Combinational Logic Circuits: Introduction, Multi-input, multi-output Combinational circuits, Code converters design and implementations

1. COMBINATIONAL
LOGIC CIRCUITS
&
ARITHMETIC
CIRCUITS
UNIT III
DIGITAL
ELECTRONICS
PROF.ARTI GAVAS-PARAB
ANNA LEELA COLLEGE OF COMMERCE AND ECONOMICS,SHOBHA
JAYARAM SHETTY COLLEGE FOR BMS
CHAPTER I

2. UNIT III: CONTENTS
 Combinational Logic Circuits:
 Introduction
 Multi-input Combinational circuits
 multi-output Combinational circuits
 Code converters design and implementations
 Arithmetic Circuits:
 Introduction,
 Adder
 BCD Adder, Excess – 3 Adder
 Binary Subtractors,BCD Subtractors
 Multiplier
 Comparator
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3. COMBINATIONAL LOGIC CIRCUIT
 Combinational circuit is a circuit in which we combine the
different gates in the circuit, for example encoder, decoder,
multiplexer and demultiplexer. Some of the characteristics
of combinational circuits are following −
 The output of combinational circuit at any instant of
time, depends only on the levels present at input
terminals.
 The combinational circuit do not use any memory.The
previous state of input does not have any effect on the
present state of the circuit.
 A combinational circuit can have an n number of inputs
and m number of outputs.

4. CLASSIFICATION OF COMBINATIONAL LOGIC
 Design procedure of combinational Logic circuit:
1. The problem is stated
2. The number of available input variables and
required output variables is determined
3. The input and output variables are assigned letter
symbols
4. The truth table that defines the required
relationship between inputs and outputs is derived
5. The simplified Boolean function for each output is
obtained
6. The logic diagram is drawn.

5. CODE CONVERTERS DESIGN AND IMPLEMENTATIONS
 Numbers are usually coded in one form or another so as to represent or use it as required.For instance, a number
‘nine’ is coded in decimal using symbol (9)d. Same is coded in natural-binary as (1001)b.While digital computers all deal
with binary numbers, there are situations wherein natural-binary representation of numbers in in-convenient or in-
efficient and some other (binary) code must be used to process the numbers.
 One of these other code is gray-code, in which any two numbers in sequence differ only by one bit change.This code is
used in K-map reduction technique.The advantage is that when numbers are changing frequently,the logic gates are
turning ON and OFF frequently and so are the transistors switching which characterizes power consumption of the
circuit; since only one bit is changing from number to number,switching is reduced and hence is the power
consumption.
 Binary-to-Gray
 Gray-to-Binary
 Decimal to BCD
 BCD-to-excess-3
 Excess-3-to-BCD

6. BINARY-TO-GRAY
 The table that follows shows natural-binary numbers (upto 4-bit) and corresponding gray codes.
G3 = B3

7. GRAY-TO-BINARY
 Once the converted code (now in Gray form) is processed,we want the processed data back in binary representation.So we need a
converter that would perform reverse operation to that of earlier converter.This we call a Gray-to-Binary converter.
B3 = G3

8. DECIMALTO BCD
 The decimal to binary encoder usually consists of 10 input lines and 4 output lines. Each input line
corresponds to the each decimal digit and 4 outputs correspond to the BCD code.This encoder accepts the
decoded decimal data as an input and encodes it to the BCD output which is available on the output lines.
Logical expression for A3,A2,A1 and A0 :
A3 =Y9 +Y8
A2 =Y7 +Y6 +Y5 +Y4
A1 =Y7 +Y6 +Y3 +Y2
A0 =Y9 +Y7 +Y5 +Y3 +Y1
Block diagram TruthTableCircuit diagram

9. BCD-TO-EXCESS-3 CODE CONVERTER
 The bit combinations of both the BCD (Binary Coded Decimal)
and Excess-3 codes represent decimal digits from 0 to 9.
 Therefore each of the code systems contains four bits and so
there must be four input variables and four output variables.
Figure provides the list of the bit combinations or truth table
and equivalent decimal values.
 The symbolsA, B, C,and D are designated as the bits of the BCD
system,andW,X,Y, and Z are designated as the bits of the
Excess-3 code system.It may be noted that though 16
combinations are possible from four bits,both code systems use
only 10 combinations.
 The rest of the bit combinations never occur and are treated as
don’t-care conditions.
For the BCD-to-Excess-3 converter,A, B, C, and D are the input variables andW,X,Y, and Z are the output variables.The
simplified Boolean expressions ofW, X,Y, and Z are given as: W = A + BC + BD
X = B′C + B′D + BC′D′
Y = CD + C′D′
Z = D′

10. BCD-TO-EXCESS-3 CODE CONVERTER
 A good designer will always look forward to reduce
the number and types of gates. It can be shown that
reduction in the types and number of gates is possible
to construct the BCD-to-Excess-3 code converter
circuit if the above Boolean expressions are modified
as follows.
 W = A + BC + BD
= A + B(C + D)
 X = B′C + B′D + BC′D′ = B′(C + D) + BC′D′
= B′(C + D) + B(C + D)′
 Y = CD + C′D′
= CD + (C + D)′
 Z = D′

11. EXCESS-3-TO-BCD
 In this case,W, X,Y, and Z are considered as input variables
and A, B, C, and D are termed as output variables.
 The Boolean expressions of the outputs are
 A =WX +WYZ
 B = X′Y′ + X′Z′ + XYZ
 C =Y′Z +YZ′
 D = Z′.
 Equations can be further modified as following on the
above Boolean expressions.
 A =WX +WYZ = W(X +YZ)
 B = X′Y′ + X′Z′ + XYZ = X′(Y′ + Z′) + XYZ = X′(YZ)′ + XYZ
 C = Y′Z +YZ′
 D = Z′

12. THANKYOU!