18Elec 326 Gates and Logic NetworksGatesGate: A simple electronic circuit (a system) that realizes a logical operation.  The direction of information flow is from the input terminals to the output terminal. The number of input and output terminals is finite and they carry binary-valued signals (i.e, 0 and 1). The transformation of input signals to output signals can be modeled as a logical operation.Reading assignment Sections 2.1—2.5 from the textXYZ = f(X,Y)Gate9Elec 326 Gates and Logic NetworksTruth Tables Since there is a finite number of input signal combinations, we can represent the behavior of a gate by simply listing all of it possible input configurations and the corresponding output signal. Such a list is called a truth table. For example, the following gate could have the behavior given by the following truth tables. The use of the symbols L and H usually correlates with the high and low voltages.GATEXYZXY ZL L L L H H H L H H H HXY Z0 0 0 0 1 1 1 0 1 1 1 1XY Z1 1 1 1 0 0 0 1 0 0 0 023Elec 326 Gates and Logic NetworksSome standard gates and their symbols and truth tables:XZ0 01 1X Za. BufferXZ0 11 0X Zb. InverterXYZX Y Z0 0 00 1 01 0 01 1 1c. AND GateXYZX Y Z0 0 00 1 11 0 11 1 1d. OR GateX Y Z0 0 10 1 11 0 11 1 0ore. NAND GateXYZXYZorX Y Z0 0 10 1 01 0 01 1 0f. NOR GateXYZXYZX Y Z0 0 00 1 11 0 11 1 0g. Exclusive OR GateXYZX Y Z0 0 10 1 01 0 01 1 1h. Equivalence GateXYZ4Elec 326 Gates and Logic Networks Gates with more than 3 inputs:AND gates: The output is 1 if and only if … ?OR gates: The output is 1 if and only if … ?NAND gates: The output is 0 if and only if … ?NOR gates: The output is 0 if and only if … ?EXCLUSIVE OR gates: The output is 1 if and only if … ?EQUIVALENCE gates: The output is 1 if and only if … ?35Elec 326 Gates and Logic NetworksLogical Expressions We can also represent the behavior of gates with a logical expressions constructed from variables and logical operations symbols. The following table gives the most common ones.C is 1 iff Both A and B are 1 or both A and B are 0.C = A≡BEQUIVALENCEC is 1 iff It is not the case that either A or B is 1.C = A↓BNORC is 1 iff It is not the case that A and B are both 1. C = A↑BNANDC is 1 iff A or B is 1, both not both.C = A⊕BEXCLUSIVE ORC is 1 iff A is 1 or B is 1.C = A+BORC is 1 iff A is 1 and B is 1.C = A•BANDC is 1 iff A is 0.C = A'NEGATIONMeaningExampleConnective6Elec 326 Gates and Logic Networks Comments on the logical symbols The NAND and NOR symbols are not very useful. There are several different symbols that have been used for the logical connectives. Exercise:Determine how many different two-input gates there can be?How many three-input gates?AND : •, &, ∧OR : +, |, ∨NOT : ', ,~,¬N 22n2163 2564 65,5365 4, 294,967,3966 1.84 x 101947Elec 326 Gates and Logic NetworksGate NetworksA gate network is a finite collection of interconnected gates, network input terminals, and network output terminals with the following restrictions: No gate output terminal or network input terminal is connected to another gate output terminal or network input terminal. Every network output terminal or gate input terminal is wired (via one or more wires) to a constant value, a network input terminal, or a gate output terminal.ExampleNetworkInputTerminalsNetwork Output TerminalsABCXY8Elec 326 Gates and Logic Networks Types of networks A combinational gate network is one in which the values of the signals present on its input terminals uniquely determine the signal values at its output terminals. A gate network that is not combinational is called a sequential gate network.  A loop in a gate network is a path that starts at a gate terminal, passes along wires and through gates, does not pass any wire or gate more than once, and terminates back at the starting gate terminal.  Networks without loops are combinational. z We call a gate network without loops a logic network, since we can describe its behavior with a logical expression.Sequential networks have loops. Combinational networks may have loops.Loop59Elec 326 Gates and Logic NetworksExercise:Which of the following networks are combinational and which are sequential?Net 1Net 2Net 3Net 410Elec 326 Gates and Logic NetworksAnalysis & Synthesis of Logic NetworksOverviewLogicNetworkTruthTableLogicalExpression1234567AnalysisSynthesis1. For a given logic network, find a truth table that describes its behavior.2. For a given logic network, find a set of logical expressions that describes its behavior.3. Transform a logical expression into the equivalent truth table representation.4. Transform a truth table into an equivalent logical expression representation.5. Transform a logical expression into an equivalent (and possibly simpler) logical expression.6. Design a logic network to have the behavior specified by a given set of logical expressions.7. Design a logic network to have the behavior specified by a given truth table.611Elec 326 Gates and Logic NetworksAnalysis of Logic Networks Logic Network Truth Table Behavioral DescriptionA B C Z1 Z2 Z3 Z4 X Y0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 16ABC1Z1Z2Z3Z42345XY11111100001111000000010111111100000000110000010012Elec 326 Gates and Logic Networks Logical Expression Behavioral Description6ABC1Z1Z2Z3Z42345XYX = Z4'X = (Z1+Z2)'X = ((A•B)'+(A⊕B))'Y = Z2•Z3•Z4Y = (A ⊕B)•(A•C)•(Z1+Z2)Y = (A ⊕B)•(A•C)•((A•B)'+(A ⊕B))713Elec 326 Gates and Logic Networks ExampleX = A•BY = A•B'•C The networks in these two examples (slides 2.19 and 2.20) are equivalent because they have the same truth table. The logical expressions for X and Y are also equivalent, but very different structurally. Graph levelization for analysisA B C X Y0 0 0 0 00 0 1 0 00 1 0 0 00 1 1 0 01 0 0 0 01 0 1 0 11 1 0 1 01 1 1 1 0ABCXY14Elec 326 Gates and Logic Networks Notation: We can represent a truth table by simply listing the indices of the rows that have value 1 or listing those that have value 0. Examplez List of 1’s:X = ΣA,B,C(6, 7) Y = ΣA,B,C(5)z List of