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EE4800 CMOS Digital IC Design & Analysis
Lecture 10 Combinational Circuit Design
Zhuo Feng
10.1
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Outline
■ Bubble Pushing
■ Compound Gates
■ Logical Effort Example
■ Input Ordering
■ Asymmetric Gates
■ Skewed Gates
■ Best P/N ratio
■ Pseudo-nMOS Logic
■ Dynamic Logic
■ Pass Transistor Logic
10.2
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Example 1
1) Sketch a design using NAND, NOR, and NOT gates.
Assume ~S is available.
D0
S
Y
D1
S
10.3
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Bubble Pushing
■ Start with network of AND / OR gates
■ Convert to NAND / NOR + inverters
■ Push bubbles around to simplify logic
► Remember DeMorgan’s Law
Y
Y
(a)
(b)
Y
(c)
10.4
D
Y
(d)
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Example 2
2) Sketch a design using one compound gate and
one NOT gate. Assume ~S is available.
D0
S
D1
S
10.5
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
Compound Gates
■ Logical Effort of compound gates
unit inverter
AOI21
YA
Y  A BC
A
Y
A
B
C
A
A
2
1
Y
AOI22
Y  A BC D
Y
4 B
C
A
2
B
2
4
4
C
Y
1
Complex AOI
A
B
C
D
Y
A
4 B
4
C
4 D
4
A
2 C
2
B
2 D
2
Y
Y  A B  C  D E
D
E
A
B
C
Y
B
6
C
6
A
3
D
6
E
6
E
2
A
2
D
2
2
C
gA = 3/3
gA = 6/3
gA =
gA =
p = 3/3
gB = 6/3
gB =
gB =
gC = 5/3
gC =
gC =
p = 7/3
gD =
gD =
p=
gE =
p=
10.6
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
B
Y
2
Compound Gates
■ Logical Effort of compound gates
unit inverter
AOI21
YA
Y  A BC
A
Y
A
B
C
A
A
2
1
Y
AOI22
Y  A BC D
Y
4 B
C
A
2
B
2
4
4
C
Y
1
Complex AOI
A
B
C
D
Y
A
4 B
4
C
4 D
4
A
2 C
2
B
2 D
2
Y
Y  A B  C  D E
D
E
A
B
C
Y
B
6
C
6
A
3
D
6
E
6
E
2
A
2
D
2
2
C
B
gA = 3/3
gA = 6/3
gA = 6/3
gA = 5/3
p = 3/3
gB = 6/3
gB = 6/3
gB = 8/3
gC = 5/3
gC = 6/3
gC = 8/3
p = 7/3
gD = 6/3
gD = 8/3
p = 12/3
gE = 8/3
p = 16/3
10.7
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
2
Example 3
■ The multiplexer has a maximum input capacitance of 16
units on each input. It must drive a load of 160 units.
Estimate the delay of the NAND and compound gate
designs.
D0
S
Y
D1
S
D0
S
D1
S
H = 160 / 16 = 10
B=1
N=2
10.8
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
NAND Solution
P 224
G  (4 / 3) (4 / 3)  16 / 9
F  GBH  160 / 9
fˆ  N F  4.2
D0
S
Y
D1
S
D  Nfˆ  P  12.4
10.9
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Compound Solution
P  4 1  5
G  (6 / 3) (1)  2
F  GBH  20
fˆ  N F  4.5
D0
S
D1
S
D  Nfˆ  P  14
10.10
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
Example 4
■ Annotate your designs with transistor
sizes that achieve this delay.
8
8
8
8
25
25
25
8
8
Y
25
10
10
10
10
24
6
6
12
6
6
Y
8
8
16
10.11
160 * (4/3) / 4.2 = 50
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
16
160 * 1 / 4.5 = 36
Input Order
■ Our parasitic delay model was too simple
► Calculate parasitic delay for Y falling
▼ If A arrives latest? 2
▼ If B arrives latest? 2.33
2
10.12
2
A
2
B
2x
Y
6C
2C
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Inner & Outer Inputs
■ Outer input is closest to rail (B)
■ Inner input is closest to output (A)
■ If input arrival time is known
► Connect latest input to inner terminal
10.13
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
2
2
A
2
B
2
Y
Asymmetric Gates
■ Asymmetric gates favor one input over another
■ Ex: suppose input A of a NAND gate is most critical
► Use smaller transistor on A (less capacitance)
► Boost size of noncritical input
► So total resistance is same
A
reset
Y
2
A
■ gA = 10/9
■ gB = 2
reset
2
Y
4/3
4
■ gtotal = gA + gB = 28/9
■ Asymmetric gate approaches g = 1 on critical input
■ But total logical effort goes up
10.14
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Symmetric Gates
■ Inputs can be made perfectly symmetric
10.15
2
2
A
1
1
B
1
1
Y
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Skewed Gates
■ Skewed gates favor one edge over another
■ Ex: suppose rising output of inverter is most critical
► Downsize noncritical nMOS transistor
HI-skew
inverter
unskewed inverter
(equal rise resistance)
2
A
unskewed inverter
(equal fall resistance)
2
Y
1/2
A
1
Y
A
1
Y
1/2
■ Calculate logical effort by comparing to unskewed inverter
with same effective resistance on that edge.
► gu = 2.5 / 3 = 5/6
► gd = 2.5 / 1.5 = 5/3
10.16
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
HI- and LO-Skew
■ Def: Logical effort of a skewed gate for a particular
transition is the ratio of the input capacitance of that gate
to the input capacitance of an unskewed inverter
delivering the same output current for the same transition.
■ Skewed gates reduce size of noncritical transistors
► HI-skew gates favor rising output (small nMOS)
► LO-skew gates favor falling output (small pMOS)
■ Logical effort is smaller for favored direction
■ But larger for the other direction
10.17
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Catalog of Skewed Gates
Inverter
NAND2
2
unskewed
1
Y
gu = 1
gd = 1
gavg = 1
A
2
B
2
2
HI-skew
1/2
Y
gu = 5/6
gd = 5/3
gavg = 5/4
1
B
1
1
LO-skew
10.18
A
1
Y
gu = 4/3
gd = 2/3
gavg = 1
2
B
2
4
1
1
B
4
A
4
1/2
gu = 1
gd = 2
gavg = 3/2
1
A
A
gu = 5/3
gd = 5/3
gavg = 5/3
Y
Y
1
4
gu = 4/3
gd = 4/3
gavg = 4/3
2
A
B
Y
Y
2
A
2
Y
2
A
NOR2
1/2
B
2
A
2
gu = 3/2
gd = 3
gavg = 9/4
Y
gu = 2
gd = 1
gavg = 3/2
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
1
1
gu = 2
gd = 1
gavg = 3/2
Asymmetric Skew
■ Combine asymmetric and skewed gates
► Downsize noncritical transistor on unimportant input
► Reduces parasitic delay for critical input
A
reset
Y
1
A
reset
10.19
2
Y
4/3
4
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Best P/N Ratio
■ We have selected P/N ratio for unit rise and fall
resistance (m = 2-3 for an inverter).
■ Alternative: choose ratio for least average delay
■ Ex: inverter
► Delay driving identical inverter
P
► tpdf = (P+1)
A
► tpdr = (P+1)(m/P)
► tpd = (P+1)(1+m/P)/2 = (P + 1 + m + m/P)/2
► Differentiate tpd w.r.t. P
► Least delay for P =
10.20
m
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
1
P/N Ratios
■ In general, best P/N ratio is sqrt of that giving
equal delay.
► Only improves average delay slightly for inverters
► But significantly decreases area and power
Inverter
NAND2
2
fastest
P/N ratio
10.21
A
1.414
Y
1
gu = 1.15
gd = 0.81
gavg = 0.98
NOR2
2
Y
A
2
B
2
B
2
A
2
Y
gu = 4/3
gd = 4/3
gavg = 4/3
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
1
1
gu = 2
gd = 1
gavg = 3/2
Observations
■ For speed:
► NAND vs. NOR
► Many simple stages vs. fewer high fan-in stages
► Latest-arriving input
■ For area and power:
► Many simple stages vs. fewer high fan-in stages
10.22
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
■ What makes a circuit fast?
-> tpd  (C/I) DV
► low capacitance
► high current
► small swing
► I = C dV/dt
B
4
A
4
■ Logical effort is proportional to C/I
■ pMOS are the enemy!
► High capacitance for a given current
■ Can we take the pMOS capacitance off the
input?
■ Various circuit families try to do this…
10.23
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
1
1
Pseudo-nMOS
■ In the old days, nMOS processes had no pMOS
► Instead, use pull-up transistor that is always ON
■ In CMOS, use a pMOS that is always ON
► Ratio issue
► Make pMOS about ¼ effective strength of pulldown network
1.8
1.5
load
P/2
1.2
P = 24
Ids
Vout 0.9
Vout
16/2
Vin
0.6
P = 14
0.3
P=4
0
0
0.3
0.6
0.9
Vin
10.24
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
1.2
1.5
1.8
Pseudo-nMOS Gates
■ Design for unit current on output
to compare with unit inverter.
■ pMOS fights nMOS
Y
inputs
f
Inverter
Y
A
10.25
NAND2
gu
gd
gavg
pu
pd
pavg
=
=
=
=
=
=
A
B
gu
g
Y gd
avg
pu
pd
pavg
NOR2
=
=
=
=
=
=
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
A
B
gu
gd
gavg
Y pu
pd
pavg
=
=
=
=
=
=
Pseudo-nMOS Gates
■ Design for unit current on output
to compare with unit inverter.
■ pMOS fights nMOS
Y
inputs
f
Inverter
2/3
Y
A
4/3
10.26
NAND2
gu
gd
gavg
pu
pd
pavg
= 4/3
= 4/9
= 8/9
= 6/3
= 6/9
= 12/9
A
B
gu
2/3
g
Y gd
avg
8/3
pu
pd
8/3
pavg
NOR2
= 8/3
= 8/9
= 16/9
= 10/3
= 10/9
= 20/9
2/3
A
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
4/3 B
gu
gd
gavg
Y pu
4/3 pd
pavg
= 4/3
= 4/9
= 8/9
= 10/3
= 10/9
= 20/9
Dynamic Logic
■ Dynamic gates uses a clocked pMOS pullup
■ Two modes: precharge and evaluate
2
A
Y
1
Static


2/3
Precharge
1
Y
A
4/3
Pseudo-nMOS
Y
A
Dynamic
Evaluate
Y
10.27
1
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Precharge
The Foot
■ What if pulldown network is ON during
precharge?
■ Use series evaluation transistor to prevent fight.
precharge transistor



Y
Y
inputs
A
Y
inputs
f
f
foot
footed
10.28
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
unfooted
Logical Effort
Inverter

unfooted
NAND2
1
gd
pd
footed
2
2
10.29
2
B
2

gd
pd
= 2/3
= 3/3
1
Y
gd
pd
= 2/3
= 3/3
1
A
B
1
gd
pd
= 1/3
= 3/3
1
Y
1
Y
A
A
= 1/3
= 2/3


1
Y
1
Y
A

NOR2
A
3
B
3
3

1
Y
gd
pd
= 3/3
= 4/3
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
A
2
B
2
2
gd
pd
= 2/3
= 5/3
Monotonicity
■ Dynamic gates require monotonically rising
inputs during evaluation
► 0 -> 0
► 0 -> 1

► 1 -> 1
A
► But not 1 -> 0
violates monotonicity
during evaluation
A

Precharge
Evaluate
Precharge
Y
Output should rise but does not
10.30
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Monotonicity Woes
■ But dynamic gates produce monotonically
falling outputs during evaluation
■ Illegal for one dynamic gate to drive another!
A=1

A
Y

Precharge
Evaluate
Precharge
X
X
X monotonically falls during evaluation
Y
Y should rise but cannot
10.31
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Domino Gates
■ Follow dynamic stage with inverting static gate
► Dynamic / static pair is called domino gate
► Produces monotonic outputs

domino AND
Precharge
Evaluate
Precharge
W
W
X
Y
Z
X
A
B
C

dynamic static
NAND inverter
Y
Z

A
B
10.32


W
X
H
C
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y
H
Z
=
A
B

X
C
Z
Domino Optimizations
■ Each domino gate triggers next one, like a string
of dominos toppling over
■ Gates evaluate sequentially but precharge in
parallel
■ Thus evaluation is more critical than precharge
■ HI-skewed static stages can perform logic

S0
S1
S2
S3
D0
D1
D2
D3
H
Y

10.33
S4
S5
S6
S7
D4
D5
D6
D7
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Dual-Rail Domino
■ Domino only performs noninverting functions:
► AND, OR but not NAND, NOR, or XOR
■ Dual-rail domino solves this problem
► Takes true and complementary inputs
► Produces true and complementary outputs
sig_h
sig_l
Meaning
0
0
Precharged
0
1
‘0’
1
0
‘1’
1
1
invalid
10.34

Y_l
inputs
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
f

Y_h
f
Example: AND/NAND
■ Given A_h, A_l, B_h, B_l
■ Compute Y_h = AB, Y_l = AB
■ Pulldown networks are conduction complements

Y_l
A_h
= A*B
A_l
B_l
B_h

10.35
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Y_h
= A*B
Example: XOR/XNOR
■ Sometimes possible to share transistors

Y_l
= A xnor B
A_h
Y_h
A_l
A_l
B_l
B_h

10.36
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
A_h
= A xor B
Leakage
■ Dynamic node floats high during
evaluation
► Transistors are leaky (IOFF  0)
► Dynamic value will leak away over time
► Formerly miliseconds, now nanoseconds
■ Use keeper to hold dynamic node
► Must be weak enough not to fight evaluation
weak keeper

A
1 k
X
H
Y
2
2
10.37
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Charge Sharing
■ Dynamic gates suffer from charge sharing


A
Y
CY
x
A
Y
B=0
Cx
Charge sharing noise
x
CY
Vx  VY 
VDD
C x  CY
10.38
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Secondary Precharge
■ Solution: add secondary precharge
transistors
► Typically need to precharge every other node
■ Big load capacitance CY helps as well

Y
A
secondary
precharge
transistor
x
B
10.39
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Noise Sensitivity
■ Dynamic gates are very sensitive to noise
► Inputs: VIH  Vtn
► Outputs: floating output susceptible noise
■ Noise sources
► Capacitive crosstalk
► Charge sharing
► Power supply noise
► Feedthrough noise
► And more!
10.40
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Power
■ Domino gates have high activity factors
► Output evaluates and precharges
▼ If output probability = 0.5, a = 0.5
– Output rises and falls on half the cycles
► Clocked transistors have a = 1
■ Leads to very high power consumption
10.41
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis
Domino Summary
■ Domino logic is attractive for high-speed circuits
► 1.3 – 2x faster than static CMOS
► But many challenges:
▼ Monotonicity, leakage, charge sharing, noise
■ Widely used in high-performance
microprocessors in 1990s when speed was king
■ Largely displaced by static CMOS now that
power is the limiter
■ Still used in memories for area efficiency
10.42
Z. Feng MTU EE4800 CMOS Digital IC Design & Analysis