1 ENE 429 Antenna and Transmission lines Theory Lecture 4 Transmission lines

2 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.

3 Transmission lines (2) Properties to address:  time delay  reflections  attenuation  distortion

4 Distributed-parameter model Types of transmission lines

5 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

6 Telegraphist’s equations General transmission lines equations:

7 Telegraphist’s time-harmonic wave equations Time-harmonic waves on transmission lines After arranging we have where

8 Traveling wave equations for the transmission line Instantaneous form Phasor form

9 Lossless transmission line lossless when R’ = 0 and G’ = 0 and

10 Low loss transmission line (1) low loss when R’ <<  L’ and G’ <<  C’ Expanding in binomial series gives for x << 1

11 Low loss transmission line (2) Therefore, we get

12 Characteristic impedance or For lossless line, Characteristic impedance Z 0 is defined as the the ratio of the traveling voltage wave amplitude to the traveling current wave amplitude.

13 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 Power transmission W.

14 A convenient way to measure power ratios Power gain (dB) Power loss (dB) 1 Np = dB Power ratios on the decibel scale (1) dB

15 Representation of absolute power levels is the dB m scale Power ratios on the decibel scale (2) dB m

16 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?

17 Ex3 A 20 m length of the transmission line is known to produce a 2 dB drop in the power from end to end, a) what fraction of the input power does it reach the output? b) What fraction of the input power does it reach the midpoint of the line? c) What is the attenuation constant?

18 Wave reflection at discontinuities To satisfy boundary conditions between two dissimilar lines If the line is lossy, Z 0 will be complex.

19 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.

20 Reflection coefficient at the load (2) Reflection coefficient A reflected wave will experience a reduction in amplitude and a phase shift Transmission coefficient

21 Power transmission in terms of reflection coefficient W W W

22 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.

23 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

24 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.

25 Forms of voltage (2) let a load be position at z = 0 and the input wave amplitude is V 0, where

26 Forms of voltage (3) we can show that traveling wavestanding wave The maximum amplitude occurs when The minimum amplitude occurs when standing waves become null,

27 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 .

28 The locations where minimum and maximum voltage amplitudes occur (2) If  = 0,  is real and positive and Each z min are separated by multiples of one-half wavelength, the same applies to z max. The distance between z min and z max is a quarter wavelength. We can show that

29 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 Z 0 = 50 .