MATHEMATICAL REQUIREMENTS FOR PHYSICAL CHANNELS - Advanced Signal Integrity for High-Speed Digital Design - page 340

It is easy to see that the integral above will be positive as long as the square of
the reflection coefficient is less than or equal to 1 ( 2 ≤ 1). This puts a limitation

on the termination resistance R t .

R t − Z 0
R t + Z 0

2

2

= ≤

1

The only condition where 2 > 1 is when R t is negative. Therefore, the system
is passive as long as R t ≥ 0.

8.2.3 Stability

A model of a passive component such as a transmission line, via, connector, or
package must remain stable in order to mimic real-world behavior successfully.
For the purposes of this topic, stability will be defined so that the output of
a system y ( t ) is stable for all bounded inputs x ( t ). Using this definition, the
stability of a linear time-invariant system is guaranteed only if all the elements
in the impulse response matrix [ h ( t )] satisfy [Triverio et al., 2007]
∞
−∞ | h ij (t) |

dt< ∞ (8-25)

Example 8-6 Determine the conditions of stability of a mass-spring system
similar to that used to derive the frequency dependence of the dielectric permit-
tivity in Section 6.3.2. Assume that the system is subjected to a sharp impulse
at time t = τ . The spring equation is given:
m d 2 x
dt 2 dt + kx

dx

+ b = δ(t − τ)

SOLUTION
Step 1: Solve the differential equation and compute the impulse response. For
simplicity sake, assume initially that m 2. Since the inverse

Laplace transform of the transfer function is the impulse response, it is easiest
to solve this problem by converting to the Laplace domain:

= 1, b = 3, and k =

s 2 Y(s) + 2 Y(s) = e − τs 3 sY(s) +

= e − τs . From equa-

where the Laplace transform of the input is L [ δ(t − τ) ]

tion (8-9b), Y ( s ) is solved, where X(s) = e − τs :

Y(ω) = H (ω)X(ω) (8-9b)

Y(s) = H(s)e − τs