Logic Design (CE1111 ) Lecture 4 (Chapter 4) Combinational Logic Prepared by Dr. Lamiaa Elshenawy 1

Outlines  Combinational Circuits  Analysis Procedure  Design Procedure  Binary Adder-Subtractor  Decimal Adder  Binary Multiplier  Magnitude Comparator  Decoders  Encoders  Multiplexers 2

3 o consists of logic gates o outputs are determined from only the present combination of inputs o consists of logic gates and storage elements o outputs are determined from the present combination of inputs & past inputs Logic Circuits Combinational Circuits Sequential Circuits Logic Circuits

Combinational Circuits 4

Analysis Procedure 5  How analyze combinational circuits? 1. Label all gate inputs & outputs with arbitrary symbols 2. Determine the Boolean functions for each gate output 3. Repeat step 2 until the outputs of the circuit are obtained 4. Determine the output Boolean functions in terms of input variables

Analysis Procedure 6

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 How derive a truth table for a combinational circuit? 1. Write the input variables and their possible binary number combinations and list the binary numbers from 0 to (2n - 1) 2. Obtain the truth table for the outputs of logic gates 3. Repeat step 2 until the outputs of the circuit are obtained 8

Design Procedure  How design combinational circuits? 1. Determine the required number of inputs and outputs and assign a symbol to each 2. Derive the truth table that defines the required relationship between inputs and outputs 3. Obtain the simplified Boolean functions for each output as a function of the input variables 4. Draw the logic diagram and verify the correctness of the design (manually or by simulation) 9

Example 1  Design a combinational circuit to convert from BCD into excess-3 code.  Solution: 10

Example 1 (Contd.) 11

Example 1 (Contd.) 12

Example 1 (Contd.) 13

Binary Adder-Subtractor  What is Binary Adder-Subtractor?  A binary Adder–Subtractor is a combinational circuit that performs the arithmetic operations of addition and subtraction with binary numbers 14

Binary Adder 15 Half Adder Add 2 variables Full Adder Add 3variables

Binary Adder 16 Half Adder Implementation

Binary Adder 17 Full Adder Implementation

Binary Adder  Is there another way to implement Full Adder? 18

Binary Adder 19

Example 2 (Contd.) 20 Four-bits Adder

Binary Subtractor  How subtract two binary numbers with n-bits?  It can be done using full adder connected in cascade with X- OR logic gates  Example 2: Subtract the following two binary numbers A and B  Solution o Unsigned numbers A - B if A> B or 2’s complement of (B – A) if A < B (determine the 1’s complement and then add 1) o Signed numbers 21 A-BNo overflow

Binary Subtractor 22

Binary Subtractor 23

BCD Adder  BCD adder is a combinational circuit that add decimal numbers  BCD adder requires 1. nine inputs 2. five outputs(four bits for each decimal digit, one bit input and output carry)  Each input digit does not exceed 9, the output sum cannot be greater than = 19,  the 1 in the sum being an input carry 24

Decimal Adder 25

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BCD Adder  When C = 1, it is necessary to add 0110 to the binary sum and provide an output carry for the next stage 27

Binary Multiplier  Multiplication of binary numbers is performed in the same way as multiplication of decimal numbers 1. The multiplicand is multiplied by each bit of the multiplier, starting from the least significant bit 2. Each such multiplication forms a partial product 3. Successive partial products are shifted one position to the left 4. The final product is obtained from the sum of the partial products 28

Binary Multiplier 29 multiplicand multiplier

Binary Multiplier  For J multiplier bits and K multiplicand bits, we need: 1. (J x K) AND gates 2. (J – 1) K-bit adders to produce a product of 3. (J + K) bits 30

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Magnitude Comparator  A magnitude comparator is a combinational circuit that compares two numbers A and B. The outcome of the comparison is specified by three binary variables 1. A = B 2. A > B 3. A < B 32

Magnitude Comparator 33

Magnitude Comparator 34

Decoders 35

Decoders 36 Three-to-eight-line decoder

Decoders 37

Decoders 38 Two-to-four-line decoder with enable input

Decoders 39 4 x16 decoder constructed with two 3 x 8 decoders

Decoders 40 Implementation of a full adder with a decoder

Encoder 41

Encoder 42

Priority Encoder 43  A priority encoder is an encoder circuit that includes the priority function  the higher the subscript number, the higher the priority of the input High subscript

Priority Encoder 44 Maps for a priority encoder

Priority Encoder 45 Four- input priority encoder

Multiplexers 46

Multiplexers 47 Two-to-one-line multiplexer

Multiplexers 48 Four-to-one-line multiplexer

Multiplexers 49

Multiplexers 50 Example 1:

Multiplexers 51 Example 2:

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