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7/29/2019 Review of Electromagnetic Waves & Waveguides1.pdf
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 1
Review: TimeDependent Maxwells Equations
( ) ( )( ) ( )D t E tB t H tG G
G G
= =
( )( ) 0D tB tG
G = =
( ) ( )
( ) ( )B t
E tt
D tH t Jt
G
G
G
G G
= = +
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 2
Electromagnetic quantities:
Vectorquantitiesin space
Electric Field
Magnetic Field
Electric Flux (Displacement) DensityMagnetic Flux (Induction) Density
Current Density
Displacement Curren
E
H
D
B
J
D
t
G
G
G
G
G
G tCharge Density
Dielectric Permittivity
Magnetic Permeability
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 3
In free space:
In a material medium:
If the medium is anisotropic, the relative quantities are tensors:
[ ] [ ][ ] [ ]12
0
70
8.854 10 As/Vm or F/m
4 10 Vs/Am or Henry/m
= = = =
0 0;
relative permittivity (dielectric constant)
relative permeability
r r
r
r
= = = =
;
xx xy xz xx xy xz
r yx yy yz r yx yy yz
x zy zz zx zy zz
= =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 40
In engineering it is very important to considertime-harmonic fields
with a sinusoidal time-variation. If we assume a steady-statesituation (after all transients have died out) most physical situationsmay be investigated by considering one single frequency at a time.
This assumption leads to great simplifications in the algebra. It is
also realistic, because in practical electromagnetics applicationswe often have a dominant frequency (carrier) to consider.
The time-harmonic fields have the form
We can use the complex phasor representation
( ) ( ) ( ) ( )0 0cos cosE HE t E t H t H tG G G G= + = +
( ) { } ( ) { }0 0Re ReE Hj jj t j tE t E e e H t H e eG G G G = =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 41
We define
Maxwells equations can be rewritten for phasors, with the time-derivatives transformed into linear terms
( )( )
0
0
E phasor of
H phasor of
E
H
j
j
E e E t
H e H t
G G G
G G G
= == =
( )( )222
E phasor of
E phasor of
E tj
t
E t
t
G
G
G
G
= =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 42
In phasor form, Maxwells equations become
where all electromagnetic quantities are phasors and functions onlyofspace coordinates.
E H
H J E
D
B 0
D E
B H
j
j
G G
G G G
G
G
G G
G G
= = + = == =
( )F E Bq vG G G
G= +
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 43
Lets consider first vacuum as a medium. The wave equations for
phasors become Helmholtz equations
The general solutions for these differential equations are wavesmoving in 3-D space. Note, once again, that the two equations areuncoupled.
This means that each equation contains all the necessary
information for the total electromagnetic field and one only needs tosolve the equation forone field to completely specify the problem.The other field is obtained with a curl operation by invoking one ofthe original Maxwell equations.
2 20 0
2 2 0 0
E E 0
H H 0
G G
G G
+ = + =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 44
At this stage we assume that a wave exists, and we do not yet
concern ourselves with the way the wave is generated. So, for thesake of understanding wave behavior, we can restrict the Helmhlotzequations to a simple case:
We assume that the wave solution has an electric field which is
uniform on the {x,y}-plane and has a reference positiveorientation along the x-direction. Then, we verify that this is areasonable choice corresponding to an actual solution of theHelmholtz wave equations. We recall that the Laplacian of a
scalaris a scalar
and that the Laplacian of a vectoris a vector
2 2 22
2 2 2
f f ff
x y z
= + + 2 2 2 2 E E E Ex x y y z zi i i
G = + +
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 45
The Helmholtz equation becomes:
Only thex-component of the electric field exists (due to the chosen
orientation) and only the z-derivative exists, because the field is
uniform on the {x,y}-plane.We have now a one-dimensional wave propagation problemdescribed by the scalardifferential equation
( )22 2 20 0 0 02E E E E 0x x x xi iz
G G + = + =
2 20 02
EE 0x x
z
+ =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 46
This equation has a well known general solution
where the propagation constant is
The wave that we have assumed is a plane wave and we haveverified that it is a solution of Helmholtz equation. The generalsolution above has two possible components
For the simple wave orientation chosen here, the problem ismathematically identical to the one solved earlier for voltagepropagation in a homogeneous transmission line.
( ) ( )exp expj z B j z +
0 0c = =
( )exp zA j z Forward wave, moving along positive( )exp zB j z Backward wave, moving along negative
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 47
If a specific electromagnetic wave is established in an infinite
homogeneous medium, moving for instance along the positivedirection, only the forward wave should be considered.
A reflected wave exists when a discontinuity takes place along thepath of the forward wave (that is, the material medium changes
properties, either abruprtly or gradually).
We can also assume that the amplitude of the forward plane wavesolution is given and that it is in general a complex constant fixed
by the conditions that generated the wave
We can write at last the phasor electric field describing a simpleforward plane wave solution of Helmholtz equation as:
0j
E e=
0E ( ) j j zx xz E e e i
G =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 48
The corresponding time-dependent field is obtained by applying the
inverse phasor transformation
The phasor magnetic field is obtained directly from the Maxwellequation for the electric field curl
( ) ( ) } { }( )
0
0
, Re E Re
cos
j t j j z j tx x x x
x
E z t z e i E e e e i
E t z i
G = == +
( )( )0 0
0
0
E H
H
j j zx
j j zx
E e e i j
E e e i
j
G G
G
= = =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 49
We then develop the curl as
( ) ( )( ) ( )
0
0 0
0
det
E 0 0
x y z
j j zx
x
j jj z j z
y z
j j zy
i i i
E e e i x y z
z
E e e E e ei i
z y
j E e e i
= = = =
= = 0
( )Ex z
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 50
The final result for the phasormagnetic field is
We define
( )
( )
0
0 00
0
0 000 0
H
E
j j z
y y
j j zy
j j z y x y
j E e ez i
j
E e e i
E e e i z i
G
= = = = = =
00
0
377 Intrinsic impedance of vacuum = =
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 51
We have found that the fields of the electromagnetic wave are
perpendicular to each other, and that they are also perpendicular(ortransverse) to the direction of propagation.
x
z
y
E
H
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 52
Electromagnetic power flows with the wave along the direction of
propagation and it is also constant on the phase-planes. Thepower density is described by the time-dependent Poynting vector
The Poynting vector is perpendicular to both field components, andis parallel to the direction of wave propagation.
When the wave propagates on a general direction, which does notcoincide with one of the cartesian axes, the propagation constantmust be considered to be a vector with amplitude
and direction parallel to the Poynting vector.
( ) ( ) ( )P t E t H tG G G=
| | G
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 53
The condition of mutual orthogonality between the field
components and the Poynting vector is general and it applies toany plane wave with arbitrary direction of propagation. The mutualorientation chosen for the reference directions of the fields followsthe right hand rule.
( , , )E x y zG
( , , )H x y zG
, PG
G
x
y
z
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 54
So far, we have just verified that electromagnetic plane waves arepossible solutions of the Maxwell equations for time-varying fields.One may wonder at this point if plane waves have practical physicalrelevance.
First of all, we should notice that plane waves are mathematicallyanalogous to the exponential basis functions used in Fourieranalysis. This means that a general wave, with more than onefrequency component, can always be decomposed in terms ofplane waves.
Forperiodic signals, we have a discrete set of waves which areharmonics of the fundamental frequency (analogy with Fourierseries).
For general signals, we must consider a continuum offrequencies in order to decompose in terms of elementary planewaves (analogy with Fourier transform).
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 55
From a physical point of view, however, the properties of a plane
wave may be somewhat puzzling.
Assume that a steady-state plane wave is established in an idealinfinite homogeneous medium. On any plane perpendicular to thedirection of propagation (phase-planes), the electric and magnetic
fields have uniform magnitude and phase.
The electromagnetic power, flowing with a phase-plane of the wave,is obtained by integrating the Poynting vector, which is alsouniform on each phase-plane. For a plane where the Poyntingvector is non-zero, the total power carried by the wave is infinite
In many practical cases, we approximate an actual wave with aplane wave on a limited region of space, thus considering anappropriate finite power.
( ) ( ) ( )plane plane
P t E t H t G G G
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 56
Review of Boundary Conditions
Consider an electromagnetic field at the boundary between twomaterials with different properties. The tangent and the normalcomponent of the fields must me examined separately, in order tounderstand the effects of the boundary.
Medium 11 ; 1
Medium 22 ; 2
boundary1tH
G
2tHG
2nHG
1nHG 1H
G
2HG
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 57
Tangential Magnetic Field
Ampres law for the boundary region in the figure can be written as
Medium 11 ; 1
Medium 22 ; 2
boundary
1HtG
2HtG
3Hn
G
a4Hn
G
b
.x
y
H HH Ey x
zJ jx y
= + G
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 58
In terms of finite differences approximation for the derivatives
If one lets the boundary region shrink, with a going to zero fasterthan b,
4 3 1 2H H H H En n t t zJ jb a
= +
t
t
z
t
t sa
J a Jfor perfect conducto
for materials wi
rs
(sur
th finite co
face cur
nducti
ren
v ty
)
i
t2 10
2 1
H H lim
( )
H 0
H
= = = Tangential components are conserved
3 42 1
0
H HH H lim ( E )n nt t z z
a
J a j a a
b
= + +
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 59
For a general boundary geometry
In the case of a perfect conductor, the electromagnetic fields goimmediately to zero inside the material, because the conductivity is
infinite and attenuates instantly the fields. The surface current isconfined to an infinitesimally thin skin, and it accounts for thediscontinuity of the tangential magnetic field, which becomesimmediately zero inside the perfect conductor.
For a real medium, with finite conductivity, the fields can penetrateover a certain distance, and there is a current distributed on a thin,but not infinitesimal, skin layer. The tangential field components onthe two sides of the interface are the same. Nonetheless, theperfect conductor is often a good approximation for a real metal.
t t sn J1 2 (H H ) =G Gn unit vector normal to the su e rfac
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 60
Tangential Electric Field
Faradays law for the same boundary region can be written as
Medium 11 ; 1
Medium 22 ; 2
boundary
1EtG
2EtG
3En
G
a4En
G
b
.x
y
E EE H
y x jx y
= G
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 61
In terms of finite differences approximation for the derivatives
If one lets the boundary region shrink, with a going to zero fasterthan b,
For a general boundary geometry
4 3 1 2E E E E Hn n t t jb a
=
t t2 1 E E 0 = Tangential components are conserved3 4
2 1
0
E EE E lim ( H )n nt t z
a
j a a
b
= +
t tn 1 2 (E E ) 0 =G
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 62
Normal components
Consider a small box that encloses a certain area of the interface
with
Medium 11 ; 1
Medium 22 ; 2
boundary
1 1D Bn nG G
2 2D Bn nG G
w
Area
.x
y
+ + + + + + s
s interface charge density=
El t ti Fi ld
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 63
Integrate the divergence of the fields over the volume of the box:
VoluVolume
Surface
me
dd
ds
rr
Divergence theorem
Flux of D out of the box
D
D n
=
=
G
G G
G G
G
w
Volume
Surface
dr
ds
Divergence theorem
Flux of B out of the box
B 0
B n
= =
G
G G
G Gw
El t ti Fi ld
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 64
If the thickness of the box tends to zero and the charge density is
assumed to be uniform over the area, we have the following fluxes
The resulting boundary conditions are
The discontinuity in the normal component of the displacementfield D is equal to the density of surface charge.
The normal components of the magnetic induction field B arecontinuous across the interface.
n n
s
n n
Area
Area
Area
G
G
1 2
1 2
= (D D )
= T
D-Flux out of box
B-Flux out of bo
otal interface charge =
= (B ) 0x B
= =
n n s n n1 2 1 2D D B B 0= =
Electromagnetic Fields
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 65
For isotropic and uniform values ofandin the two media
Even when the interface charge is zero, the normal components of
the electric field are discontinuous at the interface, if there is achange of dielectric constant .
The normal components of the magnetic field have a similardiscontinuity at the interface due to the change in the magnetic
permeability. In many practical situations, the two media may havethe same permeability as vacuum, 0, and in such cases the normalcomponent of the magnetic field is conserved across the interface.
n n n n s
n n n n
1 2 1 1 2 2
1 2 1 1 2 2
D D E E
B B H H 0
= = = =
G G G GG G G G
Electromagnetic Fields
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 66
SUMMARYIf medium 2 is
perfect conductor
t
G
1H
t
G
2
H
t
G
1E
t
G
2E
n
G
1H
n
G
2H
n
G
1E
n
G
2E
1, 1
1, 1
1, 1
1, 12, 2
2, 2
2, 2
2, 2 t t t s
t
n1 2 1
2
H J
H H
H 0
==
= G GG
G G
t t t
t
1 2 1
2
E 0
E E
E 0
= ==
GG
G G
n n n
n
11
2
2 H 0
H H1 2
H 0
==
G G G
1 12
2
1 E
E E1 2
E 0
s
+ n
n
n n s ==
GG
G G
Electromagnetic Fields
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 67
Examples:
An infinite current sheet generates a plane wave (free space onboth sides)
The E.M. field is transmitted on both sides of the infinitesimally thinsheet of current.
x
y
+ z- z
Js
H
s
( ) cos( )
Phasor J
s so x
so x
J t J t i
J i
=
G
G
Electromagnetic Fields
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Electromagnetic Fields
Amanogawa, 2006 Digital Maestro Series 68
BOUNDARY CONDITIONS
1 2 (H H ) Jt t sn =G G
1 2
1 2
1 0 1
1 2
1 2
H H
E E
E H
Symmetry H H
H H2 2
t t so x
t t
t t
t t
so so
J i
J J
=== == =
G GG GG G
G G
Electromagnetic Fields
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g
Amanogawa, 2006 Digital Maestro Series 69
A semi-infinite perfect conductor medium in contact with free space
has uniform surface current and generates a plane wave
The E.M. field is zero inside the perfect conductor. The wave is onlytransmitted into free space.
x
y
+ z- z
Js
H
J cos( )s so x J t iG
Perfect
Conductor
Free Space
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 70
BOUNDARY CONDITIONS
1 2 (H H ) Jt t sn =G G
1 2 1
2
1 2
1 2
H H H 0
E 0
Asymmetry H H
H H 0
t t t so x
t
t t
t so t
J i
J
= ==
= =
G G GG
G G
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 75
Electromagnetic Waves in Material Media
In a material medium free charges may be present, which generatea current under the influence of the wave electric field. The current
Jc is related to the electric field E through the conductivity as
The material may also have specific relative values of dielectricpermittivity and magnetic permeability
J Ec =
r o r o= =
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 76
Maxwells equations become
In phasor notation, it is as if the material conductivity introduces an
imaginary part for the dielectric constant . The wave equation forthe phasor electric field is given by
We have assumed that the net charge density is zero, even if aconductivity is present, so that the electric field divergence is zero.
E H
H E E ( )E
j
j j j
= = + =
2
c
2
E E E H
(J E)
E ( )E
j
j j
j j
= = = + = +
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 77
In 1-D the wave equation is simply
with general solution
These resemble the voltage and current solutions in lossytransmission lines.
22
2
E( )E Ex x xj j
z
= + =
( )
( )
E ( ) exp( ) exp( )
1( ) exp( ) exp( )
1exp( ) exp( )
x
xy
z A z B z
E jH z A z B zj z j
A z B z
= + + = =
=
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 78
The intrinsic impedance of the medium is defined as
For the propagation constant, one can obtain the real and imaginaryparts as
j je
j
= = +
1 / 22
1 / 22
( )
1 12
1 12
j j j= + = + = + = + +
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 79
Phase velocity and wavelength are now functions of frequency
The intrinsic impedance of the medium is complex as long as theconductivity is not zero. The phase angle of the intrinsicimpedance indicates that electric field and magnetic field are out ofphase. Considering only the forward wave solutions
( )( )
1 / 22
1 / 22
21 1
2 21 1
pv
f
= = + + = = + +
E ( ) exp( ) exp( ) exp( )
1 1H ( ) exp( ) exp( ) exp( )
x
y
z A z A z j z
A z j A z j z j
= = = =
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 80
In time-dependent form
where the integration constant has been assumed to be in general acomplex quantity as
{ }
{ }1
( , ) Re exp( )exp( )exp( )
exp( )cos( )
1( , ) Re exp( )exp( )exp( )
exp( )
exp( )
exp( )cos( )
x
y
E z t z j z j t
z t z
H z t A z j z j j t
A j
A
j
A z t z
= +
=
+
=
exp( )A j
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 81
Classification of materials
Perfect dielectrics - For these materials = 0Propagation constant
0
r o r o= =Medium Impedance
= r o
r o
j
j
=
Phase velocity
1p
r o r o
v= =
Wavelength
2 1p
r o r o
v
f f
= = =
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 82
Imperfect dielectrics For these materials 0 but (/)
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Amanogawa, 2006 Digital Maestro Series 83
If (/)
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Amanogawa, 2006 Digital Maestro Series 84
Good conductors For these materials 0 but (/)>>1( )
1 1exp( ) (1 )
4 2 2
4 2 4
exp( )4
1 1
2 2(1 )
p
j j j j
j j f j
f
v f
j jj
f
fjj
f
j
= + = = = + = +
= = =
= + = + =
+
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 85
The simple rule of thumb is that approximations for good conductorcan be applied when
Note that for a good conductor the attenuation constant and thepropagation constant are approximately equal.The medium impedance has nearly equal real and imaginaryparts, therefore its phase angle is approximately 45.This means that in a good conductor the electric and magnetic
fields have always a phase difference = 45 = /4.
10
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 86
Also, in a good conductor the fields attenuate very rapidly. The
distance over which fields are attenuated by a factor exp(1.0) is
A typical good conductor is copper, which has the following
parameters:
1 1Skin depth
f= = =
75.80 10 [S/m]
o
o
=
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 87
Copper remains a good conductor at extremely high frequencies.Another good conductor example is sea water at relatively lowfrequencies
At a frequency of25 kHz
4.0 [S/m]80 o
o
36,000
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 88
Perfect conductor - For this ideal material For this material, the attenuation is also infinite and the skin depthgoes to zero. This means that the electromagnetic field must go tozero below the perfect conductor surface.
General medium - When a material is not covered by one of the limitcases, the complete formulation must be used. We can classify a
material for which the conditions (/)10 areinvalid as a general medium.
The simple rule of thumb for general medium is
10 0.1> >
Electromagnetic Fields
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Power Flow in Electromagnetic Waves
The time-dependent power flow density of an electromagnetic waveis given by the instantaneous Poynting vector
Fortime-varying fields it is important to consider the time-average
power flow density
where Tis the period of observation.
( ) ( ) ( )P t E t H tG G G=
0 0
1 1( ) ( ) ( ) ( )
T TP t P t dt E t H t dt
T T
G G G G= =
Electromagnetic Fields
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Considertime-harmonic fields represented in terms of their phasors
The time-dependent Poynting vectorcan be expressed as the sumof the cross-products of the components
(Note that: 1cos sin sin 22
t t t = )( )
2
2
( ) ( ) Re{E} Re{H} cos
Im{E} Im{H} sin
Re{E} Im{H} Im{E} Re{H} cos sin
E t H t t
t
t t
G G G G
G G
G G G G
= + +
{ }{ }( ) Re E exp( ) Re{E} cos Im{E} sin
( ) Re H exp( ) Re{H} cos Im{H} sin
E t j t t t
H t j t t t
= = = = G G G G
G G G G
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 91
The time-average power flow density can be obtained by integrating
the previous result over a period of oscillation T . The pre-factorscontaining field phasors do not depend on time, therefore we haveto solve for the following integrals:
2
00
20
0
2
00
1 1 sin 2cos2 4
1 1 sin 2sin2 4
1 1 sincos sin2
1
12
0
2
TT
T
T
T
T
t tt dtT T
t tt dtT T
tt t dt T T
= + = = =
= =
Electromagnetic Fields
Th fi l lt f th ti fl d it i i b
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Amanogawa, 2006 Digital Maestro Series 92
The final result for the time-average power flow density is given by
Now, consider the following cross product ofphasor vectors
( )0
1( ) ( ) ( )
1
Re{E} Re{H} Im{E} Im{H}2
TP t E t H t dt
T
G G G
G G G G
= = +
( )*
E H Re{E} Re{H} Im{E} Im{H}
Im{E} Re{H} Re{E} Im{H}j
G G G G G G
G G G G = +
+
Electromagnetic Fields
B bi i th i lt bt i th f ll i
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By combining the previous results, one can obtain the followingtime average rule
We also call complex Poynting vectorthe quantity
NOTE: the complex Poynting vector is not the phasor of the time-dependent powernorthat of the time-average power density!
Phasor notation cannot be applied to the product of two time-
harmonic functions (e.g.,P( t)), even if they have same frequency.
{ }*01 1( ) ( ) ( ) Re E H2TP t E t H t dtTG G G G G= = *1
P E H2
G G G=
{ } { }( ) Re P ( ) Re P exp( )P t P t j tdon't t( )ryG G G G= =
Electromagnetic Fields
Consider a 1 D electro magnetic wave moving along the z direction
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Amanogawa, 2006 Digital Maestro Series 94
Consider a 1-D electro-magnetic wave moving along the z-direction,
with a specified electric field amplitudeEo
The time-average power flow density is
Power in a lossy medium decays as exp(-2 z)!
E ( ) exp( )exp( )
H ( ) exp( )exp( )exp( )
x o
o
y
z E z j z
Ez z j z j
= =
{ }{ }
**
2 22 2
1 1( ) Re E H Re2 2
1 1
Re cos2 2
j z z j z joo
z zj
o o
EP t E e e e e e
e e
E e E
G G G
= = = =
Electromagnetic Fields
Consider the same wave with a specified amplitude for the
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Amanogawa, 2006 Digital Maestro Series 95
Consider the same wave, with a specified amplitude for themagnetic field
The time-average power flow density is expressed as
If is the attenuation constant for the electromagnetic fields 2 is the attenuation constant for power flow.
H ( ) exp( )exp( )
E ( ) exp( )exp( )exp( )
y o
x o
z H z j z
z H z j z j
= =
{ }*2 2
1( ) Re
21
cos2
j z z j z j
o o
zo
P t H e e H e e e
H e
G
= =
Electromagnetic Fields
If the wave is generated by an infinitesimally thin sheet of uniform
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Amanogawa, 2006 Digital Maestro Series 96
If the wave is generated by an infinitesimally thin sheet of uniform
currentJso (embedded in an infinite material with conductivity )we have for propagation along the positive z-direction (normal tothe plane of the current sheet):I
For this ideal case, an identical wave exists, propagating along the
negative z-direction and carrying the same amount of power.
22
2 2
( ) cos
8
so soo o
zso
J JH E
JP t e
G = =
=
Electromagnetic Fields
Poynting Theorem
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Amanogawa, 2006 Digital Maestro Series 97
Poynting Theorem
Consider the divergence of the time-dependent power flow density
The curls can be expressed by using Maxwells equations
This is the differential form ofPoynting Theorem.
( )( ) ( ) ( ) ( ) ( ) ( ) ( )P t E t H t H t E t E t H tG G G G G G G = =
2 2 2
( ) ( ) ( ) ( ) ( )
1 1( ) ( ) ( )
2 2
H EP t H t E t E t E t
t t
E t E t H tt t
G GG G G G G =
= Density of
dissipatedpower
Rate of change
of stored electricenergy density
Rate of change
of stored magneticenergy density
Electromagnetic Fields
Now integrate the divergence of the time-dependent power over a
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Amanogawa, 2006 Digital Maestro Series 98
Now, integrate the divergence of the time-dependent power over a
specified volume V to obtain the integral form ofPoynting theorem
2 2 2
Power Flux through S( ) ( )
1 1( ) ( ) ( )
2 2
V S
V V V
P t dV P t ds
E t dV E t dV H t dVt t
G G
w = = =
Power dissipated
in volumeRate of change
of electric energystored in volume
Rate of changeof magnetic energy
stored in volume
Electromagnetic Fields
Typical applications
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Amanogawa, 2006 Digital Maestro Series 99
Typical applications
L
inP t( )G
outP t( )G
= ?
1 m2
2
Watts( ) ( ) exp( 2 )
m
( )1 Nepersln2 ( ) m
out
in
out inP t P t
P tL P t
L
G
G G
G = =
Electromagnetic Fields
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Example:
Pay attention to the logarithms:
2 2
Watts Watts( ) 30 ; ( ) 5 ; 20 m
m m
Nepers= 0.0448
m
in out P t P t LG G = = =
( ) ( )ln ln( ) ( )
out inin out
P t P t
P t P t
G G
G G
=
Electromagnetic Fields
SURFACE A SURFACE B
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Amanogawa, 2006 Digital Maestro Series 101
Area = Area(A) = Area(B)
Power IN ( ) ( ) Area
Power OUT ( ) ( ) Area
( ) ( ) exp( 2 )
= Power IN Power OUPower dissipated T
A A
A
B B
B
B A
P t dS P t
P t dS P t
P t P t L
G G
G G
G G
= = = =
=
L
out
B
P t( ) Power OUT GinA
P t( ) Power IN GPower dissipated
between A and B?
Electromagnetic Fields
Example
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Amanogawa, 2006 Digital Maestro Series 102
p
2Area = 5 m
2
8.2244637 General Lossy medium
130.88 0.725rad 130.88 41.5
; 1.0 cm; 1.0 GHz; 10 V/m
; ; 0.45755 S/m
34
40.0 Ne/m; ( ) 0.286 W/m ;
( ) ( ) exp( 2 )
in
out i
o
nB A
o o
P t
P t P t L
L f E
D
G
G G
= = =
= = = = = =
= == = 2Power IN Area ( )
Power OUT Area ( )
= Power IN PowerPower dissipat Te OUd
0.12845 W/m ;
1.43 W
0.6423 W
0.7876 W
in
B
P t
P t
G
G
= =
=
==
Electromagnetic Fields
Incidence on Perfect Conductor
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Amanogawa, 2006 Digital Maestro Series 124
Consider first normal incidence at an interface between a dielectricand a perfect conductor. Total reflection occurs, as in a short-circuited transmission line.
Medium 11 = r1 o1 = r1oMedium 2Perfect
Conductor
2Incident wave
Reflected wave
z0
x
y
Interface{x,y}-plane
0
E
H
0
==G
G
E 1.0=
Electromagnetic Fields
Because ofinterference between incident and reflected wave, there
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is a standing wave in medium 1.
Medium 11 = r1 o1 = r1oMedium 2
Perfect
Conductor
2
z0
x
y
0
E
H
0
==G
G
EG
HG
oE2
o
E2
/ 2 /
Electromagnetic Fields
Consider now incidence at an angle. We choose an electric field
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Amanogawa, 2006 Digital Maestro Series 126
perpendicular to the plane of incidence.
Medium 11 = r1 o1 = r1o
Medium 2
Perfect
Conductor
2
z0
x
y
0
E
H
0==G
G
E
G
HG
Gx
EG
HG
Electromagnetic Fields
Only the normal component, corresponding to zis reflected.
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Amanogawa, 2006 Digital Maestro Series 127
Note: zz >
Medium 11 = r1 o1 = r1oMedium 2Perfect
Conductor
2
z0
x
y
0
E
H
0
==G
G
EG
HG
oE2
oE2
z/ / 2 =2 / z z =
Electromagnetic Fields
2 2 2
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Amanogawa, 2006 Digital Maestro Series 128
max
max
max
m
max
in
First minimum
First maximum
4 4 cos
45 0.35
15 0.259
2 2 co
2 2
s
2
; cos
Exa
0 0.25
mples:
z
z
z
z
z
z
z
z
z
D
D
D
=
= = =
= =
=
= = =
Medium 2
Perfect
Conductor
2
z0
x
y
0
E
H
0
==G
G
EG
H
G
G
x
EG
HG
Electromagnetic Fields
If we place a second perfect conductor interface, parallel to thei th i id d l th di ti b fl ti
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Amanogawa, 2006 Digital Maestro Series 129
previous one, the wave is guided along the x-direction by reflection.
z0
x
y
0EH
0==G
G
E
G
H
G
G
EG
HG
Perfect
Conductor
2
Perfect
Conductor
2
0EH
0==G
G
Electromagnetic Fields
Parallel Plate Waveguide
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Amanogawa, 2006 Digital Maestro Series 130
Assume uniform waves along they-direction ( )y
0 =
Assume no fringing effects w a>> Propagation along thez-direction
0
a
x
y
z
w
a
Electromagnetic Fields
Maxwells equations
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Amanogawa, 2006 Digital Maestro Series 131
Maxwell s equations
E E H
det E
(1)
(2E H
E E EE (E
)
E
3)
H
H
z y xx y z
x z y
x y z y x z
ji i i
j
y z
jx y z z x
jx y
= = =
=
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 132
H (4)
(5
H E
det H H )
(6)
E
E
E
H
H H HH H
z y xx y z
x z y
x y zy x z
ji i i y z
j
x y z z x
jy
j
x
= = =
=
Electromagnetic Fields
From (1) & (2) & (5)
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Amanogawa, 2006 Digital Maestro Series 133
From (1) & (2) & (5)
Wave equation forTransverse Electric (TE) modes
2
2
2
2
2 2 22 2
E H
E H
E E E
(1)
(3)
(5)
H
From
H
E
y x
y z
y y yx z
y
jz zz
jx xx
jz x z x
j
= =
+ = =
Electromagnetic Fields
From (4) & (6) & (2)
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Amanogawa, 2006 Digital Maestro Series 134
From (4) & (6) & (2)
Wave equation forTransverse Magnetic (TM) modes
2
2
2
2
2 2 22 2
E E
H E
H E
H H H
Fro
(4)
(6)
(2)m H
y x
y z
y xy z y
y
jz zz
jx
z x
j
xx
jz x
= = + = =
Electromagnetic Fields
Transverse Electric (TE) modes
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This solution satisfies the boundary conditions:
E
H
E
H
y
x
x a
0E 0
== =Boundary Conditions
( ) ( )E sin 2 zx xj z j zj x j xoy o x EE x e j e e e = = x
z
H
E
Electromagnetic Fields
We have
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Amanogawa, 2006 Digital Maestro Series 136
We have
and from boundary conditions at the conductor plates
22 2 2 2
2
4x z
= = + =
( )0
sin 0
0
1,
)
2, 3
) y
x x
x E
a a m
m
x a
= = =
===
1 / 22 22
cos
sin 12
x
z
m
a
m m
a a
= = = = =
Electromagnetic Fields
For each possible index m we have a mode of propagation. Modes
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Amanogawa, 2006 Digital Maestro Series 137
For each possible index m we have a mode of propagation. Modesare labeled TE10 , TE20 , TE30 , .
The first index gives the periodicity (number of half sinusoidaloscillations) between the plates, along the x-direction. The second
index is zero to indicate uniform solution along the y-direction.
Note that the solution m = 0 (or mode TE00) is not acceptable,because it would require a field configuration with uniform electricfield tangent to the metal plates. This is an unphysical boundary
condition, which is possible only for the case of trivial solution ofzero field everywhere.
E
H
x z
m00TE 0
0
Unphysical !!!
& ==
Electromagnetic Fields
A mode can propagate only if the frequency is sufficiently high, so
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Amanogawa, 2006 Digital Maestro Series 138
p p g y q y y g ,
that z > 0.We have the cut-off condition when
Exactly at cut-off the wave would bounce between the plates,without propagation along the wave guide axis.
12 2 2
2
2 2
1 02
2
x cc
z
pc
c
m a
a m
m m
a a
f mv m
a
Cut - off frequency for mode
= = = = = = =
= =
Electromagnetic Fields
When the frequency is below the cut-off value
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Amanogawa, 2006 Digital Maestro Series 139
The mode attenuates entering the guide as an evanescent wave.
12
22
12 2
2
( )
1
1
2
12
2
j j
z
z
c
z
caf
j e
fm
m
a
mj
a
m
a
e
>
= =
= =
< > =
=
Electromagnetic Fields
Transverse Magnetic (TM) modes
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Amanogawa, 2006 Digital Maestro Series 140
The magnetic field can be tangent to the conductor plates. In fact, it
is maximum at the plates, since the reflection coefficient is H= 1.The solution is of the form:
( ) ( )H cos2
z zx xj z j zj x j xoy o x
HH x e e e e
= = +x
z
E
H
E
H
E
H
Electromagnetic Fields
At the metal plates
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Amanogawa, 2006 Digital Maestro Series 141
Modes are labeled TM00 , TM10 , TM20 , TM30 ,
Note that the solution m = 0 (or mode TM00) is acceptable, becausethe magnetic field can be uniform and tangent to the metal plates.
( )H Hcos 1
0))
0, 1, 2, 3
y o
x x
xx a a a m
m
= = = = =
=
E
H
x
m00TM 0
0
Physical !!!
& =
Electromagnetic Fields
The TM00 mode is like a portion of a uniform plane wave slidingbetween the plates of the waveguide.
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Amanogawa, 2006 Digital Maestro Series 142
Both the electric and the magnetic field are transverse (normal tothe guide axis) therefore this mode is usually known as TransverseElectro Magnetic mode (TEM). For this mode we have
The TEM mode is the fundamental mode. It can propagate at anyfrequency.
All other TM modes have the same cut-off frequency condition asthe TE modes with identical indices.
2 20
0
z x cc
pc
c
v
f Cut - off frequency for TEM mode
= = = = = =
Electromagnetic Fields
The apparent wavelength along the guide axis is also called theguide wavelength
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Amanogawa, 2006 Digital Maestro Series 143
( ) ( )
2
2 2
2 2
sin
2cos
cos sin
1 /
1
2
1 /
g zz
xc c
cc
g
c c
c
f f
ma
f
f
Since :
= = = = = = =
= = =
=
=
Electromagnetic Fields
There is a corresponding apparent velocity along the guide axis, orguide phase velocity
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Amanogawa, 2006 Digital Maestro Series 144
The expressions for guide wavelength and guide velocity are alsoidentical for TE and TM modes.
( ) ( )2 2s
1 / /
n
1
i
p
p
p
z
pz
c c
z
v v
v
v
f f
= =
= =
Electromagnetic Fields
Consider a TE wave with electric field amplitude Eo. The totalamplitude of the magnetic field is
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Amanogawa, 2006 Digital Maestro Series 145
The magnetic field has two components with amplitude
oo
EH =
2
sin sin
H sin sin
H cos cos
2
cos
o ox o
g
o oz
g
o
c
c
E E
m
E EH
a
H
since :
since :
= = =
= = = = =
= =
Electromagnetic Fields
Consider a TM wave with magnetic field amplitude Ho. The totalamplitude of the electric field is
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Amanogawa, 2006 Digital Maestro Series 146
The electric field has two components with amplitude
o oE H
0
2
sin sin
E sin sin
E co
2
s c
os
o
c
s
x o og
z o o o
g
c
c
E H H
E
a
H H
m
since :
since :
= = =
= = =
= =
= =
Electromagnetic Fields
The xcomponent of the magnetic field for the TE wave isassociated with the wave moving along the zdirection (axis of the
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Amanogawa, 2006 Digital Maestro Series 147
waveguide). The guide impedance for the TE modes is defined as
The xcomponent of the electric field for the TM wave is associatedwith the wave moving along the zdirection (axis of the waveguide).The guide impedance for the TM modes is defined as
( ) ( )2 2g 1 / 1 /c cTMg
f f
= = =
( ) ( )g
2 2
1 1
1 / 1 /
g
TE
c cf f
= = =
Electromagnetic Fields
If there is a discontinuity along the guide axis (e.g., a change indielectric medium), one can use transmission line theory to analyze
th d b h i i di id ll i t f t i i d
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Amanogawa, 2006 Digital Maestro Series 148
the mode behavior individually in terms of transmission andreflection. Sections of the guide can be replaced by a transmissionline, with the guide impedance as the characteristic impedance.
Note that the guide impedance is a function of frequency for all
modes, except for the fundamental TEM mode
The reflection coefficient at a discontinuity is of the usual form
The power reflection coefficient is again ||2 and the powertransmission coefficient is 1||2.
( )2g 1 0( ) 0 /TEMcf TEM f
= =
2 1
2 1
g g
g g
= +
Electromagnetic Fields
The phasor fields for TE modes are summarized as follows
El t i Fi ld i l t t
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Electric Field: a single transverse component
Magnetic Field: two components, obtained from Faradays law:
( ) E sin sinz zj z j zo x y o ymE x e i E x e ia
= =
E E
H si
E ( )
n
cos
z
z
y x x zy x y z
j zo x
j
z
zo
xz
j Hi i
z xm
E x e ia
mjE x e
i H i
ia
= +
+ = =
+
Electromagnetic Fields
The following relationships are useful to introduce the mediumimpedance in the TE field expressions above
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Amanogawa, 2006 Digital Maestro Series 150
Note once again that there is no allowed solution form = 0 in the
case of TE modes. The first allowed TE mode is the TE10.
g
sin
cos
1
x
cc
z
TEg g
= = = = = =
=
Electromagnetic Fields
The phasor fields for TM modes are summarized as follows
Magnetic Field: a single transverse component
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Magnetic Field: a single transverse component
Electric Field: two components, obtained from Amperes law:
( ) H cos cosz zj z j zy o x y o ymH x e i H x e ia
= =
H H
E c
H ( )
os
sin
z
z
y x x zy x y z
j zo x
x
z
j zo z
z
i i
z xm
H x e ia
mjH x
j E E
e ia
i i
+ = =
+
= +
Electromagnetic Fields
The following relationships are useful to introduce the mediumimpedance in the TM field expressions above
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Amanogawa, 2006 Digital Maestro Series 152
The field expressions simplified for the TEM mode resemble a
uniform plane wave propagating along the axis of the guide
Remember, the TM00 or TEM mode is the fundamental mode.
gsin
cos
z
Tg
c
Mg
x
c
= = = = = =
=
H
E
z
z z
j zy o y
j z j zx o x o x
H e i
H e i E e i
== =
Electromagnetic Fields
Wave Dispersion
A plane wave by itself does not carry information For transmissionf i f ti it i t h f t f fi it
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Amanogawa, 2006 Digital Maestro Series 153
A plane wave by itself does not carry information. For transmissionof information it is necessary to have a frequency spectrum of finitesize, as obtained by modulation of a wave, for instance.
Information does not travel at the guide phase velocity, but itpropagates according to the group velocity
To illustrate the nature of the group velocity, consider the simple
case of an amplitude modulated signal (assume >> )
pz
g
v
dvd
group ve
guide phase veloc
lo
ity
city
= =
( )) ( )( ) 1 cos cosy o oE t E m t t+
Electromagnetic Fields
This signal has three components
( ) cos cos cosE t E t tm E t
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Amanogawa, 2006 Digital Maestro Series 154
( ) ( ) ( )( )
( )( )
cos
co
( )
cos2
cos cos cos
s
2
o
o
y o
o
o o o
o
o o
E t
mE
mE t t
t t
E t t
E
m
t
E t
= +
+
=
++
o o+o
Electromagnetic Fields
The line at angular frequency o is the carrier. The modulationinformation is contained in the two side frequency lines at o
d +
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Amanogawa, 2006 Digital Maestro Series 155
and o+.Now, consider an amplitude modulated wave propagating in a
parallel plate wave guide. The zcomponents of the propagationfactor depend on frequency and are different for the two sidefrequencies. In general, we have
21 z mm z z z z= = +
1 m 2
oo+m
Electromagnetic Fields
The dispersion relation () is approximately linear when
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Amanogawa, 2006 Digital Maestro Series 156
Under this assumption, we can write
( )( ) ( )( ) ( )cos
2
cos
(
2
, cos)
o o z
o o z z
zy o
z
o
mE t
E z t E t
z
z
mE t z
+ +
+
+
1
zm oz( ) z2
o +mo
z
Electromagnetic Fields
( )( , ) scoy o o zE z tm
zE t=
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Amanogawa, 2006 Digital Maestro Series 157
The modulation envelope travels at the group velocity
( ) ( )
( ) ( )( )( ) ( )
( ) ( )cos
cos cos
1 cos cos
cos
2
2
coso o
o o z
o o
o o
z z
o z o z
z z
z z
E t z
mE
mE
mE t
t z t z
E m t z
t z t z
z
z
t
tz
modulated amplitude
++
= +
= +
+
/g zv =
Electromagnetic Fields
15 0 v
MODULATION ENVELOPE
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Amanogawa, 2006 Digital Maestro Series 158
4.000.00 0.50 1.50 2.00 3.00
-10.0
-5.0
0.0
5.0
15.0
-15.0
10.0
gz
v =
pzz
v=
CARRIER
Electromagnetic Fields
For the parallel plate wave guide
1 / 2 1 / 22 2f
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Amanogawa, 2006 Digital Maestro Series 159
( ) ( )( ) ( )
2
2 2
2 2
2 2
1 1
1 / 1 /
1 / 1 /
cz
c
p ppz
zc c
g p c p cz
pz g p
pz p g p
f
f
v vv
f f
d
v v
v v
v f fd
v
v v v vSince
= = = = = = = =
=
Electromagnetic Fields
Information travels at the group velocity, which is always less thanthe corresponding phase velocity in the given medium.
The group and phase velocities for each mode propagating in the
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Amanogawa, 2006 Digital Maestro Series 160
The group and phase velocities for each mode propagating in the
wave guide are frequencydependent. This means that frequencycomponents of a broadband signal travel at different speed andchange their phase relationship as they propagate along the wave
guide. The group and phase velocities of the modes are alsomodedependent. This means that if a signal is distributed over anumber of different modes, the components spread out over timeduring propagation.
This phenomenon is called dispersion. Wave guides are in generaldispersive media.
Note: For the fundamental TEM mode in parallel plate wave guide
0c pz p g f v v v no dispersion = =
Electromagnetic Fields
gv
Slope
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Amanogawa, 2006 Digital Maestro Series 161
Dispersion diagram
z1 2
1c
2
g
pz
v
v
0 at cutoff
pv1
=Slope
pzvSlope
g
Electromagnetic Fields
The power flow follows the Poynting vector, with the same directionas the propagation vector. The group velocity accounts for the
effective motion of the power flow in the direction parallel to theaxis of the wave guide
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Amanogawa, 2006 Digital Maestro Series 162
axis of the wave guide.
P
gL v t2 sin = L L
pL v t2 =
22 si
si
n
npp gg
L
vt vv
L
v= = =
Electromagnetic Fields
The guide phase velocity corresponds to the apparent motionillustrated by the following diagrams
L v t/ sin / 2
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Amanogawa, 2006 Digital Maestro Series 163
P
Lp
L v t/ 2
P
L
pzL v t/ sin / 2=
pzL v t/ sin / 2=
pL v t/ 2
Electromagnetic Fields
Therefore, we obtain for the guide phase velocity
22 pvLL
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Amanogawa, 2006 Digital Maestro Series 164
From the results above, we have again
2
s n
2
siin
p
ppz
pz
vL
vt
v
Lv= = =
2
sinsin
p
p
p
p p
pz
g
pz g
vv
v v
vv
v
v v
= =
Electromagnetic Fields
Rectangular Wave Guide
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Amanogawa, 2006 Digital Maestro Series 240
Assume perfectly conducting walls and perfect dielectric filling thewave guide.
a wiCon derventio is always the side of the wave gun : ide.
a
b
x z
y
Electromagnetic Fields
It is useful to consider the parallel plate wave guide as a startingpoint. The rectangular wave guide has the same TE modes
corresponding to the two parallel plate wave guides obtained byconsidering opposite metal walls
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Amanogawa, 2006 Digital Maestro Series 241
considering opposite metal walls
TEm0
E
TE0n
E
a
b
Electromagnetic Fields
The TE modes of a parallel plate wave guide are preserved ifperfectly conducting wallsare added perpendicularly to the electric
field.
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Amanogawa, 2006 Digital Maestro Series 242
On the other hand, TM modes of a parallel plate wave guidedisappear if perfectly conducting walls are added perpendicularly tothe magnetic field.
EThe added metal plate doesnot disturb normal electricfield and tangent magneticfield.H
HThe magnetic field cannotbe normal and the electric
field cannot be tangent to aperfectly conducting plate.
E
Electromagnetic Fields
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Amanogawa, 2006 Digital Maestro Series 243
The remaining modes are TE and TM modes bouncing off each wall,all with non-zero indices.
TEmn
TMmn
Electromagnetic Fields
We have the following propagation vector components for themodes in a rectangular waveguide
x y z2 2 2 2 2 = = + +
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Amanogawa, 2006 Digital Maestro Series 244
At cut-offwe have
x y
x yz g
m n
a b
m n
a b
222 2 2 2
2 22
;
2 2
= = = = = =
( )z c m nfa b
2 222 0 2
= =
Electromagnetic Fields
The cut-off frequencies for all modes are
c m nfa b
2 2
12
= +
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Amanogawa, 2006 Digital Maestro Series 245
with cut-off wavelengths
with indices
a b2
TmTE M
m
mn
nn
0, 1, 2, 3, 1, 2, 3,0, 1, 2, 3, 1, 2, 3,
(but
m mode
not allowed)
o s
0
sde
== == ==
c
m n
a b
2 2
2 = +
Electromagnetic Fields
The guide wavelengths and guide phase velocities are
g zz 2 2
2 2 = = = =
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Amanogawa, 2006 Digital Maestro Series 246
pz
z c
c
v
ff
2 2
1 1 1 1
11
= = =
z
c
c
m n
a b
f
f
2
2 2
11
= =
Electromagnetic Fields
The fundamental mode is the TE10with cut-off frequency
( )c mf TE a10 2 =
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Amanogawa, 2006 Digital Maestro Series 247
The TE10 electric field has only the y-component. From Ampereslaw
z
x y z
x y z
yi i i
x y
j
z
E
det
E = 0 E E = 0
E H
=
y x
x
j
z
z
E H
E
=
zxE
y
y x
j
x y
H 0
E E
= zj H
Electromagnetic Fields
The complete field components for the TE10mode are then
E sin z zy o jxEa
e =
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Amanogawa, 2006 Digital Maestro Series 248
with
H sin
H cos
E1E
E1
z
z
j zz
x o
j zz o
y z
y
x
xj
jE e
a
j xE
z j
a
aj ze
a
= = =
=
=
22
za =
Electromagnetic Fields
The time-average power density is given by the Poynting vector
{ } ( )1 1
Re Re2 2
*( ) E H { sin z yj z
oP t ixE e
= =
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Amanogawa, 2006 Digital Maestro Series 249
{ } ( )
( ) ( )
( ) ( ) ( )
*
* *
2 2
E
H
2 21 2Re2
sin co( )}
sin s ni c
s
os
z z
y
xj z zj
o z o
z
z
zo o
x j xE e E e
a ai i
E Ex x xi ja a a
a
a
a
=
( )
22
sin2
x
o z
z
i
E x
ia
=
Electromagnetic Fields
The resulting time-average power density flow is space-dependent
on the cross-section (varying along x, uniform along y)
22( ) sin
oE xzP t i =
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Amanogawa, 2006 Digital Maestro Series 250
The total transmitted power for the TE10 mode is obtained byintegrating over the cross-section of the rectangular wave guide
( ) sin2
P t iza =
) ( )2 2
2
2 2( ) sin0 0 02 2
1 1
sin 22 2 4
sino o
b
o
E Ex aa b z zP t btot a
E abzb u u
dx dy u du
=
= = =
=
2
2 2
0area
average 1
|E( , )|
1
4 2 2
TE
o o z
x y
E Ezab ab
= =
Electromagnetic Fields
The rectangular waveguide has a high-pass behavior, since signalscan propagate only if they have frequency higher than the cut-off
for the TE10 mode.
For mono mode (or single mode) operation only the fundamental
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Amanogawa, 2006 Digital Maestro Series 251
For mono-mode (or single-mode) operation, only the fundamentalTE10 mode should be propagating over the frequency band ofinterest.
The mono-mode bandwith depends on the cut-off frequency of thesecond propagating mode. We have two possible modes toconsider, TE01and TE20
( )
( ) ( )
01
20 10
1
2
1 2
c
c c
f TEb
f TE f TEa
=
= =
Electromagnetic Fields
( ) ( ) ( )c c cf TE f TEa
ab f TE01 20 102
2
1
= = =If
Mono-mode bandwidth
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Amanogawa, 2006 Digital Maestro Series 252
0 ( )cf TE10 ( )cf TE20( )cf TE01f
( ) ( ) ( )c c cf TE f TE fab TEa 10 01 202
< IfMono-mode bandwidth
0 ( )cf TE10 ( )cf TE20)cf TE01 f
Electromagnetic Fields
Mono-mode bandwidth
( ) ( )c cab f TE f TE 20 012
<
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Amanogawa, 2006 Digital Maestro Series 253
In practice, a safety margin of about 20% is considered, so that theuseful bandwidth is less than the maximum mono-mode bandwidth.This is necessary to make sure that the first mode (TE10) is well
above cut-off, and the second mode (TE01 or TE20) is stronglyevanescent.
0 ( )cf TE10 ( )cf TE01)cf TE20
Useful bandwidth
0
f
( )cf TE10 ( )cf TE01)cf TE20
Safet mar in
Electromagnetic Fields
( ) ( )10 01c cf TE f TEa b =(square wave guide)If
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Amanogawa, 2006 Digital Maestro Series 254
In the case of perfectly square wave guide, TEm0 and TE0n modes
with m=n are degenerate with the same cut-off frequency.
Except for orthogonal field orientation, all other properties ofdegenerate modes are the same.
0 ( )cf TE10 ( )cf TE20( )cf TE01
f
( )02cf TE
Electromagnetic Fields
Example - Design an air-filled rectangular waveguide for thefollowing operation conditions:
a) 10 GHz is the middle of the frequency band (single-modeoperation)
b) b = a/2
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Amanogawa, 2006 Digital Maestro Series 255
b) b = a/2
The fundamental mode is the TE10 with cut-off frequency
For b=a/2, TE01 and TE20 have the same cut-off frequency.
co o
cf TEa aa
810
1 3 10( ) Hz2 22 = =
co o
co o
c c cf TE
b a a ab
cf TE
a aa
8
01
820
1 2 3 10( ) Hz
2 22
1 3 10( ) Hz
= = = = = =
Electromagnetic Fields
The operation frequency can be expressed in terms of the cut-offfrequencies
01 10( ) ( )( ) c cf TE f TE
f f TE
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Amanogawa, 2006 Digital Maestro Series 256
10
10 01
8 89
2 2
( )2
( ) ( ) 10.02
1 3 10 3 1010.
2.25 10 1.125 1
0
02
10
2 2
c
c c
a
a
f f TE
f TE f TE GHz
m b
a a
m
= ++= =
= +
= = =
Electromagnetic Fields
Maxwells equations forTE modes
Since the electric field must be transverse to the direction ofpropagation for a TE mode, we assume
E 0=
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Amanogawa, 2006 Digital Maestro Series 257
In addition, we assume that the wave has the following behavioralong the direction of propagation
In the general case of TEmn modes it is more convenient to startfrom an assumed intensity of the z-component of the magnetic field
zj ze
( ) ( )H cos cos
cos cos
z
z
j zz o x y
j zo
H x y e
m nH x y e
a b
=
=
E 0z =
Electromagnetic Fields
Faradays law for a TE mode, under the previous assumptions, is
E Hj =
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Amanogawa, 2006 Digital Maestro Series 258
E E H
det E E H
E E
(1)
(2)
0E E H (3
E H
)
y z y xx y z
x z x y
x yy x z
j ji i i z
j jx y z z
jx y
j
= =
= =
=
Electromagnetic Fields
Amperes law for a TE mode, under the previous assumptions, is
H Ej =
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Amanogawa, 2006 Digital Maestro Series 259
(4)
(
H H E
det H H E
H H HH H E 0
5)
(6)
z z y xx y z
z x z y
x y zy x z
j ji i i y
j jx y z x
jx y
+ =
=
= =
Electromagnetic Fields
From (1) and (2) we obtain the characteristic wave impedance for
the TE modesEE yx
TE
= = =
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Amanogawa, 2006 Digital Maestro Series 260
At cut-off
H HTE
y x z
= = =
2 2
2 2
2
0 2
1
c
pc
c c
z
c
m nf
a bv
f
m n
a b
= +
= = =
+
=
Electromagnetic Fields
In general,
( )
2 2 222 2
2 41
2z
m n
a b
= =
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Amanogawa, 2006 Digital Maestro Series 261
and we obtain an alternative expression for the characteristic waveimpedance ofTE modes as
( )
22 1
2
zc
ca b
=
1 22
1TE oz c
= =
Electromagnetic Fields
From (4) and(5) we obtain
H H E H
1 H 1 HH
z z y x TE y
z z
j j jy
+ = =
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Amanogawa, 2006 Digital Maestro Series 262
2
2 2
2
2 2
H HH
2
H HH
H
2
H
H H E
z zy
TE z
zz
z x z y T
cz z zy z
z
E x
cz z zx zz
jj
j j y yj j
j j
y y
jx
jx
jx
= =
= =
= =
= =
Electromagnetic Fields
We have used
2
2 2 2 2 2 2
1 1 1
2cz x y m n
b
= = = + +
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Amanogawa, 2006 Digital Maestro Series 263
The final expressions for the magnetic field components of TEmodes in rectangular waveguide are
2
2
H sin cos2
H cos sin2
H cos cos
z
z
z
j zc
x z o
j zcy z o
j zz o
m m nj H x y e
a a b
n m nj H x y e
b a b
m nH x y ea b
=
=
=
a b
Electromagnetic Fields
The final electric field components for TE modes in rectangularwave guide are
E H
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Amanogawa, 2006 Digital Maestro Series 264
2
2
E H
cos sin2
E H
sin cos2
E 0
z
z
x TE y
j zcTE z o
y TE x
j zcTE z o
z
n m nj H x y e
b a b
m m nj H x y e
a a b
=
=
=
=
=
Electromagnetic Fields
Maxwells equations for TM modes
Since the magnetic field must be transverse to the direction ofpropagation for a TM mode, we assume
H 0z =
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Amanogawa, 2006 Digital Maestro Series 265
In addition, we assume that the wave has the following behavioralong the direction of propagation
In the general case of TMmn modes it is more convenient to startfrom an assumed intensity of the z-component of the electric field
zj ze
( ) )E cos cos
cos cos
z
z
j zz o x y
j zo
E x y e
m nE x y e
a b
=
=
z
Electromagnetic Fields
Faradays law for a TM mode, under the previous assumptions, is
E Hj =
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Amanogawa, 2006 Digital Maestro Series 266
E E H
det E E H
E E
(1)
(2
EE H
)
(3)E
z z y xx y z
z x z y
x y zy x z
j ji i i y
j jx y z x
jx y
+ =
=
=
Electromagnetic Fields
Amperes law for a TMmode, under the previous assumptions, is
H Ej
=
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Amanogawa, 2006 Digital Maestro Series 267
(4)
(5)
(6
H E
det H E
H H 0 H H E )
x y zz y x
z x y
x y y x z
i i i j j
j jx y z
jx y
= =
=
Electromagnetic Fields
From (4) and (5) we obtain the characteristic wave impedance forthe TM modes
EE
H H
yx zTM
y x
= = =
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Amanogawa, 2006 Digital Maestro Series 268
We can finally express the characteristic wave impedancealternatively as
Note once again that the same cut-off conditions, found earlier forTE modes, also apply forTM modes.
y
2
1zTM oc
= =
Electromagnetic Fields
From (1)and(2) we obtain
EE E Hy
z z y xTM
j j jy
+ = =
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Amanogawa, 2006 Digital Maestro Series 269
2
2 2
2
2 2
1 E 1 EE
/
EE
E E
E 2
E E
E H
E2
z z
y TM zz
z
xz x z y
cz z z
y zz
T
c
M
z z zx z
z
j
j
j j y yj j
j j
y y
jj
x
jx
x
= =
= =
= =
= =
Electromagnetic Fields
The final expressions for the electric field components ofTM modesin rectangular waveguide are
2
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Amanogawa, 2006 Digital Maestro Series 270
2
2
E cos sin
2
E sin cos2
E sin sin
z
z
z
j zcx z o
j zcy z o
j zz o
m m nj E x y e
a a b
n m nj E x y e
b a b
m nE x y ea b
=
=
=
Electromagnetic Fields
The final magnetic field components forTM modes in rectangularwave guide are
H E /x y TM=
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Amanogawa, 2006 Digital Maestro Series 271
Note: all the TM field components are zero if either x=0 or y=0.This proves that TMmo or TMon modes cannot exist in therectangular wave guide.
2
2
sin cos2
H E /
cos sin2
H 0
z
z
j zczo
TM
y x TM
j zcz oTM
z
n m n
j E x y eb a b
m m nj E x y ea a b
= =
= =
Electromagnetic Fields
Field patterns for the TE10mode in rectangular wave guide
x
zSide view
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Amanogawa, 2006 Digital Maestro Series 272
Cross-section
y
x
E
H
y
x
zTo view
E
H
Electromagnetic Fields
The simple arrangement below can be used to excite the TE10 in a
rectangular waveguide.
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Amanogawa, 2006 Digital Maestro Series 273
The inner conductorof the coaxial cable behaves like an antennaand it creates a maximum electric field in the middle of the cross-section.
Closed end
TE10
Electromagnetic Fields
Waveguide Cavity Resonator
d
xm
a
=
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Amanogawa, 2006 Digital Maestro Series 274
The cavity resonatoris obtained from a section of rectangular waveguide, closed by two additional metal plates. We assume againperfectly conducting walls and loss-less dielectric.
x
y
z
a
b
y
z
a
nb
l
d
==
Electromagnetic Fields
The addition of a new set of plates introduces a condition for
standing waves in the zdirection which leads to the definition ofoscillation frequencies
2 2 21
2c
m n lf
a b d
= + +
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Amanogawa, 2006 Digital Maestro Series 275
The high-pass behavior of the rectangular wave guide is modified
into a very narrow pass-band behavior, since cutoff frequencies ofthe wave guide are transformed into oscillation frequencies of theresonator.
2 a b d
In the wave guide, each mode isassociated with a band of frequencieslarger than the cut-off frequency.
In the resonator, resonant modes canonly exist in correspondence ofdiscrete resonance frequencies.
0 0f f1cf 2cf 1rf 2rf
Electromagnetic Fields
The cavity resonator will have modes indicated as
The value of the index corresponds to periodicity (number of halfsine or cosine waves) in the three directions. Using z-direction as
m lmnl nTMTE
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Amanogawa, 2006 Digital Maestro Series 276
sine or cosine waves) in the three directions. Using z direction asthe reference for the definition of transverse electric or magnetic
fields, the allowed indices are
The mode with lowest resonance frequency is called dominant
mode. In the case ad> b the dominant mode is the TE101.
0, 1, 2, 3 1, 2, 3
0, 1, 2, 3 1, 2, 3
1, 2, 3 0, 1, 2, 3
m n
m m
n n
l
MT T
l
E
= === =
=
with only one zero indexor allowed
Electromagnetic Fields
Note that for a TM cavity mode, with magnetic field transverse tothe z-direction, it is possible to have the third index equal to zero.This is because the magnetic field is going to be parallel to the thirdset of plates, and it can therefore be uniform in the third direction,with no periodicity.
Th l t i fi ld t ill h th f ll i f th t
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The electric field components will have the following form that
satisfies the boundary conditions for perfectly conducting walls
E cos sin sin
E sin cos sin
E sin sin cos
x ox
y oy
z oz
m n lE x y z
a b d
m n lE x y z
a b d
m n lE x y za b d
=
=
=
Electromagnetic Fields
The magnetic field intensities are obtained from Amperes law
H sin cos cosz y y zx E E m n lx y zj a b d
=
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Similar considerations for modes and indices can be made if the
other axes are used as reference for transverse fields, leading toanalogous resonant field configurations.
H cos sin cos
H cos cos sin
x z z x
y
y x x yz
E E m n l
x y zj a b d
E E m n lx y z
j a b d
=
=
Electromagnetic Fields
A cavity resonatorcan be coupled to a wave guide through a smallopening. When the input frequency resonates with the cavity,
electromagnetic radiation enters the resonator and a lowering in theoutput is detected. With carefully tuned cavities, this scheme canbe used forfrequency measurements.
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OUTPUT
INPUT
Movable piston changesthe resonance frequencies
Electromagnetic Fields
Examples of resonant cavity excited by using coaxial cables.
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The termination of the inner conductor of the cable acts like an
elementary dipole (left) or an elementary loop (right) antenna.
E
H
E
H