Digital Electronics Lecture 2 (4CS015) PDF

Lecture 2: 4CS015 Digital Electronics Prepared by: Uttam Acharya 1 Coverage ◆ Boolean Logic and Logic Gates Truth Table ◆ Boolean Algebra Laws ◆ Combinational Circuit 2 Boolean Logic History George Boole (1815-1864) “An Investigation i...

Lecture 2: 4CS015 Digital Electronics Prepared by: Uttam Acharya 1 Coverage ◆ Boolean Logic and Logic Gates Truth Table ◆ Boolean Algebra Laws ◆ Combinational Circuit 2 Boolean Logic History George Boole (1815-1864) “An Investigation into the Laws of Thought” Defined an algebra for solving logical problems Limited to dealing with facts True or False Now known as Boolean Algebra 3 Basic Logic Definitions In a Logic System a variable can have one of two possible states. Single capital letters are used to represent variables. The bits 1 and 0 are also used as constants. TRUE ON CLOSED ‘1’ Yes 5v FALSE OFF OPEN ‘0’ No 1v 4 Logic States If a switch is closed: The light will be ON. This can represent Logic TRUE. If a switch is open: The light will be OFF. This can represent Logic FALSE. The switch is a Logic variable. 5 Boolean Operators Boole defined three basic operations that could be used with these Boolean variables. AND OR NOT All logical expressions can be built from these three. 6 Boolean Operators Logical AND Logical OR Logical NOT 7 Logical AND How can we switch the light on? 8 Logical AND Boolean Expression: F = A AND B or F= A∙B Gate Diagram: A B F Truth Table: Input A Input B Output F 0 0 0 0 1 0 1 0 0 1 1 1 9 AND Relationship Boolean representation. (Period) If F,A and B are Boolean variables. Then the expression F = A∙B means F is only true when A AND B are both true. As A is capable of being 1 or 0 and B is capable of being 1 or 0 there are 4 possible states. 00, 01, 10 or 11 Input A Input B Output F 0 0 0 0 1 0 1 0 0 1 1 1 10 Logical OR How can we switch on the light? Switch A Switch B Light Battery 11 Logical OR Boolean Expression: F = A OR B alternatively F =A+B Gate Diagram: Truth Table: Input A Input B Output F 0 0 0 0 1 1 1 0 1 1 1 1 12 OR Relationship Boolean representation + (Plus) If F,A and B are Boolean variables. Then the expression F = A+B means F is only true when A OR B, OR both, are true. As A is capable of being 1 or 0 and B is capable of being 1 or 0 there are 4 possible states. 00, 01, 10 or 11 Input A Input B Output F 0 0 0 0 1 1 1 0 1 1 1 1 13 Logical Not How can we switch the light off? Switch A Light Boolean Expression: F = NOT A or F=!A or F=A’ Battery Gate Diagram: A F Truth Table: Input A Output F 0 1 1 0 14 NOT Relationship The NOT relationship reverses the value. NOT True is False etc… The Symbol ( ) used is usually a bar above the variable or expression to be reversed E.g if A= true then A’ = false In some circumstances we use ! E.g. if B = true !B = false (easier to type) 15 Integrated Circuit and Logic Gates 16 Other Logic Gates To make life a little easier the basic logical functions are expanded to include: NAND This is an AND with a NOT output. NOR This is an OR with a NOT output. XOR This is the Exclusive OR function. XNOR This is Complement of XOR. 17 NAND Boolean Expression: F = NOT(A AND B) or F = A∙B Gate Diagram: Truth Table: Input A Input B Output F 0 0 1 0 1 1 1 0 1 1 1 0 18 NOR Boolean Expression: F = NOT(A OR B) or F = A+B Gate Diagram: Truth Table: Input A Input B Output F 0 0 1 0 1 0 1 0 0 1 1 0 19 XOR Boolean Expression: F = A XOR B or F = A⊕B Gate Diagram: A F B Truth Table: Input A Input B Output F 0 0 0 0 1 1 1 0 1 1 1 0 20 XNOR Boolean Expression: F = A XNOR B or F = A ☉ B Gate Diagram: Truth Table: Input A Input B Output F 0 0 1 0 1 0 1 0 0 1 1 1 21 Boolean Algebra Laws The operations +,. And ’ consequently satisfy the basic laws 1, 2 and 3 of Boolean algebra. That is: A+B≡B+A A⋅B≡B⋅A Commutative Laws (A + B) + C ≡ A + (B + C) Associative Laws (A ⋅ B) ⋅ C ≡ A ⋅ (B ⋅ C) A ⋅ (B + C) ≡ (A ⋅ B) + (A ⋅ C) A + (B ⋅ C) ≡ (A + B) ⋅ (A + C) Distributive Laws 22 Boolean Algebra Laws Identity Law Negation Law A+0=A low = high A.1=A (0=1) Idempotent Law A+A=A Double Negation Law A.A=A A = A Complement Law Domination Law A. A’ = 0 A + A’ = 1 A + high = high A. low = low (A + 1 = 1) A. 0 =0 Absorption Law A + (A.B) = A A. (A + B) = A 23 DeMorgan’s Laws A+B = A.B A.B = A+B 24 Universal Gate The repeated use of a NOR or a NAND gate alone can produce all the three basic logic gates. 25 Question 1 : NAND and NOR gates are called universal gates, why? 26 AOI using NAND and NOR gate AOI using NOR gate? 27 Precedence of operators As with normal mathematics when working out the value of a function it is very important to do it in the right order. NOT AND OR Parenthesis (brackets) override in the normal way. When a bar goes above more than 1 symbol it becomes a bracket that reverses. 28 Example 29 Circuit Design 30 Digital Component The main thing to remember is that combinations of gates implement Boolean functions. The circuit below implements the Boolean function: We simplify our Boolean expressions so that we can create simpler circuits. 31 Truth Table 32 Combinational Logic We have designed a circuit that implements the Boolean function: This circuit is an example of a combinational logic circuit. Combinational logic circuits produce a specified output (almost) at the instant when input values are applied. In a later section, we will explore circuits where this is not the case. 33 Combinational Circuit Examples Example 1 Circuit after Simplification 34 Combinational Circuit Examples Example 2 35 Combinational Circuit Examples Example 3 Simplify: X = (A.B.C) +(A.B'.C) +(A'.B.C) 36 Combinational Circuit Examples Example 4: Write the output functions of the following circuits. Fig: Multiplexor Fig: Decoder 37 Exercises Simplify and construct the logic circuit: 1.A’.B’ + (A.B)’ 2.(A + B).(A + B) + A.(A + B’) 3.(A.B’.C’ + A’.B’.C+A.B.C+A’.B.C’) 38 Summary We have looked at the basic logic gates: Identifying OR, AND, NOT, NAND, NOR and XOR. We have seen that gates can be joined together to form Combinatorial Logic. 39 Thank you… 40