AN INNOVATIVE WAY FOR COMPUTERIZED SMITH CHART GENERATION AND TRANSMISSION LINE PROBLEM SOLVING
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eISSN: 23191163  pISSN: 23217308
AN INNOVATIVE WAY FOR COMPUTERIZED SMITH CHART
GENERATION AND TRANSMISSION LINE PROBLEM SOLVING
Sangita Choudhury1, Namrata Kataki2
1Assistant Professor, EE, GIMT, Guwahati, Assam, India
2Assistant Professor, ECE, GIMT, Guwahati, Assam, India
Abstract
The Smith chart is one of the most useful graphical tools for high frequency circuit applications. It provides a clever way to
visualize complex functions and solving problems with transmission lines and matching circuits. Here a computerassisted
learning technique is presented using C language for making lively and dynamic problem solving interface, by which one can
implement the smith chart and solve problems easily related to transmission line and matching circuit. The concept of explaining
the use of an important old tool like the Smith Chart, using modern tool like C to explain the use of the Smith Chart represents a
prime example of the melding of the traditional and the leading edge. Here, we have solved some of the most common
transmission line problems using this software and have compared the results of both experimental calculations as well as manual
calculations.
Keywords: Smith chart, Transmission line, Impedance matching Reflection coefficient, VSWR, Normalized
Impedance, Normalized admittance
***
1. INTRODUCTION
nomographs used in the practice of electrical engineering
[5].
Smith chart is a graphical tool for solving transmission line
problems which was developed by P. H. Smith [1]. It reveals
graphically the complex impedance anywhere along a line.
The smith chart is very useful when solving transmission
problems. The real utility of the Smith chart, it can be used
to convert from reflection coefficients to normalized
impedances (or admittances), and vice versa [2]. The
mathematics of transmission lines becomes cumbersome at
times, especially when dealing with complex impedances
and nonstandard situations. In 1939, Phillip H. Smith
published a graphical device for solving these problems, the
Smith Chart. It consists of a series of overlapping
orthogonal circles that intersect each other at right angles.
These sets of orthogonal circles make up the basic structure
of the Smith chart and are shown in Fig. 1[3]. When
illustrating fundamental concepts the Smith chart is the ideal
tool, especially at the level where concepts are being
Fig1: A Smith chart
explained to students and when they are exposed to these
concepts for the first time. The Smith chart provides an
2. TRANSMISSION LINE PARAMETERS
extremely useful way of visualizing transmission line
The following transmission line parameters are associated
phenomenon, and so is also important for pedagogical
with this method
reasons. Microwave and RF engineers can develop intuition
about transmission line impedance matching problems by
2.1 Reflection Coefficient
learning to think in terms of the Smith chart [4]. In the
modern age, using the Smith chart to perform calculations
The reflection coefficient is used in physics and electrical
has gone the way of the slide rule; such calculations are best
engineering when wave propagation in a medium containing
done using a computer program. However, the utility of the
discontinuity is considered. It is designated by (gamma)
Smith chart to enhance the understanding of complex
and is defined as
impedances has not diminished. Indeed, use of the Smith
chart has grown over the decades since its invention. The
= reflected voltage or current/incident voltage or current
Smith chart remains one of the most widely used
= V ref /V inc
= I ref /I inc
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
2.2 Transmission Coefficient
1 +
+
+ j xL =
A
transmission
line
terminated
in
its
characteristic
1  
impedance
Z
2
2
o
is called a properly terminated
line.
1  
According to the principle of conservation of energy, the
=
2
2
incident power minus the reflected power must be equal to
1 
+
the power transmitted to the load. This can be expressed as
2
+
1  2 + 2
1
r = (Zo/Zr ).T
where r ,
where T is the transmission coefficient and is defined as
i are real and imaginary part of reflection co
efficient respectively and , are real and imaginary part
of normalized impedance respectively
T=transmitted voltage or current / incident voltage or current
= Vtr / Vinc
(6) Equations for drawing constant resistance and constant
=
Itr / Iinc
reactance circle
2
1
2
2.3 Voltage Standing Wave ratio

+
 0 2 =
+ 1
+ 1
The ratio of the maximum of the voltage standingwave
1 2
1 2
pattern to the minimum is defined as the voltage standing
 1 2 + 
=
wave ratio ( ) and is given by
=VSWR=Vmax/Vmin
4. SOFTWARE IMPLEMENTATION STEPS
4.1 Preparation of Smith Chart
Also, = [( 1 +) /(1)]
A circle is drawn with a horizontal line passing
2.4 Normalized Impedance
through its centre
The circles of constant resistance and conductance
The normalized impedance with characteristic impedance
are drawn
(Zo) may be calculated by dividing the load impedance (ZL)
The lines of constant reactance and susceptance are
by the characteristic impedance i.e. zL= ZL / Z0
drawn
Two more outer circles (inner wavescale and outer
For a lossless transmission line
wavescale) are drawn and marked
The maximum normalized impedance, Zmax=1 +  /1  
4.2 To Determine Reflection Coefficient
2.5 Normalized Admittance
The input values are load resistance, load reactance
and characteristic impedance
The normalized is the reciprocal of normalized impedance.
After that the normalized resistance and normalized
i.e. y = ZO / ZL = 1/ zL
reactance are calculated
Then the constant resistance and constant reactance
Actual admittance = y/Zo
circles are drawn
The
magnitude
of
reflection
coefficient
is
2.5 Stub
calculated
from the
origin to
the
point
of
Impedance matching can be achieved by inserting a section
intersection of the two circles
of short circuited transmission line (in shunt with the main
The angle is calculated from the slope of the line
line) in between the load and source is called stub. This
process is called stubmatching.
4.3 To Determine VSWR
3. MATHEMATICAL FORMULATION
Repeat the steps of determination of reflection co
efficient as given in (4.2)
The following formulas are used in implementation of the
Taking the reflection coefficient as the radius, the
software [6]
VSWR circle is drawn with origin at
the centre of
(1) characteristic impedance Z0 = R0 + j X0
the screen
(2) load impedance ZL = RL + j XL
Wherever the circle cuts the horizontal line in the
(3)
right hand side, it is the location of VSWR and in
L
ZZ
0
zL
1
=
the left hand side, it is 1/ VSWR
L
ZZ
0
zL 1
where normalized load impedance, zL= ZL / Z0
4.4 To calculate Admittance Point
(4) = r + j =  e j
i
(5) normalized load impedance,
Repeat the steps of calculation of VSWR as given
in (4.3)
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
The line drawn from the origin to the point of
The input admittance point is determined as
intersection of the two circles to find magnitude of
mentioned above. Location of this point is found
reflection coefficient is extended in opposite
from wavescale
direction and finds the point where it cuts
the
The point at which VSWR circle cuts the unit
VSWR circle
conductance circle is determined. The location of
Then the normalized conductance and normalized
the above point is found from the wavescale
susceptance
is
calculated.
This
gives
the
The difference between the abobe two points gives
normalized admittance point
the location nearest to the load at which the real
part of the admittance equals the line characteristic
4.5 To Calculate Zmin and Zmax
admittance
Repeat the steps of determination of reflection co
4.10 To Continue or Exit
efficient as described in (4.2)
Taking the reflection coefficient as the radius, the
The program asks the user if he wants to continue
VSWR circle is drawn with origin at
the centre
with with the program or not by choosing the
of the screen
option y (to continue) and n (to exit)
Wherever the circle cuts the horizontal line in the
right hand side, it is the location of
Zmax and
5. SOFTWARE APPLICATIONS
in the left hand side, it is Zmin
The implemented software can be used in the following
cases
4.6 To Calculate Voltage Minimum (Vmin)
Determination of reflection co efficient from a
Repeat the steps of determination of reflection co
given load impedance
efficient as described in (4.2)
Determination of VSWR from a given load
The location of the reflection coefficient point is
impedance
found from the wavescale and deducted from the
Determination of
load admittance from a given
total wavescale value i.e. 0.5 . This gives the
load impedance
location of voltage minimum
Determination of Zmax and Zmin (maximum
impedance and minimum impedance respectively)
4.7 To Calculate Input Impedance and Admittance
Determination of distance of first voltage minimum
of the standing wave pattern from a given load
Repeat the steps of determination of reflection co
impedance
efficient as described in (4.2)
Determination of input impedance and input
The reflection coefficient as the radius, the VSWR
admittance from a given load impedance at a given
circle is drawn with origin at the centre of the
distance
screen
Determination of location of the point at which real
The location of the above point is found from the
part of the line admittance is equal to the line
wavescale and the given distance is added to this
characteristic impedance
from a given load
value and new location is found. Now a line is
impedance
drawn from this point to the centre of the VSWR
Determination of location of load and stub from a
circle. The point of intersection of the line with
given load impedance and operating frequency
VSWR circle is determined. Now, resistance and
reactance of the point are found. This gives the
6. EXPERIMENTAL RESULTS
input impedance
Similarly input admittance is also determined
In this column various transmission line related problem
solving
results
(found
by
both
experimentally
and
4.8 To Calculate Location of Stub
manually) along with software outputs are listed in table (1
8) and figure (29).
The location of the admittance point is found as
done in step 4.4
Table 1
The location of the admittance point is found from
Determination of Reflection Coefficient
the wavescale. This gives the location of the load
Given Data
load Impedance, Z= (27.5+ j 80)
The point at which VSWR circle cuts the unit
Characteristic impedance, Zo= 50
conductance circle is determined. The location of
Experimental
the above point is found from the wavescale. This
results
0.746 < 59.80
gives the location of the stub
Manually
calculated
0.75 < 600
4.9 To Calculate Location at which Real Part of the
value
Admittance
Equals
the
Line
Characteristic
Figure No
Fig 2
Impedance
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
Table 2
Table 6
Determination of VSWR
Determination of input impedancee and input admittance
from a given load impedance at a given distance
Given Data
Load Impedance, Z= (50 j 100)
Given Data
load Impedance, Z= (150 j 20)
Characteristic impedance, Zo= 75
characteristic impedance, Zo= 50
Experimental
given distance = 0.05
results
of
the
4.616
Experimental
normalized impedance = (0.257j 0.08)
software
results
of
the
normalized admittance = (3.4 + j 1.19)
Manually
software
calculated value
4.6
actual impedance = (12.89 j 4.44)
Figure No
Fig 3
actual admittance = (0.069 + j 0.023)
mho
Table 3
Manually
normalized impedance = (0.22j 0.1)
Determination of Load admittance
calculated value
normalized admittance = (3.5 + j 1.1)
Given Data
load Impedance, Z= (50+ j 100)
actual impedance = (13.5 j 5)
characteristic impedance, Zo= 50
actual admittance = (0.065 + j 0.022)
Experimental
normalized admittance = (0.2j 0.4)
mho
results of the
actual admittance = (0.004 j 0.008) mho
Figure No
Fig 7
software
Manually
normalized admittance =( 0.2j 0.4)
calculated
actual admittance=(0.004 j0.008) mho
Table 7
value
Determination of location of the point at which real part of
Figure No
Fig 4
line admittance is equal to the line characteristic
impedance from a given load impedance
Table 4
Given Data
load Impedance, Z= (15 j 20)
Determination of maximum and minimum impedance
characteristic impedance, Zo= 50
(Zmax and Zmin)
Experimental
0.1396
results
of
the
Given Data
load Impedance, Z= (50 j 100)
software
characteristic impedance, Zo= 75
Manually
0.14
Experimental
Zmax = 4.61
calculated value
results
of
the
Zmin = 0.216
Figure No
Fig 8
software
Manually
Zmax = 4.6
calculated value
Zmin = 0.22
Table 8
Determination of location of load and stub and also the
Figure No
Fig 5
input susceptance of the stub from a given load impedance
and operating frequency
Table 5
Determination of distance of 1st voltage minimum of the
Given Data
load Impedance, Z= (30 j 40)
standing wave pattern from a given load impedance
characteristic impedance, Zo= 50
Experimental
location of the load = 0.1249
Given Data
load Impedance, Z= (150 j 20)
results of the
location of the stub = 0.166
characteristic impedance, Zo= 50
software
location of the stub from load= 0.0083 m
Experimental
distance of first voltage minimum =
input susceptance of the stub= 0.0230
results
of
the
0.065
mho
software
Manually
location of the load = 0.125
Manually
distance of first voltage minimum =
calculated
location of the stub = 0.165
calculated value
0.065
value
location of the stub from load = 0.008 m
Figure No
Fig 6
input susceptance of the stub= 0.0236
mho
Figure No
Fig 9
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Volume: 03 Issue: 09  Sep2014, Available @ http://www.ijret.org
293
IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
Fig 2: Software output for determining reflection co
Fig 5: Software output for determining maximum and
efficient
minimum impedance
Fig 6: Software output for determining distance of first
Fig 3: Software output for determining VSWR
voltage minimum from a given load impedance
Fig 7: Software output for determining input impedance
and input admittance from a given load impedance at a
Fig 4: Software output for determining load admittance
given distance
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IJRET: International Journal of Research in Engineering and Technology
eISSN: 23191163  pISSN: 23217308
to handle. It also requires very less time to execute.
Furthermore, the computations are more accurate and the
results can be displayed and interpreted in a more effective
manner.
REFERENCES
[1]. P. H. Smith Transmission line calculator//Electronics,
1939Vol.12No.1P.3942.
[2]. R. Rajoria, Patch Antenna using Metamaterial
Structure for Performance Specification, IJESRT, 3(4),
April2014.
[3]. J. J. Carr, Practical Antenna Handbook, fourth edition,
McGrawHill, New York, 2001.
[4]. David M. Pozar, Microwave and RF Design of Wireless
Systems, First Edition. New York: John Wiley & Sons
Fig 8: Software output for determining location of the point
(Publishers), 2001.
at which real part of line admittance is equal to the line
[5]. Ward HarrimanAE6TY, A New Program for the
characteristic impedance from a given load impedance
Venerable Smith Chart, The QRP Quarterly, Winter 2011,
page 4753, www.qrparci.org.
[6]. Samuel Y. Liao, Microwave devices and circuits, third
edition, Prentice Hall Publication, 1996.
[7]. G. Fikioris, Analytical studies supplementing the Smith
Chart, IEEE Transaction, Vol. 47, Issue.2, May2004.
[8]. Jose R. Pereira and Pedro Pinho, Using the Smith
Chart
in
an
ELearning
Approach,
ELearning
Organizational Infrastructure and Tools for Specific Areas,
Prof. Adilson Guelfi (Ed.), 2012, ISBN: 978 953 51 0053
9.
[9]. Fong Mak and Ram Sundaram, A MatlabBased
Teaching Of The TwoStub Smith Chart Application For
Electromagnetics Class, 38th ASEE/IEEE Frontiers in
Education Conference, October 22  25, 2008, Saratoga
Springs, NY.
Fig 9: Software output for determining location of load and
stub and also the input susceptance of the stub from a given
load impedance and operating frequency
7. CONCLUSIONS
The study of transmission lines is basic in undergraduate
electrical and electronics engineering education. In relevant
courses, a key subject is the Smith chart , which is a
graphical device providing insight [7]. Understanding the
Smith chart is still important nowadays, despite the present
generalization
of
personal
computers
and
powerful
calculators. It is easy to plug a few numbers into a program
and have it spit out solutions. When the solutions are
complex and multifaceted, having a computer to do the
grunt work is especially handy. However, knowing the
underlying theory and principles that have been ported to
computer platforms, and where they came from, makes the
engineer or designer a more wellrounded and confident
professional, and makes the results more reliable. Moreover
it is interesting to note that these kinds of graphical tools are
still useful now adays [8]. This software provides a powerful
computation and display platform by means of which
fundamental and advanced concepts relating to transmission
lines can be understood and implemented [9]. This software
is very simple for beginners, cost effective, reliable and easy
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Volume: 03 Issue: 09  Sep2014, Available @ http://www.ijret.org
295