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Combinational Logic Circuits Simplification Guide

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Learn to simplify combinational logic circuits using Sum Of Products form, trial and error techniques, and algebraic simplification steps. Master circuit design from truth tables with step-by-step examples.

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Combinational Logic Circuits Simplification Guide

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  1. Chapter 4 Combinational Logic Circuits

  2. Sum Of Products Form (SOP) • Two or more AND terms ORed together • An inversion sign over more than one variable not allowed A B C + A B C A B C + A B C

  3. Algebraic Simplification of Circuits • Trial and Error! • Not obvious which theorems should be applied (or in what order) • No way to determine when the expression is in it’s simplest form

  4. Algebraic Simplification of Circuits Two essential steps: • Put the original expression into SOP form • Check the product terms for common factors

  5. Simplify the Following Circuit A AC AB(AC) B C ABC + AB(AC) ABC

  6. Simplify the Circuit Z = ABC + AB(AC) Z = ABC + AB(A+C) Break up the bar Z = ABC + AB(A+C) Discard double bars Z = ABC + ABA + ABC Distribute terms Z = ABC + AB + ABC A * A = A Factor with a goal of Trying to group something That can be removed Z = AC(B + B) + AB Z = AC(1) + AB B + B’ = 1 Z = AC + AB AC * 1 = AC Z = A(C + B)

  7. Simplify the Circuit Z = A(C + B)

  8. Simplifying Expressions Z = ABC + ABC + ABC Factor with a goal of Trying to group something That can be removed Z = AB(C + C) + ABC Z = AB(1) + ABC C’ + C = 1 Z = AB + ABC AB’ * 1 = AB’ Z = A(B + BC) Theorem 15 Z = A(B + C)

  9. Simplifying Expressions Z = ABC + ABC + ABC X + X = X Z = ABC + ABC + ABC + ABC Z = AB(C + C) + AC(B + B) Z = AB(1) + AC(1) Z = AB + AC Z = A(B + C)

  10. Simplifying Expressions Z = (A + B)(A + B + D)D Z = AAD + ABD + ADD + BAD + BBD + BDD Distribute terms Z = ABD + BAD + BBD X * X’ = 0 0 * Y = 0 Z = ABD + BAD + BD X * X = X Factor with a goal of Trying to group something That can be removed Z = BD(A + A + 1) Z = BD(1) X + X’ = 1 1 + Y = 1 Z = BD

  11. Simplifying Expressions Z = AC(ABD) + ABCD + ABC Z = AC(A+B+D) + ABCD + ABC Z = ACA + ACB + ACD + ABCD + ABC Z = ACB + ACD + ABCD + ABC Z = BC(A + A) + AD(C + BC) Z = BC(1) + AD(C + B) Z = BC + AD(C + B)

  12. Circuit Design from Truth Tables A B X 0 0 0 0 1 0 1 0 0 1 1 1 Output goes high only when both of the inputs to thecircuit are HIGH

  13. Circuit Design from Truth Tables A B X 0 0 0 0 1 1 1 0 0 1 1 0

  14. Circuit Design from Truth Tables A B X 0 0 0 0 1 1 1 0 0 1 1 1

  15. Circuit Design from Truth Tables • Create the truth table showing the circuit’s action • Look for positions in the truth table having a 1 foroutput • Write a term for each one as a product of the input variables (a zero input is an inverted term) • OR all of the products • Simplify the expression

  16. Circuit Design Example Design a logic circuit having three inputs.The output should go HIGH whenever a majority of inputs is HIGH. Step 1: Create a truth table Step 2: Write a product of input variables for each “1” A B C X 0 0 0 0 0 1 0 1 0 1 0 0 ABC 0 0 0 ABC 0 1 1 1 ABC 0 1 0 1 1 1 1 0 1 ABC 1 1 1 1

  17. Circuit Design Example Design a logic circuit having three inputs.The output should go HIGH whenever a majority of inputs is HIGH. Step 3: OR the products Step 2: Write a product of input variables for each “1” A B C X 0 0 0 0 0 1 0 1 0 1 0 0 ABC 0 0 0 ABC ABC + ABC + ABC + ABC 0 1 1 1 ABC 0 1 0 1 1 1 1 0 1 ABC 1 1 1 1

  18. Circuit Design Example Design a logic circuit having three inputs.The output should go HIGH whenever a majority of inputs is HIGH. Step 4: Simplify the expression Step 3: OR the products ABC + ABC + ABC + ABC

  19. Circuit Design Example ABC + ABC + ABC + ABC ABC + ABC + ABC + ABC + ABC + ABC BC(A + A) + AC(B + B) + AB(C + C) BC(1) + AC(1) + AB(1) BC + AC + AB

  20. Circuit Design Example • A 12 volt battery exists on a particular spaceship • The output (voltage) from the battery is an analog value (constantly changing voltage values) • An analog-to-digital converter is being used to convert the analog voltage to a digital value • The converter’s output is a 4-bit binary number corresponding to the battery voltage in 1 volt increments • This value is fed into a logic circuit that will start a battery charger • The battery charger is designed to operate (start charging) whenever it receives a digital LOW value • We need to begin charging the battery whenever the battery’s voltage drops below 7 volts

  21. Interpretation A B C D MSB LSB + Analog To Digital Converter Logic Circuit 12V _ Z Battery Charger

  22. Interpretation • Battery charger starts on a LOW and is inactive on a HIGH • Battery charger needs to begin operating when voltage drops below 7 volts • 0V to 6V = LOW7V to 12V = HIGH

  23. Create Truth Table & Generate Terms A B C D Z 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 ABCD ABCD ABCD ABCD ABCD ABCD ABCD ABCD ABCD

  24. Create Truth Table & Generate Terms Z = ABCD + ABCD + ABCD + ABCD + ABCD + ABCD + ABCD + ABCD + ABCD Z = ABCD + ABC(D + D) + ABC(D + D) + ABC(D + D) + ABC(D + D)

  25. Create Truth Table & Generate Terms Z = ABCD + ABC + ABC + ABC + ABC Z = ABCD + AB(C + C) + AB(C + C) Z = ABCD + AB + AB Z = ABCD + A(B + B)

  26. Create Truth Table & Generate Terms Z = ABCD + A Theorem 15: X + XY = X + Y X = A Y = BCD Z = BCD + A HIGH if A = 1 OR B = C = D = 1

  27. Create Truth Table & Generate Terms A B C D Z 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 A Z = + BCD HIGH if A = 1 OR B = C = D = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  28. Create Truth Table & Generate Terms

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