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JIT 1
WaveguidesMr. HIMANSHU DIWAKAR
Assistant ProfessorGETGI
JIT 2
JIT 3
Basic waveguides 1. Waveguide
Rectangular waveguide Circular waveguide Coaxial line
Optical waveguide Parallel-plate waveguide
JIT
Transverse Electro Magnetic (TEM) wave:
Here both electric and magnetic fields are directed
components. (i.e.) E z = 0 and Hz = 0
Transverse Electric (TE) wave: Here only the electric field is purely transverse to the
direction of propagation and the magnetic field is not purely transverse. (i.e.) E z = 0, Hz ≠ 0
Transverse Magnetic (TM) wave: Here only magnetic field is transverse to the direction of
propagation and the electric field is not purely transverse. (i.e.) E z ≠ 0, Hz = 0.
Hybrid (HE) wave: Here neither electric nor magnetic fields are purely transverse to the
direction of propagation. (i.e.) E z ≠ 0, Hz ≠ 0.
4
Types of Modes
JIT 5
Transmission line
Voltage applied between conductors(E: vertically between the conductors)
Interior fields: TEM (Transverse ElectroMagnetic) wave (wave vector indicates the direction of wave propagation as well as the direction of power flow)
1. Waveguide
JIT 6
Waveguide A waveguide is a structure that guides waves, such as
electromagnetic waves or sound waves They enable a signal to propagate with minimal loss of energy by restricting expansion to one dimension or two
Zigzag reflection, waveguide mode, cutoff frequency
k|||| du kk
JIT 7
JIT 8
The electric and magnetic wave equations in frequency domain is given by
For a loss less dielectric or perfect conductor
The above equations are like Helmholtz equations
Let =X(x).Y(y).Z(z)Be the solution of above equations
JIT 9
Cont’dSo separation equation-= On Solving the above equations
The propagation of wave in guide is conventionally assumed in +ve Z direction.
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Here the propagation constant in guide differs from intrinsic propagation constant
Let And
Where is cutoff wave number.For a lossless dielectric
So So there are three cases….
Cont’d
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Case 1
If no propagation
This is critical condition for cutoff propagation
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Case 2
If
This shows that operating frequency should be greater than critical frequency to propagate the wave in wave in wave guide
JIT 13
Case 3If
This shows that if operating frequency is below the cutoff frequency the wave will decay exponentially wrt a factor - and there will no wave
propagationThere for the solution of Helmholtz equation in rectangular co-ordinates
is given by
JIT 14
Rectangular waveguide
WR (Waveguide Rectangular) series- EIA (Electronic Industry Association) designation
WR-62- Size: 1.58 cmx0.79 cm- Recommended range: 12.4-18.0 GHz- Cutoff: 9.486 GHz
2/cm/100 54262inch/100 62
ab.a
JIT 15
Waveguide modes TE (Transverse Electric) mode- E parallel to the transverse plane of the waveguide- In waveguide Wave propagates in +ve Z direction- TEmn in characterized by Ez=0- In other words the z component of magnetic field must exist
in order to have energy transmission in the guide. TM (Transverse Magnetic) mode- H is within the transverse plane of the waveguide
JIT 16
TE modes in rectangular waveguides
Therefore the magnetic field in +ve Z direction ie. The solution of above partial differential equations
H0z is the amplitude constant, so field equations in rectangular waveguides
mode:)sin()sin(),(
mode:)cos()cos(),(
mnzj
nmz
mnzj
nmz
TMeybxayxE
TEeybxayxHmn
mn
222 and,, where nmmnnm bakb
nba
ma
zjnmzz
mneybxaHH )cos()cos(0
JIT 17
Utilization
Transmission of power
3. Waveguide
JIT 18
TE and TM modes
Hz and Ez fields: TE and TM modes Non-TEM modes: Hz = Ez = 0 Concept of a dominant mode: TE10 mode
mode:)sin()sin(),(
mode:)cos()cos(),(
mnzj
nmz
mnzj
nmz
TMeybxayxE
TEeybxayxHmn
mn
Boundary condition enforcements: PEC (Perfect Electric Conductor)
222 and,, where nmmnnm bakb
nba
ma
3. Waveguide
JIT 19
Cutoff wave no kc =Where = and =
kc =fc =
Propagation constant as discussed earlier
So from case-1 and case-2 propagation constant or phase constant
JIT 20
And attenuation constant
We know that So cutoff frequency
JIT 21
Dominant mode: TE10 mode
ak 2
10
.0 and 1 where)/cos(
)cos()cos(),(10
1001
nmeax
eybxayxHzj
zjz
3. Waveguide
JIT 22
Parallel-plate waveguide 2. Parallel-plate
Phase front: out of phase
Phase front: in phase (guided mode)
JIT 23
Wavenumbers
22mm kk
medium cnonmagneti and Lossless
cn
ck r
rooo
index refrective a is where n
2. Parallel-plate
JIT 24
mdkdk mm 2d
mkm
Reflections 2. Parallel-plate
JIT 25
0
TE and TM modes 2. Parallel-plate
TM modeTE mode
JIT 26
Cutoff frequency
2222
1111
11
2coscoscoscoscos
ndcmk
kdmkkk
ndm
ndcm
kdm
kk
kk
mm
mmmm
ndcmm cm : mode,for frequency cutoff
2
1
cm
m cn
propagate.not does mode theandimaginary is , Ifpropagate. willmode theand constant phase valued-real , If
mcm
mcm
2. Parallel-plate
JIT 27
mndc
cmcm
22 :h wavelengtcutoff
2
12
cmm
n
Cutoff wavelength 2. Parallel-plate
JIT 28
TE mode representation
waves.plane downward and upward theofionsuperposit thefrom resultingpattern ceinterferen theis field mode TE The
cutoff) above mode (TE
)cos(sin)Re(),(
sinsin2)(
, and
'0
'000
r0
r0
ztxkEeEtzE
exkEexkjEeeeEE
zxkk
eEeEE
mmtj
ysy
zjm
zjm
zjxjkxjkys
zxzmxmzmxm
jjys
mmmmm
aaraakaak du
kk du
2. Parallel-plate
JIT 29
)cutoff below mode TE(
)cos(sin),( and sin
121 , If
'0
'0
22
texkEtzEexkEE
nc
nj
zmy
zmys
cm
cmcm
cmmmcm
mm
TE mode representation
infinity. approaches as 90 gapproachin
increases, angle wave the,decreased) is or ( cutoff beyond increased is Asguide. down the progress forward no making arethey
forth; andback reflectingjust are wavesplane theand 0 ),( cutoffAt
o
cm
m
cm
cmm
cos
2. Parallel-plate
JIT 30
Phase and group velocity
mm cnk sinsinm
mpm n
cv
sin
velocity Phasem
mcm
mgm n
cnc
ddv
sin1 : velocityGroup
2
.relativity special a of principle theenot violat :medium in thelight of speed theexceedmay This
2. Parallel-plate
JIT 31
Field analysis
cnkk / where22 ss EE
):variation- ,0( 0
) ofcomponent a(only modes TE
2
22
2
2
2
2
2
2zj
ysysysysysmezE
yEkE
zE
yE
x
y
E
zjmys
mexfEE )(0
0)()()( 2
22
2
xfkdx
xfdmm
m 0)()(
, 2
2
222
2 xfkdx
xfdkk mm
mmm
dxmxfdxxE
xkxkxf
my
mmm
sin)( . and 0at zero bemust :BC
),sin()cos()(
2. Parallel-plate
JIT 32
Characteristics of TE mode
cavityresonant ldimensiona One
2sinsin cutoff,At
22 cutoff,At /2 and 0
.2 isshift phase trip-roundNet walls.conducting ebetween thdown and up bouncessimply waveThe
zero. is guide in the incidence of angle waveplane thecutoff,At
00
cmysys
cm
cmcmmm
xnEEd
xmEE
nm
dnd
mnkk
m
zjys
med
xmEE
sin0
2. Parallel-plate
JIT 33
Field representations
xzj
mmzzj
mmxys
zys mm exkEjexkEk
zE
xE
yj
zx
aaaaE
EHE
H
s
s
ss
s
)sin()cos(
, ofcomponent aOnly
mode TE afor of components and
00
zjm
mzs
zjm
mxs
mm exkEk
jHexkEH
)cos( ,)sin( 00
2. Parallel-plate
JIT 34
1cossin and
)(cos)(sin||
||
22222
00222/1220
**
AAkk
EkExkxkk
E
HHHH
mm
mmmm
zszsxsxsxs*
s
sss
H
HHH
Intrinsic impedance
2. Parallel-plate
JIT 35
Circular waveguide
WC (Waveguide Circular) series Hz and Ez fields: TE and TM modes
3. Waveguide
JIT 36
A microstrip is constructed with a flat conductor suspended over a ground plane. The conductor and ground plane are separated by a dielectric.
The surface microstrip transmission line also has free space (air) as the dielectric above the conductor.
This structure can be built in materials other than printed circuit boards, but will always consist of a conductor separated from a ground plane by some dielectric material.
Microstrip transmission line 4. Tx line
JIT 37
Circular waveguide
Circular waveguides offer implementation advantages over rectangular waveguide in that installation is much simpler.
When forming runs for turns and offsets - particularly when large radii are involved - and the wind loading is less on a round cross-section, meaning towers do not need to be as robust.
JIT 38
For a circular waveguide of radius a, we can perform the same sequence of steps in cylindrical coordinates as we did in rectangular coordinates to find the transverse field components in terms of the longitudinal (i.e. Ez, Hz) components.
JIT 39
The scalar Helmholtz equation in cylindrical co-ordinate is given by
Using the method of separation of variables, the solution of above equation is assumed
Substituting (1) into (a) and solving this equation for So here also
This is also called as characteristic equation of Bessel’s equations.For a lossless guide
(a)
JIT 40
The total solution of Helmholtz equation in cylindrical co-ordinate
JIT 41
TE Modes in Circular Waveguides
It is commonly assumed that the waves in a circular waveguide are propagating in the positive z direction. Here in this mode , so
After substituting boundary conditions the final solution is
For a lossless dielectric, Maxwell’s equations
JIT 42
In cylindrical co-ordinates, the components of E and H fields can be expressed as
JIT 43
When the differentiation is replaced by ()and the z component by zero, the TE mode equations in terms of in circular waveguide are expressed as
Where
JIT 44
The permissible value of kc can be written as
Where is a constantAnd from above table =1.841 for TM11 MODE
JIT 45
The final equations for the E and H fields can be written as
JIT 46
Where Zg = Er/ , = - / Hr has been replaced for the wave impedance in the guide and where n = 0,1,2,3,... And p = 1, 2, 3, 4,....
The first subscript n represents the number of full cycles of field variation in one revolution through rad of .
The second subscript p indicates the number of zeroes of .
The mode propagation constant is determined by
JIT 47
The cutoff wave number of a mode is that for which the mode propagation constant vanishes. Hence
So
And the phase velocity for TE modes is
JIT 48
The wavelength and wave impedance for TE modes in a circular guide are given, respectively, by
𝒁 𝒈=𝑬 𝒙
𝑯 𝒚=¿
JIT 49
TM Modes in Circular Waveguides
The TMnp modes in a circular guide are characterized by Hz = 0. However, the z component of the electric field E, must exist in order to have energy transmission in the guide.Consequently, the Helmholtz equation for Ez in a circular waveguide is given byIts solution is given in Eq.
Which is subject to the given boundary conditions.
(A)
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JIT 51
Similarly
On differentiating equation (A) wrt z and substituting the result in above equations yield the field equations of TMnp modes in a circular waveguide:
JIT 52
JIT 53
Some of the TM-mode characteristic equations in the circular guide are identical to those of the TE mode, but some are different. For convenience, all are shown here:
JIT 54
It should be noted that the dominant mode, or the mode of lowest cutoff frequency in a circular waveguide, is the mode of TEnp that has the smallest value of theproduct, kc .a = 1. 841, as shown in above Table.
JIT 55
Thank you