Logic Circuit Modules 1

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LOGIC CIRCUITS and SWITCHING THEORY

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Learning Objectives:

At the end of this topic you will be able to;

1.9.1 Introduction.

Recognize high/low, 1/0, as two state logic levels;

1.9.2 Truth Tables.

Draw symbols and construct truth tables for AND, OR, NOT, NOR,

and NAND gates;Produce a truth table for a system of up to five gates;

Devise a system of gates from a truth table;

Design simple systems using logic gates to solve a given problem;

Use Boolean notation as a shorthand method of expressing a truth

table;

1.9.3 Use of data sheets.

Use data sheets to;o Select a logic IC for given applications;

o Identify pin connections of logic gates;

1.9.4 NAND gate implementation.

Show how other gates can be made up from NAND gates;

Implement a given logic circuit using NAND gates;Remove double inversions;

1.9.5 Pull up/down resistors.

Recognise the use of pull up/down resistors to provide the correct

logic levels at a gate input.


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Unit E1 : Discovering Electronics

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Combinational Logic Systems

1.9.1 Introduction

In this topic we will be concentrating on the basics of digital logic circuits

which will then be extended in Module E2. We should start by ensuring that

you understand the difference between a digital signal and an analogue signal.

An analogue signal

This is a signal

that can have

any value

between the

zero and

maximum of

the power

supply. Changesbetween values

can occur slowly or rapidly depending on the system involved.

A digital signal

This is a signal

that can only

have twofinitevalues, usually at

zero and

maximum of the

power supply.

Changes between

these two values

occur instantaneously.

Voltage (V)

time (s)

Max

Min

Voltage (V)

time (s)

Max

Min0V

0V


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For this part of the course we will concentrate on digital systems.

Recap of work covered in Sub-systems (topic 1.2)

When an input or output signal is at the minimum power supply voltage (usually

0V) this is referred to as a LOWsignal or LOGIC 0 signal.

When an input or output signal is at the maximum power supply voltage this is

referred to as a HIGHsignal or LOGIC 1 signal.

Remember then that a digital signal is a two state system with input and

output signals being either referred to as high/low, 0/1, on/off

depending on the application.

We will nowlook at the basic building block of all digital systems, the logic

gate, and their associated truth tables.

NoteLogic gates are available with up to 8 inputs per gate which may be useful for

project work later on in the course, but for this introductory section and for

the purposes of the examination questions we will only consider 2 input logic

gates.


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A

BQ

1.9.2 Truth Tables

Here is a summary of the three logic gates you have already studied

GATE SYMBOL TRUTH TABLE FUNCTION

NOT(INVERTER)

Input OutputA Q

0 1

1 0

Signal out of

gate is theopposite of

the signal in

i.e. it inverts

the input

signal

AND

Inputs Output

A B Q

0 0 0

0 1 0

1 0 0

1 1 1

The output Q

is only at alogic 1 when

input AAND

input Bare at

a logic 1

OR

Inputs OutputA B Q

0 0 0

0 1 1

1 0 1

1 1 1

The output Q

is at a logic 1

when input A

ORinput BOR

both are at a

logic 1

A Q

A

BQ


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We will now look at two additional logic gates:

The NAND gate

The symbol for a 2 input NAND gate is:

The truth table for the 2 input NAND gate is shown below.

Inputs OutputA B Q

0 0 1

0 1 1

1 0 1

1 1 0

If you compare this truth table with that for the AND gate, you will

find that the output Qis the exact opposite of the AND.

A

BQ


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The NOR gate

The symbol for a 2 input NOR gate is:

The truth table for the 2 input NOR gate is shown below.

Inputs OutputA B Q

0 0 1

0 1 0

1 0 0

1 1 0

If you compare this truth table with that for the OR gate, you will find

that the output Qis the exact opposite of the OR.

Now let us see what you can remember !

A

BQ


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Exercise 1

1. Look at the following logic symbols labelled A E.

A B C D E

i. Which is the correct symbol for an AND gate.

ii. Which is the correct symbol for a NOT gate.

iii. Which is the correct symbol for a NOR gate.

iv. Which is the correct symbol for a NAND gate?

v. Which is the correct symbol for an OR gate.

2. Complete the following truth tables.

i. AND gate.

Inputs OutputA B Q

0 0

0 11 0

1 1


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ii. NOR gate.

Inputs OutputA B Q

0 0

0 1

1 0

1 1

iii. NAND gate.

Inputs OutputA B Q

0 0

0 1

1 0

1 1

iv. OR gate.

Inputs OutputA B Q

0 0

0 1

1 0

1 1


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Practical Logic Gates

Logic gates are usually supplied in plastic d.i.l. (dual in line) packages

containing multiple copies of one type of logic gate. The following diagram

shows a picture of this type of package.

There are two common types available, TTL or 74xx series and CMOS or

4xxx series. It is likely that you will come across both types in your practicalwork, so whats the difference between them?

The key differences are outlined in the table below:

Parameter TTL (74xx family) CMOS (4xxx family)

Supply Voltage 5V 0.25V only 3V to 18V

Logic 0 range 0 to 0.8V

Below 30% of

supply voltage

Logic 1 range 2.0 to 5.0VAbove 70% of

supply voltage

Frequency of operation (Max) 50 MHz 4 MHz

Power consumption 10mW / gate 0.1mW / gate

This information will be important in practical work, as you will need to know

which type of logic gate you are using. You will also need to be careful howyou connect each logic gate into your circuit.

Pin 1 identification

Pin 1


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To be able to identify which leads are connected to which gate you need to

look at a data sheet for the actual logic gate you are using. Here are two data

sheets from the TTL (74xx) family.

It is important that you check the connections every time you use a logic gate

as connecting these incorrectly can result in the whole logic chip being

destroyed.

You will not be required to know the difference between TTL and CMOS

devices in the examination. This is required for any practical tests that

you carry out, and will be particularly important for your project work.

You will however need to be able to identify the output pin of a logic gate

given its symbol. For example if you are given the pinout of the 7432 device

shown above you can be asked to identify the pin numbers of the outputs of

the logic gates. In this case the relevant pin numbers are; 3, 6, 8 & 11.

Alternatively you might be asked to identify the power supply connections, in

which case the answer would be Pin 14 for the positive supply and Pin 7 for

the negative of the supply.


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Exercise 2

The pin out diagrams for a logic IC is shown below.

a) How many logic gates are contained in this IC? ..................

b) How many inputs does each gate have? .......................

c) Give the number of the pin connected to the output of gate G?

d) Which twopins should be connected to the power Supply? ...............

e) What is the name given to the type of logic gate contained in this

IC?

Choose from the following list:

AND OR NOT NAND NOR

Answer:


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Analysis of simple logic circuits

In the examination you will have to recognise truth tables for these basicgates individually for some of the easier questions in the examination.

However, it is much more likely later on in the paper that these gates will be

linked together in simple combinations and you will be asked to complete a

truth table for a larger system. We will now consider a couple of examples of

these systems.

1. Study the following logic system carefully and then complete the truth

table that follows:

Inputs OutputsA B C Q

0 0

0 1

1 0

1 1

In this problem, the output of the NOT gate has been labelled C. The

first stage is to complete the output column for C which is the NOT ofA as shown below.


0 0 1

0 1 1

1 0 0

1 1 0


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Now we need to complete the final column Qwhich is the output of the AND

gate with B and C as the inputs.


0 0 1 0

0 1 1 1

1 0 0 0

1 1 0 0

Do notfall into the trap of writing the answer to the Qcolumn in the orderyou would normally do for the truth table for an AND gate. Because the

inputs to the AND gate are Band Crather than Aand B,the logic 1 in the Q

column appears in the row where Band Care both 1 rather than when Aand

Bequals 1.


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table that follows:

Inputs OutputsA B C F G Q0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

You can see that the truth table for a 3 input logic system

contains 8 possible input combinations. Notice the way the logic state

of each input changes as you move down the table.

First complete the output column for the NOT gate (Column F)

{Remember the input is B.}

Then complete the output column for the AND gate (Column G)

{Remember the inputs are Fand C.}

Finally complete the final output from the NOR gate (Column Q)

{Remember the inputs are Aand G}

A solution to this problem will be found at the end of this chapter.

Heres a couple for you to try:


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Exercise 3

1. Study the following logic system carefully and then complete the truthtable that follows:

Inputs OutputsA B K Q

0 0

0 1

1 0

1 1


table that follows:

Inputs OutputsA B C F G Q

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 01 1 1


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table that follows:

Inputs OutputsA B C D E F G Q

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1


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Transferring a truth table into a Logic Diagram

In the previous section we looked at how a system of logic gates could beused to complete a truth table to illustrate the conditions needed for the

output to operate. We will now consider how we can reverse this process and

construct a logic circuit diagram from a truth table. This is best done by

looking at a couple of examples.

Note

In the following examples the outputs have been chosen so that they are not

the output of one of the five logic gates considered previously.

Examples:

1. The following truth table represents a particular logic function. Use the

information in the table to draw a corresponding logic gate system that

will produce this function.

Inputs OutputA B Q

0 0 0

0 1 0

1 0 1

1 1 0

We first have to identify all the combinations of the inputs that causethe output to come on. In this case it only occurs once, when input Ais

on andinput Bis noton.

The description of what is required to cause the output to operate gives

a very good clue as to the logic gates required in this example. In this

case two logic gates are required, a NOT gate and an AND gate.

The NOT gate is used to invert the Binput, as shown below.


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The output of this NOT gate is then connected to the AND gate with input A

to provide the full solution, as follows:

Quick Rule

In any 2-input logic system, for every row of the truth table for which

the output is logic 1, this output can be written in terms of the following

input conditions: A, NOT A, B, NOT Bdepending whether there is a 0

or a 1 in that cell. The 2 inputs are linked with an AND gate.

Going back to our example we identify the output row where Qis a logic 1 and

note that A= 1 and B= 0. Because Bis 0 we write it down as NOTBas

shown:

Output Q= AANDNOTB

This gives the same answer as the longer method.

Inputs Output

A B Q0 0 0

1 0 0

1 0 1

1 1 0


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So far we have two separate logic systems providing the output Q. We

need to link the two systems together so that either system can

produce the output.

This is achieved by using an OR gate as shown below:

We have some duplicated input terminals here now so the circuit

diagram can be simplified by linking these together as shown below.


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Truth tables with multiple outputs

Quite often a logic system will have more than one output. For example a setof traffic lights might have 3 outputs.

For this type of system we can follow a simple set of rules.

For each output column of the truth table ask yourself the following

questions in the order listed below

1. Is the output column pattern the same as one of the input columnpatterns?

If the answer is yes then Q= The Input (e.g. Q= B)

2. Is the output column pattern the inverse of the input column pattern?

If the answer is yes then Q= NOT The Input(e.g. Q= NOT C)

3. Is the output column pattern the same as a logic gate output?

If the answer is yes then Q= logic gate expression (e.g. Q= AOR B)

4. Is the output column pattern the inverse of one of the other output

patterns already identified?

If the answer is yes then Q= NOT Other Output (e.g. Q3= NOT Q1)

5. Use the Quick rule by labelling rows of the outputs which are logic 1

and link with an OR gate

e.g. Q= [NOT AAND NOT B] OR [AAND B]


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Example

The following truth table shows the outputs required for three LEDsLEDs used to represent the operation of a set of traffic lights.

Determine the combination of logic gates required to produce the

output pattern shown.

Inputs OutputsA B Red Yellow Green

0 0 1 0 0

0 1 1 1 0

1 0 0 0 1

1 1 0 1 0

Here we have three separate outputs to be produced by just two inputs,

to solve this we just treat each individual output as a separate problem.

If you examine the input Acolumn and Redoutput column carefully whatdo you notice? They are reproduced below with these columns

highlighted.

Comparing the two highlighted columns we can see that the Redoutput

is the exact opposite of the Ainput column. This means that if we

simply invert the input Asignal, this will produce the Redoutput.

i.e. Red= NOT A


0 0 1 0 0

0 1 1 1 0

1 0 0 0 1

1 1 0 1 0


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Now for theYellowoutput, again check the truth table carefully.

The solution is that theYellowoutput follows the Binput exactly, and

therefore to produce theYellowoutput no logic gates are required. Itis simply a case of connecting theYellowoutput to the Binput.

i.e. Yellow= B

Here is the solution for theYellowoutput:

Finally we have to consider the Greenoutput. A check of the truth

table shows there is no simple relationship to the inputs as was the case

with the RedandYellowoutputs. Neither does the output correspond

to the output of a logic gate. We have no choice therefore other than to

use the Quick ruleto solve this part of the problem. You should be

able to produce the system as shown below.

Green = AAND NOT B

This gives;


0 0 1 0 0

0 1 1 1 0

1 0 0 0 1

1 1 0 1 0


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If we connect all three sections together the final system design will

look like this:

Note: If we were very observant we could have noticed that the Greenoutput can be obtained from a NOR gate connected to the Redand

Yellowoutputs.

i.e. Green= RedNORYellow

The final system would then become:

It is left to you to check that both solutions produce the correct output

pattern.

Do not worry if you cannot understand how the second solution wasobtained as you would receive full marks for the first solution.


0 0 1 0 0

0 1 1 1 0

1 0 0 0 1

1 1 0 1 0


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2. The following truth table represents a particular logic function. Use the

information in the table to draw a corresponding logic gate system that

will produce this function.

Inputs OutputA B Q

0 0 0

0 1 1

1 0 1

1 1 0

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3. An electronic system has two input sensors Aand B, and three outputs

P, Qand R.

The truth table showing how the input sensors control the outputs is

shown below.

Inputs Outputs

A B P Q R

0 0 1 0 1

0 1 1 0 01 0 0 0 0

1 1 0 1 0

(a) Study the Poutput. It is the inverse of one of the inputs.

Write down an expression to describe this output.

P = ...................................................................................

(b) Study the Qoutput. There is one type of logic gate that will provide

this.

What is the name of this gate? .............................................................

(c) Study the Routput. There is one type of logic gate that will provide

this.

What is the name of this gate? .............................................................

(d) You have a selection of AND, OR, NOT, NAND and NOR gates

available. Draw a labelled diagram to show how the logic system can

be made.


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4. The following truth table shows the outputs required for three LEDs

used to represent the operation of a set of traffic lights. Determinethe combination of logic gates required to produce the outputs required.


0 0 0 1 0

0 1 0 0 1

1 0 1 1 0

1 1 1 0 0

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A

BQ

Boolean Notation (Higher Level Topic)

There is also a shorthand way of writing down the function of logic gates,using a special type of algebra called Boolean Algebra. This is used

extensively for advanced work in digital electronics.

We shall briefly consider how to express the output of a truth table and logic

gates in Boolean notation. We will start by looking at the five basic gates we

have introduced previously.

There are 3 basic things to remember

1. A dot . between twoinput labels is read as AND

2.A plus +between two input labels is read as OR

3.A bar _ over the top of aninput label is read as NOT

Gate Symbol Boolean Notation

NOT (read as Q= NOT A)

AND (read as Q= AAND B)

OR (read as Q= AOR B)

NAND (read as Q= ANAND B)

NOR (read as Q= ANOR B)

In addition to the five Boolean notations shown above, each line of a truthtable for which the output is a 1 can also be written in Boolean notation

A Q

AQ

A

BQ B.AQ

BAQ

A

BQ BAQ .

A

BQ

BAQ


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Consider the solution to example 2 on page 19

Output = NOT AAND NOT B

Output = AAND B

Using Boolean notation the outputs can be labelled as follows

Output = BANDA

Output = BANDA

Remember that these expressions need to be linked together with an ORgate

to produce the output Q, so the full Boolean expression for Qcan be written

as B.AB.AQ

BA ANDORBANDAQ

Inputs OutputA B Q

0 0 1

0 1 0

1 0 0

1 1 1

Inputs OutputA B Q

0 0 1

0 1 0

1 0 0

1 1 1


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Exercise 5

1. The Boolean equations labelled A E, below are to be used to answer thefollowing questions.

A) BAQ .

B) BAQ

C) BAQ

D) AQ

E) BAQ .

i. Which expression is correct for an AND gate.

ii.

Which expression is correct for a NOT gate.

iii. Which expression is correct for a NOR gate.

iv. Which expression is correct for a NAND gate.

v. Which expression is correct for an OR gate.

2. Write down the Boolean expressions for outputs X,Yand Z:

X=

Y=

Z=

A B X Y Z

0 0 1 0 0

0 1 0 0 1

1 0 1 0 0

1 1 0 1 0


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Logic System Design

In the previous two sections we have considered the function of a number oflogic gates that are available for us to use in electronic system design. We

have derived a truth table from a logic circuit, and we have constructed a

logic circuit from a truth table.

In this section we will be completing the design process by converting a

design brief of a problem into a truth table. Once this has been achieved

then we can use the techniques used in the last section to complete the logic

circuit design.

Design Problems

1. A logic system hastwo input sensors Aand Band two outputs. Output 1

is high when sensor Ais high and sensor Bis high. Output 2 is high

either when sensor Ais low and sensor Bis high or when sensor Ais

high and sensor Bis high.

a) Complete the truth table to satisfy these conditions

b) Draw the circuit diagram for the logic system.

Solution:

a) O/p 1 is high only when A= 1 and B=1. Identify this cell in the o/p

1 column at the truth table and place a 1 in it. Place zeros in the

three other cells in the o/p 1 column.

O/p 2, is high when A= 0 and B= 1 or when A= 1 and B= 1.

Identify these two cells in the o/p 2 column of the truth table.

Place a 1in these two cells and zeros in the other two.

Inputs OutputsA B O/p 1 O/p 2

0 0

0 11 0

1 1


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b) You should have obtained the following truth table.

Inputs OutputsA B O/p 1 O/p 2

0 0 0 0

0 1 0 1

1 0 0 0

1 1 1 1

Examine the o/p 1 pattern. You should realise that it is the same

pattern as for an AND gate.

Examine the o/p 2 pattern. You should realise that it is the same as

input B.

The circuit diagram can then be drawn.


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2. A system is required that will monitor a cars cooling system. When the

water level in the radiator is below a certain level a LED will light up.

When the engine temperature is above a pre-determined value and thewater level is too low a buzzer should sound in addition to the LED

lighting up. The positioning and signals out of the sensors used are

shown below.

a) Complete the

following truth table for thesystem.

Inputs OutputsA B LED buzzer

0 0 0 00 1

1 0 1

1 1

b) Study the LED output and compare it with the inputs. What do you

notice?

Sensor A (moisture)state logic level

Wet 0

Dry 1Sensor B (temperature)

state logic level

Cool 0

Hot 1

A

B

Moisture Sensor

Temperature Sensor

Radiator


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c) Study the buzzer output. There is one type of gate that will

provide this output pattern.

What type of logic gate is required?

d) Complete the following diagram showing how the system can be

made up.

3. Before take off, the pilot and co-pilot of an aircraft carry out

preflight safety checks. When all checks have been completed they

each move a switch from the up to the down position.

When both switches are up, a red indicator on the instrument panel is on.

This changes to yellow when at least one of them operate their switch.

When both have operated their switches, a green indicator comes on.

The engines can only be started when the green indicator is on.

Assume that the switches provide logic level 0 in the up position andlogic level 1 in their down position. The LED indicators operate on logic

level 1.

a) Complete the following truth table for the system. The yellow ,Y

column has been completed for you.


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The complete system therefore is:

4. Two sensors Aand Bare used to monitor a chemical process. Output Q1

is a heater, output Q2is a motor and output Q3is a bell.

The heater is on either when sensor Ais low and sensor Bis high or

when both sensors are low.

The motor is on when either Ais low and Bis high or when both sensors

are high.

The bell comes on when both sensors are high.

a) Complete the following truth table for the system.

Inputs Outputs

A BQ1

(heater)Q2

(motor)Q3

(bell)

0 0 0 0

0 1 1

1 0 11 1 1

b) Write down an expression to describe Q1and Q2by comparing

them with the inputs.

Q1=

Q2=


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c) Which type of gate will provide the Q3output?


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d) Draw the circuit for the system

Now its time for you to have a go.

Exercise 6

1. A logic system hastwo input sensors Aand Band three outputs.

Output 1 is high when sensor Ais low.

Output 2 is high when sensor Ais low and sensor Bis low

Output 3 is high when sensor Ais high and sensor Bis low.

a) Complete the truth table to satisfy these conditions

Truth Table

Inputs OutputsA B O/p 1 O/p 2 O/p 3

0 0

0 1

1 0

1 1


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b) i) Examine the O/p 1 pattern. This can be generated from one

of the input signals. Write down the logic function required

to generate this output.

ii) Examine the O/p 2 pattern. This can be generated from one

of the standard logic gates. Write down the logic function

required to generate this output.

iii) Examine the O/p 3 pattern. This cannot be generated from

the inputs using one of the standard logic gates. Write down

the logic function required to generate this output.

c) Draw the circuit diagram for the logic system.


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Inputs OutputsA B R Y G

0 0

0 1

1 0

1 1

b. Study the Routput. There is one type of gate which will provide

the required output.

What type of gate is it?

c. Study theYoutput. Write down an expression to describe it.

Y =

d. Study the Goutput. There is one type of gate which will provide

the required output.

What type of gate is it?

e. Complete the following diagram showing how the system can be

made up.


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4. Two sensors Aand Bare used to control the paint mixing process at a

local DIY store. Three output valves control the flow of cyan, magenta,

and yellow pigment. Valve V1is the cyan, Valve V2 is the magenta, andValve V3 is the yellow. Mixing occurs according to the following

sequence. A logic 1 operates the valve.

Valve 1 operates when input Ais high and input Bis high.

Valve 2 operates when input Ais low and input Bis low or when input Ais high and

input Bis low.

Valve 3 operates when input Ais low and input Bis high or when input Ais low and

input Bis low.


Inputs Outputs

A BV1

(cyan)V2

(magenta)V3

(yellow)

0 0

0 11 0

1 1

b) Which type of gate will provide the V1output?

c) Write down an expression to describe V2and V3by comparing

them with the inputs.

V2=

V3=


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d) Draw the circuit for the system


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The following pin outs are from an alternative family of logic gates called the

CMOS 4xxx series.

You could be asked to use these diagrams to answer a series of questions

about the logic gates in each particular package.

Here are some typical questions based on the symbols on page 45/46.


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Exercise 7

1. How many logic gates are contained in the 7408 package?

2. What type of logic gate is in the 7402 i.c. package?

3. What type of logic gate is contained in the 4011 package?

4. How many inputs do the logic gates in the 4072 package have ?

5. What are the output pins of the 7408 package?

6. Which package contains NOT gates?

7. How many logic gates are there in the 4002 package?

8. Which 2 packages have some pins that are not connected to anything ?

9. What pin number is the positive supply for a 4081 package ?

10. What pin number(s) are the inputs of the logic gate, whose output is

connected to pin 13 of the 4072 package?

11. What type of logic gate is contained in the 7408 package?

12. Which 74xx family package has an output pin connected to pin 1?


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1.9.4 NAND gate implementation (Higher Level Topic)

In section 1.9.2 we found out how to construct logic systems from a truthtable. This often resulted in logic systems that required a number of

different types of logic gate (e.g. NOT, AND and OR) in order to fulfil the

function required.

In some of the designs we have looked at we ended up with just one of three

different types of logic gate needed in the final design. As we have seen from

the last section only one type of logic gate is built inside each package, and

there could be as many as six of these logic gates in the package of which we

are only going to use one.

This is very wasteful not only in terms of unused devices but also in the space

needed on circuit boards to accommodate three different logic gate

packages.

The inverted gates, NAND and NOR are special because the function of allother gates can be made from various combinations of NAND or NOR gates.

In this syllabus only NAND gate alternatives of the other logic functions will

be discussed.You may find some reference to NOR gate logic in some

text books but these will not be asked for in the examination.

By using just one type of logic gate we may be able to reduce the number of

types of logic gate required to make any particular design. This has a number

of advantages:

i. There will be less confusion about which type of gate goes where in the

circuit as they are all the same.

ii.There will be no need to keep stocks of all the different types of logic

gate, therefore saving money.

iii.Larger quantities of a single type of gate can be purchased, which

makes cost lower.


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We will now look at an example to show you how making this change can

improve the situation. Consider the two logic circuits below, which perform

the samelogic function.

System 1 : Mixture of gates.

System 2 : NAND gates only

When system 2 is compared to system 1, you may think that we have made

the circuit more complicated as we have more logic gates in system 2,

however, in system 1 threedifferent types of gates are required NOT, OR

and AND.

To construct system 1 using these gates would require 3 different logic i.cs,

and many of the logic gates on these i.cs would not be used.

Using system 2, however, whilst there are four logic gates required these are

all of the same type, and only onelogic i.c. would be required where all gates

in the i.c. are used.

Q

A

B

C

A

B

Q

C


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A Q A Qis the same as

This would provide a considerable cost saving compared to the design in

system 1.

In industry if such systems are to be mass produced such savings can be

considerable, and it is up to the engineers making the systems to use this

technique as much as possible to enable more profit to be made.

Now that we know why NAND gate logic is used lets find out how to carry out

this procedure. We need to understand the combination of NAND gates

required to replace each of our standard gates.

NAND gate equivalent circuits for the four other gates

1. The NOT gate

This is the simplest of the standard gates to form from NAND gates.

Complete the truth table below for the NAND equivalent circuit.

Input Output

A Q0

1

NoteThe NAND equivalent of a NOT gate is sometimes referred to as a

NAND Inverter. You will need to remember this for later on.


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A

BQ

A

BQX

is the same as

A

Q

B

A

BQ

X

Y

2. The AND gate

This is the inverse of a NAND gate, and is simply a NAND gate followedby an inverter (NOT Gate).


Inputs OutputA B X Q

0 0

0 1

1 0

1 1

3. The OR gate

The OR gate is a little more complicated, and requires three NAND

gates as shown below.


Inputs Intermediate Outputs OutputA B X Y Q

0 0

0 11 0

1 1


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is the same as

A

B

A

BQ

Q

X

Y

Z

4. The NOR gate

The NOR gate is the inverse of the OR gate, so just one more gate isneeded as shown below.


Inputs Intermediate Outputs OutputA B X Y Z Q

0 0

0 11 0

1 1


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Converting Logic Diagrams to NAND gates

The process for converting logic system diagrams into NAND gate format isquite straight forward if you work logic ally through the circuit. Each gate is

replaced in turn by its NAND equivalent, and connected up in the same way.

We will look at an example to show how this is done.

Example 1: Convert the following logic system into NAND gates only.

In this case we need to replace a NOT gate, OR gate and an AND gate.

Stage 1: Redraw the NAND equivalent circuits of the gates shown above,

where possible retain the position of these gates so that you can

identify the connections afterwards.

Drawing a box around each gate and its corresponding NAND equivalent

will allow you to check that you have replaced every gate in the circuit.

A

Q

B

C

QA

B

C


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Stage 2: It is then just a matter of connecting the equivalent circuits

together.

This circuit is now the equivalent circuit to that using in NOT, OR and

AND gate given earlier, however there is one further simplification we

can make.

Stage 3 : Consider the circuit again as shown below.

If you look carefully at the two NAND gates labelled 1 & 2, we can see

that these are both configured to be inverters or NOT gates. If we

consider what happens to signal A as it passes through these two gates

we have the following:

A logic 1 at A, becomes a 0 after gate 1 and then a 1 again after gate 2

A logic 0 at A, becomes a 1 after gate 1 and then a 0 again after gate 2

Therefore gates 1 and 2, serve no useful purpose in this circuit, and are

known as redundant gates and can be removed. We call this double

inversionand it occurs commonly when creating NAND gate circuits

from other logic systems. Remember a double inversion onlyoccurs when2 NAND Inverters are directlyconnected to one another.

A

Q

B

C

A

Q

B

C

1 2


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Example 2 : Convert the following logic diagram into NAND gates only.

First of all we will replace all of these gates with their NAND equivalent and

connect them together.

Finally we check for any redundant gates, and identify these.

Note the way in which different pairs of redundant gates are marked.

C

Q

A

B

C

QB

A

C

Q

A

B


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A

B

C

Q

Now here are a couple for you to try.

Exercise 8

1. (a) Redraw the following logic circuit using 2 input NAND gates only.

(b) Identify any redundant gates.


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A

BQ

1.9.5 Pull Up / Pull Down Resistors (Higher Level Topic)

Up until now we have shown the input connections to a logic gate either as awire with a label, or connected to a logic input:

These diagrams are called schematic circuit diagram which help us to

concentrate on what is happening to the logic signals withinthe logic circuit

without worrying to much how the inputs are wired up.

If we want to build a logic circuit we have to provide the logic gate with a

suitable input sub-system to provide the correct logic levels.

The input to a logic gate can come from a number of different sources but

for the purposes of this unit we are going to concentrate on mechanical

switches.

Whichever type of switch we use, they have to be used along with a series

resistor as part of a voltage divider circuit.

We have to be careful which way around the resistor and switch are

connected in the voltage divider circuit to produce either a Logic 0 signal or a

Logic 1 signal when the switch is pressed.


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Two input sub-system circuits using a push to make switch are shown below.

Signal at point X is at Logic 0

when switch is pressed

Circuit A

Signal at point Y is at Logic 1

when switch is pressed

Circuit B

The resistor used in Circuit A is called a pull up resistor and the resistor

used in Circuit B is called a pull down resistor. This is because of theirbehavior in the circuit, either pulling up the voltage at the input to Logic 1 or

pulling down the voltage to Logic 0 when the switch is not pressed.

In Circuit A, before the switch is pressed, there is no connection to the 0V

line, and the input to the logic gate is pulled-up to 5V, giving a Logic 1 input

to the logic system. When the switch is operated, the input to the logic

system is connected to the 0V line through the switch and the logic level falls

to Logic 0.

In Circuit B, before the switch is pressed, there is no connection to the 5V

line, and the input to the logic gate is pulled-down to 0V, giving a Logic 0

input to the logic system. When the switch is operated, current flows through

the resistor, causing the voltage across it to rise to 5V, changing the Logic

level into the logic system to Logic 1.

Now try these


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Exercise 9

1. Study the circuits below and complete the statements that follow:

a) In circuit C with the switch openthe input to the logic system is at logic

b) In circuit C with the switch closedthe input to the logic system is at logic

c) In circuit D with the switch openthe input to the logic system is at logic

d) In circuit D with the switch closedthe input to the logic system is at logic

2. Study the circuit below and complete the statements that follow:

a) Resistor R1 is a pull resistor and R2 is a pull resistor.

b) When switch SW2 is pressed input B is at logic .......

c) When both switches are pressed output Q is at logic .......


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Solutions to Exercises

Exercise 1

1.

i. The correct symbol for an AND gate is D.

ii. The correct symbol for a NOT gate is A.

iii. The correct symbol for a NOR gate is E.

iv. The correct symbol for a NAND gate is B.

v. The correct symbol for an OR gate is C.

2.

i. AND gate.

Inputs OutputA B Q

0 0 00 1 0

1 0 0

1 1 1

ii. NOR gate.

Inputs Output

A B Q0 0 1

0 1 0

1 0 0

1 1 0


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iii. NAND gate.

Inputs OutputA B Q

0 0 1

0 1 1

1 0 1

1 1 0

iv. OR gate.

Inputs OutputA B Q

0 0 0

0 1 1

1 0 1

1 1 1

Exercise 2

a) 4.

b) 2.

c) Pin 10.

d) 7 & 14.

e) NOR


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Solution to Problem on Page 14:


0 0 0 1 0 1

0 0 1 1 1 0

0 1 0 0 0 1

0 1 1 0 0 1

1 0 0 1 0 0

1 0 1 1 1 0

1 1 0 0 0 0

1 1 1 0 0 0

Exercise 3

1.

Inputs OutputsA B K Q

0 0 1 00 1 0 1

1 0 0 1

1 1 0 1

2.


0 0 0 1 1 1

0 0 1 0 0 1

0 1 0 1 1 1

0 1 1 0 1 1

1 0 0 1 1 0

1 0 1 0 0 1

1 1 0 1 1 0

1 1 1 0 1 0


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3.

Inputs Outputs

A B C D E F G Q0 0 0 1 1 0 1 1

0 0 1 1 0 0 1 1

0 1 0 1 1 1 0 1

0 1 1 1 0 1 1 1

1 0 0 0 1 0 1 1

1 0 1 0 0 0 1 1

1 1 0 0 1 0 0 0

1 1 1 0 0 0 1 1


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Exercise 4

1. Output Q is on when input A is high and input B is low.(i.e. Q = A AND NOT B)

The logic circuit required is as follows:

2. Output Q is on when input A is low and input B is high or when input A is

high and input B is low.

(i.e. Q = [NOT A AND B] OR [A AND NOT B])

The logic circuit required is as follows:


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3. (a) P = Inverse of input A(or P = NOT A)

(b) The name of this gate is AND.

(c) R= ANOR B.

(d)

4. Red = Input A

Yellow = NOT B

Green = NOT AAND B


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Exercise 5

1.i. Expression Ais correct for an AND gate.

ii. Expression Dis correct for a NOT gate.

iii.Expression Bis correct for a NOR gate.

iv.Expression Eis correct for a NAND gate.

v. Expression Cis correct for an OR gate.

2. X = B

Y = B.A

Z = B.A


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Exercise 6

1. a)

Truth Table

Inputs OutputsA B O/p 1 O/p 2 O/p 3

0 0 1 1 0

0 1 1 0 0

1 0 0 0 1

1 1 0 0 0

b) i) O/p 1 = NOT A, or A

ii) O/p 2 = ANOR B, or BA

iii) O/p 3 = AAND NOT B, or B.A

c)


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2. a) Complete the following truth table for the system.

Inputs OutputsA B LED buzzer

0 0 1 1

0 1 1 0

1 0 0 0

1 1 0 0

b) The led output is the inverse of input A.

c) Buzzer = A NOR B, so a NOR logic gate is required.

d)

3. a)

Inputs OutputsA B R Y G

0 0 0 1 1

0 1 0 1 0

1 0 0 1 0

1 1 1 0 0

b) An AND gate is required to provide the R output.

c) Y = A NAND B ( or Y = NOT R)


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d) A NOR gate is required to provide the G output.

e)

Or


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4. a)

Inputs Outputs

A BV1

(cyan)V2

(magenta)V3

(yellow)

0 0 0 1 1

0 1 0 0 1

1 0 0 1 0

1 1 1 0 0

b) An AND gate will provide the V1output.

c) V2= NOT B (or B )

V3= NOT A (or A )

d)


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Exercise 7

1. 4.

2. NOR gates.

3. NAND gates.

4. 4.

5. 3, 6, 8 and 11.

6. 7404.

7. 2.

8. 4002 & 4072

9. 14.

10. 9, 10, 11 & 12.

11. AND gates.

12. 7402.


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A

Q

B

C

A

QB

C

Exercise 8

1.

2.

Exercise 9

a) In circuit C with the switch openthe input to the logic system is at logic 1

b) In circuit C with the switch closedthe input to the logic system is at logic 0

c) In circuit D with the switch openthe input to the logic system is at logic 0

d) In circuit D with the switch closedthe input to the logic system is at logic 1

2. a) Resistor R1 is a pull DOWNresistor and R2 is a pull UPresistor.

b) When switch SW2 is pressed input B is at logic 0

c) When both switches are pressed output Q is at logic 0


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(b) Complete the truth tables for the following logic gates:

(i) a NOT gate

Input Output

0

1

[1]

(ii) an AND gate

Input Output

0 0

0 1

1 0

1 1

[1]

(c) A NOTgate and an ANDgate are connected together as shown in the block diagram.

(i) Complete the following truth table for this system:

L M X Q

0 0

0 1

1 0

1 1

[2]

(ii) Name the single logic gate which produces the same effect as this logic system.

........................................

[1]


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4. The following diagram shows a logic system.

(a) Complete the following truth table for this system.

Input A Input B X Q

0 0

0 1

1 0

1 1

[2]

(b) (i) Complete the diagram to show how a NAND gate can be made to behave as a NOT

gate.

[1]

(ii) Draw a diagram to show the NAND gate equivalent of an AND gate.

[1](iii) Here is a logic system built using only NAND gates.

Cross out all redundant gates.

[1]


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5. The diagram shows the pin-out for an IC (Integrated circuit).

(a) How many logic gates are in this IC? ...........................................

(b) How many inputs does each gate have? ...........................................

(c) Pin 1 is labelled.

(i) What is the pin number for the 0V pin? ...........................................

(ii) What is the pin number for the output of gate X? ...........................................

(d) Choose the type of logic gate found on this IC from the following list:

AND OR NOT NAND NOR

Answer : ...............................................

[5]

6.

Complete the statements:

(a) The signal at P will be logic 0 only when input A is logic .......................

[1]

(b) Output Q will be logic 1 only when the signal at P is logic ......................, and input B is

logic ......................

[1]


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7. Here are five logic gate symbols:

(a) Which symbol, V, W, X, Y orZ, is the symbol for a NOT gate? ...........................

[1]

(b) Which symbol, V, W, X, Y, orZ, is the symbol for the logic gate which has the following

truth table?

Input A Input B Q

0 0 0

0 1 1

1 0 1

1 1 1

Answer : ....................................

[1]

(c) Which one of the following logic gate systems, R, S, T, orU, has the same output as an

AND gate?

Answer : ....................................

[1]


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8. Here is part of the block diagram for a system that tells the assistant when someone enters a shop.

The switch unit is used to arm (switch on) the system. It outputs logic 1 when switched on.

The buzzer sounds when someone stands on the pressure pad, but only of the system is armed. The pressure pad outputs logic 1 when someone stands on it.

The transistor switch needs a logic 1 input to make the buzzer sound.

(a) Which of the following truth tables, C, D, E, or F, gives the required output for logic gate

X?

Answer : ................................

[1]

(b) What type of logic gate is required in block X?

Choose your answer from the following list:

AND gate NAND gate NOT gate OR Gate

Answer : ................................

[1]


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(c) Which oneof the following, P, Q, R, orSis a suitable circuit for the switch unit?

It outputs logic 1 (12V) when switched on.

Answer : .................................

[1]


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9. (a) Complete the truth table for the following logic system:

A B X Y Q

0 0

0 1

1 0

1 1

[3]

(b) Redraw the system replacing each of the three gates with its equivalent NAND gate

arrangement.

[3]


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10. The diagram shows the pin-out for an IC (integrated circuit).



(c) Pin 1 is labelled.

(i) What is the pin number connected to 0V? ...........................................

(ii) What is the number of the pin labelled X? ...........................................


AND OR NOT NAND NOR

Answer : ...............................................

[5]


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11. (a) Here are five logic gates:

(i) Which one, A, B, C, D, orE, is an AND gate?

Answer : .............................................

[1]

(ii) Which one, A, B, C, D, orE, has the following truth table?

Inputs Q

0 0 1

0 1 1

1 0 1

1 1 0

Answer : .............................................

[1]

(b) The following logic system gives the same output as one of the logic gates in part (a).

(i) Complete the following truth table for this logic system.

A B X Q

0 0

0 1

1 0

1 1

[2]

(ii) Which single logic gate gives the same output as this system?

Answer : ...........................................

[1]


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12. The following logic system is built using three logic gates.

(a) (i) Complete the following truth table for the logic system above.

A B X Y Q

0 00 1

1 0

1 1

[2]

(ii) Which single logic gate gives the same output as this system?

Answer : ....................................

[1]

(iii) Redraw the system showing the NAND equivalent of each gate.

[2]

(b) Here is another system of NAND gates.

(i) Simplify it by crossing out any redundant gates.

[2]

(ii) Give one reason why it is cheaper to convert a logic system into its NAND gate

equivalent.

.................................................................................................................................

.................................................................................................................................

[1]


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13. Here are five logic gate symbols:

(a) Which symbol, V, W, X, Y orZ, is the symbol for a NOT gate? ...........................

[1]

(b) Complete the truth table for the logic gate W.

Input A Input B Q

0 0

0 1

1 0

1 1

[1]

(c) Which one of the following logic gate systems, P, Q, R, orS, has the same output as logic

gate W?

Answer : ....................................

[1]


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14. The diagram shows the pin-out for an IC (integrated circuit).



(c) Label Pin 1 of the IC.

(d) What is the number of the pin connected to the output of gate A?

...........................................

(e) Choose the type of logic gate found on this IC from the following list:

AND OR NOT NAND NOR

Answer : ...............................................

[5]


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15. (a) Here is a list of logic gates:

AND OR NOT NAND NOR

(i) Which of the gates has the following symbol?

Answer : ....................................

[1]

(ii) Which of the gates has the following truth table?

A B Q

0 0 0

0 1 0

1 0 0

1 1 1

Answer : ...................................

[1]

(ii) Which of the gates has the opposite effect to (inverts) an OR gate?

Answer : ....................................

(b) Complete the truth table for the following logic system:

A B P Q

0 0

0 1

1 0

1 1

[2]


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17. (a) Here are five logic gates symbols:

Which symbol, A, B, C, D orE, is the symbol for:

(i) a NOT gate; ...........................................

(ii) an OR gate; ...........................................

(iii) a NAND gate? ........................................... [3]

(b) Here are five truth tables:

Which table, A, B, C, D orE, is the truth table for:

(i) a NOT gate; ...........................................

(ii) an OR gate; ...........................................

(iii) a NAND gate? ...........................................

[3]


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(c) (i) Complete the truth table for the following logic system:

A B X Y Q

0 0

0 1

1 0

1 1

(ii) Name the single logic gate which produces the same effect as this logic system.

..................................................... [1]


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18. (a) Complete the truth table for the following logic gates.

(i) OR

(ii) AND

[4]

(b) Complete the truth table for the following system of logic gates.

A B C D Q

0 0

0 1

1 0

1 1

[3]


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(c) (i) Redraw the system replacing each of the three gates with its equivalent NAND gate

arrangement.

[3]

(ii) Draw a line through each redundant gate.

[1]


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19. (a) Write the name of each logic gate in the spaces provided.

[3]

(b) The three gates are arranged in the following logic system. Complete the truth table.

A B Q R S

0 0

0 1

1 0

1 1

[3]


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20. (a) Complete the truth table for the following NAND gate.

[1]

(b) The NAND gate is used along with a NOT gate and an OR gate as part of a logic systemshown below.

Complete the truth table for the logic system.

A B C D Q2

0 0

0 1

1 0

1 1

[3]


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21. A logic gate system is required to switch on three different lights P, Q and R. input switches A and

B control the lights. The following truth table shows how the lights come on for various switching

conditions.

A B P Q R

0 0 0 1 1

0 1 1 0 1

1 0 0 1 0

1 1 0 0 0

(a) (i) Output Q can be obtained by inverting one input. Write down the Boolean

expression for Q obtained in this way.

Q = ......................................................................

[1]

(ii) Using the truth table write down the Boolean expressions for outputs P, and R.

P = ......................................................................

R = .....................................................................

[2]

(b) Draw the logic circuit needed to produce output P.

[2]


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(c) Here is another logic system.

Draw the NAND gate equivalent of this circuit.

[3]


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22. Here is the pin-out for an IC (integrated circuit).



(c) Which pin, A, B, Cor Dis Pin 1 of this IC. ...........................................


AND OR NOT NAND NOR

Answer : ...............................................

[4]


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Self Evaluation Review

Learning ObjectivesMy personal review of these objectives:

1.9.1 Introduction.

Recognise high/low, 1/0, as two state logic

levels;

1.9.2 Truth Tables.

Draw symbols and construct truth tables for

AND, OR, NOT, NOR, and NAND gates;Produce a truth table for a system of up to

five gates;

Devise a system of gates from a truth table;

Design simple systems using logic gates to

solve a given problem;

Use Boolean notation as a shorthand method

of expressing a truth table;

1.9.3 Use of data sheets.

Use data sheets to;

Select a logic IC for given applications;Identify pin connections of logic gates;

1.9.4 NAND gate implementation.

Show how other gates can be made up from

NAND gates;

Implement a given logic circuit using NAND

gates;

Remove double inversions;

1 9 5 Pull up/down resistors

Documents

Logic Circuit Modules 1