in Lossy Dielectrics Previously, in lossless dielectric; the wave velocity: ν  =

ω 

(m/s)

k 

In lossy dielectric; the wave velocity: ν  =

ω   β 

(m/s)

In conclusion: for wave propagation in lossy dielectrics, two important observations can be made: (i) Both electric and magnetic field waves will be attenuated by e - z  (ii)

E  leading H  by θ η

Example A lossy dielectric has an intrinsic impedance of  200 ∠30 o Ω at the particular frequency. If at that particular frequency a plane wave that propagate in a medium has a magnetic field given by  : -α x

 H  = 10 e

Find  E  and α 

.

cos( ω t  - x/2 ) yˆ  A / m.

Wave Power Calculation From previous lecture, the  E plane wave and H  plane wave were found to be perpendicular to each other. Hence the wave power:

P =  E × H 

(W/m2)

(14)

Equation (14) can also provide the wave propagation direction. Equation (14)



Poynting vector 

(i) For lossless dielectrics:

 E  =  E  x+ cos (ω t  − kz ) xˆ +

 H  =  H  y cos (ω t  − kz ) yˆ =

 E  x+

η 

cos (ω t  − kz ) yˆ

Wave Power Calculation From (14); The wave power:

( E  )

+ 2

P=

x

η 

cos

2

(ω t  − kz ) zˆ

2 W  / m

 To find the time average power density:

Pavg =

1

T 

T  ∫

P dt 

0

⇒ Pavg =

1

T 

T  ∫

( E  )

0

( E  )

+ 2

 x

η 

cos

2

(ω t  − kz ) dt 

+ 2

⇒ Pavg =

 x

2η 

(15)

Wave Power Calculation (ii) For lossy dielectrics:

 E  =  E  x+ e −

α  z

cos (ω t  −  β  z ) xˆ

 H  =  H  y+ cos (ω t  −  β  z − θ η  ) yˆ = + (  E  ) P =

 E  x+

2

 x

η 

e−

2 α z

η 

e −α  z cos (ω t  −  β  z − θ η  ) yˆ

(

cos (ω t  −  β  z ) cos ω t  −  β  z

− θ η  ) zˆ

 The time average power density:

Pavg =

1

T 

T  ∫

P dt 

0

⇒ Pavg =

1

T 

T  ∫ 0

( E  )

+ 2

 x

η 

( E  ) e − + 2

cos

2

(ω t  − kz ) dt  =

 x

2η 

2 α  z

cos θ η  (16)

Example At frequencies of 1, 100 and 3000 MHz, the dielectric constant of ice made from pure water has values of 4.15, 3.45 and 3.20 respectively, while the loss tangent is 0.12, 0.035 and 0.0009, also respectively. If a uniform plane wave with an amplitude of 100 V/m at z=0 is propagating through such ice, find the time average power density at z=0 and z=10 m for each frequency.

Next Lecture

Please have a preliminary reading on the following topic: 1) Propagation of Plane Waves in Good Conductors

Reference: Hayt & Buck (page 416-423)