UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer SciencesElad Alon Homework #4 - Solutions EECS141 Due Thursday, September 24, 5pm, box in 240 Cory PROBLEM 1: Inverter Chain In this problem you will optimize the delay of a chain of four inverters. The load capacitance is CL=64*Cin, where Cinrepresent the capacitance of the first inverter in the chain. Assume that the input capacitance of the first inverter is Cunit, γ=0.8, and tinvis the unit delay of an inverter as defined in lecture ( i.e., tp= tinv(γ+f) ). a)Size the inverters (with respect to Cin) to minimize the delay. Solution:From the lectures we know that the optimal way to size the inverter chain for minimal delay is to size every inverter for the same fanout f. Since we are given the input and the output capacitance and the number of stages, we can find the total fanout (F) and the fanout of each stage (f) as: F=64CinCin=64f=644=22≈2.82. Now we know that the optimal sizing for the chain is (starting from the beginning of the chain):Cin,fCin,f2Cin,f3Cin. The exact numbers are shown on figure below.
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b)What is the optimal delay? Solution:Every stage has the same fanout and therefore the same delay: td,1=tinv(γ+f). The total delay is therefore: td=4td,1=4tinv(γ+f)≈14.5tinv. c)Now add an additional load of 500*Cinafter the 3rdinverter in the chain. With the same sizing as in part a), now what is the delay of the chain? Solution:With the sizing from part a), the fanout of the third inverter in the chain will change due to the added capacitive load. The delay of this inverter is now: td,3=tinv(γ+f3)=tinv(γ+500+22.78)=tinv(γ+65.3)and therefore the total delay is: td=3td,1+td,3=3tinv(γ+f)+tinv(γ+f3)≈77tinv. d)[BONUS] How could you modify design of the chain (i.e., change sizes, add or remove stages, etc.) to improve the delay of the circuit from part c)? Solution:The large fixed load from this capacitor is now going to be the dominant factor for the overall delay of the chain. This means that we should probably treat the chain of the first 3 inverters as a new sizing problem, still with the same Cinfor the first stage (and leaving the last inverter the same size), but with the final load equal to 522.7 Cin. (We’ll see next week that this heuristic will indeed get us close to the true optimal results.) Assuming we stick with just 3 inverters for this new chain, the new overall fanout will be: