Chapter Fourteen:Transmission Lines

Introduction

• Signals can be delivered from the transmitter to the receiver using a variety of means:– Metallic cable

– Optical fiber

– Radio transmission

Coaxial Lines• Two conductors are

concentric, separated by an insulating dielectric

• Coaxial cables are unbalanced because of their lack of symmetry with regard to ground

Parallel Lines• Parallel lines are

typically balanced lines, the impedance to ground from each of the wires being equal

• Balanced refers to the signals being the same level but opposite in polarity

Electrical Model of a Transmission Line

• The electrical characteristics of a transmission line become increasingly critical as the frequency of transmission increases

• Factors influencing transmission lines:– Resistance– “Skin effect”– Conductance of the dielectric– Impedance– Capacitance– Inductance

• These factors are distributed rather than lumped

Model Transmission Line

Step and Pulse Response of Lines• In a line of infinite length, a stepped input signal

will surge forever because of the capacitance of the line

• The characteristic impedance of the line is also know as the surge impedance

• The impedance is a real number for a line with no losses; for example, a 50-ohm line does not refer to the resistance of the wire in the line, but the voltage/current ratio as seen by the source

CjGLjRZ ω

ω0

Characteristic Impedance of a Line• A terminated transmission line that is matched in

its characteristic impedance is called a matched line

• The characteristic impedance depends upon the electrical properties of the line, according to the formula:

Characteristic Impedance• The characteristic impedance for any type of

transmission line can be calculated by calculating the inductance and impedance per unit length– For a parallel line with an air dielectric the impedance

is:

– For a coaxial cable:

d

DZ

r

log138

0

r

DZ log2760

Coaxial Cable Applications• In practice, it is usually unnecessary to find the impedance of coaxial

cable since the impedance is part of the cable specification

• As indicated in the table, there are standard impedances for coaxial cable

Impedance

(ohms)

Application Typical type numbers

50 Radio Transmitters

Communications Receivers

RG-8/U

RG-58/U

75 Cable Television

TV Antenna feedlines

RG-59/U

93 Computer networks RG-62/U

Velocity Factor• A signal moves down a transmission line at a finite rate, i.e.

somewhat less than the speed of light• The propagation velocity of a signal, compared to the speed

of light, varies as follows:– Coaxial cable with polyethylene dielectric: 66%– Coaxial cable with polyethylene foam dielectric: 78%– Air-dielectric cable: 95%

• Rather than specify the actual velocity, manufacturers specify the velocity factor

• The velocity factor for a transmission line depends almost entirely upon the dielectric

Reflections• In a line where the termination is equal to the

impedance of the line, the reflections are zero

• A line that is terminated other than Z0 is said to be mismatched and will have reflections

• The reflection coefficient is found by:

i

r

V

V

Wave Propagation on Lines• If a sine wave is applied to a transmission line, the signal

moves down the line and disappears into the load

• Such a signal is called a traveling wave• This process also takes time• A time delay of one period causes a phase shift of 360º,

which is indistinguishable from the original• The length of a line L that causes a delay of one period is

known as a wavelength

Traveling Waves

Standing Waves• The interaction of incident

and reflected waves in a transmission line results in standing waves

• When a reflected wave is present but has lower amplitude than the incident, there will be no point on the line where the voltage or current remains zero over the whole cycle

Variation of Impedance Along a Line• A matched line presents its impedance to a source

located any distance from the load• An unmatched line impedance can vary greatly

with its distance from the load• At some points mismatched lines may look

inductive, other points may look capacitive, at still other points it may look resistive

Impedance on a Lossless Line

• The impedance on a lossless transmission line is given by the formula:

θsinθcos

θsinθcos

0

00

L

L

jZZ

jZZZZ

Characteristics of Open and Shorted Lines

• An open or shorted line can be used as an inductive, capacitive, or even a resonant circuit

• In practice, short-circuited sections are more common because open-circuited lines radiate energy from the open end

• The impedance of a short-circuited line is:

θtan0jZZ

Variation of Impedance

Transmission Line Losses

• No real transmission line is completely lossless• However, approximation is often valid assuming

lossless lines

Loss Mechanisms• The most obvious loss in a transmission line is due to

the resistance of the line, called I2R loss • The dielectric can also cause loss, with the conductance

becoming higher with increasing frequency• Open-wire systems can radiate energy

– Loss becomes more significant as the frequency increases – Loss becomes worse as spacing between conductors increases

Loss in Decibels• Transmission line losses are usually given in

decibels per 100 feet or 100 meters• When selecting a transmission line, attention must

be paid to the losses• A 3-dB loss equates to 1/2 the power being

delivered to the antenna• Losses are also important in receivers where low

noise depends upon minimizing the losses before the first stage of amplification

Mismatched Lossy Lines• When a transmission line is lossy, the Standing-

Wave Ratio (SWR) at the source is lower than that at the load

• The reflection coefficient and standing-wave ratio both have larger magnitudes at the load

• Computer programs and Smith Charts are available to calculate losses and mismatches in transmission lines

Power Ratings• The maximum power that can be applied to a

transmission line is limited by one of two things:– Power dissipation in the line– A maximum voltage, which can break down the

dielectric when exceeded

• A compromise is often achieved in power lines between voltage and line impedance

Impedance Matching• Impedance mismatches are deleterious in transmission lines• Mismatches result in power being reflected back to the source and

in higher-than-normal voltages and currents that can stress the line• Best results are obtained when the load is matched to the

characteristic impedance of the transmission line• Impedance matching can be accomplished by matching networks

using:– Lumped constants (inductors, capacitors, transformers)– Waveguide components– Transmission line sections

The Smith Chart• The Smith Chart has been used since 1944 to

indicate complex impedances and admittances and the way in which they vary along a line

• Computer programs are now available that make use of the functions formerly relegated to the Smith Chart

Matching Using a Transformer

• A transformer can be used for impedance matching provided the load impedance is real at the point where the transformer is inserted

• Transformers are also used for connecting balanced and unbalanced lines. These transformers are called balun transformers

Series Capacitance and Inductance• When the resistive part of the load is correct, the

reactive part of the load impedance can be corrected by adding a series of reactances of the opposite type

• Stub Matching– Shorted transmission line stubs are often used instead

of capacitors or inductors at VHF and above– In these cases, admittance is calculated for, rather than

impedance

Transmission-Line Measurements

• Specialized test equipment is available to measure and evaluate transmission lines using these techniques:– Time-Domain Reflectometry

– The Slotted Line

– Standing-Wave-Ratio Meters and Directional Wattmeters