Digital Circuits and Logic

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# Digital Circuits and Logic - PowerPoint PPT Presentation

Digital Circuits and Logic. Gates Combinatorial logic adder ALU Sequential logic flip-flop memory CPU fetch-decode-execute cycle. How to make a computer. Computers are electronic equipment contain many connected electronic components memory Central Processing Unit ( CPU )

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Presentation Transcript
Digital Circuits and Logic
• Gates
• Combinatorial logic
• ALU
• Sequential logic
• flip-flop
• memory
• CPU
• fetch-decode-execute cycle

CMPUT 229

How to make a computer
• Computers are electronic equipment
• contain many connected electronic components
• memory
• Central Processing Unit (CPU)
• specialized devices (network, video, etc.)
• components largely made of circuits containing wires and gates

CMPUT 229

Logic Level
• A Boolean logical signal always takes one of two logic levels.
• on / off
• high (H) / low (L)
• one (1) / zero (0)
• true (T) / false (F)
• positive / negative
• Why two levels ?
• Why not use, say, 4 levels?
• For example, electrical voltage as
• 0 v, 5v, 10v, and 20v

CMPUT 229

Gates
• Tiny electronic switches
• implement Boolean logic
• 0/1  low/high voltage
• digital
• for given inputs, produce known output
• according to truth table
• represented in circuit diagrams by symbols

CMPUT 229

Gates

NOT

In

Out

0

1

1

0

CMPUT 229

AND

In 1

In 2

Out

0

0

0

0

1

0

1

0

0

1

1

1

Gates

CMPUT 229

Gates

NAND

In 1

In 2

Out

0

0

1

0

1

1

1

0

1

1

1

0

CMPUT 229

In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

Gates

OR

CMPUT 229

XOR

In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

0

Gates

CMPUT 229

Implementation of Logic Gates
• The NAND gate, the basic logic gate
• easier to manufacture
• any other logic gate can be made from a combination of NAND gates

CMPUT 229

Implementation of Logic Gates
• CMOS
• The standard, 4000 series (4011)

CMPUT 229

Implementation of Logic Gates
• TTL
• The 4011's TTL counterpart is the 7400

CMPUT 229

Logic Circuits
• Combinatorial Logic
• Output depends solely on the input values
• Example:
• arithmetic logic unit (ALU)
• Sequential logic
• output depends on
• the present input, and
• the history of the input
• Examle:
• Flip-flop, Memory

CMPUT 229

Combinatorial logic
• What about more complex truth tables?
• Made by combining many gates together
• wires connect inputs and outputs
• each wire can carry only one bit, 0 or 1

CMPUT 229

Combinatorial logic
• Circuits made from collections of gates
• outputs depend only on inputs
• not on prior state
• characterized by truth table
• comparable to Boolean expression

CMPUT 229

0

A

1

B

X

1

C

Combinatorial logic

boolean expression

X = (A & ~B) | (B & C)

truth table

logic circuit

0

0

0

0

1

1

1

1

1

all three of forms

are equivalent; each can be converted to the other two.

CMPUT 229

In 1

In 2

Out

0

0

0

0

1

1

1

0

1

1

1

1

CMPUT 229

Combinatorial logic
• Many parts of a computer are constructed of combinatorial logic
• arithmetic logic unit (ALU)
• performs many kinds of arithmetic and bitwise logical operations
• multiplexer and decoder
• direct bit traffic between components

CMPUT 229

• A combinatorial circuit that adds binary values
• according to this truth table

Carry is equivalent to AND

Sum is equivalent to XOR

CMPUT 229

Carry

In 1

In 2

Sum

This circuit can add two binary digits and is called a half-adder

CMPUT 229

1

5

3

+

4

3

8

1

9

1

. . . because to add multi-digit numbers each column requires two additions

CMPUT 229

in 1

This circuit is called a full adder and can add three binary digits

in 2

carry out

carry in

This OR gate combines the carries from the two half-adders

sum

CMPUT 229

A3

A2

A1

A0

B3

B2

B1

B0

out3

out2

out1

out0

CMPUT 229

0

1

0

1

0

1

1

0

A3

A2

A1

A0

0101

+ 0110 = 1011

B3

B2

B1

B0

0

1

0

0

0

0

1

0

1

0

0

0

0

1

0

1

1

out3

out2

out1

out0

CMPUT 229

• So named because gate results (including carries) propagate (“ripple”) from LSB to MSB
• right to left
• corresponds to how humans add numbers with pen and paper
• More sophisticated, faster, adder circuits exist that can create the higher order carries more quickly

CMPUT 229

A3-A0

B3-B0

Adders (and other arithmetic circuits) are usually drawn like this in block diagrams

inputs

+

output

collections of parallel, related wires like this are known as buses; they carry multi-bit values between components

out3-out0

CMPUT 229

Arithmetic
• Computers need to do more than just addition
• arithmetic: + – * / %
• logic: & | ~ << >>
• Need a circuit that can select operation to perform

CMPUT 229

Multiplexer (Mux)

B

A

A

B

S

S

C

C

CMPUT 229

Arithmetic Logic Unit (ALU)

A

B

more operations here

. . .

op 0

op 1

op 2

op 3

+

*

&

<<

Multiplexer: a combinatorial circuit which selects exactly one input

0

1

2

3

..

MUX

op

op selects operation:

0 = add, 1 = multiply, ...

out

CMPUT 229

Arithmetic Logic Unit (ALU)

A = 15

B = 2

for example: compute 15 << 2

more operations here

. . .

op 0

op 1

op 2

op 3

+

*

&

<<

other results also computed but ignored by multiplexer

0

1

2

3

..

MUX

op = 3

out = 60

CMPUT 229

How to design a logic circuit for a given truth table?
• For each row in the truth table whose output value is 1, construct a logic expression of the conjunction of all input column values
• Form the disjunction of all conjunctive formulas obtained in step 1
• Simplify the disjunction obtained from Step 2
• Construct the logic circuit from the simplified logical expression in Step 3

CMPUT 229

Memory
• Computers need memory for storage
• Different kinds of memory distinguished by speed, size, cost and proximity to CPU
• main memory
• slowish, huge, cheap, far from CPU
• typical size 109 bits
• cache
• fast, medium-sized, expensive, near to CPU
• typical size 106 bits
• registers
• extremely fast, tiny, very expensive, located on CPU
• in MIPS (32 GPRs)×32 bits = 1024 bits

CMPUT 229

Memory
• The smallest piece of memory is a single binary digit (bit)
• can hold 0 or 1 only
• A one-bit memory is called a flip-flop or a data latch
• because its value can flip and flop between 0 and 1
• because it can latch onto a data value and store it

CMPUT 229

Flip-flop
• Flip-flop needs two operation modes
• write: store (memorize) a value
• Also need
• data in
• for telling flip-flop what value to store (0 or 1)
• used only when writing
• data out
• for finding out what value flip-flop currently contains (0 or 1)

CMPUT 229

Flip-flop
• Flip-flop can be implemented with gates
• Not combinatorial logic
• because current output may depend on previous state
• Example of sequential logic
• current output depends on inputs and prior output

CMPUT 229

Flip-flop

NOR gate: OR gate followed by NOT gate

data in

data out

CMPUT 229

Flip-flop: writing

Try changing data in to 0 and watch data out

data in = 1

1

1

0

1

1

0

1

0

1

0

data out = 1

1

when read/write = 1, data out = data in

CMPUT 229

when read/write = 0, no signals in box can change, data out holds value regardless of data in

data in = ?

?

0

0

0

1

0

?

?

1

0

data out = 1

0

CMPUT 229

Flip-flop

D = data in

Q = data out

Flip-flops are often drawn like this in block diagrams

D

Q

CK

CK is read/write (“clock” because this input is often connected to computer’s processor clock)

CMPUT 229

Flip-flop using NAND Gates

Data In

Data Out

CMPUT 229

Memory
• Memory can store many bits independently
• many flip-flops
• Need to identify which bit (flip-flop) to read or write
• Give each flip-flop a unique number (address)

CMPUT 229

Memory

Decoder: feeds input to selected output, 0 to all others

data in

D

Q

CK

D

Q

data out

0

0

rd/wr

1

1

CK

DEC

MUX

2

2

3

3

D

Q

...

...

CK

D

Q

CK

. . . millions more flip-flops . . .

CMPUT 229

Memory: writing

Writing value 1 to flip-flop at address 2

data in

?

D

Q

1

0

CK

?

D

Q

data out

0

0

rd/wr

0

1

1

CK

DEC

MUX

2

2

1

1

3

3

D

Q

1

...

...

1

CK

?

D

Q

0

CK

2

. . . millions more flip-flops . . .

CMPUT 229

data in

D

Q

0

CK

D

Q

data out

0

0

rd/wr

0

1

1

CK

DEC

MUX

2

2

0

1

1

3

3

D

Q

1

...

...

0

CK

D

Q

0

CK

2

. . . millions more flip-flops . . .

CMPUT 229

Memory
• Memory usually operates in terms of bytes (8 bits), not single bits
• Repeat memory circuit eight times
• connect each memory circuit to one of the eight lanes of the data bus
• reads and writes occur in parallel for each bit in byte

CMPUT 229

Central Processing Unit (CPU)
• Coordinates all computer’s components according to program being run
• Contains
• registers
• ALU
• program counter (PC)
• instruction register (IR)
• copy of current instruction
• control logic
• Runs programs using fetch-(decode)-execute cycle

CMPUT 229

CPU

IR

Registers

out

Memory

in

Control

logic

rd/wr

ALU

PC

CMPUT 229

Fetch-execute cycle: fetch

IR

Registers

out

Stage 1 of fetch-execute cycle: instruction at address pointed to by PC is fetched into IR

Memory

in

Control

logic

rd/wr

ALU

PC

CMPUT 229

Fetch-execute cycle: decode

IR

Registers

out

Memory

Stage 2 of fetch-execute cycle: instruction (now in IR) is decoded by control logic to determine which operation to perform

in

Control

logic

rd/wr

ALU

PC

CMPUT 229

Fetch-execute cycle: execute

IR

Registers

out

Stage 3 of fetch-execute cycle: CPU performs the operation (for example, arithmetic on two registers)

Memory

in

Control

logic

rd/wr

ALU

PC

CMPUT 229

Fetch-execute cycle: update PC

IR

Registers

out

Stage 4 of fetch-execute cycle: PC’s value is updated to point to the next instruction

Memory

in

Control

logic

rd/wr