plane waves in lossy dielectric
TRANSCRIPT
Lecture #9
Topics to be covered:
(i) Wave Propagation in Lossy Dielectrics
(ii)Wave Power – Poynting Vector
Reference: Hayt & Buck (page 404-416)
1. To study the wave propagation behaviour in lossydielectrics 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 PoyntingTheorem
Objectives
Recall: Lossy Dielectrics
The concept has been discussed in the topic of displacement current.
The lossy dielectric permittivity is complex
( )m/Fj'j' or ωσεε
ωσεε −=−= (1)
Where;
ωσε ' = dielectric permittivity
= dielectric conductivity
= frequency in rad/s
Plane Wave in Lossy DielectricsRecall: For wave propagation in lossless dielectric;
Propagation constant,
µεω=k (rad/m) (2)
However, in lossy dielectric;
ε complex
∴ k complex
Thus, from (2):
⎟⎠⎞
⎜⎝⎛ −=
ωσεµω jk '
'1'
ωεσµεω jk −= (3)
Plane Wave in Lossy Dielectrics
Let:βα jjk +=
( )22 βα jk +=−⇒
( ) αββα 2222 jk −−−=⇒ (4)
From (3):ωµσµεω jk −= '22 (5)
From (4) and (5):
( ) '222 µεωβα −=− (real)
ωµσαβ =2 (imag)
(6)
(7)
Plane Wave in Lossy Dielectrics
(6) and (7) are used to determine the and β:
)8( / 1-'
12
'22
2
mNp⎥⎥⎦
⎤
⎢⎢⎣
⎡+=
εωσµεωα
)9( /1'
12
'22
2
mrad⎥⎥⎦
⎤
⎢⎢⎣
⎡++=
εωσµεωβ
Please prove (8) and (9) on your own !!!
is an attenuation constant: A measure of wave attenuation while travelling in a medium.
β is a phase constant. A measure of phase change while travelling in a medium.
Recall: From the Lecture #22; the wave propagation equation can be written as:
Plane Wave in Lossy Dielectrics
( ) zjkx
zjkxx
oo eEeEzE +−−+ +=
Where ko is a wave propagation constant in free space.
In lossy dielectrics, replace jko with . (10) becomes: βα j+
(10)
( ) ( ) ( )zjx
zjxx eEeEzE βαβα ++−+−+ +=
Consider a wave propagation in +z direction:
( ) zjzxx eeEzE βα −−+=
(11)
(12)
In real form:( ) ( )ztcoseEzE z
xx βωα −= −+ (13)
Plane Wave in Lossy Dielectrics
From (13); wave will be attenuated by e-z when propagate in the lossy dielectrics.
( ) ( )ztcoseEzE zxx βωα −= −+
( ) ( )zktcosEzE oxx −= + ω ( ) ( )kztcosEzE xx −= + ω(free space) (lossless dielectrics)
(lossy dielectrics)
z
x
z
x
e-z
Plane Wave in Lossy Dielectrics
In lossless dielectric; the intrinsic impedance:
εµη = (Ω)
In lossy dielectric; the intrinsic impedance:
ηθη ηθη
ωεσεµ
ωεσε
µη jejj
=∠=⎟⎠⎞
⎜⎝⎛ −
=⎟⎠⎞
⎜⎝⎛ −
=
'1
'
'1'
(Ω)
Where:
412
1/
'
'/
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
=
ωεσ
εµη
'tan
ωεσθη =2
From:
ηθη
η
∠=⇒
=
++
+
+
xy
y
x
EH
HE
Plane Wave in Lossy Dielectrics
Thus, consider only the +z propagation of the magnetic field wave:
( ) ( )ηα θβω
η−−= −
+
zteEzH zxy cos
ηθη
η
∠−=⇒
−=
−−
−
−
xy
y
x
EH
HE
Ex
Hy
x
z
y
Plane Wave in Lossy Dielectrics
Previously, in lossless dielectric; the wave velocity:
kων = (m/s)
In lossy dielectric; the wave velocity:
βων = (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 θη
Example
A lossy dielectric has an intrinsic impedance of at the particular frequency. If at that particular frequency a plane wave that propagate in a medium has a magnetic field given by :
Ωo30∠200
./ˆ)x/2-cos(10 - mAyteH x ωα=
Find and .E α
Wave Power Calculation
From previous lecture, the plane wave and plane wave were found to be perpendicular to each other.
E H
Hence the wave power:
HEP ×= (W/m2) (14)
Equation (14) can also provide the wave propagation direction.
Equation (14) Poynting vector
(i) For lossless dielectrics:
( )
( ) ( )ykztEykztHH
xkztEE
xy
x
ˆcosˆcos
ˆcos
−=−=
−=+
+
+
ωη
ω
ω
Wave Power Calculation
From (14); The wave power:
( ) ( ) 222
/ˆcos mWzkztEP x −=+
ωη
To find the time average power density:
( ) ( )
( )η
ωη
2
cos1
1
2
2
0
20
+
+
=⇒
−=⇒
=
∫
∫
xavg
Tx
avg
T
avg
EP
dtkztET
P
dtPT
P
(15)
Wave Power Calculation
(ii) For lossy dielectrics:
( )
( ) ( )yzteEyztHH
xzteEE
zxy
zx
ˆcosˆcos
ˆcos
ηα
η
α
θβωη
θβω
βω
−−=−−=
−=
−+
+
−+
( ) ( ) ( ) zztzteEP zx ˆcoscos22
ηα θβωβω
η−−−= −
+
( ) ( ) ( )η
α
θη
ωη
cos2
cos1
1
222
0
20
zx
Tx
avg
T
avg
eEdtkztET
P
dtPT
P
−++
=−=⇒
=
∫
∫
The time average power density:
(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)