Part 3 The Smith Chart and its Applications

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• Part 3 - The Smith Chart and its Applications
• Lec. Topics
• 8. Introduction to the Smith Chart
• Principle of Operation
• Construction of the Smith Chart
• Key Points on the Smith Chart
• Using Smith Chart with Load and Line
  Combinations
• Smith Chart and General Transmission Lines
  
• Effect of Variation in Frequency
• 9. Smith Chart and VSWR
• Using the Smith Chart and VSWR to Find ZL
• Adding Components Using a Smith Chart
  
• Matching with Smith Chart and Series
  Components
• Admittance Using a Smith Chart
• Single Stub Matching

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normalised reactance, x
normalised resistance, r
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Step 1 Plot ?
Example (page 2) Measurements on a slotted
line with Zo 50O give ? 0.707 and ?
45º. Find ZL
Unit Circle in Reflection Coefficient Plane
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Step 2 Superimpose Smith Chart grid
Example (page 2) Measurements on a slotted
line with Zo 50O give ? 0.707 and ?
45º. Find ZL
Unit Circle in Reflection Coefficient Plane
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Step 3 Read off normalised values for r and x
Example (page 2) Measurements on a slotted
line with Zo 50O give ? 0.707 and ?
45º. Find ZL
, x 2
r 1
Unit Circle in Reflection Coefficient Plane
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Step 4 Calculate R and X by denormalizing ZL
Zo(r jx) 50 j100O
Example (page 2) Measurements on a slotted
line with Zo 50O give ? 0.707 and ?
45º. Find ZL
, x2
r1
Unit Circle in Reflection Coefficient Plane
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RATIO OF V-/V AT AN ARBITRARY DISTANCE l
FROM THE LOAD ZL

Zo
ZL
x -l
x 0
If the line is lossless gl jbl , hence
r-l re-2 g l re-j 2b l r ejf e-j
2b l r ej(f- 2b l) i.e. the magnitude of
the reflection coefficient at x -l is the
same as at x 0 but its phase changes from f to
f - 2b l . An additional phase change of -2b l
has been added by the introduction of the length
of line, l .
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l
ZL
Load
Transmission line
Generator
1
0.5
2
3
z rjx
? of ZL only
0.2
5
?
?
10
x
2ßl
0 0.2 0.5
1 2 3
5 10
1?0
2ßl
r
0
-10
-5
? of ZL plus line of length l
-0.2
z of load and line of length l
-3
-2
-0.5
Point rotates clockwise by 2ßl radians (2x360 l/?
degrees) at a constant radius
-1
Reflection Coefficient Plane
Smith Chart
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l
ZL
Clockwise towards generator
Load
Transmission line
Generator
1
0.5
2
3
z rjx
? of ZL only
0.2
5
?
?
10
x
2ßl
0 0.2 0.5
1 2 3
5 10
1?0
2ßl
r
0
-10
-5
? of ZL plus line of length l
-0.2
z of load and line of length l
-3
-2
-0.5
-1
Point rotates clockwise by 2ßl radians (2x360 l/?
degrees)
Reflection Coefficient Plane
Smith Chart
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Tutorial C, Question 2 Solution by Smith Chart
To find Zin

• zL ZL/Zo
• (100j50)/75
• 1.33j0.67

zL
3. 2?l 2x2?l/? 2x2?fl/v 1.26 radians 72º
4. Read z(l) from chart z(l) 1.75-j0.45
72º
2. Plot zL on Smith Chart
z(l)
5. Denormalise to find Zin Zin Zo x z(l)
75(1.75-j0.45) 131-j34 O (cf. 126-j36 O by
exact calculation)
Zo 75 O
ZL 100 j50 O
l 2.2 m, f 100 MHz v 2 x 108 ms-1
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The Smith Chart General Transmission Lines
r(l) r(0)e-2 g l r(0) e-2ale-j2bl
r(0)ejf e-2al e-j2bl
r(0)e-2alej(f-2bl)
? ? j?
? - propagation constant ? - attenuation
constant ? - phase constant
1
0.5
2
z rjx
?(0)
3
0.2
5
10
x
0 0.2 0.5
1 2 3
5 10
1?0
r
0
?(l)
-10
z of load and line of length l
-5
-0.2
-3
-2
-0.5
-1
Reflection Coefficient Plane
Smith Chart
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The Smith Chart and Variation of Frequency
r(l) r(0)e-j2bl for a lossless
line rotation angle is -2?l -2.(2?/?).l
-2.(2?.f/v).l
-4?fl/v or constant x f
1
0.5
2
?dc
zdc
3
?f1
zf1
0.2
5
10
x
0 0.2 0.5
1 2 3
5 10
1?0
?f2
r
0
zf2
-10
-5
-0.2
?f3
zf3
-3
-2
-0.5
frequency f3 gt f2 gt f1
-1
Reflection Coefficient Plane
Smith Chart
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Summary The Smith Chart is used as a graphical
aid for converting between a load impedance, Z,
and a reflection coefficient, ?. (This can be
done with or without sections of line being
present.)
? ZL
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• To avoid having to use a different Smith Chart
  for every value of Zo, the normalised impedance,
  z, is used
• z ZL/Zo
• (z r jx)
• z can then be denormalised to obtain the load
  impedance ZL, by multiplying by Zo
• ZL z.Zo
• The Smith Chart shows how the complex impedance
  plane maps on to the reflection coefficient
  circle of unit radius.
• The circles correspond to lines of constant
  normalised resistance, r.
• The arcs correspond to lines of constant
  normalised reactance, x.

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• Adding a length, l, of lossless line to a load,
  ZL, corresponds on the Smith Chart to rotating at
  constant radius from zL CLOCKWISE through an
  angle 2?l.
• (CLOCKWISE TOWARDS GENERATOR)
• If the line is not lossless, the radius
  decreases as we rotate around the centre.
• Increasing the signal frequency causes zL to
  rotate clockwise around Smith Chart at constant
  radius.

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• THE SMITH CHART AND VSWR
• The impedance of a load and line combination is
  a maximum, zmax, when
• ? - 2?l ( ?) 0
• zmax VSWR
• Intersection of circle through load point, zL,
  with right-hand half of resistance axis gives
  VSWR and zmax.
• Adding components in series with load
• Adding inductors move clockwise around constant
  r circle
• Adding capacitors move anticlockwise around
  constant r circle
• Adding resistors move along arc of constant x
  towards r-axis