Logic Gates

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Introduction

Logic gates process signals which represent true or false. Normally the positive supply voltage +Vs represents true and 0V represents false. Other terms used for the true and false states are shown in the table, it is best to be familiar with them all.

Gates are identified by their function: NOT, AND, NAND, OR, NOR, EX-OR and EX-NOR. Capital letters are normally used to make it clear that the term refers to a logic gate.

Note that logic gates are not always required because simple logic functions can be performed with switches or diodes, for example:

Logic states
TrueFalse
10
HighLow
+Vs0V
OnOff

Logic gate symbols

There are two series of symbols for logic gates. The traditional symbols have distinctive shapes making them easy to recognise, they are widely used in industry and education. The IEC (International Electrotechnical Commission) symbols are rectangles with a symbol inside to show the gate function. They are rarely used despite their official status but you may need to know them for an examination.

AND gate traditional symbol
AND gate IEC symbol

Traditional

IEC

Inputs and outputs

AND gate with inputs and output labelled

Gates have two or more inputs, except a NOT gate which has only one input. All gates have only one output. Usually the letters A, B, C and so on are used to label inputs, and Q is used to label the output. On this page the inputs are shown on the left and the output on the right.

The inverting circle (o)

NAND gate showing inverting circle

Some gate symbols have a circle on their output which means that their function includes inverting of the output. It is equivalent to feeding the output through a NOT gate. For example the NAND (Not AND) gate symbol shown on the right is the same as an AND gate symbol but with the addition of an inverting circle on the output.


Truth tables

A truth table is a good way to show the function of a logic gate. It shows the output states for every possible combination of input states. The symbols 0 (false) and 1 (true) are usually used in truth tables. The example truth table shows the inputs and output of an AND gate.

There are summary truth tables below showing the output states for all types of 2-input and 3-input gates. These can be helpful if you are trying to select a suitable gate.

Input AInput BOutput Q
000
010
100
111

Summary truth tables

These summary truth tables below show the output states for all types of 2-input and 3-input gates. Note that EX-OR and EX-NOR gates can only have 2 inputs.

Summary for all 2-input gates
InputsOutput of each gate
AB ANDNANDORNOREX-OREX-NOR
00 010 101
01 011 010
10 011 010
11 101 001
Summary for all 3-input gates
InputsOutput of each gate
 A  B  C  ANDNAND OR NOR
000 01 01
001 01 10
010 01 10
011 01 10
100 01 10
101 01 10
110 01 10
111 10 10

Logic ICs

Logic gates are available on ICs (chips) which usually contain several gates of the same type, for example the 4001 IC contains four 2-input NOR gates. There are several families of logic ICs and they can be split into two groups: the 4000 series and the 74 Series

To compare the different families see the ICs page.

The 4000 and 74HC families are the best for battery powered projects because they will work with a good range of supply voltages and they use very little power. However, if you are using them to design circuits and investigate logic gates please remember that all unused inputs MUST be connected to the power supply (either +Vs or 0V), this applies even if that part of the IC is not being used in the circuit!

4001 and other quad 2-input gates

NOT gate (inverter)

A NOT gate can only have one input and the output is the inverse of the input. A NOT gate is also called an inverter.

The output Q is true when the input A is NOT true: Q = NOT A

traditional NOT gate symbol

Traditional symbol

IEC NOT gate symbol

IEC symbol

Input AOutput Q
01
10

AND gate

An AND gate can have two or more inputs, its output is true if all inputs are true. The output Q is true if input A AND input B are both true: Q = A AND B

AND gate traditional symbol

Traditional symbol

AND gate IEC symbol

IEC symbol

Input AInput BOutput Q
000
010
100
111

NAND gate

NAND = Not AND. This is an AND gate with the output inverted, as shown by the 'o' on the symbol output. A NAND gate can have two or more inputs, its output is true if NOT all inputs are true. The output Q is true if input A AND input B are NOT both true: Q = NOT (A AND B)

traditional NAND gate symbol

Traditional symbol

IEC NAND gate symbol

IEC symbol

Input AInput BOutput Q
001
011
101
110

OR gate

An OR gate can have two or more inputs, its output is true if at least one input is true. The output Q is true if input A OR input B is true (or both of them are true): Q = A OR B

traditional OR gate symbol

Traditional symbol

IEC OR gate symbol

IEC symbol

Input AInput BOutput Q
000
011
101
111

NOR gate

NOR = Not OR. This is an OR gate with the output inverted, as shown by the 'o' on the symbol output. A NOR gate can have two or more inputs, its output is true if no inputs are true. The output Q is true if NOT inputs A OR B are true: Q = NOT (A OR B)

traditional NOR gate symbol

Traditional symbol

IEC NOR gate symbol

IEC symbol

Input AInput BOutput Q
001
010
100
110

EX-OR gate

EXclusive-OR. This is like an OR gate but excluding both inputs being true. The output is true if inputs A and B are DIFFERENT. EX-OR gates can only have 2 inputs. The output Q is true if either input A is true OR input B is true, but not when both of them are true: Q = (A AND NOT B) OR (B AND NOT A)

traditional EX-OR gate symbol

Traditional symbol

IEC EX-OR gate symbol

IEC symbol

Input AInput BOutput Q
000
011
101
110

EX-NOR gate

EXclusive-NOR. This is an EX-OR gate with the output inverted, as shown by the 'o' on the symbol output. EX-NOR gates can only have 2 inputs. The output Q is true if inputs A and B are the SAME (both true or both false): Q = (A AND B) OR (NOT A AND NOT B)

traditional EX-NOR gate symbol

Traditional symbol

IEC EX-NOR gate symbol

IEC symbol

Input AInput BOutput Q
001
010
100
111

Combinations of logic gates

Logic gates can be combined to produce more complex functions.

For example to produce an output Q which is true only when input A is true and input B is false, we can combine a NOT gate and an AND gate as shown.

A AND NOT B

Q = A AND NOT B

Working out the function of a combination of gates

Truth tables can be used to work out the function of a combination of gates such as the system shown below:

Combination of NOR, AND and OR gates

Begin by creating a table showing all possible combinations of inputs (A, B and C in this example) with enough extra columns for each intermediate output (D and E in this example) as well as the final output (Q). Then work out all the intermediate output states, filling in the table as you do go. These intermediate outputs form the inputs to the next gate (or gates) so you can use these to work out the next output(s), in this example that is for the final output (Q).

D = NOT (A OR B)
E = B AND C
Q = D OR E = (NOT (A OR B)) OR (B AND C)

The truth table shows the intermediate outputs D and E as well as the final output Q.

InputsOutputs
 A  B  C   D  E  Q 
000101
001101
010000
011011
100000
101000
110000
111011

Substituting one type of gate for another

Logic gates are available on ICs which usually contain several gates of the same type, for example four 2-input NAND gates or three 3-input NAND gates. This can be wasteful if only a few gates are required unless they are all the same type. To avoid using too many ICs you can reduce the number of gate inputs or substitute one type of gate for another.

Reducing the number of inputs

3-input AND gate operating as a 2-input AND gate

The number of inputs to a gate can be reduced by connecting two (or more) inputs together. The diagram shows a 3-input AND gate operating as a 2-input AND gate.

Making a NOT gate from a NAND or NOR gate

making a NOT gate from a NAND gate

Reducing a NAND or NOR gate to just one input creates a NOT gate. The diagram shows this for a 2-input NAND gate.

Any gate can be built from NAND or NOR gates

As well as making a NOT gate, NAND or NOR gates can be combined to create any type of gate! This enables a circuit to be built from just one type of gate, either NAND or NOR. For example an AND gate is a NAND gate then a NOT gate (to undo the inverting function). Note that AND and OR gates cannot be used to create other gates because they lack the inverting (NOT) function.

To change the type of gate, such as changing OR to AND, you must do three things:

For example an OR gate can be built from NOTed inputs fed into a NAND (AND + NOT) gate.


NAND gate equivalents

The arrangements below show how to use NAND gates to make NOT, AND, OR and NOR gates:

NOT made from one NAND gate:

NOT gate made from a NAND gate

AND made from two NAND gates:

AND gate made from NAND gates

OR made from three NAND gates:

OR gate made from NAND gates

NOR made from four NAND gates:

NOR gate made from NAND gates


Substituting gates in an example logic system

This system has 3 different gates (NOR, AND and OR) so it requires three ICs, one for each type of gate.

Combination of NOR, AND and OR gates

To re-design this system using NAND gates only begin by replacing each gate with its NAND gate equivalent, as shown below:

Equivalent NAND gate system

Then simplify the system by deleting adjacent pairs of NOT gates (marked X above). This can be done because the second NOT gate cancels the action of the first:

Simplified NAND gate system

The final system has five NAND gates and requires two ICs (with four gates on each IC). This is better than the original system which required three ICs (one for each type of gate).

Substituting NAND (or NOR) gates does not always increase the number of gates, but when it does (as in this example) the increase is usually only one or two gates. The real benefit is reducing the number of ICs required by using just one type of gate.


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