9781424453863/09/$26.00 ©2009 IEEE
Interactive Smith Chart for Microwave Engineering
Students
Francisco Ramos
Departamento de Comunicaciones. ETSI Telecomunicación. Universidad Politécnica de Valencia.
Camino de Vera, s/n. 46022 Valencia (Spain)
framos@upvnet.upv.es
Abstract— An electronic Smith chart for the design and analysis
of transmission line circuits is shown. To our knowledge, this is
the first interactive Smith chart fully developed in Flash. The
applet is free available through Internet. The successful feedback
received from both students and instructors confirms that it is a
powerful educational resource.
I. I
NTRODUCTION
The Smith chart is one of the most important tools for the
microwave engineer [1]. The great advantage of the Smith
chart is that it is essentially a “graphical calculator” that
allows the relatively complicated mathematical calculations
involved in the design of transmission line circuits, which use
complex algebra and numbers, to be replaced with
geometrical constructs. This allows impedancematching
solutions to be found by drawing on the Smith chart rather
than by performing lengthy calculations. In addition, it allows
us to see at a glance what the effects of altering the
transmission line characteristics and the lumped elements will
be. Originally the Smith chart was a paperbased design aid.
The design engineer would then perform his impedance
matching exercises using pencil, ruler, and compass. However
this calculation method is not efficient and leads to limited
accuracy because the errors are accumulated with each step.
Recent advances in multimedia technologies provide an
exciting opportunity to significantly enhance the teaching
procedures. Nowadays there are lots of digital learning objects
stored in numerous repositories as well as plenty of simulators
available through Internet either commercially or in the form
of public domain tools. These developments are usually based
on Java applets or Flash animations. Java is usually employed
when you need speed or have a very complex simulation.
However, Flash has more advantages for elearning
applications. It can be executed on more machines (Flash
player has more than a 95% penetration among Internet users),
requires a smaller plugin, allow vector graphics and their
userfriendly graphics tools provide an easier way to program
2D and 3D animations using the last version of the powerful
programming language called ActionScript 3.0.
In this paper, an electronic Smith chart for the design and
analysis of transmission line circuits is shown. To our
knowledge, this is the first interactive Smith chart fully
developed in Flash. More sophisticated computeraided design
tools or software packages are also available [2]. However,
the main advantages of our approach are its simplicity,
usability and the possibility of being executed from a web
browser without the need of installing any software. The
advantages for both students and instructors are clear. On one
hand, students have a quicker method to asses the theory and
validate the solutions to the problems prepared by the
instructor. On the other hand, instructors have an useful tool to
explain the theory in a smart way, avoiding static slides or
complex chalk diagrams to show inclass examples and
practical exercises. The successful feedback received from all
the parties confirms that we are on the correct way.
II. S
MITH
C
HART
B
ASICS
The Smith chart was devised and developed by Philip H.
Smith [3]. During his career working as a transmission line
engineer at Bell Telephone Laboratories in New Jersey, Smith
published two key papers [45], which described his work.
The wellknown Smith chart is shown in Fig. 1. Basically, it is
a conformal mapping between the normalized complex
impedance plane and the complex reflection coefficient plane.
The normalized impedance is given as
jxr
Z
jXR
Z
Z
Z
LLL
L
+=
+
==
00
, (1)
where Z
0
is the characteristic impedance of the transmission
line, and r and x are the normalized resistance (real part) and
reactance (imaginary part), respectively. On the other hand,
the reflection coefficient is calculated as
jvu
Z
Z
ZZ
ZZ
L
L
L
L
+=
+
−
=
+
−
=Γ
1
1
0
0
, (2)
where u and v are its real and imaginary parts, respectively. So
we also can obtain that
Γ−
Γ+
=
1
1
L
Z
. (3)
The Smith chart is obtained by performing a transformation
of the positive real impedance plane into a reflection
coefficient circle of radius 1, according to (1)(3) [1]. These
mathematical relations imply the following properties:
1) Constant resistance or reactance lines in the impedance
plane are transformed into circles in the reflection
coefficient plane (see Fig. 1).
2) The normalized admittance is defined as the inverse of the
normalized impedance:
jbg
Z
Y
L
L
+==
1
, (4)
where g and b are the normalized conductance (real part)
and susceptance (imaginary part), respectively. Therefore,
the reflection coefficient can also be calculated as
L
L
Y
Y
+
−
=Γ
1
1
. (5)
By comparing (2) and (5), we see that (g, b) admittance
values can be read on the Smith chart from (r, x)
impedance values by adding a π phase shift to the
reflection coefficient.
3) Displacements through the transmission line (toward
generator or load) are represented on the Smith chart by
movements over a circle centred at the origin (e.g. only the
phase of the reflection coefficient is changed, assuming a
lossless transmission medium). A displacement L gives a
change in the phase of the reflection coefficient of:
L
λ
π
±=θ∆
4
, (6)
where λ is the wavelength and the sign is positive or
negative for a displacement toward load or generator,
respectively. Therefore, according to the previous property
an impedance inverter (impedance to admittance converter)
can be implemented by a λ/4 transmission line (π phase
shift). Additionally, the complex reflection coefficient and
impedance repeats every half wavelength along the
transmission line.
III. I
NTERACTIVE
S
MITH
C
HART
The Flashbased interactive Smith chart developed for
educational purposes [6] is shown in Fig. 2. As it can be seen,
the application window has 4 zones. The upperleft one is an
schematic diagram of the Smith chart where the reflection
coefficients are placed dynamically on it and the calculated
data values are shown on the information panel. On the other
hand, the lowerright zoom window shows a more detailed
view of the inset placed on the Smith chart. The data values
for impedances, admittances, reflection coefficients or
displacements along the transmission line can be set in the
input data boxes shown at the upper right corner. Additionally,
several buttons are available for doing some calculations and
transformations on the Smith chart diagram. All the
previouslyexplained properties may be checked by using
these input data boxes and buttons. Finally, the application has
the possibility of calculating more complex transmission line
circuits comprised of serial and parallel connections of
lumped elements and transmission lines, with the aim of
solving transmission line and impedance matching exercises.
The problem solving tool is shown at the lowerleft corner of
the application window. Without any doubt this is the most
powerful tool.
A. Problem solving and impedance matching examples
In order to address the capabilities of the application, an
exercise is proposed. We want to calculate the input
impedance of the transmission line circuit shown in Fig. 3.
The data values are: Z
0
= 50 Ω, Z
L
= 35 Ω, Y
P
= 0.05 – j0.1 S
and L = 0.3λ.
As a first step, we need to normalize the impedance and
admittance values:
Fig. 1 The conventional normalized impedance Smith chart
Fig. 2 Interactive Smith chart
Z
0
Z
L
L
Z
in
Z
b
Y
P
Z
a
Fig. 3 Proposed exercise
( )
⎪
⎩
⎪
⎨
⎧
−=−×==
===
55.21.005.050
7.0
50
35
0
0
jjYZY
Z
Z
Z
PP
L
L
The
L
Z
value can be placed on the Smith chart by filling the
input boxes values and pressing the “Place Z” button. We see
that (r, x) = (0.7, 0) corresponds to a reflection coefficient of Γ
= 0.176
180º
. The problem solving tool is activated by choosing
the “toward generator” option from the combo box. Now a
transmission line of 0.3λ length can be added by filling the
input box value and pressing the “Generator” button. A
normalized
b
Z
= 1.299 – j0.278 can be read from the
information panel. Finally, a parallel admittance,
P
Y
, is placed
by filling the input boxes values and pressing the “Place Y”
button. After that, the information panel gives an input
impedance value of
in
Z
= 0.095 + j0.143 (see Fig. 4), which is
equivalent to a denormalized value of Z
in
= 4.75 + j7.15.
Suppose that you now want to calculate impedance
matching conditions. This can simply be done by clicking the
“Matching” checkbox. Then a window opens showing the
length of transmission line needed (see Fig. 4). In this case,
we can choose the shortest length toward generator. Therefore,
impedance matching can be done with a 0.18λ−length
transmission line span and a serial lumped element of a value
x = −2.965 (see Fig. 5). In a similar way, problem solving
toward load can also be selected.
B. Programming
The interactive Smith chart was implemented in Flash
using Adobe
®
Flash
®
CS3 Professional. This software
package is the industryleading authoring environment for
creating engaging interactive experiences. The programming
was based on ActionScript 3.0, a powerful, objectoriented
programming language that signifies an important step in the
evolution of the capabilities of the Flash Player runtime.
ActionScript 3.0 consists of two parts: the core language and
the Flash Player API. The core language defines the basic
building blocks of the programming language, such as
statements, expressions, conditions, loops, and types. The
Flash Player API is made up of classes that represent and
provide access to Flash Player specific functionality.
Moreover, ActionScript 3.0 contains a host of powerful new
features that can greatly speed the development process. The
new Display List API makes working with visual objects far
more straightforward and consistent. The standardized DOM
event model cements the way those objects talk and respond
to each other at runtime. These are only a few of the many
new capabilities of ActionScript 3.0.
As an example, a piece of code is shown below, where the
simplicity of programming can be observed:
...
rho.angle = 4 * Math.PI * desp;
rho.real = rho.modulo * Math.cos(rho.angle);
rho.imag = rho.modulo * Math.sin(rho.angle);
rho.angle = Math.atan2(rho.imag, rho.real);
if (rho.angle<0) { rho.angle += 2 * Math.PI; }
...
situa_rho();
muestra_datos();
...
In addition, the file size for the full application was around
1 MB, so it can be fast executed online from a web browser.
IV. C
ONCLUSIONS
An interactive Smith Chart which is fullydeveloped in
Flash has been shown. This educational resource, aimed to
microwave engineering students, allows for the design and
calculation of transmission line circuits and impedance
matching exercises. The students found this application
attractive, innovative, userfriendly and with big advantages in
testing the level of knowledge achieved. The applet is free
available through Internet and it can be simply executed from
a web browser. As the feedback received from both students
and instructors has been very positive, we are continuously
working on updating the application with new features.
Fig. 4 Problemsolving example using the interactive Smith chart
Fig. 5 Impedancematching example using the interactive Smith chart
R
EFERENCES
[1] C.W. Davidson, Transmission Lines for Communications, London:
MacMillon, 1989.
[2] Agilent Technologies. Smith Chart Utility. [Online]. Available:
http://eesof.tm.agilent.com/products/e5600a_smith_chart_util.html
[3] R. Rhea, Philip H. Smith: A Brief Biography, New York: Noble, 1995.
[4] P.H. Smith, “Transmissionline calculator,” Electronics, vol. 12, no. 1,
pp. 29–31, 1939.
[5] P.H. Smith, “An improved transmissionline calculator,” Electronics,
vol. 17, no. 1, p. 130, 1944.
[6] F. Ramos. Interactive Smith Chart. [Online]. Available:
http://www.objetosinteractivos.es/smith/
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