Digital Signal Processing Reference
In-Depth Information
high frequencies (at least 50 GHz), as discussed in Chapters 5, 6, 8, and 10.
Consequently, the methodologies in this topic for modeling transmission lines
are based on a quasistatic approach, and corrections are applied in later chapters
to recover the frequency-dependent behavior.
3.4.1 Laplace and Poisson Equations
In Section 2.4 we show that the behavior of an electrostatic field can be described
by the differential equations
∇·
D
=
ρ
(2-3)
E
=−∇
(2-65)
the latter coming from the fact that Ampere's law is zero for an electrostatic field
(
∇×
E
=
0). If (2-65) is substituted into (2-3),
ρ
ε
∇·
(
−∇
)
=
we arrive at
Poisson's equation
:
ρ
ε
2
∇
=−
(3-38)
In a medium that lacks any charge density, Poisson's equation is reduced to
Laplace's equation
,
2
∇
=
0
(3-39)
Poisson's and Laplace's equations are particularly useful for solving quasistatic
transmission-line problems, as demonstrated in the remainder of this chapter.
3.4.2 Transmission-Line Parameters for a Coaxial Line
Assume that a pair of long, coaxial, circular conductors are statically charged
with the inner conductor at the potential
=
V
relative to the outer conductor,
which is held at zero potential (ground), where the cross section is as shown in
Figure 3-17. The region between the conductors has a dielectric permittivity of
ε
=
ε
0
ε
r
and a relative magnetic permeability of
µ
r
=
1.
The first step is to calculate the capacitance per unit length. If we make the
assumption that the dielectric is charge free (very reasonable; see Chapter 6), the
capacitance can be derived by solving Laplace's equation (3-39) in cylindrical
coordinates. From Appendix A we get
∇
=
a
r
∂
1
r
∂
∂φ
+
a
z
∂
∂r
+
a
φ
∂z
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