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C H A P T E R 2
Transmission Line Analysis
As we already know, higher frequencies imply
decreasingwavelengths. The consequence for an RF circuit is that
voltages and currents no longer
remain spatially uniform when compared to the geometric size of
the discrete circuit
elements: They have to be treated as propagating waves. Since
Kirchhoffs voltage and
current laws do not account for these spatial variations, we
must significantly modify the
conventional lumped circuit analysis.
The purpose of this chapter is to outline the physical reason
for transitioning from
lumped to distributed circuit representation and, in the
process, develop one of the most
useful equations: the spatially dependent impedance
representation of a generic RFtransmission line configuration. The
application of this equation to the analysis and design of
high-frequency circuits is going to assume central importance in
subsequent chapters.
Developing the background of transmission line theory in this
chapter, we have purposely
attempted to minimize (albeit not eliminate) the reliance on
electromagnetics. The motivated
reader who would like to delve deeper into the concepts of
electromagnetic wave theory is
referred to a host of excellent books listed at the end of this
chapter.
2.1 Why Transmission Line Theory?
Let us once again return to the wave field representation
(1.1a): .
Here we have an x-directed electric field propagating in the
positive z-direction. For
propagation in free space the orthogonality between electric
field and direction of
propagation is always assured. If, on the other hand, we assume
that the wave is confined to
a conducting medium that is aligned with the z-axis, we will
find that the electric field has a
longitudinal component Ez that, when integrated in z- direction,
gives us a voltage
drop is the line element in the
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38 Chapter 2 Transmission Line Analysis
z-direction). Let us now consider more closely the argument of
the cosine term in (1.1a). It
couples space and time in such a manner that the sinusoidal
space behavior is characterized
by the wavelengthXalong the z-axis. Moreover, the sinusoidal
temporal behavior can be
quantified by the time period T = l/falong the time-axis. In
mathematical terms this leadsto the method of characteristics,
where the differential change in space over time denotes
the speed of evolution, in our case the constant phase velocity
in the form vp.
For a frequency of, let us say, / = 1 MHz and medium parameters
of er= 10 and |j,r= 1
(vp = 9.49xl07 m/s), a wavelength ofK = 94.86 m is obtained.
This situation is spatially
and temporally depicted in Figure 2-1 for the voltage wave
We next direct our attention to a simple electric circuit
consisting of load resistorRL
and sinusoidal voltage source VG with internal resistance RG
connected to the load by
means of 1.5 cm long copper wires. We further assume that those
wires are aligned along
the z-axis and their resistance is negligible. If the generator
is set to a frequency of 1 MHz,
then, as computed before, the wavelength will be 94.86 m. A 1.5
cm long wire connecting
source with load will experience spatial voltage variations on
such a minute scale that they
are insignificant.
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Why Transmission Line Theory? 39
When the frequency is increased to 10 GHz, the situation becomes
dramatically
different. In this case the wavelength reduces to X = vp/1010 m
= 0.949 cm and thus is
approximately two-thirds the length of the wire. Consequently,
if voltage measurements
are now conducted along the 1.5 cm wire, location becomes very
important in determining
the phase reference of the signal. This fact would readily be
observed if an oscilloscopewere to measure the voltage at the
beginning (location A), at the end (location B), or
somewhere along the wire, where distance A-B is 1.5 cm measured
along the z-axis in
Figure 2-2.
We are now faced with a dilemma. A simple circuit, seen in
Figure 2-2, with a
voltage source VG
and source resistanceRG
connected to a load resistorRL
through a two-
wire line of length /, whose resistance is assumed negligible,
can only be analyzed with
Kirchhoffs voltage law
when the line connecting source with load does not possess a
spatial voltage variation, as
is the case in low-frequency circuits. In (2.2) V i (i= I , . .
. , N ) represents the voltage
drops overN discrete components. When the frequency attains such
high values that the
Figure 2-2 Amplitude measurements of 10 GHz voltage signal atthe
beginning (location A ) and somewhere in between a wire
connecting load to source.
