ELEC 7770 Advanced VLSI Design Spring 2010 Zero-Skew Clock Routing

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ELEC 7770 Advanced VLSI Design Spring 2010 Zero-Skew Clock Routing Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr10/course.html Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Zero-Skew Clock Routing FF FF FF FF FF FF FF FF CK FF FF FF FF FF FF FF FF Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Zero-Skew: References H-Tree A. L. Fisher and H. T. Kung, “Synchronizing Large Systolic Arrays,” Proc. SPIE, vol. 341, pp. 44-52, May 1982. A. Kahng, J. Cong and G. Robins, “Hig-Performance Clock Routing Based on Recursive Geomrtric Matching,” Proc. Design Automation Conf., June 1991, pp. 322-327. M. A. B. Jackson, A. Srinivasan and E. S. Kuh, “Clock Routing for High-Performance IC’s,” Proc. Design Automation Conf., June 1990, pp. 573-579. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Zero-Skew Routing Build clock tree bottom up: Leaf nodes are all equal loading flip-flops. Two zero-skew subtrees are joined to form a larger zero-skew subtree. Entire clock tree is built recursively. R.-S. Tsay, “An Exact Zero-Skew Clock Routing Algorithm,” IEEE Trans. CAD, vol. 12, no. 2, pp. 242-249, Feb. 1993. J. Rubenstein, P. Penfield and M. A. Horowitz, “Signal Delay in RC Tree Networks,” IEEE Trans. CAD, vol. 2, no. 3, pp. 202-211, July 1983. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Balancing Subtrees (1) xL r1 A t1 c1/2 c1/2 C1 Subtree 1 Tapping point (1 – x)L r2 B t2 c2/2 c2/2 C2 Subtree 2 Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Balancing Subtrees (2) Subtrees 1 and 2 are each balanced (zero-skew) trees, with delays t1 and t2 to respective leaf nodes. Total capacitances of subtrees are C1 and C2, respectively. Connect points A and B by a minimum-length wire of length L. Determine a tapping point x such that wire lengths xL and (1 – x)L produce zero skew. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Balancing Subtrees (3) Use Elmore delay formula: 0.69 r1(C1 + c1/2) + t1 = 0.69 r2(C2 + c2/2) + t2 Substitute: r1 = axL, r2 = a(1 – x)L c1 = bxL, c2 = b(1 –x)L abL2x + aL(C1+C2)x = 1.45 (t2 – t1) + aL(C2+bL/2) Then solve for x: 1.45 (t2 – t1) + aL (C2 + bL/2) x = ─────────────────── aL(bL + C1 + C2) Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Balancing Subtrees Example 1 Subtree parameters: Subtree 1: t1 = 5ps, C1 = 3pF Subtree 2: t2 = 10ps, C2 = 6pF Interconnect: L = 1mm Wire parameters: a = 100Ω/cm, b = 1pF/cm Tapping point: 1.45(t2 – t1) + aL (C2 + bL/2) 1.45(10–5) + 100×0.1(6 + 1×0.1/2) X = ────────────────── = ────────────────────── aL (bL + C1 + C2) 100×0.1(1×0.1+3+6) = 1.45(5 + 60.5)/(10×9.1) = 0.7445 Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Example 1 FF t1 = 5ps, C1 = 3pF FF 0.77445mm Subtree 1 FF FF To next level FF 0.2555mm t2 = 10ps, C2 = 6pF FF Subtree 2 FF Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Balancing Subtrees, x > 1 Tapping point set at root of tree with larger loading (C2, t2). Wire to the root of other tree is elongated to provide additional delay. Wire length L is found as follows: Set x = 1 in abL2x + aL(C1+C2)x = 1.45(t2 – t1)+aL(C2+bL/2) i.e., L2 + (2C1/b)L – 2.9 (t2 – t1)/(ab) = 0 Wire length is given by: [(aC1)2 + 2.9 ab(t2 – t1)]½ – aC1 L = ──────────────────── a b R.-S. Tsay, “An Exact Zero-Skew Clock Routing Algorithm,” IEEE Trans. CAD, vol. 12, no. 2, pp. 242-249, Feb. 1993. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Balancing Subtrees Example 2 Subtree parameters: Subtree 1: t1 = 2ps, C1 = 1pF Subtree 2: t2 = 15ps, C2 = 10pF Interconnect: L = 1mm Wire parameters: a = 100Ω/cm, b = 1pF/cm Tapping point: 1.45(t2 – t1) + aL (C2 + bL/2) 1.45(15–2) + 100×0.1(10 + 1×0.1/2) x = ─────────────────── = ────────────────────── aL (bL + C1 + C2) 100×0.1(1×0.1+1+10) = (18.85 + 100.5)/(10×11.1) = 1.0752 Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Example 2, x = 1.0752 Setting x = 1.0, [(aC1)2+2.9ab(t2 – t1)]½ – aC1 L = ──────────────────── a b [(100×1)2 + 290 (15 – 2)]½ – 100×1 = ─────────────────────── 100×1 = 0.1735cm For a wire of 1.735mm length, place the clock feed at one end. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Example 2, L = 1.735mm FF t1 = 2ps, C1 = 1pF FF Subtree 1 FF L = 1.7355mm FF FF To next level t2 = 15ps, C2 = 10pF FF Subtree 2 FF Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Balancing Subtrees, x < 0 Tapping point set at root of tree with smaller loading (C1, t1). Wire to the root of other tree is elongated to provide additional delay. Wire length L found as follows: Set x = 0 in abL2x + aL(C1+C2)x = 1.45(t2 – t1)+aL(C2+bL/2) i.e., L2 + (2C2/b)L – 2.9 (t1 – t2)/(ab) = 0 Wire length is given by: [(aC2)2 + 2.9 ab(t1 – t2)]½ – aC2 L = ──────────────────── a b R.-S. Tsay, “An Exact Zero-Skew Clock Routing Algorithm,” IEEE Trans. CAD, vol. 12, no. 2, pp. 242-249, Feb. 1993. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

Balancing Subtrees Example 3 Subtree parameters: Subtree 1: t1 = 15ps, C1 = 10pF Subtree 2: t2 = 2ps, C2 = 1pF Interconnect: L = 1mm Wire parameters: a = 100Ω/cm, b = 1pF/cm Tapping point: 1.45(t2 – t1) + aL (C2 + bL/2) 1.45(2–15) + 100×0.1(1 + 1×0.1/2) x = ─────────────────── = ────────────────────── aL (bL + C1 + C2) 100×0.1(1×0.1+1+10) = ( – 18.85 + 10.5)/(10×11.1) = – 0.0752 Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Example 3, x = – 0.0752 Setting x = 0.0, [(aC2)2+2.9ab(t1 – t2)]½ – aC2 L = ────────────────── a b [(100×1)2+290 (15 – 2)]½ – 100×1 = ─────────────────────── 100×1 = 0.1735cm Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Example 3, L = 1.255mm FF FF To next level t1 = 15ps, C1 = 10pF FF FF L = 1.735mm Subtree 1 FF t2 = 2ps, C2 = 1pF FF Subtree 2 FF Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Zero-Skew Design Delay =75ns Delay = 50ns FF A Comb. FF B Comb. FF C CK CK time Tck = 75ns Single-cycle path delay Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Nonzero-Skew Design Delay =75ns Delay = 50ns FF A Comb. FF B Comb. FF C CK Delay = 25ns CK time Tck = 50ns Single-cycle path delay Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Conclusion Zero-skew design is possible at the layout level. Zero-skew usually results in higher clock speed. Nonzero clock skews can improve the design with reduced hardware and/or higher speed. Spring 2010, Mar 5 ELEC 7770: Advanced VLSI Design (Agrawal)