LOGIC GATES

All digital systems are made from a few basic digital circuits that we call logic gates.

These circuits perform the basic logic functions that we will describe in this session.

The physical realization of these logic gates has changed over the years from mechanical relays to electronic vacuum tubes to transistors to integrated circuits containing thousands of transistors.

In this appendix you will learn:

Definitions of the basic gates in terms of truth tables and logic equations DeMorgan’s Theorem

How gates defined in terms of positive and negative logic are related To use multiple-input gates

How to perform a sum of products and a product of sums design from a truth table specification

The Three Basic Logic Gates

Much of a computer’s hardware is comprised of digital logic circuits.

Digital logic circuits are built from just a handful of primitive elements, called logic gates, combined in various ways.

In a digital logic circuit, only two values may be present.

The values may be −5 and + 5 volts Or the values may be 0.5 and 3.5 volts Or the values may be you get the picture.

To allow consideration of all of these possibilities, we will say that digital logic circuits allow the presence of two logical values: 0 and 1.

So, signals in a digital logic circuit take on the values of 0 or 1.

Logic gates are devices which compute functions of these binary signals.

AND Gate

Consider the circuit below which consists of a battery, a light, and two switches in series:

When will the light turn on? It should be clear that the light will turn on only if both switch S1 and switch S2 are shut.

It is quite likely that you encounter the and operation in some shape or form hundreds of times each day.

Consider the simple action of withdrawing funds from your checking account at an ATM.

You will only be able to complete the transaction if you have a checking account and you have money in it.

The ATM will only permit the transaction if you have your ATM card and you enter your correct 4-digit PIN.

To enter the correct PIN, you have to enter the first digit correctly and enter the second digit correctly and enter the third digit correctly and enter the fourth digit correctly.

Returning to the circuit above, we can represent the light’s operation using a table:

The switch is a binary device: it can be open or closed.

Let’s represent these two states as 0 and 1.

Likewise, the light is a binary device with two states: off and on, which we will represent as 0 and 1.

Rewriting the table above with this notation, we have:

This table, which displays the output for all possible combinations of the input, is termed the truth table for the AND operation.

In a computer, this and functionality is implemented with a circuit called an AND gate.

The simplest AND gate has two inputs and one output and is represented pictorially by the symbol:

where the inputs have been labeled a and b, and the output has been labeled c.

If both inputs are 1 then the output is 1. Otherwise, the output is 0.

We represent the and operation by using either the multiplication symbol (i.e., “ ∙ “) or by writing the inputs together.

Thus, for the AND gate shown above, we would write the output c as c = a b or as c = ab.

This would be pronounced: “c = a and b.”

The truth table for the AND gate is shown below. The output c = ab is equal to 1 if and only if (iff) a is 1 and b is 1.

Otherwise, the output is 0.

AND gates can have more than one input (however, an AND gate always has just a single output).

Let’s consider a three-input AND gate:

OR Gate

Now consider the circuit shown below, that has 2 switches in parallel.

It is evident that the light will turn on when either switch S₁ is shut or switch S₂ is shut or both are shut.

It is quite likely that you encounter the or operation in some shape or form hundreds of times each day.

Consider the simple action of sitting on your couch at home at two in the morning studying for your Digital Logic class.

Your phone will ring if you get a call from Alice or from Bob.

Your home’s security alarm will go off if the front door opens or the back door opens.

You will drink a cup of coffee if you are drowsy or you are thirsty.

We can represent the light’s operation using a table

Changing the words open and off to 0 and the words shut and on to 1 and the table becomes:

This is the truth table for the OR operation.

This or functionality is implemented with a circuit called an OR gate.

The simplest OR gate has two inputs and one output and is represented pictorially by the symbol:

If either or both inputs are 1, the output is 1.

Otherwise, the output is 0.

We represent the or operation by using the addition symbol.

Thus, for the OR gate above, we would write the output c as c = a + b. This would be pronounced: “c = a or b.”

The truth table for the OR gate is shown below. The output is 1 if a is 1 or b is 1; otherwise, the output is 0.

NOT Gate

The last of our basic logic gates is the NOT gate.

The NOT gate always has one input and one output. If the input is 1, the output is 0.

If the input is 0, the output is 1. This operation— chaging the value of the binary input—is called complementation, negation or inversion.

The mathematical symbol for negation is an apostrophe.

If the input to a NOT gate is P, the output, termed the complement, is denoted as P’.

The pictorial symbol for a NOT gate is intended to depict an amplifier followed by a bubble, shown below.

Sometimes the NOT operation is represented by just the bubble, without the amplifier.

The truth table for the NOT gate is shown below:

Three New Gates

Three new gates, NAND, NOR, and Exclusive-OR, can be formed from our three basic gates: NOT, AND, and OR.

NAND Gate

The logic symbol for a NAND gate is like an AND gate with a small circle (or bubble) on the output.we see that the output of a NAND gate is 0 (low) only if both inputs are 1 (high) .

The NAND gate is equivalent to an AND gate followed by an inverter (NOT-AND).

NOR Gate

The logic symbol for a NOR gate is like an OR gate with a small circle (or bubble) on the output.

From the truth table .we see that the output of a NOR gate is 1 (high) only if both inputs are 0 (low).

The NOR gate is equivalent to an OR gate followed by an inverter (NOT-OR), as shown by the two truth tables.

Exclusive OR Gate

The XOR gate logic symbol is like an OR gate symbol with an extra curved vertical line on the input.

From the truth table .we see that the output Z of an XOR gate is 1 (true or high) if either input, X or Y, is 1 (true or high), but not both.

The output Z will be zero if both X and Y are the same (either both 1 or both 0).

The equation for the XOR gate is given as Z = X ^ Y. In this book we will use the symbol ^ as the XOR operator.

Sometimes the symbol or the dollar sign $ is used to denote Exclusive-OR.

We will use the symbol ^ because that is the symbol recognized by the Verilog software used to program a CPLD.

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Construction of DC GENERATOR

Bipolar Junction Transistor(BJT)

Binary Number System and Conversion

Hexadecimal Numbers

ASCII Character Encoding

Boolean Algebra

Adder & Flip Flop

ASCII Character Encoding

ASCII Character Encoding the name ASCII is an acronym for: American Standard Code for Information Interchange.

It is a character encoding standard developed several decades ago to provide a standard way for digital machines to encode characters.

The ASCII code provides a mechanism for encoding alphabetic characters, numeric digits, and punctuation marks for use in representing text and numbers written using the Roman alphabet.

As originally designed, it was a seven bit code.

The seven bits allow the representation of 128 unique characters.

All of the alphabet, numeric digits and standard English punctuation marks are encoded.

The ASCII standard was later extended to an eight bit code (which allows 256 unique code patterns) and various additional symbols were added, including characters with diacritical marks (such as accents) used in European languages, which don’t appear in English.

There are also numerous non-standard extensions to ASCII giving different encoding for the upper 128 character codes than the standard.

For example, The character set encoded into the display card for the original IBM PC had a non-standard encoding for the upper character set.

This is a non-standard extension that is in very wide spread use, and could be considered a standard in itself.

Some important things to points about ASCII code:

The numeric digits, 0-9, are encoded in sequence starting at 30h

The upper case alphabetic characters are sequential beginning at 41h The lower case alphabetic characters are sequential beginning at 61h

The first 32 characters (codes 0-1Fh) and 7Fh are control characters.

They do not have a standard symbol (glyph) associated with them. They are used for carriage control, and protocol purposes.

The include 0Dh (CR or carriage return), 0Ah (LF or line feed), 0Ch (FF or form feed), 08h (BS or backspace).

Most keyboards generate the control characters by holding down a control key (CTRL) and simultaneously pressing an alphabetic character key.

The control code will have the same value as the lower five bits of the alphabetic key pressed.

So, for example, the control character 0Dh is carriage return. It can be generated by pressing CTRL-M.

To get the full 32 control characters a few at the upper end of the range are generated by pressing CTRL and a punctuation key in combination.

For example, the ESC (escape) character is generated by pressing CTRL-[ (left square bracket).

Conversions Between Upper and Lower Case ASCII Letters

ASCII code chart that the uppercase letters start at 41h and that the lower case letters begin at 61h.

In each case, the rest of the letters are consecutive and in alphabetic order.

The difference between 41h and 61h is 20h. Therefore the conversion between upper and lower case involves either adding or subtracting 20h to the character code.

To convert a lower case letter to upper case, subtract 20h, and conversely to convert upper case to lower case, add 20h.

It is important to note that you need to first ensure that you do in fact have an alphabetic character before performing the addition or subtraction.

Ordinarily, a check should be made that the character is in the range 41h–5Ah for upper case or 61h-7Ah for lower case.

Conversion Between ASCII and BCD

ASCII code chart that the numeric characters are in the range 30h-39h.

Conversion between an ASCII encoded digit and an unpacked BCD digit can be accomplished by adding or subtracting 30h.

Subtract 30h from an ASCII digit to get BCD, or add 30h to a BCD digit to get ASCII.

Again, as with upper and lower case conversion for alphabetic characters, it is necessary to ensure that the character is in fact a numeric digit before performing the subtraction.

The digit characters are in the range 30h-39h.

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Other links 

Construction of DC GENERATOR

Bipolar Junction Transistor(BJT)

Binary Number System and Conversion

Hexadecimal Numbers

Logic Gates

Boolean Algebra

Adder & Flip Flop