Digital Electronics Fundamentals: Combinational Logic

CHAPTER 5
COMBINATIONAL LOGIC

LEARNING OUTCOMES

After completing the unit, students should be able to:
1. Utilize Boolean algebra laws and rules for simplifying

combinational logic circuits.
2. Distinguish between half adder and full adder.
3. Implement a basic binary encoder
4. Understand and explain the function of encoder, decoder and

comparator
5. Identify the function of multiplexer and de-multiplexer.

5.1 BASIC COMBINATIONAL LOGIC CIRCUITS &
IMPLEMENTATION

• Cbaosmicbilnoagticiognaatlelsogtoicfeomrmplaoyms otrhee uusseefuolf, tcwoomoprlemxofruencotfiothne. This equation can be stated as B is HIGH if K
• Fitourseexsacmopmleb,inianttiohneadl elosigginc.of an automobile warning buzzer, and D are HIGH or if H and D are HIGH.
• ftTaohnhledelodtcwhrrisievt:eedtrrho’isoeonrdboifusoozorrzpetisehrneoaepcadetc.intveiavdta,etsioorinfiftohftehtehheekaewdyaliirgsnhiinntsgtahbreueziOzgenNritaiisonndas
• BTciloohlumoesltbelroiaangntaiectedifqousunnyamocttfiboiolononlgiicnifcoatrlfhluytenhicefnotaifroimugntusormteho5ab.t1ilc.eaTnwhaberenfiwignurgirtebteuinlzlzuaessrtraiastes a Key in Ignition K

• B = ( K and D ) or ( H and D ) Door Opened D
Headlights on H
• which can also be written as
Door Opened D
• B = KD + HD
Figure 5.1: An example of the application of
combinational logic – automobile warning
buzzer

CONTINUE

• Closely looking at the system, this warning Door Opened D Warning Buzzer
buzzer can also be activated whenever the Headlights on H
door is opened and either the key is in the K
ignition or the headlights are on. This is a Key in Ignition D
simplification method for Boolean algebra.

• The new Boolean equation becomes B = D and
( K = H ) also written as B = D (K +H).

• The new equation represents the same logic
operation but is simplified implementation.
Because it requires only two logic gates as
shown in figure 5.2.

Example 5.1 :

• Write the Boolean logic function and draw the logic circuit that
represent the following function:

• A bank burglar alarm (A) is to activate if it is after banking hours (H)
and the front door (F) is opened or if it is after banking hours (H) and
the vault door is opened (V).

Solution :

• A= HF + HV. The logic circuit is shown below

Key in Ignition H

Front Door Opened F
After Hours H

Vault Door Opened V

Example 5.2 :

• Using common reasoning, reduce the logic function described in
Example 5-1 to a simpler form.

Solution :

The alarm is activated if it is after banking hours and if either the front door is
opened or the vault is opened (see below). The simpler equation is written as

A = H (F+ V)

After Hours H Warning Buzzer
Front Door Opened F

Vault Door Opened V

5.2 BASIC ADDER

• Computer executes adding operation on two binary numbers only
at a time. Every number consist a few bits. Adding operation
starts from LSB to MSB. In some cases, ‘Carry’ is produced.

• There are 3 types of adder circuit

1. Half adder
2. Full adder
3. RIPPLE CARRY ADDER / PARALLER

5.2.1 HALF ADDER

5.2.2 FULL ADDER

5.2.3 RIPPER CARRY ADDER

UNIT 5 :COMBINATIONAL LOGIC

COMPARATORS

Learning Outcome

At the end of the class student should be
able to:-
Understand and explain the function of
comparator

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COMPARATOR

Basic function
 To compare the magnitude of two binary

quantities to determine the relationship
of those quantities.
 Comparator determines whether two
numbers are equal or unequal.

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COMPARATOR - equality

 EX-OR gate can be used as a 2-bit
comparator. Why?
Because o/p is HIGH if two i/p are not equal
& o/p is LOW if i/p are equal.

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COMPARATOR- equality

 To compare binary numbers containing
two bits each:

 LSB are compared by gate G1
 MSB are compared by gate G2
 If number equal, o/p is HIGH & if number

not equal, o/p is LOW.

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COMPARATOR - equality

 Remember!!!

Equality – o/p is 1
Inequality – o/p is 0

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COMPARATOR - equality

Example:
Determine the output by applying the
following sets of binary numbers to the
comparator.

a) 10 and 10
b) 11 and 10

Answer:
a) Output is 1 which is equal
b) Output is 0 which is not equal

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COMPARATOR - inequality

 Many IC comparators provide additional
outputs that indicate which of the two
binary numbers being compared is the
larger.

 Comparison is made in terms of :
◦ Equal to ‘=’
◦ Less than ‘<’
◦ Greater than ‘>’

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COMPARATOR - inequality

 If number A is greater than number B
(A > B )

 If number A is less than number B
(A < B )

 If number A equal to number B
(A = B )

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COMPARATOR - inequality

Equal to ‘=’

For two 1-bit quantities For two 2-bit quantities For two n-bit quantities
An-1 COMP
A B OA=B A1 COMP
00 1 A0 A0 OA=B
01 0 Bn-1
10 0 OA=B
11 1 BB01 B0

COMP
A OA=B
B

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COMPARATOR - inequality

Inputs Outputs

A B A<B A=B A>B

00 0 1 0

01 1 0 0

10 0 0 1

11 0 1 0

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COMPARATOR - inequality
Logic symbol for a 4 bit comparator.

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COMPARATOR - inequality

To determine an inequality of binary
numbers A and B, you first
examine the highest-order bit in
each number:

◦ If A3=1 and B3=0  number A is
greater than number B

◦ If A3=0 and B3=1  number A is
less than number B

◦ If A3=B3  you must examine the
next lower bit position for an
equality

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COMPARATOR - inequality
Remember !!!!!!

Please start with
highest-order bits

(MSBs)

When any inequality found, the relationship of
the two numbers is established and any
inequalities in lower-order bit positions
must ignored.

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COMPARATOR - inequality

Example:
Determine the A=B ,A>B & A<B o/p for
the i/p given.

0
1
1
0

1
1
0
0

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COMPARATOR - inequality

Answer :

0
1
11
0

0
1
10
0
0

A > B output is HIGH and other outputs are
LOW

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COMPARATOR - inequality
A 4-bit magnitude comparator ( 74HC85 )

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COMPARATOR - inequality
Any differences ?????

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COMPARATOR - inequality
 Cascading inputs allow several comp to be

cascaded of any number of bits greater than
four.
 Output of lower-order comp are
connected to the cascading inputs of next
higher-order comparator.
 The lowest-order comparator must have
high on the A=B input and LOW on the
A<B and A>B inputs.

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COMPARATOR - inequality
How to connect two 74HC85

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ありがとう
ございました

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UNIT 5 :COMBINATIONAL LOGIC

DECODERS &
ENCODERS

Learning Outcome

At the end of this class, students
should be able to :

 Implement a basic binary encoder.
 Understand and explain the function

of encoder & decoder.

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MEDIUM SCALE INTEGRATION
(MSI) LOGIC CIRCUIT

Digital systems get binary-coded data and
information in the many ways.The
operation of getting the data include:

 Decoding and Encoding
 Multiplexing
 Demultiplexing
 Code conversion

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Decoders

 Circuit that

◦ accepts a set of inputs (binary number/code)
and activates only the output that
corresponds to that particular input number.

◦ Only one output is activated at one time,
correspond to the input number.All other
outputs remain inactive.

◦ Has n i/p line and from one to 2n o/p line.

◦ Active-HIGH or Active-LOW input and
output.

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Decoders

Basic Binary decoder

If we need binary 1001 on the input.AND
gate can be used because only HIGH output
will produce when all input is HIGH.

X=A3 A2 A1 A0

Logic equation for the decoder of figure above

is X=A3 A2 A1 A0 DFE 1013 MOHD RADZI

Decoders

Example of decoder: 3-line-to-8-line

(or 1-of-8) binary decoder.

A0 Y0
Y1

A1 1-of-8
DECODER

A2 Y6
Y7

Active-HIGH input and output.

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Decoders

 Truth table for 1-of-8 Binary Decoder

A B C Y0 Y1 Y2 Y3 Y4 Y5 Y6 Y7

0001

001 1

010 1

011 1

100 1

101 1

110 1

11 1 1

 Y0 = ABC 7
 Y1 = ABC

Decoders

 1-of-8 Binary Decoder logic circuit
ABC

Y0
Y1

Y6
Y7

8

Decoders

 The 74138 Decoder (1-of-8 Decoder)
 Enable the input pin.

E3 Y0
E2
Y1
E1 74 LS 138 Y2

Y3

Y4

A Y5
B Y6
C Y7

9

Decoders

 74138 Decoder

E1 E2 E3

10

The 4-Bit Decoder

 In order to decode all possible combinations
of four bits, 16 decoding gates are required
(24=16).

 This type of decoder is commonly called
either:

◦ A 4-line-to-16-line decoder, or
◦ A 1-of-16 decoder

 Decoding functions and truth table for a 4-
line-to-16-line decoder with active-LOW
outputs.

The 4-Bit Decoder

The 4-Bit Decoder

74HC154: 1-of-16 decoder

Decoders

The BCD-to-Decimal Decoder
The BCD-to-decimal converts BCD code

(8421 code) into one of ten possible decimal
digit indications.
 Known as 4-line-to-10-line or 1-of-10
decoder.
The BCD code represents only the ten
decimal digits 0 through 9.

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Decoders

BCD to Decimal Decoder

Decimal Digit BCD CODE
A3 A2 A1 A0
0
1 0 00 0
2 0 00 1
3 0 01 0
4 0 01 1
5 0 10 0
6 0 10 1
7 0 11 0
8 0 11 1
9 1 00 0
1 00 1

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Decoders

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Decoders

BCD to 7 Segment Decoder

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