coaxial cable modeling and verification

COAXIAL CABLE MODELING AND

VERIFICATION

by

Luyan Qian Zhengyu Shan

A Thesis for the Degree of

BACHELOR OF SCIENCE

in

ELECTRICAL ENGINEERING

Blekinge Institute of Technology

Karlskrona, Sweden

2012

Supervisor: Anders Hultgren

Blekinge Institute of Technology, Sweden

3

ABSTRACT

In this paper, analysis of coaxial cable is used to reveal how an electromagnetic wave

propagates in an electrical conductor, and a new modeling language, MODELICA is

introduced. Some transmission line properties, such as propagation delay, reflection

coefficient, attenuation, are all verified by comparing the results from MATLAB and

MODELICA. The models we simulated are different types of coaxial cables,

including lossless cables and lossy cables. It can be shown that MODELICA, a very

powerful and convenient tool, can process complex physical systems.

5

NOTATION

Capacitance

Inside diameter of the shield

Outside diameter of the enter conductor

Relative dielectric constant

Free space dielectric constant

Dielectric constant of the insulator

Inductance

Relative permeability

Permeability of free space

Magnetic permeability of the insulator

Resistance

Length of the conductor

Cross-section area of the conductor

Electrical resistivity of the material

Conductance

Voltage

Current

Wavenumber

Angular frequency

Function represents a wave traveling from left to right

Function represents a wave traveling from right to left

Characteristic impedance

Position in transmission line

Time

Propagation speed

Velocity factor

The speed of light

State vector

Output vector

Input vector

State matrix

Input matrix

Output matrix

Feedthrough matrix,

The differential equation of

Reflection coefficient

Electric field strength of the reflected wave

Electric field strength of the incident wave

Impedance toward the load

Magnitude of reflection coefficient

Transmitted power

Reflected power

6

The time signal enters cable

The time signal exits cable

Propagation time

Voltage of incident wave

Voltage of reflected wave

Signal attenuation constant

Phase constant

Propagation constant

Wavelength

ABBREVIATION

PVC Polyvinyl chloride

AC Alternating current

KCL Kirchhoff’s current law

KVL Kirchhoff’s voltage law

VSWR Voltage stand wave ratio

RL Return loss

RF Radio frequency

7

TABLE OF CONTENTS

ABSTRACT ..................................................................................................................................... 3

NOTATION ..................................................................................................................................... 5

ABBREVIATION ............................................................................................................................ 6

TABLE OF CONTENTS ................................................................................................................ 7

1 INTRODUCTION ........................................................................................................................ 9

2 BACKGROUND ........................................................................................................................ 13

2.1 Cable Background ............................................................................................................. 13

2.2 Technique Background ...................................................................................................... 14

3 THEORIES ................................................................................................................................. 19

3.1 Transmission Line Theory ................................................................................................. 19

3.1.1 The structure of cable ..................................................................................................... 19

3.1.2 Fundamental electrical parameters ............................................................................... 20

3.1.3 Telegrapher’s equation ................................................................................................... 21

3.1.4 Characteristic impedance .............................................................................................. 24

3.1.5 Wave propagation........................................................................................................... 26

3.1.6 Attenuation in transmission line ..................................................................................... 27

3.2 Methods Used to Solve Circuits ........................................................................................ 30

3.2.1 Kirchhoff’s circuit laws .................................................................................................. 30

3.2.2 State space form ............................................................................................................. 30

3.3 Reflection Theory.............................................................................................................. 28

4 MODELING METHODS .......................................................................................................... 33

4.1 Simple Circuit Solution ..................................................................................................... 33

4.1.1 Lossless transmission line terminated in open-circuit ................................................... 34

4.1.2 Lossless transmission line terminated in short-circuit ................................................... 36

4.1.3 Lossless transmission line terminated in matched load ................................................. 38

4.1.4 Lossy transmission line .................................................................................................. 40

4.1.5 Two different lossless cables connected ......................................................................... 41

4.2 MATLAB Modeling and Simulat1ion............................................................................... 45

4.3 MODELICA Modeling and Simulation ............................................................................ 47

5 VERIFICATION AND ANALYSIS .......................................................................................... 53

5.1 Lossless Coaxial Cable ..................................................................................................... 53

5.1.1 Propagation Time ........................................................................................................... 53

5.1.2 Reflection Coefficient and Analysis ................................................................................ 58

5.2 Lossy Coaxial Cable ......................................................................................................... 65

5.2.1 Propagation constant ..................................................................................................... 65

5.2.2 Lossy coaxial cable verification for 2 conditions ........................................................... 67

5.2.3 Analysis for lossy cable in other conditions ................................................................... 71

6 CONCLUSION .......................................................................................................................... 73

REFERENCE ................................................................................................................................ 75

APPENDICES ............................................................................................................................... 77

9

1 INTRODUCTION

How does signal propagates through a transmission line? Can we know it before

doing the measurements? The answer is yes, and in this report, you could get the

answer.

Transmission line is widely used to transport signals and electric power so that the

research on it is important, since it could help people to understand thoroughly

characteristics of transmission lines and how they behave in the data and energy

delivery. According to this, we can make the response measures in order to improve

the transmission efficiency, which plays a significant role in modern technological

and sustainable world.

Figure 1.1, Transmission Lines in some Applications

Our thesis is developed based on the coaxial cable project of course “Modeling and

Verification”, which is an experiment performed on an electrical cable to reveal how

an electromagnetic wave travels in an electrical conductor. And in that project, we just

need to use MATLAB to model one of several conditions.

When we reviewed that course, we are interested in accomplishing all tasks of the

cable connection situations in that project to see what will happen as the result.

Additionally, as some neoteric modeling software come out such as MODELICA and

Scilab, all of which are developing very quickly, we also desire to try one by

ourselves which is totally new for us.

Therefore, in this paper, we introduced the modeling language MODELICA, which

can simulate the electrical circuits in a more convenient way. We built different

models and analyzed the results from OpenModelica, comparing them with the results

given by MATLAB.

10

Figure 1.2, The open windows of MATLAB(2009a) and OpenModelica

The following conditions of coaxial cable are made:

1) a RG58 coax cable and the terminator end is open

2) a RG58 coax cable with short end

3) a RG58 coax cable terminated in matched load

4) a RG58 coax cable with RG59 coax cable in the end

5) lossy coax cable terminated with open circuit.

To verify the propagation delay and reflection coefficient for lossless cables and

attenuation for lossy cables, we introduced some transmission line theories such as

wave propagation, characteristics impedance, reflection coefficient…, applied some

powerful method to model the system.

In Chapter 2, we will tell BACKGROUND of the Cables and Techniques that are

used in this project.

The necessary THEORIES are discussed in Chapter 3 including Transmission Line

Theory, Methods Used to Solve Circuits and Reflection Theory.

In Chapter 4 MODELING METHODS, the detailed modelling solutions are shown

in terms of Simple Circuit, MATLAB and MODELICA Modeling and Simulation

separately which describes how we did this software computation.

Then we lead the reader to Chapter 5 VERIFICATION AND ANALYSIS, in which

part, the characteristics of Lossless Coaxial Cable and Lossy Coaxial Cable are

analyzed and the results from MATLAB and MODELICA are compared using

theories.

After these, we will make discussion over all of this report in Chapter 6

CONLUSION.

11

Figure 1.3, Overview of the report’s structure

Chapter 2

Background

Transmission

Line

Circuit

Calculation

Reflection

Matlab

Modelica

Simple

Circuit

Lossless

Cable

Lossy

Cable

Chapter 3

Theories

Chapter 3

Modeling

Methods

Chapter 5

Verification

Analysis

Chapter 6

Conclusion

13

2 BACKGROUND

2.1 Cable Background

There are several types of transmission lines whose losses are small: coaxial cable,

microstrip, stripline, balanced line, single-wire line, waveguide, optical fiber. One

advantage of coax over other types of radio transmission line is that in an ideal

coaxial cable can be installed next to metal objects such as gutters without the power

losses that occur in other types of transmission lines. It has a large frequency range

which allows it to carry multiple signals. Coaxial cable also provides protection of the

signal from external electromagnetic interference. However, coaxial cable is more

expensive to install, and it uses a network topology that is prone to congestion. [1]

In recent years, coaxial cables have become an essential component of our

information superhighway. They are applied in a wide variety of residential,

commercial and industrial installations. Coaxial cables serve as transmission line for

radio frequency signals. They are applied in feedlines connecting radio transmitters

and receivers with their antennas, computer network connections, and distributing

cable television signals. Short lengths of coaxial cables are also used for connecting

devices with test equipment, like signal generator. [1]

Coaxial cable is perhaps the most commonly used transmission line type for RF and

microwave measurements and applications. In 1894 Heaviside, Tesla and others

received patents for coaxial line and related structures. A development of coax theory

is often provided as part of basic physics and engineering equation, which are

generally used for transmission line and macroscopic electromagnetic analysis.

Accordingly, the analysis, measurement and application of coax are usually

considered to be quite mature and complete. [2]

Coaxial cable is typically identified or classified based on its impedance or RG-type.

Coaxial cables that conform to U.S. Government specifications are identified with an

RG designation.

Figure 2.1, Meaning of some letters

The RG series was originally used to describe the types of coax cables for military use,

and the specification took the form RG plus two numbers. The RG designation stands

for Radio Guide, the U designation stands for Universal. The current military standard

14

is MIL-SPEC MIL-C-17.MIL-C-17 numbers. However, the RG-series designations

were so common for generations that they are still used. [1]

In this paper we emphasis on modeling coaxial cable RG-58 and RG-59.

RG-58 is a coaxial cable that is used for wiring purpose. The insulation surrounding

the RG-58 cable carries a low impedance of around 50 or 52 ohms. It generally serves

for generating signal connections that are of low power. The RG-58 cable is most

often used for the Thin Ethernet when the maximum length required is about 185

meters. The RG-58 cable frequency acts as a generic carrier of power signals. These

signals are generated in physical laboratories. The RG-58 cable is specially designed

to work with most two-way radio systems. This communication system is different

from the usual broad cast receiver since the latter can receive data from one end only.

In case of the two-way radio system, it can be generated by the RG-58 cable, where

content travels in both directions. The radio can receive and transmit data at the same

time. The RG-58 can also be used for higher frequencies. The range, however,

remains fairly moderate. The Ethernet wiring for which the RG-58 cable is used is

sometimes termed “cheapernet”, since it draws low-power signal connections. [3]

The RG-59 cable is a type of coaxial cable that is used to generate low power video

connections. The RG-59 cable conducts video and radio frequency at an impedance of

around 75 ohms. The RG-59 cable is used for generating short-distance

communication. The cable can be applied in baseband video frequencies, which are

measured from the lowest count of zero and continue to the highest signal frequency.

Baseband refers to a collection of signals and frequencies varying over a wide range.

The RG-59 cable cannot be used over long distance due to its high-frequency power

losses. The RG-59 cables are comparatively less expensive than other cables. One of

the greatest uses of the RG-59 cable is synchronization between two digital audio

devices. The coaxial cable coordinates between the digital signals that are responsible

for producing sound. The digital audio devices are used for storage, conversion, and

transmission of the auto signals. The RG-59 cable maintains a unique coordination

between these devices. The RG-59 cable undergoes a small amount of signal

reduction, which is owing to the shielding on the cable. The low cost of the RG-59

coaxial cable has made it easily accessible and usable. [4]

2.2 Technique Background

MATLAB

MATLAB is a programming language for technical computing. MATLAB is used for

algorithm development, model prototyping, data analysis and exploration of data,

visualization and numeric computation.

MATLAB was first conceived as a teaching tool by Moler who was at the University

of New Mexico in the late 1970s. Moler wanted his students to have access to

Linpack and Eispack matrix software without having to use the Fortan programming

language, which was complex; he came up with the MATLAB system to solve this

15

problem. [5] The original MATLAB was designed specifically to handle computations

with matrices and mathematics. Little and Steve Bangert developed PC MATLAB by

porting Moler’s code from FORTRAN to C, adding user-defined functions, improved

graphics, and libraries of MATLAB routines, the toolboxes.

There is general agreement in the technical computing community that the main

reasons for MATLAB’s success are its intuitive, concise syntax, the use of complex

matrices as the default numeric data object, the power of the built-in operators, easily

used graphics, and its simple and friendly programming environment, allowing easy

extension of the language. [6]

It has been widely used by engineers, mathematicians and scientists. MATLAB boats

more than 1 million users around the word. MATLAB now has been used in such

varied areas as automobiles, airplanes, hearing aids, cellphones, financial derivative

pricing and academics. [5]

Figure 2.2, MATLAB window environment

MODELICA

Object-Oriented modeling is a fast-growing area of modeling and simulation that

provides a structured, computer-supported way of doing mathematical and

equation-based modeling. MODELICA is today the most promising modeling and

simulation language in that it effectively unifies and generalized previous

object-oriented modeling languages and provides a sound basis for the basic concepts.

[7]

16

The MODELICA design effort was initiated in September 1996 by Hilding Elmqvist.

The goal was to develop an object-oriented language for modeling of technical

systems in order to reuse and exchange dynamic system models in a standardized

format. [8]

The four most important features of MODELICA are: [9]

MODELICA is based on equation instead of assignment statements. This permits a

causal modeling that gives better reuse of classes since equations do not specify a

certain data flow direction. Thus a MODELICA class can adapt to more than one

data flow context.

MODELICA has multi-domain modeling capability, meaning that model

components corresponding to physical objects from several different domains

such as electrical, mechanical, thermodynamic, hydraulic, biological and control

applications can be described and connected.

MODELICA is an object-oriented language with a general class concept that

unifies classes, generics―known as templates in C++, and subtyping into a single

language construct. This facilitates reuse of components and evolution of models.

MODELICA has a strong software component model, with constructs for creating

and connecting components. Thus the language is ideally suited as an architectural

description language for complex physical systems and to some extent for

software systems.

OpenModelica

The OpenModelica environment is an open-source environment for modeling,

simulation, and development of MODELICA applications. The current version of the

OpenModelica environment allows most of expression, algorithm and function parts

of MODELICA to be executed interactively, as well as equation models and

MODELICA functions to be compiled into efficient C code. The generated C code is

combined with a library of utility functions, a run-time library, and a numerical DAE

solver. An external function library interfacing a LAPACK subset and other basic

algorithms is under development. [10]

The OpenModelica environment has several goals: [10]

Providing an efficient interactive computational environment for the MODELICA

language.

Development of a complete reference implementation of MODELICA in an

extended version of MODELICA itself.

Providing an environment for teaching modeling and simulation.

Language design to improve abstract properties such as expressiveness,

orthogonality, declarativity, reuse, configurability, architectural properties, etc.

17

Improved implementation techniques, e.g. to enhance the performance of

compiled MODELICA code by generating code for parallel hardware.

Improved debugging support for equation based languages such as MODELICA,

to make them even easier to use.

Easy-to-use specialized high-level user interfaces for certain application domains.

Visualization and animation techniques for interpretation and presentation of

results.

Application usage and model library development by researches in various

application areas.

Figure 2.3, OpenModelica window environment

19

3 THEORIES

3.1 Transmission Line Theory

In communications and electronic engineering, a transmission line is a specialized

cable designed to transfer alternating current of radio frequency, that is, currents with

a frequency high enough that their wave nature must be taken into account.

Transmission lines are used for purposes such as connection radio transmitters and

receivers with their antennas, distributing cable television signals, and computer

network connections. Transmission lines can be realized in number of ways. Common

examples are the coaxial cable and the parallel-wire line. [11]

3.1.1 The structure of cable

Figure 3.1, Inner structure of the cable

Coaxial cables are the interconnections that transmit pulses from one end to another,

protecting the information in the signal. A cable can be treated as a transmission line if

the length is greater than 1/10 of the wave length.

Coaxial cable has a core wire, surrounded by an insulation jacket which is a PVC

material. Normally the shield is kept at ground potential. Then it is surrounded by a

copper mesh which is often constituted by braided wires. The inner dielectric

separates the core and the shielding apart. The central wire carries the RF signal and

the outer shield is considered to prevent the RF signal from radiating to the

atmosphere and to keep outside signals from interfering with the signal carried by the

core. The electrical signal always travels along the outer layer of the central conductor,

and as a result, the larger the central conductor, the better signal will flow. Coaxial

cable is a good choice for carrying weak signals that cannot tolerate interference from

the environment or for higher electrical signals that must not be allowed to radiate or

couple into adjacent structures or circuits. [12]

20

Table 3.1, Physical parameters for typical cables

Cable Type Core (mm)

Dielectric

(mm)

Shield (mm) Jacker(mm)

RG-58

0.9

2.95

3.8

4.95

RG-213

2.26

7.24

8.64

10.29

LMR-400

2.74

7.24

8.13

10.29

3/8” LDF

3.1

8.12

9.7

11

3.1.2 Fundamental electrical parameters

Generally, a transmission line has these four parameters: capacitance, resistance,

conductance and inductance.

Shunt capacitance C per unit length, in farads per meter. [13]

Where: d is the outside diameter of the enter conductor (millimeters)

D is the inside diameter of the shield (millimeters)

is the relative dielectric constant

is the free space dielectric constant

is the dielectric constant of the insulator, which equal to

Series inductance L per unit length, in henrys per meter. [13]

Where: is the relative permeability, it almost always be 1

is the permeability of free space

is the magnetic permeability of the insulator, which equal to

Series resistance R per unit length, in ohms per meter. This parameter is the resistance

of the inner conductor and the shield. Resistance primarily depends upon two factors:

the material it is made of, and its shape. Another factor, which affects this parameter,

is the skin effect, wherein the propagating microwave signal is intend to confine itself

on the top layer or the 'skin' of the conductor, thus increasing the effective resistance.

Assume the current density is totally uniform in the conductor, the resistance R can be

computed as: [14]

21

Where: is the length of the conductor (meters)

is the cross-section area of the conductor (square meters)

is the electrical resistivity of the material (ohm-meters)

Shunt conductance G per unit length, in siemens per meter. The shunt conductance

happens due to the dielectric loss of the insulator used. An insulating material with

good dielectric properties will have a low shunt conductance.

Assume the current density is totally uniform in the conductor, the conductance G can

be computed as: [14]

3.1.3 Telegrapher’s equation

Telegrapher’s equations are a pair of linear differential equations which characterize

the voltage and current on an electrical transmission line with distance and time. We

can derive characteristic impedance and wave speed from the telegrapher’s equation.

Lossless transmission model

Figure 3.2, Equivalent circuit model of a lossless transmission line

In lossless transmission line, it possesses a certain series inductance . If is

the current through the wire, the voltage across the inductance is

,

denotes the voltage at position and time . We have that the charge in

voltage between the ends of the piece of wire is:

(3.1)

Further that current can escape from the wire to ground through the capacitance .

Because the charge of capacitor is , the amount of the current escapes

from the capacitor is

. We have the charge in current is:

(3.2)

Both side of equation (3.1) and (3.2) are divided by , get the difference equation:

22

(3.3)

(3.4)

From

and

(3.5)

(3.6)

Putting

to equation (3.5)

(3.7)

To get similar equation for the current, using

and

(3.8)

(3.9)

Putting

to equation (3.9)

(3.10)

So, the telegraph’s equations for the lossless transmission line are:

(3.11)

(3.12)

Lossy transmission model

Figure 3.3, Equivalent circuit model of a lossy transmission line

The components for the model of a lossy transmission line are the series

23

inductance , shunt capacitance , series resistance , and shunt

conductance . For a homogeneous transmission line, those parameters are

distributed evenly along the length of the line.

The change in voltage between the ends of the piece of wire is:

(3.13)

We have the charge in current is:

(3.14)

Both side of equation (3.13) and (3.14) are divided by , get the difference equation:

(3.15)

(3.16)

From

and

get:

(3.17)

(3.18)

Putting

,

to equation (3.17)

(3.19)

(3.20)

(3.21)

To get similar equation for the current, using

and

(3.22)

(3.23)

Putting

24

to equation (3.23)

(3.24)

(3.25)

(3.26)

So, the telegraph’s equations for the lossless transmission line are:

(3.27)

(3.28)

3.1.4 Characteristic impedance

Characteristic impedance refers to the equivalent resistance of a transmission line if it

were infinitely long, it is due to distributed capacitance and inductance as the voltage

and current waves flow along its length at a propagation velocity equal to some large

fraction of light speed. The inductance increases with increasing spacing between the

conductors, and the capacitance decreases with increasing spacing between the

conductors. Hence a line with closely spaced large conductors has low characteristic

impedance. [12]

Characteristic impedance for lossless transmission line can be derived by lossless

telegraph’s equation

.

There are two solutions for the traveling wave: one forward and one reverse. The

solution for the wave equation can be written as: [15]

Where: k is the wavenumber (radians/meter)

is the angular frequency (radians/second)

and can be any function, represents a wave traveling from left to

right in positive x direction, while represents a wave traveling from right to left

Since the current is related to the voltage by the telegrapher’s equations, we can write:

[15]

The differential equation for

25

(3.29)

And the first order differential equation for

:

(3.30)

Comparing telegraph’s equation

with the result of

equation (3.29) divided by equation (3.30), we can get the characteristic impedance:

(3.31)

We have calculated the relationship between and , putting

to

equation (3.31)

Thus, the expression of characteristic in lossless transmission line is:

To calculate the characteristic impedance for lossy transmission line, we replace each

time derivative by a factor for lossy telegraph’s equation (3.27) and express them

in frequency domain, the equations become:

(3.32)

(3.33)

Where , and

Mathematically, we can solve the equations for a lossy transmission line in exact the

same way as we did for lossless line. The characteristic impedance for lossy

transmission line is:

Matched load

A line terminated in a purely resistive load equal to the characteristic impedance is

said to be matched. In a matched transmission line, all the power is transmitted over a

transmission line. It minimizes signal distortion in transmission lines, prevents wave

from reflections and pulse. [12]

26

3.1.5 Wave propagation

Propagation speed for lossless transmission line can be derived by lossless telegraph’s

equation

.

We have mentioned the solution for the wave equation can be written as:

We can get the first differential equation by using

(3.34)

Using

of equation (3.34) to get secondary differential equation

(3.35)

Using the same method to get secondary differential equation for

(3.36)

One can easily show by comparing telegraph’s equation

with the result of equation (3.35) divided by equation (3.36), the velocity with which

the electromagnetic energy propagates along this lossless line is given by:

The propagation speed for lossy cable can be calculated with the similar solution

which used to solve the characteristic impedance for lossy cable by replacing

and :

Velocity of propagation

The velocity factor is the speed at which RF signal travels through a material

compared to the speed the same signal travels through a vacuum. The higher the

velocity factor, the lower the loss through a coaxial cable. Velocity factor is a

parameter that characterizes the speed at which an electrical signal passes through a

medium. It varies from 0 to 1. The velocity of light is the speed limit for electrical

signals and is never reached in coaxial cable, the range of velocity factor is from 66

percent to 86 percent for typical flexible coaxial cable. The type of dielectric material,

determines the dielectric constant, which is the primary determinant of the velocity of

the cable. [16]

27

Where: is the velocity factor

Dielectric materials

Dielectric material is the material between the center and outer conductors. There is a

variety of materials that can be successfully used as dielectrics in coax cables. Each

has its own dielectric constant, and as a result, coax cables that use different dielectric

materials will exhibit different velocity factors.

Table 3.2, Dielectric constants and velocity factors of some common dielectric materials used in

coax cables

MATERIAL

DIELECTRIC

CONSTANT

VELOCITY

FACTOR

Polyethylene

2.3

0.659

Foam polyethylene

1.3 – 1.6

0.88 – 0.79

Solid PTFE

2.07

0.695

For a lossless transmission line: [17]

Where c is the speed of light (meters/second)

3.1.6 Attenuation in transmission line

Every transmission has some losses, since the resistance of the conductors and power

is consumed in the dielectric which used for insulating the conductors. Power lost in a

transmission line is not directly proportional to the line length, but varies

logarithmically with the length. And line losses are usually presented in terms of

decibels per unit length. Losses in transmission line arise from sources: radiation,

dielectric loss, skin effect loss. [18]

Skin effect loss

Skin effect occurs in conductors carrying an AC current. As the frequency increases,

the current tends to be concentrated near the surface of the conductor, and the skin

effect becomes more pronounced and the loss in conductors increases dramatically.

Skin effect loss is the resistance aggravated by the inhomogeneous current

distribution that caused by the skin effect. For a perfect coaxial cable, the skin

28

resistance is proportional to the square root of the frequency. [18]

Dielectric loss

Dielectric loss is due to the electric absorbing energy as it is polarized in each

direction. It occurs when the conductance is non-zero. Dielectrics have losses increase

when increasing the voltage on the conductors. Dielectric losses also increase with the

frequency since the shunt conductance increase approximately linearly with frequency.

[18]

Radiation loss

Radiation loss occurs in two wire lines since the fields from one line do not completed

cancel out those from the other line. If the conductors form a tight electromagnetic

system with the outer conductor have a thickness greater than 5 times the skin depth

then radiation is negligible. If outer conductor is a loose braid, it will result in

radiation. Special types of coax with multiple braids, or a solid outer conductor have

no measureable radiation losses. [18]

3.2 Reflection Theory

A signal travelling along an electrical transmission line will be partly, or wholly,

reflected back in the opposite direction when the travelling signal encounters a

discontinuity in the transmission line, or when a transmission line is terminated with

other than its characteristic impedance. [19]

Reflection Coefficient

Reflection coefficient describes the ratio of reflected wave to incident wave at point

of reflection, where circuit parameter has sudden change. This value varies from -1

(for short load) to +1 (for open load), and becomes 0 for matched impedance load.

The reflection coefficient is defined as:

[20]

Where: is the electric field strength of the reflected wave

is the electric field strength of the incident wave

The reflection coefficient may also be established using circuit quantities:

Where: is the impedance toward the load

is the impedance toward the source

http://en.wikipedia.org/wiki/Transmission_line

http://en.wikipedia.org/wiki/Reflection_(physics)

http://en.wikipedia.org/wiki/Discontinuity_(mathematics)

http://en.wikipedia.org/wiki/Transmission_parameters

http://en.wikipedia.org/wiki/Electronic_circuit

29

Figure 3.4, Simple circuit configuration showing measurement location of reflection coefficient

Voltage Standing Wave Ratio (VSWR)

Voltage standing wave ratio is the ratio of maximum to minimum voltage amplitude

in standing wave pattern. It varies from 1 to plus infinite. VSWR is used as an

efficiency measure for transmission lines, electrical cables that conduct radio

frequency signals, used for purposes such as connecting radio transmitters and

receivers with their antennas, and distributing cable television signals. Impedance

mismatches in the cable causes radio waves to reflect back toward the source end of

the cable. VSWR measures the relative size of these reflections. An ideal transmission

line would have a VSWR of 1:1, with all the power reaching the destination and no

reflection. An infinite VSWR represents complete reflection, with all the power

reflected back down the cable. [21]

VSWR is related to the reflection by:

Where , the magnitude of reflection coefficient

Return Loss

Return loss is the reflection of signal power resulting from the inserting of a device in

a transmission line or optical fiber. Return loss is a convenient way to characterize the

input and output of signal sources. Return loss is a measure of how well devices or

lines are matched. A large positive return loss indicates the reflected power is small

relative to the incident power, which indicates good impedance match from source to

load. This loss value become 0 for 100% reflection and become infinite for ideal

connection.

It is usually expressed as a ratio in dB relative to the transmitted signal power:

Where: is the power transmitted by the source

is the power reflected by the source

Return lose also is the negative of the magnitude of the reflection coefficient in dB.

http://en.wikipedia.org/wiki/Radio_frequency

http://en.wikipedia.org/wiki/Radio_frequency

http://en.wikipedia.org/wiki/Transmitter

http://en.wikipedia.org/wiki/Antenna_(radio)

http://en.wikipedia.org/wiki/Cable_television

http://en.wikipedia.org/wiki/Impedance_matching

http://en.wikipedia.org/wiki/Impedance_matching

30

Since power is proportional to the square of the voltage, it is given by:

[22]

3.3 Methods Used to Solve Circuits

In order to use Simulink in MATLAB to model the systems, we should at first

calculate the ABC matrices using Kirchhoff’s Laws and State Space Form.

3.3.1 Kirchhoff’s circuit laws

Kirchhoff’s circuit laws are two equations that deal with the conservation of charge

and energy in electrical circuits. [23]

Kirchhoff’s current law (KCL)

KCL: At any node (junction) in an electrical circuit, the sum of currents introducing

into that node is equal to the sum of currents extracted from that node, or the algebraic

sum of currents in a network of conductors meeting at a point is zero.

This principle can be stated as: [23]

Where n is the total number of branches with currents flowing towards or away from

the node

Normally, current is signed positive when its direction towards the node.

Kirchhoff’s voltage law (KVL)

KVL: The directed sum of the electrical potential differences around any closed loop

is zero, in other words, the algebraic sum of the products of the resistances of the

conductors and the currents in them in a closed loop is equal to the total emf available

in that loop. [23]

3.3.2 State space form

State space refers to the space whose axes are state variables. A state space form

provide the dynamics as a set of first-order differential equations in a set of internal

variables known as state variables, together with a set of algebraic equations that

combine the state variables into physical output variables. To extract from the number

of inputs, outputs and states, the variables are expressed as vectors. Additionally, if

the dynamical system is linear and time invariant, the differential and algebraic

equations may be presented in matrix form. The state space representation provides a

convenient and compact way to model and analyze systems with multiple inputs and

outputs. The state variables are an internal interpretation of the system which

completely characterizes the system state at any time . [24]

31

The most general state-space representation of a linear system with inputs,

outputs and state variables is written in the following form: [24]

Where: is called the state vector

is called the output vector

is called the input vector

is the state matrix

is the input matrix

is the output matrix

is the feedthrough matrix, in cases where the system model does

not have a direct feedthrough, Is the zero matrix

is the differential equation of ,

33

4 MODELING METHODS

4.1 Simple Circuit Solution

To find out the ABC-matrix which will be used in MATLAB, we need to apply state

space form to solve the transmission line.

Transmission line can be modeled based on state space method. It provides a method

with the exact accuracy to effectively calculate the state space models. In this case,

the number of state variables is equal to the number of independent energy storage

elements in the system. In the following circuits, except the last one, there are two

independent energy storages, the capacitor which stores energy in an electric field and

the inductor which stores energy in magnetic field. The state variables are and .

The energy storage elements of a system make the system dynamic. The flow of

energy into or out of a storage element occurs at a finite rate and is presented by a

differential equation.

So the vector of the inductor’s current and capacitor’s voltage can be expressed as the

state vector , denotes the vector of source voltage and is the vector of

output voltage. The matrices and are properties of the system and determined

by the system structure and elements. The matrices and are determined by the

particular choice of output variables.

Damped harmonic oscillation phenomenon

When we used MATLAB and OpenModelica to model the lossless cable, we applied

the LC-circuit to these modeling languages. And there will be a special phenomenon

appears in the results.

Figure 4.1, One section of lossless cable in model

In the results, electric charge oscillates back and forth just like the position of a mass

on a spring oscillates, in other words, damped harmonic oscillation, the amplitude

vibrates at its eigenfrequency.

34

Figure 4.2, Damped harmonic oscillation

Angular oscillation frequency can be calculated by: [37]

Where is the inductance in each section,

is the capacitance in each section,

The value of eigenfrequency will be influenced by the number of sections, the greater

the number of sections, the greater the eigenfrequency will be. So we prefer to use a

large set of sequences to achieve more precise results when making the models in

MATLAB and OpenModelica.

4.1.1 Lossless transmission line terminated in open-circuit

Suppose we have three sections in this circuit, and for convenience, we assumed the

value of inductors and capacitors are constant along the line.

Figure 4.3, Circuit of 3-section transmission line terminated in open

35

First, we applied Kirchhoff’s current law (KCL) to three nodes to get equations which

are related to current. And assume the direction current flow toward the node is

positive. KCL says that the net current outflow vanishes at any vertex of the graph.

The current of capacitor is equal to

.

At node ①:

(4.1)

At node ②:

(4.2)

At node ③:

(4.3)

Then we applied Kirchhoff’s voltage law (KVL) to three loops to get equations

related to voltage. The voltage of capacitor is equal to

.

In loop I:

(4.4)

In loop II:

(4.5)

In loop III:

(4.6)

Rearrange equations (4.1), (4.2), (4.3), (4.4), (4.5), (4.6) to put the derivative of the

state variables , on the left side. ,

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

We can also write as state space representation:

36

The results matrices A, B, C, D are:

From the above matrixes, it can be concluded that nth elements has:

A =

B =

C = D = 0

4.1.2 Lossless transmission line terminated in short-circuit

Figure 4.4, Circuit of 3-section transmission line terminated in short

37

In this circuit, the current flow out will not pass , it will directly enter the short

line. This circuit can be transforms to the following equivalent circuit.

Figure 4.5, Circuit of 3-section transmission line terminated in short

Then we used the same method to get A, B, C, D matrix.

At node ①:

(4.13)

At node ②:

(4.14)

In loop I:

(4.15)

In loop II:

(4.16)

In loop III:

(4.17)

Rearrange equations (4.13), (4.14), (4.15), (4.16), (4.17) to put the derivative of the

state variables , on the left side. ,

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)


38


A =

B =

C = D = 0

4.1.3 Lossless transmission line terminated in matched load

Figure 4.6, Circuit of 3-section transmission line terminated in matched load

At node ①:

(4.23)

At node ②:

(4.24)

At node ③:

(4.25)

In loop I:

(4.26)

39

In loop II:

(4.27)

In loop III:

(4.28)

Rearrange equations (4.23), (4.24), (4.25), (4.26), (4.27), (4.28) to put the derivative

of the state variables , on the left side. ,

(4.29)

(4.30)

(4.31)

(4.32)

(4.33)

(4.34)



A =

B =

C = D = 0

40

4.1.4 Lossy transmission line

Suppose the circuit consists of these components: source voltage V, inductance ,

, , resistance , , , , capacitance , , and conductance ,

, .

Figure 4.7, Circuit of 3-section lossy transmission line terminated in open

The total current at node ② is equal to the sum of current at node ①, and the

direction of current are opposite:

(4.35)

The same situation for node ③ and ④

(4.36)

At node ⑤:

(4.37)

In loop I:

(4.38)

In loop II:

(4.39)

In loop III:

(4.40)

Rearrange equations (4.35), (4.36), (4.37), (4.38), (4.39), (4.40) to put the derivative

of the state variables , on the left side. , ,

,

(4.41)

(4.42)

(4.43)

(4.44)

41

(4.45)

(4.46)



A =

B =

C = D = 0

4.1.5 Two different lossless cables connected

Figure 4.8, Circuit of two 3-section lossless transmission lines connected

There are six sections in this circuit, the form three sections have the same elements

42

and they are different with the last three sections. Suppose ,

, , . The state variables are ,

, ,

.

At node ①:

(4.47)

At node ②:

(4.48)

At node ③:

(4.49)

At node ④:

(4.50)

At node ⑤:

(4.51)

At node ⑥:

(4.52)

In loop I:

(4.53)

In loop II:

(4.54)

In loop III:

(4.55)

In loop IV:

(4.56)

In loop V:

(4.57)

In loop VI:

(4.58)

Rearrange equations (4.51) (4.58) to put the derivative of the state variables ,

on the left side

(4.59)

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

(4.65)

(4.66)

43

(4.67)

(4.68)

(4.69)

(4.70)

We can also write as state space representation:


44

When the output voltage is the voltage of the last capacitor of the first cable:

When the output voltage is the voltage of the last capacitor of the last cable:


A=

B =

45

D = 0

4.2 MATLAB Modeling and Simulat1ion

MATLAB

MATLAB is a software package for high performance computation and visualization.

The combination of analysis capabilities, flexibilities, reliability and powerful

graphics makes MATLAB the premier software package for engineers and scientists.

MATLAB provides an iterative environment with mathematical functions. These

functions provide solution to a broad range of mathematical problems including:

Matrix Algebra, Complex Arithmetic, Linear Systems, Differential Equations, Signal

Processing, Optimization and other types of scientific computations. [23]

Simulink

Simulink is an environment for multidomain simulation and Model-Based Design for

dynamic and embedded systems. The system may be both linear and nonlinear; they

can also be continuous or discrete. It provides an interactive graphical environment

and a customizable set of block libraries that let you design, simulate, implement, and

test a variety of time-varying systems, including communications, controls, signal

processing, video processing, and image processing. [24]

In this paper, we used Simulink® which is offered as a toolbox in the MATLAB to

simulate different type of cables. And we modeled these transmission lines with the

state space parameters which we have calculated.

Figure 4.9, Normal electrical circuit model

46

Figure 4.10, Parameters for step voltage

In this model, we assumed the input voltage as step-voltage and its final value is 1 .

It connected with two state-space blocks which transfer the original signal to input

signal and output signal with different value of C. Since C is decided according to

which output voltage we choose. The Clock block outputs the current simulation time

at each simulation step. It displays and provides the simulation time. Normally, the

time period we use is between 0 and 2 10-6

s.

Then we combined this model with the MATLAB codes. We defined the

representation of matrixes A, B, C, D and set stop time to make the specified Simulink

model to be executed. Last, we plotted the figure with the signals transmitted with

time in voltage amplitude.

For lossless cable RG58, the capacitance equals to 101 10-12

F/m and the inductance

equals to 252 10-10

H/m. And for lossless cable RG59, capacitance is 67 10-12

F/m

and inductance is 376 10-9

H/m.

For lossy cable in different conditions, values we set the same capacitance and

inductance as cable RG58. In Heaviside condition, the value of resistance and

conductance are 0.2Ω and

respectively. In low loss condition, resistance and

inductance are equal to 252 10-6

Ω and 101 10-8

S. Furthermore, we run all the

models with the number of sections 200.

Figure 4.11, Solver options in MATLAB

47

For numerical method in Simulink, MATLAB has several for different systems. As we

did in course Modeling and Verification, ode45, the default solver in MATLAB, is

good enough to calculate this system.

Ode45 is automatic step size Runge-Kutta-Fehiberg integration methods, using a 4th

and 5th

order pair for higher accuracy. [38]

4.3 MODELICA Modeling and Simulation

MODELICA

MODELICA is a non-proprietary, object-oriented, equation based language to

conveniently model complex physical systems containing, e.g., mechanical, electrical,

electronic, hydraulic, thermal, control, electric power or process-oriented

subcomponents. MODELICA is a modeling language rather than a conventional

programming language. MODELICA is designed to be domain neutral and, as a result,

is used in a wide variety of applications, such as fluid system, automotive applications

and mechanical systems. [25]

OPENMODELICA

OpenModelica is an open-source MODELICA-based modeling and simulation

environment intended for industrial and academic usage. The goal of the

OpenModelica project is to create a complete MODELICA modeling, compilation

and simulation environment based on free software distributed in binary and source

code form. [26]

In OpenModelica, there exist many electrical components. We can connect them and

form the circuit.

Figure 4.12, Oline in MODELICA

Figure 4.13, Inner components of Oline

48

As can be seen from Figure 4.13, the lossy transmission line Oline consists of series

of resistances, inductances, conductance and capacitances. To get a symmetric line

model, there are resistors and inductors in both beginning and end positions. Since

the inside components of Oline are terminated with an inductance, we need to connect

a capacitance to node p2 when connecting circuit for Lossless cable. So we can treat it

as a cable by setting some parameters to it.

Following are the circuits we connected for different cables in OpenModelica.

Figure 4.14, Circuit model for open-terminated coaxial cable

Figure 4.15, Circuit model for short-terminated cable

49

Figure 4.16, Circuit model for matched-load

Figure 4.17, Circuit model for two coaxial cables

As we want to compare the results from MATLAB and OpenModelica, we should

make them in same situations. Therefore, the properties of step voltage are same as

that in MATLAB. Here are the basic parameters of inductance and capacitance in

Olines in MODELICA, which are also exactly the same as what we used in MATLAB.

The first one is for RG58 and second is for RG59 cable.

Figure 4.18, Properties of the Olines in Model of MODELICA for Lossless Transmission Line

50

As an additional capacitor, which is also regarded as an element, is terminated at the

end, we should use 199 elements in Oline to make the sections the same as MATLAB.

For Lossy Transmission Line, as to make it the same as the circuit model we used in

MATLAB, we should add a capacitor and conductor parallel across the Oline and

ground like the graph below.

Figure 4.19, Circuit model for lossy cable

The following two tables describe the properties of the components we used in two

conditions. The left one is for Heaviside condition and the right is the one in Low loss

condition, both of which we will explain in details in the Analysis part later.

Figure 4.20, properties for Lossly Cable modeling in Oline at MODELICA

In Simulation on OpenModelica, we tried different Integration Methods such as dassl,

dassl2, rungekutta and euler. By comparing the results, we choose the differential

algfebraic system solver, dassl, as the numerical method for OpenModelica modeling.

51

Because we think the figures got from this default method is good enough.

Figure 4.21, Simulation method in OpenModelica

In order to get model more smooth curve instead of zigzag ones, we changed the

value of Tolarance to 0.000001, which can be seen in Figure 4.21. Besides, from the

previous modeling, we know that time period 2 10-6

s is sufficient and on the other

hand it also should be same as that of MATLAB simulation.

53

5 VERIFICATION AND ANALYSIS

In this part, the verification and analysis on results from MATLAB and MODELICA

will be discussed in terms of Lossless Coaxial Cable and Lossy Coaxial Cable

separately via different perspectives.

5.1 Lossless Coaxial Cable

For Lossless Cables, We analyze the simulation results on Propagation Time and

Reflection Coefficient, which will be explained in detail within this section on the

basis of theories and graphs.

5.1.1 Propagation Time

Supposed a RG58 cable is connected by a RG59 cable in a circuit, MATLAB and

MODELICA are used to model this circuit condition to plot the voltage figures

corresponding to different capacitance, considered as the three specific nodes in the

transmission process, to check the signal propagation time in these two types of

cables.

Theoretical Calculation of Propagation Time

Coaxial cable serves as the transmission line to carry RF signals, the time it takes for

a signal to travel from one end of the cable to the other is usually presented as smaller

units such as milliseconds, microseconds, or nanoseconds, since RF signals travel so

fast. The time delay can be considered as following:

Figure 5.1, Signal propagation time through a cable

The propagation time, in other words, the difference between and is:

Where: is the time signal enters cable (seconds)

is the time signal exits cable (seconds)

is the length of cable (meters)

and is the velocity of factor (meters/second)

54

c is the speed of light (meters/second)

v is the phase velocity (meters/second), which we have mentioned in

previous part.

These two tables below show the parameters and calculated velocities of RG58 and

RG59, in whose light, the theoretical values of propagation time can be calculated

with the formula given above.

Table 5.1, Parameters for lossless cable RG58

Length

( )

Inductance

(L)

Capacitance

(C)

Impedance

( )

100 m

252 10-9

H/m

101 10-12

F/m

50 Ω

1.98216 108

m/s

Time delay for this type of cable (RG58) can be regarded approximately as

504.5 ns

Table 5.2, Parameters for lossless cable RG59

Length

( )

Inductance

(L)

Capacitance

(C)

Impedance

( )

25 m

376 10-9

H/m

67 10-12

F/m

70 Ω

1.99236 108 m/s

According to the data in Table 5.2, we can get the propagation time of RG59 is

125.48 ns

Time Delay Analysis in MATLAB

In MATLAB, we simulated a two connected cables model and obtained graphics of

the voltage for the very beginning and the last capacitors of the first cable RG58 and

that for the very last capacitor in the second cable RG59 in time direction. We can

mark points where waves begin to flow into the corresponding capacitors so that the

values of time for each line are easily to find.

To show the wave propagation in different nodes of cables more clearly, we plotted

the outcome figures respectively.

55

Figure 5.2, Simulation result of the input signal (x:Voltage, y:Time)

Figure 5.3, Simulation result of signal at middle point (x:Voltage, y:Time)

Figure 5.4, Simulation result of signal at the end (x:Voltage, y:Time)

The red line represents the signal passing the very first capacitor in RG58 cable,

considered to be the input signal. And the green curve describes the voltage crossing

56

the terminator end of RG58, which is come out from the connection node between

RG58 and RG59, while the blue wave shows signal in the last capacitor of RG59

cable, known as the signal output at the end.

In the model, when two coaxial cables connect together, the incident wave (red line)

will occur two times reflection, the green line reflects once, and the amplitude and

time interval for reflection is similar to the second reflection of the red one. The red,

green and blue lines will finally concentrate to 1V which is the value of source

voltage.

Figure 5.5, Result for the voltage of capacitors in different positions (x:Voltage, y:Time)

We marked the first impulse points of these three curves to see the time (x-axis) when

the signal arrives at them the first time.

x1 = 3.797 ns, x2 = 515 ns, x3 = 640.7 ns,

Δt1 = 515 – 3.797 = 511.203 ns, Δt2 = 640.7 – 515 = 125.7 ns

Δt1 is the time delay for cable RG58 and Δt2 is the propagation time for cable RG59.

Time Delay Analysis in MODELICA

In MODELICA, we cannot mark the point in the figure like what we did in MATLAB

part. Although, since we knew the voltage values of the points we took from

MATLAB, we can zoom in the area of these points to show more precise value of the

voltage and time.

The following curves gotten from MODELICA show input signal, signal come out

from the connection node between RG58 and RG59 cables and the signal out of from

end with the same colors as those in MATLAB.

57

Figure 5.6, Modeling result from MODELICA (x:Voltage, y:Time)

According to Figure 5.6, we estimated time values (x-axis) of the points:

x1 = 6.87 ns, x2 = 518.28 ns, x3 = 649.1 ns,

Δt1 = 518.28 – 6.87 = 511.41ns, Δt2 = 649.1 – 518.28 = 130.82 ns,

where Δt1 is the time delay for cable RG58 and Δt2 is the propagation time for cable

RG59.

Comparison and Verification

We made a form to compare the values of Propagation Time in Theoretical

Calculation, MATLAB and MODELICA like this.

Table 5.3, Propagation time of different cable calculated by different methods

Time

Delay

Cable

Theoretical MATLAB MODELICA

RG58

504.5ns

511.203ns

511.41ns

RG59

125.48ns

125.7ns

130.82ns

In line with Table 5.3, we can see it is very clear that the values of time delay, given

by Theoretical Calculations via cables’ parameters, simulation in MATLAB and

modeling in MODELICA, are very close. Although there are some slight differences,

generally they are so small that the errors can be neglected. Thus, validated, the

results are proved to be correct.

58

5.1.2 Reflection Coefficient and Analysis

We use RG58 as an example; consider that a RG58 cable is terminated in some typical

conditions. With the aid of MATLAB and MODELICA, we can just plot the incident

waves to verify their reflection coefficients and discuss the wave propagations. It is

possible to find the amplitude of initial signal and voltage after reflection from the

following figures.

RG58 cable in open circuit

The graphs of the input signal at the very beginning of cable RG58, given by

MATLAB and MODELICA, are brought forward.

Figure 5.7, Input signal in open-circuit condition simulated by MATLAB (x:Voltage, y:Time)

Figure 5.8, Input signal in open-circuit condition simulated by MODEILCA (x:Voltage, y:Time)

We have mentioned eigenfrequency in chapter 4.1.

In RG58 cable, H/m, F/m, the length of cable

59

equal to 100m, and there are 200 elements inside the cable, then we can calculate the

eigenfrequency:

rad/s

rad/s

In MATLAB and MODELICA, the incident waves float near 0.5V, and then they

jump and fluctuate near 1V, which is caused by the reflection from end point. The

reflection coefficient is:

Where represents the voltage of incident wave and is the voltage of reflected

wave.

In this condition, reflection coefficient equals to 1, VSWR is 0 and return loss is 0,

which means all the energy is be reflected and it causes maximum losses.

Then we took the time periods before the waves begin to reflect back towards in light

of the figures above:

Table 5.4, Propagation time it takes before the wave reflected back

Reflection Starting

Time MATLAB

MODELICA

Δt

1016.954 ns

1019.935 ns

The time it takes before the reflection starting is equal to that for signal to travel round

the cable, which is double of delay time. And in RG58, time delay is 504.5 ns, so as a

result, the theoretical result is about 1009 ns, which is very close to both the results

from MATLAB and MODELICA.

RG58 cable in short circuit

Here shows the figures of the input signal coming into cable RG58, which is

connected to a short end.

60

Figure 5.9, Input signal in short-circuit condition simulated by MATLAB (x:Voltage, y:Time)

Figure 5.10, Input signal in short-circuit condition simulated by MODELICA (x:Voltage, y:Time)

According to the graphics, incident wave floats near 0.5V, and then it diminishes to

zero.

The reflection coefficient is:

In this condition, reflection coefficient equals to -1, VSWR equals to 0 and return loss

is 0. It is similar to the open-circuit condition, all the power is reflected and it has

maximum losses.

When the transmission line is terminated in open circuit or with a short end, the power

reaching the end of the line is reflected back toward the source. In both of these two

conditions, the reflected voltage amplitudes are equal to 0.5 V. And in open circuit,

the reflected voltage wave is in phase with the incident voltage wave at the plane of

the load.

Besides, in short-circuit condition, voltage at the end of the line goes to zero, and the

incident voltage disappears at the short. The reflected voltage wave is equal in

61

magnitude to the incident voltage wave and be 180 degrees out of phase with it at the

plane of the load.

RG58 cable with matched load

Figure 5.11 and Figure 5.12 represent that signal waves at beginning of RG58

connected with a 50 Ohm load.

Figure 5.11, Input signal in matched load condition simulated by MATLAB (x:Voltage, y:Time)

Figure 5.12, Input signal in matched load condition in MODELICA (x:Voltage, y:Time)

In above figures, curves always float near 0.5V. It is very clear that the reflection

coefficient is 0 and VSWR is 1, while return loss will be infinite. It indicates there is

no reflection in matched load. All the power is transmitted.

When the transmission line is linked to its characteristic impedance, no reflected

signal occurs, as what we can see from the figure above, and the power is transferred

outward from the source until it reaches the load at the end, where it is completely

absorbed. As a result, although there is some impulse and noise, no standing waves

will be developed along the line. The voltage through the line remains a constant, half

of the source.

62

RG58 cable connected by RG59

The figures of this situation have already been showed in previous part 5.1.1. Now we

just separate the input signal lines from both MATLAB and MODELICA.

Figure 5.13, Input signal simulated by MATLAB (x:Voltage, y:Time)

Figure 5.14, Input signal simulated by MODELICA (x:Voltage, y:Time)

As can be seen from the figures, in this case, the incident wave jumps twice. And the

same as what we found before, the periods it takes before the reflections begin are

double of these two cable’s delay time.

In the course Modeling and Verification, we did the measurement on the

two-cable-connected condition in laboratory. By using CSV format files to save the

data from oscilloscope, we plot the result of input signal in the system in MATLAB as

following. When we focus on the reflections, they increase smoothly instead of

jumping immediately. So we mark the time points when they start to rise. The first

point represents the step impulse time.

63

Figure 5.15, Input signal result from the experiment (x:Voltage, y:Time)

In the measurement result above, the curve is almost smooth any time even if it has

some noises while there are severe vibrations at every impulse in the previous two

graphs from MATLAB and OpenModelica simulations. The reason is that we separate

the circuit into many sections of inductor and capacitor which may lead to eigen

frequency, which we have already explained in chapter 4.1 and chapter 5.2.1.

When calculating the first reflection coefficient, we can regard the characteristic

impedance of the second cable which is 75 Ω as a load. Then the reflection coefficient

will be:

Since the last cable is terminated in open-condition, the second reflection coefficient

equals to 1.

Then we can process the data values and calculate time periods before the first

reflections and second reflections, as well as the first and second reflection

coefficients in Theoretical way, MATLAB and MODELICA.

Table 5.5, data calculated by different methods

Method First reflection

interval

Second reflection

interval

First reflection

coefficient

Sencond

reflection

coefficient

Theoretical 1009 ns 250.96 ns 0.2 1

MATLAB 1018.203 ns 250 ns 0.218 1.02

MODELICA 1022.37 ns 275.38 ns 0.218 1.02

Experiment 1033 ns 275 ns 0.15 0.67

64

It is obvious that all these values at the same line of first three rows are very similar to

each other so that we can conclude that modeling results from MATLAB and

OpenModelica are almost correct.

Moreover, in Experiment results, the time periods are close to the Theoretical values

although there are some disparities, since they are just about 20 ns (10-9

s) which are

such small.

However, the first and second reflection coefficients of measurement results are both

around 30% less than the theoretical answers respectively. We think this phenomenon

may be caused by the loss in the real cables for they are not ideal as well as

interferences around.

RG58 with different resistor load

Next, we want to study how the load the cable is terminated with affects on the

reflection. We illustrate two conditions which the RG58 cable is end with 20 ohm

resistor and 70 ohm resistor.

Figure 5.16, Input signal when RG58 cable terminated with 20 ohm resistor (x:Voltage, y:Time)

Figure 5.17, Input signal when RG58 cable terminated with 70 ohm resistor (x:Voltage, y:Time)

65

From these two curves, we can easily find that, when the terminating resistance is not

equal to its characteristic impedance which is 50 Ohm here, the termination absorbs

only part of the power reaching it. And the remainder goes back along the line toward

the source. By comparing Figure 5.16 and Figure 5.17, we found the amplitude

decreased more in Figure 5.16 than it jumped in Figure 5.17. It indicates that the more

the terminating resistance differs from characteristic impedance, the larger the

percentage of the incident power that is reflected. When the terminating resistance is

less than the characteristic impedance, the reflected wave is 180 degrees out of phase

with the incident wave at the plane of the load, and it is in phase with the incident

voltage wave at the plane of the load in opposite way.

5.2 Lossy Coaxial Cable

For the lossy cables, we will assume some parameters which are derived based on

RG58 cable and analyze the propagation constant in some specific conditions. To

simplify the modeling system, we assumed the values of both additional resistance

and conductance are constant.

5.2.1 Propagation constant

The propagation constant of an electromagnetic wave is a measure of the change

undergone by the amplitude of the wave as it flows in a given direction. The quantity

measured, such as voltage, is expressed as a sinusoidal phasor. The phase of the

sinusoid varies with distance which contributes the propagation constant being a

complex number, the imaginary part being caused by the phase change. [29]

The general propagation constant of a lossy line is:

Where describes the signal attenuation, and describes the wave propagation

along the line.

From the definition of wavenumber [29]:

Where is wavelength

Then the wave phase velocity can also be expressed as:

The propagation constant will have the following solutions when the values of

resistance and conductance are under these two conditions.

Low loss transmission line

In low loss transmission line, assume and and we can get the

value of and in this condition.

66

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

Since and , then we can ignore the value of

:

(5.6)

Then we can make a Taylor series expansion, which is:

(5.7)

When is tend to zero, then it can be expressed as

We set

, and apply the Taylor series equation to equation (5.6):

(5.8)

(5.9)

(5.10)

(5.11)

Where

is the characteristics impedance when

And equate real and imaginary parts in equation (5.11) to give:

(5.12)

(5.13)

The propagation velocity of the wave

Distortionless transmission line

If the different frequencies that comprise a signal travel at different velocities, that

signal will arrive at the end of a transmission line distorted. We call this phenomenon

67

signal dispersion. In an opposite way, if the phase velocity is independent of

frequency, then no dispersion will occur.

Heaviside found that a transmission line would be distortionless if the line parameters

exhibited the following ratio [30]:

The complex propagation constant can be expressed as:

(5.14)

(5.15)

Since

, propagation constant can be rewritten as:

(5.16)

Thus real and imaginary parts are:

(5.17)

(5.18)

The propagation velocity of the wave

The propagation velocity is independent of frequency, so this lossy transmission line

is not dispersive.

Typically

, and to make a line meet the Heaviside condition the four primary

constants need to be adjusted. G could be increased, but this is highly undesirable

since G will have significant influence in the loss. Decreasing R is sending the loss in

right direction, but this is still not a satisfactory solution since it makes the cable much

more bulky and cost much. Decreasing C also makes the cable more bulky but is not

so costly as increasing the copper content. This leaves increasing L which is the usual

solution adopted. It is achieved by adding series inductors periodically along the

transmission line. [30]

5.2.2 Lossy coaxial cable verification for 2 conditions

We use MATLAB and MODELICA to analyze lossy cable in 2 specific conditions:

Heaviside condition and low loss approximation.

Heaviside condition

Consider a cable has the following parameters.

68

Table 5.6, Parameters for a lossy cable which fulfill Heaviside condition

Length

( )

Inductance

(L)

Capacitance

(C)

Impedance

( )

Resistance

(R)

Conductance

(G)

Velocity

100 m

252 10-9

H/m

101 10-12

F/m

50 Ω 0.2 Ω

C

1.98216 108 m/s

Propagation Time for this kind of cable can be calculated,

504.5 ns

Figure 5.18, Simulation result from MATLAB in Heaviside condition (x:Voltage, y:Time)

Figure 5.19, Modeling result from MODELICA in Heaviside condition (x:Voltage, y:Time)

The upper figures indicate there is no dispersion occurs when C, L, G, R has this

relation:

69

Unfortunately, this ideal property lossy cable does not exist since the R, L, G and C

are sufficiently frequency dependent.

The figures show that the speed of propagation is the same for all angular frequency.

Time delay for this condition is equal to lossless condition which is 504.5ns.

From the figure, we can see that the output signal arrive about 0.7V in the end and the

final amplitude of the input signal is little higher than the output signal. There are

some losses in this kind of cable. The shape of the signal with respect to position

remains constant although it gradually gets smaller with the attenuation. And we can

calculate attenuation through attenuation coefficient :

.

Further, we can calculate the value of gain and the number of dB loss of that cable

over a length l00m:

In MATLAB and MODELICA, the source voltage is 1V.

Then we put all the results that have been calculated above into the form to contrast

outcomes in different method.

Table 5.7, Results concluded in Heaviside condition

Methods

Theoretical MATLAB MODELICA

Time delay

504.5 ns

469.6 ns

469.3 ns

LossdB

3.474 dB

3.462 Db

3.533 dB

In the line with Table 5.7, the numbers in each rows are close to each other which can

be considered as they are mostly correct.

Low loss coaxial cable ( and )

Consider a lossy cable has the following parameters:

70

Table 5.8, Parameters for a lossy cable which fulfill Low loss approximation

Length

( )

Inductance

(L)

Capacitance

(C)

Impedance

( )

Resistance

(R)

Conductance

(G)

Velocity

100 m

252 10-9

H/m

101 10-12

F/m

50 Ω

252 10-6

Ω

101 10-8

S

1.98216 108 m/s

Time delay for this kind of cable can be calculated,

Figure 5.20, Simulation result from MATLAB in low loss condition (x:Voltage, y:Time)

Figure 5.21, Modeling result from MODELICA in low loss condition (x:Voltage, y:Time)

In upper two figures, the amplitude of the output signal almost arrives about 1V. The

losses are so slight that we cannot find from the figures. Since the value of resistance

and conductance are very small, the effect of these two components is very slight on

the wave propagation and signal attenuation.

71

Under and condition, the attenuation coefficient:

.

The same as what we did before, we made a table to integrate the data.

Table 5.9, Results concluded for low loss condition

Methods

Mathematics MATLAB MODELICA

Time delay

504.5 ns

465.6 ns

465.33 ns

0.024 dB

0 dB

0 dB

It is clear that every row has three similar values. Therefore, we can obtain the

conclusion that according to verification, the results are almost correct.

5.2.3 Analysis for lossy cable in other conditions

Following conditions are dispersion phenomenon, which signal arrives at the end of

transmission line distorted. Dispersion can be a problem if the lines are very long and

just a small difference in phase velocity can result in significant difference in

propagation delay. The values of length, L and C we set to run MODELICA are the

same as RG58 cable.

Figure 5.22, Result from MODELICA when R=0.5 and G is negligible (x:Voltage, y:Time)

When R is not very small and neglects the value of G, this kind of loss is result from

the skin effect. This causes sharp edged pulses to become rounded and distorted. We

72

can find that the attenuation is very slight in this condition since the output signal

finally arrive 1V in the end.

Figure 5.23, Simulation result from MODELICA when R=0.5 and G=0.00005 (x:Voltage, y:Time)

When conductance is added, the line has significant losses even the value of G is very

small since it has both skin-effect losses and dielectric losses. The contribution of

addition conductance to the losses is very obvious. The output signal does not overlap

the input signal at the end. And it also has the dispersion phenomenon.

Figure 5.24, Simulation result from MODELICA when R=0.05 and G=0.0002 (x:Voltage, y:Time)

In this case, the cable has both resistor and conductor components, and the value of

conductor is a little bigger than the former conditions, but the waves have a huge

difference with the former conditions. The former condition waves increase gradually,

then stop at some point and propagate smoothly. In this condition, the waves decrease

gradually, then stop at some point and propagate smoothly. Even the value of

resistance is less than the former conditions, it has more losses. It reveals that the

dielectric losses influence the attenuation more easily.

73

6 CONCLUSION

At first, comparing the results from modeling simulation between MATLAB and

OpenModelica we have shown in previous parts, they are very closed to each other

separately. And we achieve our goals and requirements on studying on the

propagation time of the voltage waves, the signal amplitudes and reflections in the

coaxial cables. Thus, we can say the project has been finished properly as the

behaviors shown by the modeling from two types of software and theoretical answers

are essentially the same.

Secondly, about convenience and time used. The way that MATLAB and Simulink

model the cable is more complicated when compared with MODELICA.

In MATLAB and Simulink, we need to first solve the transmission line circuit and

find the matrix of ABC-model. But in MODELICA, we can connect the components

to form the circuit and plot the figure which describe the performance of wave in

different components, which is a more simple and convenient than MATLAB. Though

the results MODELICA achieved is not exactly the same as the results from

MATLAB, the difference between them is very tiny and normally we can neglect it.

So the precision of MODELICA is reliable.

For the time the software took to run the simulation, e.g. when we modeled the

lossless cable terminated in open-circuit. MATLAB took 13.948832 seconds to run

the program, while OpenModelica took 5 minutes and 34.7 seconds to process the

model. It is very obvious that OpenModelica spent much more time on running the

program than MATLAB. Even though, as what we discussed before, it also took much

time for us on calculating the ABC matrices, making simulink and typing the code to

build models when we applied MATLAB. So we can regard the time MATLAB and

OPENMODELICA spend are similar.

Since a cable consists of a high number of lumped elements, we need to set the

number of element in these two kinds of software. We can set the number up to 1000

for MATLAB, while it will be hard for computer to run the program when the number

is set beyond 250 for MODELICA. MODELICA is a very new modeling language

and it is just developed recently, who still has a great room to improve.

For the version “OpenModelica-1.8.0”, it can only run 12 elements as maximum,

whereas for the latest version which was issued 6 months later, it can run up to 250

elements. We think the developers will make it more and more in the future.

Price is also very important for users. For the official price of MATLAB & Simulink

Student Version is 89 USD while OpenModelica is totally free and all things that we

mentioned before, OpenModelica is a software that worthy looking forward to.

In our thesis project, we only study on a special transmission line: coaxial cable with

short length. But when the lines have long distance especially the bus structure and a

higher frequency signal line with large intensity, crosstalk may occur. In addition,

74

since there are various kinds of transmission line as what we have mentioned in

Background part, all of which have distinct features to separate with each other, as

well as some other common one which are also proverbially applied such as 3 -phase

transmission lines.

75

REFERENCE

[1] Wikipedia, “Coaxial cable”

http://en.wikipedia.org/wiki/Coaxial_cable

[2] Glenn Elmore, “Introduction to the Propagation Wave on a Single Conductor”,

Corridor System Inc., 20090727.

[3] Tech-FAQ, “RG-58”

http://www.tech-faq.com/rg-58.html

[4] Tech-FAQ, “RG-59”


[5] Tom Gresham, “History of Matlab”, eHow Contributor, June 29, 2011

http://www.ehow.com/info_8665330_history-matlab.html

[6] Rob Schreiber, “MATLAB”, Scholarpedia, 2007

http://www.scholarpedia.org/article/MATLAB

[7] Peter Fritzson, “Principles of object-Oriented Modeling and Simulation with

Modelica 2.1”, ISBN-0-471-47163-1, February 2004, Wiley-IEEE Press

[8] Wikipedia, “Modelica”

http://en.wikipedia.org/wiki/Modelica

[9] Peter Fritzson, “Introduction to Modelica”, September 3, 2001

[10] Peter Fritzson, Peter Aronsson, Håkan Lundvall, Kaj Nyström, Adrian Pop,

Levon Saldamli, David Broman, “The OpenModelica Modeling, Simulation

and Development Environment”, Linköing University, Computer Science Dept.

[11] Wikipedia, “Transmission line”


[12] Prof.Dr.Sandro M.Radicella, Pro.Dr.Ryszard Struzak, “Radio Laboratory

Handbook 2004”, chapter 2: Transmission Line

[13] “Transmission line analysis for a coaxial system”

http://www.rfcables.org/articles/14.html

[14] Wikipedia, “Electrical resistance and conductance”

http://en.wikipedia.org/wiki/Electrical_resistance_and_conductance

[15] Wikipedia, “Telegrapher’s equation”

http://en.wikipedia.org/wiki/Telegrapher's_equations

[16] P.Coiner, “Calculating the Propagation delay of coaxial cable”, S&M

department, 20110125

[17] Wikipedia, “Wave propagation speed”

http://en.wikipedia.org/wiki/Wave_propagation_speed

[18] “Electrical Characteristics of Transmission lines”, 20060126

[19] Wikipedia, “Reflections of signals on conducting lines”

http://en.wikipedia.org/wiki/State_space_(controls)

[20] Wikipedia, “Reflection coefficient”

http://en.wikipedia.org/wiki/Reflection_coefficient

[21] Wikipedia, “Standing wave ratio”

http://en.wikipedia.org/wiki/Standing_wave_ratio

http://en.wikipedia.org/wiki/Coaxial_cable



http://www.ehow.com/info_8665330_history-matlab.html

http://www.scholarpedia.org/article/MATLAB



http://www.rfcables.org/articles/14.html

http://en.wikipedia.org/wiki/Electrical_resistance_and_conductance

http://en.wikipedia.org/wiki/Telegrapher's_equations

http://en.wikipedia.org/wiki/Wave_propagation_speed


http://en.wikipedia.org/wiki/Reflection_coefficient

http://en.wikipedia.org/wiki/Standing_wave_ratio

76

[22] “VSWR, Reflection coefficient, Return loss, s11/s22”, Signal Processing

Group Inc., Technical memorandumRF-0909

[23] Wikipedia, “Kirchhoff’s laws”

http://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws

[24] Wikipedia, “State space (controls)”


[25] “Matlab”, Mathworks

http://www.mathworks.se/products/matlab/

[26] “Simulink”, Mathworks

http://www.mathworks.se/products/simulink/

[27] Wikipedia, “Modelica”


[28] “OpenModelica”

http://www.openmodelica.org/

[29] Wikipedia, “Propagation constant”

http://en.wikipedia.org/wiki/Propagation_constant

[30] Wikipedia, “Heaviside condition

http://en.wikipedia.org/wiki/Heaviside_condition

[31] Murray Thompson, “Transmission Lines”, Physics 623, Sept. 1,1999.

[32] H. Riege, “HIGH-FREQUENCY AND PULSE RESPONSE OF COAXIAL

TRANSMISSION CABLES WITH CONDUCTOR, DIELECTRIC AND

SEMICONDUCTOR LOSSES”, European organization for nuclear research,

Proton Synchrotron Department, 4 Feb, 1970

[33] P. Fonseca, A.C.F. Santos and E.C. Montenegro, “A very simple way to

measure coaxial cable impedance”, Instituto de Fisica, Universidade Federal do

Rio de Janeiro

[34] “Transmission lines”, University of Liverpool, PHYS370- Advanced

Electromagnetism

[35] Mohazzab JAVED, Hussain AFTAB, Muhammad QASIM, Mohsin SATTAR,

“RLC Circuit Response and Analysis (Using State Space Method)”, IJCSNS

International Journal of Computer Science and Network Security, VOL.8 NO.4,

April 2008

[36] Eric Bogatin, Mike Resso, Steve Corey, “Practical Characterization and

Analysis of Lossy Transmission Lines”, DesignCon 2001, 2002 Agilent

Technologies, Inc.

[37] Richard Fitzpatrick, “Oscillations and Waves”, Professor of Physics, The

Univeristy of Texas at Austin

[38] “MATLAB Reference Guide”, COPYRIGHT 1984-93 by The MathWorks,

October 1992

http://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws


http://www.mathworks.se/products/matlab/

http://www.mathworks.se/products/simulink/


http://www.openmodelica.org/

http://en.wikipedia.org/wiki/Propagation_constant

http://en.wikipedia.org/wiki/Heaviside_condition

77

APPENDICES

Matlab file to model lossless RG58 cable terminated in open-circuit clear all

close all

n=200;

CC=(101e-12)*100/n;

L=(252e-9)*100/n;

R=50;

A=zeros((2*n),(2*n));

B=zeros((2*n),1);

C=zeros(1,(2*n));

B((n+1),1)=1/L;

C(1,n)=1;

D=0;

for i=1:n;

A(i,(i+n))=1/CC;

A((i+n),i)=-1/L;

end;

for i=1:(n-1);

A(i,(i+n+1))=-1/CC;

A((i+n+1),i)=1/L;

end;

A((n+1),(n+1))=-R/L;

C0=zeros(1,2*n);

C0(1,1)=1;

sim('short',0.000002);

figure(1)

plot(time,u0,'r');

hold on

plot(time,u1,'g');

grid on,title('open circuit');

xlabel('time(s)');

ylabel('voltage(V)');

legend('input signal','output signal');

Matlab file to model lossless RG58 cable terminated in matched load:

clear all

78

close all

n=200;

CC=(101e-12)*100/n;

L=(252e-9)*100/n;

R=50;

A=zeros((2*n),(2*n));

B=zeros((2*n),1);

C=zeros(1,(2*n));

B((n+1),1)=1/L;

C(1,n)=1;

D=0;

for i=1:n;

A(i,(i+n))=1/CC;

A((i+n),i)=-1/L;

end;

for i=1:(n-1);

A(i,(i+n+1))=-1/CC;

A((i+n+1),i)=1/L;

end;

A(n,n)=-1/(R*CC);

A((n+1),(n+1))=-R/L;

C0=zeros(1,2*n);

C0(1,1)=1;

sim('short',0.000002);

figure(1)

plot(time,u0,'r');

hold on

plot(time,u1,'g');

grid on,title('matched circuit');

xlabel('time(s)');



Matlab file to model lossless RG58 cable terminated in short-circuit: clear all

close all

n=200;

79

CC=(101e-12)*100/n;

L=(252e-9)*100/n;

R=50;

A=zeros((2*n+1),(2*n+1));

B=zeros((2*n+1),1);

C=zeros(1,(2*n+1));

B((n+1),1)=1/L;

C(1,n)=1;

D=0;

for i=1:n;

A(i,i+n)=1/CC;

A(i,i+n+1)=-1/CC;

A(i+n,i)=-1/L;

A(i+n+1,i)=1/L;

end;

A((n+1),(n+1))=-R/L;

C0=zeros(1,2*n+1);

C0(1,1)=1;

sim('short',2e-6);

figure(1)

plot(time,u0,'r');

hold on

plot(time,u1,'g');

grid on,title('short circuit');

xlabel('time(s)');



Matlab file to model a RG58 cable terminated in matched load clear all

close all

n=200;

CC=(101e-12)*100/n;

L=(252e-9)*100/n;

R=50;

A=zeros((2*n),(2*n));

B=zeros((2*n),1);

C=zeros(1,(2*n));

80

B((n+1),1)=1/L;

C(1,n)=1;

D=0;

for i=1:n;

A(i,(i+n))=1/CC;

A((i+n),i)=-1/L;

end;

for i=1:(n-1);

A(i,(i+n+1))=-1/CC;

A((i+n+1),i)=1/L;

end;

A(n,n)=-1/(R*CC);

A((n+1),(n+1))=-R/L;

C0=zeros(1,2*n);

C0(1,1)=1;

sim('short',0.000002);

figure(1)

plot(time,u0,'r');

hold on

plot(time,u1,'g');

grid on,title('matched circuit');

xlabel('time(s)');



Matlab file to model a circuit which RG58 cable connected with RG59 cable: clc

clear all

close all

n=400;

A=zeros(2*n,2*n);

CC1=(101e-12)*100/n*2;

L1=(252e-9)*100/n*2;

CC2=(67e-12)*25/n*2;

L2=(376e-9)*25/n*2;

R=50;

for i=1:1:(n/2-1);

81

A(i,i+n/2+1)=-1/CC1;

end;

for i=1:1:n/2;

A(i,i+n/2)=1/CC1;

end;

for i=(n/2+1):1:n;

A(i,i-n/2)=-1/L1;

end;

for i=(n/2+2):1:n;

A(i,i-n/2-1)=1/L1;

end;

for i=(n+1):1:(3*n/2-1);

A(i,i+n/2+1)=-1/CC2;

end;

for i=(n+1):1:3*n/2;

A(i,i+n/2)=1/CC2;

end;

for i=(3*n/2+1):1:2*n;

A(i,i-n/2)=-1/L2;

end;

for i=(3*n/2+2):1:2*n;

A(i,i-n/2-1)=1/L2;

end;

A((n/2+1),(n/2+1))=-R/L1;

A(n/2,(3*n/2+1))=-1/CC1;

A((3*n/2+1),n/2)=1/L2;

B=zeros(2*n,1);

B((n/2+1),1)=1/L1;

C0=zeros(1,2*n);

C0(1,1)=1;

C1=zeros(1,2*n);

C1(1,n/2)=1;

C2=zeros(1,2*n);

C2(1,3*n/2)=1;

D=0;

clear CC1

clear CC2

clear L1

clear L2

82

clear R

clear i

clear n

sim('coax',2e-6);

figure(1)

plot(time,input,'r');

hold on

plot(time,y1,'g');

plot(time,y2,'b');

grid on,title('two coaxial cable');

xlabel('time(s)');


legend('input signal','signal at middele','output signal');

Matlab file to model a lossy cable: clear all

close all

n=100;

CC=(101e-12)*100/n;

L=(252e-9)*100/n;

R=50;

R1=252e-6*100/n;

G=101e-8*100/n;

A=zeros((2*n),(2*n));

B=zeros((2*n),1);

C=zeros(1,(2*n));

B((n+1),1)=1/L;

C(1,n)=1;

D=0;

for i=1:n;

A(i,i)=-G/CC;

A(i,(n+i))=1/CC;

A((i+n),i)=-1/L;

end;

for i=1:(n-1);

A(i,(i+n+1))=-1/CC;

A((i+n+1),i)=1/L;

end;

83

for i=(n+2):2*n;

A(i,i)=-R1/L;

end;

A((n+1),(n+1))=-(R+R1)/L;

C0=zeros(1,2*n);

C0(1,1)=1;

sim('endwith',0.000006);

figure(1)

plot(time,u0,'r');

hold on

plot(time,u1,'b');

grid on,title('lossy cable');

grid on,title('lossy cable');

xlabel('time(s)');



MODELICA notebook to model lossless RG58 cable terminated in open-circuit

model coaxcable

Modelica.Electrical.Analog.Lines.OLine oline1(r = 0, l = 2.52e-007, g = 0, c = 1.01e-010,

length = 100, N = 199)

annotation(Placement(visible = true, transformation(origin = {13.0751,50.3632}, extent =

{{-12,-12},{12,12}}, rotation = 0)));

Modelica.Electrical.Analog.Basic.Resistor resistor1(R = 50) annotation(Placement(visible

= true, transformation(origin = {-38.2567,50.8475}, extent = {{-12,-12},{12,12}}, rotation

= 0)));

Modelica.Electrical.Analog.Basic.Ground ground1 annotation(Placement(visible = true,

transformation(origin = {-72.6392,-39.7094}, extent = {{-12,-12},{12,12}}, rotation = 0)));

84

Modelica.Electrical.Analog.Sources.StepVoltage stepvoltage1(V = 1)

annotation(Placement(visible = true, transformation(origin = {-72.6392,11.6223}, extent

= {{-12,12},{12,-12}}, rotation = -90)));

Modelica.Electrical.Analog.Basic.Capacitor capacitor1(C = 1.01e-010)


{{-12,12},{12,-12}}, rotation = -90)));

equation

connect(oline1.p3,ground1.p) annotation(Line(points =

{{13.0751,38.3632},{13.5593,38.3632},{13.5593,-28.5714},{-72.6392,-28.5714},{-72.6392,

-27.7094}}));

connect(capacitor1.n,ground1.p) annotation(Line(points =

{{60.5327,-0.861985},{61.0169,-0.861985},{61.0169,-28.5714},{-72.6392,-28.5714},{-72.6

392,-27.7094}}));

connect(oline1.p2,capacitor1.p) annotation(Line(points =

{{25.0751,50.3632},{61.0169,50.3632},{61.0169,23.138},{60.5327,23.138}}));

connect(resistor1.n,oline1.p1) annotation(Line(points =

{{-26.2567,50.8475},{0.484262,50.8475},{0.484262,50.3632},{1.07506,50.3632}}));

connect(stepvoltage1.n,ground1.p) annotation(Line(points =

{{-72.6392,-0.377724},{-72.6392,-0.377724},{-72.6392,-27.7094},{-72.6392,-27.7094}}));

connect(resistor1.p,stepvoltage1.p) annotation(Line(points =

{{-50.2567,50.8475},{-73.1235,50.8475},{-73.1235,23.6223},{-72.6392,23.6223}}));

end coaxcable;

MODELICA notebook to model lossless RG58 cable terminated in short-circuit:

model short





= 0)));

85


length = 100, N = 199) annotation(Placement(visible = true, transformation(origin =

{-2.41546,36.715}, extent = {{-12,-12},{12,12}}, rotation = 0)));


annotation(Placement(visible = true, transformation(origin = {28.9855,-8.21256}, extent

= {{-12,12},{12,-12}}, rotation = -90)));


annotation(Placement(visible = true, transformation(origin = {-84.058,-6.28019}, extent

= {{-12,12},{12,-12}}, rotation = -90)));

equation

connect(stepvoltage1.p,resistor1.p) annotation(Line(points =

{{-84.058,5.71981},{-84.058,5.71981},{-84.058,37.1981},{-64.657,37.1981},{-64.657,36.7

15}}));


{{-84.058,-18.2802},{-84.058,-18.2802},{-84.058,-51.7681},{-84.058,-51.7681}}));


{{9.58454,36.715},{67.6329,36.715},{67.6329,-52.657},{-84.058,-52.657},{-84.058,-51.76

81},{-84.058,-51.7681}}));


{{28.9855,-20.2126},{28.9855,-20.2126},{28.9855,-52.1739},{-84.058,-52.1739},{-84.058,

-51.7681}}));


{{9.58454,36.715},{29.4686,36.715},{29.4686,3.78744},{28.9855,3.78744}}));


{{-2.41546,24.715},{-2.41546,24.715},{-2.41546,-51.6908},{-84.058,-51.6908},{-84.058,-

51.7681}}));


{{-40.657,36.715},{-14.4928,36.715},{-14.4928,36.715},{-14.4155,36.715}}));

end short;

MODELICA notebook to model a RG58 cable terminated in matched load

86

model matched





= 0)));



{-16.9082,28.5024}, extent = {{-12,-12},{12,12}}, rotation = 0)));


annotation(Placement(visible = true, transformation(origin = {-82.1256,-12.5604}, extent

= {{-12,12},{12,-12}}, rotation = -90)));


annotation(Placement(visible = true, transformation(origin = {12.0773,-7.24638}, extent

= {{-12,12},{12,-12}}, rotation = -90)));


= true, transformation(origin = {46.8599,-4.34783}, extent = {{-12,12},{12,-12}}, rotation

= -90)));

equation

connect(resistor2.n,ground1.p) annotation(Line(points =

{{46.8599,-16.3478},{45.8937,-16.3478},{45.8937,-47.343},{-81.6425,-47.343},{-81.6425,

-47.4203}}));

connect(oline1.p2,resistor2.p) annotation(Line(points =

{{-4.90821,28.5024},{47.343,28.5024},{47.343,7.65217},{46.8599,7.65217}}));


{{12.0773,-19.2464},{12.0773,-19.2464},{12.0773,-46.8599},{-81.6425,-46.8599},{-81.642

5,-47.4203}}));


{{-4.90821,28.5024},{12.0773,28.5024},{12.0773,4.75362},{12.0773,4.75362}}));

87


{{-16.9082,16.5024},{-16.4251,16.5024},{-16.4251,-47.343},{-81.6425,-47.343},{-81.6425

,-47.4203}}));


{{-42.5894,28.5024},{-30.4348,28.5024},{-30.4348,28.5024},{-28.9082,28.5024}}));

connect(stepvoltage1.p,resistor1.p) annotation(Line(points =

{{-82.1256,-0.560386},{-82.1256,-0.560386},{-82.1256,28.5024},{-66.5894,28.5024},{-66.

5894,28.5024}}));


{{-82.1256,-24.5604},{-81.6425,-24.5604},{-81.6425,-47.4203},{-81.6425,-47.4203}}));

end matched;

MODELICA notebook to model a circuit which RG58 cable connected with

RG59 cable:

model bicoax



= 0)));



{-5.32688,35.8354}, extent = {{-12,-12},{12,12}}, rotation = 0)));



{53.2688,35.3511}, extent = {{-12,-12},{12,12}}, rotation = 0)));

88




annotation(Placement(visible = true, transformation(origin = {-77.9661,-8.88178e-016},

extent = {{-12,12},{12,-12}}, rotation = -90)));


annotation(Placement(visible = true, transformation(origin = {21.3075,-2.66454e-015},

extent = {{-12,12},{12,-12}}, rotation = -90)));


annotation(Placement(visible = true, transformation(origin = {79.4189,0.968523}, extent

= {{-12,12},{12,-12}}, rotation = -90)));

equation


{{79.4189,-11.0315},{78.9346,-11.0315},{78.9346,-35.8354},{-77.4818,-35.8354},{-77.481

8,-35.9419}}));


{{53.2688,23.3511},{53.753,23.3511},{53.753,-35.8354},{-77.4818,-35.8354},{-77.4818,-3

5.9419}}));


{{21.3075,-12},{21.7918,-12},{21.7918,-35.8354},{-77.4818,-35.8354},{-77.4818,-35.9419

}}));


{{-5.32688,23.8354},{-4.84262,23.8354},{-4.84262,-35.8354},{-77.4818,-35.8354},{-77.48

18,-35.9419}}));


{{-77.9661,-12},{-77.4818,-12},{-77.4818,-35.9419},{-77.4818,-35.9419}}));


{{65.2688,35.3511},{79.9031,35.3511},{79.9031,12.9685},{79.4189,12.9685}}));

connect(oline1.p2,oline2.p1) annotation(Line(points =

{{6.67312,35.8354},{41.1622,35.8354},{41.1622,35.3511},{41.2688,35.3511}}));


{{6.67312,35.8354},{21.3075,35.8354},{21.3075,12},{21.3075,12}}));


{{-33.0363,36.3196},{-8.23245,36.3196},{-8.23245,35.8354},{-17.3269,35.8354}}));


{{-57.0363,36.3196},{-77.4818,36.3196},{-77.4818,12},{-77.9661,12}}));

end bicoax;

MODELICA notebook to model a lossy cable:

89

model lossy





= 0)));

Modelica.Electrical.Analog.Lines.OLine oline1(r = 0.005, l = 2.52e-007, g = 2e-006, c =

1.01e-010, length = 100, N = 199) annotation(Placement(visible = true, transformation(origin

= {2.90557,49.8789}, extent = {{-12,-12},{12,12}}, rotation = 0)));


annotation(Placement(visible = true, transformation(origin = {-74.092,8.71671}, extent =

{{-12,12},{12,-12}}, rotation = -90)));



{{-12,12},{12,-12}}, rotation = -90)));

Modelica.Electrical.Analog.Basic.Conductor conductor1(G = 2e-006)


{{-12,12},{12,-12}}, rotation = -90)));

equation

connect(conductor1.n,ground1.p) annotation(Line(points =

{{72.6392,8.33898},{72.6392,8.33898},{72.6392,-31.477},{-73.6077,-31.477},{-73.6077,-3

1.0993}}));


{{44.0678,8.33898},{44.0678,8.33898},{44.0678,-31.9613},{-73.6077,-31.9613},{-73.6077,

-31.0993}}));

connect(oline1.p2,conductor1.p) annotation(Line(points =

{{14.9056,49.8789},{72.6392,49.8789},{72.6392,32.339},{72.6392,32.339}}));

90


{{2.90557,37.8789},{3.38983,37.8789},{3.38983,-31.477},{-73.6077,-31.477},{-73.6077,-3

1.0993}}));


{{14.9056,49.8789},{44.0678,49.8789},{44.0678,32.339},{44.0678,32.339}}));


{{-32.0678,49.8789},{-9.20097,49.8789},{-9.20097,49.8789},{-9.09443,49.8789}}));


{{-56.0678,49.8789},{-74.092,49.8789},{-74.092,20.7167},{-74.092,20.7167}}));


{{-74.092,-3.28329},{-73.6077,-3.28329},{-73.6077,-31.0993},{-73.6077,-31.0993}}));

end lossy;

coaxial cable modeling and verification

Documents