Digital Signal Processing Reference
In-Depth Information
APPENDIX E
CAUSAL RELATIONSHIP
BETWEEN SKIN EFFECT
RESISTANCE AND INTERNAL
INDUCTANCE FOR ROUGH
CONDUCTORS
To derive a causal relationship between the skin effect resistance calculated with
rough copper and the internal inductance, we start with the equation for the
surface impedance, which was derived in equation (5-29):
Z
s
(ω)
=
R
ac
(ω)
+
jωL
internal
(ω)
(5-29)
For a function to be causal, it must be zero for negative time, as described in
Section 8.2.1.
†
. To be causal, the function must obey the relationship.
h(t)
=
h
e
(t)
+
sgn
(t)h
e
(t)
(8-17)
From Section 8.2.1, equation (8-20) shows that the function
H
(
ω
) [which is the
Fourier transform of
h(t)
] can be written in terms of only the even part:
H
e
(ω)
H(ω)
=
H
e
(ω)
−
j
(8-20)
where
H
e
(ω)
is the Hilbert transform of the even part of the function.
†
For practical purposes, it can also be said that a system is not causal if a portion of energy is
traveling faster than the propagation velocity in the board allows
v
p
=
c/
√
ε
r
. This means that
time t
=
0isdefined by the length of the interconnect and the speed of light in the material.
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