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. 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:

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:

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:

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:

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