Digital Logic Design 1. Truth Tables. Truth Tables. OR Operation With OR Gates

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1 2007 oolean Constants and Variables K TP.HCM Tran Ngoc Thinh HCMC University of Technology oolean algebra is an important tool in describing, analyzing, designing, and implementing digital circuits. oolean 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, ND, and NOT. Truth Tables truth table describes the relationship between the input and output of a logic circuit. The number of entries corresponds to the number of inputs. For eample a 2 input table would have 2 2 = 4 entries. 3 input table would have 2 3 = 8 entries. Truth Tables Eamples of truth tables with 2, 3, and 4 inputs. OR Operation With OR Gates The oolean epression for the OR operation is X = + This is read as equals or. X = 1 when = 1 or = 1. Truth table, circuit symbol and timing diagram for a two input OR gate: OR Operation With OR Gates The OR operation is similar to addition but when = 1 and = 1, the OR operation produces = 1. In the oolean epression =1+1+1=1 We could say that is true (1) when is true (1) OR is true (1) OR C is true (1). In general, the output of an OR gate is HIGH whenever one or more inputs are HIGH C D

2 OR Operation With OR Gates There are many eamples of applications where an output function is desired when one of multiple inputs is activated. ND Operations with ND gates The oolean epression for the ND operation is X = This is read as equals and. = 1 when = 1 and = 1. Truth table and circuit symbol for a two input ND gate are shown. Notice the difference between OR and ND gates. ND Operation With ND Gates The ND operation is similar to multiplication. In the oolean epression X = C X = 1 only when = 1, = 1, and C = 1. The output of an ND gate is HIGH only when all inputs are HIGH. NOT Operation The oolean epression for the NOT operation is X = This is read as: equals NOT, or equals the inverse of, or equals the complement of Truth table, symbol, and sample waveform for the NOT circuit. C Describing Logic Circuits lgebraically The three basic oolean operations (OR, ND, NOT) can describe any logic circuit. Eamples of oolean epressions for logic circuits: Describing Logic Circuits lgebraically The output of an inverter is equivalent to the input with a bar over it. Input through an inverter equals. Eamples using inverters.

3 Evaluating Logic Circuit Outputs Rules for evaluating a oolean epression: Perform all inversions of single terms. Perform all operations within parenthesis. Perform ND operation before an OR operation unless parenthesis indicate otherwise. If an epression has a bar over it, perform the operations inside the epression and then invert the result. Evaluating Logic Circuit Outputs Evaluate oolean epressions by substituting values and performing the indicated operations: = 0, = 1, C = 1, and D = 1 = C( + D) = (0 + 1) = (0 + 1) = (1) = = 0 Evaluating Logic Circuit Outputs Output logic levels can be determined directly from a circuit diagram. Technicians frequently use this method. The output of each gate is noted until a final output is found. Implementing Circuits From oolean Epressions It is important to be able to draw a logic circuit from a oolean epression. The epression = C could be drawn as a three input ND gate. more comple eample such as y = C + C + C could be drawn as two 2-input ND gates and one 3-input ND gate feeding into a 3-input OR gate. Two of the ND gates have inverted inputs. NOR Gates and NND Gates Combine basic ND, OR, and NOT operations. The NOR gate is an inverted OR gate. n inversion bubble is placed at the output of the OR gate. The oolean epression is = + NOR Gates and NND Gates The NND gate is an inverted ND gate. n inversion bubble is placed at the output of the ND gate. The oolean epression is =

4 oolean Theorems The theorems or laws below may represent an epression containing more than one variable. 0 = 0 = + 1 = 1 + = 1 1 = 1 = 0 + = Laws of oolean lgebra Commutative Laws ssociative Laws Distributive Law Commutative Laws of oolean lgebra + = + ssociative Laws of oolean lgebra + ( + C) = ( + ) + C = ( C) = ( ) C Distributive Laws of oolean lgebra ( + C) = + C ( + C) = + C Rules of oolean lgebra ( + C) = + C ( + C) = + C ( C) = ( ) C

5 Rules of oolean lgebra ( + C) = + C ( + C) = + C Rules of oolean lgebra ( + C) = + C ( + C) = + C Rules of oolean lgebra Rules of oolean lgebra Rule 11: + = + Rule 10: + = Rule 12: ( + )( + C) = + C DeMorgan s Theorems Theorem 1: When the OR sum of two variables is inverted, it is equivalent to inverting each variable individually and NDing them. + =. Theorem 2: When the ND product of two variables is inverted, it is equivalent to inverting each variable individually and ORing them.. = + DeMorgan s Theorems NOR gate is equivalent to an ND gate with inverted inputs. NND gate is equivalent to an OR gate with inverted inputs. For N variables, DeMorgan s theorem is epressed as: and

6 Universality of NND and NOR Gates Universality of NND and NOR Gates NND or NOR gates can be used to create the three basic logic epressions (OR, ND, and INVERT) lternate 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 eist on the standard symbol. Change a standard OR gate to and ND gate, or an ND gate to an OR gate. lternate Logic-Gate Representations Standard and alternate symbols for various logic gates and inverter lternate Logic-Gate Representations lternate Logic-Gate Representations The equivalence can be applied to gates with any number of inputs. No standard symbols have bubbles on their inputs. ll of the alternate symbols do. The standard and alternate symbols represent the same physical circuitry. ctive high an input or output has no inversion bubble. ctive low an input or output has an inversion bubble. n ND gate will produce an active output when all inputs are in their active states. n OR gate will produce an active output when any input is in an active state.

7 lternate Logic-Gate Representations Interpretation of the two NND gate symbols. lternate Logic-Gate Representations Interpretation of the two OR gate symbols. Which Gate Representation to Use Which Gate Representation to Use 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. When a logic signal is in the active state (high or low) it is said to be asserted. When a logic signal is in the inactive state (high or low) it is said to be unasserted. bar over a signal means asserted (active) low. The absence of a bar over a signal means asserted (active) high. IEEE/NSI Standard Logic Symbols (a) Original circuit using standard NND symbols; (b) equivalent representation where output Z is active- HIGH; (c) equivalent representation where output Z is active- LOW; (d) truth table. Rectangular symbols represent logic gates and circuits. Dependency notation inside symbols show how output depends on inputs. small triangle replaces the inversion bubble.