Smith Chart

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Smith Chart

• Impedance measured at a point along a
  transmission line depends not only on what is
  connected to the line, but also on the properties
  of the line, and where the measurement is made
  physically, along the transmission line, with
  respect to the load (possibly an antenna).

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Smith Chart

• The SMITH chart is a graphical calculator that
• allows the relatively complicated mathematical
• calculations, which use complex algebra and
• numbers, to be replaced with geometrical
• constructs, and it allows us to see at a glance
• what the effects of altering the transmission
• line (feed) geometry will be. If used regularly,
  it
• gives the practitioner a really good feel for the
• behaviour of transmission lines and the wide
• range of impedance that a transmitter may see
• for situations of moderately high mismatch
• (VSWR).

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Smith Chart

• The SMITH chart lets us relate the complex
  dimensionless number gamma at any point P along
  the line, to the normalised load impedance zL
  ZL/Zo which causes the reflection, and also to
  the distance we are from the load in terms of the
  wavelength of waves on the line

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Smith Chart

• Traveling Waves

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Smith Chart
It is a polar plot of the complex reflection
coefficient (called gamma herein), or also known
as the 1-port scattering parameter s or s11, for
reflections from a normalised complex load
impedance z r jx the normalised impedance is
a complex dimensionless quantity obtained by
dividing the actual load impedance ZL in ohms by
the characteristic impedance Zo (also in ohms,
and a real quantity for a lossless line) of the
transmission line.
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Smith Chart

• How it works
• ? (ZR Z0) / (ZR Z0)
• For Open circuited line, ZR ? , Hence
• (1-Z0/ZR) / (1ZR/Z0)1
• For Short circuited line, ZR0, Hence
• ? -Z0/Z0 -1

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Smith Chart

• How it works
• ?Vmax? A B A(1B/A)
• ?Vmin?A B A(1- B/A)
• Standing Wave Ratio ?Vmax ? to ?Vmin?
• s ?Vmax ?/ ?Vmin? (1B/A)/(1-B/A)
• Reflection Factor ?? ?B/A

1 ?? ?
s -1
?? ?
S
s 1
1- ?? ?
ZR Z0
Since ?
ZR Z0
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Smith Chart
1 ?? ?
s -1
S
?? ?
1- ?? ?
s 1
S ?
S3
??? 1
??? 0.5
0,?
1,0
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Smith Chart

• How it works

The Smith chart resides in the complex plane of
reflection coefficient G Gr Gi G ejq
G /u. At point A, G 0.6 j0.3 0.67/26.6.
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Smith Chart

• How it works

Points of constant resistance form circles on the
complex reflection-coefficient plane. Shown here
are the circles for various values of load
resistance.
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Smith Chart
Values of constant imaginary load impedances xL
make up circles centered at points along the blue
vertical line. The segments lying in the top half
of the complex-impedance plane represent
inductive reactances those lying in the bottom
half represent capacitive reactances. Only the
circle segments within the green circle have
meaning for the Smith chart
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Smith Chart

• How it works

The circles (green) of and the segments (red) of
lying within the G 1 circle combine to form
the Smith chart, which lies within the complex
reflection-coefficient (G) plane, shown in
rectangular form by the gray grid.
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Smith Chart

• How it works

With a Smith chart, you can plot impedance values
using the red and green circles and circle
segments and then read reflection-coefficient
values from the gray grid. Many Smith charts
include a scale (yellow) around their
circumference that lets you read angle of
reflection coefficient.
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Smith Chart

• How it works

Standing waves, which repeat for every half
wavelength of the source voltage, arise when (b)
a matched generator and transmission line drive
an unmatched load. (c) Time-varying sine waves of
different peak magnitudes appear at different
distances along the transmission line as a
function of wavelength.
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Smith Chart

• How it works

point L represents a normalized load impedance zL
2.5 j1 0.5/18. If point L corresponds to
G 0.5 and SWR 3, then any point in the
complex reflection-coefficient plane equidistant
from the origin must also correspond to G
0.5 and SWR 3, and a circle centered at the
origin and whose radius is the length of line
segment OL represents a locus of constant-SWR
points.
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Smith Chart

• How it is drawn
• Normalized Impedances RJ?LRjX
• Normalized Resistance rR/Z0
• Normalized Reactance xX/Z0
• Normalized Impedance zZ/Z0rjx
• Normalized Conductance ygjb
• Resistance Circles on Horizontal lines are at
  centre r/(r1),0 with radius of 1/(r1)
• Reactance Circles on Vertical lines are at centre
  1, 1/x with radius of 1/x

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Smith Chart

• Problem 1 Load Impedance
• ZR 50 j100 ? on a 50 ? line.
• Solution-
• Normalize the Impedance z 1 j2.0 ?
• On horizontal Resistance line move from zero to
  1.0 on the right
• Follow resistance circle upward(ve immag)
• Locate point where crosses reactance circle j2.0
• Point of intersection is z 1.0 j2.0 ?

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Smith Chart

• Problem-Voltage Standing Wave Ratio
• ZR100 j50? on a 50? line. Find VSWR?
• Solution
• Normalize ZzR 2 j1.0 ?
• Plot it on the chart
• Draw a circle with center at point (1,0)through
  point zR
• VSWR is 2.6

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Smith Chart

• Problem Reflection Coefficient-
• ZR 100 j75? on a 50? line Find ?? ??
• Solution
• Normalize Load zR 2 j1.5 ?
• Plot the zR on the chart
• Draw VSWR circle through zR, read VSWR 3.3
• ?? ? (s -1)/(s 1) ?
• ?? ? 0.535
• Draw a radial line through zR to meet the phase
  line. ? 0.535?300

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Smith Chart

• Problem Admittance
• ZR150 j75? on 50? line Find YR?
• Solution
• Normalize zR
• plot zR
• draw VSWR through zR from the center
• Point of intersection yR 0.27 0.14
• Admittance YR yR/50 0.0054 j0.0028S