In high-speed signaling with embedded clock a few ps in-pair skew may cause serious signal degradations. Point-to-point topologies and long PCB traces put the emphases on local speed variations due to bends, glass-weave, deterministic and random asymmetries. In this paper, we analyze the contributors to the delay in single-ended traces. Length in differential pairs varies due to bends and turns, so, at each turn, the outer trace has a little extra length. We show that using the center-line trace length can give a delay estimation error up to several ps. We show how different turns (right angle, double 45-degree or arced turn) impact delay. Next, we look at practical ways of compensating skew and compare their performances. We consider a few statistical contributors to skew and establish a limit below which compensation makes no sense. The simulated data is illustrated by the measured performance of a few simple structures.

**Introduction and Background**

Skew is the deviation of propagation delay from required reference timing. It is important in traditional parallel synchronous busses between the data lines of the parallel bus and a separate timing signal. In such cases, skew was usually between the transition of the timing signal and the transition of loaded single-ended signals. The main source of skew was the length difference and electrical loading of the devices along the lines and only to a lesser extent the variations of the propagation characteristics of the printed-circuit lines. Beyond a few hundred Mbps, skew has to be reduced by using a forwarded or embedded clock. Because of the embedded clock, in high-speed signaling the lane-to-lane skew has low importance. The differential signaling used in most high-speed serial design, however, requires a tight skew management between the positive and negative legs of a lane. Multi-Gbps data rates may allow only a few ps skew before signal degradation, such as mode conversion, reduction of signal magnitude or EMC problems show up [1], [2], [14], [15]. With point-to-point topologies and long PCB traces, new emphasis must be placed on local variations of the propagation speed due to bends and turns [3], [4], glass-weave effects [5], [6], [7] and similar deterministic and random asymmetries between the positive and negative legs of a differential pair.

Skew is the time-of-arrival error of a signal with respect to some reference time. It is inherently a time domain parameter, relying on the measurement of propagation delay. Whether we measure skew in the time or frequency domain, already the definition of the propagation delay raises questions. When we measure the time of arrival of a waveform, we need both a reference time and a reference level. As *Figure 1* illustrates, as soon as the transition waveforms are not identical in shape at the source and destination points, the (propagation) delay value depends on the reference level we select, introducing further uncertainty.

*Figure 1:** Illustration of delay definitions with ideal edges (on the left) and with different reference and target waveforms (on the right).*

The selection of reference level becomes important when the signal propagates through a dispersive, lossy media. The signal magnitude and the signal wave shape both will change.

When we measure delay in the frequency domain, we can calculate a phase delay or group delay (the derivative of phase delay) for the reference and target signals, but these delay numbers will be a function of frequency. *Figure 2* illustrates the phase and group delays calculated from causal models of single uniform transmission lines.

*Figure 2:** Phase delay and group delay for an ideal 50-ohm uniform transmission line (left) and a causal lossy 50-ohm transmission line of the same length (right).*

Note that the phase delay and group delay both decay with increasing frequency, and both are steeper at low frequencies. This behavior is related to the facts that inductance as well as capacitance will go down as frequency goes up and this also means that the group delay line is running below the phase delay line.

Data in *Figure 2* assumed matched interconnects, in other words the reference impedance for the scattering matrix equals the characteristic impedance of interconnect. When the reference impedance does not equal the characteristic impedance, the delay values will also depend on reflections, as shown in *Figure 3*. This eventually convolves the effects of locally changing propagation velocity and characteristic impedance [8].

*Figure 3:**Phase delay and group delay calculated with 40-ohm reference impedance for an ideal 50-ohm uniform transmission line with 40 ps delay (left) and a causal lossy 50-ohm transmission line of the same length (right).*

When reflections are present, the steady-state (frequency-domain) and transient time-domain delays will be different: while the transient, incident-wave delay remains the same, the steady-state phase and group delays show periodic fluctuation with frequency. The fact that reflections have an effect on steady-state delay suggests that in *Eq. (2)* we should calculate the unwrapped phase from the actual transfer function of interest (say *V _{out} / V_{source}*) rather than from the

**parameters, which do not carry information about the source and load reflections. On the plot of**

*S**Figure 3*we should notice that both the phase delay and group delay have values lower than the 40 ps delay of the line. A similar trend is seen also on

*Figure 4*, but because of the dispersion introduced by losses this effect is not so obvious. With group delay this is less surprising since we know that at resonances it can even go negative. With phase delay, on the other hand, we may suspect that values lower than the ideal propagation delay of the loss less line indicates non-causality.

To understand what is going on, it is useful to look at extreme cases. *Figures 4* and *5* show the same loss less interconnect we had on *Figures 2 *and* 3*, except we now assume much larger mismatch. The first minimum of phase delay with a 10:1 mismatch is 30.8 ps. With 10,000:1 mismatch the first minimum bottoms out at 20 ps, which is half of the ‘normal’ 40 ps value. The first minimum occurs at 12.5 GHz, which corresponds to a half-wavelength resonance calculated with the 40 ps delay value. Note that after the 12.5 GHz resonance, the phase delay jumps up to 60 ps and eventually at much higher frequencies it settles at 40 ps. The reason for this large deviation from the expected 40 ps value is explained by the unwrapped phase curve: it is now essentially a staircase starting at -90 degrees. This is happening because the chosen values create an almost perfect integrator. A transmission line terminated in high impedances can be simplified as its static capacitance driven by a current source, which is an integrator circuit with constant phase shift. Though with such extreme termination values we could not practically use the interconnect for signal transmission at frequencies other than narrow bands around the half-wave resonances, this figure illustrates and explains the physical reason why phase delay with mismatches can be lower than matched phase delay.

*Figures 4 and 5:**Phase delay and group delay calculated with 500-ohm reference impedance (on the left) and 500 kOhm reference impedance (on the right) for an ideal 50-ohm uniform transmission line with 40 ps delay.*

Any asymmetry between positive and negative legs of a differential pair causes a difference in phase delay, which in turn results in signal distortion in the time-domain response. Consider two 12 inches long differential pairs, of which the measured intra-pair difference in phase delay is 3 ps and 18 ps, respectively.

*Figure 6:** Distortion created by skew. Output voltages at the receiver show a difference between a board with 3ps (left) and 18ps (right) of skew.*

This differential pair is connected to a differential source, which is defined as a step function transitioning between -500 mV and +500 mV with a rise time of 50 ps. The resulting simulated waveforms at the receiver are shown in *Figure* *6*. Comparing the two figures, the existing intra-pair skew actually shows up in the time-domain response and the skew numbers are well preserved. To further investigate the time-domain response, the rise time of the differential input signal has been swept from 1ps to 100ps. As a result, the resultant skew number for single transient edges stays the same regardless of the rise time, suggesting that the skew can affect the differential signal quality in a wide data-rate range. In contrast, when there are multiple reflections along the signal path, the resulting frequency-dependent phase delay creates rise-time dependent skew.

Geometrically there is a deterministic element of length variations when we route a complex board and need to bend or turn traces several times, which in differential trace pairs will result in a length difference. At each turn the outer trace has a little bit of extra length.

*Figure 7:** Center-line length differences of differential pairs based on layout geometry.*

*Figure 7* shows the top views of four cases: a) is a straight etch for reference, b), c) and d) are differential pairs making a sharp 90-degree turn, double 45-degree turns and an arced 90-degree turn, respectively. The expression above each sketch shows the center-line length difference. We will look at these structures again in more detail in the paper.

To determine the extra length in the outer traces of a pair, we face an important question: what is the length of a trace with bends and turns. No matter how narrow traces we use, the inner perimeter of the turning shape is always shorter than the outer perimeter. Not knowing better, we may use the centerline length of each trace. If we assume w = s = 6 mils and t_{pd} = 150 ps/”, we get 24-mil, 19.9-mil and 18.8-mil center-line difference and 3.6 ps, 2.99 ps and 2.82 ps calculated skew for cases b), c) and d), respectively. Note that for all three geometries, the length difference is proportional to the sum of trace width and trace separation, suggesting that in general, narrower and tighter coupled traces at bends incur less skew.

Still deterministic, but somewhat harder to quantify is the change of delay at bends and turns due to the resulting discontinuities. Discontinuities of this sort are small localized fields, which we may be approximated as small lumped capacitive or inductive loads at the location of the turns. The reactive loading changes the overall propagation delay and it also changes the impedance of the trace. The changing impedance creates reflections, which also has an impact on the delay. The bends and turns in this respect are somewhat similar to vias, which are vertical right-angle turns with additional features (pads and antipads), potentially creating more discontinuities and variations of the propagation delay. And finally there are a number of statistical effects contributing to the uncertainty of the delay. Part-to-part and layer-to-layer variations of the dielectric constant due to glass-weave effects and due to the slower potential variations of glass-resin ratio along the area of the board.

Dowload the entire paper in PDF format. Additional topics include detailed discussion of: Layout and Material-Dependent Skew, Skew Due to Glass Weave and its Statistical Analysis, Skew Compensation, and Measurement Data

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