Download MOS Inverter II

Digital Integrated
Circuits
A Design Perspective
Jan M. Rabaey
Anantha Chandrakasan
Borivoje Nikolic
The Inverter
Revised from Digital Integrated Circuits, © Jan M. Rabaey el, 2003
© Digital Integrated Circuits2nd
Inverter
Propagation Delay
© Digital Integrated Circuits2nd
Inverter
CMOS Inverter Propagation Delay
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
© Digital Integrated Circuits2nd
t
Inverter
MOS transistor model for simulation
G
CGS
CGD
D
S
CGB
CSB
CDB
B
© Digital Integrated Circuits2nd
Inverter
Computing the Capacitances
Consider each capacitor individually is almost impossible for
manual analysis
VDD
VDD
M2
Vin
Cg4
Cdb2
Cgd12
M4
Vout
Cdb1
Cw
M1
Vout2
Cg3
M3
Interconnect
Fanout
Simplified
Model
© Digital Integrated Circuits2nd
Vin
Vout
CL
Inverter
Computing the Capacitances
 NMOS and PMOS transistor are either in cutoff or
saturation mode during at least the first half (50%) of the
output transient.
 So, the only contributions to Cgd are the overlap
capacitance, since channel capacitance occurs between
either Gate-Body for transistors in cutoff region or GateSource for transistors in saturation region.
© Digital Integrated Circuits2nd
Inverter
The Miller Effect
Cgd1
V
Vout
Vout
V
Vin
M1
V
2Cgd1
M1
V
Vin
“A capacitor experiencing identical but opposite voltage swings
at both its terminals can be replaced by a capacitor to ground,
whose value is two times the original value.”
© Digital Integrated Circuits2nd
Inverter
The Miller Effect

Consider the situation that an impedance is connected
between input and output of an amplifier
 The same current flows from (out) the top input terminal if an
impedance Z in,Miller is connected across the input terminals
 The same current flows to (in) the top output terminal if an
impedance Z out, Miller is connected across the output terminal
 This is know as Miller Effect
 Two important notes to apply Miller Effect:
 There should be a common terminal for input and output
 The gain in the Miller Effect is the gain after connecting feedback
impedance Z f
Inverter
© Digital Integrated Circuits2nd
Graphs from Prentice Hall
Computing the Capacitances
© Digital Integrated Circuits2nd
Inverter
CMOS Inverters
VDD
ADp=45 λ 2
PDp=19 λ
PMOS
9λ/2λ
In
Out
Metal1
Polysilicon
ADn=19 λ 2
PDn=15 λ
3λ/2λ
NMOS
© Digital Integrated Circuits2nd
GND
Inverter
Junction Capacitance
© Digital Integrated Circuits2nd
Inverter
Linearizing the Junction Capacitance
Replace non-linear capacitance by
large-signal equivalent linear capacitance
which displaces equal charge
over voltage swing of interest
© Digital Integrated Circuits2nd
Inverter
Capacitance parameters in 0.25mm
CMOS process
Cdb will be slightly
different in L-to-H and Hto-L, why?
(the pn junction
reverse bias
voltage range)
© Digital Integrated Circuits2nd
Cgd1
Cgd2
Cdb1
Cdb2
Cg3
Cg4
HL(fF)
0.23
0.61
0.66
1.50
0.76
2.28
LH(fF)
0.23
0.61
0.90
1.15
0.76
2.28
Inverter
Transient Response
Cgd directly couples the steep input change before the circuit can even start
to react to the changes at input (potential forward bias the pn junction)
3
2.5
?
Vout(V)
2
tp = 0.69 CL (Reqn+Reqp)/2
1.5
1
tpLH
tpHL
0.5
0
-0.5
0
0.5
1
1.5
t (sec)
© Digital Integrated Circuits2nd
2
2.5
-10
x 10
Inverter
Low-to-High and High-to-Low delay
 It
is desired to have identical propagation
delays for both rising and falling inputs.
 Equal
delay requires equal equivalent onresistance, thus equal current IDAST
(neglecting the channel length modulation)
 This
demands almost the same
requirements for a Vm at VDD/2. Why?
© Digital Integrated Circuits2nd
Inverter
Requirements for equal delay
2

VDSAT 
 (VDD  VT )VDSAT 

I DSAT

2 

V


' W
I Dn  k n ( ) n VDSATn  (VDD  VTn )  DSATn 
L
2 

VDSATp 

' W

I Dp  k p ( ) p VDSATp  (VDD  VTp ) 
L
2 

W
 k'
L
Assume VDD  VTp  VDSATp / 2
'
then
k p VDSATp (W / L) p
'
k n VDSATn (W / L) n

k pVDSATp
k nVDSATn
1
This is exactly the formerly defined parameter r (last lecture)
© Digital Integrated Circuits2nd
Inverter
Design for delay performance
 Keep capacitances small
• careful layout, e.g. to keep drain
diffusion as small as possible
 Increase transistor sizes
• watch out for self-loading! When
intrinsic capacitance starts to
dominate the extrinsic ones
 Increase VDD (????)
© Digital Integrated Circuits2nd
Inverter
Delay as a function of VDD
5.5
For fixed (W/L)
5
tp(normalized)
4.5
Recall a range
of low voltage
is able to give
even better
voltage transfer
characteristic
4
3.5
3
2.5
2
1.5
1
0.8
1
1.2
1.4
1.6
V
1.8
2
2.2
2.4
(V)
DD
© Digital Integrated Circuits2nd
Inverter
Device Sizing
-11
3.8
x 10
(for fixed load and VDD)
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
6
© Digital Integrated Circuits2nd
8
S
W/L
10
12
14
Inverter
NMOS/PMOS ratio
-11
5
x 10
tpHL
tpLH
4.5
tp(sec)
(Average)
tp
b= Wp/Wn
4
Lp=Ln
3.5
3
1
1.5
2
2.5
3
3.5
4
4.5
5
b
Widening PMOS improves the L-H delay by increasing the charge current,
but it also degrades the H-L by giving a larger parasitic capacitance.
Considering average is more meaningful!!
© Digital Integrated Circuits2nd
Inverter
Delay Definitions
© Digital Integrated Circuits2nd
Inverter
Impact of Rise Time on Delay
0.35
tpHL(nsec)
0.3
0.25
0.2
0.15
0
© Digital Integrated Circuits2nd
0.2
0.4
0.6
trise (nsec)
0.8
1
Inverter
Custom design
process:
An inverter design
example
© Digital Integrated Circuits2nd
Inverter
1. Schematic design
© Digital Integrated Circuits2nd
Inverter
2. Layout design
© Digital Integrated Circuits2nd
Inverter
Step 1: define Nwell (for Pmos)
© Digital Integrated Circuits2nd
Inverter
Step 2: define pselect (for Pmos)
© Digital Integrated Circuits2nd
Inverter
Step 3: define active region (for Pmos)
© Digital Integrated Circuits2nd
Inverter
Step 4: define poly (gate for Pmos)
© Digital Integrated Circuits2nd
Inverter
Step 5: define contacts (for Pmos)
© Digital Integrated Circuits2nd
Inverter
Step 6: define Vdd and connect source (of Pmos) to it
© Digital Integrated Circuits2nd
Inverter
Step 7: make at least one Nwell contact
© Digital Integrated Circuits2nd
Inverter
Step 8: create Nmos (repeat similar steps before
except you do not need make Nwell)
© Digital Integrated Circuits2nd
Inverter
step9: make input and output connections
© Digital Integrated Circuits2nd
Inverter
2. DRC (Design Rule Check)
© Digital Integrated Circuits2nd
Inverter
Correct error if there is any
© Digital Integrated Circuits2nd
Inverter
After correction
© Digital Integrated Circuits2nd
Inverter
3. LVS (Layout versus Schematic)
© Digital Integrated Circuits2nd
Inverter
4. Extract Layout parasitics and post-layout simulation
© Digital Integrated Circuits2nd
Inverter
CMOS Inverter
N Well
VDD
VDD
PMOS
2l
Contacts
PMOS
In
Out
In
Out
Metal 1
Polysilicon
NMOS
NMOS
GND
© Digital Integrated Circuits2nd
Inverter
Two Inverters
Layout preference:
Share power and ground
Abut cells
VDD
Connect in Metal
© Digital Integrated Circuits2nd
Inverter
Impact of Process Variations (DFM)
2.5
2
Good PMOS
Bad NMOS
1.5
out
(V)
Nominal
V
Good NMOS
Bad PMOS
1
0.5
0
0
0.5
1
1.5
2
2.5
V (V)
© Digital Integrated Circuits2nd
in
Inverter
Inverter Sizing
for delay
© Digital Integrated Circuits2nd
Inverter
Inverter with Load
W means the size is increased by a factor of W with
respect to the minimum size
Delay
3W
W
Cint
CL
Load
Delay = kRW(Cint + CL) = kRWCint + kRWCL
= Delay (Internal) + Delay (Load)
= kRW Cint(1+ CL /Cint)
© Digital Integrated Circuits2nd
Inverter
Delay as function of size
Delay = kRW Cint(1+ CL /Cint)
= Delay (Internal) + Delay (Load)
RW = Runit / W ; Cint = W Cunit
t p  t p 0 (1  C L /(WCunit ))
tp0 = 0.69RunitCunit
• Intrinsic delay is fixed and independent of sizeW
• Making W large yields better performance gain, eliminating the
impact of external load and reducing the delay to intrinsic only. But
smaller gain at penalty of silicon area if W is too large!
© Digital Integrated Circuits2nd
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 for modern technology (see page199
Cgin : input gate capacitance
CL = f Cgin - effective fanout
This formula maps the intrinsic capacitor and load
capacitor as functions of a common capacitor,
which is the gate capacitance of the minimum-size
inverter
© Digital Integrated Circuits2nd
Inverter
Single inverter versus inverter chain
 Gate sizing for an isolated gate is
not really meaningful. Realistic chips
always have a long chain of gates.
 So, a more relevant and realistic
problem is to determine the optimal
sizing for a chain of gates.
© Digital Integrated Circuits2nd
Inverter
Inverter Chain
In
Out
CL
If CL is given:
- How many stages are needed to minimize the delay?
- How to size the inverters?
© Digital Integrated Circuits2nd
Inverter
Apply to Inverter Chain (fixed N stages)
Unit size (minimum size) inverter
Size ?
In
1
2
Size ?
N
Out
CL
tp = tp1 + tp2 + …+ tpN
 C gin, j 1 

t pj ~ RunitCunit 1 
 C

gin
,
j


N
N 
C gin, j 1

t p   t p, j  t p0  1 
 C
j 1
i 1 
gin, j
© Digital Integrated Circuits2nd

, C gin, N 1  C L


Inverter
Optimal Tapering for Given N
 Delay equation has N - 1 unknowns, Cgin,2 – Cgin,N
 Minimize the delay, find N - 1 partial derivatives equated to 0
 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
- each stage has the same delay
© Digital Integrated Circuits2nd
Inverter
Optimum Delay for fixed N stages
When each stage is sized by f and has same effective
fanout f:
f N  F  CL / Cgin,1
Effective fanout of each stage:
effective fanout
of the overall
circuit
f NF
Minimum path delay
N
t p   t p, j
j 1
© Digital Integrated Circuits2nd
 C gin, j 1
 t p 0  1 
 C
i 1 
gin, j
N

  Nt p 0 (1  N F /  )


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
© Digital Integrated Circuits2nd
Inverter
Optimum Number of Stages
For a given load CL and given input capacitance Cin
Find optimal sizing f
CL  F  Cin  f Cin
N
t p  Nt p 0 F
1/ N
ln F
with N 
ln f
t p 0 ln F  f


/   1 

  ln f ln f



t p
t p 0 ln F ln f  1   f


0
2
f

ln f
For  = 0, f = e, N = lnF
© Digital Integrated Circuits2nd
f  exp 1   f 
Inverter
Optimum Effective Fanout f
Optimum f for given process defined by 
f  exp 1   f 
© Digital Integrated Circuits2nd
fopt = 3.6 for =1
Inverter
Normalized delay function of F

t p  Nt p 0 1  N F / 
© Digital Integrated Circuits2nd

fopt = 4
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
1
N
64
© Digital Integrated Circuits2nd
22.6
Inverter