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Wave Propagation on Lines

Traveling waves on a matched line

(a) t = 0 (b) t = T/4

(c) t = T/2 (d) t = 3T/4

Traveling Waves

Example: What length of standard RG- 8/U coaxial cable would be required to obtain a 45 degrees phase shift at 200 MHz?

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

When a reflected signal is present but has a lower amplitude than the incident wave, there will be standing waves of voltage and current, but there will be no point on the line where the voltage and current remains zero over the whole cycle. SWR = Vmax/Vmin

For matched line, SWR =1

• Relationship bet Reflection coefficient (Г) and SWR

l Г l = SWR - 1 SWR + 1

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

0 0

*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:θtan0*jZZ *

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

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