Bài giảng Digital System - Chapter 2 -Tran Ngoc Thinh

What is the only input combination that will produce a
HIGH at the output of a five-input AND gate?
– all 5 inputs = 1
• 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
• True or false: An AND gate output will always differ from
an OR gate output for the same input conditions.
– False
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  1. dce dce 2016 2016 Boolean Constants and Variables • Boolean algebra is an important tool in describing, analyzing, designing, and implementing digital circuits. • Boolean algebra allows only two values; 0 Digital Systems and 1. • Logic 0 can be: false, off, low, no, open Tran Ngoc Thinh switch. BK TP.HCM HCMC University of Technology • Logic 1 can be: true, on, high, yes, closed switch. • Three basic logic operations: OR, AND, and NOT. 2 dce dce 2016 Truth Tables 2016 Truth Tables • A truth table describes the relationship • Examples of truth tables with 2, 3, and 4 inputs. between the input and output of a logic circuit. • The number of entries corresponds to the number of inputs. For example a 2-input table would have 22 = 4 entries. A 3-input table would have 23 = 8 entries. 3 4 Digital Systems 1
  2. dce dce 2016 AND Operations with AND gates 2016 AND Operation With AND Gates • The Boolean expression for the AND operation is • The AND operation is similar to multiplication. X = A • B • In the Boolean expression – This is read as “x equals A and B.” X = A • B • C – x = 1 when A = 1 and B = 1. X = 1 only when A = 1, B = 1, and C = 1. • Truth table and circuit symbol for a two input AND gate are • The output of an AND gate is HIGH only when all inputs shown. Notice the difference between OR and AND gates. are HIGH. A A B B C x x 9 10 dce dce 2016 Review Questions 2016 NOT Operation • The Boolean expression for the NOT operation is • What is the only input combination that will produce a X A X A' HIGH at the output of a five-input AND gate? • This is read as: – all 5 inputs = 1 – x equals NOT A, or – x equals the inverse of A, or • What logic level should be applied to the second input of – x equals the complement of A • Truth table, symbol, and sample waveform for the NOT a two-input AND gate if the logic signal at the first input circuit. is to be inhibited(prevented) from reaching the output? – A LOW input will keep the output LOW • True or false: An AND gate output will always differ from an OR gate output for the same input conditions. – False 11 12 Digital Systems 3
  3. dce dce 2016 Evaluating Logic Circuit Outputs 2016 Evaluating Logic Circuit Outputs • Evaluate Boolean expressions by • Output logic levels can be determined substituting values and performing the directly from a circuit diagram. indicated operations: • Technicians frequently use this method. A 0, B 1, C 1, and D 1 • The output of each gate is noted until a x ABC(A D) final output is found. 0 11(0 1) 111(0 1) 111(1) 1110 0 17 18 dce dce 2016 Implementing Circuits From Boolean Expressions 2016 Example • It is important to be able to draw • Draw the circuit diagram to implement the expression a logic circuit from a Boolean expression. x (A B)(B C) • The expression x A  B C could be drawn as a three input AND gate. • A more complex example such as y AC BC ABC could be drawn as two 2-input AND gates and one 3-input AND gate feeding into a 3-input OR gate. Two of the AND gates have inverted inputs. 19 20 Digital Systems 5
  4. dce dce 2016 Commutative Laws of Boolean Algebra 2016 Associative Laws of Boolean Algebra A + B = B + A A + (B + C) = (A + B) + C A • B = B • A A • (B • C) = (A • B) • C 25 26 dce dce 2016 Distributive Laws of Boolean Algebra 2016 Rules of Boolean Algebra A • (B + C) = A • B + A • C A • (B + C) = A • B + A • C A (B + C) = A B + A C A (B + C) = A B + A C A • (B • C) = (A • B) • C 27 28 Digital Systems 7
  5. dce dce 2016 Examples 2016 DeMorgan’s Theorems • Simplify the expression • Theorem 1: When the OR sum of two y ABD ABD variables is inverted, it is equivalent to inverting each variable individually and y AB ANDing them. z (A B)(A B) A B A.B z B • Theorem 2: When the AND product of two x ACD ABCD variables is inverted, it is equivalent to inverting each variable individually and x ACD BCD ORing them. y AC ABC A.B A B y AC 33 34 dce dce 2016 DeMorgan’s Theorems 2016 Implications of DeMorgan’s Theorems • A NOR gate is equivalent to an AND gate with inverted inputs. • A NAND gate is equivalent to an OR gate with inverted inputs. For N variables, DeMorgan’s theorem is expressed as: and 35 36 Digital Systems 9
  6. dce dce 2016 Exercises 2016 Universality of NAND and NOR Gates • Simplify the expressions • NAND or NOR gates can be used to create the three basic logic expressions – a) (OR, AND, and INVERT) – b) • De Morgan’s 41 42 dce dce 2016 Universality of NAND and NOR Gates 2016 Alternate Logic-Gate Representations • To convert a standard symbol to an alternate: – Invert each input and output (add an inversion bubble where there are none on the standard symbol, and remove bubbles where they exist on the standard symbol. – Change a standard OR gate to and AND gate, or an AND gate to an OR gate. 43 44 Digital Systems 11
  7. dce dce 2016 Alternate Logic-Gate Representations 2016 Which Gate Representation to Use • Interpretation of the two OR gate symbols. • Using alternate and standard logic gate symbols together can make circuit operation clearer. • When possible choose gate symbols so that bubble outputs are connected to bubble input and nonbubble outputs are connected to nonbubble inputs. 49 50 dce dce 2016 Which Gate Representation to Use 2016 • When a logic signal is in the active state (high or low) it is said to be asserted. (a) Original circuit using standard NAND • When a logic signal is in the inactive state symbols; (b) (high or low) it is said to be unasserted. equivalent representation where • A bar over a signal means asserted (active) output Z is active- low. HIGH; (c) equivalent • The absence of a bar over a signal means representation where output Z is active- asserted (active) high. LOW; (d) truth table. 51 52 Digital Systems 13
  8. dce dce 2016 Summary of Methods to Describe Logic Circuits 2016 Summary • Boolean Algebra: a mathematical tool used in the analysis • The three basic logic functions are AND, and design of digital circuits OR, and NOT. • OR, AND, NOT: basic Boolean operations • Logic functions allow us to represent a • OR: HIGH output when any input is HIGH decision process. • AND: HIGH output only when all inputs are HIGH – If it is raining OR it looks like rain I will take an • NOT: output is the opposite logic level as the input umbrella. • NOR: OR with its output connected to an INVERTER • NAND: AND with its output connected to an INVERTER – If I get paid AND I go to the bank I will have • Boolean theorems and rules: to simplify the expression of money to spend. a logic circuit and can lead to a simpler way of implementing the circuit • NAND, NOR: can be used to implement any of the basic Boolean operations 57 58 Digital Systems 15