m.4- Skew Formulation
III.5- An Upper Bound Result on Clock Skew
A necessary condition for the skew sl2 to be maximum is that:
Let us express the partial derivatives of the first and second orders relative to the gradient orientation 0.
as12
--
ade - O for 00 = arctan- b
By combining Eqs. 3.14 and 3.20, we get a skew upper bound:
= asin (arctan
(i)) +
b cos (arctan(i))
which depends on the value of the N-tuple (x,,y,) (Eqs. 3.17 and 3.18), i.e., the pair of H-tree leaves we are considering. In the following, we address the issue of which pair of H-tree ieaves leads to the highest possible skew.
Fig. 3.5 shows the effect of varying parameters a and b on skew. We can look at this figure from two angles, with different consequences. If we consider variables a and b as continuous, it is obvious that the skew diverges when a and b increase, and the center of Fig. 3.5 is a kind of inflection point in three dimensions that does not lead to an extremum. However, if we limit the a and b values to those illiistrated in Fig. 3.5, function SI* has obviously a maximum and a minimum. This interpretation corresponds to the case of an H-tree whose depth is finite, and where parameters a and b are limited
to a finite range detemiîned by those of the N-tuple of parameters (xv, y,).
Fig. 3. 5. Skew variation venus the configuration of direction variables
At this level, the next hurdle to overcorne is to determine, for an H-tree of depth N, the two leaves, and therefore the orientation of the time constant gradient, which lead to a maximum skew.
Fig. 3.5 shows that the maximum skew is reached in the region where b is at its most positive value. Let us consider E as the set of pairs (xj,yi), where ( i j ) belongs to [1,
$1
2, such that b is maximum Then, al1 the elements of E verify the relation<
O and they lead to a maximum of the sl2 function. However, only two pairs will produce the upper bound. The first pair corresponds to the parameter a, at its most positive value,and the second corresponds to the parameter a, at its most negative value. Because a11 the pu and A, coefficients of Eqs. 3.17 and 3.18 are positive, then the two pairs descnbed above are obtained for (xv,y,)=(2,2) and (xv,yu)=(2,-2). The first pair depicts the case of leaves L1 and L2 (Fig. 3.6) and the second corresponds to leaves L3 and L4 (Fig. 3.6).
The above result supports the strong intuition which States:
There cannot be a larger skew than the one observed for the huo leaves Iocafed nt the ends of the$rst diagonal of the H-iree, where the displacement along al1 the branches that constitute the path leading tu the fVst leaf occurs in the ascending direction of the gradient, while the displacement along al1 the branches that cons titute the path leading to the second leaf occurs in the descending direction of the gradient
Indeed, we shall not expect more skew than what is obtained when sending the same signal along two paths, one path following the direction of monotonically increasing process variations and the other path following the direction of monotonically ciecreasing process variations.
Due to the perspective view, Fig. 3.5 seems to present two different skew upper bounds reached at the two corners of the surface. These corners correspond to the two pairs (xj,yi) described so far. However, another geometrical point of view reveals that these two pairs lead to an identical skew upper bound. Indeed, when substituting the values of (a,b) corresponding to these two pairs into Eq. 3.21, we arrive at the same value. In the following, due to the symmetry in the problem, we can then focus only
on the first pair.
Thus, we can describe the location of the two leaves located at the ends of the first diagonal (Fig. 3.6) of the H-tree as:
Fig. 3. 6. The two pairs of leaves that produce the skew uppcr bound
Then, the upper bound of si2 is the following:
amas
m a x ( s 1 2 ) = amazsin
(
arctan-2:) +
bmaZcos (arctanb_; (3.23) where, a,, and b,, corne from the combination of Eqs. (3.15), (3.16), (3.22a) and (3.22b). Thus:and the corresponding gradient orientation is:
amas do = arctan-
bmas
We now have the expression of the skew upper bound between H-tree leaves, as well as the corresponding orientation of the time constant gradient, as a function of the H-tree topology only.
From Appendix A of [61], we get:
where:
0 t is a parameter that depends on the H-tree topology, such that:
t = 1.43 for the l(2k
+
1) = l(2k+
2) =2k+l
U topology* r o is the transistor time constant when using a 1.2pm CMOS technology.
*DinVo is the length of the basic inverter with which the H-tree is built.
* /3 is the scaling factor from a 1.2pm CMOS technology.
In these calculations, we assumed that the size of the chip D >> Di,,.
Getting some concrete numerical results from the previous analysis is of interest.
Using the following parameters:
r0=24ps,
p=1
(1.2pm CMOS technology), M=4, Dinvo=12.4pm and t=1.43 (topology of Fig. 3.1),Eq. (3.27) predicts that, on a silicon die of l c m x lcm, we can propagate a lûûMHz clock with a skew upper bound not exceeding one tenth of clock period, that is to say,
lns. Moreover, for the same silicon area, when the technology is scaled down, this skew upper bound is reduced in proportion to the scaling factor
P,
thus allowing a clock rate that increases in the sarne proportion. Also, we will see in section III.6 that the clock rate can be doubled for neighbor-to-neighbor communications between processors.The gradient orientation which mâximizes the skew (Eq. 3.28) is about 10"
off from the intuitive 45' one might expect due to the presence of the trigonometric functions sin0 and cos0 in the expression of the transistor time constant gradient (Eq.
3.5). This phenornenon, which is thoroughly characterized in [fil], is associated with the step increase of the transistor time constant variation between two successive colinear segments of the tree.
We cm now state important results:
In the case of multi-dimensional arrays, where there exist direct communications between the leaves at opposite sides of the array (side-to-side communications), Eq. (3.27) shows that the skew upper bound grows like the square of the H-tree size. Note that the skew between 2 processors communicating through a third processor is not relevant in the determination of the skew upper bound.
In a logic-based H-tree subject to a uniforrn gradient of the transistor time constant, the leaf-to-leaf skew grows faster than the root-to-leaf delay according to Eq. (3.27). This quadratic growth of the skew relative to the H-tree size is due to the space dimension added by the time constant gradient (Eq. 3.1).