Digital Logic

Boolean Algebra to Logic Gates Logic circuits are built from components called logic gates. The logic gates correspond to Boolean operations +, *, ’. Binary operations have two inputs, unary has one OR + AND * NOT’

AND A B A*B Logic Gate: Series Circuit: ABABA*B Truth Table: A*B

A B A+B Logic Gate: Parallel Circuit: A B ABA+B Truth Table: A+B OR

NOT A A’ or A Logic Gate: (also called an inverter) Single-throw Double-pole Switch: A aA Truth Table: A’ or A

n -input Gates Because + and * are binary operations, they can be cascaded together to OR or AND multiple inputs. A B C A B C A+B+C A B A B C ABC

n -bit Inputs For convenience, it is sometimes useful to think of the logic gates processing n -bits at a time. This really refers to n instances of the logic gate, not a single logic date with n -inputs

Logic Circuits ≡ Boolean Expressions All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit. AND-OR logic circuits are equivalent to sum-of-products form. Consider the following circuits: A C B abc aBc Ab y=abc+aBc+Ab y A B C Y y=aB+Bc

NAND and NOR Gates NAND and NOR gates can greatly simplify circuit diagrams. As we will see, can you use these gates wherever you could use AND, OR, and NOT. NAND NORAB ABABABAB AB ABABABAB

XOR and XNOR Gates XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both. XOR XNOR AB ABABABAB ABA B

Properties of NAND AND NOR NAND and NOR have special properties, but neither satisfies the distributive or associative laws. NANDNOR x  1=X x  0=X x  0=1 x  1=0 x  x=X x  x=X x  y=X+Y x  y=XY X  Y=x+y X  Y=xy not (x  y)=xy not (x  y)=x+y It should be clear by looking at these properties that NAND and NOR are duals.

NAND and NOR as Universal Logic Gates Any logic circuit can be built using only NAND gates, or only NOR gates. They are the only logic gate needed. Here are the NAND equivalents:

NAND and NOR as Universal Logic Gates (cont) Here are the NOR equivalents: NAND and NOR can be used to reduce the number of required gates in a circuit.

Example Problem A hall light is controlled by two light switches, one at each end. Find (a) a truth function, (b) a Boolean expression, and (c) a logic network that allows the light to be switched on or off by either switch. xy f(x,y) (What kind of gate has this truth table? Let x and y be the switches:

Example (cont) One possible equation is the complete sum-of-products form: f(x,y) = xY + Xy Use The Most Complex Machine xLogicCircuit Module to implement the equation. xy f(x,y)