1. Elmore Delay, Logical Effort

2. Modern Interconnect
© Rabaey, ch4Wire.ppt, slide 22

3. Example: Intel 0.25 micron Process
5 metal layers
Ti/Al - Cu/Ti/TiN
Polysilicon dielectric
© Rabaey, ch4Wire.ppt, slide 23

4. Modern Interconnect
90nm process
© Chris Kim (image from Intel?)

5. The Lumped RC-Model, The Elmore Delay
(result *0.69)
© Rabaey, ch4Wire.ppt, slide 27

6. Example: The Elmore Delay
Shared Paths:
R44 = R1+R3+R4
Rii = R1+R3+Ri
Ri4 = R1+R3
Ri2 = R1
Ti = C1R1 + C2R1 + C3(R1+R3) + C4(R1+R3) + Ci(R1+R3+Ri)

7. The Elmore Delay RC Chain
© Rabaey, ch4Wire.ppt

8. The Distributed RC-line
Diffusion Equation
© Rabaey, ch4Wire.ppt

9. Deriving the Diffusion Eq

10. Step-response of RC wire as a function of time and space
© Rabaey, ch4Wire.ppt

11. RC-Models
© Rabaey, ch4Wire.ppt

12. Driving an RC-line
© Rabaey, ch4Wire.ppt

13. Designing Fast CMOS Gates
Slides from chapter6.ppt of Rabaey’s page

14. Fan-In ConsiderationsB C D CL A
Distributed RC model
(Elmore delay)
tpHL = 0.69 Reqn(C1+2C2+3C3+4CL)
Propagation delay deteriorates rapidly as a function of fan-in – quadratically in the worst case. C3 B C2 C
While output capacitance makes full swing transition (from VDD to 0), internal nodes only transition from VDD-VTn to GND
C1, C2, C3 on the order of 0.85 fF for W/L of 0.5/0.25 NMOS and 0.375/0.25 PMOS
CL of 3.2 fF with no output load (all diffusion capacitance – intrinsic capacitance of the gate itself).
To give a 80.3 psec tpHL (simulated as 86 psec) C1 D

15. tp as a Function of Fan-In
tpHL
quadratic
linear tp
Gates with a fan-in greater than 4 should be avoided.
tp (psec)
tpLH
Fixed fan-out (NMOS 0.5 micrcon, PMOS 1.5 micron)
tpLH increases linearly due to the linearly increasing value of the diffusion capacitance
tpHL increase quadratically due to the simultaneous incrase in pull-down resistance and internal capacitance
fan-in

16. tp as a Function of Fan-In and Fan-Out
Fan-in: quadratic due to increasing resistance and capacitance
Fan-out: each additional fan-out gate adds two gate capacitances to CL
tp = a1FI + a2FI2 + a3FO
a1 term is for parallel chain, a2 term is for serial chain, a3 is fan-out

17. Fast Complex Gates: Design Technique 1
Transistor sizing
as long as fan-out capacitance dominates
Progressive sizing CL
Distributed RC line
M1 > M2 > M3 > … > MN
(the fet closest to the
output is the smallest)
InN MN
M1 have to carry the discharge current from M2, M3, … MN and CL so make it the largest
MN only has to discharge the current from MN (no internal capacitances) C3
In3 M3 C2
In2 M2
Can reduce delay by more than 20%; decreasing gains as technology shrinks C1
In1 M1

18. Fast Complex Gates: Design Technique 2
Transistor ordering
critical path
critical path
01 CL CL
charged
charged 1
In1
In3 M3 M3 1 C2 1 C2
In2
In2 M2
discharged M2
charged
For lecture.
Critical input is latest arriving signal
Place latest arriving signal (critical path) closest to the output 1 C1 C1
In3
discharged
In1
charged M1 M1
01
delay determined by time to discharge CL, C1 and C2
delay determined by time to discharge CL

19. Fast Complex Gates: Design Technique 3
Alternative logic structures
F = ABCDEFGH
Reduced fan-in -> deeper logic depth
Reduction in fan-in offsets, by far, the extra delay incurred by the NOR gate (second configuration).
Only simulation will tell which of the last two configurations is faster, lower power

20. Fast Complex Gates: Design Technique 4
Isolating fan-in from fan-out using buffer insertion CL CL
Reduce CL on large fan-in gates, especially for large CL, and size the inverters progressively to handle the CL more effectively

21. Slides from chapter6.ppt of Rabaey’s page
Logical Effort
Slides from chapter6.ppt of Rabaey’s page

22. Transistor Sizing D=1+f D=2+4/3 f D=2+5/3 f Cg= Cint= Cg= Cint= Cg=
Assumes Rp = Rn

24. Normalized Space

25. Parasitic Term P
NOTE: p is a gate parameter  function(W)

26. Logical Effort Term g
NOTE: g is a gate parameter  function(W)

27. Transistor Sizing a Complex CMOS GateB 8 6 4 3 C 8 6 D 4 6
OUT = D + A • (B + C)
For class lecture.
Red sizing assuming Rp = Rn
Follow short path first; note PMOS for C and B 4 rather than 3 – average in pull-up chain of three – (4+4+2)/3 = 3
Also note structure of pull-up and pull-down to minimize diffusion cap at output (e.g., single PMOS drain connected to output)
Green for symmetric response and for performance (where Rn = 3 Rp)
Sizing rules of thumb
PMOS = 3 * NMOS
1 in series = 1
2 in series = 2
3 in series = 3
etc. A 2 D 1 B 2 C 2

28. Logical Effort
From Sutherland, Sproull

29. Logical Effort of Gates

30. tp as a Function of Fan-Out
All gates have the same drive current.
tpNOR2
tpNAND2
tpINV
tp (psec)
Slope is a function of “driving strength”
slope is a function of the driving strength
eff. fan-out

31. Buffer Example In Out CL 1 2 N (in units of tinv)
For given N: Ci+1/Ci = Ci/Ci-1
To find N: Ci+1/Ci ~ 4
How to generalize this to any logic path?

32. Delay in a Logic Gate Gate delay: d = h + p effort delay
intrinsic delay
Effort delay:
h = g f
logical effort
effective fanout = Cout/Cin
Logical effort is a function of topology, independent of sizing
Effective fanout (electrical effort) is a function of load/gate size

37. Add Branching Effort
Branching effort:

39. Multistage Networks Stage effort: hi = gifi
Path electrical effort: F = Cout/Cin
Path logical effort: G = g1g2…gN
Branching effort: B = b1b2…bN
Path effort: H = GFB
Path delay D = Sdi = Spi + Shi

40. Optimum Effort per Stage
When each stage bears the same effort:
Stage efforts: g1f1 = g2f2 = … = gNfN
Effective fanout of each stage:
Minimum path delay

41. Optimal Number of Stages
For a given load,
and given input capacitance of the first gate
Find optimal number of stages and optimal sizing
Substitute ‘best stage effort’

42. Example: Optimize Path
g = 1 f = a
g = 5/3 f = b/a
g = 5/3 f = c/b
g = 1 f = 5/c
Effective fanout, F = 5
G =
H =
h =
a =
b =
c =

43. Example: Optimize Path
g = 1 f = a
g = 5/3 f = b/a
g = 5/3 f = c/b
g = 1 f = 5/c
Effective fanout, F = 5
G = 25/9
H = 125/9 = 13.9
h = 1.93
a = 1.93
b = ha/g2 = 2.23
c = hb/g3 = 5g4/f = 2.59

45. Example – 8-input AND

46. Method of Logical Effort
Compute the path effort: F = GBH
Find the best number of stages N ~ log4F
Compute the stage effort f = F1/N
Sketch the path with this number of stages
Work either from either end, find sizes: Cin = Cout*g/f
Reference: Sutherland, Sproull, Harris, “Logical Effort, Morgan-Kaufmann 1999.

47. Summary
Sutherland,
Sproull
Harris