11/17/2014
EE   4347
Lecture   #5
Lecture   6
notes   may   contain   copyrighted   material   obtained   under   fair   use   rules.
Distribution   of   these   materials   is   strictly   prohibited
Slide   1
• Construction   of   the   Smith   Chart
• Admittance   and   impedance
• Circuit   theory
• Determining   VSWR   and

• Impedance   transformation
• Impedance   matching
Lecture   6 Slide   2
1

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2

The   Smith   chart   is   based   on   a   polar   plot   of   the   voltage   reflection   coefficient    .
The   outer   boundary   corresponds   to   |  |   =   1.
The   reflection   coefficient   in   any   passive   system   must   be|

|  ≤  1.
    e j 
  radius on Smith chart
  angle measured CCW from right side of chart
Smith   Charts

5
All   impedances   are   normalized.
This   is   usually   done   with   respect   to   the   characteristic   impedance   of   the   transmission   line   Z
0
.
z 
Z
Z
0
Smith   Charts 6
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3

We   can   write   the   reflection   coefficient   in   terms   of   normalized   impedances.
 
Z
L
 Z
0
Z
L
 Z
0

Z
L
Z
0
Z
L
Z
0

Z
0

Z
0
Z
0  z
Z
0 z
L
L
 1
 1
Smith   Charts
Derivation   of   Smith   Chart:
Solve for Load Impedance
Solving   the   previous   equation   for   load   impedance,   we   get
  z
L z
L
  z
L
1
 z
L



1
1
1 z
L
    z
L
 1 z
L z
L z
L

1
 z
L

1  
1   z
L

1  
1  
Smith   Charts 8
7
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4

Derivation   of   Smith   Chart:
Real and imaginary parts
The   load   impedance   and   reflection   coefficient   can   be   written   in   terms   of   real   and   imaginary   parts.
z
L r jx
L j i
Substituting   these   into   the   load   impedance   equation   yields z
L
 r
L
 jx
L
 r
L
 jx
L

1  
1
1
1
 




1
1

 j j j j i i

 i i
Smith   Charts 9
Derivation   of   Smith   Chart:
Solve for r
L and x
L
We   solve   or   previous   equation   for   r
L imaginary   parts   equal.
and   x
L by   setting   the   real   and   r
L
 jx
L







1
1

 j j
1
1

 j j i i i i



1
1

 j j i i

1   r

1   
 j
1
1
  r
 2

  i
2 j i

1
 i
2

1 j r i i
2

1 r j i

1 j r i
  r
 2 j
  i i
2

1 r
  r

2 i
2 j
 
2 i
2 i


1
1

2 r r i
  2   i
2
 j

1  
2  r
 2 i
  i
2
Smith   Charts r
L
 x
L


1
1
  r r
 2 i
2
  i
2  i

1   r
 2   i
2
2
10
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5

Derivation   of   Smith   Chart:
Rearrange equation for r
L
We   rearrange   the   equation   for   r
L so   that   it   has   the   form   of   a   circle.
r
L


1
1

2 r r i
  2   i
2

1   r
 2   
1 r r
L i
2

1   r
 2    r
1
L

 r
L
2 r 
 r
L i
2
 0
2 r r
 r
L
2 r   
 r
L i
2
1
1 r
L
 0
2
2 r r
L r
L r
 r 2
L r r
L
1 r
2 r r
L
L

 can be factored r
1
  i
2

2 r

2 r r
L r
L r
L i
2
 r
L
 1
 1
1
 0
 i
2 i
2 r
L r
L
1 0









 r
L r
L
 1


2


 r
L r
L
 1
 2 r
L r
L
 1

2 i

 r
L r
L
 1

2 r
L r
L
 1
 2 i r
L
2
 r
L
 1
 2 r
L r
L
 1
 1
 0
 r
L r
L
 1
 1

 r
L

 r
L
1
 r
L
 1
 2
 1
 r
L r
L
 1
 2
 r
L r
L
2
 1
  r
L
2 r
L

 1
1
 2 r
L r
L
 1
 2
1
 r
L
 1
 2
Smith   Charts 11
Derivation   of   Smith   Chart:
Rearrange equation for x
L
We   rearrange   the   equation   for   x
L so   that   it   has   the   form   of   a   circle.
x
L


1  
2  i r
 2   i
2

1   r
 2   
2  i x
L

1   r swap terms
 2    x
2
L
  can be factored
0
   1
 2   


 1 x
L



2
 x
1
2
L
 0
Smith   Charts 12
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6

Derivation   of   Smith   Chart:
Two families of circles
Constant   Resistance   Circles


 r
L r
L
 1



2 i
These   have   centers   at r
L r
L
 1
  0
Radii
1 
1 r
L


 1 
1 r
L



2
Constant   Reactance   Circles
   1
 2   


 1 x
L



2
 

 x
1
L



2
These   have   centers   at
1  
1 x
L
Radii
1 x
L
Smith   Charts 13
Derivation   of   Smith   Chart:
Putting it all together
Lines   of   constant   resistance
Lines   of   constant   inductive   reactance
+
Lines   of   constant   capacitive   reactance
+
Lines   of   constant   reflection   coefficient
=
Supe rpo sit ion
We   ignore   what   is   outside   the   |

|   =   1   circle.
We   don’t   draw   the   constant   |  |   circles.
This   is   the   Smith   chart !
Smith   Charts 14
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7

Alternate   Way   of   Visualizing   the   Smith
Chart
Lines of constant resistance Lines of constant reactance Reactance Regions short circuit
L
C open circuit
Smith   Charts 15
The   3D   Smith   Chart   unifies   passive   and   active   circuit   design.
2D 3D
11/17/2014
EE3321  ‐‐ Final   Lecture
8

Smith   Charts
17
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9

We could have derived the Smith chart in terms of admittance.
You can make an admittance Smith chart by rotating the standard
Smith chart by 180  .
Smith   Charts
Impedance/Admittance
Conversion
The Smith chart is just a plot of complex numbers. These could be admittance as well as impedance.
To determine admittance from impedance (or the other way around)…
1. Plot the impedance point on the Smith chart.
2. Draw a circle centered on the Smith chart that passes through the point (i.e. constant VSWR).
3. Draw a line from the impedance point, through the center, and to the other side of the circle.
4. The intersection at the other side is the admittance.
impedance admittance
19
Smith   Charts 20
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10

Visualizing   Impedance/Admittance
Conversion
11/17/2014
Smith   Charts
Example   #1   – Step   1
Plot   the   impedance   on   the   chart z  0.2
 j 0.4
21
Smith   Charts 22
11

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Example   #1   – Step   2
Draw   a   constant   VSWR   circle z  0.2
 j 0.4
Smith   Charts
Example   #1   – Step   3
Draw   line   through   center   of   chart z  0.2
 j 0.4
23
Smith   Charts 24
12

Example   #1   – Step   4
Read   off   admittance z  0.2
 j 0.4
y  1.0
 j 2.0
Smith   Charts
Example   #2   – Step   1
Plot   the   impedance   on   the   chart z  0.5
 j 0.3
25
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Smith   Charts 26
13

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Example   #1   – Step   2
Draw   a   constant   VSWR   circle z  0.5
 j 0.3
Smith   Charts
Example   #2   – Step   3
Draw   line   through   center   of   chart z  0.5
 j 0.3
27
Smith   Charts 28
14

Example   #2   – Step   4
Read   off   admittance z  0.5
 j 0.3
y  1.0
 j 2.0
Smith   Charts

29
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15

1. Plot the normalized load impedance on the Smith chart.
2. Draw a circuit centered on the Smith chart that intersections this point.
3. The VSWR is read where the circle crosses the real axis on right side.
Example: 50  line connected to 75+ j 10  load impedance.
z 
Z
L 
Z
0
75  j 10
50
 1.5
 j 0.2
impedance
1
VSWR
VSWR = 1.55
31 Smith   Charts
3.3157 nH
50 
1.9894 pF
Z in z in

 20  j 40 
Z
Z in
0

20  j 40 
5 0 
 0.4
 j 0.
8
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Smith   Charts 32
16

Example   #1   –What   is   the   reflection   coefficient?
Smith   Charts
  0.62
33
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17

Normalized   Impedance   Transformation
Formula
Our   impedance   transformation   formula   was
Z in
 Z
0
Z
L
 jZ
Z
0
 jZ
L
0 tan tan
 
 
We   can   write   this   in   terms   of   the   reflection   coefficient.
Z in
 Z
0
Z
L
Z
0 cos cos


 
  jZ
0 jZ
L sin  sin 


 Z
0
 Z
0
1 
 Z
0
1  j   j  


 j  
 j  




Z
Z
L
L


Z e

  j  j 




Z
Z
L
L


Z e

  j  j 

 j   j  


0.5
Z e
0.5

 j  j 


 j  
 j  
 Z
0

 e  e
 j   j  



 0.5
0.5




Z
Z
L
L


Z e

 j   j  

 j   j  

 e  e
 j   j  




Z
L
 Z e
Z
L
 
  j  
 j  
 Z
0
1   e

1   e
 j 2   j 2  
We   normalize   by   dividing   by   Z
0
.
z in

1   e  j 2  
1   e  j 2  
Smith   Charts
The   normalized   impedance   transformation   formula   was z in

1   e 
1   e  j 2   j 2  
Recognizing   that
 = |  | e j 
,   this   equation   can   be   written   as z in

1  
1  
 j 2   e e  j 2  

1  
1   e j
   2  
 e j
   2  

Thus   we   see   that   traversing   along   the   transmission   line   simply   changes   the   phase   of   the   reflection   coefficient.
As   we   move   away   from   the   load   and   toward   the   source,   we   subtract   phase   from

.
On
Smith   chart,   we   rotate   clockwise   (CW)   around   the   constant   VSWR   circle   by   an   amount   2 the
 l .
A   complete   rotation   corresponds   to

/2.
Smith   Charts 36
35
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18

Impedance   Transformation on   the   Smith   chart
1. Plot the normalized load impedance on the Smith chart.
2. Move clockwise around the middle of the Smith chart as we move away from the load (toward generator). One rotation is  /2 in the transmission line.
3. The final point is the input impedance of the line.
11/17/2014
Smith   Charts
Example   #2   – Impedance   Trans.
Normalize   the   parameters
0.67

Z
0
 50  Z
L
 50  j 25 
0.67
 z
L
  j 0.5  z in
 z
L
 j
1  jz
L tan tan






1  j 0.5
  j tan 2   0.67
 

1  j

1  j 0.5 tan 2   0.67


 1.299
 j 0.485
Smith   Charts
37
38
19

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Example   #2   – Impedance   Trans.
Plot   load   impedance
0.67
 z
L
  j 0.5 
Smith   Charts
Example   #2   – Impedance   Trans.
Walk   away   from   load   0.67

0.67
 z
L
  j 0.5 
Since   the   Smith   chart   repeats   every   0.5

,   traversing   0.67
 is   the   same   as   traversing   0.17

.
Here   we   start   at   0.145
on   the   Smith   chart.
We   traverse   around   the   chart   to
0.145
+   0.17
=   0.315.
Smith   Charts
0.145
39
40
20

Example   #2   – Impedance   Trans.
Determine   input   impedance
0.67

Z in z
L
  j 0.5 
Reflection   at   the   load   will   be   the   same   regardless   of   the   length   of   line.
Therefore   the   VSWR   will   the   same.
The   input   impedance   must   lie   on   the   same   VSWR   plane.
z in
 1.3
 j 0.5
Smith   Charts
Example   #2   – Impedance   Trans.
Denormalize
Z in
0.67
 z
L
  j 0.5 
To   determine   the   actual   input   impedance,   we   denormalize.
Z in
 Z z      j 0.5
  65  j 
Smith   Charts
41
42
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21