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ECE 134
Introductory Field Theory
Fall 2007
Reading Assignment:
Cheng:
Chapter 6
Cartoon Guide: Chapters 1718
Homework #5
Due:
Monday 5 November 2007
1)
In this problem we will use a numerical PDE
solver to explore the reduction in resistance
due to a corner in a thinfilm resistor as shown
here.
This is a very thin sheet of conducting
material with contacts covering each end.
Five
squares of material are shown (
2
w
=
A
).
Find
the true resistance, which is less than 5
sh
r
for
reasons we discussed in class. From this,
deduce the contribution of a corner.
Include a
plot of the equipotentials with your solution.
Note that you will need to compute the total
current by integrating the current density
across the device width; one possible integration path is shown in the figure. This
problem can also be solved analytically using a technique called conformal mapping,
0
y
∂Φ
=
∂
1 V
Φ=
0
x
∂Φ
=
∂
x
y
(0,0)
(3,0)
(2,1)
(0,1)
(2,3)
0 V
Φ=
A possible
integration path
for finding the
current
ℓ
w
w
ℓ
(3,3)
2
2
ln cosh
1
2
sh
r
R
w
π
⎡
⎤
⎛
=
⎜
⎞
+
⎟
⎢
⎥
⎝⎠
⎣
⎦
A
(1.1)
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 Fall '08
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