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      ADDER CIRCUITS
      HALF ADDER AND FULL ADDER CIRCUIT


      Learning Objective:

      • To make Electronic Adder Circuits.
      • Use of Gates such as AND,OR,NAND and XOR
      • Use of Bread board and IC.

      Introduction
      An Integrated circuit (IC) is a small electronic device made out of a semiconductor material. An integrated circuit, commonly referred to as a IC, is a microscopic array of electronic circuits and components that has been diffused or implanted onto the surface of a single crystal, or chip, of semiconducting material such as silicon. It is called an integrated circuit because the components, circuits, and base material are all made together, or integrated, out of a single piece of silicon, as opposed to a discrete circuit in which the components are made separately from different materials and assembled later. ICs range in complexity from simple logic modules and amplifiers to complete microcomputers containing millions of elements.

      Activity 1


      Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of physical logic gates. A sum-of-products expression can always be implemented using AND gates feeding into an OR gate, and a product-of-sums expression leads to OR gates feeding an AND gate. Karnaugh maps can also be used to simplify logic expressions in software design. Boolean conditions, as used for example in conditional statements, can get very complicated, which makes the code difficult to read and to maintain. Once minimized, canonical sum-of-products and product-of-sums expressions can be implemented directly using AND and OR logic operators.

      The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones
      • Groups may not include any cell containing a zero

      • Groups may be horizontal or vertical, but not diagonal.

      • Groups must contain 1, 2, 4, 8, or in general 2n cells.
      That is if n = 1, a group will contain two 1's since 21 = 2.
      If n = 2, a group will contain four 1's since 22 = 4.

      • Each group should be as large as possible.

      • Each cell containing a one must be in at least one group.

      • Groups may overlap.

      • Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.

      • There should be as few groups as possible, as long as this does not contradict any of the previous rules.

      Summary:

      1. No zeros allowed.
      2. No diagonals.
      3. Only power of 2 number of cells in each group.
      4. Groups should be as large as possible.
      5. Every one’s must be in at least one group.
      6. Overlapping allowed.
      7. Wrap around allowed.
      8. Fewest number of groups possible.



      Activity2

      The Quine–McCluskey algorithm (or the method of prime implicants) is a method used for minimization of boolean functions. It is functionally identical to Karnaugh mapping, but the tabular form makes it more efficient for use in computer algorithms, and it also gives a deterministic way to check that the minimal form of a Boolean function has been reached. It is sometimes referred to as the tabulation method.

      Make a table of four columns. In first column we have to write group i.e. the number of ‘1’s present in given minterms. For example first group having zero times ‘1’s i.e. there is no ‘1’ in that row (see column of variables), where as in second group that is group having one time ‘1’ , there are two possible cases first is 1=0001 and second is 8=1000 in both cases ‘1’ is present only one time. Just follow it for all the given minterms. We will discuss remark column in second table.



      Now here is second table. In second table we have to do the same thing only the difference is that, we have to refer first table. For first group i.e. group ‘0’ (0,1) and (0,8) have only one digit different (see third column of table1). So put ‘-’ in that place.(keep in mind that group ‘0’ means there should not any ‘1’ in that row). Similarly for group ‘1’ there should present ‘1’ only ones. Now check the remark column of first table. When we take (0,1) for first group, we have to fill remark column.


      Finally, the following table is of prime implicants. If you observe last table (table 3) carefully, the minterms for each group are same only the position is different, for example for first group ‘0’ there are 0,1,8,9 which is nothing but 0,8,1,9. So we have to fill prime implicants with corresponding variables of third table. There is (-00-) it means B’C’ since A and D are absent. Similarly (-0-1) means B’D and so on


      Activity3

      All digital circuits can be implemented in VHDL(Very high speed hardware description language), HDL (Hardware Description Language) based design has established itself as the modern approach to design of digital systems, with VHDL (VHSIC Hardware Description Language) .


      Half adder:

      ENTITY half_adder IS --- Half Adder
      PORT(a,b:IN BIT; s,c :OUT BIT);
      END half_adder;
      ARCHITECTURE half_adder_beh OF half_adder IS
      BEGIN
      s <= a XOR b; -- Implements Sum for Half Adder
      c <= a AND b; -- Implements Carry for Half Adder
      END half_adder_beh;
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