Considering that all high-speed differential interfaces are defined in specifications and standards these documents may have useful information for the design of various deviations, like connectors requirements or return loss of the packages. With increase of the signal bandwidth the PCB structures that could be neglected at lower data rates have become quite important. For example, the parameters of the packages should be considered (and very often the models of ICs do not include them). Or the PCB area under packages and connectors (breakout areas) are so densely populated with vias that traces routed in this area may have parameters quite different than in the free routing area. Another example of such deviations could be a via in a trace for transitions between different layers. When we are dealing with data rates above 3 Gbps the accurate analysis of these deviations is imperative. For the lower data rates the interface analysis may utilize a simplified approach replacing the actual line characteristics with lumped parameters like capacitance or inductance or if needed by impedance variation.
Of course the proliferation of signal integrity simulation tools allows an engineer to incorporate the lumped parameters into the model and verify the effect of these parameters. However, sometimes an engineer may want to understand the reason for the behavior of certain structure in order to determine the course of actions. Quite often one may want to understand what kind of parameter may cause certain signal behavior. For example, an engineer may want to figure out the effect of adding via capacitance or adding some inductance by clearing the reference plain around via and so on. Another situation may call for a quick analysis without a simulation tool or verification of simulation tool performance. In all these cases it may be beneficial to understand how to calculate an addition of small lumped parameters to an interface line.
Let us consider an example of addition of extra capacitance in the middle of the trace. Adding it at one of the line ends is too simple because the parameters of addition are just added to the lumped circuit parameters of driver or receiver.
To begin with we may check the reflection coefficient of this transmission line with added lumped parameter C and its effect upon the signal. Using the Laplace transform it is easy to show that:
r(s) = (1/sCIIZ0) – Z0)/(1/sCIIZ0) + Z0) = – s/(s + 2/CZ0)
Consider an incident signal ramp: Vi(t) = (V/tr)[t*U(t) – (t-tr)*U(t-tr), which may be expressed using the Laplace transform as: Vi(s) = V/trs2 – Ve-trs/trs2 = (V/trs2)*(1 – e-trs)
Where:
II – defines parallel circuits, for example: 1/sCIIZ0 means value of capacitor parallel to transmission line impedance
t – time domain argument
s – Laplace transform argument
C – lumped capacitance
Z0 – characteristic impedance of the line
V – amplitude of the incident wave
tr – transition time
U(t), U(t-tr) – unity functions
Vi(t) – incident signal
Vr(t) – reflected signal
Therefore, the reflection amplitude may be estimated as:
Vr(s) = r(s) * Vi(s) = –(V/tr)*[s/(s + 2/Z0C)]*(1/s2)*(1 – e-trs)] =
–(V/tr)/[s(s + 2/Z0C)]+(V/tr)*e-trs/[s(s + 2/Z0C)]
Using the inverse Laplace transform:
L-1{–(V/tr)/[s(s + 2/Z0C)]} = V*(Z0C/2tr)*[e–(2/Z0C)*t – 1]
L-1{(V/tr)*e-trs/[s(s + 2/Z0C)]} = –V*(Z0C/2tr)*[e–(2/Z0C)*(t-tr) – 1]
Therefore:
Vr(t) = V(Z0C/2tr)*[e–(2/Z0C)*tu(t) – e–(Z0/2C)*(t-tr)u(t-tr))]
The result has shown a small negative pulse, which those who are used to deal with Time Domain Reflectometry (TDR) measurements may recognize for the case of additional capacitance on the line. This pulse will be added to the edge of the incident signal and affect the rise time of the signal that, of course, will be seen as an additional deterministic jitter.
Similar analysis methodology may be used if additional inductance or combination of inductance-capacitance-resistance is added anywhere on the line. Of course, I am using the Laplace transforms just as a convenient way to simplify the differential equations (and transform tables are readily available). Other solution methods are equally usable.
As it was mentioned above this analysis is a simplified method that is sufficient to the applicable data rates and it may allow engineer a quick evaluation of the effect of any circuit addition to the transmission line. Of course, the topic of how to determine the parasitic lumped parameters added to the line is not covered in this paper.
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