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FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
FUNDAMENTALS OF
ELECTROMAGNETICS
WITH MATLAB®
Karl E. Lonngren | Sava V. Savov | Randy J. Jost
Chapter 7: Transmission Lines
Flexibility... Affordability... and MATLAB!
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-1.
1-1. Illustration
Three common
of thetransmission
scalar product
lines.
of two
a) Coaxial
vectorscable.
A 5 2ub)
0uy 1 0uz line.
and
x 1Microstrip
B
1ux 1 2uline
product
of lead).
the two vectors is equal to 2.
y 1 0u
z. The scalar
c) 5Two-wire
(sometimes
called
a twin
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-2. a) Distributed transmission line. b) Equivalent circuit of this transmission
line. The circuit elements are given in their per-unit-length values. In each section,
ˆ and C 5 CDz,
ˆ respectively. Hence transmission line models
the values are L 5 LDz
can be easily constructed in the laboratory.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-3. The lossless transmission line model comprises a number of distinct
sections. The length of each section is Dz, and each section contains an inductance
ˆ 5 L/Dz and Cˆ 5 C/Dz.
and a capacitance. The values are L
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-4. A semi-infinite transmission line that is terminated in a load
impedance ZL.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-5. Transmission line that has a characteristic impedance Zc. The line is excited at
z 5 2` with a sinusoidal voltage and the voltage is depicted at various times.a) Standing
voltage wave if ZL 5 0. b) Standing voltage wave if ZL 5 `. The maximum amplitude of the
voltage standing wave is 2A1.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-6. A transmission line with a characteristic impedance Zc ? ZL is joined to
the load with a quarter-wave transformer.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-7. Input impedance, which in a lossless line is a reactance, of a) a shortcircuited transmission line and b) an open-circuited transmission line. The vertical
lines are repeated for equal intervals of l/4.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-8. a) The input admittance of a transmission line at a distance d1 from the load is equal to
Yin = Yc + jB. b) The addition of a susceptance whose value is equal to 2jB at a distance d1 from the
load admittance causes the input admittance at that point to be equal to Yc. c) The addition of a shortcircuited transmission line whose length is d2 at the location d1 will match the parallel combination of
the transformed load admittance and the matching transmission line.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-9. A Smith chart created with MATLAB.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-10. The transformation of an impedance to an admittance using the
Smith chart. A semicircle whose radius is equal to the magnitude of the reflection
coefficient is drawn. This corresponds to a distance of l/4 on the transmission line.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-11. The battery is connected to the transmission line along with a switch
that is closed at t = 0.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-12. The bounce diagram. The magnitude of the slope of each line is equal to 1. The amplitude of
each individual component is usually specified. A vertical line at a certain position on the transmission line,
which in the figure is at the midpoint of the transmission line, indicates the location of an oscilloscope probe.
The intersection of this vertical line and the trajectory marks the times when the voltage will change. These
points are indicated with the short horizontal lines, and the subsequent voltage during that interval is given.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-13. A pulse generator Vg that has an internal impedance of Zg is
connected to a transmission line that has a characteristic impedance Zc and is
terminated in a load impedance ZL.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-14. An integrated circuit transmission line element.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-15. Snapshots of a voltage pulse crossing a discontinuity in a transmission line. The
pictures are taken at equal intervals in time, and the velocity of propagation has vz>0 > vz<0. In
addition, the characteristic impedances of the two lines have the relative values Zc(z>0) > Zc(z<0).
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-16. Model of a section whose length is Dz of a transmission line that
includes loss terms. The units of all of the elements are per unit length.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-17. The evolution in space of a time-harmonic voltage signal at an instant
in time as a function of space. From this figure, one can determine the complex
propagation constant a 5 b 1 jb 5 21/2 2 j 2p/2 5 21/2 2 jp
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-18. A model of a section of a transmission line, whose length is Dz, that
includes dispersion.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-19. The normalized propagation characteristics of a dispersive transmission
line. a) The solid line is the dispersion relation (7.87). The dotted line is a nondispersive
propagation constant. b) The phase velocity is a function of frequency.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-20. Dispersion curves.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-21. The linear summation of two cosine waves with slightly different
frequencies of oscillation is depicted in the figure. There is constructive and destructive
interference between these two signals that is indicated in the bottom figure.
FUNDAMENTALS OF ELECTROMAGNETICS WITH MATLAB® SECOND EDITION
Figure 7-22. The propagation of a signal in a dispersive medium. The signals are detected at
two locations. A point of constant phase and the peak of the envelope are followed. The point of
constant phase propagates with the phase velocity, and the modulation envelope propagates
with the group velocity. In this figure, the phase velocity is greater than the group velocity.