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DATE: 24/07/06 28/07/06. ENE 429 Antenna and Transmission lines Theory. Lecture 4 Transmission lines. Transmission lines (1). Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies. Examples: Transmitter and antenna
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DATE: 24/07/06 28/07/06 ENE 429Antenna and Transmission lines Theory Lecture 4 Transmission lines
Transmission lines (1) • Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies. • Examples: • Transmitter and antenna • Connections between computers in a network • Interconnects between components of a stereo system • Connection between a cable service provider and aTV set. • Connection between devices on circuit board • Distances between devices are separated by much larger order of wavelength than those in the normal electrical circuits causing time delay.
Transmission lines (2) • Properties to address: • time delay • reflections • attenuation • distortion
Distributed-parameter model • Types of transmission lines
Distributed-parameter model • The differential segment of the transmission line R’ = resistance per unit length L’= inductance per unit length C’= capacitor per unit length G’= conductance per unit length
Telegraphist’s equations • General transmission lines equations:
Telegraphist’s time-harmonic wave equations • Time-harmonic waves on transmission lines After arranging we have where
Traveling wave equations for the transmission line • Instantaneous form • Phasor form
Lossless transmission line • lossless when R’ = 0 and G’ = 0 and
Low loss transmission line (1) • low loss when R’ << L’ and G’ << C’ Expanding in binomial series gives for x << 1
Low loss transmission line (2) Therefore, we get
Characteristic impedance • Characteristic impedance Z0is defined as the the ratio of the traveling voltage wave amplitude to the traveling current wave amplitude. or For lossless line,
Power transmission • Power transmitted over a specific distance is calculated. • The instantaneous power in the +z traveling wave at any point along the transmission line can be shown as • The time-averaged power can be shown as W.
Power ratios on the decibel scale (1) • A convenient way to measure power ratios • Power gain (dB) • Power loss (dB) • 1 Np = 8.686 dB dB dB
Power ratios on the decibel scale (2) • Representation of absolute power levels is the dBm scale dBm
Ex1 A 12-dB amplifier is in series with a 4-dB attenuator. What is the overall gain of the circuit?Ex2 If 1 W of power is inserted into a coaxial cable, and 1 W of power is measured 100m down the line, what is the line’s attenuation in dB/m?
Ex3 A 20 m length of the transmission line is known to produce a 2 dB drop in the power from end to end, • what fraction of the input power does it reach the output? • What fraction of the input power does it reach the midpoint of the line? • What is the attenuation constant?
Wave reflection at discontinuities • To satisfy boundary conditions between two dissimilar lines • If the line is lossy, Z0will be complex.
Reflection coefficient at the load (1) • The phasor voltage along the line can be shown as • The phasor voltage and current at the load is the sum of incident and reflected values evaluated at z = 0.
Reflection coefficient at the load (2) • Reflection coefficient • A reflected wave will experience a reduction in amplitude and a phase shift • Transmission coefficient
Total power transmission (matched condition) • The main objective in transmitting power to a load is to configure line/load combination such that there is no reflection, that means.
Voltage standing wave ratio • Incident and reflected waves create “Standing wave”. • Knowing standing waves or the voltage amplitude as a function of position helps determine load and input impedances Voltage standing wave ratio
Forms of voltage (1) • If a load is matched then no reflected wave occurs, the voltage will be the same at every point. • If the load is terminated in short or open circuit, the total voltage form becomes a standing wave. • If the reflected voltage is neither 0 nor 100 percent of the incident voltage then the total voltage will compose of both traveling and standing waves.
Forms of voltage (2) • let a load be position at z = 0 and the input wave amplitude is V0, where
Forms of voltage (3) we can show that traveling wave standing wave • The maximum amplitude occurs when • The minimum amplitude occurs when standing waves become null,
The locations where minimum and maximum voltage amplitudes occur (1) • The minimum voltage amplitude occurs when two phase terms have a phase difference of odd multiples of . • The maximum voltage amplitude occurs when two phase terms are the same or have a phase difference of even multiples of .
The locations where minimum and maximum voltage amplitudes occur (2) • If = 0, is real and positive and • zmin and zmaxare separated by multiples of one-half wavelength. The distance between zmin and zmax is a quarter wavelength. • We can show that
Ex4 Slotted line measurements yield a VSWR of 5, a 15 cm between successive voltage maximum, and the first maximum is at a distance of 7.5 cm in front of the load. Determine load impedance, assuming Z0 = 50 .