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6.976
High Speed Communication Circuits and Systems
Lecture 2
Transmission Lines
Michael Perrott
Massachusetts Institute of Technology
Copyright 2003 by Michael H. Perrott
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Maxwells Equations
General form:
Assumptions for free space and transmission line propagation- No charge buildup = 0- No free current J = 0
Note: well only need Equations 1 and 2
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Assumptions
Orientation and direction
- E field is in x-direction and traveling in z-direction- H field is in y-direction and traveling in z-direction-
In freespace:
For transmission line (TEM mode)
y
x
z
Ex
Hy
direction
of travel
x
z
ExHy
b
aydirection
of travel
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Solution
Fields change only in time and in z-direction
- Assume complex exponential solution
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Solution
Fields change only in time and in z-direction
- Assume complex exponential solution
Implications:
But, what is the value of k ?
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Evaluate Curl Operations in Maxwells Formula
Definition
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Evaluate Curl Operations in Maxwells Formula
Definition
Given the previous assumptions
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Now Put All the Pieces Together
Solve Maxwells Equation (1)
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Now Put All the Pieces Together
Solve Maxwells Equations (1) and (2)
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Connecting to the Real World
Current solution is complex
But the following complex solution is also valid
And adding them together is also a valid solution that
is now real-valued
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Calculating Propagation Speed
The resulting cosine wave is a function of time AND
position
Consider riding one part of the wave
Velocity calculation
yx
z
direction
of travel
z
tEx(z,t)
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Freespace Values
Constants
Impedance
Propagation speed
Wavelength of 30 GHz signal
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Voltage and Current
Definitions:
x
y
a
b
H
t
w
x
z
ExHy
b
ay
I
E
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Parallel Plate Waveguide
E-field and H-field are influenced by plates
x
z
ExHyb
ay
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Current and H-Field
Assume that (AC) current is flowing
x
z
ExHyb
ay
I
I
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Current and H-Field
Current flowing down waveguide influences H-field
x
z
ExHyb
ay
x
y
I
I
H
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Current and H-Field
Flux from one plate interacts with flux from the other
plate
x
z
ExHyb
ay
x
y
I
I
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Current and H-Field
Approximate H-Field to be uniform and restricted to lie
between the plates
x
z
ExHyb
ay
x
y
a
b
I
I
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Voltage and E-Field
Approximate E-field to be uniform and restricted to lie
between the plates
x
z
ExHyb
ay
a
b
J
J
x
y
EV
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Back to Maxwells Equations
From previous analysis
These can be equivalently written as
Where
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Wave Equation for Transmission Line (TEM)
Key formulas
Substitute (2) into (1)
Characteristic impedance (use Equation (1))
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Connecting to the Real World
Current solution is complex
But the following solution is also valid
And adding them together is also a valid solution
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Calculating Propagation Speed
The resulting cosine wave is a function of time AND
position
Consider riding one part of the wave
Velocity calculation
yx
z
direction
of travel
z
tEx(z,t)
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LC Network Analogy of Transmission Line (TEM)
LC network analogy
Calculate input impedance
L
C
L
C
L
C
L
Zin
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How are Lumped LC and Transmission Lines Different?
In transmission line, L and C values are infinitely
small
- It is always true that
For lumped LC, L and C have finite values
- Finite frequency range for
L
C
L
C
L
C
L
Zin
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Lossy Transmission Lines
Practical transmission lines have losses in their
conductor and dielectric material
- We model such loss by including resistors in the LCmodel
The presence of such losses has two effects on
signals traveling through the line
- Attenuation- Dispersion (i.e., bandwidth degradation)
See Chapter 5 of Thomas Lees book for analysis
Zin 1/G
L
C
R
1/G
L
C
R
1/G
L
C
R LR
