1 Microwave Physics and Techniques UCSB –June 2003 •Maxwell’s equations •Grad, div and curl •Wave equations •Plane Waves •Boundary conditions Lecture 2 A. Nassiri
1Microwave Physics and Techniques UCSB –June 2003
•Maxwell’s equations•Grad, div and curl•Wave equations•Plane Waves•Boundary conditions
Lecture 2
A. Nassiri
2Microwave Physics and Techniques UCSB –June 2003
Maxwell’s EquationsMaxwell’s Equations
The general form of the time-varying Maxwell’s equations can be written in differential form as:
0=⋅∇=⋅∇
∂∂
−=×∇
∂∂
+=×∇
BD
BE
DJH
ρt
t
3Microwave Physics and Techniques UCSB –June 2003
What is Maxwell’s theory?What is Maxwell’s theory?
“I cannot give any clearer or briefer answer than the following:Maxwell’s theory is the system of Maxwell’s equations”
-Heinrich Rudolph Hertz(1857-94)
4Microwave Physics and Techniques UCSB –June 2003
Physical interpretation of curlPhysical interpretation of curl
River
velocity
position
Centre of river flows at maximum speed.Now put the “curl meter”into the river and seewhat happens!
Rotating paddle wheelor “curl-meter”
5Microwave Physics and Techniques UCSB –June 2003
No rotation!
River RiverRiver
Three different positions Three different positions (centre left and right)(centre left and right)
Anti-clockwiserotation.
Clockwiserotation.
6Microwave Physics and Techniques UCSB –June 2003
ConclusionsConclusions
• Curl-meter only indicates rotation if there is non-uniformity inthe vector field.
• The amount of rotation is proportional to the degree of non-uniformity.
• The rotation cannot be described just with a scalar. Directionshould also be given. Rotation is therefore a vector quantity.
Curl therefore describes the variation across the field. The abovephysical interpretation is concerned with only one dimension orone component. For electromagnetic fields we must imaginethe concept applying to three dimensions.
7Microwave Physics and Techniques UCSB –June 2003
A few other fundamental relationshipsA few other fundamental relationships
H/m104 F/m,108548with
ity)(permeabil and ity)(permittiv here
ips"relationsh veconstituti"
equation" continuity"
law" sOhm'"
70
120
00
−− ×=×=
==
==
∂∂
−=⋅∇
=
πµε
µµµεεεµε
ρσ
.
HBED
J
EJ
rr
t
Note this well!!!
8Microwave Physics and Techniques UCSB –June 2003
Physical interpretation of divergencePhysical interpretation of divergence
Flux in = flux outso no sources orsinks inside V.
Flux out > flux inPositivedivergence.Must be a sourceinside V.
Flux out < flux inNegativedivergence.Must be a sink ordrain inside V.
V VV
9Microwave Physics and Techniques UCSB –June 2003
ExampleExample
tJ V
∂∂
−=⋅∇ρr
The current density fluxflowing out of the closed surfaceequals the rate of decrease ofthe positive charge density.
10Microwave Physics and Techniques UCSB –June 2003
Maxwell’s EquationsMaxwell’s Equations– All (macroscopic) electromagnetic phenomena can be
explained in terms of Maxwell’s equations, the continuity equation and the Lorentz force equation.
–We also note that Maxwell’s equations are not independent. In fact, the two Divergence equations can be derived from the two Curl equations.
– We have only listed the differential form of Maxwell’s equations. There are corresponding integral forms of these equations as well.
Microwave Physics and Techniques UCSB –June 2003
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18Microwave Physics and Techniques UCSB –June 2003
Integral form of the equationsIntegral form of the equations ρ=⋅∇ D
r∫ =⋅⇒S
QsdD rr
tBE
∂∂
−=×∇r
r∫∫ ⋅
∂∂
−=⋅⇒SC
sdtBdE rr
lrr
0=⋅∇ Br
∫ =⋅⇒S
sdB 0rr
tDJH
∂∂
+=×∇r
rr∫∫ ⋅
∂∂
+=⋅⇒SC
sdtDJdH rr
rlrr
19Microwave Physics and Techniques UCSB –June 2003
Wave EquationsWave Equations
In any problem with unknown E, D, B, H we have 12 unknowns. To solve for these we need 12 scalar equations. Maxwell’s equations provide 3 each for the two curl equations. and 3 each for both constitutive relations (difficult task).Instead we anticipate that electromagnetic fields propagate as waves. Thus if we can find a wave equation, we could solve it to find out the fields directly.
20Microwave Physics and Techniques UCSB –June 2003
Maxwell’s equations not Maxwell’s equations not independent!independent!
E.g.
[ ]B0E
BE
⋅∇∂∂
−==×∇⋅∇
∂∂
−=×∇
t
t
[ ]
tt
t
∂∂
+∂∂
−=
⋅∇∂∂
+⋅∇=×∇⋅∇
ρρ0
DJH
0=⋅∇ BThis implies
This implies
Use the identity0≡×∇⋅∇ A
r
21Microwave Physics and Techniques UCSB –June 2003
Wave equationsWave equations
Take the curl of the first Maxwell:
( )
( )
2
2
t
tt
t
t
∂∂
−×∇=
∂∂
−∂∂
+×∇=
×∇∂∂
+×∇=
∂∂
×∇+×∇=×∇×∇
HJ
HJ
EJ
EJH
µε
µε
ε
ε
( ) LHS on the use Now 2 HHH ∇−⋅∇∇≡×∇×∇
0
22Microwave Physics and Techniques UCSB –June 2003
Wave EquationsWave Equations
The result is:
JHHr
rr
×−∇=∂
∂−∇ 2
22
tµε
Similarly, the same process for the second Maxwell produces
ερµµε ∇+
∂∂
=∂∂
−∇ttJEErr
r2
22
Note how in both case we have a wave equation (2nd order PDE) for both E and H with fields to the left of the = sign and sourcesto the right. These two wave equations are completely equivalentto the Maxwell equations.
23Microwave Physics and Techniques UCSB –June 2003
Solutions to the wave equationsSolutions to the wave equations
Consider a region of free space (σ = 0) where there are no sources(J = 0). The wave equations become homogeneous:
02
22 =
∂∂
−∇tEE µε
02
22 =
∂∂
−∇tHH µε
Actually there are 6 equations; we will only consider one component:e.g. Ex(z,t)
2
00
22
2
22
2 1 e wher01 cvtE
vzE xx ===
∂∂
−∂
∂εµ
24Microwave Physics and Techniques UCSB –June 2003
Solutions to the wave equationSolutions to the wave equation
Try a solution of the form f(z-vt) e.g. sin[β(z-vt)]. By differentiatingtwice and substituting back into the scalar wave equation, we find that it satisfies!
f(z) t=0z
f(z-vt1) t=t1 z
f(z-vt2) t=t2 z
25Microwave Physics and Techniques UCSB –June 2003
Plane WavesPlane Waves• First treat plane waves in free space.• Then interaction of plane waves with media.• We assume time harmonic case, and source free situation.
020
2 =+∇ EkErv
We require solutions for E and H (which are solutions to thefollowing PDE) in free space
No potentials here!(no sources)
Note that this is actually three equations:
zyxiEkzE
yE
xE
iiii ,,==+
∂∂
+∂∂
+∂∂ 02
02
2
2
2
2
2
26Microwave Physics and Techniques UCSB –June 2003
How do we find a solution?How do we find a solution?Usual procedure is to use Separation of Variables (SOV). Take one component for example Ex.
( ) ( ) ( )
0
0
20
20
=+′′
+′′
+′′
=+′′+′′+′′
=
khh
gg
ff
fghkhfggfhfgh
zhygxfEx
Functions of a single variable ⇒sum = constant = -k0
2
ckkkkk
khhk
ggk
ff
zyx
zyx
ωλπ
===++
−=′′
−=′′
−=′′
2 with so and
020
222
222 ;;
27Microwave Physics and Techniques UCSB –June 2003
Mathematical SolutionMathematical Solution
We note we have 3 ODEs now.
zjkz
yjky
xjkx
z
y
x
ehhkdz
hd
egkdy
gd
effkdx
fd
±
±
±
==+
==+
==+
issolution 0
g issolution 0
issolution 0
22
2
22
2
22
2
( )zkykxkjx
zyxeAE ++±=
28Microwave Physics and Techniques UCSB –June 2003
But, what does it mean physically?But, what does it mean physically?
( )zkykxkjx
zyxeAE ++±=
This represents the x-component of the travelling wave E-field(like on a transmission line) which is travelling in the direction of the propagation vector, with Amplitude A. The direction ofpropagation is given by
zkykxkk zyx ˆˆˆ ++=r
29Microwave Physics and Techniques UCSB –June 2003
Physical interpretationPhysical interpretation
The solution represents a wave travelling in the +z direction withvelocity c. Similarly, f(z+vt) is a solution as well. This latter solution represents a wave travelling in the -z direction.So generally,
( ) ( )( )( )[ ]vtzvtyvtxftzEx ±±±=,
In practice, we solve for either E or H and then obtain theother field using the appropriate curl equation.
What about when sources are present? Looks difficult!
30Microwave Physics and Techniques UCSB –June 2003
Generalize for all componentsGeneralize for all components
If we define the normal 3D position vector as:
rkjz
rkjy
rkjx
zyx
CeE
BeE
AeE
zkykxkrk
zzyyxxr
vv
vv
vv
rr
r
⋅−
⋅−
⋅−
=
=
=
++=⋅
++=
similarly
so
then
ˆˆˆ
zCyBxAEeEE rkj ˆˆˆ ++== ⋅−00 wherevrr vr
General expressionfor a plane wave
+sign droppedhere
31Microwave Physics and Techniques UCSB –June 2003
Properties of plane wavesProperties of plane waves
For source free propagation we must have ∇·E = 0. If we satisfythis requirement we must have k·E0= 0. This means that E0is perpendicular to k.
The corresponding expression for H can be found by substitution of the solution for E into the ∇×E equation. The result is:
EnkHrr
×= ˆ0
0
ωµWhere n is a unit vector in the k direction.
32Microwave Physics and Techniques UCSB –June 2003
Transverse Electromagnetic (TEM) Transverse Electromagnetic (TEM) wavewave
Note that H is also perpendicular to k and also perpendicular toE. This can be established from the expression for H.
E
H
Direction of propagationk,n
Note that:
knHE ˆˆˆˆ or =×E and H lie on theplane of constantphase (k·r = const)
33Microwave Physics and Techniques UCSB –June 2003
Plane waves at interfaces (normal Plane waves at interfaces (normal incidence)incidence)
Consider a linearly polarized (in x-direction) wave travelling inthe +z direction with magnitude Ei
Ei
Hi
Er
Hr
Incident
Reflected
µ2ε2σ2µ1ε1σ1
Transmitted
z
x Arbitraryorientation!
Et
Ht
34Microwave Physics and Techniques UCSB –June 2003
Boundary conditionsBoundary conditions
We deal with a general dielectric interface and two specialcases. First the general case. For convenience we consider the boundary to be planar.
Maxwell’s equations in differential form require known boundaryvalues in order to have a complete and unique solution. The so called boundary conditions (B/C) can be derived by consideringthe integral form of Maxwell’s equations.
nε1µ1σ1
ε2µ2σ2
35Microwave Physics and Techniques UCSB –June 2003
General caseGeneral case
Equivalent
Et1 nε1µ1σ1
ε2µ2σ2 Et2
Tangential E continuous
nε1µ1σ1
ε2µ2σ2 Ht2
Ht1
n x (H1-H2)=Js
nε1µ1σ1
ε2µ2σ2
Bn1
Bn2Normal B continuous
nε1µ1σ1
ε2µ2σ2D2n
D1n
n·(D1-D2)=ρs
36Microwave Physics and Techniques UCSB –June 2003
Special case (a) Lossless dielectricSpecial case (a) Lossless dielectric
nε1µ1σ1=0
ε2µ2σ2=0
Bn1=Bn2 normal B fields continuous
Ht1=Ht2 tangential H fields continuous (no current)
Dn1=Dn2 normal D fields continuous (no charge)
Et1=Et2 tangential E fields continuous)
37Microwave Physics and Techniques UCSB –June 2003
Special case (b) Perfect ConductorSpecial case (b) Perfect Conductornε1µ1σ1=0
∞→2σ Perfect Electric Conductor Et2=Ht2=0
Bn1= 0 Normal B(H) field is zero on conductor.
Et1= 0 Tangential Electric field on conductor is zero.
n × H1=Js H field is discontinuous by the surface current
n . D1= ρ Normal D(E) field is discontinuous by surface charge
38Microwave Physics and Techniques UCSB –June 2003
Boundary conditionsBoundary conditions
Continuity at the boundary for the tangential fields requires:
(2) (1)
tri
tri
HHHEEE
=+
=+ Fix signs whendefining impedance!
211 :define Now ZHEZ
HEZ
HE
t
t
r
r
i
i =−==
Substituting into (1) and (2) and eliminating Er gives
21
22t coefficienon TransmissiZZ
ZEE
i
t
+==τ
39Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaRecall the Maxwell’s equations:
vDB
JDjBBjE
ρ=⋅∇
=⋅∇
+ω=×∇
ω−=×∇
rr
rr
rrrr
rrr
0
( ) ( )
( )
( ) BjEBEj
tBez,y,xE
tBE
ez,y,xEt;x,y,xE
tj
tj
rrrrrr
rr
rr
rr
ω−=×∇⇒=×∇ω
∂∂
−=×∇
∂∂
−=×∇
=
∫∫ ω
ω
1
40Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive Media• So far, for lossless media, we considered J=0, and ρv=0 but, there are actually two types of current and one of them should not be ignored. • Total current is a sum of the Source current and Conduction current.
oo
c
oc
JEjjJEEjH
JDjH
EJ
JJJ
rrrrrrr
rvrr
rr
rrv
+
ωσ
−εω=+σ+ωε=×∇
+ω=×∇
σ=
+=
set
41Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaDefining complex permittivity
ωσ
−ε=ε j
Maxwell’s equations in a conducting media (source free) can be written as
00
=⋅∇
=⋅∇
ωε=×∇
ωµ−=×∇
EH
EjHHjE
rr
rr
rrr
rrr
42Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaWe have considered so far:
Plane Wavesin Free space
Plane Wavesin Isotropic Dielectric
Plane Wavesin anisotropic Dielectric
Plane Wavesin Dissipative Media
00
0
0
=⋅∇
=⋅∇
ωε=×∇
ωµ−=×∇
EH
EjH
HjE
rr
rr
rrr
rrr
00
0
=⋅∇
=⋅∇
ωε=×∇
ωµ−=×∇
HE
EjHHjE
vr
rr
rrr
rrr
00
=⋅∇
=⋅∇
ωε=×∇
ωµ−=×∇
BD
EjHHjE
vr
rr
rrr
rrr
00
=⋅∇
=⋅∇
ωε=×∇
ωµ−=×∇
HE
EjHHjE
rr
rr
rrr
rrr
43Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaWave equation for dissipative media becomes:
( )( ) ( )
HH
EE
EjjEE
HjE
rrr
vrr
vvrrrr
rrrrr
µεω−=∇
µεω−=∇
ωεωµ−=∇−⋅∇∇
×∇ωµ−=×∇×∇
22
22
2
The set of plane-wave solutions are:
zj
zj
eEyH
eExE
κ−
κ−
η
=
=
0
0r
r
44Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaSubstituting into and
yields the dispersion relationEEvrr
µεω−=∇ 22 HHrrr
µεω−=∇ 22
εµ
=η
µεω=κand
22
Is the complex intrinsic impedance of the isotropic media.
45Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaDenoting the complex values:
( )
( ) ( ) φ−κ−κ−κ−κ−
κ−κ−κ−κ−κ−
φ
η=
η
=
====
η=η
κ−κ=κ
jzjjzjj
xzzjzjjzj
jIR
eeEyeEyH
ExeeExeExeExE
e
j
IRIR
IRIR
00
000
r
rthen,
46Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive Media
47Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaLoss tangent is defined from
tangent loss as defined is
ωεσ
ωεσ
−µεω=
ωσ
−εµω=εµω=κ−κ=κ
j
jj IR
1
ε′ε ′′
=δ
ε ′′−ε′=
ωεσ
−ε=ωσ
−ε=ε
tan
jjj 1
48Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaSlightly lossy case: 1<<
ωεσ
εµσ
=ωεσ
µεω=κ
µεω=κ
ωεσ
−µεω=
ωεσ
−µεω=κ
22
211
I
R
jj
µε
σ=
2pd
49Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive MediaHighly lossy case: 1>>
ωεσ
ωεσ
−µεω=
ωεσ
−µεω=κ2
1 jj
( )j−σ
ωµ= 12
δ≈ωµσ
= 2pd Skin depth
50Microwave Physics and Techniques UCSB –June 2003
Plane Wave in Dispersive MediaPlane Wave in Dispersive Media
51Microwave Physics and Techniques UCSB –June 2003
Reflection & TransmissionReflection & Transmission
Similarly, substituting into (1) and (2) and eliminating Et
12
12t coefficien ReflectionZZZZ
EEρ
i
r
+−
==
We note that τ = 1+ρ, and that the values of the reflection and transmission are the same as occur in a transmission linediscontinuity.
Z1 Z2
τρ
Not 1-ρ
52Microwave Physics and Techniques UCSB –June 2003
Special case (1)Special case (1)
(1) Medium 1: air; Medium 2: conductor
iitt
t
iit
m
HEZ
HZEH
EZZEE
jZZZ
22 usethen
2 So
1 377
12
1
2
21
≈=⇒=
≈=
+==>>Ω=
τ
σδ
This says that the transmitted magnetic field is almost doubledat the boundary before it decays according to the skin depth. On the reflection side Hi ≈ Hr implying that almost all theH-field is reflected forming a standing wave.
53Microwave Physics and Techniques UCSB –June 2003
Special case (2)Special case (2)
(2) Medium 1: conductor; Medium 2: air
Reversing the situation, now where the wave is incidentfrom the conducting side, we can show that the wave isalmost totally reflected within the conductor, but that the standing wave is attenuated due to the conductivity.
54Microwave Physics and Techniques UCSB –June 2003
Special Case (3)Special Case (3)
(2) Medium1: dielectric; Medium2: dielectric
2
02
1
01
εµ
εµ
== ZZ ,1
1
2
1
2
1
+
−=⇒
εεεε
ρ
This result says that the reflection can be controlled by varyingthe ratio of the dielectric constants. The transmission analogycan thus be used for a quarter-wave matching device.
55Microwave Physics and Techniques UCSB –June 2003
λλ/4 Matching Plate/4 Matching Plate
Z0 Zp Z2
λ/4
Air: εr=1 Plate εr'=? Dielectric εr=4
Transmission line theory tells us that for a match
20ZZZ p =
2 and 266 So
1882
7376 7376
2
0
020
==Ω=
Ω===Ω=
ZZZ
ZZZ
rp
r
'
.,.
ε
ε
We will see TL lectures later
56Microwave Physics and Techniques UCSB –June 2003
Plane of IncidencePlane of Incidence
Surfacenormal
Dielectric interface in x-z plane
x
y
z
Plane of incidence containsboth direction of propagation vector and normal vector.
Direction ofpropagation
57Microwave Physics and Techniques UCSB –June 2003
ApplicationsApplications
The principle of λ/4 matching is not only confined to transmissionline problems! In fact, the same principle is used to eliminate reflections in many optical devices using a λ/4 coating layer onlenses & prisms to improve light transmission efficiency.
Similarly, a half-wave section can be used as a dielectric window.Ie. Full transparency. In this case Z2=Z0 and thematching section is λ/2. Such devices are used to protect antennasfrom weather, ice snow, etc and are called radomes.
Note that both applications are frequency sensitive and that thematching section is only λ/4 or λ/2 at one frequency.
58Microwave Physics and Techniques UCSB –June 2003
Oblique IncidenceOblique Incidence
The transmission line analogy only works for normal incidence.When we have oblique incidence of plane waves on a dielectric interface the reflection and transmission characteristics becomepolarization and angle of incidence dependent.
We need to distinguish between the two different polarizations.We do this by first, explaining what a plane of incidence is, then we will point out the distinguishing features of each polarization. We are aiming for expressions for reflection coefficients.
We note again that we are only dealing with plane waves
59Microwave Physics and Techniques UCSB –June 2003
Parallel & Perpendicular IncidenceParallel & Perpendicular Incidence
HE
x
EH
x
y y
Plane of incidence is the x-y plane
E is Perpendicular to theplane of incidence
E is Parallel to theplane of incidence
60Microwave Physics and Techniques UCSB –June 2003
Perpendicular incidencePerpendicular incidence
y
Hi EiEr
Hr
x
θr θi
θtHtEt
ε2µ2
ε1µ1
61Microwave Physics and Techniques UCSB –June 2003
Write math expression for fields!Write math expression for fields!
( )[ ]
( ) ( )[ ]iiiii
iii
yxjZEyxH
yxjEzE
θθβθθ
θθβ
cossinexpsinˆcosˆ
cossinexpˆ
11
0
10
++−=
+=
( )[ ]
( ) ( )[ ]rrrrr
rrr
yxjZEyxH
yxjEzE
θθβρθθ
θθβρ
cossinexpsinˆcosˆ
cossinexpˆ
11
0
10
−+=
−=
⊥
⊥
( )[ ]
( ) ( )[ ]ttttt
ttt
yxjZEyxH
yxjEzE
θθβτθθ
θθβτ
cossinexpsinˆcosˆ
cossinexpˆ
22
0
20
++−=
+=
⊥
⊥
62Microwave Physics and Techniques UCSB –June 2003
Say What!!! How did you get that?Say What!!! How did you get that?
Within the exponential: This tells the direction of propagationOf the wave. E.g. for both the incident Ei and Hi
( )ii yxj θθβ cossin1 +A component in the – x direction
Another component in the –y direction
PropagatingIn medium 1
Outside the exponential tells what vector components of the fieldAre present. E.g. for Hr
( )1
0sinˆcosˆZ
Eyx rr⊥+
ρθθ+x and +y components of Hr
Perpendicular reflectioncoefficient
E0/Z1 converts E to H
63Microwave Physics and Techniques UCSB –June 2003
Apply boundary conditionsApply boundary conditions
Tangential E fields (Ez) matches at y=0Tangential H fields (Hx) matches at y=0
( ) ( ) ( )tri xjxjxj θβτθβρθβ sinexpsinexpsinexp 211 ⊥⊥ =+
We know that τ =1+ ρ, so then the arguments of the exponents must be equal. Sometimes called Phase matchingin optical context. It is the same as applying the boundaryconditions.
tri jjj θβθβθβ sinsinsin 211 ==
64Microwave Physics and Techniques UCSB –June 2003
Snell’s laws and Fresnel coefficientsSnell’s laws and Fresnel coefficients
The first equation gives
and from the second using it θεµεµθ
λπβ sinsin
22
11 2==
ir θθ =
By matching the Hx components and utilizing Snell, we can obtain the Fresnel reflection coefficient for perpendicular incidence.
ti
ti
ZZZZ
θθθθρ
coscoscoscos
12
12
+−
=⊥
65Microwave Physics and Techniques UCSB –June 2003
Alternative form Alternative form
Alternatively, we can use Snell to remove the θt and write it interms of the incidence angle, at the same time assuming non-magnetic media (µ= µ0 for both media).
ii
ii
θεεθ
θεεθ
ρ2
1
2
2
1
2
sincos
sincos
−+
−−=⊥
Note how both formsreduce to the transmissionline form when θi=0
This latter form is the one that is most often quoted in texts,the previous version is more general
66Microwave Physics and Techniques UCSB –June 2003
Some interesting observationsSome interesting observations
• If ε2 > ε1 Then the square root is positive,• If ε1> ε2 i.e. the wave is incident from more dense to
less denseAND
⊥ρ Is real
1
22
εεθ ≥isin
Then is complex and ⊥ρ 1=⊥ρ
This implies that the incident wave is totallyinternally reflected (TIR) into the more dense medium
67Microwave Physics and Techniques UCSB –June 2003
Critical angle Critical angle
When the equality is satisfied we have the so-called criticalangle. In other words, if the incident angle is greater than or equal to the critical angle AND the incidence is from more dense to less dense, we have TIR.
1
21
εεθ −= sinic
For θi> θic Then as noted previously.1=⊥ρ
68Microwave Physics and Techniques UCSB –June 2003
Strange resultsStrange results
Now
1A where
imaginary! is 1
! 1 since so
2
2
1
2
212
1
−=
=−=
>⇒>=
i
ttt
tit
jA
θεε
θθθ
θεεθεεθ
sin
cossincos
sinsinsin
What is the physical interpretation of these results? To seewhat is happening we go back to the expression for thetransmitted field and substitute the above results.
69Microwave Physics and Techniques UCSB –June 2003
Transmitted field Transmitted field
( )[ ]
[ ] [ ]
1 where 2
2
1222
20
20
−==
−=
+=
⊥
⊥
i
t
ttt
Aα
yxjEz
yxjEzE
θεεεµωβ
αθβτ
θθβτ
sin
expsinexpˆ
cossinexpˆpreviously
cos θt=jA
Physically, it is apparent that the transmitted field propagatesalong the surface (-x direction) but attenuates in the +y directionThis type of wave is a surface wave field
70Microwave Physics and Techniques UCSB –June 2003
ExampleExample
Assume:εr = 81σ = 0µr = 1
Let θi = 45°evaluate TIRicic ⇒>== °− θθθ i
1 so 386811 .sin
y
airwater
Ei
Hi
21 x
71Microwave Physics and Techniques UCSB –June 2003
Example (Example (ctdctd))
°−∠=
−+
−−+=
+=
===
+=−°±=
==
⊥
°
6.4442.1
5.0811707.0
5.0811707.0
1
1
/5.3928.6228.6145sin81cos
38.645sin181sin Snell Using
002
2
ρτλλ
πβα
θ
θ
mNepA
jjt
t
This means that ifthe field strength onthe surface is1Vm-1, then
-1Vm421.== it EE τ
Choose + signto allow forattenuationin +y direction
72Microwave Physics and Techniques UCSB –June 2003
Evaluate the field just above the Evaluate the field just above the surfacesurface
Lets evaluate the transmitted E field at λ/4 above the surface.
dB
VmEt
8854211027320
2734
4939421
6
10
0
..
.log
..exp.
−=
×=
=
−=
−
−µλλ
This means that the surface wave is very tightly bound to thesurface and the power flow in the direction normal to the surface is zero.
73Microwave Physics and Techniques UCSB –June 2003
What about the factorWhat about the factor ??0
0
kωµ
0
0
00000
0 122µε
µωµπ
ωµλπ
ωµ====
ccfk
This term has the dimensions of admittance, in fact
0
0
000
11µε
η===
ZY
Ω≈= 377space free of impedance Zwhere 0
EnHrr
×= ˆ0
1η
And now
74Microwave Physics and Techniques UCSB –June 2003
Propagation in conducting mediaPropagation in conducting media
We have considered propagation in free space (perfect dielectricwith σ = 0). Now consider propagation in conducting media whereσ can vary from a finite value to ∞.
ερµµε ∇+
∂∂
=∂∂
−∇ttJEE 2
22Start with
Assuming no free charge and the time harmonic form, gives
22
22
22
where0
µεωωµσγ
γ
ωµσµεω
−=
=−∇
=+∇
jEE
EjEErr
rrr
Complex propagationcoefficient due tofinite conductivity
75Microwave Physics and Techniques UCSB –June 2003
Conduction current and Conduction current and displacement currentdisplacement current
In metals, the conduction current (σE) is much larger than the displacement current (jωε0E). Only as frequencies increase tothe optical region do the two become comparable.
E.g. σ = 5.8x107 for copperωε0 = 2πx1010x 8.854x10-12 = 0.556
So retain only the jωµσ term when considering highly conductivematerial at frequencies below light. The PDE becomes:
002 =−∇ EjE
rrσωµ
76Microwave Physics and Techniques UCSB –June 2003
Plane wave incident on a conductorPlane wave incident on a conductor
Consider a plane wave entering a conductive medium at normalincidence.
zx
Ex
Hy
Free space Conducting medium
Some transmittedMostly reflected
77Microwave Physics and Techniques UCSB –June 2003
Mathematical solutionMathematical solution
The equation for this is:002
2
=−∂
∂x
x EjzE σωµ
The solution is: zjx eEE σωµ0
0−=
We can simplify the exponent: ( )2
1 00
σωµσωµγ jj +==
So now γ has equal realand imaginary parts.
2 with 0
0σωµβαβα === −− zz
x eeEE
Alternatively write δδjzz
x eeEE −−= 0
78Microwave Physics and Techniques UCSB –June 2003
Skin DepthSkin Depth
The last equation
gives us the notion of skin depth:βασωµ
δ 112
0
===
On the surface at z=0 we have Ex=E0at one skin depth z=δ we have Ex=E0/e
δδjzz
x eeEE −−= 0
field has decayed to 1/eor 36.8% of value on thesurface.
79Microwave Physics and Techniques UCSB –June 2003
Plot of field into conductorPlot of field into conductor
δ 2δ …….
z
E0
E0/e
80Microwave Physics and Techniques UCSB –June 2003
Examples of skin depthExamples of skin depth
f
2
0
106162 −×==
.σωµ
δ
at 60Hz δ=8.5x10-3 mat 1MHz δ=6.6x10-5 mat 30GHz δ=3.8x10-7 m
σ = 5.8x107 S/mCopper
Seawaterf
210522 ×=
.δ
at 1 kHz δ=7.96m
σ = 4 S/m
Submarine comms!
81Microwave Physics and Techniques UCSB –June 2003
Submarine communicationSubmarine communication
82Microwave Physics and Techniques UCSB –June 2003
Characteristic or Intrinsic Impedance Characteristic or Intrinsic Impedance ZZmm
Define this via the materialas before:
ωσε
µεµ
jZ
cm
−== 00
But again, the conduction current predominates, which means the second term in the denominator is large. With thisapproximation we can arrive at:
( )σδσ
ωµ jjZm+
=+=1
21 0
For copper at 10GHz Zm= 0.026(1+j) Ω
83Microwave Physics and Techniques UCSB –June 2003
Reflection from a metal surfaceReflection from a metal surface
So a reflection coefficient at metal-air interface is
00
0 since 1 ZZZZZZ
mm
m <<−≈+−
=ρ
We also note that as σ→ ∞, Zm→ 0 and that ρ= -1 for the caseof the perfect conductor. Thus the boundary condition for a PECis satisfied in the limit.
The transmission coefficient into the metal is given by τ = 1+ρ
84Microwave Physics and Techniques UCSB –June 2003
Conductors and dielectricsConductors and dielectrics
Materials can behave as either a dielectric or a conductor depending on the frequency.
EjEH ωεσ +=×∇ recall
Conduction current density
Displacement current density
3 choicesωε >> σ displacement current >> conductor current ⇒ dielectricωε ≈ σ displacement current ≈ conductor current ⇒ quasi conductorωε << σ displacement current << conductor current ⇒ conductor
85Microwave Physics and Techniques UCSB –June 2003
A rule for determining whether dielectric or A rule for determining whether dielectric or conductorconductor
ωεσωεσ
ωεσ
<
<<
<
100 Conductors
100100
1 Conductors Quasi
1001 sDielectric
M
21
0-1-2
8 9 10 11
dielectric
quasi conductor
conductor
ground seawater
copper
N Freq=10N
ωεσ
= 10M
86Microwave Physics and Techniques UCSB –June 2003
General case: (both conduction & General case: (both conduction & displacement currents)displacement currents)
+−=−=
ωεσµεωµεωωµσγj
j 1 222
From 4.4, we now retain both terms:
If we now let γ = α+jβ, square it and equate real and imaginaryparts and then solve simultaneously for α and β. We obtain:
rad/m 1121
Np/m 1121
21
2
21
2
+
+=
−
+=
ωεσµεωβ
ωεσµεωα
87Microwave Physics and Techniques UCSB –June 2003
ApproximationsApproximations
By taking a binomial expansion of the term under the radicaland simplifying, we can obtain:
( )jZw +12
2
2
2
σωµ
εµ
ωµσµεωβ
ωµσεµσα
Good conductorGood dielectric
88Microwave Physics and Techniques UCSB –June 2003
Example Problem 1:
An FM radio broadcats signal traveling in the y-dirrection in air has a magnetic field given by the phasor
( ) ( )
( ) ingcorrespond the Find (b)m). (in h wavelengtand MHZ) (infrequency theetermine (a)
.
ˆˆ. .
yE D
mAjzxeyH yj 1680310922 −π−− −+−×=
( ) ( ) 1680
1
11
1022
680
−π−
−
−−−≈⇒
ωε=∂
∂−
∂∂
=×∇
≈π
ω=
−π=εµω=β
mVzjxeyE
Ejy
Hzy
HxH
MHzf
mrad
yj
oxz
oo
ˆˆ.
ˆˆ
.
.
whichfrom
have we(a)
89Microwave Physics and Techniques UCSB –June 2003
Example Problem 2:A uniform plane wave of frequency 10 GHz propagates in a sufficiently large sample of gallium arsenide (GaAs, εr≈12.9,µr ≈1, tanδc ≈5x10-
5),which is a commonly substrate material for high-speed solid-state devices. Find (a) the attenuation constant α in np-m-1,(b) phase velocity νpin m-s-1,and (c) intrinsic impedance ηc in Ω.
18
410
00410
410
4
18809121032
1051022
1051022
10510222
1105
−−
−
−
−
−≈××
×××π=
εµεµ×××π=
εµ×××π
=εµδωε
=εµσ
≈α
<<×=δ
mnp
rr
c
c
have We(a).dielectric good a forapprox the use can we Since
..
tan
,tan
90Microwave Physics and Techniques UCSB –June 2003
Example Problem 2:
air. in that that smaller times ~ is impedanceintrinsic the that Note
impedanceintrinsic The (c)
air. the in that slower times ~ isvelocity phase
the that Note
have we where
velocity hase Since (b)
p
p
593
105912
377593
10358912
1031 178
.
..
.
...
,
Ω≈≈εµ
≈η
−×≈×
≈µε
≈ν
µεω≈β
βω
=ν
−
c
sm
p
91Microwave Physics and Techniques UCSB –June 2003
Example Problem3:A recent survey conducted in USA indicates that ~50% of the population is exposed to average power densities of approximately 0.005 µW-(cm)-
2due to VHF and UHF broadcast radiation. Find the corresponding amplitude of the electric and magnetic fields.
( )
( )
( ) ( )[ ]ztE
zztE
EzHΕ
ztEH
ztEΕ
y
x
β−ω+η
=β−ω
η
=×=Ρ
εµ=ηµεω=β
β−ωη
=
β−ω=
212
1
2020
0
0
0
cosˆcosˆ
.
cos
cos
by given is wavethis for vector Poynting The and where
:mediumlosslessa in gpropagatinwaveplaneuniform the Consider
92Microwave Physics and Techniques UCSB –June 2003
Example Problem3:
( ) ( )[ ]
η=⇒
β−ω+η
=Ρ= ∫∫
2
212
11
20
20
00
EzS
dtztE
zT
dttzT
S
av
T
p
T
pav
pp
ˆ
cosˆ,
( )
11
00
1490
22
0
515377
194
194101053772
00502
−−
−−−
−
−µ=Ω−
=η
=
−≈×××≈
−µ=η
=
mAmmVE
H
mmVE
cmWE
zSav
so
.ˆ