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Digital Integrated
Circuits
A Design Perspective
Jan M. Rabaey
Anantha Chandrakasan
Borivoje Nikolic
The Inverter
July 30, 2002
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1
Inverter
The CMOS Inverter: A First Glance
VDD
Vin
Vout
CL
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2
Inverter
CMOS Inverters
VDD
PMOS
2l
In
Out
Metal1
Polysilicon
NMOS
GND
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Inverter
Voltage Transfer
Characteristic
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Inverter
CMOS Inverter VTC
NMOS off
PMOS res
2.5
Vout
2
NMOS s at
PMOS res
1
1.5
NMOS sat
PMOS sat
0.5
NMOS res
PMOS sat
0.5
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1
1.5
2
NMOS res
PMOS off
2.5
Vin
5
Inverter
Switching Threshold as a function
of Transistor Ratio
1.8
1.7
1.6
1.5
M
V (V)
1.4
1.3
1.2
1.1
1
0.9
0.8
10
0
10
W /W
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p
n
1
6
Inverter
Impact of Process Variations
2.5
2
Good PMOS
Bad NMOS
Vout(V)
1.5
Nominal
1
Good NMOS
Bad PMOS
0.5
0
0
0.5
1
1.5
2
2.5
Vin (V)
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Inverter
Propagation Delay
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CMOS Inverter Propagation Delay
Approach 1
VDD
tpHL = CL Vswing/2
Iav
CL
Vout
~
CL
Iav
kn VDD
Vin = V DD
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Inverter
CMOS Inverter Propagation Delay
Approach 2
VDD
tpHL = f(Ron.CL)
= 0.69 RonCL
Vout
ln(0.5)
Vout
CL
Ron
1
VDD
0.5
0.36
Vin = V DD
RonCL
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t
10
Inverter
Transient Response
3
2.5
?
Vout(V)
2
tp = 0.69 CL (Reqn+Reqp)/2
1.5
1
tpHL
tpLH
0.5
0
-0.5
0
0.5
1
1.5
t (sec)
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2
2.5
-10
x 10
11
Inverter
Design for Performance
 Keep
capacitances small
 Increase transistor sizes
 watch out for self-loading!
 Increase
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VDD (????)
12
Inverter
Delay as a function of VDD
5.5
5
tp(normalized)
4.5
4
3.5
3
2.5
2
1.5
1
0.8
1
1.2
1.4
1.6
V
1.8
(V)
DD
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2
2.2
2.4
13
Inverter
Device Sizing
-11
3.8
x 10
(for fixed load)
3.6
3.4
tp(sec)
3.2
3
2.8
Self-loading effect:
Intrinsic capacitances
dominate
2.6
2.4
2.2
2
2
4
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Circuits2nd
8
S
10
12
14
14
Inverter
NMOS/PMOS ratio
-11
5
x 10
tpHL
tpLH
tp(sec)
4.5
b = Wp/Wn
tp
4
3.5
3
1
1.5
2
2.5
3
3.5
4
4.5
5
b
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Inverter
Inverter Sizing
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Inverter
Inverter Chain
In
Out
CL
If CL is given:
- How many stages are needed to minimize the delay?
- How to size the inverters?
May need some additional constraints.
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Inverter
Delay Formula
Delay ~ RW Cint  C L 
t p  kRW Cint 1  C L / Cint   t p 0 1  f /  
Cint = Cgin with   1
f = CL/Cgin - effective fanout
R = Runit/W ; Cint =WCunit
tp0 = 0.69RunitCunit
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Inverter
Apply to Inverter Chain
In
Out
1
2
N
CL
tp = tp1 + tp2 + …+ tpN
 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

j 1
i 1 
gin, j 
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Inverter
Optimal Tapering for Given N
Delay equation has N - 1 unknowns, Cgin,2 – Cgin,N
Minimize the delay, find N - 1 partial derivatives
Result: Cgin,j+1/Cgin,j = Cgin,j/Cgin,j-1
Size of each stage is the geometric mean of two neighbors
C gin, j  C gin, j 1C gin, j 1
- each stage has the same effective fanout (Cout/Cin)
- each stage has the same delay
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Inverter
Optimum Delay and Number of
Stages
When each stage is sized by f and has same eff. fanout f:
f N  F  CL / Cgin,1
Effective fanout of each stage:
f NF
Minimum path delay

t p  Nt p 0 1  N F / 
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
21
Inverter
Example
In
C1
Out
1
f
f2
CL= 8 C1
CL/C1 has to be evenly distributed across N = 3 stages:
f 38 2
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Inverter
Optimum Number of Stages
For a given load, CL and given input capacitance Cin
Find optimal sizing f
ln F
N
CL  F  Cin  f Cin with N 
ln f
t p 0 ln F  f


t p  Nt p 0 F /   1 

  ln f ln f
t p t p 0 ln F ln f  1   f


0
2
f

ln f
1/ N
For  = 0, f = e, N = lnF
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


f  exp 1   f 
23
Inverter
Optimum Effective Fanout f
Optimum f for given process defined by 
f  exp 1   f 
fopt = 3.6
for =1
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Inverter
Impact of Self-Loading on tp
No Self-Loading, =0
With Self-Loading =1
u/ln(u)
60.0
40.0
x=10,000
x=1000
20.0
x=100
x=10
0.0
1.0
3.0
5.0
7.0
u
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Inverter
Normalized delay function of F

t p  Nt p 0 1  N F / 
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
26
Inverter
Buffer Design
1
f
tp
1
64
65
2
8
18
64
3
4
15
64
4
2.8
15.3
64
1
8
1
4
16
2.8
8
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1
N
64
22.6
27
Inverter
Power Dissipation
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Inverter
Where Does Power Go in CMOS?
• Dynamic Power Consumption
Charging and Discharging Capacitors
• Short Circuit Currents
Short Circuit Path between Supply Rails during Switching
• Leakage
Leaking diodes and transistors
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Inverter
Dynamic Power Dissipation
Vdd
Vin
Vout
CL
Energy/transition = CL * Vdd2
Power = Energy/transition * f = CL * Vdd2 * f
Not a function of transistor sizes!
Need to reduce CL, Vdd, and f to reduce power.
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Inverter
Modification for Circuits with Reduced Swing
Vdd
Vdd
Vdd -Vt
CL
E0
1
= CL  Vdd   Vdd – Vt 
Can exploit reduced sw ing to low er power
(e.g., reduced bit-line swing in memory)
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Inverter
Adiabatic Charging
2
2
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32
Inverter
Adiabatic Charging
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Inverter
Node Transition Activity and Power
Consider switching a CMOS gate for N clock cycles
E N = CL  V dd2  n N 
EN : the energy consumed for N clock cycles
n(N ): the number of 0->1 transition in N clock cycles
EN
2
n N 
P avg = lim --------  fclk =  lim ----------- C  Vdd  f clk
N   N 
N N
L
0  1 =
n N 
lim -----------N N
P avg = 0 1  C  Vdd 2  f clk

L
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Inverter
Transistor Sizing for Minimum
Energy
In
Out
Cg1
 Goal:
1
f
Cext
Minimize Energy of whole circuit
 Design parameters: f and VDD
 tp  tpref of circuit with f=1 and VDD =Vref

f 
F 
t p  t p 0  1    1   
f  
   
VDD
t p0 
VDD  VTE
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Inverter
Transistor Sizing (2)

Performance Constraint (=1)
tp
t pref



F
 2  f  
f  VDD Vref  VTE


3  F 
Vref VDD  VTE
t p0
t p 0 ref

F
 2  f  
f 

1
3  F 
Energy for single Transition
2
E  VDD
C g1 1   1  f   F 
2
 VDD   2  2 f  F 
E
 




Eref  Vref   4  F 
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Inverter
Transistor Sizing (3)
VDD=f(f)
E/Eref=f(f)
4
1.5
3.5
F=1
normalized energy
3
2
vdd (V)
2.5
5
2
1.5
1
10
0.5
20
0
1
2
3
4
f
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5
6
7
1
0.5
0
1
2
3
4
5
6
7
f
37
Inverter
Short Circuit Currents
Vd d
Vin
Vout
CL
IVDD (mA)
0.15
0.10
0.05
0.0
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1.0
2.0
3.0
Vin (V)
4.0
5.0
38
Inverter
How to keep Short-Circuit Currents Low?
Short circuit current goes to zero if tfall >> trise,
but can’t do this for cascade logic, so ...
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Inverter
Minimizing Short-Circuit Power
8
7
6
Vdd =3.3
Pnorm
5
4
Vdd =2.5
3
2
1
Vdd =1.5
0
0
1
2
3
4
5
t /t
sin sout
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Inverter
Leakage
Vd d
Vout
Drain Junction
Leakage
Sub-Threshold
Current
Sub-threshold current one of most compelling issues
Sub-Threshold
in low-energy
circuitCurrent
design!Dominant Factor
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Inverter
Reverse-Biased Diode Leakage
GATE
p+
p+
N
Reverse Leakage Current
+
V
- dd
IDL = JS  A
2
JS = JS
1-5pA/
for a 1.2
technology
= 10-100
at 25
degCMOS
C for 0.25m
CMOS
mpA/m2
m
JS doubles for every 9 deg C!
Js double with every 9oC increase in temperature
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Inverter
Subthreshold Leakage Component
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Inverter
Static Power Consumption
Vd d
Istat
Vout
Vin =5V
CL
Pstat = P(In=1) .Vdd . Istat
Wasted •energy
… over dynamic consumption
Dominates
Should be avoided in almost all cases,
• Not a function of switching frequency
but could help reducing energy in others (e.g. sense amps)
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Inverter
Principles for Power Reduction
 Prime
choice: Reduce voltage!
 Recent years have seen an acceleration in
supply voltage reduction
 Design at very low voltages still open
question (0.6 … 0.9 V by 2010!)
 Reduce
switching activity
 Reduce physical capacitance
 Device Sizing: for F=20
– fopt(energy)=3.53, fopt(performance)=4.47
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Inverter