1

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/micro-

architectural– Logical– Circuit and– Layout levels.

• Delay estimation is essential in the design of critical paths.

• Some parameters of note in delay estimation include:– 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.

2

Delay Time and Gate Delays

• 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.

• 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:

321

1111

ppp

eff

βββ

β++

=

3

Gate Delays

• 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.

βeff =βn 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:

• The gain factor of the three series p-type transistors is given by:

• The delay through this series connection is therefore given by:

• If all three parallel n-type devices are conducting we have that:

n

Lf

DDp

Lr

Ckt

V

Ckt

ββ == and

3

3p

series

ββ =

DDp

Lseries

V

Ck

3

βτ =

3f

f

tt =

4

RC Delays

• Transistors have complex non-linear 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.

5

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.

6

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.

7

Delay Estimation (RC Models)

8

The Elmore Delay Model

• Transistors that are conducting must be viewed as resistors.

• The Elmore delay estimates the delay of 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.

VDD

VinR3R1 R2 RN

C2 C3C1 CN

9

AND Gate Intrinsic Capacitance

10

Two Input NAND Gate

11

CMOS-Gate Transistor Sizing

• 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.

• If Wp = 2Wn the delay response for an inverter pair:

• The above expression can be compared to one for equal sized inverter devices. The inverter pair delay becomes:

eqpairinv

eqeqpairinv

risefallpairinv

RCt

CR

CRt

ttt

6

32

23

_

_

_

∝

+∝

+∝

eqpairinv

eqeqpairinv

risefallpairinv

RCt

CRCRt

ttt

6

222

_

_

_

∝

+∝

+∝

12

Stage Ratio

• 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)

• 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.