Digital Signal Processing Reference
In-Depth Information
6.7.1 Why Does the Signal Rise Time Matter But Not the Pulse Width?
We have seen how the relationship between the length of line and the signal rise
time determines if the interconnect acts as a transmission line, but we have not ex-
plained why the pulse width is not a factor.
Assuming that the transmission line resistance
R
is very small, circuit theory
shows that the voltage drop across the inductor is determined by the amount of
time required by the current to change. This is illustrated in:
change in current
di
Vl
=×
L
=×
L
(6.6)
change in time
dt
For a given change in current (
di
), the voltage drop
Vl
is affected by the total
inductance
L
and the change in time (
dt
). As the voltage drop across the inductors
becomes smaller, the individual capacitors in Figure 6.4 increasingly act as if they
are connected in parallel. In this situation the line is no longer a distributed RLC
circuit, but is a single lumped RLC (or, if the voltage drop across the inductor and
resistor is very small, simply a lumped C circuit).
However, the capacitors become separated from one another by the inductor's
voltage drop when it is high enough to no longer be negligible. In this situation the
inductors and capacitors are distributed along the line, and the line exhibits distrib-
uted RLC (transmission line) behavior.
This means that, because a short piece of interconnect does not have much in-
ductance, the signal must have a very fast rise time before the voltage across the in-
ductors becomes noticeable and the line acts like a transmission line. On the other
hand, if the interconnect is long, the inductance will be high, so even a slow rise
time signal can cause a noticeable voltage drop across the total inductance, causing
transmission line behavior. This matches our intuition. A short trace will not act
like a transmission line unless the signal rise time is very fast, but a long trace will
act like one even to slow rise time signals.
6.7.2 Why Does Propagation Delay Time Depend on the Line Length, But
Impedance Does Not?
It may seem natural to assume that the impedance and the propagation delay time
should depend on the line's length, but, in fact, the impedance does not depend on
the length of line, while the propagation delay does. This comes about because (as
illustrated in the Problems) in (6.1) the impedance is the ratio of the inductance to
capacitance, while the propagation delay in (6.2) is the product of them. Because of
this, the length units cancel when calculating the impedance, but they do not in the
propagation delay calculations.
6.8 Main Points
A transmission line is comprised of two or more conductors: the signal and
its return.
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