Introduction to Logic Design Lecture 3 Dr. Chuck rown Engineering and Computer Information Science Folsom Lake College
Outline for Todays Lecture Logic Circuits SOP / POS oolean Theorems DeMorgan s Theorem <2>
Recap - Circuits Nodes Inputs:,, C Outputs:, Z E1 n1 Internal: n1 E3 Circuit elements C E2 Z E1, E2, E3 Each a circuit <3>
Recap - Types of Logic Circuits Combinational Logic Memoryless Outputs determined by current values of inputs Sequential Logic Has memory Outputs determined by previous and current values of inputs inputs functional spec timing spec outputs <4>
Rules of Combinational Composition Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths Example: <5>
oolean Equations Functional specification of outputs in terms of inputs Example: S = F(,, C in ) C out = F(,, C in ) S C L C C out in S = C in C out = + C in + C in <6>
Some Definitions Complement: variable with a bar over it,, C Literal: variable or its complement,,,, C, C Implicant: product of literals C, C, C Minterm: product that includes all input variables C, C, C Maxterm: sum that includes all input variables (++C), (++C), (++C) <7>
Sum-of-Products (SOP) Form ll equations can be written in SOP form Each row has a minterm minterm is a product (ND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE Thus, a sum (OR) of products (ND terms) 0 0 0 0 1 1 1 0 0 1 1 1 minterm minterm name m 0 m 1 m 2 m 3 = F(, ) = <8>
Sum-of-Products (SOP) Form ll equations can be written in SOP form Each row has a minterm minterm is a product (ND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE Thus, a sum (OR) of products (ND terms) 0 0 0 0 1 1 1 0 0 1 1 1 minterm minterm name m 0 m 1 m 2 m 3 = F(, ) = <9>
Sum-of-Products (SOP) Form ll equations can be written in SOP form Each row has a minterm minterm is a product (ND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where the output is TRUE Thus, a sum (OR) of products (ND terms) 0 0 0 0 1 1 1 0 0 1 1 1 minterm minterm name m 0 m 1 m 2 m 3 = F(, ) = + = Σ(1, 3) <10>
Product-of-Sums (POS) Form ll oolean equations can be written in POS form Each row has a maxterm maxterm is a sum (OR) of literals Each maxterm is FLSE for that row (and only that row) Form function by NDing the maxterms for which the output is FLSE Thus, a product (ND) of sums (OR terms) 0 0 0 0 1 1 1 0 0 1 1 1 maxterm + + + + = F(, ) = ( + )( + ) = Π(0, 2) maxterm name M 0 M 1 M 2 M 3 <11>
oolean Equations Example ou are going to the cafeteria for lunch ou won t eat lunch (E) If it s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E). O C E 0 0 0 1 1 0 1 1 <12>
oolean Equations Example ou are going to the cafeteria for lunch ou won t eat lunch (E) If it s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E). O C E 0 0 0 1 1 0 1 1 0 0 1 0 <13>
SOP & POS Form SOP sum-of-products O C E 0 0 0 1 1 0 1 1 minterm O C O C O C O C POS product-of-sums O C E 0 0 0 1 1 0 1 1 maxterm O + C O + C O + C O + C <14>
SOP & POS Form SOP sum-of-products O C E 0 0 0 0 1 0 1 0 1 1 1 0 minterm O C O C O C O C E = OC = Σ(2) POS product-of-sums O C E 0 0 0 1 1 0 1 1 0 0 1 0 maxterm O + C O + C O + C O + C E = (O + C)(O + C)(O + C) = Π(0, 1, 3) <15>
oolean Theorems <16>
oolean Theorems ( ) Note: T8 differs from traditional algebra: OR (+) distributes over ND ( ) <17>
Simplifying oolean Equations Example 1: = + <18>
Simplifying oolean Equations Example 1: = + = ( + ) T8 = (1) T5 = T1 <19>
DeMorgan s Theorem The complement of the product of all the terms is equal to the sum of the complement of each term. That is, a NND gate is equivalent to an OR gate with inverted inputs = = + Likewise, the complement of the sum of all the terms is equal to the product of the complement of each term Or, a NOR gate is equivalent to an ND gate with inverted inputs = + = <20>
DeMorgan s -> ubble Pushing ackward: Gate changes dds bubbles to inputs Forward: Gate changes dds bubble to output <21>
DeMorgan s - > ubble Pushing What is the oolean expression for this circuit? C D <22>
DeMorgan s -> ubble Pushing What is the oolean expression for this circuit? C D = + CD <23>
DeMorgan s -> Procedure egin at output, then work toward inputs Push bubbles on final output back Draw gates in a form so bubbles cancel C D <24>
DeMorgan ubble Pushing Example C D <25>
DeMorgan ubble Pushing Example no output bubble C D <26>
DeMorgan ubble Pushing Example no output bubble C D bubble on input and output C D <27>
DeMorgan ubble Pushing Example no output bubble C D C D C D no bubble on input and output = C + D bubble on input and output <28>