Transcript Delay Time and Gate Delays

Delay Estimation
• Most digital designs have
multiple data paths some of
which are not critical.
• The critical path is defined as
the path the offers the worst
case delay.
• Several factors affect a design’s
delay analysis. These could be
at:
– Architectural/microarchitectural
– Logical
– Circuit and
– Layout levels.
• Delay estimation is essential in
the design of critical paths.
• Some parameters of note in
delay estimation include:
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Rise time
Fall time
Average delay (edge rate).
Propagation delay
Contamination delay
• When input changes the output
maintains its old value for a
duration called the
contamination time.
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Delay Time and Gate Delays
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In most CMOS circuits the delay of
a single gate is dominated by the
rate at which the output node can
be charged and discharged.
The delay can be approximated by:
tdr = tr/2 and tdf = tf/2
The average delay for rising and
falling output transitions is:
tav = (tdf+tdr)/2
The delay equations presented use
only first order MOS equations for
the calculations of drain currents
and thus do not account for second
order effects.
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The delay of simple gates maybe
approximated by constructing an
equivalent inverter. WHY?
Consider a 3 input NOR gate with
Wp = Wn for all transistors:
– When there is a path in the pull-up
network from the output to VDD
the effective gain factor of the
series p-type transistors is:
 eff 
1
 p1

1
1
 p2

1
 p3
2
Gate Delays
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
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For the pull-down network only
one n-type transistor need to be
on in order for us to have a path
from the output node to ground.
eff =n and this gain factor is
improved by a factor of three if
all the n-type devices conduct
simultaneously.
For the NOR gate example we
can thus estimate the rise and fall
times as follows:
tr  k
CL
p
3
VDD
and t f  k
CL
n
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The gain factor of the three series
p-type transistors is given by:
 series 
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p
3
The delay through this series
connection is therefore given by:
 series  k
CL
p
3
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VDD
If all three parallel n-type devices
are conducting we have that:
tf 
tf
3
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RC Delays
• Transistors have complex nonlinear current-voltage
characteristics, but can be fairly
approximated as a switch in
series with a resistor.
• The effective resistance is
chosen to match the amount of
current delivered by the
transistor.
• The transistor gates and the
diffusion nodes have
capacitance.
• An nMOS with and effective
width of one unit has resistance
R. The unit-width pMOS has
higher resistance that depends
on the mobility of holes and we
will say 2R.
• If we double the unit-width of
the pMOS so that it delivers the
same current as a unit-width
nMOS we end up with
resistance R for the pMOS.
• Parallel and series transistors
combine like resistors.
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Effective Capacitance and Resistance
• Wider transistors have lower
resistance.
• When multiple transistors are in
series their resistance is the sum
of each individual resistance.
• When transistors are in parallel
and are ON the resistance is
lowered.
• The capacitance consists of gate
capacitance (Cg) and
source/drain capacitance (Cdiff)
• We can approximate the
capacitances to be Cg=Cdiff=C.
• Cg and Cdiff are proportional to
the transistor width.
• The contacted diffusion has
higher capacitance than the
uncontacted diffusion. A 3-input
NAND gate has 2 uncontacted
diffusion terminals for the
series devices and a single
contact for the parallel pMOS
devices.
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RC Delay Model
• Our model will assume
minimum device sizes for
delay estimation
– A minimum sized nMOS
has resistance R
– Recall that in general the
mobility of electrons is
twice that of holes.
– We have thus designed
pMOS devices to have
twice the widths of nMOS
devices to attain symmetric
rise and fall times.
– This fact allows us to estimate
the resistance of a pMOS to be
2R.
• Transistors with increased
widths have reduced resistance
i.e. increase a minimum width
transistor by k the resistance
reduces to R/k.
• A pMOS device of double
width therefore has a resistance
value of 2R/2 = R.
• Parallel and series transistors
combine just like resistors in
parallel and resistors in series.
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Delay Estimation (RC Models)
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The Elmore Delay Model
• Transistors that are conducting must be viewed as resistors.
VDD
Vin
R1
R2
R3
RN
C2
C
• The Elmore delay estimates
the delayCof an RC ladder as
the sum over each node in the ladder resistance Rn-i
between that node and the source multiplied by C the
capacitance on the node.
C1
3
N
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AND Gate Intrinsic Capacitance
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Two Input NAND Gate
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CMOS-Gate Transistor Sizing
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It has been shown that to have
symmetric switching in an inverter
we need to make the width of the
p-type device (Wp) at least 2->3
times that of the n-type device
(Wn).
This approach increases the area
occupied by the p-type devices and
dynamic power dissipation.
Some structures can be cascaded to
use minimum or equal sized
devices without compromising the
switching response.
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If Wp = 2Wn the delay response for
an inverter pair:
tinv _ pair  t fall  t rise
tinv _ pair  R3Ceq  2
R
3Ceq
2
tinv _ pair  6 RC eq
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The above expression can be
compared to one for equal sized
inverter devices. The inverter pair
delay becomes:
tinv _ pair  t fall  t rise
tinv _ pair  R 2Ceq  2 R 2Ceq
tinv _ pair  6 RC eq
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Stage Ratio
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To drive large load capacitances
such as long buses, I/O buffers,
pads and off chip capacitive loads a
chain of inverters can be used.
With this configuration each
successive gate is made large than
the previous one until the last
inverter in the chain can drive the
large load within the required time.
To maintain shorter delays between
input and output, minimal area and
maintain power dissipation to a
minimum as well we can use the
stage ratio approach (increase each
stage)
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A cascade of n inverters with stage
ratio a driving a load capacitance
CL, with inverter 1 having
minimum-sized devices, driving
inverter 2 which is a times the size
of inverter 1.
Inverter 2 drives inverter 3 which is
a2 the size of inverter 1.
The delay through each stage is atd
with td being the delay of the
minimum sized inverter.
The delay through n stages is natd
If CL/Cg = R, then an = R, where Cg
is the gate capacitance of the
minimum sized inverter.
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