1. 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. 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: t dr = t r /2 and t df = t f /2 The average delay for rising and falling output transitions is: t av = (t df +t dr )/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 W p = W n for all transistors: –When there is a path in the pull-up network from the output to V DD the effective gain factor of the series p-type transistors is:

3. 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.  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: 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:

4. 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. 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 (C g ) and source/drain capacitance (C diff ) We can approximate the capacitances to be C g =C diff =C. C g and C diff 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. 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. 7 Delay Estimation (RC Models)

8. 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 R n-i between that node and the source multiplied by C the capacitance on the node. V DD V in R3R3 R1R1 R2R2 RNRN C2C2 C3C3 C1C1 CNCN

9. 9 AND Gate Intrinsic Capacitance

10. 10 Two Input NAND Gate

11. 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 (W p ) at least 2->3 times that of the n-type device (W n ). 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 W p = 2W n 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:

12. 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 C L, 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 a 2 the size of inverter 1. The delay through each stage is at d with t d being the delay of the minimum sized inverter. The delay through n stages is nat d If C L /C g = R, then a n = R, where C g is the gate capacitance of the minimum sized inverter.