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Logic Gates

Digital Logic andSoftware Principles


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History and development

Thebinary number systemwas refined byGottfried Wilhelm Leibniz(published in 1705) and he also established that by using the binary system, the principles

of arithmetic and logic could be combined. In an 1886 letter,Charles Sanders Peirce

described how logical operations could be carried out by electrical switching circuits.[7]

Eventually,vacuum tubesreplaced relays for logic operations.Lee De Forest's modification,

in 1907, of theFleming valvecan be used as AND logic gate.Ludwig Wittgenstein

introduced a version of the 16-rowtruth tableas proposition 5.101 ofTractatus Logico-

Philosophicus (1921).Walther Bothe, inventor of thecoincidence circuit, got part of the

1954Nobel Prizein physics, for the first modern electronic AND gate in 1924.Konrad

Zusedesigned and built electromechanical logic gates for his computerZ1(from 193538).Claude E. Shannon introduced the use of Boolean algebra in the analysis and design of

switching circuits in 1937. Active research is taking place inmolecular logic gates.
http://en.wikipedia.org/wiki/Binary_number_systemhttp://en.wikipedia.org/wiki/Binary_number_systemhttp://en.wikipedia.org/wiki/Binary_number_systemhttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://en.wikipedia.org/wiki/Logic_gate#cite_note-P2M-7http://en.wikipedia.org/wiki/Logic_gate#cite_note-P2M-7http://en.wikipedia.org/wiki/Logic_gate#cite_note-P2M-7http://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Lee_De_Foresthttp://en.wikipedia.org/wiki/Lee_De_Foresthttp://en.wikipedia.org/wiki/Lee_De_Foresthttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Ludwig_Wittgensteinhttp://en.wikipedia.org/wiki/Ludwig_Wittgensteinhttp://en.wikipedia.org/wiki/Ludwig_Wittgensteinhttp://en.wikipedia.org/wiki/Truth_tablehttp://en.wikipedia.org/wiki/Truth_tablehttp://en.wikipedia.org/wiki/Truth_tablehttp://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicushttp://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicushttp://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicushttp://en.wikipedia.org/wiki/Walther_Bothehttp://en.wikipedia.org/wiki/Walther_Bothehttp://en.wikipedia.org/wiki/Walther_Bothehttp://en.wikipedia.org/wiki/Coincidence_circuithttp://en.wikipedia.org/wiki/Coincidence_circuithttp://en.wikipedia.org/wiki/Coincidence_circuithttp://en.wikipedia.org/wiki/Nobel_Prizehttp://en.wikipedia.org/wiki/Nobel_Prizehttp://en.wikipedia.org/wiki/Nobel_Prizehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Z1_(computer)http://en.wikipedia.org/wiki/Z1_(computer)http://en.wikipedia.org/wiki/Z1_(computer)http://en.wikipedia.org/wiki/Claude_E._Shannonhttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Claude_E._Shannonhttp://en.wikipedia.org/wiki/Z1_(computer)http://en.wikipedia.org/wiki/Z1_(computer)http://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Konrad_Zusehttp://en.wikipedia.org/wiki/Nobel_Prizehttp://en.wikipedia.org/wiki/Nobel_Prizehttp://en.wikipedia.org/wiki/Coincidence_circuithttp://en.wikipedia.org/wiki/Coincidence_circuithttp://en.wikipedia.org/wiki/Walther_Bothehttp://en.wikipedia.org/wiki/Walther_Bothehttp://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicushttp://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicushttp://en.wikipedia.org/wiki/Truth_tablehttp://en.wikipedia.org/wiki/Truth_tablehttp://en.wikipedia.org/wiki/Ludwig_Wittgensteinhttp://en.wikipedia.org/wiki/Ludwig_Wittgensteinhttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Lee_De_Foresthttp://en.wikipedia.org/wiki/Lee_De_Foresthttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Logic_gate#cite_note-P2M-7http://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://en.wikipedia.org/wiki/Charles_Sanders_Peircehttp://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttp://en.wikipedia.org/wiki/Binary_number_systemhttp://en.wikipedia.org/wiki/Binary_number_system


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Logic

Formal logic is a branch of mathematics

that deals with true and false valuesinstead of numbers.

In 1840s, George Boole developedmany Logic ideas.

A logic gateperforms alogical operationon one or more logic inputs and produces asingle logic output.

3
http://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operation


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Introduction

A logic gateis an idealized or physical device implementing a

Boolean function; that is, it performs alogical operationon one or

more logical inputs, and produces a single logical output. Depending

on the context, the term may refer to an ideal logic gate, one

that has for instance zerorise timeand unlimitedfan-out

, or it may refer to a non-ideal physical device. Logic gates areprimarily implemented usingdiodesortransistorsacting as

electronic switches, but can also be constructed using

electromagneticrelays(relay logic),fluidic logic,pneumatic logic,

optics,molecules, or evenmechanicalelements. With

amplification, logic gates can be cascaded in the same way that

Boolean functions can be composed, allowing the construction ofa physical model of all ofBoolean logic, and therefore, all of the

algorithms andmathematicsthat can be described with Boolean

logic.
http://en.wikipedia.org/wiki/Boolean_functionhttp://en.wikipedia.org/wiki/Boolean_functionhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Fan-outhttp://en.wikipedia.org/wiki/Fan-outhttp://en.wikipedia.org/wiki/Fan-outhttp://en.wikipedia.org/wiki/Diodehttp://en.wikipedia.org/wiki/Diodehttp://en.wikipedia.org/wiki/Diodehttp://en.wikipedia.org/wiki/Transistorhttp://en.wikipedia.org/wiki/Transistorhttp://en.wikipedia.org/wiki/Transistorhttp://en.wikipedia.org/wiki/Switch#Electronic_switcheshttp://en.wikipedia.org/wiki/Switch#Electronic_switcheshttp://en.wikipedia.org/wiki/Relayhttp://en.wikipedia.org/wiki/Relayhttp://en.wikipedia.org/wiki/Relayhttp://en.wikipedia.org/wiki/Relay_logichttp://en.wikipedia.org/wiki/Relay_logichttp://en.wikipedia.org/wiki/Relay_logichttp://en.wikipedia.org/wiki/Fluidic_logichttp://en.wikipedia.org/wiki/Fluidic_logichttp://en.wikipedia.org/wiki/Fluidic_logichttp://en.wikipedia.org/wiki/Pneumatics#Pneumatic_logichttp://en.wikipedia.org/wiki/Pneumatics#Pneumatic_logichttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Analytical_enginehttp://en.wikipedia.org/wiki/Analytical_enginehttp://en.wikipedia.org/wiki/Analytical_enginehttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Analytical_enginehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Molecular_logic_gatehttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Pneumatics#Pneumatic_logichttp://en.wikipedia.org/wiki/Pneumatics#Pneumatic_logichttp://en.wikipedia.org/wiki/Fluidic_logichttp://en.wikipedia.org/wiki/Fluidic_logichttp://en.wikipedia.org/wiki/Relay_logichttp://en.wikipedia.org/wiki/Relay_logichttp://en.wikipedia.org/wiki/Relayhttp://en.wikipedia.org/wiki/Relayhttp://en.wikipedia.org/wiki/Switch#Electronic_switcheshttp://en.wikipedia.org/wiki/Switch#Electronic_switcheshttp://en.wikipedia.org/wiki/Transistorhttp://en.wikipedia.org/wiki/Transistorhttp://en.wikipedia.org/wiki/Diodehttp://en.wikipedia.org/wiki/Diodehttp://en.wikipedia.org/wiki/Fan-outhttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Boolean_functionhttp://en.wikipedia.org/wiki/Boolean_function


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Logic Signals

There are a number of different systems for representing binaryinformation in physical systems.

A voltage signal with zero (0) corresponding to 0 volts and one (1)corresponding to five or three volts.

A sinusoidal signal with zero corresponding to some frequency,and one corresponding to some other frequency.

A current signal with zero corresponding to 4 milliamps and onecorresponding to 20 milliamps.

And one last way is to use switches, OPEN for "0" and CLOSED for"1".

( And there are more ways !)


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AND gatee.g. I get up ifit is 8-00 a.m. ANDit is a weekday he said if A = 8-00 a.m. B = weekday

and Y = get up then he said you can write:

BAY

where the dot represents logical AND.

He went on to say that if 1 represents TRUE

and 0 represents FALSE

then the function can be defined in a truth table.


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Logic Gates

Truth TableThe truth table has an entry for each possible combination of inputs.

For n inputs there will be 2nentries 2 inputs = 4 entries.

We can have more than two inputs in which

case the only time we would have a 1 out is whenall the inputs are true.

A B Y

0 0 0

0 1 0

1 0 0

1 1 1


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Logic Gates

SymbolThe symbol adopted for the AND function ( gate) is shown below

American (MIL-STD-806) British (IEC 617:12)

A

A

B

B

Y

Y &


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Logic Gates

We can have more than two

inputs in which case the only

time we would have a 0 out iswhen all the inputs are false.


A B Y

0 0 0

0 1 1

1 0 1

1 1 1

AA

B

B

YY 1


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Logic Gates

NOT gatee.g. I turn on the heating ifit is NOThot

if A = hot and Y = Heating on then:

AY

where the bar represents logical NOT.

We can only have one input and the output is always the opposite sign.


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Logic Gates

American (MIL-STD-806)

British (IEC 617:12)

Exclusive OR EXOR gate

A Y

0 1

1 0

1


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Logic Gates

BAY

Note that the normal OR includes the case where we have both inputs true. The EXOR does notinclude this case.

For more than two inputs the gate is defined as:

The output is TRUE if we have an odd number of inputs TRUE

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

where the sign represents logical EXOR.

The symbol adopted for the EXshown below

American (MIL-STD-806)


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Logic Gates

Not AND NAND gate

where the dot and

bar representslogical NAND.

BAY We can have morethan two inputs inwhich case the onlytime we would have a 0out is when all theinputs are true.


A

B

Y&

A

B

Y

Logic Gates


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Logic Gates

Not OR NOR gate

where the + sign

and bar representslogical NAND.

We can have morethan two inputs inwhich case the onlytime we would have a 1out is when all theinputs are false.


BAY

A A

B

B

Y

Y

1

Logic Gates


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Logic Gates

Universal Gates

NAND and NOR gates are referred to as

universal gates as the three basic gates can

be constructed using either one of the two.

This therefore implies that all logic circuits can

be constructed using either of the gates.

The notes show this process for NAND only but

it can be shown for NOR also.


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Logic Gates

NOT using NANDs only

The Truth Table is for a NAND gateIf we tie the inputs of a NAND together then welimit the possible input combinations to two, 1 1 and0 0. These are shown on the table now if the input is

0 the output is 1 and vice versa a NOT gate

A

Y

Logic Gates


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Logic Gates

AND using NANDs only

As a NAND is simply an AND followed by a NOTgate (inverter) we can simply use a NANDfollowed by NOT.

A

B

Y

Note more than one NAND gate toproduce the desired AND gate.

Logic Gates


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Logic Gates

OR using NANDs only

A B BA

This is our desired OR gate


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Logic Gates

OR using NANDs only

A B BA A B

0 0 0 1 1

0 1 1 1 0

1 0 1 0 1

1 1 1 0 0

If we now add NOT A and NOT Binto our table

OR using NANDs only


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Logic Gates

A B BA A

B BA

0 0 0 1 1 1

0 1 1 1 0 0

1 0 1 0 1 0

1 1 1 0 0 0

If these are now ANDed together

OR using NANDs only


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Logic Gates

A B BA A

B

BA

BA

0 0 0 1 1 1 0

0 1 1 1 0 0 1

1 0 1 0 1 0 1

1 1 1 0 0 0 1

Finally if we invert our result we see that the 3rdand 7thcolumn are

identical. This means that if we invert the inputs then NAND then

we will end up with the OR function.

OR using NANDs onlyA


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Logic Gates

Let us examine the way in which logic gates can be used to

realise logic circuits:

Example


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Logic Gates

A drill (D) is to operate if we are in automatic (A) and the system (S)

is running or if we are in manual (M) and a button (B) is pressed or

if an override (O) input is not operated.

The boolean (logic) expression for this can be written in the

following way:

OBMSAD

This can be constructed in the following way:A

S

M


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Logic Gates


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Logic Gates

At this point let us examine different logic integrated circuits (I.C.s) families which

can be used to construct logic circuits.

Logic Families

Transistor-Transistor Logic(TTL) is a class of digital circuits built from bipolarjunction transistors (BJT), and resistors. It is called transistor-transistor logic

because both the logic gating function (e.g. AND) and the amplifying function are

performed by transistors (contrast this with RTL and DTL).

Transistor Transistor Logic TTL prefix 74 e.g. 7400Quad 2-input NAND

More Specifically MM74XXX00P

MM Manufacturers codes

e.g. SN Texas Instruments

CD Harris Semiconductors

DM Fairchild Semiconductors

M SGS-Thomson Microelectronics


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Logic Gates

MC Motorola

Transistor Transistor Logic TTL

XXX variants

e.g. L Low powerS Schottky high speed

LS Low power Schottky

ALS Advanced low power Schottky

Voltage range Speed Power

LS +5V 5% 10nS 2 mWALS +5V 5% 7nS 1 mW


Prefix 54 not 74 is used for higher specifications( normally military )

Temperature Voltage supply


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Logic Gates

range toleranceCommercial 74 0 - 70C 5 %

families

Military 54 families -55 - +125C 10 %


Most TTL families

An input is recognised as 1 if the input is >2VAn input is recognised as 0 if the input is < 0.8V

Noise immunity is the difference = 1.2V

A low output has a maximum output of 0.2V A high output has a minimum output of 3.3V

Available TTL Gate Packages

Quad 2-input gates


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Logic Gates

7400 quad 2-input NAND

7403 quad 2-input NAND with open collector outputs

7408quad 2-input AND

7409quad 2-input AND with open collector outputs7432 quad 2-input OR

7486 quad 2-input EX-OR

74132 quad 2-input NAND with Schmitt trigger inputs

7402 quad 2-input NOR

Triple 3-input gates

7410triple 3-input NAND7411 triple 3-input AND7412triple 3-input NAND with open collector outputs

7427 triple 3-input NORDual 4-input gates

7420dual 4-input NAND7421 dual 4-input AND


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Logic Gates

Others

7430 8- input NAND gate

Hex NOT gates

7404hex NOT7405hex NOT with open collector outputs7414 hex NOT with Schmitt trigger inputs

Complementary Metal Oxide Semiconductor Logic CMOS

Number sequence originally from 4000 upwards but not the same as TTL

Characteristics

Delay 50nS Power 1 W Voltage 3-18V

Input

Logic 1 is recognised above 2/3 Supply

Logic 0 is recognised below 1/3 Supply

Output

The minimum for logic 1 is Supply 0.01V

The maximum for logic 0 0.01V


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Logic Gates

Available CMOS Gate Packages

Quad 2-input gates

4001 quad 2-input NOR

4011 quad 2-input NAND

4070quad 2-input EX-OR

4071 quad 2-input OR

4077 quad 2-input EX-NOR

4081 quad 2-input AND

4093 quad 2-input NAND with Schmitt trigger inputs

CMOS Gate Packages

Triple 3-input gates

4023 triple 3-input NAND


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Logic Gates

4025 triple 3-input NOR 4073

triple 3-input AND

4075 triple 3-input ORDual 4-input gates

4002 dual 4-input NOR

4012 dual 4-input NAND

4072 dual 4-input OR4082 dual 4-input AND

4069 hex NOT (inverting buffer)

Developments in TTL and CMOS

Often there are different pin-outs in the two family

types.


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Logic Gates

CMOS chips are available which are the same numbers

due to the popularity of TTL.

74HC High speed CMOS operating 2V to 6V

74HCT High speed CMOS with TTL compatible

supplies

74ACT Advanced CMOS with TTL compatible levels

and pin-outs

74AC Advanced CMOS with CMOS compatible

levels and TTL pin-outs

Logic Problem.


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Logic Gates

Getting back to our example, we can see that we

would require:

2 x 2-input AND 7408 (4 x 2-input AND)

4081 (4 x 2-input AND )

1 x inverter (NOT) 7404 (6 x inverter)

4069 (6 x inverter )1 x 3-input OR Not available?

4075 (3 x 3-input OR )

This is a total of 3 chips and we end up not using

9 gates within the packages.


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Logic Gates

8 x 2-input NAND 2 x 7400


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Logic Gates

1x 3-input NAND 1 x 7410

again 3 chips.

BUTBy observation we can see that NANDs 2 and 3

simply invert 1s output then invert it again.

This means that they cancel each other outand can be removed.

This is also true for NANDs 5 and 6 and

NANDs 8 and 9, leaving us with


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Logic Gates

This requires:

2x 2-input NAND 1 x 7400

1 x 3-input NAND 1 x 7410

With a little understanding of logic gates we

can reduce the requirements to only one chip

by using the fact that:


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Logic Gates

So we need:

3 x 3-input NANDs 1 x 7410

Note.

Conversions from AND, OR, NOT to NAND only

rarely produce a less complex circuit butnormally the complexity is similar. The

advantage lies in the fact that NAND chips


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Logic Gates

are readily available and are inexpensive due

to the number sold and that any gates left

over can be used in other circuits as allcircuits use the same gate types.

Logic Circuits TTL and CMOS


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Logic Gates

Complementary Metal Oxide Semiconductor CMOS NOR

Transistor Transistor Logic (TTL) NAND Gate.

R1 R2

R3

R4

Q1Q2

Q3

Q4

D

F

ab

c


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Logic Gates

gate

Output

Input A

Input B

Vs+

Q1

Q2

Q3

Q4


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Logic Gates


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Logic Gates

This resource was created by the University of Wales Newport and released as an open educational resource

through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open

Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.

2009 University of Wales Newport

This work is licensed under aCreative Commons Attribution 2.0 License.

The JISC logo is l icensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK:

England & Wales Licence. All reproductions must comply with the terms of that licence.

The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes

only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original

publisher.

The name and logo of University of Wales Newport is a trade mark and all rights in it are reserved. The name and logo should not

be reproduced without the express authorisation of the University.
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