Chapter 5-Transmission Lines

CHAPTER 5: TRANSMISSION LINES

Chp 5/Transmission Lines Puan Norliza Mohamed 2

INTRODUCTION

Transmission lines in communication systems carry signals from a transmitter to an antenna and from an antenna to a receiver, carry telephone signals, computer data in LANs, TV signals and many more.

Transmission lines are critical links in any communication system.

Their electrical characteristics are critical and must be matched to the equipment for successful communication to take place.

Chp 5/Transmission Lines Prepared By: Noor Hamizah Hussain 3

The TWO primary requirements of a transmission line are :

1. The line should introduce minimum attenuation to the signal.

2. The line should not radiate any of the signal as radio energy.

All transmission lines and connectors are designed with these requirements in mind.

TRANSMISSION LINE BASICS


TYPE OF TRANSMISSION LINES

PARALLEL-WIRE LINES – is made of two parallel wire spaced from one another by an insulator. Also called twin lead. These line are rarely used today.

COAXIAL CABLE – consists of a solid center conductor surrounded by a dielectric material. Over the insulator is a second conductor, a tubular braid made of fine wires. An outer plastic sheath protects and insulates the braid. Most widely used.


TYPE OF TRANSMISSION LINES


BALANCED & UNBALANCED LINE

BALANCED LINE – means that the same current flows in each wire with respect to ground, although the direction of current in one wire is 180o out of phase with the current in the other wire. Neither wire is connected to ground. Example is twin lead.

UNBALANCED LINE – one conductor is connected to ground. The current in one conductor is referenced to the other conductor which is connected to ground. Example is coaxial cable.


ELECTRICAL MODEL OF A TRANSMISSION LINE

R = resistance of the wireG = conductance of the dielectricL = series inductanceC = shunt capacitance



At dc and low frequenciesThe inductance has no effect because the line reactance is very small compared with the resistance of the line.Reactance of the shunt capacitance is very large, so the effect of the capacitance is negligible.The line is characterized by its resistance.



At high frequencyAs the frequency increases, the inductance and capacitance begin to have an effect.The higher the frequency, the larger the series inductive reactance and the lower the parallel capacitive reactance.Therefore only the inductance and capacitance of the line is considered and the resistance can be neglected.Such line is called lossless, since the inductive and capacitive reactance store energy but do not dissipate it.


CHARACTERISTIC IMPEDANCE ZO

Characteristic impedance define the impedance at the input of an infinite length transmission line.

Where Zo = characteristic impedance of the line in ohms R = conductor resistance in ohms per unit length L = inductance in henrys per unit length G = dielectric conductance in siemens per unit length C = capacitance in farads per unit length ω = operating frequency in radians per second

CjGjωR

oZ

L


HIGH FREQUENCY MODEL OF A TRANSMISSION LINE

At high frequency, as ω gets larger, the values of R and G becomes less significant in comparison with L and C. The characteristic impedance for a finite length of line is is purely resistive .

EXAMPLE A coaxial cable has a capacitance of 90 pF/m and a characteristic

impedance of 50 Ω. Find the inductance of a 1 m length. Answer = 225

nH/m

ΩCLoZ


ZO FOR A PARALLEL LINE WITH AN AIR DIELECTRIC

Where :Zo = characteristic impedance of the line

D = spacing between the centers of the conductorsr = conductor radius

ΩrDlog276oZ


ZO FOR COAXIAL CABLE

Where :Zo = characteristic impedance of the lineD = inside diameter of the outer conductord = diameter of the inner conductorr = relative permittivity of the dielectric, compared with that of free space (also known as dielectric constant)

ΩdDlog

rε138

oZ


EXAMPLE

Find the Zo for each of the following lines

(a) an open-wire line with conductors 3 mm in diameter separated by 10 mm.

(b) a coaxial cable using a solid polyethylene dielectric having r = 2.3, with an inner conductor 2 mm in diameter and an outer 8 mm in inside diameter.

Answer : 277 , 54.78


COAXIAL CABLE APPLICATIONS

Impedance Application Typical Type

Numbers

50 Radio transmitters RG-8/U

50 Communications receivers RG-58/U

75 Cable television, tv antenna feedlines RG-59/U

93 Computer network RG-62/U


TRANSMISSION LINE FUNDAMENTALS

WAVELENGTH OF CABLESAt LOW frequencies, the transmission line acts simply as a carrier of the ac signal. The only characteristics of the cable is its resistive loss.

However when such cables are used to carry RF energy, it appears to be complex interconnections of inductors, capacitors and resistors.

fc


WAVELENGTH OF CABLES

Whenever the length of a transmission line is the same order of magnitude of greater than the WAVELENGTH of the transmitted signal, now the transmission line takes on special characteristics that require a more complex analysis.

A conductor is not considered a transmission line unless it’s length is at least 0.1 l long at the signal frequency

Example : Calculate the wavelength of 50 MHz signal.


VELOCITY FACTOR

The speed of the signal in the transmission line is slower than the speed of a signal in free space.

The velocity of propagation, VP of a signal in a cable is less than the velocity of propagation of light in free space.This is caused by a fraction called the velocity factor, VF.

8PP

103V

cVVF


VF vary from 0.5 to 0.9.

VF can also be computed using

Where is the dielectric constant of the insulating material.

The VF must be taken into consideration when computing the length of a transmission line in wavelengths,

VELOCITY FACTOR

ε1VF


VELOCITY FACTOR

Dielectric VF

Solid Polyethelene 0.66

Foam Polyethelene 0.78

Air 0.95

Teflon

Example:Find the velocity factor and propagation velocity for a cable with a Teflon dielectric with er =2.1.

Answer: 0.69, 2.07 x 108 m/s


DIELECTRIC CONSTANTS

Material Dielectric Constant, e

Vacuum 1

Air 1.00006

Teflon 2.1

Polyethelene 2.27

Paper 2.5

Rubber 3.0

Polyvinyl chloride, PVC 3.3

Mica 5.0

Glass 7.5


TERMINATED TRANSMISSION LINE

source loadTransmission line

Incident wave

Reflected wave

An ordinary transmission line is bidirectional, power can propagate equally well in both direction.Incident voltage is the voltage that propagate towards the load.Voltage that propagates from the load toward the source is called reflected voltage.


MATCHED LINE

A matched line is a transmission line that is terminated with a load equal to the characteristics impedance.The energy will propagate down the line, and the ratio between voltage and current at the source will be equal to the Zo.The transmission goes smoothly and maximum power being transferred to the load or radiated by the antenna.The line can be of any length.The ac voltage (rms) at any point on a matched line is a constant value, therefore is said to be FLAT.


THE VOLTAGE ON A MATCHED LINE


SHORT CIRCUIT LINE

An open circuit line is a transmission line that is terminated with an SHORT circuit.The energy will propagate down the line but cannot be fully dissipated or absorbed since there is no load.At the end of an short-circuited line, the voltage is zero when the current is maximum.All of the energy is reflected from the end of the line and moves back toward the generator.This reflected voltage adds to the incident voltage and forms a composite voltage that is distributed along the line.This pattern of voltage and its related current constitutes what is called a STANDING WAVE.


STANDING WAVE ON ASHORT-CIRCUITED LINE


OPEN CIRCUIT LINE

A short circuit line is a transmission line that is terminated with an OPEN circuit.The energy will propagate down the line but cannot be fully dissipated or absorbed since there is no load.At the end of an open-circuited line, the voltage is maximum when the current is zero.All of the energy is reflected from the end of the line and moves back toward the generator.This reflected voltage adds to the incident voltage and forms a composite voltage that is distributed along the line.This pattern of voltage and its related current constitutes what is called STANDING WAVE.


STANDING WAVE ON AN OPEN-CIRCUITED LINE


MISMATCHED LINE

A MISMATCH circuit line is a transmission line that is terminated with a load where the impedance differ from Zo.

The energy will propagate down the line but cannot be fully dissipated or absorbed by the load.Some of the power is reflected from the end of the line and moves back toward the generator.This reflected voltage adds to the incident voltage and forms a composite voltage that is distributed along the line.This pattern of voltage and its related current constitutes what is called STANDING WAVE.


STANDING WAVE ON A MISMATCHED LINE


STANDING WAVE RATIO

The magnitude of the standing wave on a transmission line is determined by the ratio of the maximum current to the minimum current, or the ratio of the maximum voltage to the minimum voltage along the line.The ratio is referred to as STANDING WAVE RATIO (SWR).

min

max

min

maxVV

IISWR


STANDING WAVE RATIO

Under the shorted and open conditions, the current or voltages minimum is zero, thus produces SWR of infinity.It means that no power is dissipated in the load, all of it is reflected.On a matched line, or ideal case, the V and I are constant, therefore the SWR is 1.On a mismatched line, the V and I is changing with respect to the location on the line, and the SWR is lower than the SWR for a short and open circuited.


STANDING WAVE RATIO

The SWR can also be computed if the characteristic impedance of the transmission line and the actual impedance of the load are known.

L

oL

o

LL

ZZSWR;oZZ If

ZZSWR;oZZ If


STANDING WAVE & REFLECTION COEFFICIENT

Since the SWR is really the composite of the original incident wave added to the reflected, the SWR can also be defined in terms of those waves.The ratio of the reflected voltage wave, Vr to the incident voltage wave, Vi is called the reflection coefficient, G.

Matched line, G = 0, open-circuited line, G = 1, short-circuited line, G = -1.

i

rVV


STANDING WAVE & REFLECTION COEFFICIENT

1SWR1SWR

VVVV

VV

minmax

minmax

i

r

11SWR

oL

oLZZZZ

SWR concerns with magnitude only and thus it is a real number. It must be positive and greater than or equal to 1.For a matched line SWR =1The closer the line is to be matched, the lower its SWR.


POWER REFLECTION COEFFICIENT

The presence of standing waves causes the voltage at some points on the line to be higher than it would be with a matched line, while at other points the voltage is low but the current is higher than with a matched line.This situation results in increased losses.In a transmitting application, standing waves put additional stress on the line and can result in failure of the line or of equipment connected to it.Reflections cause power dissipated in the load to be less than it would be with a matched line for the same source, because some of the power is reflected back to the source.


POWER REFLECTION COEFFICIENT

Since power is proportional to the square of the voltage, the fraction of the power that is reflected is

i2

r PP

The amount of power absorbed by the load, PL is the difference between the incident power, Pi and the reflected power, Pr.

)2ii

2iriL (1PPPPPP


EXAMPLE

A generator sends 50 mW down a 50 W line. The

generator is matched to the line, but the load is not. If the

reflection coefficient is 0.5, how much power is reflected

and how much is dissipated in the load?

Answer : 12.5 mW, 35.7 mW


CHARACTERISTICS OF A SHORTED & OPENED LINES

Length of short-circuited line Impedance

Inductive

Open Circuit(Parallel Resonant)

Short Circuit(Series Resonant)

Capacitive

Cycle repeats

4l

4l

24 l

2l

2l

Length of open-circuited line Impedance

Inductive

Open Circuit(Parallel Resonant)

Short Circuit(Series Resonant)

Capacitive

Cycle repeats

4l

4l

24 l

2l

2l

To understand this tables clearly, please Refer to Smith Chart.


STUBS

Stubs is a short transmission lines section.Stubs can be substitute for capacitors, inductors, or tuned circuits in applications where lumped-constant components (conventional inductors and capacitors) would be inconvenient or impractical.For example, it is difficult to build a 1pF capacitor, therefore a short circuit stubs can be used as a substitution.A short circuited stubs is preferred than an open circuited stubs since open circuited stubs tend to radiate energy from the end of the line.Short circuited stubs can be adjusted by means of a moving the location of the short circuit on the original transmission line.


TRANSMISSION LINES LOSSESCOPPER LOSS

I2R loss or copper loss is due to the resistance of the conductors.Also called copper loss since copper is the most common material for conductors. I2R loss increases with frequency due to the skin effect, which causes the resistance of the conductors to rise as the frequency increases.Any current flow in a conductor is associated with a magnetic field, both within the conductor and in the space surrounding it.At high frequencies, the magnetic field within the conductor causes most of the current to flow near its surface.As the frequency increases, the region of high current density becomes thinner, reducing the effective cross-sectional area and increasing the resistance of the conductor. Because most of the current flows in a thin region resembling a ‘skin’ near the surface of the wire, this phenomenon is called the SKIN EFFECT.


TRANSMISSION LINES LOSSES

DIELECTRIC LOSS

The dielectric of a transmission line also has some loss.The conductance of the dielectric increases with frequency.Dry air and air-dielectric lines has low loss.Therefore open-wire line and air-dielectric coaxial cable have better performance than solid-dielectric coaxial cable.However, open-wire is exposed to weather, it does not always have lower loss that might be expected.


RADIATION LOSSRadiation of energy not only contributes to cable losses but also can be a source of interference and crosstalk.Open-wire lines can radiate energy. Radiation is not a problem for coaxial cables that are properly terminated and grounded. 3 dB loss in a feedline between a transmitter and its antenna means that only one-half the transmitter power actually reaches the antenna.For a lossy line, the SWR at the source is lower than that at the load.When using SWR measurements to aid in matching a load to a line, it is better to take the measurements at the load, rather than at the source, particularly if the line is long.

TRANSMISSION LINES LOSSES


POWER RATINGS

Both power dissipation and maximum voltage increase with the SWR, so a transmission line is capable handling more power when it is properly matched than when it is mismatched.Published power ratings for commercial cables generally specify a maximum SWR at which they are valid.The two most common impedances for coaxial cables in RF circuits are 50 and 75 W.50 W is more common in transmitting application, and has greater power handling capability. 75 W is used for television antennas and cable TV systems and it has lower power loss since the power levels its handling are low.


IMPEDANCE MATCHING

In most transmission-line applications, it is highly desirable that the load matched to the impedance of the line, in order to avoid the problems associated with reflections and high SWR.

Another requirement for transmission line operation is that balanced and unbalanced lines should be terminated by balanced and unbalanced loads, respectively.


BALUN TRANSFORMER

Balanced and unbalanced lines, loads, and sources can be connected together by using a transformer called a “balanced and unbalanced transformer” or BALUN.The balun can have a 1:1 turns ratio, or it can also be used to match impedances.For example, television-receiver balun; transformer balun is used to match a 75 W unbalanced coaxial cable to an antenna input designed for 300 W balanced twin-lead.


QUARTER-WAVE TRANSFORMER

A section of transmission line that is ¼l long can act as an impedance transformer.The idea is to have the input impedance of the transformer equal to the characteristic impedance of the main line.A match can be obtained only over a small range of frequencies, since the length of the matching section will no longer be a quarter-wavelength if the frequency changes.The impedance of the quarter-wave transformer section is given by the equation below where Zo is the line impedance, Zo

’ is the impedance of the transformer section, and ZL is the load impedance.

LZoZ'oZ


STUB MATCHING

Stub matching makes use of two properties of transmission lines; (i) the impedance varies with position along a mismatched line, and (ii) a shorted section of line, called a stub, can simulate a capacitance or an

inductance, depending on its electrical length.

ZLSource

(a) Single Stub

ZLSource

(b) Double Stub


Consist of 2 steps:

(i) Locate a position where the resistive component of the impedance looking into a line, toward the load, is equal to the characteristic impedance of the line. In general the impedance at the point is complex but the reactive component can be cancelled by adding the appropriate reactance to the line.

(ii) Then calculate the length of a stub with the correct susceptance. The calculations can be done using a Smith Chart, a calculator or a computer.

SINGLE STUB MATCHING


SMITH CHART

Developed by P. H. Smith in 1938. Smith Chart is a graphical transmission line calculator.The impedance of a lossless transmission line repeats every ½ l, that is the impedance ½ l from the load is the same as that of the load itself.Clockwise rotation represents movement toward the generator.Counterclockwise rotation represents progress toward the load.Smith Chart uses normalized impedance and admittance. The normalized load impedance is

o

LZZz


SMITH CHART


Using a Smith Chart, find the normalized admittance of a load impedance of 100 + j25 W on a 50 W line.

Find the impedance and length of a quarter-wave transformer to match a 75 W load to a 50 W line at a frequency of 300 MHz.

Match a load with impedance ZL = 120 - j100 W to a line with ZO = 72 W, using a shorted stub.

EXAMPLE


EXAMPLE

A 50 W line operating at 100 MHz has a velocity factor of 0.7. It is 6 m long, and is terminated with a load impedance of 50 + j50 W. Find:

(i) the load admittance YL

(ii) the SWR, and

(iii) the input impedance of the line Zi.


The measured input impedance of a 10 m length of 300 W line is 120 – j650 W. The velocity factor is 0.9, and the frequency is 50 MHz. Find:

(a) the load impedance,

(b) a suitable place to install a quarter-wave transformer, and

(c) the characteristic impedance that the transformer section

should have.

EXAMPLE

Chapter 5-Transmission Lines

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