TELECOMMUNICATIONS

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TELECOMMUNICATIONS

• Dr. Hugh Blanton
• ENTC 4307/ENTC 5307

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Background

• An ATT Bell Lab engineer, Philip Smith,
  developed a graphical tool in 1933 to simplify
  the task of plotting impedance variation in
  passive transmission line circuits.
• Although Smiths original paper has been rejected
  by the IRE (predecessor of the IEEE) it has
  become one of the most popular design aids for RF
  and microwave engineers.
• It is estimated that over 70,000,000 copies have
  been distributed throughout the world during the
  past sixty years.

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• Recognizing that passive transmission circuit
  impedances may vary through a very wide range
  (zero to infinity),
• Smith decided to plot reflection coefficient that
  has a limited magnitude range (zero to one).
• To translate reflection coefficient to
  impedance, he created a unique overlay that
  became the Impedance Smith Chart.
• Later, a second chart was created to provide
  conversion between reflection coefficient and
  admittance, and finally the two charts were
  superimposed to form the Impedance-Admittance
  (Immittance) Chart.

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• Although initially the charts were used for
  passive circuit impedance manipulation only,
  later additional applications were developed for
  active circuit design.
• Constant-gain, constant-noise, constant power
  output, and RF stability plots are now
  traditionally shown on the Smith Chart.
• Modern RF/MW test equipment and CAE software can
  also display their output on the Chart.
• Therefore, anyone involved with development,
  production, or test of RF/MW components, circuits
  and systems will benefit from a thorough
  understanding of this powerful graphical tool.

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Smith Chart

• Slide rule of the RF engineer
• Circuit matching
• Impedance Admittance transformation
• Conversion between ? and ZL
• What is it
• ? is complex
• Phasor diagram or Argand diagram of ?

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• The Impedance Smith Chart is a result of a
  mathematical transformation of the rectangular
  impedance Z, to a polar reflection coefficient G,
  where

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• This transformation places all impedances with
  positive real parts (R ? 0) inside of a circle.
• The center of the circle refers to Zo, which is
  called the characteristic impedance.
• Zo is generally resistive and equal to 50W.

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Ideal Inductors Pure
Positive Reactance No Resistive
Component
90
jwXL
j75
Ideal Resistors No
Reactive Component
j50
j25
25
50
75
100
0
180
jwXC
-90
Ideal Capacitors Pure
Negative Reactance No Resistive
Component
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• Impedances of passive circuits may vary from zero
  to infinity and are difficult to plot due to
  their wide range.
• Converting impedances to reflection coefficients
  limits the magnitudes to be between 0 and 1.
• Referencing the impedances to Z0 and plotting in
  a polar coordinate system, a small manageable
  chart is created that includes all points of the
  right-hand side of the rectangular impedance
  system (0 ? R ? ?).
• At the center of the chart, the reflection
  coefficient is zero (G0), and the impedance is
  the characteristic impedance (Z Z0).

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• Sample transformations using Z0 50 W.
• Infinite impedance has three possible
  combinations
• ? ? jX
• R j?
• R ? j?

?
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• The top half of the Impedance Smith Chart
  represents inductive terminations, while the
  lower half represents capacitive terminations.
• Ideal resistors (X 0) are located on the
  horizontal centerline,
• ideal inductors (R 0) on the upper half of the
  charts circumference, and
• ideal capacitors on the lower half of the
  circumference.

Ideal Inductor
j50
?j50
Ideal Capacitor
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Formation of Smith Chart
125º
?0.73
?0.20.5j
For a passive system ? lt 1 so ? must be within
the unit circle. The area marked on diagram. We
know what the axis are so we can miss them out
Plot ? as either a phasor or complex number
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• In the rectangular impedance system,
• Vertical lines represent constant resistances
  with varying reactances.
• Horizontal lines are the loci of impedances with
  constant reactance and varying resistance.

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• The Smith Chart transformation changes both
  vertical and horizontal lines to circles.
• The families of constant resistance and constant
  reactance circles form the impedance Smith Chart.

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Formation of Smith Chart
Mark on the diagram contours of constant
normalized resistance. The resistance has been
normalized against the characteristic impedance
of the transmission line
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• The lower part of the commercially available 50 W
  Smith Chart includes several scales, including
  G, Return Loss, Mismatch Loss, and VSWR.

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• Generally, Zo is 50 W and the center of the chart
  refers to that impedance.
• However at times the characteristic impedance is
  other than 50 W and the Smith Chart must be
  labeled accordingly.
• Instead of creating a different Smith Chart for
  every characteristic impedance, the
  transformation equation may be normalized by
  dividing all terms by Zo.

Letting
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Lossless Series Inductors

• Series additions of components are most
  conveniently handled in the impedance system.
• Beginning with a component of reflection
  coefficient G1, we first convert to impedance z1.
• A lossless series inductor adds inductive
  reactance, keeping the real part (resistance)
  constant.

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• The sum of the two impedances, zT, is computed as

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• On the impedance Smith Chart, addition of a
  series lossless inductor represents an upward
  movement on the constant resistance circle,
  toward x j?.

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• Insert a 80 nH series inductor to a one port of
  G1 0.45?-116?.
• Find the new combined input impedance, zT, at 0.1
  GHz. Use Zo 50 W for normalization.

1. Locating G1 on the normalized Impedance Smith
   Chart, convert
2. The reactance of the 80 nH inductor
   jXL j0.126FGHzLnH ? j1
3. Move from z1 on the constant resistance circle of
   r 0.5, upward j1 unit.
4. Read zT 0.5 j0.5.

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The transformation moves on the appropriate
constant resistance circle that represents the
resistance of the termination.
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• Series capacitors are also handled most
  conveniently in the impedance system.
• Beginning with an arbitrary impedance z1, a
  lossless series capacitor adds capacitive
  reactance, keeping the real part (resistance)
  constant.
• The sum of the impedances, zT, is computed as

On the impedance Smith Chart, addition of a
series lossless capacitor represents a downward
movement on the constant resistance circle,
toward x -?.
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• Insert a 32 pF series inductor to a one port of
  G1 0.45?-116?.
• Find the new combined input impedance, zT, at 0.1
  GHz. Use Zo 50 W for normalization.

1. Locating G1 on the normalized Impedance Smith
   Chart, convert
2. The reactance of the 32 pF capacitor
   -jXC 3.18/(jFGHzCpF) ? -j1
3. Move from z1 on the constant resistance circle of
   r 0.5, upward -j1 unit.
4. Read zT 0.5 j1.5.

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The transformation moves on the appropriate
constant resistance circle that represents the
resistance of the termination.