EEE104
LOGIC CIRCUIT DESIGN
Objectives and Outcomes of
EEE104 Logic Circuit Design
The aim of this course for students is to learn digital design
principles, digital logic gates, design of combinational logic
and circuits, circuit design with flip-flops, and flip-flops,
circuit design with counters.
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Comprehend importance of digital systems in application and the
computer architecture
Comprehend the number systems and binary arithmetic
operations
Comprehend the basic theorems and axioms of boole algebra
Comprehend the operations of the basic logic gates
Gain experience on the realization of the boole algebra functions
with basic logic circuits
Comprehend structures of the basic storage elements
Comprehend operation of the different type counters and to gain
experience on the design
Gain experience of the design of the combinational and sequential
networks
2
1. Week
2. Week
3. Week
4. Week
5. Week
6. Week
7. Week
8. Week
9. Week
10. Week
11. Week
12. Week
13. Week
14. Week
Schedule
of
the
Course
Introduction to computers and digital systems
Binary Systems
Boolean Algebra
Logic Gates
Simplification of Boolean Functions
The Map Method, Don’t Care Conditions.
The tabulation method, determination of prime
Combinational Logic, Design procedure
Adders, Subtractors, Code Conversion, Analysis Procedure
Multilevel NAND circuits, multilevel NOR circuits, Exclusive-OR
functions.
Combinational Logic with MSI and LSI
Decoders and encoders, multiplexer, read-only memory
(ROM), programmable logic array (PLA).
Synchronous Sequential Logic: Flip-flops, triggering of flipflops, analysis of clocked sequential circuits. State reduction
and assignment.
Synchronous Sequential Logic: Flip-flop excitation tables,
design procedure, design of counters, design with state
implements.
3
Assessment of the Course
Quantity
Midterm Exams
Homework
Percent of In-term Studies
Percentage of Final Exam to Total
Score
1
2
Total Weighting
(%)
90
10
60
40
4
Text Book of The Course
 Digital Design, M. Morris Mano, Prentice-Hall, Inc.,
Third Edition .
 Foundamentals of Digital Systems Design, V. T. Rhyne,
Prentice-Hall, Inc.
 Digital Foundamentals, Thomas L. Floyed, A. Bell &
Howell Company.
 Principles of Digital Design, Daniel D. Gajski,
Prentice-Hall, Inc.
5
Useful Softwares for the Course
Online Simulators
https://circuitverse.org/simulator
https://logic.ly/demo
Programs
Isis (Proteus)
6
Motivation
 Microelectronic technologies have revolutionized
our world: computers, cell phones, internet ect.
 The semiconductor industry has grown from $21
billion in 1985 to $433.3 billion in 2020.
7
The Digital Revolution
ENIAC
(1.6 x 11.1 mm)
Integrated Circuit
First PC
Commodore PET
1947
1949
1983
8
Building Digital Circuits
9
Robert Noyce, 1927 - 1990
• Nicknamed “Mayor of Silicon
Valley”
• Cofounded Fairchild
Semiconductor in 1957
• Cofounded Intel in 1968
• Co-invented the integrated
circuit
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Gordon Moore
• Cofounded Intel in
1968 with Robert
Noyce.
• Moore’s Law: the
number of transistors
on a computer chip
doubles every 1.5 years
(observed in 1965)
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Technology Trends: Moore’s Law
• Since 1975, transistor counts have doubled every two years.
12
Scope
 The purpose of this section is:





Learning the fundamentals of digital design
Designing and building of digital systems
Numbers and Number Systems
Bitwise operations
Introduction to Boelean Algebra
13
Number Systems
Used number systems:
 binary, octal, and hexadecimal
Conversions:
 combinations of binary, octal, and hexadecimal
Signed/Unsigned Numbers & Characters:
 negative binary numbers
 floating point numbers
 character codes
Dr. Uğurhan KUTBAY
14
Decimal-Octal-Binary-Hexadeciamal
Number Systems
Decimal: 10 symbols, from 0 to 9
 Reached 9, add 1 to next digit and make 0 the digit:
0,1,2,3,4,5,6,7,8,9, 10,11,12,13, …, 19
Octal: 8 symbols, from 0 to 7
 Reached 7, add 1 to next digit and make 0 the digit:
0,1,2,3,4,5,6,7, 10,11,12,13, …, 17
Binary: Only 2 distinct symbols: 0, 1
 0,1, 10,11
Hexadecimal: 16 symbols: 0-9,A-F
0,1, …, 9,A,B,C,D,F, 10,11, …, 19,1A,1B,1C,1D,1E,1F
Dr. Uğurhan KUTBAY
15
Why Different Number Systems
 Binary:
 Digital computer uses 0 and 1
 Octal:
 Ease of use better for human
 Easily convert to/from binary
 1 hex digit = 3 binary digits
 Hexadecimal
 Easily convert to/from binary
 1 hex digit = 4 binary digits (power of 2: 1 byte = 2 hex)
Dr. Uğurhan KUTBAY
16
Number System Conversions
 convert binary  octal
 3 bits = 1 octal digit
 use conversion table:
O
0
1
2
3
B
000 001 010 011
4
5
6
7
100
101
110
111
 Example:
101110100001102 = 10 111 010 000 1102 = 272068
 It also works with a decimal point, e.g.:
50.38 = 101 000.0112
Dr. Uğurhan KUTBAY
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Number System Conversions
 convert binary  hex
 4 bits = 1 hex digit
 use conversion table:
H
0
B
1
4
5
6
7
0000 0001 0010 0011
0100
0101
0110
0111
H
8
C
D
E
F
B
1000 1001 1010 1011
1100
1101
1110
1111
9
2
A
3
B
 Example:
101110100001102 = 10 1110 1000 01102 = 2E8616
 It also works with a decimal point, e.g.:
50.316 = 101 0000.00112
Dr. Uğurhan KUTBAY
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Number System Conversions
 to/from decimal
 use basic principle of radix numbers:
each digit corresponds to a power of the radix
 in decimal:
3205.3 = 3*1000 + 2*100 + 0*10 + 5*1 + 3*0.1
= 3*103 + 2*102 + 0*101+ 5*100 + 3*10-1
In general, a decimal number can be decomposed into:
…d3*103 + d2*102 + d1*101 + d0*100 + d-1*10-1 + d-2*10-2…
Dr. Uğurhan KUTBAY
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Number System Conversions
 a binary number can be decomposed into:
… +d3*23 + d2*22 + d1*21 + d0*20 + d-1*2-1 + d-2*2-2 + …
… +d3*8 + d2*4 + d1*2 + d0*1 + d-1*0.5 + d-2*0.25 +…
 convert binary  decimal
 learn powers of 2:
 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
 for each binary digit that is 1 add corresponding power
 Example:
1101012 = 25 + 24 + 22 + 20 = 32 + 16 + 4 + 1 = 5310
1.012 = 20 + 2-2 = 1 + ¼ = 1.2510
Dr. Uğurhan KUTBAY
20
Number System Conversions
 decimal  binary:
 decompose number into powers of 2
 How: repeatedly subtract the largest possible power
 for each component write a 1 in corresponding digit
 Example: convert 5310
53 – 25 = 53 – 32 = 21
21 – 24 = 21 – 16 = 5
5 – 22 = 5 – 4 = 1
1 – 20 = 1 – 1 = 0
 the powers are: 5, 4, 2, 0
 the binary number is: 1101012
Dr. Uğurhan KUTBAY
21
Number System Conversions
 summary
 binary  octal/hex:
grouping of 3 or 4 bits
 octal  hex:
via binary
 binary  decimal:
sum up powers of 2
 decimal  binary:
decompose into powers of 2
 octal/hex  decimal:
use powers of 8 or 16
easier: via binary
Dr. Uğurhan KUTBAY
22
Negative Binary Numbers
 signed magnitude:
 left-most bit is a sign bit
 Example with 8 bits:
5 = 0000 0101, –5 = 1000 0101
11=0000 1011, -11=1000 1011
45=0010 1101, -45=1010 1101
0=0000 0000, -128=1000 0000
Dr. Uğurhan KUTBAY
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ONE’S COMPLEMENT:




flip all bits (0 becomes 1, 1 becomes 0)
Example: 5 = 0000 0101, –5 = 1111 1010
Advantage: subtract by adding negative number
Example: 12 – 5 = 12+(–5)
 add left-most carry to sum
 Problem: 2 forms of zero:
 0000 0000 = 1111 1111
Dr. Uğurhan KUTBAY
24
Negative Binary Numbers
 two’s complement:
 one’s complement + 1
 Example: 5 = 0000 0101, –5 = 1111 1010+1 = 1111 1011
 add/subtract as before, but left-most carry is discarded
12
0000 1100
–5
+ 1111 1011
7 1 0000 0111
Discarded
 drawback of two’s complement: extra negative number
 with 8 bits we have 28 = 256 possible combinations:
 27 = 128 combinations have left-most bit = 0:
represent zero and 127 positive integers
 27 = 128 combinations have left-most bit = 1:
represent 128 negative integers
Dr. Uğurhan KUTBAY
25
Finite-Precision Numbers
 number of bits determines the max/min number
 Problem:
 assume 4 bits
 4+5=9
4
0100
5
0101
?
1001
 1001 = -1  wrong result!
9 is too large for 4 bits (largest positive integer: 0111=7)
 overflow – must compare sign bits to detect:
A and B have the same sign and A+B has a different sign
Dr. Uğurhan KUTBAY
26
Negative Binary Numbers
 excess N representation
 basic idea: add N to every number (N=2m-1 or 2m-1-1)
 Example: 8 bits
 can represent 28 = 256 different combinations (0-255)
 add 27 = 128 (or 127) to every number to be represented
number
excess 128
representation
excess 127
representation
-128 -127
0
…
-1
0
1
…
127
1
…
127
128
129
…
255
0
…
126
127
128
…
254
128
255
 used in floating point formats
Dr. Uğurhan KUTBAY
27
Floating-Point Numbers
 needed for very large or very small numbers
 express numbers as n = f * 10e
 Example: +213.1 = +2.131 * 102 = +0.2131 * 103
 two forms of normalized notation
 represent number as:
sign bit
exponent e
fraction f
 base is implicit, normally 2
 fraction (mantissa) is normalized; common choices:
 0.1 …
 1. …
 sign bit applies to fraction (signed magnitude)
 exponent: signed integer (2’s complement, excess 2m-1)
Dr. Uğurhan KUTBAY
28
Floating-Point Numbers
 exponent size vs. fraction size:
max/min size of expressible numbers vs. accuracy
 Example: IEEE Standard
 single precision
 double precision
Dr. Uğurhan KUTBAY
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Floating-Point Numbers
 single precision standard: 32 bit word
 1 sign bit (0 is positive, 1 is negative)
 8 bits in exponent
 excess 127 system used: exponent ranges from -126 to +127
 23 bits in fraction
 normalized to 1. …
 leading 1 is always present and thus implied (not represented)
 I.e.: precision is 24 bits
 max. positive number:
 fraction: 1.111 1111 … 1111 = 20 + 2-1 + 2-2 + 2-3 + … + 2-23
= 1 + 1/2 + 1/4 + 1/8 + …
= ~2
 max = +2 * 2127 = +2128
 min. negative number: –2128
Dr. Uğurhan KUTBAY
30
Floating-Point Numbers
 min. positive number:
 smallest fraction: 1.000 … 0000 = 20 = 1
 smallest exponent: -126
 min pos number = 1 * 2-126 = 2-126
 max. negative number: –2-126
expressible
negative
numbers
negative
overflow
–2128
Dr. Uğurhan KUTBAY
negative
underflow
–2-126
positive
underflow
0
expressible
positive
numbers
+2-126
positive
overflow
+2128
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Floating-Point Numbers
 Examples: show 0.5 as floating point (hex bit pattern)




change to binary and normalize: 0.510 = 0.12 = 1.0 * 2-1
leading 1 is implied, fraction: 000 0000 … 00002
exponent: -1 in decimal is -1+127 in excess 127, i.e. 126
126 = 64 + 32 + 16 + 8 + 4 + 2 = 0111 11102
1
8 bits
0 0111 1110
23 bits
000 0000 0000 0000 0000 0000
 0011 1111 0000 0000 … 00002 = 3F00000016
 show 42E48000 in decimal





42E4800016 = 0100 0010 1110 0100 1000 0000 0000 00002
exponent = 1000 01012 = 128 + 4 + 1 = 13310
133 is in excess 127 notation; in decimal: 133 – 127 = 6
mantissa: 1.110010012
result = 1.110010012 * 26 = 1110 010.012 = 114.2510
Dr. Uğurhan KUTBAY
32
Representing Characters
 ASCII:
American Standard Code for Information Interchange
 7-bit code: 128 characters
 Examples
 do not confuse chars with
numbers, e.g.
 6: 0011 0110 (char)
 6: 0000 ... 0000 0110 (int)
 UNICODE
 16-bits: 65,536 chars (symbols)
 cover foreign languages
Dr. Uğurhan KUTBAY
33
Combinational Logic vs Sequential Network
x1
.
.
.
xn
fi(x)
Combinational logic:
yi = fi(x1,..,xn)
x1
.
.
.
xn
si
fi(x,s)
CLK
Sequential Networks
1. Memory
2. Time Steps (Clock)
yit = fi (x1t,…,xnt, s1t, …,smt)
sit+1 = gi(x1t,…,xnt, s1t,…,smt)
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Scope
Subjects
Building Blocks Theory
Combinational
Logic
AND, OR,
NOT, XOR
Sequential
Network
AND, OR, NOT Finite State
Machine
Standard
Modules
Operators,
Interconnects,
Memory
Data Paths,
Control Paths
System Design
Boolean Algebra
Arithmetics,
Universal Logic
Methodologies
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What is a combinational circuit?
• No memory
• Realizes one or more functions
• Inputs and outputs can only have two discrete values
• Physical domain (usually, voltages) (Ground 0V, Vdd 1V)
• Mathematical domain : Boolean variables (True, False)
Differentiate between different representations:
• physical circuit
• schematic diagram
• mathematical expressions
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Boolean Algebra
A branch of algebra in which the values of the variables
belong to a set B (e.g. {0, 1}), has two operations {+, .}
that satisfy the following four sets of laws.
•Commutative laws: a+b=b+a,
a·b=b·a
•Distributive laws: a+(b·c)=(a+b)·(a+c),
a·(b+c)=a·b+a·c
•Identity laws: a+0=a, a·1=a
•Complement laws: a+a’=1, a·a’=0
(x’: the complement element of x)
<37>
Representations of combinational circuits:
The Schematic
A
Y
B
• What is the simplest combinational circuit that you
know?
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Representations of combinational circuits
Truth Table: Enumeration of all combinations
Example: AND
id
A
B
Y
0
1
2
0
0
1
0
1
0
0
0
0
3
1
1
1
A
Y=AB
B
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Boolean Algebra
Similar to regular algebra but defined on sets with only
three basic ‘logic’ operations:
1. Intersection: AND (2-input);
2. Union: OR (2-input);
3. Complement: NOT ( 1-input);
Operator: . ,&
Operator: + ,|
Operator: ‘ ,!
“&, |, !” Symbols in C Programming
<40>
Boolean algebra and switching functions
Two-input AND ( ∙ )
AND A B
0 0
0 1
1 0
1 1
OR
Y
0
0
0
1
One-input NOT
(Complement, ’ )
Two-input OR (+ )
AB
0 0
0 1
1 0
1 1
Y
0
1
1
1
NOT A Y
0 1
1 0
A
1
A
A
1
1
A
0
0
A
0
A
For an AND gate,
0 at input blocks the other inputs
and dominates the output
1 at input passes signal A
For an OR gate,
1 at input blocks the other inputs
and dominates the output
0 at input passes signal A
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So, what is the point of representing gates as
symbols and Boolean expressions?
• Given the Boolean expression, we can draw the
circuit it represents by cascading gates (and vice
versa)
a
b
c
d
ab
ab + cd
y=e (ab+cd)
cd
e
Logic circuit vs. Boolean Algebra Expression
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