UNIT -1

Digital Computers

A Digital computer can be considered as a digital system that performs various computational tasks.

 The first electronic digital computer was developed in the late 1940s and was used primarily for
numerical computations.
 The digital computers use the binary number system, which has two digits: 0 and 1. A binary digit
is called a bit.
 A computer system is subdivided into two functional entities:
 Hardware and
 Software.

The hardware consists of all the electronic components and electromechanical devices that comprise the
physical entity of the device.

The software of the computer consists of the instructions and data that the computer manipulates to
perform various data-processing tasks.
1.Explain about block diagram of digital computers and its working Principles/operations with neat
diagram?

A computer can process data, pictures, sound and graphics. They can solve highly complicated problems
quickly and accurately. A computer as shown in Fig. performs basically five major computer operations
or functions irrespective of their size and make. These are

1) It accepts data or instructions by way of input,

2) It stores data,

3) It can process data as required by the user,

4) It gives results in the form of output, and

5) It controls all operations inside a computer.

Working principles:

1. Input:
This is the process of entering data and programs in to the computer system. It takes as inputs
raw data and performs some processing giving out processed data. Therefore, the input unit takes
data from us to the computer in an organized manner for processing.

2. Storage:

The process of saving data and instructions permanently is known as storage. The data is
first stored in the storage unit for faster access and processing. This storage unit or the primary storage of
the computer system is designed to do the above functionality. It provides space for storing data and
instructions.

3. Processing:

The task of performing operations like arithmetic and logical operations is called processing.
The Central Processing Unit (CPU) takes data and instructions from the storage unit and makes all sorts
of calculations based on the instructions given and the type of data provided. It is then sent back to the
storage unit.

4. Output:

This is the process of producing results from the data for getting useful information. The
output produced by the computer after processing must also be kept somewhere inside the computer
before being given to you in human readable form. Again the output is also stored inside the computer for
further processing.

5. Control:

The manner how instructions are executed and the above operations are performed.
Controlling of all operations like input, processing and output are performed by control unit. It takes care
of step by step processing of all operations inside the computer.
Functional Units

The computer system is divided into three separate units for its operation. They are

Arithmetic Logical Unit (ALU)

After you enter data through the input device it is stored in the primary storage unit. The actual
processing of the data and instruction are performed by Arithmetic Logical Unit.

The major operations performed by the ALU are addition, subtraction, multiplication, division,
logic and comparison. Data is transferred to ALU from storage unit when required. After processing the
output is returned back to storage unit for further processing or getting stored.

Control Unit (CU)

The Control Unit, which acts like the supervisor seeing that things are done in proper fashion.

Control Unit is responsible for coordinating various operations using time signal.

The control unit determines the sequence in which computer programs and instructions are
executed. It coordinates the activities of computer’s peripheral equipment as they perform the input and
output.

Central Processing Unit (CPU)

The ALU and the CU of a computer system are jointly known as the central processing unit.

CPU as the brain of any computer system. It is just like brain that takes all major decisions, makes all
sorts of calculations and directs different parts of the computer functions by activating and controlling the
operations.
 2.Logic Gates and its types

o The logic gates are the main structural part of a digital system.
o Logic Gates are a block of hardware that produces signals of binary 1 or 0 when input logic
requirements are satisfied.
o The seven basic logic gates includes: AND, OR, XOR, NOT, NAND, NOR, and XNOR.
o Each gate has one or two binary input variables designated by A and B and one binary output
variable designated by x.

AND GATE:

The AND gate is an electronic circuit which gives a high output only if all its inputs are high. The AND
operation is represented by a dot (.) sign.

Logic diagram

Truth Table
OR GATE:

The OR gate is an electronic circuit which gives a high output if one or more of its inputs are high. The
operation performed by an OR gate is represented by a plus (+) sign.

NOT GATE:

The NOT gate is an electronic circuit which produces an inverted version of the input at its output. It is
also known as an Inverter.
NAND GATE:

The NOT-AND (NAND) gate which is equal to an AND gate followed by a NOT gate. The NAND gate
gives a high output if any of the inputs are low. The NAND gate is represented by a AND gate with a
small circle on the output. The small circle represents inversion.

NOR GATE:

The NOT-OR (NOR) gate which is equal to an OR gate followed by a NOT gate. The NOR gate gives a
low output if any of the inputs are high. The NOR gate is represented by an OR gate with a small circle
on the output. The small circle represents inversion.
Exclusive-OR/ XOR GATE:

The 'Exclusive-OR' gate is a circuit which will give a high output if one of its inputs is high but not both
of them. The XOR operation is represented by an encircled plus sign.

EXCLUSIVE-NOR/Equivalence GATE:

The 'Exclusive-NOR' gate is a circuit that does the inverse operation to the XOR gate. It will give a low
output if one of its inputs is high but not both of them. The small circle represents inversion.
 Minimization Of Boolean Expressions-
There are following two methods of minimizing or reducing the boolean expressions-

Karnaugh Map:

 The Karnaugh Map also called as K Map is a graphical representation that provides a systematic
method for simplifying the boolean expressions.
 For a boolean expression consisting of n-variables, number of cells required in K Map = 2n cells.

Two Variable K Map-

Two variable K Map is drawn for a boolean expression consisting of two variables.

 The number of cells present in two variable K Map = 22 = 4 cells.

 So, for a boolean function consisting of two variables, we draw a 2 x 2 K Map.
Two variable K Map may be represented as-

Here, A and B are the two variables of the given boolean function.
 Three Variable K Map-

Three variable K Map is drawn for a boolean expression consisting of three variables.

 The number of cells present in three variable K Map = 2 3 = 8 cells.

 So, for a boolean function consisting of three variables, we draw a 2 x 4 K Map.
Three variable K Map may be represented as-

Here, A, B and C are the three variables of the given boolean function.

Four Variable K Map-

Four variable K Map is drawn for a boolean expression consisting of four variables.

 The number of cells present in four variable K Map = 2 4 = 16 cells.

 So, for a boolean function consisting of four variables, we draw a 4 x 4 K Map.
Four variable K Map may be represented as-

Here, A, B, C and D are the four variables of the given boolean function.
 Karnaugh Map Simplification Rules-
To minimize the given boolean function,

 We draw a K Map according to the number of variables it contains.

 We fill the K Map with 0’s and 1’s according to its function.
 Then, we minimize the function in accordance with the following rules.
Rule-01:

We can either group 0’s with 0’s or 1’s with 1’s but we can not group 0’s and 1’s together.

 X representing don’t care can be grouped with 0’s as well as 1’s.

Rule-02:

Groups may overlap each other.

Rule-03:

We can only create a group whose number of cells can be represented in the power of 2.

 In other words, a group can only contain 2n i.e. 1, 2, 4, 8, 16 and so on number of cells.
Example-

Rule-04:

Groups can be only either horizontal or vertical.

 We can not create groups of diagonal or any other shape.

 Rule-05:

Each group should be as large as possible.

Example-

Rule-06:

Opposite grouping and corner grouping are allowed.

 The example of opposite grouping is shown illustrated in Rule-05.

 The example of corner grouping is shown below.
Example-

Rule-07:

There should be as few groups as possible.

KARNAUGH MAP

1: Minimize the following boolean function- F(A, B, C, D) = Σm(0, 1, 2, 5, 7, 8, 9, 10, 13, 15)

F(A, B, C, D) = (A’B + AB)(C’D + CD) + (A’B’ + A’B + AB + AB’)C’D + (A’B’ + AB’)(C’D’ + CD’)

= BD + C’D + B’D’

F(A, B, C, D) = BD + C’D + B’D’

2: Minimize the following boolean function-F(A, B, C, D) = Σm(0, 1, 3, 5, 7, 8, 9, 11, 13, 15)

F(A, B, C, D) = (A’B’ + A’B + AB + AB’)(C’D + CD) + (A’B’ + AB’)(C’D’ + C’D)

= D + B’C’

F(A, B, C, D) = B’C’ + D
3.Minimize the following boolean function F(A, B, C, D) = Σm(3, 4, 5, 7, 9, 13, 14, 15)

F(A, B, C, D) = A’B(C’D’ + C’D) + (A’B’ + A’B)(CD) + (AB + AB’)(C’D) + AB(CD + CD’)

= A’BC’ + A’CD + AC’D + ABC

F(A, B, C, D) = A’BC’ + A’CD + AC’D + ABC

4: Minimize the Boolean function using k-map

F(W, X, Y, Z) = Σm(1, 3, 4, 6, 9, 11, 12, 14)

F(W, X, Y, Z) = (W’X + WX)(Y’Z’ + YZ’) + (W’X’ + WX’)(Y’Z + YZ)

= XZ’ + X’Z

=X⊕Z

F(W, X, Y, Z) = X ⊕ Z
 Combinational Circuits

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 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.

Block diagram
Half Adder

 Half adder is a combinational logic circuit with two inputs and two outputs.

 The half adder circuit is designed to add two single bit binary number A and B.

 It is the basic building block for addition of two single bit numbers.

 This circuit has two outputs carry and sum.

Block diagram

Circuit Diagram

Truth Table
Full Adder

 Full adder is developed to overcome the drawback of Half Adder circuit.

 It can add two one-bit numbers A and B, and carry c.

 The full adder is a three input and two output combinational circuit.

Block diagram

Circuit Diagram

Truth Table
Half Subtractor

Half subtractor is a combination circuit with two inputs and two outputs (difference and
borrow).

It produces the difference between the two binary bits at the input and also produces an output
(Borrow) to indicate if a 1 has been borrowed.

In the subtraction (A-B), A is called as Minuend bit and B is called as Subtrahend bit.

Circuit Diagram

Truth Table
Full Subtractor

The full subtractor is a combinational circuit with three inputs A,B,C and two output D and C'.

A is the 'minuend', B is 'subtrahend', C is the 'borrow' produced by the previous stage, D is the difference
output and C' is the borrow output.

Circuit Diagram

Truth Table
 Parallel Adder

 The Full Adder is capable of adding only two single digit binary number along with a carry input.
But in practical we need to add binary numbers which are much longer than just one bit.

 To add two n-bit binary numbers we need to use the n-bit parallel adder.

 It uses a number of full adders in cascade.

 The carry output of the previous full adder is connected to carry input of the next full adder.

4 Bit Parallel Adder

Block diagram

.
 FLIP-FLOP

Flip Flop

 A flip-flop is a sequential digital electronic circuit having two stable states that can
be used to store one bit of binary data.
 Flip-flops are the fundamental building blocks of all memory devices.

Types of Flip–Flops

 S-R flip-flop
 J-K flip-flop
 D flip-flop
 T flip-flop
 S-R Flip-flop

 This is the simplest flip-flop circuit. It has a set input (S) and a reset input (R).
 When in this circuit when S is set as active, the output Q would be high and the Q’ will be low.
 If R is set to active then the output Q is low and the Q’ is high.
 Once the outputs are established, the results of the circuit are maintained until S or R get changed,
or the power is turned off.

Truth table

S R Q State

0 0 0 No Change

0 1 0 Reset

1 0 1 Set

1 1 X
 J-K Flip-flop

 The operation of the JK flip-flop is similar to the SR flip-flop.

 When the input J and K are different then the output Q takes the value of J at the next clock edge.
 When J and K both are low then NO change occurs at the output. If both J and K are high, then at
the clock edge, the output will toggle from one state to the other.

Truth table :

J K Q State

0 0 0 No Change

0 1 0 Reset

1 0 1 Set

1 1 Toggles Toggle
 D Flip-flop

 The D flip-flops are generally used for shift-registers and counters.

 The change in output state of D flip-flop depends upon the active transition of clock.
 The output (Q) is same as input and changes only at active transition of clock

Truth table :

D Q

0 0

1 1
T Flip-flop

 A T flip-flop (Toggle Flip-flop) is a simplified version of JK flip-flop.

 The T flop is obtained by connecting the J and K inputs together.
 The flip-flop has one input terminal and clock input.
 These flip-flops are said to be T flip-flops because of their ability to toggle the input state.
 Toggle flip-flops are mostly used in counters.

Truth Table:

T Q(t) Q(t+1)

0 0 0

0 1 1

1 0 1

1 1 0
Master-Slave JK Flip Flop

 The master-slave flip flop is constructed by combining two JK flip flops.

 These flip flops are connected in a series configuration.
 In these two flip flops, the 1st flip flop work as "master", called the master flip flop, and the 2nd
work as a "slave", called slave flip flop.
 The master-slave flip flop is designed in such a way that the output of the "master" flip flop is passed
to both the inputs of the "slave" flip flop. The output of the "slave" flip flop is passed to inputs of the
master flip flop.
 In simple words, when CP set to false for "master", then CP is set to true for "slave", and when CP
set to true for "master", then CP is set to false for "slave".