chapter 2: transmission lines and waveguides 2.1 generation solution for tem, te and tm waves 2.2...
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Chapter 2: Transmission lines and waveguides
2.1 Generation solution for TEM, TE and TM waves2.2 Parallel plate waveguide2.3 Rectangular waveguide2.4 Circular waveguide2.5 Coaxial line2.6 Surface waves on a grounded dielectric slab2.7 Stripline2.8 Microstrip2.9 Wave velocities and dispersion2.10 Summary of transmission lines and waveguids
Transmission lines and waveguides
2.1 Generation solution for TEM, TE and TM waves
Two-conductor TL
Closed waveguide
Electromagnetic fields (time harmonic eiωt and propagate along z axis):
where e(x,y) and h(x,y) represent the transverse (x,y) E and H components, while ez and hz are the longitudinal E and H components.
zjz
zjz
eyxhzyxhzyxH
eyxezyxezyxE
)],(),([),,(
,)],(),([),,(
In the case without source, the harmonic Maxwell’s equations can be written as:
With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz:
Wavevector in xy plane: (kc
2=kx2 + ky
2)
(1) TEM waves (Ez = Hz = 0)
Propagation constant:
k
The Helmholtz equation for Ex:
0)( 2222
222
xEkzyx
For a e-jz dependence, and the above equation can be simplified
xxx EkEEz 2222 )/(
0)(22
22
xEyx
Similarly, 0)(22
22
yEyx
0),(2 yxet
0),(2 yxht Transverse magnetic field:
Laplace equations, equal to static fields
(kc = 0, no cutoff)
0
0
Note:
1.Wave impedance, Z relates transverse field components and is dependent only on the material constant for TEM wave. For TEM wave
2.Characteristic impedance of a transmission line, Z0:relates an incident voltage and current and is a function of the line geometry as well as the material filling the line.For TEM wave: Z0= V/I
),(1
),( yxezZ
yxhTEM
(V incident wave voltage I incident wave current)
Wave impedance:
x
y
y
xTEM H
E
H
EZ
0
0
k
(2) TE waves (Ez = 0 and Hz 0) The field components can be simplified as:
is a function of frequency and TL/WG structure
Solve Hz from the Helmholtz equation
0)( 2222
222
zHkzyx
Because , then
zjzz eyxhzyxH ),(),,(
0)( 222
22
zc hkyx
TE wave impedance:
where . Boundaries conditions will be used to solve the above equation.
222 kkc
k
H
E
H
EZ
x
y
y
xTE
Solve Hz first and then obtain Hx, Hy, Ex, Ey
Solve Hz first and then obtain Hx, Hy, Ex, Ey
(3) TM waves (Hz = 0 and Ez 0) The field components can simplified:
is a function of frequency and TL/WG structure
Solve Ez from Helmholtz equation:
0)( 2222
222
zEkzyx
Because , then
zjzz eyxezyxE ),(),,(
0)( 222
22
zc ekyx
TM wave impedance:
kH
E
H
EZ
x
y
y
xTM
where . Boundaries conditions will be used to solve the above equation.
222 kkc
k
H
E
H
EZ
x
y
y
xTE
Solve Ez first and then obtain Hx, Hy, Ex, Ey
Solve Ez first and then obtain Hx, Hy, Ex, Ey
(4) Attenuation due to dielectric loss
Total attenuation constant in TL or WG = c + d.
c: due to conductive loss; calculated using the perturbation method; must be evaluated separately for each type.
d: due to the dielectric loss; calculated from the propagation constant.
Taylor expansion (tan << 1)
ModeDefinitio
nPropagation
constantWave
impedanceSolve the fields
TEMEz = Hz =
0
TE (H wave)
Hz 0 andEz = 0
TM(E wave)
Ez 0 and Hz = 0
0),(2 yxet
0),(2 yxht
0)( 222
22
zc ekyx
Ez Hx, Hy, Ex, Ey
Hz Hx, Hy, Ex, Ey
kH
E
H
EZ
x
y
y
xTM
x
y
y
xTEM H
E
H
EZ
w >> d (fringing fields and any x variation could be ignored)
Formed from two flat plates or
strips
Probably the simplest type of
guide
Support TEM, TE and TM
modes
Important for practical reasons.
2.2 Parallel plate waveguide
(a) TEM modes (Ez = Hz = 0)
Laplace equation for the electric potential (x,y)
The transverse field , so that we have
dyWxyxt 0,0,0),(2for
Characteristic impedance:
1
pv
Boundary conditions:
0),(0)0,( Vdxandx
dyVyx /),( 0
Phase velocity:
The transverse ez(x,y) satisfies
0),(22
2
yxeky zc )( 222 kkc
(b) TMn modes (Hz = 0)
General solutions:
Boundary condition 1: Bn = 0
Boundary condition 2:
Solutions of TMn modes:
The components of TMn:
Example: TM1
(b) TMn mode (Hz = 0) Propagation constant:
2222 )(d
nkkk c
k > kc Traveling wavek = kc Tunneling? k < kc Evanescent waveCutoff frequency:
@ (k = kc)
Power flow:
Frequency and geometry dependent
The TMn mode cannot propagate at f < fc!
(c) TEn mode (Ez = 0)
The transverse hz(x,y) satisfies
And boundary Ex(x,y) = 0 at y = 0, d.
0),(22
2
yxhky zc
zjn
cy
zjn
cx
zjnz
ed
ynB
k
jH
ed
ynB
k
jE
ed
ynByxH
sin
sin
cos),(
• Cutoff frequency :
where the propagation constant
2222 )/( dnkkk c
d
nkf cc
22
k
H
EZ
y
xTE
• Wave impedance:
An = 0 TE1
Parallel plate waveguide
TEM TM1 TE1
Homework:1. Derive the field solutions of TE1 mode for a parallel-plate metallic waveguide and plot the field pattern of each component roughly if possible.
Require: start from the first two Maxwell’s equations
Substitute Eq. (1) into (2) by eliminating H and we have
(no source, isotropic)
In the case without source, the harmonic Maxwell’s equations can be written as:
With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz:
Wavevector in xy plane: (kc
2=kx2 + ky
2)