1. Chapter 3 Combinational Design Digital Logic Design III
وزارة التعليم العالي والبحث العلمي
جامعة الكوفة - كلية التربية – قسم علوم الحاسوب
Digital Logic Design III
Chapter 3
Combinational Design
Dr. Wissam Hasan Mahdi Alagele

2. Combinational Circuits
Princess Sumaya University
Combinational Circuits
Digital Logic Design
Output is function of input only
i.e. no feedback
When input changes, output may change (after a delay)
Combinational
Circuits •
n inputs
m outputs 
Dr. Bassam Kahhaleh

3. Combinational Circuits
Princess Sumaya University
Digital Logic Design
Combinational Circuits
Analysis
Given a circuit, find out its function
Function may be expressed as:
Boolean function
Truth table
Design
Given a desired function, determine its circuit ? ?
Dr. Bassam Kahhaleh

4. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Boolean Expression Approach
T2=ABC
T1=A+B+C
F2=AB+AC+BC
F’2=(A’+B’)(A’+C’)(B’+C’)
T3=AB'C'+A'BC'+A'B'C
F1=AB'C'+A'BC'+A'B'C+ABC
F2=AB+AC+BC
Dr. Bassam Kahhaleh

5. Princess Sumaya University
Analysis Procedure
Digital Logic Design
Truth Table Approach
A B C F1 F2
= 0 1
Dr. Bassam Kahhaleh

6. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Truth Table Approach
A B C F1 F2
= 0
= 1 1 1 1 1
Dr. Bassam Kahhaleh

7. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Truth Table Approach
A B C F1 F2 1
= 0
= 1 1 1 1 1
Dr. Bassam Kahhaleh

8. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Truth Table Approach
A B C F1 F2 1
= 0
= 1 1 1
Dr. Bassam Kahhaleh

9. Princess Sumaya University
Analysis Procedure
Digital Logic Design
Truth Table Approach
A B C F1 F2 1
= 1
= 0 1 1 1 1
Dr. Bassam Kahhaleh

10. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Truth Table Approach
A B C F1 F2 1
= 1
= 0 1 1
Dr. Bassam Kahhaleh

11. Princess Sumaya University
Analysis Procedure
Digital Logic Design
Truth Table Approach
= 1
= 0
A B C F1 F2 1 1 1
Dr. Bassam Kahhaleh

12. Princess Sumaya University
Digital Logic Design
Analysis Procedure
Truth Table Approach
A B C F1 F2 1
= 1 1 1 1 B 1 A C B 1 A C
F1=AB'C'+A'BC'+A'B'C+ABC
F2=AB+AC+BC
Dr. Bassam Kahhaleh

13. Princess Sumaya University
Digital Logic Design
Design Procedure
Given a problem statement:
Determine the number of inputs and outputs
Derive the truth table
Simplify the Boolean expression for each output
Produce the required circuit
Example:
Design a circuit to convert a “BCD” code to “Excess 3” code
4-bits
0-9 values
Value+3 ?
Dr. Bassam Kahhaleh

14. Princess Sumaya University
Digital Logic Design
Design Procedure
BCD-to-Excess 3 Converter C 1 B A x D C 1 B A x D
A B C D
w x y z
x x x x
w = A+BC+BD
x = B’C+B’D+BC’D’ C 1 B A x D C 1 B A x D
y = C’D’+CD
z = D’
Dr. Bassam Kahhaleh

15. Princess Sumaya University
Digital Logic Design
Design Procedure
BCD-to-Excess 3 Converter
A B C D
w x y z
x x x x
w = A + B(C+D)
y = (C+D)’ + CD
x = B’(C+D) + B(C+D)’
z = D’
Dr. Bassam Kahhaleh

16. Seven-Segment Decoder
Princess Sumaya University
Seven-Segment Decoder
Digital Logic Design f e g d a b c ? w x y z a b c d e f g
BCD code
w x y z
a b c d e f g
x x x x x x x 1 x
a = w + y + xz + x’z’
e = x’z’+yz’
b = x’+yz+y’z’
c = x+y’+z
f = w+x+y’z’
d = x’z’+yz’+x’y+xy’z
g = w+x’y+xy’+x’y+xz’
Dr. Bassam Kahhaleh

20. Problem
Design a circuit which displays the letters A through J on a seven-segment indicator. The circuit has four inputs W, X, Y, Z which represent the last 4 bits of the ASCII code for the letter to be displayed. For example, if WXYZ = 0001, “A” will be displayed. The letters should be displayed in the following form: Design your circuit using only two-, three-, and four-input NOR gates and inverters. Any solution with 22 or fewer gates and inverters (not counting the four inverters for the inputs) is acceptable.

21. Princess Sumaya University
Binary Adder
Digital Logic Design
Half Adder
Adds 1-bit plus 1-bit
Produces Sum and Carry HA x y S C x
+ y
───
C S
x y
C S
0 0
0 0
0 1
0 1
1 0
1 1
1 0 x y S C
Dr. Bassam Kahhaleh

22. Princess Sumaya University
Digital Logic Design
Binary Adder
Full Adder
Adds 1-bit plus 1-bit plus 1-bit
Produces Sum and Carry FA x y z S C x
+ y
+ z
───
C S y 1 x z
x y z
C S
0 0
0 1
1 0
1 1
S = xy'z'+x'yz'+x'y'z+xyz = z’(xy’+x’y)+z(x’y’+xy)
=z’(x  y) + z(x  y)’
= x  y  z y 1 x z
C = xy + xz + yz
Dr. Bassam Kahhaleh

23. Princess Sumaya University
Digital Logic Design
Binary Adder
Full Adder
S = xy'z'+x'yz'+x'y'z+xyz=x  y  z
C = xy + xz + yz x y z S C x y z S C
Dr. Bassam Kahhaleh

24. Princess Sumaya University
Binary Adder
Digital Logic Design
Full Adder HA x y z S C x y z S C
Dr. Bassam Kahhaleh

25. QUIZ ? Q#1- This combinational logic circuit is described as a(n) ___.
Q#2- What are the sum and carry out outputs of this
full-adder circuit?
Q#3- What are the sum and carry out outputs of this
full-adder circuit?
Q#4- What are the sum and carry out outputs of this
full-adder circuit?
Q#5- What are the sum and carry out outputs of this
full-adder circuit?
Q#6- What are the sum and carry out outputs of this
full-adder circuit?
ANS: full-adder
ANS: Sum=0, Carry out=0
ANS: Sum=1, Carry out=0
ANS: Sum=0, Carry out=1
ANS: Sum=1, Carry out=1
ANS: Sum=0, Carry out=1
Cin = 0
A = 0
B = 0
Cin = 0
A = 0
B = 1
Cin = 1
A = 0
B = 1
Cin = 1
A = 1
B = 1 ?
Cin = 0
A = 1
B = 1

26. Princess Sumaya University
Digital Logic Design
Binary Adder
c3 c2 c
+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────
Cy S3 S2 S1 S0 FA
x x x x0
y y y y0
S S S S0
C C C C1
Binary Adder
x3x2x1x y3y2y1y0
S3S2S1S0 C0 Cy
Carry Propagate Addition
Dr. Bassam Kahhaleh

27. 74x283 4-bit adder
Uses carry look-ahead internally

28. Princess Sumaya University
Binary Adder
Digital Logic Design
Carry Propagate Adder
c3 c2 c
x3 x2 x1 x0
y3 y2 y1 y0
────────
Cy S3 S2 S1 S0
c7 c6 c5 Cy .
x7 x6 x5 x4
y7 y6 y5 y4
Cy S7 S6 S5 S4
CPA
A3 A2 A1 A0
B3 B2 B1 B0
S3 S2 S1 S0 C0 Cy
x3 x2 x1 x0
y3 y2 y1 y0
x7 x6 x5 x4
y7 y6 y5 y4
S3 S2 S1 S0
S7 S6 S5 S4
Dr. Bassam Kahhaleh

29. Subtractors Half Subtractor Full Subtractor Adder/Subtractor - 1

30. Half Subtractor Input Output Logic Symbol: Logic Diagram: A Di B Bo Bo
Di (difference)
B0 (borrow out) B Bo A B Di A B Di Bo -1 1 2
Logic Diagram:

31. QUIZ ? Q#1- What is the difference and borrow outputs from
This half-subtractor circuit?
Q#2- What is the difference and borrow outputs from
this half-subtractor circuit?
Q#3- What is the difference and borrow outputs from
this half-subtractor circuit?
Q#4- What is the difference and borrow outputs from
this half-subtractor circuit?
ANS: Di= 0, Bo= 0
ANS: Di= 1, Bo= 0
ANS: Di= 0, Bo= 0
ANS: Di= 1, Bo= 1
(A – B)
A = 0
B = 0
(A – B)
A = 1
B = 0
(A – B)
A = 1
B = 1
(A – B)
A = 0
B = 1 ?

32. Full Subtractor Input Output Logic Symbol: Logic Diagram: Full
Di (difference)
B0 (borrow out)
Full
Subtractor
Input Output
Bin A B Di B0
H. S.
Bin
Logic Diagram:
half subtractor Bo

33. Full Subtractor Di Di Same as S in full adder Bo 1 1 Ci A B 00 01 1110 Di
Ci A B Di Bo
Di Same as S in full adder Bo 00 01 11 10 1 1

34. HINT: truth table from textbook (Fig. 10-10) is helpful
QUIZ
Q#1- What are the Difference and Borrow out output from
this full-subtractor circuit?
Q#2- What are the Difference and Borrow out output from
this full-subtractor circuit?
Q#3- What are the Difference and Borrow out output from
this full-subtractor circuit?
Q#4- What are the Difference and Borrow out output from
this full-subtractor circuit?
Q#5- What are the Difference and Borrow out output from
this full-subtractor circuit?
Q#6- What are the Difference and Borrow out output from
this full-subtractor circuit?
HINT: truth table from textbook (Fig ) is helpful
ANSWER: Di = 0, Bo = 0
ANSWER: Di = 1, Bo = 1
ANSWER: Di = 1, Bo = 1
ANSWER: Di = 0, Bo = 1
ANSWER: Di = 1, Bo = 0
ANSWER: Di = 0, Bo = 0
(A – B - Bin)
Bin = 0
A = 0
B = 0
(A – B - Bin)
Bin = 1
A = 0
B = 0 ?
(A – B - Bin)
Bin = 1
A = 1
B = 1
(A – B - Bin)
Bin = 1
A = 0
B = 1
(A – B - Bin)
Bin = 0
A = 1
B = 0
(A – B - Bin)
Bin = 0
A = 1
B = 1

35. Adder/Subtractor - 1 E = 0: Half adder E = 1: Half subtractorDi Bo A B S C
Half adder
Half subtractor i
E = 0: Half adder
E = 1: Half subtractor o

36. Adder/Subtractor-1 E = 0: Full adder E = 1: Full subtractor C A B Bo EDi C Bo i E
E = 0: Full adder
E = 1: Full subtractor

37. Adder/Subtractor-2
E = 0: 4-bit adder
E = 1: 4-bit subtractor