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ENGG 1203 Tutorial Combinational Logic (I) 1 Feb Learning Objectives
Recall Boolean Algebra (SoP/PoS, DeMorgan's Theorem, grouping, redundant) Simplify logic expressions News Lab, TA office hour Tutorial: Ack.: HKU ELEC1008, Wikimedia Commons
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Quick quiz What is the only set of input conditions that will produce a LOW output for any OR gate? Any one of the input is LOW Any one of the input is HIGH All inputs are LOW All inputs are HIGH C
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Quick quiz What logic level should be applied to the second input of a two-input AND gate if the logic signal at the first input is to be inhibited (prevented) from reaching the output? A LOW input will keep the output LOW A LOW input will keep the output HIGH A HIGH input will keep the output LOW A HIGH input will keep the output HIGH A
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Quick quiz What is the only input combination that will produce a HIGH at the output of a five-input AND gate? Any one of the input is LOW Any one of the input is HIGH All inputs are LOW All inputs are HIGH D
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Quick quiz What is the output expression of the following logic-circuit diagram? D
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Boolean Algebra A + B = B + A AB = BA A + (B + C) = (A + B) + C
More questions in Appendix A + B = B + A AB = BA A + (B + C) = (A + B) + C A (BC) = (AB) C A + BC = (A + B) (A + C) A (B+C) = AB + AC A + AB = A A (A + B) = A NOT (NOT (A)) = A
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De Morgan’s Theorem De Morgan's theorem
Bubble pushing via De Morgan's theorem AND NOT NOT OR NOT OR AND NOT OR NOT NOT AND NOT AND OR NOT 7
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Solution 2 Use DeMorgan's Theorem for simplification
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Boolean Algebra Simplification
Sum of Products Find out the “1”s 𝑌= 𝐴 ∙𝐵 + 𝐴∙ 𝐵 Better if less “1” Products of Sum Find out the “0”s 𝑌= 𝐴 + 𝐵 ∙ 𝐴+𝐵 Better if less “0”
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Boolean Algebra Using SOP and POS
Find an expression for F and
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Solution Sum of Products for F Product of Sums for F
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Solution Sum of Products for Product of Sums for
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Representing logic operations
Each function can be represented equivalently in 3 ways: Truth table – Try every combinations of every input variables Boolean logic expression – SOP/POS + Simplify the expression Schematics – Construct from Boolean expressions
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From logic equations Boolean expressions Truth table and logic circuit (AND/OR/NOT) 1 1 1
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From logic equations Boolean expressions Truth table and logic circuit (AND/OR/NOT)
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From truth tables Derive the Boolean expression of the output x in terms of the input Construct the logic circuit using AND gates, OR gates, and INVERTERs.
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Solution Extra redundant Terms 1. Construct A/B/C 2. Construct not A/B/C 3. Construct AND gates 4. Construct OR gate 1 2 3 4
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From truth tables Truth table Boolean expressions and logic circuit
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From schematics Truth table first? SOP/POS first?
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From schematics XOR
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Circuit representation of logic equations
Show how can be implemented with one two-input NOR and one two-input NAND gate. (How to convert ?) We need to apply De Morgan’s Theorem
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Circuit representation of digital logic
a) Simplify the circuit shown in the figure using Boolean algebra. b) Change each NAND gate in the circuit of the figure to a NOR gate, and simplify the circuit using Boolean algebra.
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Solution (a) Procedure:
1) Obtain the Boolean expression from the circuit 2) Check if we need NAND/NOR gate 3) Simplify the expression by Boolean algebra Less gate (power and resource) Shorter “longest path”
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Solution (b) First, we convert the circuit
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Solution (b) Then, we simplify the Boolean expression
(DeMorgan's Theorem) (Expand) (Simplify) (Group, Group)
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Solution (b) (Group, Group) (Simplify) (Expand) (Simplify) (Simplify)
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Conversion of three representations
Describe the function using Boolean expressions Draw the truth table and describe the function using sum of product
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Solution Approach 1: Boolean simplification Find TT Approach 2: Construct TT Find POS (De Morgan) (XOR expansion) (De Morgan) (De Morgan) (expansion) (grouping, expansion) (cancellation) POS:
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Karnaugh map Draw the table Fill in 0s and 1s Grouping
Group one/two/four/eight/ sixteen ‘1’(s) only Use the least number of groups to group all numbers To group as many numbers as possible in every group
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Karnaugh map Yellow: Redundant 𝐴 𝐵𝐶 𝐷 +𝐴𝐵𝐶 𝐷 =𝐵𝐶 𝐷
𝐴 𝐵 𝐶𝐷+ 𝐶 𝐷+𝐶 𝐷 + 𝐶 𝐷 =𝐴 𝐵 𝐴𝐵 𝐶 𝐷 +𝐴𝐵 𝐶 𝐷+𝐴 𝐵 𝐶 𝐷 +𝐴 𝐵 𝐶 𝐷=A 𝐶 F(x) = 𝐵𝐶𝐷 + A 𝐶 + 𝐴 𝐵
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Examples of Karnaugh maps
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(Appendix) Questions for Boolean algebra
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Solutions
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(Appendix) Determining output level from a diagram
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(Appendix) From logic equations
Draw the circuit diagram to implement the expression Draw the circuit diagram that implements the expression using gates having no more than three inputs.
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(Appendix) Circuit representation of digital logic
Construct the given circuit using NAND gates only Top down approach: ? Bottom up approach: ?
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Solution (a) Top down: Expanding the Boolean expression
By DeMorgan’s Theorems,
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Solution (b) Bottom-up: Construct NOT gate, AND gate and OR gate from NAND gate iii) i) ii)
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Solution (b) Top-down and Bottom-up: Same number of gate, same configuration, different approach (cancelled)
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