Assignment: Combinational Logic
Your logic function for this assignment is to be derived from your own student number. The number 1224583 will be used as an example as to how you should derive your function and examples are given in brackets for this number. You should however use YOUR OWN NUMBER.
Take your student number (1224583), and add your date of birth, using 2 digits for the day, 2 for the month and the last 2 digits of the year. If your birthday is 21st January 1993, then the number you should add, is 210193, giving you 1434776. If you do not wish to use your own birthday date, use a friend’s. Place the different digits in ascending order (1, 3, 4, 6, 7).
Add 5 to each of the individual digits of your ascending set (6, 8, 9, 11, 12,) and include any new numbers, starting from the highest in your set of ascending numbers until you have a set of eight numbers. In this example you need three more numbers (9, 11, 12) to give the set (1, 3, 4, 6, 7, 9, 11, 12,).
YOU need a set of EIGHT ascending numbers in the range 0 to 15. If your set still contains LESS than eight numbers, add 2 to the individual digits of your original ascending set, look for new digits, and continue until you have a set of eight numbers. This is the set of terms in the 1st canonical form of your function. In this example, the resulting 4-variabled function would be:
F = f(ABCD) = ∑(1, 3, 4, 6, 7, 9, 11, 12)
1) Write down the shorthand 1st canonical form equation of your own personal function derived as above.
2) Obtain the full 1st canonical form Boolean equation of your function in AND/OR/NOT form and draw its gate-level circuit diagram.
3) Obtain the shorthand equation of the 2nd canonical form of your function.
4) Obtain the full 2nd canonical form Boolean equation of your function in AND/OR/NOT form and draw its gate-level circuit diagram.
5) Enter your function on a fully labelled K-Map.
6) Obtain the minimal 1st canonical form (AND/OR/NOT) of your function and draw its circuit diagram.
7) Obtain the minimal 2nd canonical form (AND/OR/NOT) of your function and draw its circuit diagram.
8) Use truth table equivalence to show that your minimal 1st and 2nd canonical forms do perform the same function.
9) Obtain the minimal NAND version of your function and draw its circuit diagram.
10) Obtain the minimal NOR version of your function and draw its circuit diagram.
11) Select at random, 4 terms NOT included in your original 1st canonical form shorthand equation in Question 1, to be don’t care states. (NOTE: Once these don’t cares have been defined they remain don’t care inputs for questions 12, 13 and 14 of this assignment). Obtain the minimal 1st canonical form (AND/OR/NOT) using the original terms and, where appropriate, the don’t care conditions. Draw the circuit diagram.
12) Obtain the minimal 2nd canonical form (AND/OR/NOT) using the original terms and where appropriate, the don’t care conditions. Draw the circuit diagram.
13) Obtain the minimal NAND form using the original terms and where appropriate, the don’t care conditions. Draw the circuit diagram.
14) Obtain the minimal NOR form using the original terms and, where appropriate, the don’t care conditions. Draw the circuit diagram.
15) Assess all the circuits you have designed so far, and make a recommendation as to which circuit would be best for mass production.
16) Assuming that all the logic you have been using so far, is positive coded logic (0 volts represents binary 0 and 5 volts represents binary1), determine the logic function the minimal 1st canonical form circuit obtained in question 6 performs if negative logic coding is used. (0 volts represents binary 1 and 5 volts represents binary 0). You should obtain the voltage truth table of the circuit, recode it for negative logic and obtain its minimal 1st canonical form. Draw the circuit diagram.
Devise a rule to obtain the truth table of the negative logic coded system directly from the positive coded truth table WITHOUT having to generate the voltage truth table.
17) Compare the layouts of the circuits derived in question 6 and question 16 and comment on any anomalies. (If in the unlikely event that they have identical shapes, see me, as you have a symmetrical circuit).
18) Compare the circuits derived in question 16 and question 7 and derive a design rule to obtain the exact equation, the circuit performs when the logic coding changes using ONLY the original equation and NO truth tables.
19) Check the circuits in question 6 and question7 for timing hazards and if necessary modify the circuits to remove them, giving the Boolean equations for the hazard-free circuits
20) Use Boolean algebra to convert the minimal 1st canonical form circuit in question 6 to a NOR only logic circuit. Draw its circuit diagram and give two reasons why this design strategy would NOT be used in custom logic design.