microwave lecture

1. TRANSMISSION LINES
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• There is a story about four men named Everybody, somebody,
Anybody, and Nobody. There was an important job to be done,
and Everybody was asked to do it.
• Everybody was sure that Somebody would do it. Anybody could
have done it, but Nobody did it.
• Somebody got angry about that, because it was Everybody's job.
Everybody thought that Anybody could do it, and Nobody
realized that Everybody wouldn't do it.
• It ended up that Everybody blamed Somebody, when actually
Nobody did what Anybody could have done.
—ANONYMOUS

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1.1 INTRODUCTION
• The previous Lecture was essentially on wave propagation in unbounded
media. Such wave propagation is said to be unguided
• Another means of transmitting power or information is by guided structures.
Guided structures serve to guide (or direct) the propagation of energy from
the source to the load.
• Typical examples of such structures are transmission lines and waveguides.
• Transmission lines are commonly used in power distribution (at low
frequencies) and in communications (at high frequencies)
• A transmission line basically consists of two or more parallel conductors
used to connect a source to a load. The source may be a hydroelectric
generator, a transmitter, or an oscillator; the load may be a factory, an
antenna, or an oscilloscope, respectively
• Transmission line problems are usually solved using EM field theory and
electric circuit theory, the two major theories on which electrical engineering
is based
• we use circuit theory because it is easier to deal with mathematically

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11.2 TRANSMISSION LINE PARAMETERS
• It is customary and convenient to describe a
transmission line in terms of its line parameters,
which are its :
• Resistance per unit length R,
• Inductance per unit length L,
• Conductance per unit length G,
• Capacitance per unit length C.

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• Let us consider how an EM wave propagates through a two-conductor
transmission line For example, consider the coaxial line connecting the
generator or source to the load as in Figure 11.4(a).
• When switch S is closed the inner conductor is made positive with respect to
the outer one so that the E field is radially outward as in Figure 11.4(b)
• According to Ampere's law, the H field encircles the current carrying
conductor as in Figure 11.4(b)
• The Poynting vector (E X H) points along the transmission line. Thus,
closing the switch simply establishes a disturbance, which appears as a
transverse electromagnetic (TEM) wave propagating along the line
• This wave is a nonuniform plane wave and by means of it power is
transmitted through the line

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11.3 TRANSMISSION LINE EQUATIONS
• A two-conductor transmission line supports a TEM wave
• An important property of TEM waves is that the fields E and
H are uniquely related to voltage V and current I,
respectively:
• In view of this, we will use circuit quantities V and I in
solving the transmission line problem instead of solving field
quantities E and H (i.e., solving Maxwell's equations and
boundary conditions).

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• Let us examine an incremental portion of length Δz of a two-conductor
transmission line
• We intend to find an equivalent circuit for this line and derive the line
equations
• From Figure 11.3, we expect the equivalent circuit of a portion of the line to
be as in Figure 11.5. The model in Figure 11.5 is in terms of the line
parameters R, L, G, and C, and may represent any of the two-conductor lines
of Figure 11.3
• The model is called the L-type equivalent circuit
• we assume that the wave propagates along the +z-direction, from the
generator to the load.

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By applying Kirchhoff's voltage law to the outer loop of the circuit in
Figure 11.5, we obtain

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Similarly, applying Kirchoff's current law to the main node of the circuit in
Figure 11.5 gives

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The solutions of the linear homogeneous differential equations (11.10) and
(11.12) are:

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• The characteristic impedance Zo of the line is the ratio of
positively traveling voltage wave to current wave at any point
on the line.
The transmission line considered thus far in this section is the
lossy.

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Lossless Line (R = 0 = G)
• A transmission line is said lo be lossless if the conductors of
the line are perfect (σc ≈∞ ) and the dielectric medium
separating them is lossless (σ ≈0 )

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Distortionless Line (R/L = G/C)
• A distortionless line is one in which the attenuation constant α
is frequency independent while the phase constant β is linearly
dependent on frequency

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• A lossless line is also a distortionless line, but a distortionless
line is not necessarily lossless. Although lossless lines are
desirable in power transmission, telephone lines are required
to be distortionless

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EXAMPLE 11.1
An air line has characteristic impedance of 70 Ω and phase constant of 3
rad/m at 100 MHz. Calculate the inductance per meter and the capacitance
per meter of the line?

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EXAMPLE 11.2
A distortionless line has Zo = 60 Ω, α = 20 mNp/m, u = 0.6c, where c is the
speed of light in a vacuum. Find R, L, G, C, and λ at 100 MHz.

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11.4 INPUT IMPEDANCE, SWR, AND POWER
• Consider a transmission line of length l, characterized by
γ and Zo, connected to a load ZL as shown in Figure 11.6.
• Looking into the line, the generator sees the line with the
load as an input impedance Zin.
• It is our intention to determine the
• input impedance
• the standing wave ratio (SWR),

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• Let the transmission line extend from z = 0 at the
generator to z = l at the load
• First of all, we need the voltage and current waves in
eqs. (11.15) and (11.16), that is

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• Now define ΓL as the voltage reflection coefficient (at the load)
• ΓL is the ratio of the voltage reflection wave to the incident wave
at the load, that is

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• Define the standing wave ratio, s ( SWR) as

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EXAMPLE 11.3

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11.8 MICROSTRIP TRANSMISSION LINES
• Belong to a group of lines known as parallel-plate transmission lines.
• Widely used in present-day electronics : transmission lines for microwave
integrated circuits, circuit components such as filters, couplers, resonators,
antennas, and so on.
• In comparison with the coaxial line, the microstrip line allows for greater
flexibility and compactness of design
• A microstrip line consists of a single ground plane and an open strip
conductor separated by dielectric substrate as shown in Figure 11.38.

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• Analytical derivation of the characteristic properties of the line is
cumbersome. We will consider only some basic, valid empirical formulas
necessary for calculating the phase velocity, impedance, and losses of the
line
• Due to the open structure of the microstrip line, the EM field is not
confined to the dielectric, but is partly in the surrounding air as in Figure
11.39. Provided the frequency is not too high, the microstrip line will
propagate a wave that, for all practical purposes, is a TEM wave

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Because of the fringing, the effective relative permittivity εeff is
less than the relative permittivity εr of the substrate. If w is the
line width and h is the substrate thickness, an a approximate
value of εeff is given by

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EXAMPLE 11.10