Digital Signal Processing Reference
In-Depth Information
a
transmit
driver
receive
amplifier
R
R
L
R
L
T
C
C
inpu
t
ou
tput
FR-4
FR-4
C
T
C
R
Z
ST
b
Δ
V
(
x
,
t
)
R
L
C
G
Δ
I
(
x
,
t
)
Δ
x
Fig. 3
Back-plane channel model: (
a
) a circuit model of the link, and (
b
) a lossy transmission line
element
∂
V
(
x
,
t
)
L
∂
I
(
x
,
t
)
∂
=
RI
(
x
,
t
)+
(3)
t
∂
x
(
,
)
∂
I
x
t
C
∂
V
(
x
,
t
)
∂
x
=
GV
(
x
,
t
)+
(4)
t
∂
where
V
(
x
,
t
)
and
I
(
x
,
t
)
are the voltage and current signals, respectively, at time
t
from an initial time
t
0, and distance
x
from the source, and
R
,
L
,
G
,and
C
,are
the series resistance, series inductance, shunt conductance and shunt capacitance
per unit length. The elements
R
and
G
result in loss in the transmitted signal
energy. In addition, the back-plane channel suffers from frequency-dependent loss
mechanisms at frequencies above few GHz owing to
skin effect
and
dielectric losses
.
Skin effect arises due to crowding of high-frequency current towards the surface
of the conductor leading to the resistance
R
becoming frequency-dependent, i.e.,
R
=
=
R
(
f
)
above a certain frequency. The skin depth
δ
is given by
1
δ
=
√
π
(5)
f
μσ
where
μ
and
σ
are the permeability and resistivity, respectively, of the conductor.
≥
δ
=
For
f
f
s
,where
f
s
is the frequency at which
r
w
(
r
w
is the radius of the FR-
(
)
≈
(
)(
.
(
/
δ
)+
.
4 trace), skin-effect kicks in and the resistance
R
f
R
0
0
5
r
w
0
26).
