2. TRANSMISSION LINES
Transcript of 2. TRANSMISSION LINES
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2. TRANSMISSION LINES
7e Applied EM by Ulaby and Ravaioli
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Chapter 2 Overview
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Transmission Lines
A transmission line connects a generator to a load
Transmission lines include:• Two parallel wires
• Coaxial cable
• Microstrip line
• Optical fiber
• Waveguide
• etc.
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Transmission Line Effects
Is the pair of wires connecting the voltage
source to the RC load a transmission line?
Yes.
The wires were ignored in circuits courses.
Can we always ignore them? Not always.
Delayed by l/c
At t = 0, and for f = 1 kHz , if:
(1) l = 5 cm:
(2) But if l = 20 km:
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Dispersion
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Types of Transmission Modes
TEM (Transverse
Electromagnetic):
Electric and
magnetic fields
are orthogonal to
one another, and
both are
orthogonal to
direction of
propagation
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Example of TEM Mode
Electric Field E is radial
Magnetic Field H is azimuthal
Propagation is into the page
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Transmission Line Model
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Expressions
will be
derived in
later
chapters
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Transmission-Line Equations
ac signals: use phasors
Telegrapher’s
equations
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Derivation of Wave Equations
Combining the two equations leads to:
Second-order differential equation
complex
propagation constant
attenuation
constant
Phase constant
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Solution of Wave Equations (cont.)
Proposed form of solution:
Using:
It follows
that:
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Solution of Wave Equations (cont.)
In general:
wave along +z because coefficients of t and z
have opposite signs
wave along –z because coefficients of t and z have
the same sign
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Example 2-1: Air Line
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Lossless Microstrip Line
Phase velocity in dielectric:
Phase velocity for microstrip:
Quasi-TEM
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Microstrip (cont.)
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Microstrip (cont.)
Inverse process:
Given Z0, find s
The solution formulas are based on
two numerical fits, defined in terms
of the value of Z0 relative to that
of the effective permittivity.
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Lossless Transmission Line
If
Then:
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Voltage Reflection Coefficient
At the load (z = 0):
Reflection
coefficient
Normalized load
impedance
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Voltage Reflection Coefficient
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Current Reflection Coefficient
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Standing Waves
voltage magnitude
current magnitude
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Standing-Wave Pattern
Voltage magnitude is maximum
When voltage is a maximum, current
is a minimum, and vice versa
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Standing Wave Patterns for 3 Types of Loads
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Maxima & Minima
Standing-Wave Pattern
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Maxima & Minima (cont.)
S = Voltage Standing Wave Ratio
For a matched load: S = 1
For a short, open, or purely reactive load:
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Example 2-6: Measuring ZL with a Slotted Line
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Wave Impedance
At a distance d from the load:
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Input Impedance
At input, d = l:
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Cont.
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(cont.)
Cont.
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(cont.)
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Short-Circuited Line
For the short-circuited line:
At its input, the line appears like
an inductor or a capacitor
depending on the sign of
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Open-Circuited Line
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Short-Circuit/Open-Circuit Method
For a line of known length l, measurements of its
input impedance, one when terminated in a short
and another when terminated in an open, can be
used to find its characteristic impedance Z0 and
electrical length
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Instantaneous Power Flow
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Average Power
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Tech Brief 3: Standing Waves in Microwave Oven
The stirrer or rotation
of the food platform
is used to constantly
change the standing
wave pattern in the
oven cavity
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Tech Brief 3: Role of Frequency
At low frequencies,
absorption rate is small,
so it would take a long
time for the food to cook
At very high frequencies,
the food cooks fast, but
only its the surface layer
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The Smith Chart
Developed in 1939 by P. W.
Smith as a graphical tool to
analyze and design
transmission-line circuits
Today, it is used to
characterize the
performance of microwave
circuits
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Complex Plane
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Smith Chart Parametric Equations
Equation for a circle
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Smith Chart Parametric Equations
rL circles
xL circles
rL circles are contained inside the unit circle
Only parts of the xL circles are contained
within the unit circle
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Complete Smith Chart
rL Circles
Positive xL Circles
Negative xL Circles
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Reflection coefficient at the load
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Input Impedance
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Maxima and Minima
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Impedance to Admittance Transformation
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(a)
(b)
(c)
(d)
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Given:
S = 3
Z0 = 50 Ωfirst voltage min @ 5 cm from load
Distance between adjacent minima = 20 cm
Determine: ZL
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Matching Networks
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Examples of Matching Networks
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Lumped-Element MatchingChoose d and Ys to achieve a match at MM’
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Cont.
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Single-Stub Matching
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Transients
Rectangular pulse is equivalent to the sum of two
step functions
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Transient Response
Initial current and voltage
Reflection at the load
Second transient
Load reflection coefficient
Generator reflection coefficient
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T = l/up is the time it takes the wave to
travel the full length of the line
Voltage Wave
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Current WaveReflection coefficient for current is the
negative of that for voltage
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Steady State Response
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Bounce Diagrams
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Technology Brief 4: EM Cancer Zapper
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Technology Brief 4: High Voltage Pulses
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Summary