Download Chapter 6

Designing Combinational
Logic Circuits
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Introduction
Combinational vs. Sequential Logic
In
Combinational
Logic
Circuit
In
Out
Combinational
Logic
Circuit
Out
State
Combinational
Sequential
Output = f(In, Previous In)
Output = f(In)
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Static CMOS Circuit
At every point in time (except during the switching
transients) each gate output is connected to either
VDD or Vss via a low-resistive path.
The outputs of the gates assume at all times the value
of the Boolean function, implemented by the circuit
(ignoring, once again, the transient effects during
switching periods).
This is in contrast to the dynamic circuit class, which
relies on temporary storage of signal values on the
capacitance of high impedance circuit nodes.
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Static Complementary CMOS
VDD
In1
In2
PUN
InN
In1
In2
InN
PMOS only
F(In1,In2,…InN)
PDN
NMOS only
PUN and PDN are dual logic networks
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NMOS Transistors
in Series/Parallel Connection
Transistors can be thought as a switch controlled by its gate signal
NMOS switch closes when switch control input is high
A
B
X
Y
Y = X if A and B
A
X
B
Y = X if A OR B
Y
NMOS Transistors pass a “strong” 0 but a “weak” 1
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PMOS Transistors
in Series/Parallel Connection
PMOS switch closes when switch control input is low
A
B
X
Y
Y = X if A AND B = A + B
A
X
B
Y
Y = X if A OR B = AB
PMOS Transistors pass a “strong” 1 but a “weak” 0
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Threshold Drops
VDD
PUN
VDD
S
D
VDD
D
0  VDD
VGS
S
CL
CL
VDD  0
PDN
D
VDD
0  VDD - VTn
CL
S
VGS
VDD  |VTp|
S
CL
D
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Complementary CMOS Logic Style
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Example Gate: NAND
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Example Gate: NOR
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Complex CMOS Gate
B
A
C
D
OUT = D + A • (B + C)
A
D
B
C
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Constructing a Complex Gate
VDD
VDD
C
F
SN4
F
SN1
A
SN3
D
B
C
B
SN2
A
D
A
B
D
C
F
(a) pull-down network
(b) Deriving the pull-up network
hierarchically by identifying
sub-nets
A
D
B
C
(c) complete gate
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Cell Design

Standard Cells




General purpose logic
Can be synthesized
Same height, varying width
Datapath Cells



For regular, structured designs (arithmetic)
Includes some wiring in the cell
Fixed height and width
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Standard Cell Layout
Methodology – 1980s
Routing
channel
VDD
signals
GND
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Standard Cell Layout
Methodology – 1990s
Mirrored Cell
No Routing
channels
VDD
VDD
M2
M3
GND
Mirrored Cell
GND
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Standard Cells
N Well
VDD
Cell height 12 metal tracks
Metal track is approx. 3 + 3
Pitch =
repetitive distance between objects
Cell height is “12 pitch”
2
Cell boundary
In
Out
GND
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Rails ~10
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Standard Cells
With minimal
diffusion
routing
VDD
With silicided
diffusion
VDD
VDD
M2
In
Out
In
Out
In
Out
M1
GND
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GND
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Standard Cells
2-input NAND gate
VDD
VDD
B
A
B
Out
A
GND
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Stick Diagrams
Contains no dimensions
Represents relative positions of transistors
VDD
VDD
Inverter
NAND2
Out
Out
In
GND
GND
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A B
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Stick Diagrams
Logic Graph
A
j
X
C
C
B
A
i
X
X = C • (A + B)
C
PUN
i
B
VDD
j
B
GND
A
B
C
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A
PDN
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Two Versions of C • (A + B)
A
C
B
A
B
C
VDD
VDD
X
X
GND
GND
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Consistent Euler Path
X
C
i
X
B
VDD
j
GND
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A B C
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OAI22 Logic Graph
A
C
B
D
X
C
X = (A+B)•(C+D)
C
D
A
B
D
VDD
X
B
A
B
C
D
A
GND
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PUN
PDN
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Example: x = ab+cd
x
x
c
b
VDD
x
a
c
b
VD D
x
a
d
GND
d
GND
(a) Logic graphs for (ab+cd)
(b) Euler Paths {a b c d}
VD D
x
GND
a
b
c
d
(c) stick diagram for ordering {a b c d}
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Properties of Complementary CMOS
Gates Snapshot
•High noise margins:
V OH and VOL are at VDD and GND, respectively.
•No static power consumption:
There never exists a direct path between VDD and
VSS (GND) in steady-state mode.
•Comparable rise and fall times:
(under appropriate sizing conditions)
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CMOS Properties






Full rail-to-rail swing; high noise margins
Logic levels not dependent upon the relative
device sizes; ratioless
Always a path to Vdd or Gnd in steady state; low
output impedance
Extremely high input resistance; nearly zero
steady-state input current
No direct path steady state between power and
ground; no static power dissipation
Propagation delay function of load capacitance
and resistance of transistors
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Switch Delay Model
Req
A
A
Rp
A
Rp
Rp
B
Rn
Rp
Rp
A
CL
A
Cint
A
NAND2
Cint
A
Rn
B
Rn
B
INV
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CL
Rn
Rn
A
B
CL
NOR2
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Input Pattern Effects on Delay

Rp
A
Rp

B
Delay is dependent on the
pattern of inputs
Low to high transition

Rn
CL

B
Rn
A
Cint

both inputs go low
 delay is 0.69 Rp/2 CL
one input goes low
 delay is 0.69 Rp CL
High to low transition

both inputs go high
 delay is 0.69 2Rn CL
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Delay Dependence on Input Patterns
3
Input Data
Pattern
Delay
(psec)
A=B=01
67
A=1, B=01
64
A= 01, B=1
61
0.5
A=B=10
45
0
A=1, B=10
80
A= 10, B=1
81
A=B=10
2.5
Voltage [V]
2
A=1 0, B=1
1.5
A=1, B=10
1
-0.5
0
100
200
300
time [ps]
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NMOS = 0.5m/0.25 m
PMOS = 0.75m/0.25 m
CL = 100 fF
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Transistor Sizing
Rp
2 A
Rp
B
Rn
2
B
2
Rn
Rp
4 B
2
CL
Rp
4
Cint
A
Cint
1
A
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Rn
Rn
A
B
CL
1
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Transistor Sizing a Complex
CMOS Gate
A
B
8 6
C
8 6
4 3
D
4 6
OUT = D + A • (B + C)
A
D
2
1
B
2C
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Fan-In Considerations
A
B
C
D
A
CL
B
C3
C
C2
D
C1
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.
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tp as a Function of Fan-In
1250
quadratic
tp (psec)
1000
Gates with a
fan-in
greater than
4 should be
avoided.
750
tpH
500
tp
L
250
tpL
linear
H
0
2
4
6
8
10
12
14
16
fan-in
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tp as a Function of Fan-Out
tpNOR2
tpNAND2
tp (psec)
tpINV
2
4
6

C 
Delay inv  k  RunitCunit 1  L 
 Cin 
Delay gate    p  g  f 
8
10
12
14
16
All gates
have the
same drive
current.
Slope is a
function of
“driving
strength”
eff. fan-out
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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
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Sizing Logic Paths for Speed





Frequently, input capacitance of a logic path is
constrained
Logic also has to drive some capacitance
Example: ALU load in an Intel’s microprocessor is
0.5pF
How do we size the ALU datapath to achieve
maximum speed?
We have already solved this for the inverter chain –
can we generalize it for any type of logic?
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Buffer example
In
Out
1
2
N
CL
 C gin, j 1 

t pj ~ RunitCunit 1 
 C

gin, j 

N
N 
C gin, j 1 
, C gin, N 1  C L
t p   t p , j  t p 0  1 

C gin, j 
j 1
i 1 
For given N: Ci+1/Ci = Ci/Ci-1
To find N: Ci+1/Ci ~ 4
How to generalize this to any logic path?
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Logical Effort

CL 

Delay  k  RunitCunit 1 
 Cin 
 p  g  f 
p – intrinsic delay (3kRunitCunit) - gate parameter  f(W)
g – logical effort (kRunitCunit) – gate parameter  f(W)
f – effective fanout
Normalize everything to an inverter:
ginv =1, pinv = 1
Divide everything by inv
(everything is measured in unit delays inv)
Assume  = 1.
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Delay in a Logic Gate
Gate delay:
d=h+p
effort delay
intrinsic delay
Effort delay:
h=gf
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
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Logical Effort



Inverter has the smallest logical effort and intrinsic
delay of all static CMOS gates
Logical effort of a gate presents the ratio of its input
capacitance to the inverter capacitance when sized
to deliver the same current
Logical effort increases with the gate complexity
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Logical Effort
Logical effort is the ratio of input capacitance of a gate to the input
capacitance of an inverter with the same output current
VDD
A
VDD
A
2
2
B
F
2
F
A
A
VDD
B
4
A
4
2
F
1
A
B
Inverter
g=1
1
B
1
2
2-input NAND
g = 4/3 EE141
2-input NOR
g = 5/3
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Normalized delay (d)
Logical Effort of Gates
t pNAND
g=
p=
d=
t pINV
g=
p=
d=
F(Fan-in)
1
2
3
4
5
Fan-out (h)
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Normalized delay (d)
Logical Effort of Gates
t pNAND
g = 4/3
p=2
d = (4/3)f+2
t pINV
g=1
p=1
d = f+1
F(Fan-in)
1
2
3
4
5
Fan-out (f)
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4/
3;
p
=
2
Logical Effort of Gates
=
D:
g
tN
AN
=
g
r:
e
t
er
1;
p=
1
v
in
3
pu
4
In
2-
Normalized Delay
5
Effort
Delay
2
1
Intrinsic
Delay
1
2
3
Fanout
f
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5
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Add Branching Effort
Branching effort:
b
Con path  Coff  path
Con path
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Multistage Networks
N
Delay    pi  g i  f i 
i 1
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
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Optimum Effort per Stage
When each stage bears the same effort:
hN  H
hN H
Stage efforts: g1f1 = g2f2 = … = gNfN
Effective fanout of each stage: f i  h g i
Minimum path delay
Dˆ   gi f i  pi   NH 1/ N  P
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Logical Effort
From Sutherland, Sproull
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Example: Optimize Path
1
g=1
f=a
b
a
g = 5/3
f = b/a
c
5
g = 5/3
f = c/b
g=1
f = 5/c
Effective fanout, F =
G=
H=
h=
a=
b=
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Example: Optimize Path
1
g=1
f=a
b
a
g = 5/3
f = b/a
c
5
g = 5/3
f = c/b
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
g=1
f = 5/c
fi  h gi
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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.
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Fast Complex Gates:
Design Technique 1

Transistor sizing




It decreases serial resistance
But, it increases parasit capacitance as well
as long as fan-out capacitance dominates
Progressive sizing
InN
CL
MN
In3
M3
C3
In2
M2
C2
In1
M1
C1
Distributed RC line
M1 > M2 > M3 > … > MN
(the fet closest to the
output is the smallest)
Can reduce delay by more than
20%; decreasing gains as
technology shrinks
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Fast Complex Gates:
Design Technique 2

Transistor ordering
critical path
In3 1 M3
critical path
01
In1
M3
charged
CL
In2 1 M2
C2 charged
In1
M1
01
C1 charged
delay determined by time to
discharge CL, C1 and C2
CLcharged
In2 1 M2
C2 discharged
In3 1 M1
C1 discharged
delay determined by time to
discharge CL
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Fast Complex Gates:
Design Technique 3

Alternative logic structures
F = ABCDEFGH
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Fast Complex Gates:
Design Technique 4

Isolating fan-in from fan-out using buffer
insertion
CL
CL
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Fast Complex Gates:
Design Technique 5

Reducing the voltage swing
tpHL = 0.69 (3/4 (CL VDD)/ IDSATn )
= 0.69 (3/4 (CL Vswing)/ IDSATn )




linear reduction in delay
also reduces power consumption
But the following gate is much slower!
Or requires use of “sense amplifiers” on the
receiving end to restore the signal level (memory
design)
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