## Risetime of Lossy Transmission Line

Suppose you cascade two lowpass filters having rise times ta and tb. You can estimate the rise time of the combination (ta+b) according to the root-sum-of-squares (RSS) formula: ta+b=(ta2 + tb2)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 N2.

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:

tr = ( 0.252 + 0.252 )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. 