# Discussion 2 Exercises Solution notes

### From HacDC Wiki

Problem 4, Part D was omitted from the solutions. It is presented here.

D) What is t_pd of the fastest equivalent circuit (i.e. one that implements the same function) built using only the three components listed above?

Lets start by reducing the logic equation....I'm going to use three intermediary signals, alpha, beta and gamma, and pull identity numbers from the slides

alpha=!(A+B) beta=!(alpha+A) gamma=!(alpha+!B) OUT=!(beta+gama)

OUT=!(!(alpha+A) + !(alpha+!B)) substitution OUT=!(!(!(A+B)+A) + !(!(A+B)+!B)) substitution OUT=!(!(!A*!B+A) + !(!A*!B+!B)) demorgans OUT=!!(!A*!B+A) * !!(!A*!B+!B) demorgans OUT= (!A*!B+A) * (!A*!B+!B) identity 4 elementary OUT= (!A*!B+A) * (!B) identity 10 absorption OUT= (!A*!B+A*!B) identity 8 distributivity OUT= !B (!A + A) identity 8 distributivity OUT= !B 5 elementary

Since this can be implemented with a single inverter, the smallest propagation delays is 1ns.