TELECOMMUNICATIONS

TELECOMMUNICATIONS

Dr. Hugh Blanton

ENTC 4307/ENTC 5307

Dr. Blanton - ENTC 4307 - Smith Chart 2

Background

• An AT&T 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 Smith’s 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.


• 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.


• 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.


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


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

oL

oL

ZZZZ


• 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 50.


0

-j25

-j50

-j75

j75

j50

j2525 50 75 100

jXL

jXC

Ideal Inductors Pure Positive Reactance No Resistive Component

+90°

Ideal Capacitors Pure Negative Reactance No Resistive Component

-90°

Ideal Resistors No Reactive

Component

180° 0°


• 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 (=0),

and the impedance is the characteristic impedance (Z = Z0).


• Sample transformations using Z0 = 50 .

• Infinite impedance has three possible combinations:1. jX2. R + j3. R j

oL

oL

ZZZZ

050505050

050 jZL

0100 jZL0330

5010050100

.

025 jZL180330

50255025

.

00 jZL1801

500500

LZ 015050

500 jZL

01

50505000

50505050

50505050

50505050

22

jjj

jj

jj

500 jZL901

50505050

jj


• 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 chart’s circumference, and

• ideal capacitors on the lower half of the circumference.

010 25 50 100 250

j50

j50

Ideal Inductor

Ideal Capacitor


Formation of Smith Chart

For a passive system || < 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

=0.2+0.5j

125º||=0.73


• 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.

0

-j25

-j50

-j75

j75

j50

j2525 50 75 100


• 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.


83620

5025

.

)( jZ


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

Now plot constant reactance contours.

This finally gives the standard Smith chart.


• The lower part of the commercially available 50 Smith Chart includes several scales, including |, Return Loss, Mismatch Loss, and VSWR.


• Generally, Zo is 50 and the center of the chart refers to that impedance. • However at times the characteristic impedance is other than

50 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.

1

1

o

o

o

o

o

o

o

o

o

o

ZZZZ

ZZ

ZZ

ZZ

ZZ

ZZZZ

LettingoZ

Zz

11

zz


Lossless Series Inductors

• Series additions of components are most conveniently handled in the impedance system.• Beginning with a component of reflection

coefficient 1, we first convert to impedance z1.

• A lossless series inductor adds inductive reactance, keeping the real part (resistance) constant.


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

211

21121

xxjrjxjxrjxzzT

TTz 11 z


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


• Insert a 80 nH series inductor to a one port of 1 = 0.45-116. • Find the new combined input impedance, zT, at 0.1 GHz. Use

Zo = 50 for normalization.

TTz 11 z

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.

505011 .. jz

nH80

1jjX L 50501 .. jz


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


• 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:

211

21121

xxjrjxjxrjxzzT

On the impedance Smith Chart, addition of a series lossless capacitor represents a downward movement on the constant resistance circle, toward x = -.


• Insert a 32 pF series inductor to a one port of 1 = 0.45-116. • Find the new combined input impedance, zT, at 0.1 GHz. Use

Zo = 50 for normalization.1. Locating 1 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.

505011 .. jz 50501 .. jz 1jjX C

pF32


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

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