## Risetime of Lossy Transmission Line

Suppose you cascade two lowpass filters having rise times *t*_{a} and *t*_{b}. You can estimate the rise time of
the combination (*t*_{a+b}) according to the root-sum-of-squares
(RSS) formula: *t*_{a+b}=(*t*_{a}^{2} + *t*_{b}^{2})^{½}.
The RSS formula derives from the *central-limit* theorem, which applies
well to the sort of lowpass-filtering effects found in I/O drivers; chip
packages; scope probes; BNCs; and other simple, one-pole linear filters.

Are there cases in which the RSS formula does not apply? Yes. One prominent exception case is the lossy transmission line.

Technically, the central-limit theorem says that for large values of *N*, a cascade of *N* lowpass filters tends to produce a Gaussian
frequency response with a step-response rise time that grows in proportion to the square root of *N*.
For this result to hold true, the theorem requires that the impulse response of
each individual filter have a finite mean, a finite variance, and a monotonic
step response. In the frequency domain, these conditions imply that near dc, the
transfer response of the filter must be flat with a finite curvature. Curiously,
both the skin effect and dielectric-loss mechanisms violate the flatness
conditions; therefore, the central-limit theorem does not apply to lossy
transmission lines.

As a result, when you cascade a series of transmission-line segments, you see
that the step response does not tend toward a Gaussian shape. If purely
skin-effect losses dominate the performance of each segment, the whole
combination retains the same skin-effect appearance (a quick rise followed by a
long, sloping tail) regardless of the number of sections combined. The rise
time, instead of growing in proportion to the square root of *N*,
grows at the much faster rate of *N*^{2}.

Similarly,
a cascade of transmission-line segments dominated by purely dielectric losses
produces a rise time that grows in direct proportion to *N*, always
retaining the characteristic dielectric-loss appearance.

Lossy cables wreak havoc with the RSS formula, as the following paradox shows. Suppose you have in your hand a skin-effect limited cable with a rise time of 1 nsec. Now cut it in half. According to the rule for transmission lines dominated by the skin effect (whose rise time scales with length squared), the rise time of each half-length piece should be 0.25 nsec. Now join the two half-length segments again. If you used the RSS rule to approximate the rise time of the combination, you would compute:

*t*_{r} = ( 0.25^{2} + 0.25^{2} )^{½} = 0.353 ns

This answer is obviously wrong. You should conclude that RSS doesn't apply to cables.

RSS works reasonably well when combining a cable rise time with other, central-limit-compliant rise times, such as a driver-output rise time and a receiver noise filter. The result isn't perfect, but at least with only one portion of the system in violation of the central-limit theorem, the computed answer appears relatively paradox-free.

For your amusement, Figure 1 reproduces the normalized frequency-response curves associated with both skin effect and dielectric-loss mechanisms. As you can see, the skin-effect response near dc is not flat but becomes infinitely steep at dc, which is the artifact responsible for its peculiar behavior. The dielectric effect accomplishes a similar trick, albeit at a less dramatic pace.