This document summarizes key concepts about transmission lines. It discusses different types of transmission lines including coaxial cables and parallel lines. It covers electrical models of transmission lines and how factors like resistance, inductance and capacitance are distributed along transmission lines. It also discusses characteristic impedance, reflections, standing waves, impedance matching, and measurement techniques for transmission lines.

1. Chapter Fourteen:
Transmission Lines

2. Introduction
• Signals can be delivered from the transmitter to
the receiver using a variety of means:
– Metallic cable
– Optical fiber
– Radio transmission

3. 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

4. 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

5. 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

6. Model Transmission Line

7. 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

8. C
j
G
L
j
R
Z ω
ω
0 


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:

9. 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
D
Z
r
log
138
0


r
D
Z log
276
0 

10. 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

11. 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

12. 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



13. 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

14. Traveling Waves

15. 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

16. 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

17. Impedance on a Lossless Line
• The impedance on a lossless transmission line is
given by the formula:
θ
sin
θ
cos
θ
sin
θ
cos
0
0
0
L
L
jZ
Z
jZ
Z
Z
Z




18. 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:
θ
tan
0
jZ
Z 

19. Variation of Impedance

20. Transmission Line Losses
• No real transmission line is completely lossless
• However, approximation is often valid assuming
lossless lines

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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