COAXIAL CABLE MODELING AND VERIFICATION

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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

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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.

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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

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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

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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.

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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.

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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

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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

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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

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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]

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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.

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 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

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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]

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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]

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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:

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(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

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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

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(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

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(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]

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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]

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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

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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

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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.

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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]

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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 ,

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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.

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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

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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:

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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

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

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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)

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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)

A =

B =

C = D = 0

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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)

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(4.45)

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

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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)

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(4.67)

(4.68)

(4.69)

(4.70) We can also write as state space representation:

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

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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 =

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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

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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^-6s.

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^-12F/m and the inductance equals to 252 10^-10H/m. And for lossless cable RG59, capacitance is 67 10^-12F/m and inductance is 376 10^-9H/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^-6Ω and 101 10^-8 S. Furthermore, we run all the models with the number of sections 200.

Figure 4.11, Solver options in MATLAB

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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 4^th and 5^th 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

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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

COAXIAL CABLE MODELING AND VERIFICATION