UNIT 3

Number System And Logic Gates

3.1 Number System and conversion

Introduction to number system and conversions

A digital system understands positional number system where there are few symbols called digits and these symbol represents different values depending on their position in the number.

A value of the digit is determined by using

•     The digit
•     Its position
•     The base of the number system

Decimal Number System

• Its the system that we use in our daily life. It has a base 10 and uses 10 digits from 0 to 9.
• Here, the successive positions towards the left of the decimal point represent units, tens, hundreds, thousands and so on.
• Each and every position represents a specific power of the base (10). For example, the decimal number 4321 consists of the digit 1 in the units position, 2 in the tens position, 3 in the hundreds position, and 4 in the thousands position, and its value can be written as

• As a computer programmer, we should understand the following number systems used in computers.

3.2 Binary their conversion

• It uses two digits 0 and 1.
• It is also called as base 2 number system.
• Here, each position in any binary number represents a power of the base (2). Example: 23
• The last position represents a y power of the base (2). Example: 2y where y represents the last position.

Example

Binary Number: 101112

Calculating the Decimal Equivalent of binary number −

Note: 101112 is normally written as 10111.

3.3 BCD their conversion

BCD numbers are from 0 to 9.

3.4 Octal Number System

•     It consists of eight digits 0, 1,2,3,4,5,6,7.
•     It is also named as base 8 number system.
•     Here each position represents a power of the base (8). Example: 82
•     The last position represents a y power of the base (8). Example: 8y where y represents the last position .

Example

Octal Number − 125758

Calculating Decimal Equivalent −

Note: 125758 is normally written as 12575 in octal.

3.5 Decimal to Any Base System

It includes the following steps:

•     Firstly, divide the decimal number which is to be converted by the value of its new base.
•     Then, after getting the remainder from the above step make it the rightmost digit (least significant digit) i.e. LSD of new base number.
•     Further divide the quotient by the new base.
•     Then record the remainder from the above step as the next digit (towards the left) of the new base number.
• Repeat these steps and get remainders from right to left till the quotient becomes zero.
• The last remainder obtained will be the Most Significant Digit (MSD) of the new base number.

For example −

Decimal Number: 2710

Calculating Binary Equivalent −

Hence, the remainders are arranged in the reverse order and we get:

Decimal Number − 2710 = Binary Number − 110112.

3.6 Hexadecimal Number System

•     It uses 10 digits starting from 0,1,2,3,4,5,6,7,8,9 and 6 letters A,B,C,D,E,F.
•     These letters represents numbers as A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
•     It is also known as base 16 number system.
•     Here each position represents a power of the base (16). Example 161.
• The last position represents a y power of the base (16). Example: 16y where y represents the last position .

Example −

Hexadecimal Number: 19FDA16

Calculating Decimal Equivalent −

Note − 19FDA16 is normally written as 19FDA in hexa decimal.

Conversions

There are many techniques which are used to convert numbers from one base to another. These are as follows −

• Decimal to any other Base System
• Any Base System to Decimal
• Any Base System to Non-Decimal
• Binary to Octal
• Octal to Binary
• Binary to Hexadecimal
• Hexadecimal to Binary

Any Base System to Decimal System

•     Here, the column (positional) value of each digit (this depends on the position of the digit and the base of the number system) is multiplied with the obtained column values of the digits in the corresponding columns.
•     The sum the products is then calculated which gives the total equivalent value in decimal.

For example :

Binary Number − 111102

Calculating Decimal Equivalent −

Binary Number − 111102 = Decimal Number − 3010

Any Base System to Non-Decimal System

•     The original number is converted to its equivalent decimal number (base 10).
•     Then the decimal number so obtained is further converted to the new base number.

Example

Octal Number − 268

Calculating its Binary Equivalent −

Step 1 – Converting octal to Decimal

Octal Number − 268 = Decimal Number − 2210

Step 2 − Converting Decimal to Binary

Decimal Number − 2210 = Binary Number − 101102

Octal Number − 268 = Binary Number − 101102

Binary to Octal

•     Dividing the binary digits into groups of three starting from right to left.
•     Converting each group of three binary digits into one octal digit.

For example:

Binary Number − 101012

Its Octal Equivalent −

Binary Number − 101012 = Octal Number − 258

Octal to Binary

•     Converting each octal digit to a 3 digit binary number and they can be treated as decimal number for this conversion.
•     Then combining all the resulting binary groups into a single binary number.

For example:

Octal Number − 258

Its Binary Equivalent −

Octal Number − 258 = Binary Number − 101012

Binary to Hexadecimal

•     Dividing the binary digits into groups of four (starting from right to left).
•     Then converting each group of four binary digits into one hexadecimal number.

For example:

Binary Number − 101012

Its hexadecimal Equivalent −

Binary Number − 101012 = Hexadecimal Number − 1516

Hexadecimal to Binary

•     Converting each hexadecimal digit to a 4 digit binary number and they can be treated as decimal number.
•     Combining all the resulting binary groups into a single binary number.

For example:

Hexadecimal Number − 1516

Its Binary Equivalent −

Hexadecimal Number − 1516 = Binary Number − 101012

3.7  Binary Arithmetic:

It is an essential part of all the digital calculations.

Binary Addition

• It is a basis for binary subtraction, multiplication, division.
• There are four rules which are as follows:

                                 Fig.1 Rules of Binary addition

In fourth step, a sum (1 + 1 = 10) i.e. 0 is written in the given column and a carry of 1 over to the next column is done.

For Example −

                                 Fig.2 Binary addition

Binary Subtraction

Subtraction and Borrow, these are the two words that will be used very frequently for binary subtraction. There rules of binary subtraction are:

                                  Fig.3 Rules of Binary Subtraction

For Example

                                 Fig.4 Binary subtraction

Binary Multiplication

• It is similar to decimal multiplication.
• It is also simpler than decimal multiplication as only 0s and 1s are involved.
• There are four rules of binary multiplication which are:

                             Fig.5 Rules of Binary Multiplication (Ref. 1)

For Example

                             Fig.6 Binary Multiplication

Binary Division

• It is similar to decimal division.
• It is also called as the long division procedure.

For Example

                                       Fig.7 Binary Division

Binary system complements

• Complements are used in order to simplify the subtraction operation and the logical manipulations can be done.
• For each radix-r system there are two types of complements:

• The binary system has radix r = 2.
• Therefore, the two types of complements for the binary system are known as 1's complement and 2's complement.

Octal Number System

• It consists of eight digits 0, 1,2,3,4,5,6,7.
• It is also named as base 8 number system.
• Here each position represents a power of the base (8). Example: 82

• The last position represents a y power of the base (8). Example: 8y where y represents the last position .

Example

Octal Number − 125758

Calculating Decimal Equivalent −

Note: 125758 is normally written as 12575 in octal.

Octal Addition

The octal addition table is as follows:

                            Fig.10: Octal addition table

• To use the above table, simply follow the example:
• Add 68and 58.
• Locate 6 in the A column then locate the 5 in the B column.
• The point in 'sum' area where these two columns intersect is the 'sum' of two numbers.

68 + 58 = 138.

For Example :

                            Fig.11: Octal addition  (Ref. 1)

Octal Subtraction

• It follows the same rules as the subtraction of numbers in any other system.
• The only difference is in borrowed number.
•  In the decimal system, we borrow a group of 1010.
• In the binary system, we borrow a group of 210.
• But in the octal system we borrow a group of 810.

For Example

                            Fig.12: Octal subtraction  (Ref. 1)

3.8 De Morgan’s Theorem

• It is useful in finding the complement of Boolean function.
• It states that “The complement of logical OR of at least two Boolean variables is equal to the logical AND of each complemented variable”.
• It can be represented using 2 Boolean variables x and y as

(x + y)’ = x’.y’

• The dual of the above Boolean function is

(x.y)’ = x’ + y’

• Therefore, the complement of logical AND of the two Boolean variables is equivalent to the logical OR of each complemented variable.
• Similarly, DeMorgan’s theorem can be applied for more than 2 Boolean variables also.

Simplification of Boolean Functions

Numerical

• Simplify the Boolean function,

f = p’qr + pq’r + pqr’ + pqr

Numerical

Find the complement of the Boolean function,

f = p’q + pq’.

Solution:

Using DeMorgan’s theorem, (x + y)’ = x’.y’ we get

                                  ⇒ f’ = (p’q)’.(pq’)’

Then by second law, (x.y)’ = x’ + y’ we get

                                 ⇒ f’ = {(p’)’ + q’}.{p’ + (q’)’}

Then by using, (x’)’=x  we get

                                 ⇒ f’ = {p + q’}.{p’ + q}

                                 ⇒ f’ = pp’ + pq + p’q’ + qq’

Using x.x’=0 we get

                                   ⇒ f = 0 + pq + p’q’ + 0

                                     ⇒ f = pq + p’q’

Therefore, the complement of Boolean function, p’q + pq’ is pq + p’q’.

LOGIC GATES

The basic gates are namely AND gate, OR gate  & NOT gate.

3.9 AND gate

It is a digital circuit that consists of two or more inputs and a single output which is the logical AND of all those inputs. It is represented with the symbol ‘.’.

The following is the truth table of 2-input AND gate.

Here A, B are the inputs and Y is the output of two input AND gate.

If both inputs are ‘1’, then only the output, Y is ‘1’. For remaining combinations of inputs, the output, Y is ‘0’.

3.10 OR gate

3.11 NOT gate

Universal gates

• NAND & NOR gates are known as universal gates.
• We can implement any Boolean function by using NAND gates and NOR gates alone.

NAND gate

NOR gate

Special Gates

3.12 Ex-OR gate

3.13 Ex-NOR gate

COMBINATIONAL CIRCUIT

3.14     Half Adder

• It is a combinational circuit which has two inputs and two outputs.
• It is designed to add two single bit binary number A and B.
• It has two outputs carry and sum.

3.15  Full Adder

• It is developed to overcome the drawback of Half Adder circuit.
• It can add two one-bit numbers A and B and a carry C.
• It is a three input and two output combinational circuit.

3.16     Flip flops :

Race Around Condition In JK Flip-flop –

• For J-K flip-flop, if J=K=1, and if clk=1 for a long period of time, then output Q will toggle as long as CLK remains high which makes the output unstable or uncertain.
• This problem is known as race around condition in J-K flip-flop.
• This problem can be avoided by ensuring that the clock input is at logic “1” only for a very short time.
• Hence the concept of Master Slave JK flip flop was introduced.

Master Slave JK flip flop –
It is basically a combination of two JK flip-flops connected together in series.

• The first is the “master” and the other is a “slave”.
• The output from the master is connected to the two inputs of the slave whose output is fed back to inputs of the master.
• In addition to these two flip-flops, the circuit comprises of an inverter.
• The inverter is connected to clock pulse in such a way that an inverted clock pulse is given to the slave flip-flop.
• In other words if CP=0 for a master flip-flop, then CP=1 for a slave flip-flop and vice versa.

                                         Fig.  Master Slave Flip flop

Working of a master slave flip flop –

1. When the clock pulse goes high, the slave is isolated; J and K inputs can affect the state of the system. The slave flip-flop is isolated when the CP goes low. When the CP goes back to 0, information is transmitted from the master flip-flop to the slave flip-flop and output is obtained.
2. The master flip flop is positive level triggered and the slave flip flop is negative level triggered, hence the master responds prior to the slave.
3. If J=0 and K=1, Q’ = 1 then the master goes to the K input of the slave and the clock forces the slave to reset therefore the slave copies the master.
4. If J=1 and K=0, Q = 1 then the master goes to the J input of the slave and the Negative transition of the clock sets the slave and thus copy the master.
5. If J=1 and K=1, the master toggles on the positive transition and the slave toggles on the negative transition of the clock.
6. If J=0 and K=0, the flip flop becomes disabled and Q remains unchanged.

Timing Diagram of a Master flip flop –

1. When the CP = 1 then the output of master is high and remains high till  CP = 0 because the state is stored.
2. Now the output of master becomes low when the clock CP = 1 and remains low until the clock becomes high again.
3. Thus toggling takes place for a clock cycle.
4. When the CP = 1 then the master is operational but not the slave.
5. When the clock is low, the slave becomes operational and remains high until the clock again becomes low.
6. Toggling takes place during the whole process since the output changes once in a cycle.
7. This makes the Master-Slave J-K flip flop a Synchronous device which passes data with the clock signal.

D Flip Flop

3.17     Introduction to Microprocessor and Microcontroller

Microprocessor: Introduction

• It is a controlling unit that is fabricated on a small chip.
• It is capable to perform arithmetic logic operations and communicate with various devices connected to it.
• It comprises of an ALU (Arithmetic Logic Unit), MU (Memory unit) and CU (Control unit).
• It also consists of register array with registers named as B, C, D, E, H, L and ACC.

                                Fig.: Basic block diagram of microprocessor

Microcontroller: Introduction

• Microcontroller is a compact tiny computer that is fabricated inside a chip and is used in automatic control systems including security systems, office machines, power tools, alarming system, traffic light control, washing machine, and much more.
• It is economical programmable logic control that can be interfaced with external devices in order to control the devices from a distance.
• It was specially built for embedded system and consisted of read write memory, read only memory, I/O ports, processor and built in clock.
• C and assembly languages are used to program the microcontrollers.
• There are also other languages available to program the microcontroller but at the start learning a microcontroller programming with C and assembly language is a great choice, both are easy to learn and provide a clear concept about microcontroller.
• Technology have been evolved in an amazing way and made our lives easier more than ever before.

References

1. “Electronics Devices” by Thomas. L. Floyd, 9th Edition, Pearson (Unit I, II)

2. “Modern Digital Electronics” by R.P. Jain, 4th Edition, Tata McGraw Hill (Unit III)

3. “Electronic Instrumentation” by H.S. Kalsi, 3rd Edition, Tata McGraw Hill (Unit IV)

4. “Sensors and Transducers” by D. Patrnabis, 2nd Edition, PHI (Unit V)