Optimal wire-sizing function with fringing capacitance consideration

Abstract

In this paper, we consider non-uniform wire-sizing under the Elmore delay model. Given a wire s e gment of length L, let fx be the width of the wire a t p osition x, 0 x L .It was shown in 2, 5 that the optimal wire-sizing function which minimizes delay is an exponential tapering function fx = ae ,bx , where a 0 and b 0 are c onstants.Unfortunately, 2, 5 did not consider fringing capacitance which is at least comparable in size to area c apacitance i n deep submircon designs. As a result, exponential tapering is no longer the optimal strategy. In this paper, we show that the optimal wire-sizing function, taking fringing capaci-,n n,1 n! x n is the Lambert's W function, cf and c0 are the respective fringing capacitance and area c apacitance of wire p er unit square, a 0 and b 0 are c onstants.The optimal wire-sizing function degenerates into an exponential tapering function as cf = 0 , and degenerates into a square-root tapering function fx = p b , ax; where a 0 and b 0 a s c f ! 1 . Our experimental results show that the optimal wire-sizing function can signi cantly reduce the interconnection delay of exponentially tapered wires. In the case where lower and upper bounds on the wire widths are given, the optimal wire-sizing function is a truncated version of the above function. Finally, our optimal wire-sizing function can be iteratively applied to optimally size all the wire segments in a routing tree for objectives such as minimizing weighted sink delay, minimizing maximum sink delay, or minimizing area subject to delay bounds at the sinks.

Analytical bound for unwanted clock skew due to wire width variation

“…Same conclusion is derived in [3] for the single load case, thus we omit the derivation here. From the above equation we can obtain…”

“…When the wire width w varies in the range of £ w lo ¥ w hi¦ , we need to find a wire shaping function w § x such that the delay from the virtual driver R to the sink is maximized or minimized. The minimum delay wire shaping function for a path without branch loads is solved in [8,3]. An iterative wire shaping algorithm is provided in [4] to minimize a weighted sum of sink delays in a routing tree.…”

“…The minimum delay wire shaping for a path with single load § k ¡ 0 is shown in [8,3]. In this section, we describe the minimum delay wire shaping for a path with multiple branch loads.…”

“…Our proposed technique is based on the observation that estimating the worst case skew due to wire width variation is closely related to the non-uniform wire sizing problem in physical optimizations. The minimum delay non-uniform wire sizing problem for a 2-pin wire(single load path) has been solved in [8,3]. We derive the maximum wire shaping function for both single load path and multi-pin trees.…”

“…Previously, a more general version of this problem was solved through an iterative algorithm [4]. Similar to the works of [8,3,4], we employ Elmore delay model for our derivations because of its high fidelity and ease of computation. Besides the bound, we discovered that the bound for skew between two clock sinks depends on their common upstream path, even though the skew between them is independent of the common path.…”