4 digital electronics

8/12/2019 4 Digital Electronics

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L O G I C G A T E S

F L I P - F L O P S

R E G I S T E R S

A D D E R S

Digital Electronics

Jordan Nash - Microprocessor Course

1


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Numbers in a computer:


2

In the Microprocessor and memory Bits are storedelectronically

Each bit is represented by an electrical signal which

is either a high or low voltage level. high 1 and low 0

high 0 and low 1

NormalLogic

InverseLogic


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High and Low


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1:

High

0

:

Low

CMOS

NO MANS LAND74HCxx

0.1 1.0 Volts

3.5 4.9 Volts

Transistor TransistorLogic (TTL)

NO MANS LAND74LSxx

0.8 0.4

Volts

2.0 2.4Volts

It is actually a bit more complicated than this since there are different thresholdsfor inputs and outputs and their noise margins (indicated here in RED)..


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Making a Zero or and One


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How do you actually make a 0 or 1 ?

It is clear that dependingupon the position of the J1switch the line will be either

0or 1


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Binary Logic


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Once the bits are stored, we want to be able toperform various operations on the bits

The operations which the microprocessor carries out

can all be constructed with a few simple circuits The basic building blocks are logic gates


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Binary Logic truth tables


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A B AND

0 0 0

0 1 0

1 0 0

1 1 1

A B OR

0 0 0

0 1 1

1 0 1

1 1 1

A NOT

0 1

1 0

You mayrememberfrom the first

year lab the

gates AND,OR, NOT

Any digitaldevice can bemade out of

either ORs andNOTs or ANDsand NOTs.


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DeMorgans Theorem


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You can swap ANDs with ORs if at the same timeyou invert all inputs and outputs :

Exercise:Write truth table for both and prove thatthis is correct


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An AND out of NOTs and ORs


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Exercise: Test the claim that you can make any logicdevice exclusively out of NOTs and ORs by makingan AND out of NOTs and ORs:


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Answer:


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Show that this device has an identical truth tableas the AND gate.


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Exercise: Exclusive OR


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Construct an exclusive OR (XOR)gate using OR, ANDs, and NOT:

A B XOR

0 0 0

0 1 1

1 0 1

1 1 0


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The Exclusive OR


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Solution:


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Storing Zeros and Ones: Registers


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Registers are electronic devices capable of storing 0s or 1s

D-FLIP-FLOPsare the most elementary registers

can store one bit 8 DFFs clocked together make a one byteregister Capable of storing 8 bits


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SR Latch: Set and Reset


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Gate level designof an SR latch

S* R* Q Q*

0 1 1 0

1 0 0 1

When S* is low and R*High, Q is high (Set)

When S* is high and

R* is low Q is cleared(Reset)

Q and Q* arecomplements


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SR Latch: Hold


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S* R* Q Q*

1 1 1 0

1 1 0 1

When S* and R* arehigh both states for Qare possible (andstable)

The latch remembersthe last state it was in,and holds Q in thatstate


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SR Latch: Dont do this


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S* R* Q Q*

0 0 1 1

When S* and R* arelow Q and Q* are nolonger complementary

The latch can alsoenter a race conditionwhere it oscillates


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SR Latch: 1 bit memory


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Gate level design of an SRlatch

S R Q Q*

H H H L

H H L H

H L L H

L H H L

L L H H

S/R high is ambiguous, but stable This circuit remembers that S

went low

Ambiguousbut stable


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Gated D Latch


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Ensures that S and R are always complementary.When the Clock is high D is set on the outputs of the latch and it is held therewhen the clock is lowExercise: Fill out the truth table for this circuit.


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D latch timing


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The input Data to the gated D latch appears on the output(Q)while the clock is high.

Sometimes we want to transfer the data into the register onlyat a specified moment (for example when the clock changes)

The D Flip-Flop uses two d-latches to latch the data on theedge (depending on the design either positive or negative) ofthe input clock


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Master Slave D Flip Flop


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Data is stored inthe Master latch(Qm) when theclock is high

When the clock

goes from high tolow The data is held in

the Master The Data is stored in

the slave latch

output The Qsoutput can

only change whenthe clock goes fromhigh to low


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A real D-Flip-Flop (DFF)


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CLK D S R Q Q-bar

X X L H H L

X X H L L H

X X L L H H

H H H H L

L H H L H

When S and R are High, on therising edge of the clock the dataare transferred and stored in Q.

One can Set or Reset (Clear)the DFF using S or R


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4-bit Register


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A register thatstores 4 bits

Data coming inD0,D1,D2,D3

Data Stored on risingedge of clock inQ0,Q1,Q2,Q3


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Byte Register


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8 bit register in a singlepackage

74F574

Also contains tri-state outputs

CLK

Byte coming in

Byte Stored


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Addition with Binary Logic Gates

J. Nash - Microprocessor Course

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Example: Adding two 4 bit numbers results in a 4 bit number plusone carry bit or effectively a 5 bit numberLets construct this adder from gates

Computers carry out addition with binary addition ofbits stored in registers

We can build additions of large numbers of bits out ofunits which add two bits

$3+$4 $3+$C $D+$4 $D+$C

0011 0011 1101 1101

+ 0100 + 1100 + 0100 + 1100

= 0111 = 1111 =10001 =11001

$7 $F $11 $19


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Additions of two bits: The half adder


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A B C S

0 0 0 01 0 0 1

0 1 0 1

1 1 1 0

Adding two bitsgenerates two bits ofoutput

1 Sum Bit S

1 Carry Bit C

The truth table for thisoperation is shown along

with an implementationusing two gates


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Full Adder


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In order to add larger numbers, weneed to be able to bring the carryfrom the lower order bits, and addthis to the sum

The inputs are: the two bits to be added (A,B) The Carry Bit (C)

The outputs are: The Sum Bit (S) The Carry Out Bit (C+)

We can build this from two halfadders and an XOR gate

A B C C+ S

0 0 0 0 0

0 1 0 0 1

1 0 0 0 1

1 1 0 1 00 0 1 0 1

0 1 1 1 0

1 0 1 1 0

1 1 1 1 1


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Two bit adder


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We can construct logicthat adds more than 1 bittogether by usingmultiple full adder

circuits

Exercise:Construct a circuit that adds two two bit words (Ao,A1) and(B0,B1) and produces three Sum Bits (S0,S1,S2)


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Solution


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With the Full adderbuilding block we cangeneralize to produce acircuit which adds larger

numbers


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Four plus Four

bit addition

This can clearly begeneralized to anynumber of bits

There is a problem in

building circuits thisway as the carry bitsneed to propogatethrough the circuitbefore the answer iscorrect

There are other ways toconstruct adder circuitswhich avoid thisproblem

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Arithmetic Logic Unit (ALU)


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Centre of everycomputer

Composed of buildingblocks we have beenlearning about Adders

Flip Flops

Registers

We also need to have a

data bus to move data inand out - we will learnabout this soon

4 digital electronics

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