6.776
High Speed Communication Circuits
Lecture 3
Wave Guides and Transmission Lines
Massachusetts Institute of
Technology
February 8, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H. Perrott
Maxwell’s Equations in Free Space
Take Curl of (1):
∂
⎛ ∂H ⎞
(∇ × H )
∇ × ∇ × E = −∇ × ⎜ µ
=
−
µ
⎟
∂t
⎝ ∂t ⎠
(5)
From (2)
∂2E
∂
µ (∇ × H ) = µε
∂t
∂t 2
(6)
Vector identity + (3)
∇ × ∇ × E = ∇(∇ ⋅ E ) − ∇ 2 E = −∇ 2 E
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(7)
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Simplified Maxwell’s Equations
Putting together (5), (6) and (7):
∂2E
∇ E + µε
2
2
∂t
Similarly for H
∇ H + µε
2
=0
(8)
=0
(9)
∂2H
∂t
2
For simplicity, assume only z-direction
∇2E =
∂2E
∂z
2
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and ∇ 2 H =
∂2H
∂z
2
(10)
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Solutions to Maxwell’s Equations
(10) reduces to
∂2E
∂z
2
+ µε
∂2E
∂t
2
=0
(11)
=0
(12)
Similarly for H
∂2H
∂z
2
+ µε
∂2H
∂t
2
(11) and (12) can be satisfied by any function in the form
f ( z ± vt )
where
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v=
1
µε
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Calculating Propagation Speed
ƒ
The function f is a function of time AND position
ƒ
Velocity calculation
z ± vt = constant
∂z
= ±v
∂t
ƒ
The solution propagates in the z or –z direction with a
velocity of
v=
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1
µε
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Assume Sinusoidal Steady-State
E and H solutions are in the form
z
jω ( t ± )
v
Ae
= Ae j (ωt ± kz )
Where
k=
ω
v
= ω µε
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Assumptions
ƒ
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
Hy
y
direction
of travel
ƒ
z
For transmission line (TEM mode)
b
Hy
x
y
z
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direction
of travel
Ex
a
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Solutions
ƒ
Fields change only in time and in z-direction
ƒ
Implications:
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Evaluate Curl Operations in Maxwell’s Formula
ƒ
Definition
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Evaluate Curl Operations in Maxwell’s Formula
ƒ
Definition
ƒ
Given the previous assumptions
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Now Put All the Pieces Together
ƒ
Solve Maxwell’s Equation (1)
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Now Put All the Pieces Together
ƒ
Solve Maxwell’s Equations (1) and (2)
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Freespace Values
ƒ
Constants
ƒ
Impedance
ƒ
Propagation speed
ƒ
Wavelength of 30 GHz
signal
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Voltage and Current
ƒ
Definitions:
b
x
Ex
Hy
y
a
z
I
x
H
E
t
w
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a
y
b
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Parallel Plate Waveguide
ƒ
E-field and H-field are influenced by plates
b
x
y
Hy
Ex
a
z
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Current and H-Field
ƒ
Assume that (AC) current is flowing
I
b
x
Ex
Hy
a
y
z
I
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Current and H-Field
ƒ
Current flowing down waveguide influences H-field
I
b
Ex
Hy
x
a
y
z
I
H
x
y
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Current and H-Field
ƒ
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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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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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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Back to Maxwell’s Equations
ƒ
From previous analysis
ƒ
These can be equivalently written as
ƒ
Where
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Wave Equation for Transmission Line (TEM)
ƒ
Key formulas
ƒ
Substitute (2) into (1)
ƒ
Characteristic impedance (use Equation (1))
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Connecting to the Real World
ƒ
Typical of sinusoidal analysis usingphasors, the
solutions are complex
ƒ
Take the real part of the solution to find the real-world
solution:
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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
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Integrated Circuit Values
ƒ
Constants
ƒ
Impedance (geometry/material dependant)
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Integrated Circuit Values
ƒ
Constants
ƒ
Impedance (geometry/material dependant)
ƒ
Propagation speed (geometry independent, material
dependent)
ƒ
Wavelength of 30 GHz signal in silicon dioxide
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LC Network Analogy of Transmission Line (TEM)
ƒ
LC network analogy
L
Zin
ƒ
L
L
C
C
L
C
Calculate input impedance
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LC Network Analogy of Transmission Line (TEM)
ƒ
LC network analogy
L
Zin
ƒ
L
L
C
C
L
C
Calculate input impedance
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How are Lumped LC and Transmission Lines Different?
ƒ
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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Lossy Transmission Lines
ƒ
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 textbook for analysis
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