6.976
High Speed Communication Circuits and Systems
Lecture 2
Transmission Lines
Michael Perrott
Massachusetts Institute of Technology
Copyright © 2003 by Michael H. Perrott
Maxwell’s Equations
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General form:
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Assumptions for free space and transmission line propagation
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Note: we’ll only need Equations 1 and 2
- No charge buildup ⇒ ρ = 0
- No free current ⇒ J = 0
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MIT OCW
Assumptions
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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:
E
x
x
y
ƒ
z
Hy
direction
of travel
For transmission line (TEM mode)
b
Hy
x
y
direction
of travel
Ex
a
z
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MIT OCW
Solution
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Fields change only in time and in z-direction
- Assume complex exponential solution
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Solution
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Fields change only in time and in z-direction
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Implications:
- Assume complex exponential solution
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MIT OCW
Solution
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Fields change only in time and in z-direction
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Implications:
- Assume complex exponential solution
But, what is the value of k ?
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Evaluate Curl Operations in Maxwell’s Formula
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Definition
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Evaluate Curl Operations in Maxwell’s Formula
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Definition
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Given the previous assumptions
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Now Put All the Pieces Together
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Solve Maxwell’s Equation (1)
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Now Put All the Pieces Together
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Solve Maxwell’s Equations (1) and (2)
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Now Put All the Pieces Together
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Solve Maxwell’s Equations (1) and (2)
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Connecting to the Real World
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Current solution is complex
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But the following complex solution is also valid
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And adding them together is also a valid solution that
is now real-valued
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MIT OCW
Calculating Propagation Speed
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The resulting cosine wave is a function of time AND
position
direction
Ex(z,t)
of travel
t
x
y
z
z
ƒ
Consider “riding” one part of the wave
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Velocity calculation
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MIT OCW
Freespace Values
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Constants
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Impedance
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Propagation speed
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Wavelength of 30 GHz signal
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Voltage and Current
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Definitions:
x
b
Ex
Hy
y
z
I
x
H
E
t
w
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a
a
y
b
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Parallel Plate Waveguide
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E-field and H-field are influenced by plates
b
x
y
Hy
Ex
a
z
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MIT OCW
Current and H-Field
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Assume that (AC) current is flowing
I
b
x
Ex
Hy
a
y
z
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I
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Current and H-Field
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Current flowing down waveguide influences H-field
I
b
Ex
Hy
x
a
y
z
I
H
x
y
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MIT OCW
Current and H-Field
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Flux from one plate interacts with flux from the other
plate
I
b
Ex
Hy
x
a
y
z
I
x
y
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MIT OCW
Current and H-Field
ƒ
Approximate H-Field to be uniform and restricted to lie
between the plates
I
b
Ex
Hy
x
a
y
z
I
b
a
x
y
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MIT OCW
Voltage and E-Field
ƒ
Approximate E-field to be uniform and restricted to lie
between the plates
J
b
x
Ex
Hy
a
y
z
J
b
x
V
E
a
y
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MIT OCW
Back to Maxwell’s Equations
ƒ
From previous analysis
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These can be equivalently written as
ƒ
Where
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Wave Equation for Transmission Line (TEM)
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Key formulas
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Substitute (2) into (1)
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Characteristic impedance (use Equation (1))
M.H. Perrott
MIT OCW
Connecting to the Real World
ƒ
Current solution is complex
ƒ
But the following solution is also valid
ƒ
And adding them together is also a valid solution
M.H. Perrott
MIT OCW
Calculating Propagation Speed
ƒ
The resulting cosine wave is a function of time AND
position
direction
Ex(z,t)
of travel
t
x
y
z
z
ƒ
Consider “riding” one part of the wave
ƒ
Velocity calculation
M.H. Perrott
MIT OCW
Integrated Circuit Values
ƒ
Constants
ƒ
Impedance (geometry dependant)
ƒ
Propagation speed (geometry independent)
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Wavelength of 30 GHz signal in silicon dioxide
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MIT OCW
LC Network Analogy of Transmission Line (TEM)
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LC network analogy
L
Zin
ƒ
L
L
C
C
L
C
Calculate input impedance
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MIT OCW
How are Lumped LC and Transmission Lines Different?
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In transmission line, L and C values are infinitely
small
- It is always true that
L
Zin
ƒ
L
L
C
C
L
C
For lumped LC, L and C have finite values
- Finite frequency range for
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MIT OCW
Lossy Transmission Lines
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Practical transmission lines have losses in their
conductor and dielectric material
- We model such loss by including resistors in the LC
model
R
Zin
L
1/G
ƒ
ƒ
R
C
L
1/G
R
C
L
1/G
R
L
C
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 Lee’s book for analysis
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MIT OCW