Dynamic Voltage Drop (DVD) Analyses
Dynamic voltage drop or DVD in a chip power grid segment
is in essence transient noise. It is a complex superposition of various noise components of a typical chip power grid that reduces
available power supply differential. DVD is a key measure of chip power integrity, and requires careful inspection during chip
design. It is a signoff gating aspect of chip power delivery.
Measured at points of interest on the grid, DVD is distinct from static (IR Drop) voltage reduction. But this distinction
blurs at times depending upon the analysis method adopted. A review of such methods is thus useful  both to comprehend DVD
and to recognize its derivation at present.
It is perhaps best to begin at the beginning as I've done in Chapter 1 of [1].
In that chapter, 'Power, Delivering Power, and Power Integrity,' I delve into fundamental constituents of supply noise, introducing
the 3 R's of Interconnect: Resistance, Reactance, and Resonance. Chapter 2 details the response of power delivery networks
(PDN's) to load stimuli. A separation of transient (AC) and static (DC) noise becomes clear through such response. The analysis
described therein captures circuit behavior in the TIME domain. Figure 1 displays an example.
Figure 1: Time domain simulation of supply noise at the chippkg interface.
But such waves captured, as in Figure 1, are nodal simulations: supply voltage variation at a circuit node with
respect to an ideal ground. A chip is a two dimensional surface. It may even be a 3D stack! Voltage variations across its power grid are typically displayed
as shown in Figure 2.
Figure 2: IR Drop snapshot on segments of a chip power grid [1].
With X and Y dimensions displaying the actual chip power grid, color shows voltage variation in the IR Drop snapshot
of Figure 2. Note that spatial variation of noise provides useful information though time variance is absent. One sees the
development of areas of greater current density and corresponding voltage reduction. EM hot spots in power metal layers
may be visible. Yet, such a snapshot captured hides many inadequacies as discussed in [1].
Given the computational complexity of such analysis on a ULSI chip, software tools reduce power grid segments to their base resistance values,
for example. Such reduction in IR Drop analyses loses at least two rigors of the 3R's of Interconnect.
Frequency Domain Analyses
PDN's on printed circuit boards (PCB's) and chip packages are also analyzed in parallel in the FREQUENCY domain. Lumped
approximations for PCB aspects, and impedance measurements, facilitate such analyses. Chapter 5 in [1]
discusses this in detail and describes the target impedance design method often pursued. Through such lumped abstractions for
electrical behavior, one can also correlate frequency domain characteristics with time domain behavior. Figure 3 illustrates
such correspondence.
Note that transient noise resulting from the excitation of PDN's, by load spectral components, at impedance peaks are
called 'Droops.' The term 'Voltage Drop' is usually reserved for static reductions in voltage or to averaged noise levels.
Circuitaccurate responses and droops (1st, 2nd etc.) are shown in Figure 1. Figure 3 illustrates only the correspondence
between time domain responses and frequency domain aspects. The 1^{st} droop corresponds to the 1^{st} peak, and so on.
Figure 3: Handdrawn sketch of frequency domain impedance variation and corresponding time domain voltage noise responses [2].
Figure 4: Frequency domain design of a board PDN [2].
But frequency domain analysis abstracts electrical behavior, including resistance, reactance, and resonance,
to a single aspect: impedance. It ignores vector phase relationships and surface noise wave propagation and interference.
Besides, impedance measurements are made at a node, just as timedomain noise variance is measured at a circuit junction. Hence
the applicability of this method to a surface, attached to distributed and varied electrical components, such as a chip power
grid, is rather limited. And, as established in theory in Chapter 5 of [2],
the target impedance of a PDN is not a reliable indicator of maximum possible noise.
Nevertheless, frequency domain analyses continue to assist PCB and package design. Figure 4 illustrates a design procedure
that attempts to meet a given impedance target. A combination of the frequency response of a chip model and the characteristics
of the board PDN results in an impedance peak at ~109 MHz. Observe the impedance characteristic of the chip: it is a simple
capacitorresistor model. Its impedance falls linearly on a loglog plot as frequency increases, flattening out to its effective
series resistance at the end of the range. Further analysis in [2] shows that
package inductance plays a large role in determining the combined impedance peak as seen at the board/chip interface. An
extension of the same finding establishes that local, onchip, power grid impedance thus depends upon inductance to onchip
decoupling capacitance.
Bridging Time and Frequency Domain Analyses
In [2], we explored a method bridging traditional circuitsbased time
domain analyses with systemslevel frequency domain analyses. It is conceptually simple: derive responses to individual spectral
components of load in the frequency domain, transform these responses into the time domain, and superpose them for a complete
response. But this procedure glosses over much information lost. For one, frequency domain analysis uses lumped approximations.
Spatial and time/phase relationships between spectral components may not matter in lumped individualized responses. Abstraction
in frequency domain analysis thus loses such significant information that matters in the time domain. And, lumped modeling isn't
suited to varied and distributed circuit loads and capacitance connecting to a power grid surface.
Figure 5: A DISTRIBUTED PDN model, chip/pkg/board grids and components [1].
It's important to ask why such analysis domain bridging may be useful. Chip design employs the time domain, while some
circuits such as amplifiers and filters are verified in the frequency domain. Fullchip integration and PI analysis use time
domain simulations. Though spatial IR Drop analysis employs the time domain, the very next electrical interfaces  the package and
board  are analyzed in the frequency domain.
This disconnect in design and verification environments is cause for much concern. It is clear that package and board
components have a dominant role to play in determining onchip PI. Hence a holistic, seamless design
(Slide 10 of the linked presentation) and verification environment for interconnect and PI is essential.
PIFP is just such an analysis
environment. While IR Drop provides spatial views, the absence of reactive and resonant aspects, and spatial noise wave propagation,
renders it nonphysical. IR Drop simulations break the speed of light,
a fundamental law of physics. While frequency domain analyses include resistance, reactance, and resonance, they lose spatial
time/phase relationships and cannot readily use time domain load currents available during chip design. Besides, FD analysis
lumps system properties to impedance measurable at a junction. To derive true dynamic noise on a chip power grid, we need
spatial + temporal + 3R's + dynamic aspects captured. In essence, the best capabilities of both TD and FD analyses combined.
PIFP provides such a spatiotemporal environment, and additionally provides truephysical representation of circuits,
capacitance, and power pathways as distributed area components and grids. Figure 5 illustrates a typical PIFP PDN model,
and Figure 6 a simulation snapshot of its noise animation.
Figure 6: Spatiotemporal, dynamic chip power grid noise [3].
Why Change from Time and Frequency Domain Analyses?
I won't quote clichés such as "Change is the only constant." I'd argued in the past [4]
that simulation complexity will explode in the nanoscale regime. And it has: PI verification simulations take many hours
to days in ULSI designs today. And why is it still verification, a backend task, when it has serious performance implications?
Why is it not part of design and optimization?
A key point: we envisioned PIFP as a frontend design optimization tool. As in Figure 7, we saw PIFP entering into
the chip design flow before routing. Given the projected severity of routing congestion, optimizing the allocation of metal between
power grids and routing channels seemed to us a critical task. Besides, PI studies could revise block placements and decoupling
capacitance inclusion, changing route vectors and constraints. A chip floorplan thus optimized through early PI analyses seemed
to us the best path to a correctbydesign physical layout.
Figure 7: PIFP Methodology [1].
We wrote in [4] that: “[Complex PI] noise distribution cannot be
predicted using a static analysis or dynamic IR drop tool as it is the result of complex interactions between edge timings,
block placement, grid impedance and variable wave velocities (caused by variable capacitance per unit area in different parts
of the chip). Again, this type of analysis is best done using tools that perform realistic simulations in minutes rather than
hours or days. This makes it possible to complete the large simulation based experimental designs, covering the large parameter
space (capacitance, width, space, placement etc.) required for this type of complex power grid design, within a reasonable
timescale.”
Even more important is avoiding errors in analysis and inference arising from approximations. For instance, Peak IR Drop analysis
calculates peak noise as peak current multiplied with grid resistance. I've shown that peak noise often does not correlate with peak currents. Or, with noise estimated as i⋅r
+ L⋅di/dt, Dynamic IR Drop uses package inductance for 'L.' An egregious assumption, for di/dt onchip, at
an IP Block node, is different, often much higher, from di/dt at a package/chip junction. Besides, the local onchip
inductive effect relates to distance and interconnect to neighboring capacitance. Such local inductance varies all over the
chip power grid.
One may also assume, from L⋅di/dt, that high frequency load components produce higher noise. And thus use spectral
(FD) analysis of load waves to guide clocking in the time domain. But, as in a study of Clocking and PI, a sinusoidal load wave shape may not be the most desirable. Besides, exploration of
low noise load current wave shapes may well be IP Block specific.
A frequency domain analysis environment, as illustrated in Figures 3 and 4, may show multiple impedance peaks below
an impedance target for a PDN. This, as in [2], is no guarantee of peak noise
below the limit that defines the impedance target. Varied load excitation may stimulate the PDN at more than one impedance
peak. Spatially adjacent load blocks may manifest noise waves that propagate and interfere constructively. The resulting noise
waves, of different or even the same frequency, can superpose and grow much greater than the limit set. Such waves are often called Rogue Waves, and manifest both in
oceans and optical links. FD impedance targets met thus do not limit system noise.
A holistic, truephysical environment eliminates many approximations. Truephysical accuracy also lends greater confidence
in design choices and inferences. Would you instead settle for analysis environments with approximations and lumped component
models?
Takeaways
1) A Spatiotemporal analysis environment extends time and frequency domain capabilities and insights [1].
2) Holistic models and truephysical analyses result from incorporating key time and frequency domain aspects.
3) Early, truephysical PI analysis can/should be a frontend design and optimization procedure [4].
For more on PIFP, or holistic PI analysis/experiments, please do not hesitate to
contact us.
References
