2.3 Coaxial line

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Lecture 5
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2.3 Coaxial line
Advantage of coaxial design little
electromagnetic leakage outside the shield and a
good choice for carrying weak signals not
tolerating interference from the environment or
for higher electrical signals not being allowed
to radiate or couple into adjacent structures or
circuits. Common application video and CATV
distribution, RF and microwave transmission, and
computer and instrumentation data connections?
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TEM mode in coaxial cables
The transverse fields satisfy Laplace equations,
i.e.
and
Boundary conditions
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2.3 Rectangular waveguide
Closed waveguide, propagate Transverse electric
(TE) and/or transverse magnetic (TM) modes.
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Maxwells equations (source free)
Expansion
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TEmn mode
Hz satisfy
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TEmn mode
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TEmn mode
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TEmn mode
TE10 mode (the fundamental mode)
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TMmn mode
TM11 mode (the lowest mode)
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Mode patterns-- Rectangular waveguide
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2.5 Circular waveguide
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Mode patterns _ Circular waveguide
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2.6 Surface waves on a grounded dielectric slab

• Surface waves
• a field that decays exponentially away from the
  dielectric surface
• most of the field contained in or near the
  dielectric
• more tightly bound to the dielectric at higher
  frequencies
• phase velocity Vdielectric lt Vsurface lt Vvacuum

Geometries
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2.7 Stripline

• Stripline as a sort of flattened out coaxial
  line.
• Stripline is usually constructed by etching the
  center conductor to a grounded substrate of
  thickness of b/2, and then covering with another
  grounded substrate of the same thickness.

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Propagation constant
with the phase velocity of a TEM mode given by
Characteristic impedance for a transmission
Laplace's equation can be solved by conformal
mapping to find the capacitance. The resulting
solution, however, involves complicated special
functions, so for practical computations simple
formulas have been developed by curve fitting to
the exact solution. The resulting formula for
characteristic impedance is
with
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• Inverse design
• When design stripline circuits, one usually
  needs to find the strip width, given the
  characteristic impedance and permittivity. The
  inverse formulas could be derived as

• Attenuation loss
• (1) The loss due to dielectric filler.
• (2) The attenuation due to conductor loss (can
  be found by the perturbation method or Wheeler's
  incremental inductance rule).

with
t thickness of strip
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2.8 Microstrip
For most practical application, the dielectric
substrate is electrically very thin and so the
field are quasi-TEM.
Phase velocity
Propagation constant
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Effective dielectric constant
Formula for characteristic impedance (numerical
fitting )
Inverse waveguide design with known Z0 and ?r
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• Attenuation loss

Considering microstrip as a quasi-TEM line, the
attenuation due to dielectric loss can be
determined as
Filling factor
which accounts for the fact that the fields
around the microstrip line are partly in air
(lossless) and partly in the dielectric.
The attenuation due to conductor loss is given
approximately
For most microstrip substrates, conductor loss is
much more significant than dielectric loss
exceptions may occur with some semiconductor
substrates, however.
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2.9 Wave velocities and dispersion
So far we have encountered two types of
velocities

• The speed of light in a medium
• The phase velocity (vp ?/?)

Dispersion the phase velocity is a frequency
dependent function.
The faster wave leads in phase relative to the
slow waves.
Group velocity the speed of signal propagation
(if the bandwidth of the signal is relatively
small, or if the dispersion is not too sever)
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• Consider a narrow-band signal f(t) and its
  Fourier transform

• A transmission system (TL or WG) with transfer
  function Z(w)

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A linear transmission system

• If and
  (ie., a linear function of ?),

f0(t) is a replica of f(t) except for an
amplitude factor and time shift. A lossless TEM
line (??/v) is disperionless and leads to no
signal distortion.
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Non-linear transmission system

• Consider a narrow-band signal s(t) representing
  an amplitude modulated carrier wave of frequency
  ?0

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The output signal in time domain
For a narrowband F(?), ? can be linearized by
using a Taylor series expansion about ?0
From the above, the expression for so(t)
(a time-shifted replica of the original envelope
s(t).)
The velocity of this envelope (the group
velocity), vg
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2.10 Summary of transmission lines and waveguids
Comparison of transmission lines and waveguides
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2.10 Summary of transmission lines and waveguids
Other types of lines and guides
Ridge waveguide
Dielectric waveguide
(TE or TM mode, mm wave to optical frequency,
with active device)
(lower the cutoff frequency, increase bandwidth
and better impedance characteristics)
Coplanar waveguide
Covered microstrip
Slot line
(Quasi-TEM mode, useful for active circuits)
(electric shielding or physical shielding)
(quasi-TEM mode, rank behind microstrip and
stripline)
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Homework

• 1. An attenuator can be made using a section of
  waveguide operating below cutoff, as shown below.
  If a 2.286 cm and the operating frequency is 12
  GHz, determine the required length of the below
  cutoff section of waveguide to achieve an
  attenuation of 100 dB between the input and
  output guides. Ignore the effect of reflections
  at the step discontinuities.

2 Design a microstrip transmission line for a 100
? characteristic impedance. The substrate
thickness is 0.158 cm, with ?r 2.20. What is
the guide wavelength on this transmission line if
the frequency is 4.0 GHz?