Digital Signal Processing Reference
In-Depth Information
Therefore, for a given transmission-line geometry, the right-hand side of (2-29)
can be written in terms of circuit parameters, where
L
is the series inductance
of the transmission line per unit length:
∂
B
y
·
∂µH
y
∂t
d
s
∂ψ
∂t
=
L
∂i
→
=
∂t
∂t
are HA/m
2
Don't get confused by the fact that the units of
∂µH
y
/∂t
(henry
·
amperes per square meter) and those of
L(∂i/∂t)
are H
A/m. Equation (2-29)
is in terms of generalized electric and magnetic fields. When the fields are repre-
sented by circuit parameters, the units change. Therefore, the circuit equivalent
of (2-29) is
·
∂v(z,t)
∂z
=−
L
∂i(z,t)
∂t
(3-12)
Note that equation (3-12) is simply the classic response of an inductor from
circuit theory.
Similarly, equation (2-30) says that a time-varying electric field will produce
a magnetic field:
a
x
ε
∂E
x
∂t
∂H
y
∂z
=−
(2-30)
Note that the form of the numerator on the left-hand side of equation (2-30)
(
εE
) has units of F
V/m
2
volts per square meter), which indicates that
the equivalent-circuit form should be in terms of a capacitance and a voltage.
From the integral form of Gauss's law (2-59) we can calculate the left-hand side
of (2-30) in terms of the charge, assuming that the signal conductor is an infinite
thin strip:
·
(farad
·
QA
A
Q
A
ε E
·
s
=
→
εE
x
A
=
→
εE
x
=
d
ρdV
(3-13a)
S
V
where
dV
refers to volume. For a given transmission-line geometry, the voltage
between the conductors from point
a
on the signal conductor to point
b
on the
reference plane is calculated with (3-1), which has units of volts:
b
Q
εA
E
x
·
dl
v
=
=
d
=
E
x
d
(3-13b)
a
where
A
is the area of the conductors where the electric field is established and
d
is the distance between the signal conductor and the reference plane. Therefore,
since
Q
=
Cv
,
A
=
C
b
l
1
Q
A
→
Cv
CE
x
d
A
C
l
v
→
Cv
E
x
·
εE
x
=
d
A
=
=
(3-14)
a
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