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COMBINATIONAL CIRCUITS

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COMBINATIONAL CIRCUITS

1. Combinational

2. Sequential

LOGIC CIRCUITS:

Combinational logic circuits (circuits without a memory):

Combinational switching networks whose outputs depend only

on the current inputs.

Sequential logic circuits (circuits with memory):

In this kind of network, the outputs depend on the current inputs

and the previous inputs. These networks employ storage elements

and logic gates. [Chapters 5 and 9]

COMBINATIONAL CIRCUITS

- Most important standard combinational circuits are:
- Adders
- Subtractors
- Comparators
- Decoders
- Encoders
- Multiplexers

Available in IC’s as MSI and used as

standard cells in complex VLSI (ASIC)

- From the Specification of the circuit, determine the number of inputs and output. Assign a symbol to each
- Derive the Truth Table that defines required relationship between inputs and outputs
- Obtain Boolean function for each output as a function of the input variable
- Draw the logic diagram and verify the correctness of the design

- MIN NO OF GATES
- MIN NO OF INPUTS
- MIN PROPAGATION TIME
- LIMITATION OF DRIVING CAPABILITIES OF EACH GATE

- BASIC OPERATION IS ADDITION OF TWO BINARY DIGITS. WHEN 0+0=0, 0+1=1, 1+0=1. AND 1+1=10.
- THEN FIRST THREE OPERATIONS PRODUCE SUM OF ONE DIGIT
- FORTH PRODUCE TWO DIGITS. HIGHER SIGNIFICANT BIT IS CALLED A
- CARRY

- Most Basic arithmetic function is Addition of two binary digits
- 0+0=0, 1+0=1, 0+1=1, 1+1=10 (Carry)
- Carry is added to the next higher order pair of significant values

- A combinational circuit that performs addition of two bit is called Half Adder
- A combinational circuit that performs addition of three bits is called Full Adder (Adding two half adder)
- Binary Adder Subtractor performs addition and subtraction

X

Y

S

C

0

0

0

0

0

1

1

0

1

0

1

0

1

1

0

1

BINARY ADDER – Half Adder

Inputs

Outputs

x

y

z

S

C

0

0

0

0

0

0

0

1

1

0

0

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

0

1

0

1

1

1

0

0

1

1

1

1

1

1

BINARY ADDER - Full Adder

INPUTS

OUTPUTS

C

Full Adder in SOP

Implementation Full Adder with two half Adders

X

Y

D

B

0

0

0

0

0

1

1

1

1

0

1

0

1

1

0

0

Inputs

Outputs

x

y

z

B

D

0

0

0

0

0

0

0

1

1

1

0

1

0

1

1

0

1

1

1

0

1

0

0

0

1

1

0

1

0

0

1

1

0

0

0

1

1

1

1

1

- INPUTOUTPUT EX-3
A B C D W X Y Z

0 0 0 0 0+3 0 0 1 1

0 0 0 1 1+3 0 1 0 0

0 0 1 0 2+3 0 1 0 1

0 0 1 1 3+3 0 1 1 0

0 1 0 0 4+3 0 1 1 1

0 1 0 1 5+3 1 0 0 0

0 1 1 0 6+3 1 0 0 1

0 1 1 1 7+3 1 0 1 0

1 0 0 0 8+3 1 0 1 1

1 0 0 1 9+3 1 1 0 0

DRAW K-MAPS WITH Don,t Care for WXYZ.

DRAW DIAGRAM

- Determine the function that circuit implements
- Ensure that circuit is combinational and not sequential
- No feedback (Output of one circuit as a input to another)

- Logic diagram
- Obtain the Boolean function or truth table

ANALYSIS OF COMBINATIONAL LOGIC

ANALYSIS OF COMBINATIONAL LOGIC

Inputs

Outputs

A

B

C

F1

F2

0

0

0

0

0

0

0

1

1

0

0

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

0

1

0

1

1

1

0

0

1

1

1

1

1

1

ANALYSIS OF COMBINATIONAL LOGIC

INPUTS

OUTPUTS

From the truth table can you tell the function of the circuit?

REPLACE INVERTERS WITH NAND GATES FUNCTIONS REMAIN THE SAME

- BOOLEAN FUNCTION IMPLEMENTATION
- F= A(B+CD)+BC’
- STEP -1. From Algebric Expression draw Logic Diagram with AND ,OR. NOT Gates.
- STEP -2. Draw 2nd Logic Diagram with NAND Gates.
- STEP 3. Remove Casecade Inverters.Remove Single Inverter and complement Variable.
- F= (A+B’)(CD+E)

- STEP-1. Given Logic Diagram.
- STEP-2 Boolean Function.
- STEP-3 Truth Table.
- STEP-4 K-Map
- STEP-5 Function

SYMBOLS OF NAND GATE

BLOCK DIAGRAM TRANSFORMATION

STEP-1 Draw NAND Logic Diagram

Remove Bubbles

Step-3

COVERT NAND TO AND INVERT and INVERT OR

STEP-2

MULTI LEVEL NOR CIRCUITS

REPLACE NOT WITH NOR GATES

ANALYSIS PROCEDURE AND BLOCK DIAGRAM TRANSFORMATION IS THE SAME AS NAND CCTS

EXCLUSIVE-OR AND EQUIVALENCE FUNCTIONS ALREADY COVERED IN 3RD CHAPTER

- END OF 4TH CHAPTER

- BCD Excess 3 Code
- Digit 8
- Company
- 1 Manager
- 2 Clerk
- 1 Guards
- During office hours When manager present atleast one clerk
- During office hours When Manager missing both clerks should be present
- After Business hours guard should be present
- Input
- Manager A, Clerk B, Clerk C, Guard D, Businesshours E

- Output
- Alarm F

- Input

DESIGN OF COMBINATIONAL LOGIC

Example: Design a combinational circuit with three inputs and one

output. The output is a 1 when the binary value is less than three.

The output is 0 otherwise.

y

y z

00 01 11 10

0

x

1

z

- START OF 5TH CHAPTER

- Binary Adder is a circuit that produces sum of two binary number
- It can be constructed with full adders (FA) connected in cascade, with output carry from one connected to the input carry of the next full adder
- For Example
1011+00111110

3

2

1

0

i

CASCADE 4-BIT FULL ADDER

BINARY ADDER

3

2

1

0

i

- For Example
- 1011+001111110

Carry Propagation

Addition of two numbers in parallel implies that all bits are available

for computation.

Total propagation delay = propagation delay of a gate # gate levels

In this case no of gate levels are =8

available only after C3 has

propagated through

Carry Propagation

Q: Find the total C propagation delay in the 4-bit full adder circuit.

Although some out (1/0) will be there, it may not be correct. Stable value only after carry propagation

Carry Propagation

The carry propagation time is a limiting factor on the speed with

which two numbers are added.

The most widely technique for reducing the carry propagation time

in a parallel adder uses the principle of carry lookahead

If define two variables P & G.

Carry Propagate from Ci to Ci+1

Carry Generate from Input

Carry Lookahead Generator

Boolean Function for Carry is expressed in SOP (2 level AND and OR gate)

4-bit Adder with Carry Lookahead

XOR Generates P

AND Generates G

Overflow

Overflow occurs when two numbers of n digits are added and the

sum occupies n +1 digits.

If V = 0 no overflow: n-bit results is correct.

If V = 1 overflow: The result contains n + 1 bits, and the (n+1)th bit is the

actual sign.( It means for 8bits we need 9bits)

- Binary adder (1+1) Bit + 1 Carry= 3 bits
- BCD(4+4) Bits + 1 Carry= 9 bits input
- Max output is 9+9+1=19
- 4 bit adder
- Input 2 BCD numbers
- Sum will be in binary form
- Output binary number from 0 to 19
- AIM: Convert Binary back to BCD
- When Binary Sum >1001 Add 0110 to corresponding Binary number for correction.
- Correction is needed when K=1
- For no 1010 to1111 need correction, have 1 in Z8 and Z4 or Z8 and Z2 Thus Boolean Function C=K+Z8Z4+Z8Z2. will be used to get C=1 It will apply 0110 to second 4bit Adder and carry in the output.
- Second addition take place. If carry occurs neglect it.

Magnitude Comparator

A magnitude comparator is a combinational circuit that compares

two numbers, A and B, and then determines their relative magnitudes.

A > B

A = B

A < B

NOT REQUIRED BUT CAN BE EXPLANED IF STUDENT WISH

Consider two numbers, A and B, with four digits each:

Algorithm

XNOR (note mistake p. 133)

For equality to exist, all variables must be equal to 1:

Magnitude Comparator

To determine if A is greater than or less than B, we inspect the relative

magnitudes of significant digits.

If the two digits are equal, we compare the next lower significant pair

of digits. The comparison continues until a pair of unequal digits is

reached.

The sequential comparison can be expressed by:

Compare:

4-bit Magnitude Comparator

XNOR

DECODERS

- A decoder is a combinational circuit that converts binary information
- from n input lines to 2n unique output lines.
- i.e. Given a binary number input we want a specific output.
- Applications:
- Microprocessor memory system: selecting different banks of memory.
- Microprocessor I/O: Selecting different devices.
- Microprocessor instruction decoding: Enabling different functional
- units.
- Memory: Decoding memory addresses (e.g. in ROM).
- Bell for Mess Kitchen

3-to-8-line DECODER Truth Table

- Three inputs are decoded into eight outputs, each representing one of the minterms of the three input variable
- If the input corresponds to minterm mi then the decoder ouputi will be the single asserted output.
- Binary to OCTAL conversion IS ONE OF ITS APPLICATION

3-to-8-line DECODER

2-to-4-line DECODER with Enable Input

The decoder is enabled when E = 0. The output whose value = 0 represents the

minterm is selected by inputs A and B.

The decoder is disabled when E = 1 D0 … D3 = 1

A Decoder with enable input is called a decoder/demultiplexer.

Demultiplexer receives information from a single line and directs it to the output lines.

Multiplexer is opposite of DeMultiplexer (Concept can be used to share Network)

Complemented outputs

A 4 x 16 DECODER

- When w = 0, the top decoder is enabled and the bottom is disabled.
- Top decoder generates 8 minterms 0000 to 0111, while the bottom
- decoder outputs are 0’s.
- When w = 1, the top decoder is disabled and the bottom is enabled.
- Bottom decoder generates 8 minterms 1000 to 1111, while the top
- decoder outputs are 0’s.

Combinational Logic (Full-Adder) using Decoder

- Encoding - the opposite of decoding
- Circuits that perform encoding are called encoders
- An encoder has 2n (or fewer) input lines and n output lines which generate the binary code corresponding to the input values
- Typically, an encoder converts a code containing exactly one bit that is 1 to a binary code corres-ponding to the position in which the 1 appears.

- A Octal-to- Binary encoder
- Inputs: 8 inputs corresponding to octal digits 0 through 7, (D0, … , D7).
- Outputs: 3 bits Binary number.
- Function: If input bit Di is a 1, then the output (X,Y,Z) is the Binary number for i,

- The truth table could be formed, but alternatively, the equations for each of the three outputs can be obtained directly. Circuit is drawn on board

- Input Diis a term in equation X,Y,Z is 1 in the binary value for i.
- Equations:
Z = D1 + D3+ D5 + D7

Y = D2 + D3 + D6 + D7

X = D4 + D5 + D6 + D7

Two inputs at one time gives undefined condition in the output

We get 0 o/p for all 0 i/p and D0

- If more than one input value is 1, then the encoder just designed does not work.
- One encoder that can accept all possible combinations of input values and produce a meaningful result is a priority encoder.
- Among the 1s that appear, it selects the most significant input position (or the least significant input position) containing a 1 and responds with the corresponding binary code for that position.
- Valid Bit indicator set to 1 when one or more inputs are equal to 1 If all inputs are 0, there is no valid input and V is equal to 0.

MULTIPLEXERS/DATA SELECTORS

- A multiplexer is a combinational circuit that selects one of many input lines (2n) and steers it to its single output line.
- Like a electronic Switch to select a source (Data Selector)
- There are (2n) and n selection lines whose bit combinations determine which input is selected.
- The outputs of different AND gates are passed to a single OR Gate

4-to-1LINE MULTIPLEXER DESIGN

1

0

QUADRUPLE 2-to-1LINE MULTIPLEXER

Multiplexer can be combined with common selection inputs to provide multiple bit selection logic.

This circuit has four Multiplexers, each capable of selecting one of two input lines

Output Yo can be selected to come from either input Ao or Bo

ENABLE must be active for Normal operation

This circuit allows us to consider one of two 4 bits sets of data lines

- Function in sum of minterms
- Take truth table
- Take single variable of highest order A and complement for 0 to 3 minterms and uncomplement for remaining terms
- Draw implementation Table
- Circle all minterms given in function
- Inspect each column separately IF
- Two columns not circled apply 0 to corresponding mux input

- If two minterms are circled apply 1
- If bottom minterm is circled apply A
- If top minterm is circled apply A’ to mux
- IMPLEMNT ON BOARD
- 2nd Method. Use A and B for selection,C for input of Mux

- Decoder need additionl OR gate
- One Decoder can generate all minterms. In mux one mux needed for each o/p function.
- Decoder is used for decoding Binary info
- Mux is used to form a salected path between multipal sources and single destination

RAM IS READ/WRITE MEMORY. In which data can be written or read from any sel address in any sequence.

TYPES

Static RAM(SRAM)

Dynamic RAM(DRAM)

Types of SRAM

Asynchronous SRAM

Synchronous burst SRAM

Types of DRAM

Fast Page Mode DRAM

Extended Data Out DRAM

Burst EDO DRAM

Synchronous DRAM

Storage Cell

It is a memory device in which a fixed set of info is stored

Determined by no of words =2k & word= n bits=number of outputs

- Mask Programming
- PROM
- Erasable PROM
- Functions
- It gives output for each minterm of SOP
- It is storage unit. Store words. i/p gives address to words which is applied to the o/p
- Used in design of control unit.

- Micro Programmed control unit Use ROM to store Binary con info

- Size is specified by number of bits.It means unit has 4 o/p,9 i/p,to secify=512 words or 512*4=2048 bits.
- 32*4 ROM address is 5 bit no to 32 minterms using 32 AND gates and 5 inverters
- 4 OR gates ,each having 32 links.
- Fuse can be linked as desired.
- F1 (A1,A0 )={(1,2,3,)}
- F2 (A1,A0)={(0,2)}

3*8 ROM

- I4,I3,I2,I1,I0 A7A6A5A4A3A2A1A0
- 0 0 0 0 0 1 0 1 1 0 1 1 0
- 0 0 0 0 1 0 0 0 1 1 1 0 1
- 0 0 0 1 0 1 1 0 0 0 1 0 1
- 0 0 0 1 1 1 0 1 1 0 0 1 1
- 1 1 1 1 1 0 0 1 1 0 0 1 1

Programming the ROM as given in truth table Input=00011. output=10110010 from truthtable

EXAMPLE 5.5 ROM IMPLEMENTATION PAGE 187

Step1 Derive Truth Table for the ROM. As B0=A0 so no need

B1 is always =0 so o/p is known We need to generate 4 outputs Therefore we need 3 input and 4 outputs 3 i/p gives 8 words of 4bits

PLA

- It is similar to PROM except PLA does not provide full coding because Decoder is replaced by an array of AND gates programmed to generate any product term of i/p variables and connect to OR gates to give SOP for Boolean Function.
- Size of PLA is specified by,no of i/p(n), no of product terms(k), no of o/p (m),
- No of Links=2nxk+kxm+m
- Types. a. Mask programmable b.Field Prog array
- Design PLA with 3i/p, 3 sop and 2 o/p for F1=A,B,C,=(4,5,7) F2=(A,B,C)=(3,5,7)

- Example 5.6 P 194
- Designe PLA with 3 i/p,4 Product terms,and 2 o/p

USING XOR GATES INSTEAD INVERTERS AND LINKS T=O/P MEANS 2ND I/P OF XOR BE CONNECTED TO 0 IF C THEN CONNECT TO 1.

- END OF 5TH CHAPTER

Function implementation using 8x1multiplexer

- Complete the truth table from the SOP.
- The first n – 1 variables in the table are applied to the
- selection inputs of the multiplexer.
- For each combination of the selection variables, we evaluate
- the output as a function of the last variable.
- 4.Apply these values to the data input in proper order.

Gray

Binary

A B C

x y z

0

0

0

0

0

0

1 0

0

0

0 1

1

1 0

0 1 0

0 1 0

0 1

1

0 1

1

1 0

0

1

1

1

1 0 1

1 0 1

1

1 0

0

0 1

1

1

1

- Design a circuit to convert a 3-bit Gray code to a binary code
- The formulation givesthe truth table on theright
- It is obvious from thistable that X = C and theY and Z are more complex

Function implementation using 8x1 MUX

note the order of input lines

Design:

- Complete the truth table.
- The first n – 1 variables in the table are applied to the selection inputs of the multiplexer.
- For each combination of the selection variables, we evaluate the output as a function of the last variable.
- Apply these values to the data input in proper order.

F = C

F = C

F = C

F = C

F = C

F = C

F = C

F = C

- Rearrange the table so that the input combinations are in counting order, pair rows, and find elementary functions

C

C

C

C

- Assign the variables and functions to the multiplexer inputs:

C

C

D10

D00

D11

D01

D12

Y

Out

Z

C

D02

Out

X

C

C

D13

D03

4-to-1

4-to-1

A

A

S1

S1

MUX

MUX

S0

S0

B

B

- Implement m functions of n variables with:
- Sum-of-minterms expressions
- An m-wide 2n-to-1-line multiplexer

- Design:
- Find the truth table for the functions.
- In the order they appear in the truth table:
- Apply the function input variables to the multiplexer inputs Sn - 1, … , S0
- Label the outputs of the multiplexer with the output variables

- Value-fix the information inputs to the multiplexer using the values from the truth table (for don’t cares, apply either 0 or 1)

- Rearrange the table sothat the input combinationsare in counting order
- Functions y and z can be implemented usinga dual 8-to-1-line multiplexer by:
- connecting A, B, and C to the multiplexer select inputs
- placing y and z on the two multiplexer outputs
- connecting their respective truth table values to the inputs

0

0

D10

D00

1

D11

D01

1

1

D12

D02

1

0

D13

D03

0

D14

1

0

D04

Out

Z

Y

Out

0

1

D15

D05

0

1

D16

D06

1

D17

0

D07

A

S2

A

S2

8-to-1

8-to-1

S1

B

S1

B

MUX

MUX

S0

S0

C

C

Three State Gates

A three-state gate is a digital circuit that exhibits three states: 0, 1

and a high-impedance (high z state). The high impedance state

behaves as an open circuit Output appears to be disconnected and circuit has no significance.

Because of this feature (high z state), a large number of three-state

gate outputs can be connected to form a common line without

endangering load effects.

Multiplexers with Three State Gates

When EN = 0, decoder outputs are zero,

and the bus lines are in high z state.

When EN = 1, one of the three-state buffers

will be active depending on the binary value in

the select inputs of the decoder.

When Select is 0 upper buffer is enabled, lower when Select is 1

- Control inputs to buffer determine which input will be connected to output
- No more than one buffer may be in active state at any given time
- When Enable 0 All four outputs are 0’s and circuit in high impedance
- When Enable 1 one of the three state buffer will be active