Set_2 - Classification of Digital Circuits Combinational...

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Classification of Digital Circuits Combinational. Output depends only on current input values. Sequential. Output depends on current input values and present state of the circuit, where the present state of the circuit is the current value of the devices’ memory.
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A binary switch x 1 = x 0 = (a) Two states of a switch S x (b) Symbol for a switch
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Variables and Functions L = 1 the light is on; L = 0 the light is off; The state of the light can be described as a function of the input variable x : L(x) = x S Battery Light x
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AND Function x1 x2 L 0 0 0 0 1 0 1 0 0 1 1 1 L(x 1 ,x 2 ) = x 1 • x 2 S Power supply S Light x 1 x 2
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OR Function L(x 1 ,x 2 ) = x 1 + x 2 x1 x2 L 0 0 0 0 1 1 1 0 1 1 1 1 S Power supply S Light x 1 x 2
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Example of a Logic Network L(x 1 , x 2 , x 3 ) = ( x 1 + x 2 ) • x 3 S Power supply S Light S X 1 X 2 X 3
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Inverter Function L(x) = x’ x L 0 1 1 0 S Light Power supply R x
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Logic Gates x 1 x 2 x n x 1 x 2 x n + + + x 1 x 2 x 1 x 2 + x 1 x 2 x n x 1 x 2 x 1 x 2 x 1 x 2 x n (a) AND gates (b) OR gates x x (c) NOT gate
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Analysis of a Logic Circuit x1 x2 A B f 0 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 1
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Analysis of a Logic Circuit 1 0 1 0 1 0 1 0 1 0 x 1 x 2 A B f Time (c) Timing diagram
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Analysis of a Logic Circuit f (x 1 ,x 2 ) = x 1 ’ + x 1 • x 2
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Functionally Equivalent Circuits
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Boolean Algebra 1854, George Boole created a two valued algebraic system which is now called Boolean algebra . 1938, Claude Shannon adapted Boolean algebra to analyze and describe the behavior of circuits built with relays. This adaptation is called switching algebra .
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Switching Algebra In switching algebra the condition of a logic signal is represented by symbolic variables, such as x, y, and/or z, and these variables can only have two values, 0 or 1. Two possible conventions: Positive Logic. Where LOW = 0 and HIGH = 1. Negative Logic. Where LOW = 1 and HIGH =0.
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Axioms The axioms or postulates of a mathematical system are a minimum set of basic definitions that are assumed to be true, and from which all other information about the system can be derived.
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Axioms Logical multiplication ( , ). (1a) 0 0 = 0 (2a) 1 1 = 1 (3a) 0 1 = 1 0 = 0 Logical addition ( + , ). (1b) 1 + 1 = 1 (2b) 0 + 0 = 0 (3b) 1 + 0 = 0 + 1 = 1
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Axioms Complement. (4a) If X = 0, then X’ = 1. (4b) If X = 1, then X’ = 0. Notation. X X' X ~ X ¬
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Axioms The axioms stated below embody the “digital abstraction” by formally stating that X can take on only one of two values. X = 0 if X 1 X = 1 if X 0
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Precedence By convention, the precedence of operations in a logic expression is the following: Parentheses. Complement. Multiplication. Addition.
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Theorems Theorems are statements, known to be always true, that are used to manipulate algebraic expressions to allow simpler analysis or more efficient synthesis of circuits.
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