Logic gates summary in digital electronics

IMPLEMENTING CIRCUITS FROM BOOLEAN EXPRESSIONS
Draw the circuit diagram to implement the expression f (A,B,C) = (A + B)(B + C).
F (A,B,C) = AC + BC + ABC

Universal Gates
KEC- 101
• A universal gate is a gate which can implement any Boolean function
without need to use any other gate type.
• The NAND and NOR gates are universal gates.
• This is advantageous since NAND and NOR gates are economical and
easier to fabricate and are the basic gates used in all IC digital logic
families.

Logic Gates Using Only NAND Gates
NAND AS NOT GATE
𝒀= 𝑨
.𝑩
= 𝑨
.𝑨
𝒀= 𝑨
NAND AS OR GATE
Y=A+B
𝒀=𝑨+𝑩
𝒀=𝑨.𝑩

Logic Gates Using Only NOR Gates
A
𝐀

Boolean Function Representation
• Various way of representing a given function
1- Sum of Product Form (SOP)
2- Product of Sum Form (POS)
3- Standard or Canonical SOP Form
4- Standard or Canonical POS Form
5-Truth Table Form
6- Karnaugh Map or K- Map

Sum of Product Form (SOP) Standard or Canonical SOP Form
• The Sum of Products is abbreviated as
SOP.
• It is the logical expression in Boolean
algebra where all the input terms are
ANDed (Product) first and then ORed
(summed) together.
• SOP form: F(A,B,C)=A+BC'+A'BC
• The variables in each term are not
necessarily all the variables of the
function.
• Standard SOP term must contain all
the function variables either in
complemented form or in
uncomplemented form.
• A product term which contain all the
function variables either in
uncomplemented form is called a
minterm.
F(A,B,C)=AB’C+A’BC'+A'BC

Conversion of SOP to Canonical SOP
F(A,B,C)=A+BC'+A'BC
=A+BC'+A'BC
=A(B+B')(C+C')+BC'(A+A')+A'BC
=ABC+ABC'+AB'C+AB'C'+ ABC'+A'BC'+A'BC
=ABC+ABC'+AB'C+AB'C'+ A'BC'+A'BC (A+A=A)

Product of Sum Form (POS) Standard or Canonical POS Form
• POS form means that the inputs of
each term are Added together
using OR function then all terms are
multiplied together using AND
function.
• The variables in each term are not
necessarily all the variables of the
function.
• POS form:
F(A,B,C)=A.(B+C').(A'+B+C')
• Standard POS term must contain all the
uncomplemented form.
• A sum term which contain all the
uncomplemented form is called a
maxterm.
• F(A,B,C)=(A+B+C)(A+B+C').(A'+B+C')

Conversion of POS to Canonical POS
F(A,B,C)=A.(B+C').(A'+B+C')
=[A+(B.B')+(C.C')].[(B+C')+(A.A')].(A'+B+C') =[(A+B+C).(A+B+C').(A+B'+C).
(A+B'+C')].[(A+B+C').(A'+B+C')].(A'+B+C’)
(A.A=A)
=(A+B+C).(A+B+C').(A+B'+C).(A+B'+C').(A'+B+C')

Example 1 – Express the Boolean function F = A + B’C as standard
sum of minterms.
A = A(B + B’) = AB + AB’
A = AB(C + C’) + AB'(C + C’) = ABC + ABC’+ AB’C + AB’C’
B’C = B’C(A + A’) = AB’C + A’B’C
F = A + B’C = ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C
F = A’B’C + AB’C’ + AB’C + ABC’ + ABC
= m1 + m4 + m5 + m6 + m7
=m(1,4,5,6,7)

Example 2 – Express the Boolean function F = (A+B’)(B+C) as a product of
max-terms
• F = (A+B’)(B+C)
• I term: (A+B’)= (A+B’+CC’)
= (A+B’+C) (A+B’+C’)
• II term: (B+C)= (AA’+B+C)
= (A+B+C) (A’+B+C)
• Combining both:
• F= (A+B’+C) (A+B’+C’) (A+B+C) (A’+B+C)
= M2 * M3 * M0 * M4
= ΠM(0,2,3,4)

Example 3 – Express the Boolean function F = xy + x’z as a product
of maxterms.
• F = xy + x’z
= (xy + x’)(xy + z)
= (x + x’)(y + x’)(x + z)(y + z)
= (x’ + y)(x + z)(y + z)
• x’ + y = x’ + y + zz’
= (x’+ y + z)(x’ + y + z’) x + z
• x + z + yy’
= (x + y + z)(x + y’ + z) y + z
• y + z + xx’
= (x + y + z)(x’ + y + z)
• F = (x + y + z)(x + y’ + z)(x’ + y + z)(x’ + y + z’)
= M0*M2*M4*M5
= πM(0,2,4,5)

Example 4–Convert F(A, B, C) = m(1,4,5,6,7) to POS FORM
• Missing terms of minterms = terms of maxterms
• Missing terms of maxterms = terms of minterms
• F(A, B, C) = m(1,4,5,6,7) =πM(0,2,3)
Example 5– Convert Boolean expression in standard form
F=y’+xz’+xyz
• F=y’+xz’+xyz
• F = (x+x’)y'(z+z’)+x(y+y’)z’ +xyz
• F = xy’z+ xy’z’+x’y’z+x’y’z’+ xyz’+xy’z’+xyz
• F = m5, m4, m1, m0, m6, m4, m7
• F= m (0,1,4,5,6,7)

Example 6: Generate truth table for F= xy + x’z
INPUTS OUTPUT
A B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 1
1 1 1 1
0 Maxterms
1 Minterms
Truth Table

Logic gates summary in digital electronics

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