IDEAL TRANSMISSION-LINE FUNDAMENTALS - Advanced Signal Integrity for High-Speed Digital Design - page 132

time t = τ d , the inductor will resemble on open circuit. When a voltage step is
applied initially, no current flows across the inductor. This produces a reflection
coefficient of 1, causing an inductive spike seen as a reflection at node A in
Figure 3-41b. The value of the inductor will determine how long the reflection
coefficient will remain 1. If the inductor is large enough, the signal will double in
magnitude. Eventually, the inductor will discharge its energy at a rate dependent
on the time constant τ of an LR circuit. For the circuit depicted in Figure 3-41a,
the wave shape of the rising edge at node B is calculated:

= v ss 1 exp

(t − τ d )(Z 0 + R t )

v inductor − − t>τ d (3-110a)

L

Note that the wave shape calculated by (3-110a) will also be valid for the falling
edge of the inductive spike, shown at node B in Figure 3-41b, if τ d is adjusted to
shift the waveform to the correct position in time (2 τ d ), the waveform is inverted,
and dc is shifted to the appropriate level:

= v ss 1 exp

(t − 2 τ d )(Z 0 + R t )

v inductor + − (3-110b)

L

Filtering Effects of Reactive Components Figures 3-39b and 3-41b show how
the series inductor and the shunt capacitor affect the signal integrity. The series
inductance will cause an inductive spike, which is seen as a positive reflection,
the capacitance will cause a capacitive dip, which is seen as a negative reflection,
and both will smooth the rising and falling edges seen at the receiver (node B).
To understand why the edges are smoothed, we must explore how an inductor
or a capacitor will filter the harmonics of a digital waveform. In Chapter 8 it
will be shown that high-frequency harmonics are associated with the rising and
falling edges of a digital waveform. Consequently, if the higher-frequency har-
monics are filtered out by the capacitive or inductive loads, the rising and falling
times will be increased. Equation (3-111) shows that the impedance of the shunt
capacitor will decrease with frequency, which means that the higher-order har-
monics of the digital waveform will be shunted to ground, increasing the rise and
fall times:

1
jωC

Z cap = (3-111)

where ω = 2 πf .

Similarly, (3-112) shows that the series impedance of an inductor will increase
with frequency, which will also tend to filter out the higher harmonics because
they will experience larger impedances than the lower-frequency harmonics.

Z ind = jωL (3-112)