Basics of Digital Electronics

1. Logic Gates & Boolean Algebra
Course: B.Sc.(CS)
Sem.:1st
Subject: Basics of Digital Electronics
Unit: 2nd

2. Basic Definitions
 Binary Operators
◦ AND
z = x • y = x y z=1 if x=1 AND y=1
◦ OR
z = x + y z=1 if x=1 OR y=1
◦ NOT
z = x = x’ z=1 if x=0
 Boolean Algebra
◦ Binary Variables: only ‘0’ and ‘1’ values
◦ Algebraic Manipulation

3. Boolean Algebra Postulates
 Commutative Law
x • y = y • x x + y = y + x
 Identity Element
x • 1 = x x + 0 = x
 Complement
x • x’ = 0 x + x’ = 1

4. Boolean Algebra Theorems
 Duality
◦ The dual of a Boolean algebraic expression is
obtained by interchanging the AND and the OR
operators and replacing the 1’s by 0’s and the 0’s
by 1’s.
◦ x • ( y + z ) = ( x • y ) + ( x • z )
◦ x + ( y • z ) = ( x + y ) • ( x + z )
 Theorem 1
◦ x • x = x x + x = x
 Theorem 2
◦ x • 0 = 0 x + 1 = 1
Applied to a
valid equation
produces a
valid equation

5. Boolean Algebra Theorems
 Theorem 3: Involution
◦ ( x’ )’ = x ( x ) = x
 Theorem 4: Associative & Distributive
◦ ( x • y ) • z = x • ( y • z )( x + y ) + z = x + ( y + z
)
◦ x • ( y + z ) = ( x • y ) + ( x • z )
x + ( y • z ) = ( x + y ) • ( x + z
)
 Theorem 5: De Morgan
◦ ( x • y )’ = x’ + y’ ( x + y )’ = x’ • y’
◦ ( x • y ) = x + y ( x + y ) = x • y
 Theorem 6: Absorption
◦ x • ( x + y ) = x x + ( x • y ) = x

6. Boolean Functions
 Boolean Expression
Example: F =
x + y’ z
 Truth Table
All possible
combinations
of input variables
 Logic Circuit
x y z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1x
y
z
F

7. Algebraic Manipulation
 Literal:
A single variable within a term that may be
complemented or not.
 Use Boolean Algebra to simplify Boolean
functions to produce simpler circuits
Example: Simplify to a minimum number of
literals
F = x + x’ y ( 3 Literals)
= x + ( x’ y )
= ( x + x’ ) ( x + y )
= ( 1 ) ( x + y ) = x + y ( 2 Literals)
Distributive law (+ over
•)

8. Complement of a Function
 DeMorgan’s Theorm
 Duality & Literal Complement
CBAF 
CBAF 
CBAF 
CBAF 
CBAF 
CBAF 

9. Canonical Forms
 Minterm
◦ Product (AND function)
◦ Contains all variables
◦ Evaluates to ‘1’ for a
specific combination
Example
A = 0 A B C
B = 0 (0) • (0) • (0)
C = 0
1 • 1 • 1 = 1
A B C Minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7

10. Canonical Forms
 Maxterm
◦ Sum (OR function)
◦ Contains all variables
◦ Evaluates to ‘0’ for a
specific combination
Example
A = 1 A B C
B = 1 (1) + (1) + (1)
C = 1
0 + 0 + 0 = 0
A B C Maxterm
0 0 0 0 M0
1 0 0 1 M1
2 0 1 0 M2
3 0 1 1 M3
4 1 0 0 M4
5 1 0 1 M5
6 1 1 0 M6
7 1 1 1 M7

11. Canonical Forms
 Truth Table to Boolean Function
CBAF  CBA CBA ABCA B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1

12. Canonical Forms
 Sum of Minterms
 Product of Maxterms
ABCCBACBACBAF 
7541 mmmmF 
 )7,5,4,1(F
CABBCACBACBAF 
CABBCACBACBAF 
CABBCACBACBAF 
))()()(( CBACBACBACBAF 
6320 MMMMF 
 (0,2,3,6)F

13. Standard Forms
 Sum of Products (SOP)
ABCCBACBACBAF  BA
BA
CCBA



)1(
)(
AC
BBAC

 )(
CB
AACB

 )(
)()()( BBACCCBAAACBF 
ACBACBF 

14. Standard Forms
 Product of Sums (POS)
CABBCACBACBAF 
)( CCBA 
)( AACB 
)( BBCA 
)()()( AACBCCBABBCAF 
CBBACAF 
))()(( CBBACAF 

15. Two-Level Implementations
 Sum of Products (SOP)
 Product of Sums (POS)
B’
C
FB’
A
A
C
A
C
FB’
A
B’
C
ACBACBF 
))()(( CBBACAF 

16. Logic Operators
 AND
 NAND (Not AND)
x
y
x • y
x
y
x • y
x y AND
0 0 0
0 1 0
1 0 0
1 1 1
x y NAND
0 0 1
0 1 1
1 0 1
1 1 0

17. Logic Operators
 OR
 NOR (Not OR)
x
y
x + y
x
y
x + y
x y OR
0 0 0
0 1 1
1 0 1
1 1 1
x y NOR
0 0 1
0 1 0
1 0 0
1 1 0

18. Logic Operators
 XOR (Exclusive-OR)
 XNOR (Exclusive-NOR)
(Equivalence)
x
y
x Å y
x y + x y
x
y
x Å y
x  y
x y + x y
x y XOR
0 0 0
0 1 1
1 0 1
1 1 0
x y XNOR
0 0 1
0 1 0
1 0 0
1 1 1

19. Logic Operators
 NOT (Inverter)
 Buffer
x x
x x
x NOT
0 1
1 0
x Buffer
0 0
1 1

20. Multiple Input Gates




21. De Morgan’s Theorem on
Gates
 AND Gate
◦ F = x • y F = (x • y) F = x +
y
 OR Gate
◦ F = x + y F = (x + y) F = x • y
 Change the “Shape” and “bubble” all lines

22. Universal Gates

23. NOR Gate as an Inverter
Gate
X XZ 
XXX  (Before Bubble)
X Z
0 1
1 0
Equivalent to Inverter

24. NOR Gate as an OR Gate
X
Y
YXYXZ 
YX 
NOR Gate “Inverter”
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
Equivalent to OR Gate

25. NOR Gate as an AND Gate
YXYXYXZ 
“Inverters” NOR Gate
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
Equivalent to AND Gate

26. NOR Gate Equivalent of AOI
Gates
AND OR INVERTER

27. 1. If starting from a logic expression,
implement the design with AOI logic.
2. In the AOI implementation, identify and
replace every AND,OR, and INVERTER
gate with its NOR equivalent.
3. Redraw the circuit.
4. Identify and eliminate any double
inversions. (i.e. back-to-back inverters)
5. Redraw the final circuit.
Process for NOR Implementation

28. NAND Gate
X
Y
YXYXZ 
X Y Z
0 0 1
0 1 1
1 0 1
1 1 0

29. NAND Gate as an Inverter
Gate
XXX 
XZ 
X Z
0 1
1 0 Equivalent to Inverter

30. NAND Gate as an AND Gate
X
Y
YXYXZ 
YX
NAND Gate Inverter
X Y Z
0 0 0
0 1 0
1 0 0
1 1 1
Equivalent to AND
Gate

31. NAND Gate as an OR Gate
X
Y
YXYXYXZ 
X
NAND GateInverters
Y
X Y Z
0 0 0
0 1 1
1 0 1
1 1 1
Equivalent to OR Gate

32. NAND Gate Equivalent to AOI
Gates
AND OR INVERTER

33. 1. If starting from a logic expression,
implement the design with AOI logic.
2. In the AOI implementation, identify and
replace every AND,OR, and INVERTER
gate with its NAND equivalent.
3. Redraw the circuit.
4. Identify and eliminate any double
inversions (i.e., back-to-back inverters).
5. Redraw the final circuit.
Process for NAND
Implementation

34. References
 Book References:
 Digital Logic and Computer Design –
M. Morris Mano – Pearson
 Fundamentals of Digital Circuits – A.
Anand Kumar – PHI
 Modern Digital Electronics - R.P. Jain -
TMH
 Digital Electronics -Tokneinh – MGH

35. Web References
 www.ddegjust.ac.in/studymaterial/pg
dcal
 www.indianshout.com/digital-
electronics-notes-material/3023
 www.wiley.com/college/engin/balabani
an293512/pdf/ch04.pdf
 www.electronics-tutorials.ws
 Image References:
 http://www.youshouldgoaway.com/wp-
content/uploads/2014/11/thankyou.jpg