Digital Signal Processing Reference
In-Depth Information
becomes
a
z
1
r
∂(rB
φ
)
∂r
a
z
1
r
∇×
B
=
∂
r(µ
0
ir/
2
πa
2
)
∂r
µ
0
i
πa
2
a
z
=
for
r<a
which says that the magnetic field with a
φ
-component will be induced that
circulates around a wire when current
I
is flowing in the
z
-direction.
For
r>a
, the curl becomes
a
z
1
r
∂ (r(µ
0
i/
2
πr))
∂r
∇×
B
=
=
0
The curl of the magnetic field outside the wire is zero. This does not mean
that the magnetic field does not circulate around the wire outside the conductor
(it certainly does). The zero curl result is simply due to the fact that the area
outside the conductor does not contain any current density (
J
0), and therefore
Ampere's law states that the curl of the magnetic field must be zero.
=
2.3 WAVE PROPAGATION
When studying Maxwell's equations, it becomes apparent that Faraday's and
Ampere's laws (the two curl equations), which state, respectively, that a chang-
ing magnetic field will produce an electric field and a changing electric field
will produce a magnetic field, are responsible for the propagation of an electro-
magnetic wave. In this section we derive equations that regulate electromagnetic
wave propagation in a simple source-free medium. In the study of signal integrity,
the propagation of waves in packages, on printed circuit boards, though cables,
and between power and ground planes constitutes a very large portion of the
discipline. In fact, communication between components in a high-speed digital
design necessitates the intentional propagation of electromagnetic waves guided
by transmission lines and the prevention of energy propagation across unin-
tentional pathways (such as crosstalk) or in unwanted signal propagation modes.
Without a detailed study of wave propagation, the study of signal integrity would
become impossible.
2.3.1 Wave Equation
In subsequent chapters it will become necessary to analyze electromagnetic wave
propagation only in terms of magnetic or electric fields because they are related
directly to the voltage and current propagating on transmission lines, through
vias, or across planes. The wave equation forms the basis for calculating critical
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