978-1-4244-5386-3/09/$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 impedance-matching

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 paper-based 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 e-learning

applications. It can be executed on more machines (Flash

player has more than a 95% penetration among Internet users),

requires a smaller plug-in, allow vector graphics and their

user-friendly 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 computer-aided 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 in-class 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 [4-5], which described his work.

The well-known 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

loss-less 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 Flash-based 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 upper-left 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 lower-right 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

previously-explained 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 lower-left 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 de-normalized 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 industry-leading authoring environment for

creating engaging interactive experiences. The programming

was based on ActionScript 3.0, a powerful, object-oriented

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 fully-developed 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, user-friendly 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 Problem-solving example using the interactive Smith chart

Fig. 5 Impedance-matching 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, “Transmission-line calculator,” Electronics, vol. 12, no. 1,

pp. 29–31, 1939.

[5] P.H. Smith, “An improved transmission-line calculator,” Electronics,

vol. 17, no. 1, p. 130, 1944.

[6] F. Ramos. Interactive Smith Chart. [Online]. Available:

http://www.objetos-interactivos.es/smith/

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