Logic Circuit - Engineering Electrical Circuits - Lecture Slides, Slides of Electrical Circuit Analysis

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Logic Circuit

Synthesis

ReCall the Basic Gates

Boolean Algebra

• Boolean Logic can be Mathematically Formalized

with the use of Math Operators

• The Math Operators Corresponding to Boolean

Logic Operations:

• A and B can only be TRUE or FALSE
• TRUE represented by 1; FALSE by 0

Operator Usage Notation

AND A AND B A.B or A·B

OR A OR B A+B

NOT NOT A ~A or A

Boolean Algebraic Properties

• Commutative : A.B = B.A and A+B = B+A
• Distributive :
  • A.(B+C) = (A.B) + (A.C)
  • A+(B.C) = (A+B).(A+C)
• Identity Elements : 1.A = A and 0 + A = A
• Inverse: A.A = 0 and A + A = 1

 Associative:

• A.(B.C) = (A.B).C and A+(B+C) = (A+B)+C

 DeMorgan's Laws:

• A.B = A + B and
• A+B = A.B

Prove Distributive Law

• Constructing a Truth Table that Includes and

Expands [AB+AC] & [A(B+C)] A B C B+C AB AC AB+AC A(B+C) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 IDENTICAL

Prove XOR distributive Law

• A Truth Table AlsoProves

the EXCLUSIVE-OR Version of the Dist Law

A(B⊕C) ≡ AB⊕AC

A B C B ⊕ C AB AC AB ⊕ AC A(B ⊕ C) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 IDENTICAL

DeMorgan in MATLAB

• DeMorgan → Case-b

% case-b

y = 3 y = 3 b1 = ~((y == 2)|(y > 5)) b1 = 1 b2 = (~(y == 2))&(~(y >5)) b2 = 1

( A + B ) = A ∗ B

?

Multiple-Input Gates

• The Flexibility of

Boolean Algebra

permits straightforward

Extension to Gates with

Many Inputs

• The Number of Entries

in the TT is 2N^ , where:

• N ≡ No. of INputs

Boolean  Gates (WhiteBoard)

• Draw the Logic-Gate Circuits to Implement these Boolean Equations - Use Multiple-Input Gates as desired

Q 1 = ( A ⋅ B ⋅ C ) + A ⋅( B + C )

Q 2 = ( B + C + D ) ⋅( A + C + D ) ⋅( C + D )

Principal Boolean Algebra Laws OR expression AND expression Law Name 0 + a = a 1a = a Identity 1 + a = 1 0a = 0 a + a = a aa = a Idempotency a + a' = 1 aa' = 0 Complement (a')' = a Involution a + b = b + a ab = ba Commutativity a +(bc) = (a + b)(a + c) a(b + c) = ab + ac Distributivity a + (b + c) = (a + b) + c a(bc) = (ab)c Associativity a + ab = a a(a + b) = a Absorption a + a'b = a + b a(a' + b) = ab Simplification ( a + b)' = a' b' (ab)' = a' + b' DeMorgan's Law ab + a'c + bc = ab + a'c (a+b)(a'+c)(b+c) = (a+b)(a'+c) Consensus Thrm

NOTE: (^) a ' ≡ a b '≡ b etc.

MinTerms  Sum of Products

• The Function Operation

Requires

• The output to be 1 whenever - X=0 AND Y=1 AND Z=
• OR when
  • X=1 AND Y=0 AND Z=
• This Description can

written using Boolean

Algebra:

• Function read as:

(NOT-X AND Y AND

NOT-Z) OR (X AND

NOT-Y AND Z)

• By Way of
  • NOT-X = 0
  • NOT-Y = 0
  • Etc.

Q = X ⋅ Y ⋅ Z + X ⋅ Y ⋅ Z

X 0 Y 0 (^) Z 0 Q 0 (^00 01 10 ) (^01 10 10 ) (^11 01 10 ) 1 1 1 0

MinTerms  Sum of Products

• The Ckt Below

Implements Fcn:

• The function is

composed of two

groups of three.

• Each group of three is a

minterm.

• minterm implies that each of the groups of 3 in the expression takes on a value of 1 only for one of the 8 possible combos of X, Y and Z and their inverses

Q = X ⋅ Y ⋅ Z + X ⋅ Y ⋅ Z

Q

X 0 Y 0 (^) Z 0 Q 0 (^00 01 10 ) (^01 10 10 ) (^11 01 10 ) 1 1 1 0

Boolean Eqn from TruthTable

• It turns out that ANY Boolean function can be

constructed using minterms

• To build a Boolean fcn from minterms:
  1. Examine the truth table for the function.
     • Be sure that all possible combinations of variables and inverses are accounted for.
  2. For each entry of the truth table for which the Function takes on a value of 1 , determine the corresponding minterm expression (an AND expression)
     • Remember that EVERY variable or its inverse will appear in EVERY minterm. 3. OR (add)Together all the minterms from step-

MinTerms & SOP Sumarized

• We can sum up the MinTerm Construiction

with the following:

• A truth table gives a unique

sum-of-products function that

follows directly from expanding

the ones in the truth table as

minterms.