3.7 Binary Arithmetic | unit 3 number system and logic gates

BXE

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

(1×1000) + (2×100) + (3×10) + (4×l)

(1×103) + (2×102) + (3×101) + (4×l00)

1000 + 200 + 30 + 1

1234

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

S.N.	Number System & Description
1	Binary Number System Base 2. Digits used: 0 and 1
2	Octal Number System Base 8. Digits used: 0 to 7
3	Hexa Decimal Number System Base 16. Digits used: 0 to 9, Letters used: A- F

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 −

Step	Binary Number	Decimal Number
Step 1	101012	((1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20))10
Step 2	101012	(16 + 0 + 4 + 2 + 1)10
Step 3	101012	2310

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 −

Step	Octal Number	Decimal Number
Step 1	125758	((1 × 84) + (2 × 83) + (5 × 82) + (7 × 81) + (5 × 80))10
Step 2	125758	(4096 + 1024 + 320 + 56 + 5)10
Step 3	125758	550010

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 −

Step	Operation	Result	Remainder
Step 1	27 / 2	13	1
Step 2	13 / 2	6	1
Step 3	6 / 2	3	0
Step 4	3 / 2	1	1
Step 5	1 / 2	0	1

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 −

Step	Hexadecimal Number	Decimal Number
Step 1	19FDA16	((1 × 164) + (9 × 163) + (F × 162) + (D × 161) + (A × 160))10
Step 2	19FDA16	((1 × 164) + (9 × 163) + (15 × 162) + (13 × 161) + (10 × 160))10
Step 3	19FDA16	(65536 + 36864 + 3840 + 208 + 10)10
Step 4	19FDA16	10645810

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 −

Step	Binary Number	Decimal Number
Step 1	111102	((1 × 24) + (1 × 23) + (1 × 22) + (1 × 21) + (0 × 20))10
Step 2	111102	(16 + 8 + 4 + 2 + 0)10
Step 3	111102	3010

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

Step	Octal Number	Decimal Number
Step 1	268	((2 × 81) + (6 × 80))10
Step 2	268	(16 + 6 )10
Step 3	268	2210

Octal Number − 268 = Decimal Number − 2210

Step 2 − Converting Decimal to Binary

Step	Operation	Result	Remainder
Step 1	22 / 2	11	0
Step 2	11 / 2	5	1
Step 3	5 / 2	2	1
Step 4	2 / 2	1	0
Step 5	1 / 2	0	1

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 −

Step	Binary Number	Octal Number
Step 1	101012	010 101
Step 2	101012	28 58
Step 3	101012	258

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 −

Step	Octal Number	Binary Number
Step 1	258	210 510
Step 2	258	0102 1012
Step 3	258	0101012

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 −

Step	Binary Number	Hexadecimal Number
Step 1	101012	0001 0101
Step 2	101012	110 510
Step 3	101012	1516

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 −

Step	Hexadecimal Number	Binary Number
Step 1	1516	110 510
Step 2	1516	00012 01012
Step 3	1516	000101012

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:

Addition Table

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 −

Addition 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:

Subtraction Table

Fig.3 Rules of Binary Subtraction

For Example

Subtraction 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:

Multiplication Table

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

Division 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:

S.N.	Complement	Description
1	Radix Complement	It is referred to as the r's complement.
2	Diminished Radix Complement	It is referred to as the (r-1)'s complement.

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.

1's complement

Here, a number is obtained by changing all 1's to 0's and all 0's to 1's.
This is called as 1's complement.
For Example :

1's complement

Fig.8: 1's complement

2's complement

It is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.
Hence, 2's complement = 1's complement + 1
For Example:

2's complement

Fig.9: 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 −

Step	Octal Number	Decimal Number
Step 1	125758	((1 × 84) + (2 × 83) + (5 × 82) + (7 × 81) + (5 × 80))10
Step 2	125758	(4096 + 1024 + 320 + 56 + 5)10
Step 3	125758	550010

Note: 125758 is normally written as 12575 in octal.

Octal Addition

The octal addition table is as follows:

Octal Addition Table

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 :

Octal Addition 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

Method 1

Given

f = p’qr + pq’r + pqr’ +pqr.

In first and second term r is common and in third and fourth terms pq is common.

So, taking out the common terms by using Distributive law we get,

⇒ f = (p’q + pq’)r + pq(r’ + r)

The terms present in first parenthesis can be simplified by using Ex-OR operation.

The terms present in second parenthesis is equal to ‘1’ using Boolean postulate we get

⇒ f = (p ⊕q)r + pq(1)

The first term can’t be simplified further.

But, the second term is equal to pq using Boolean postulate.

⇒ f = (p ⊕q)r + pq

Therefore, the simplified Boolean function is f = (p⊕q)r + pq

Method 2

Given f = p’qr + pq’r + pqr’ + pqr.

Using the Boolean postulate, x + x = x.

Hence we can write the last term pqr two more times.

⇒ f = p’qr + pq’r + pqr’ + pqr + pqr + pqr

Now using the Distributive law for 1st and 4th terms, 2nd and 5th terms, 3rdand 6th terms we get.

⇒ f = qr(p’ + p) + pr(q’ + q) + pq(r’ + r)

Using Boolean postulate, x + x’ = 1 and x.1 = x for further simplification .

⇒ f = qr(1) + pr(1) + pq(1)

⇒ f = qr + pr + pq

⇒ f = pq + qr + pr

Therefore, the simplified Boolean function is f = pq + qr + pr.

Hence we got two different Boolean functions after simplification of the given Boolean function. Functionally, these two functions are same. As per requirement, we can choose one of them.

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.

A	B	Y = A.B
0	0	0
0	1	0
1	0	0
1	1	1

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’.

The figure below shows the symbol of an AND gate, which is having two inputs A, B and one output, Y.

And Gate

Fig. : AND gate (ref. 1)

Timing Diagram:

3.10 OR gate

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

The truth table of 2-input OR gate is:

A	B	Y = A + B
0	0	0
0	1	1
1	0	1
1	1	1

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

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

The figure below shows the symbol of an OR gate, which is having two inputs A, B and one output, Y.

Fig. : OR gate (ref. 1)

Timing Diagram:

3.11 NOT gate

It is a digital circuit that has one input and one output. Here the output is the logical inversion of input. Hence, it is also called as an inverter.

The truth table of NOT gate is:

A	Y = A’
0	1
1	0

Here A and Y are the corresponding input and output of NOT gate. When A is ‘0’, then, Y is ‘1’. Similarly, when, A is ‘1’, then, Y is ‘0’.

The figure below shows the symbol of NOT gate, which has one input, A and one output, Y.

Fig. : NOT gate (ref. 1)

Timing Diagram:

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

It is a digital circuit which has two or more inputs and single output and it is the inversion of logical AND gate.

The truth table of 2-input NAND gate is:

A	B	Y = (A.B)’
0	0	1
0	1	1
1	0	1
1	1	0

Here A, B are the inputs and Y is the output of two input NAND gate. When both inputs are ‘1’, then the output, Y is ‘0’. If at least one of the input is zero, then the output, Y is ‘1’. This is just the inverse of AND operation.

The image shows the symbol of NAND gate:

Fig.: NAND gate (ref. 1)

NAND gate works same as AND gate followed by an inverter.

Timing Diagram:

NOR gate

It is a digital circuit that has two or more inputs and a single output which is the inversion of logical OR of all inputs.

The truth table of 2-input NOR gate is:

A	B	Y = (A+B)’
0	0	1
0	1	0
1	0	0
1	1	0

Here A and B are the two inputs and Y is the output. If both inputs are ‘0’, then the output is ‘1’. If any one of the input is ‘1’, then the output is ‘0’. This is exactly opposite to two input OR gate operation.

The symbol of NOR gate is:

Fig.: NOR gate (ref. 1)

NOR gate works exactly same as that of OR gate followed by an inverter.

Timing Diagram:

Special Gates

3.12 Ex-OR gate

It stands for Exclusive-OR gate. Its function varies when the inputs have even number of ones.

The truth table of 2-input Ex-OR gate is:

A	B	Y = A⊕B
0	0	0
0	1	1
1	0	1
1	1	0

Here A, B are the inputs and Y is the output of two input Ex-OR gate. The output (Y) is zero instead of one when both the inputs are one.

Therefore, the output of Ex-OR gate is ‘1’, when only one of the two inputs is ‘1’. And it is zero, when both inputs are same.

The symbol of Ex-OR gate is as follows:

Fig.: XOR gate (ref. 1)

It is similar to that of OR gate with an exception for few combination(s) of inputs. Hence, the output is also known as an odd function.

Timing Diagram:

3.13 Ex-NOR gate

It stands for Exclusive-NOR gate. Its function is same as that of NOR gate except when the inputs having even number of ones.

The truth table of 2-input Ex-NOR gate is:

A	B	Y = A⊙B
0	0	1
0	1	0
1	0	0
1	1	1

Here A, B are the inputs and Y is the output. It is same as Ex-NOR gate with the only modification in the fourth row. The output is 1 instead of 0, when both the inputs are one.

Hence the output of Ex-NOR gate is ‘1’, when both inputs are same and 0, when both the inputs are different.

The symbol of Ex-NOR gate is:

Fig.: XNOR gate (ref. 1)

It is similar to NOR gate except for few combination(s) of inputs. Here the output is ‘1’, when even number of 1 is present at the inputs. Hence is also called as an even function.

Timing Diagram:

COMBINATIONAL CIRCUIT

A Combinational circuit is a circuit in which we combine the different gates. For example encoder, decoder, multiplexer, demultiplexer etc. Some of the characteristics are as follows −

The output of combinational circuit depends only on the levels present at input terminals at any instant of time.
It does not use any memory element. Therefore, the previous state of input does not have any effect on the present state of the circuit.
It can have n number of inputs and m number of outputs.

Block diagram

Block Diagram of combinational circuit

Fig. : Combinational circuit (ref.2 )

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.

Block diagram

Block Diagram of Half Adder

Fig. : Half adder (ref. 2)

Truth Table

Half Adder Truth Table

Circuit Diagram

Half Adder Circuit Diagram

Fig.: Half adder (ref. 2)

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.

Block diagram

Block Diagram of Full Adder

Fig. : Full adder (ref. 2)

Full Adder Truth Table

Circuit Diagram

Full Adder Circuit Diagram

Fig. : Full adder (ref. 2)

Sequential circuits

A Sequential circuit consists of inputs variable (X), logic gates and output variable (Z).
Combinational circuit generates an output on the basis of input variable only but Sequential circuit produces an output based on current input and previous output variables.
This means that it includes memory elements which are capable of storing binary information.
This binary information is nothing but the state of the sequential circuit at any given time. A latch capable of storing one bit information.

Fig. Combinational circuit

Fig. Sequential circuit

There are two types of input to the combinational logic in fig 2 :

External inputs which are not controlled by the circuit.
Internal inputs which are function of previous output states.

Secondary inputs are state variables that are produced by the storage elements where as secondary outputs are excitations for those storage elements.

Types of Sequential Circuits –

There are two types of sequential circuit :

Asynchronous sequential circuit –

They do not use a clock signal but instead uses the pulses as inputs.
These circuits are faster because there is clock pulse and can change their state immediately when there is a change in the input signal.
It is used when speed of operation is important and is independent of internal clock pulse.

Fig. Asynchronous Sequential circuit

But they are more difficult to design and their output is also uncertain.

Synchronous sequential circuit –

These circuit uses clock signal and level inputs.
The output pulse is received in the same duration as the clock pulse for the clocked sequential circuits.
They wait for the next clock pulse to arrive to perform the next operation, these circuits are bit slower as compared to asynchronous circuits.
Level output changes state at the start of an input pulse and remains in that state until the next input or clock pulse arrives.

Fig. Synchronous Sequential circuit

It is used in synchronous counters, flip flops, and in MOORE-MEALY machines.
It is also used to design Counters, Registers, RAM, etc

3.16 Flip flops :

SR Flip Flop

JK Flip Flop

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 –

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.
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.
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.
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.
If J=1 and K=1, the master toggles on the positive transition and the slave toggles on the negative transition of the clock.
If J=0 and K=0, the flip flop becomes disabled and Q remains unchanged.

Timing Diagram of a Master flip flop –

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

D Flip Flop

T 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)

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