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Electronic Labs_24

# Electronic Labs_24 - 28-4 Circuit Simplification The...

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Unformatted text preview: 28-4 Circuit Simplification; The Karnaugh Map is required. We will see later that equivalent gates and available gate types will be a determining factor of which function is actually used. T_he co_1_np_lement_ functicin for Example 3 is E = (A+B)(A+B+D)(A+B+C+D) and is mapped in Fig. 28-8. The resulting function is E=(A+B)(A+D)(A+C). To obtain F, =E=(A+B)(A+C)(A+D) =A+B + A+C + A+D =A§+AE+A5 =A(§+E+ﬁ) In some cases, there are more terms in a sum of the products function requiring a 1 output than a 0 output as will be shown in Example 4; this could also occur for a product of the sums function. Example 4. F=ABC+AEC+ABC+KEC+KFC The Karnaugh maps for F and F are shown in Fig. 28-9 (:1) and (b), respectively. After mapping the one terms, the result is F=KE+AG+BG+EG After mapping the zero terms, the result is E=AFG+BG The results of the one terms and zero terms mapping are equivalent since F=AEG+BC= (A+B+C)(§+O)=AF+AO+BG+FC. Mapping the zero terms thus results in a simpler, but complement, function. Complementing this function will lead to a simpler circuit giving the same result as the original function. Notice that complementing E to give the function F led directly to a circuit using NOR gates. The NAND gate will result if the same procedure is used for a product of the sum function. We will implement these circuits in our exercise procedure to show that the same output results will be obtained. Fig. 28-9 Equipment and Materials The Model A818145 Digital Logic Trainer with: - 2-input AND Gates (7408) - 2-input OR Gates (7432) - 2-input NAND Gates (7400) - 2-input NOR Gates (7402) 4-input NAND Gates (7420) - Hex Inverters (7404) - Switches - LED Displays Additional Reading See bibliography at the back of this manual for additional reading material on Boolean algebra, Karnaugh maps and logic simpliﬁcation techniques. Review the materials contained in preceding ex- ercises on logic gates and Boolean algebra. CDCDerFDJWOJrP I Objective A. Use a Karnaugh map and dual gate concept to design, implement and verify 3 VOTE SYSTEM that will indicate a majority “YES”, majority “NO” and a “TIE” vote. Preparatory Information. We will simplify and implement the vote system of the preceding ex- ercise but now use the Karnaugh map to simplify the functions. We will also use DeMorgan’s theorem to implement the circuit with dual or equivalent gates. Typically, a design is started by listing all of the output requirements 1n terms of the i_nput variables. In this prob__lem they were, X= ABCD+ABCD+ ABCD+§§QQ+é§9D= 1__fo_r a eaoﬁu_:YES” vote, Y=ABCD+ABCD+ABCD+ABCD+ABCD= —1 for a majority “NO” vote, and Z equals the remaining terms that would indicate a tie, or XY. Now we will simplify the functions for X and Y, using the Karnaugh map. FX =KBCD+A§CD+ABED+ABcﬁ+ABCD The simplified expression for X from the Karnaugh map shown in Fig. 28-10 contains four terms, each with one variable eliminated. The function for X is Fx =ABC+ABD+ACD+BCD. This can be factored to give FX=AB(C+D)+CD(A+B). Both ...
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