Digital Logic Design

1. Digital Logic Design Lecture # 5 University of Tehran

2. Outline • Minterms and Maxterms • Simplification of a Function by KM • Minimization’s Effects on a Circuit • Timing Parameters of a Circuit

3. Minterms and Maxterms • We define a Minterm to be a product that contains all variables of that particular switching function in either complemented or non-complemented form. • A simpler shorthand form of representing a SSOP is to use the number of the minterms that appear in that representation. In the following example for instance we could have written:

4. Minterms and Maxterms (continued…) • We define a Maxterm to be a sum that contains all variables of that particular switching function in either complemented or non-complemented form. • Sometimes writing an expression in a POS form is easier as seen in the following example:

5. Simplification of a Function by KM • As said in the last lecture, in a karnaugh map, we consider physically adjacent 1s to be boolean adjacent terms too. This is our main concern in the construction and use of a karnaugh map -keeping the equality of physical adjacency and boolean adjacency. Considering this equality, each two adjacent boxes in the karnaugh map should only differ in a single literal. • All of the above is exactly applied to maps with a larger number of variables.

6. Simplification of a Function by KM(continued…) • Example:

7. Simplification of a Function by KM(continued…) • Example:

8. Simplification of a Function by KM(continued…) • Observing the last example shows us that our definition of physical adjacency doesn’t satisfy our need, thus a small change is needed in this definition. Hence from this point onwards we consider the top and bottom rows of the map adjacent (the first stands for the first and last columns). • Note: There is no formal way to prove the resulted expression is minimal but it can be visually observed to be minimal.

9. Simplification of a Function by KM(continued…) • Note: A standard SOP is unique for any given function whereas a minimal SOP form may not be so. • Note: Karnaugh maps will only help with functions of less that 7 variables.

10. Simplification of a Function by KM(continued…) • We will now consider another example: • This example shows us that seeing 4 or 8 adjacent 1s must also result in a combination that only has variables that don’t differ in any of the corresponding terms.

11. Simplification of a Function by KM(continued…) • Let’s see another example of the last situation:

12. Simplification of a Function by KM(continued…) • Now we consider the simplification method for a function of 4 variables. Pay attention to the example in the next slide.

13. Simplification of a Function by KM(continued…) • The stars shown in the above karnaugh map show that the particular map containing those stars would be an essential choice, because those terms aren’t covered by any other map. • In covering the 1s of a karnaugh map we look to satisfy three aims: • Covering all minterms • Fewer maps • Larger maps (In order to cover more 1s in one map)

14. Simplification of a Function by KM(continued…) • We will now see some definitions: • Implicant: Any term that it’s being one causes the result of the function to be one, in other words any term that implies a function. In a karnaugh map, each 1 or combination of 1s is an implicant. • Prime Implicant: An implicant that is not included in any larger implicant. • Essential Prime Implicant: A prime implicant that covers at least one ‘1’ not covered by any other prime implicant.

15. Simplification of a Function by KM(continued…) • Note: When forming a minimized expression, first the essential prime implicants are written and then the prime implicants which have covered the 1s not included by the EPIs are written. • We now apply the above to the following example:

16. Simplification of a Function by KM(continued…) • The resulting circuit of the preceding example is a 2-level network, as can be seen in the figure:

17. Simplification of a Function by KM(continued…) • Two level networks are the most a karnaugh map can minimize an expression to, while more minimization may be done by the use of more boolean algebra. This doesn’t necessarily mean the number of levels will decrease but the number of transistors may. For instance the three level network shown below uses less transistors than the design shown in the last example:

18. Simplification of a Function by KM(continued…) • To realize the expression , using only nand or nor gates, we use the following method:

19. Minimization’s Effects on a Circuit • Minimization decreases our use of transistors. • Minimization affects circuit timing . • Minimization affects power consumption.

20. Timing Parameters of a Circuit • Consider the following structure: • In the situation where ‘a’ goes ‘1’, after a considerable amount of time of it being ‘0’, the capacitance on the pull-down will discharge through the resistance of the first pull-down. An instance of time in the exponential phase of this discharge will come when the PMOS can no longer conduct a ‘1’ to the output on ‘b’ and the NMOS transistor starts conducting.

21. Timing Parameters of a Circuit (continued…) • Now we consider 2 delay times, trise and tfall, for the signals, the first being the time needed for a signal to reach 90% of its total value from the instance it had 10% of that value and the second the time for transition from 90% to 10%, as seen in the following figure:

22. Timing Parameters of a Circuit (continued…) • Paying attention to the following figure, it is clear that td is considered to be the time needed for the output to reach 50% of its full amount from the instance where the input has 50% of its full value.

23. Timing Parameters of a Circuit (continued…) • In practice, usually a delay time is considered for a whole gate. For instance a 5ns delay for a NAND gate would give. • Analog analysis of the mentioned matters may be used where timing is a very important issue.