Digital Logic Lecture 5 Boolean Algebra and Logic Gates Part I By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department
Outline Introduction. Boolean variables and truth tables. Fundamental logic gates (AND, OR, and NOT gates). Logic circuits, functions, and Boolean algebra. Universal logic gates (NAND and NOR). XOR and XNOR gates. Extension to multiple inputs. The Hashemite University 2
Introduction Till now we have understood the concept of binary numbers. Next, we will study ways of describing how systems using binary logic levels make decisions. After that the definition of logic circuit will be complete. Boolean algebra is an important tool in describing, analyzing, designing, and implementing digital circuits. Logic gates are the basic hardware units used to implement logic function or circuits described by Boolean algebra. The Hashemite University 3
Boolean Constants and Variables Boolean algebra allows only two values; 0 and 1. Logic 0 can be: false, off, low, no, open switch. Logic 1 can be: true, on, high, yes, closed switch. Three basic logic operations: OR, AND, and NOT. The operation of these logic operations is best described by truth tables. The Hashemite University 4
Truth Tables I A truth table describes the relationship between the input and the output of a logic circuit. The number of entries corresponds to the number of inputs. For example a 2 input table would have 2 2 = 4 entries. A 3 input table would have 2 3 = 8 entries, and so on. The Hashemite University 5
Truth Tables II Examples of truth tables with 2, 3, and 4 inputs. The Hashemite University 6
Fundamental Logic Gates OR : gives an output of 1 when at least one of its input is 1 (Disjunction or logical addition). AND: gives an output of 1 when all of its inputs are 1 (Conjunction or logical multiplication). NOT: invert its input, if the input is 1 the output is 0 and vice versa (Negation). The Hashemite University 7
OR Logic Gate I The Boolean expression for the OR operation is X = A + B This is read as x equals A or B. X = 1 when A = 1 or B = 1 or both equal 1. Truth table and circuit symbol for a two input OR gate: The Hashemite University 8
OR Logic Gate II The OR operation is similar to addition but when A = 1 and B = 1, the OR operation produces 1 + 1 = 1 (not 10 as in arithmetic addition). In the Boolean expression x=1+1+1=1 We could say in English that x is true (1) when A is true (1) OR B is true (1) OR C is true (1). The Hashemite University 9
Timing Diagram of OR Gate The Hashemite University 10
AND Logic Gate I The Boolean expression for the AND operation is X = A B or X = AB This is read as x equals A and B. x = 1 when A = 1 and B = 1. Truth table and circuit symbol for a two input AND gate are shown. Notice the difference between OR and AND gates. The Hashemite University 11
AND Logic Gate II The AND operation is similar to multiplication. In the Boolean expression X = A B C X = 1 only when A = 1, B = 1, and C = 1. The Hashemite University 12
Timing Diagram of AND Gate The Hashemite University 13
NOT Logic Gate I The Boolean expression for the NOT operation is X A This is read as: x equals NOT A, or x equals the inverse of A, or x equals the complement of A The Hashemite University 14
NOT Logic Gate II Truth table, symbol, and sample waveform for the NOT circuit. The Hashemite University 15
Buffer Logic Gate The function performed by Buffer gate is called transfer since this gate just transfer the value of its input to its output without any change (does not perform any logical operation). Equivalent to two inverters (NOT gates) connected in cascade (the output of the first one is the input of the second one). Needed for power amplification of the signal (or inputs). A x = A 0 0 1 1 A x The Hashemite University 16
Describing Logic Circuits Algebraically I The three basic Boolean operations (OR, AND, NOT) can describe any logic circuit. Operators Precedence: If an expression contains AND, OR, and NOT gates combined with each other then follow these rules: First: implement NOT operators starting from left to right. Second: implement AND operations starting from left to right. Third: implement OR operators starting from left to right. Remember that parenthesis force priority (you start implementation of what inside a parenthesis). For nested parenthesis you start from the inner most one. If multiple parenthesis exist start implementation from left to right The Hashemite University 17
Describing Logic Circuits Algebraically II Examples of Boolean expressions for logic circuits: The Hashemite University 18
Describing Logic Circuits Algebraically III The output of an inverter is equivalent to the input with a bar over it. Input A through an inverter equals A. Examples using inverters. The Hashemite University 19
Examples The Hashemite University 20
Evaluating Logic Circuit Outputs Output logic levels can be determined directly from a circuit diagram. The output of each gate is noted until a final output is found. Rules for evaluating a Boolean expression: Perform all inversions of single terms. Perform all operations within parenthesis. Perform AND operation before an OR operation unless parenthesis indicate otherwise. If an expression has a bar over it, perform the operations inside the expression and then invert the result. The Hashemite University 21
Example Evaluate Boolean expressions by substituting values and performing the indicated operations: A 0, B 1, C 1, and x ABC(A D) 0 1 1 (0 1) 1 1 1 (0 1) 1 1 1 (1) 1 1 1 0 0 D 1 The Hashemite University 22
Implementing Circuits From Boolean Expressions It is important to be able to draw a logic circuit from a Boolean expression. Follow the rules of operators precedence (used when evaluating Boolean expressions) when defining the locations of gate used in the circuit that represent the desired logic function. Examples: Implement the following Boolean expressions: (a) y = AC+BC +A BC ( x=(a+b)(b +C (b) The Hashemite University 23
Example 1 (a) The Hashemite University 24
Example 1 (b) The Hashemite University 25
NAND Logic Gate I Combine basic AND, OR, and NOT operations. The NAND gate is an inverted AND gate. An inversion bubble is placed at the output of the AND gate. The Boolean expression is x AB NAND operator symbol is x A B The Hashemite University 26
NAND Logic Gate II Truth table and gate symbol (compare it with AND gate truth table): A B x A B 0 0 1 0 1 1 1 0 1 A B x 1 1 0 The Hashemite University 27
Timing Diagram of NAND Gate The Hashemite University 28
NOR Logic Gate I Combine basic AND, OR, and NOT operations. The NOR gate is an inverted OR gate. An inversion bubble is placed at the output of the OR gate. The Boolean expression is x A B NOR operator symbol is x A B The Hashemite University 29
NOR Logic Gate II Truth table and gate symbol (Compare it with the OR gate truth table): A B 0 0 x A B 1 0 1 0 1 0 0 1 1 0 A B x The Hashemite University 30
Timing Diagram of NOR Gate The Hashemite University 31
Exclusive OR Gate I Abbreviated as XOR gate. It gives 1 when its two inputs are of different values, otherwise it gives 0 (when they are equal). So, it is also called the binary difference operator. The Boolean expression is XOR operator symbol is x x AB A B AB The Hashemite University 32
Exclusive OR Gate II Truth table and gate symbol: A B 0 0 x A B 0 0 1 1 1 0 1 1 1 0 A B x The Hashemite University 33
Exclusive NOR Gate I Abbreviated as XNOR gate. It gives 1 when its two inputs are equal, otherwise it gives 0 (when they are different). So, it is also called the binary equivalence operator. It is the complement (or the inverter of XOR) The Boolean expression is XNOR operator symbol is x A B \ x AB A ' B ' The Hashemite University 34
Exclusive NOR Gate II Truth table and gate symbol: A B 0 0 x A B \ 1 0 1 0 1 0 0 1 1 1 A B x The Hashemite University 35
Extension to Multiple Inputs I All the logical gates that we have taken (except the unary ones) can be extended to have multiple inputs. The only condition for that is the binary operation represented by the gate must be associative and commutative. E.g. OR and AND gates. NAND and NOR gates are not associative. However, to make them extendable we define them as an inverted AND and as an inverted OR respectively. E.g.: three input NAND gate have the expression of (x.y.z) The Hashemite University 36
Extension to Multiple Inputs II XOR and XNOR are both commutative and associative can be extended to multiple inputs. Uncommon from hardware point of view. Multi-input XOR gate (more than 2 inputs) is called an odd-function since it produces 1 if and only if the number of 1 s (inputs that are 1) is odd. Otherwise, it will gives an output of 0. Multi-input XNOR gate (more than 2 inputs) is called an even-function since it produces 1 if and only if the number of 1 s (inputs that are 1) is even. Otherwise, it will gives an output of 0. The Hashemite University 37
Universality of NAND and NOR Gates I NAND or NOR gates can be used to create the three basic or fundamental logic gates (OR, AND, and INVERT). So, any circuit can be implemented totally using NAND gates alone or NOR gates alone (inverters are always allowed). For this they are called universal gates and they are used in Ics fabrication. This characteristic provides flexibility and is very useful in logic circuit design. The Hashemite University 38
Implement AND, OR using NAND The Hashemite University 39
Implement AND, OR using NOR The Hashemite University 40
Boolean Function Using only NOR or only NAND So, any boolean function can implemented totally using NAND gates or using NOR gates. Example: On board. The Hashemite University 41
Logic Symbol Interpretation When an input or output on a logic circuit symbol has no bubble on it, that line is said to be active-high. Otherwise the line is said to be active-low. When a logic signal is in its active state, it can be said to be asserted (=activated). When a logic signal is not in its active state, it is said to be unasserted (=inactive). The Hashemite University 42
Example I The Hashemite University 43
Example 2 The Hashemite University 44
Which Gate Representation to Use? If the circuit is being used to cause some action when output goes to the 1 state, then use active-high representation. If the circuit is being used to cause some action when output goes to the 0 state, then use active-low representation. The Hashemite University 45
Additional Notes This lecture covers the following material from the textbook: Chapter 1: Section 1.9 Chapter 2: Sections 2.7 and 2.8 Chapter 3: Section 3.6 The Hashemite University 46