Lecture 6:
Logical
Effort
Outline
❑
❑
❑
❑
❑
❑
Logical Effort
Delay in a Logic Gate
Multistage Logic Networks
Choosing the Best Number of Stages
Example
Summary
6: Logical Effort
CMOS VLSI Design 4th Ed.
2
Introduction
❑ Chip designers face a bewildering array of choices
– What is the best circuit topology for a function?
– How many stages of logic give least delay?
– How wide should the transistors be?
???
❑ Logical effort is a method to make these decisions
– Uses a simple model of delay
– Allows back-of-the-envelope calculations
– Helps make rapid comparisons between alternatives
– Emphasizes remarkable symmetries
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Example
❑ Ben Bitdiddle is the memory designer for the Motoroil 68W86,
an embedded automotive processor. Help Ben design the
decoder for a register file.
A[3:0] A[3:0]
32 bits
6: Logical Effort
CMOS VLSI Design 4th Ed.
16 words
4:16 Decoder
❑ Decoder specifications:
– 16 word register file
– Each word is 32 bits wide
– Each bit presents load of 3 unit-sized transistors
– True and complementary address inputs A[3:0]
– Each input may drive 10 unit-sized transistors
❑ Ben needs to decide:
– How many stages to use?
– How large should each gate be?
– How fast can decoder operate?
16
Register File
4
Delay in a Logic Gate
❑ Express delays in process-independent unit d = dabs

❑ Delay has two components: d = f + p
 = 3RC
❑ f: effort delay = gh (a.k.a. stage effort)
 3 ps in 65 nm process
– Again has two components
60 ps in 0.6 mm process
❑ g: logical effort
– Measures relative ability of gate to deliver current
– g  1 for inverter
❑ h: electrical effort = Cout / Cin
– Ratio of output to input capacitance
– Sometimes called fanout
❑ p: parasitic delay
– Represents delay of gate driving no load
– Set by internal parasitic capacitance
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Delay Plots
=f+p
= gh + p
❑ What about
NOR2?
2-input
NAND
6
Normalized Delay: d
d
Inverter
5
g=1
p=1
d=h+1
4
3
g = 4/3
p=2
d = (4/3)h + 2
Effort Delay: f
2
1
Parasitic Delay: p
0
0
1
2
3
4
5
Electrical Effort:
h = Cout / Cin
6: Logical Effort
CMOS VLSI Design 4th Ed.
6
Computing Logical Effort
❑ DEF: Logical effort is the ratio of the input
capacitance of a gate to the input capacitance of an
inverter delivering the same output current.
❑ Measure from delay vs. fanout plots
❑ Or estimate by counting transistor widths
2
Y
2
A
Y
1
Cin = 3
g = 3/3
6: Logical Effort
2
A
2
B
2
A
4
B
4
Cin = 4
g = 4/3
CMOS VLSI Design 4th Ed.
Y
1
1
Cin = 5
g = 5/3
7
Catalog of Gates
❑ Logical effort of common gates
Gate type
Number of inputs
1
2
3
4
n
NAND
4/3
5/3
6/3
(n+2)/3
NOR
5/3
7/3
9/3
(2n+1)/3
2
2
2
2
4, 4
6, 12, 6
8, 16, 16, 8
Inverter
Tristate / mux
XOR, XNOR
6: Logical Effort
1
2
CMOS VLSI Design 4th Ed.
8
Catalog of Gates
❑ Parasitic delay of common gates
– In multiples of pinv (1)
Gate type
Number of inputs
1
2
3
4
n
NAND
2
3
4
n
NOR
2
3
4
n
4
6
8
2n
4
6
8
Inverter
Tristate / mux
XOR, XNOR
6: Logical Effort
1
2
CMOS VLSI Design 4th Ed.
9
Example: Ring Oscillator
❑ Estimate the frequency of an N-stage ring oscillator
Logical Effort:
Electrical Effort:
Parasitic Delay:
Stage Delay:
Frequency:
6: Logical Effort
31 stage ring oscillator in
g=1
0.6 mm process has
h=1
frequency of ~ 200 MHz
p=1
d=2
fosc = 1/(2*N*d) = 1/4N
CMOS VLSI Design 4th Ed.
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Example: FO4 Inverter
❑ Estimate the delay of a fanout-of-4 (FO4) inverter
d
Logical Effort:
Electrical Effort:
Parasitic Delay:
Stage Delay:
6: Logical Effort
g=1
h=4
p=1
d=5
The FO4 delay is about
300 ps in 0.6 mm process
15 ps in a 65 nm process
CMOS VLSI Design 4th Ed.
11
Multistage Logic Networks
❑ Logical effort generalizes to multistage networks
❑ Path Logical Effort
G = gi

❑ Path Electrical Effort
❑ Path Effort
10
g1 = 1
h1 = x/10
6: Logical Effort
x
g2 = 5/3
h2 = y/x
Cout-path
H=
Cin-path
F =  fi = gihi
y
g3 = 4/3
h3 = z/y
z
g4 = 1
h4 = 20/z
CMOS VLSI Design 4th Ed.
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12
Multistage Logic Networks
❑ Logical effort generalizes to multistage networks
❑ Path Logical Effort
G = gi

❑ Path Electrical Effort
❑ Path Effort
Cout− path
H=
Cin− path
F =  fi = gihi
❑ Can we write F = GH?
6: Logical Effort
CMOS VLSI Design 4th Ed.
13
Paths that Branch
❑ No! Consider paths that branch:
15
G
H
GH
h1
h2
F
=1
5
= 90 / 5 = 18
= 18
= (15 +15) / 5 = 6
= 90 / 15 = 6
= g1g2h1h2 = 36 = 2GH
6: Logical Effort
CMOS VLSI Design 4th Ed.
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90
90
14
Branching Effort
❑ Introduce branching effort
– Accounts for branching between stages in path
Con path + Coff path
b=
Con path
B = bi
Note:
h = BH
i
❑ Now we compute the path effort
– F = GBH
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Multistage Delays
❑ Path Effort Delay
DF =  fi
❑ Path Parasitic Delay
P =  pi
❑ Path Delay
D = di = DF + P
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Designing Fast Circuits
D = di = DF + P
❑ Delay is smallest when each stage bears same effort
fˆ = gi hi = F
1
N
❑ Thus minimum delay of N stage path is
D = NF + P
1
N
❑ This is a key result of logical effort
– Find fastest possible delay
– Doesn’t require calculating gate sizes
6: Logical Effort
CMOS VLSI Design 4th Ed.
17
Gate Sizes
❑ How wide should the gates be for least delay?
fˆ = gh = g CCoutin
giCouti
 Cini =
fˆ
❑ Working backward, apply capacitance
transformation to find input capacitance of each gate
given load it drives.
❑ Check work by verifying input cap spec is met.
6: Logical Effort
CMOS VLSI Design 4th Ed.
18
Example: 3-stage path
❑ Select gate sizes x and y for least delay from A to B
x
x
A
8
6: Logical Effort
x
CMOS VLSI Design 4th Ed.
y
45
y
B
45
19
Example: 3-stage path
x
x
A
8
x
y
45
y
Logical Effort
Electrical Effort
Branching Effort
Path Effort
Best Stage Effort
Parasitic Delay
Delay
6: Logical Effort
B
45
G = (4/3)*(5/3)*(5/3) = 100/27
H = 45/8
B=3*2=6
F = GBH = 125
fˆ = 3 F = 5
P=2+3+2=7
D = 3*5 + 7 = 22 = 4.4 FO4
CMOS VLSI Design 4th Ed.
20
Example: 3-stage path
❑ Work backward for sizes
x
x
A
8
x
y
45
y
B
y = 45 * (5/3) / 5 = 15
x = (15*2) * (5/3) / 5 = 10
45
45
A P: 4
N: 4
6: Logical Effort
CMOS VLSI Design 4th Ed.
P: 4
N: 6
P: 12
N: 3
21
B
45
Best Number of Stages
❑ How many stages should a path use?
– Minimizing number of stages is not always fastest
❑ Example: drive 64-bit datapath with unit inverter
Initial Driver
1
1
1
1
8
4
2.8
16
8
D = NF1/N + P
= N(64)1/N + N
23
Datapath Load
N:
f:
D:
6: Logical Effort
64
1
64
65
CMOS VLSI Design 4th Ed.
64
2
8
18
64
3
4
15
Fastest
64
4
2.8
15.3
22
Derivation
❑ Consider adding inverters to end of path
– How many give least delay?
Logic Block:
n1Stages
Path Effort F
n1
D = NF +  pi + ( N − n1 ) pinv
1
N
N - n1 ExtraInverters
i =1
1
1
1
D
= −F N ln F N + F N + pinv = 0
N
❑ Define best stage effort
 =F
1
N
pinv +  (1 − ln  ) = 0
6: Logical Effort
CMOS VLSI Design 4th Ed.
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❑
Best Stage Effort
pinv +  (1 − ln  ) = 0 has no closed-form solution
❑ Neglecting parasitics (pinv = 0), we find  = 2.718 (e)
❑ For pinv = 1, solve numerically for  = 3.59
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Sensitivity Analysis
D(N) /D(N)
❑ How sensitive is delay to using exactly the best
1.6
number of stages?
1.51
1.4
1.26
1.2
1.15
1.0
( =2.4)
(=6)
0.0
0.5
0.7
1.0
1.4
2.0
N/ N
❑ 2.4 <  < 6 gives delay within 15% of optimal
– We can be sloppy!
– I like  = 4
6: Logical Effort
CMOS VLSI Design 4th Ed.
25
Example, Revisited
❑ Ben Bitdiddle is the memory designer for the Motoroil 68W86,
an embedded automotive processor. Help Ben design the
decoder for a register file.
A[3:0] A[3:0]
32 bits
6: Logical Effort
CMOS VLSI Design 4th Ed.
16 words
4:16 Decoder
❑ Decoder specifications:
– 16 word register file
– Each word is 32 bits wide
– Each bit presents load of 3 unit-sized transistors
– True and complementary address inputs A[3:0]
– Each input may drive 10 unit-sized transistors
❑ Ben needs to decide:
– How many stages to use?
– How large should each gate be?
– How fast can decoder operate?
16
Register File
26
Number of Stages
❑ Decoder effort is mainly electrical and branching
Electrical Effort:
H = (32*3) / 10 = 9.6
Branching Effort:
B=8
❑ If we neglect logical effort (assume G = 1)
Path Effort:
F = GBH = 76.8
Number of Stages:
N = log4F = 3.1
❑ Try a 3-stage design
6: Logical Effort
CMOS VLSI Design 4th Ed.
27
Gate Sizes & Delay
Logical Effort:
Path Effort:
Stage Effort:
Path Delay:
Gate sizes:
A[3] A[3]
10
10
A[2] A[2]
10
10
A[1] A[1]
10
10
G = 1 * 6/3 * 1 = 2
F = GBH = 154
fˆ = F1/3 = 5.36
D = 3 fˆ + 1 + 4 + 1 = 22.1
z = 96*1/5.36 = 18
y = 18*2/5.36 = 6.7
A[0] A[0]
10
10
y
z
word[0]
96 units of wordline capacitance
y
6: Logical Effort
z
word[15]
CMOS VLSI Design 4th Ed.
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Comparison
❑ Compare many alternatives with a spreadsheet
❑ D = N(76.8 G)1/N + P
Design
NOR4
N
1
G
3
P
4
D
234
NAND4-INV
2
2
5
29.8
NAND2-NOR2
2
20/9
4
30.1
INV-NAND4-INV
NAND4-INV-INV-INV
3
4
2
2
6
7
22.1
21.1
NAND2-NOR2-INV-INV
4
20/9
6
20.5
NAND2-INV-NAND2-INV
4
16/9
6
19.7
INV-NAND2-INV-NAND2-INV
5
16/9
7
20.4
NAND2-INV-NAND2-INV-INV-INV
6
16/9
8
21.6
6: Logical Effort
CMOS VLSI Design 4th Ed.
29
Review of Definitions
Term
Stage
Path
number of stages
1
N
logical effort
g
G = gi
electrical effort
h = CCoutin
H=
branching effort
b=
effort
f = gh
F = GBH
effort delay
f
DF =  fi
parasitic delay
p
delay
d= f +p
6: Logical Effort
Con-path +Coff-path
Con-path
CMOS VLSI Design 4th Ed.
Cout-path
Cin-path
B = bi
P =  pi
D = di = DF + P
30
Method of Logical Effort
1)
2)
3)
4)
5)
Compute path effort
Estimate best number of stages
Sketch path with N stages
Estimate least delay
Determine best stage effort
D = NF + P
ˆf = F N1
1
N
giCouti
Cini =
fˆ
6) Find gate sizes
6: Logical Effort
F = GBH
N = log4 F
CMOS VLSI Design 4th Ed.
31
Limits of Logical Effort
❑ Chicken and egg problem
– Need path to compute G
– But don’t know number of stages without G
❑ Simplistic delay model
– Neglects input rise time effects
❑ Interconnect
– Iteration required in designs with wire
❑ Maximum speed only
– Not minimum area/power for constrained delay
6: Logical Effort
CMOS VLSI Design 4th Ed.
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Summary
❑ Logical effort is useful for thinking of delay in circuits
– Numeric logical effort characterizes gates
– NANDs are faster than NORs in CMOS
– Paths are fastest when effort delays are ~4
– Path delay is weakly sensitive to stages, sizes
– But using fewer stages doesn’t mean faster paths
– Delay of path is about log4F FO4 inverter delays
– Inverters and NAND2 best for driving large caps
❑ Provides language for discussing fast circuits
– But requires practice to master
6: Logical Effort
CMOS VLSI Design 4th Ed.
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