Computer Science 210 Computer Organization Introduction to Logic Circuits (Chapter 3)

From Boolean Logic to Hardware Boolean values true and false map to binary values 1 and 0 Binary values 1 and 0 map to discrete states on and off Discrete states on and off map to voltage levels +5V and 0V on an analog device Connect these devices to perform logical operations

Transistor - Electronic Switch Collector Base Emitter Switch Base High (+5v or 1) Makes connection Base Low (0v or 0) Disconnects

AND Gate Output is 1 only if Output = Input1 AND Input2 +5v Input-1 is 1 and Input-2 is 1 Output = Input1 AND Input2 Input-1 Input-2 AND Gate A AB B Output

OR Gate Output is 1 if Output = A OR B A is 1 or if B is 1 +5v A B

NOT Gate Input High (+5v or 1) Output Low (0v or 0) Input Low (0v or 0) Output High (+5v or 1) Output is opposite of Input +5v Output Input NOT Gate A _ Ground

Basic Gates AND Gate A AB B NAND Gate A B ___ AB OR Gate A A + B B NOR Gate A B _____ A + B NOT Gate A _

Boolean Exp  Logic Circuit To draw a circuit from a Boolean expression: From the left, make an input line for each variable. Next, put a NOT gate in for each variable that appears negated in the expression. Still working from left to right, build up circuits for the subexpressions, from simple to complex.

Logic Circuit A~B + ~(A+B)B Input Lines for Variables A B

Logic Circuit A~B + ~(A+B)B NOT Gate for B A B _ B

Logic Circuit A~B + ~(A+B)B _ Subexpression AB _ AB A B _ B

Logic Circuit A~B + ~(A+B)B Subexpression A+B _ AB A B A+B _ B

Logic Circuit A~B + ~(A+B)B ___ Subexpression A+B _ AB A ____ A+B B A+B _ B

Logic Circuit A~B + ~(A+B)B ___ Subexpression (A+B)B _ AB A ____ A+B B A+B _ B ____ (A+B)B

Logic Circuit A~B + ~(A+B)B Entire Expression _ AB A B A+B ____ A+B _ B ____ (A+B)B

Designing a Circuit from a Truth Table The problem reduces to this: Given a truth table with all values for inputs. And given a column of values for the output. Find a Boolean expression that gives the column. If we can do this, we can get the circuit from the Boolean expression. The Sum of Products algorithm finds the expression.

Sum of Products Algorithm Identify each row of the output that has a 1. For each such row Make a product of all the input variables. Put bar over each variable with a 0 in this row. Make a sum of all of these product terms. Then, simplify the expression if possible.

Sum of Products Example https://www.allaboutcircuits.com/textbook/digital/chpt-7/converting-truth-tables-boolean-expressions/

The Boolean Triangle Boolean Expression Logic Circuit Truth Table