Building a Computer: Understanding Switches, Gates, and Logic Circuits, Study notes of Programming Languages

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CS 350

Computer Organization & Assembly

Language Programming

Spring 2009

Notes 04, 2009-02-

1

Basic Switches and Gates

• You need a lot of switches

• Switches are combined to implement

logic functions

- AND, OR, NOT, NAND, NOR ….

• Which are combined to build higher-level

structures

- Adder, multiplexor, decoder, register, memory

- [We'll study these later]

How to Build a Computer

3

How to Build a Computer

• These higher-level structures are used to

build a processor.

- Decode and act on instructions

+ Keep track of where next instruction is

- Do arithmetic

- Do tests ("is this = 0?")

- Go to some instruction

+ Change where next instruction comes from

CMOS (Complementary

MOS)

• The p- and n-type transistors are

complements of each other.

• CMOS circuit: Uses p- and n-type

transistors.

• We combine them to implement logical

functions AND, OR, NOT

7

Inverter (NOT Gate)

• NOT^ gate: Device that takes an input

voltage and produces an output voltage

so that the voltages of the input/output

are opposite.

- Input: 0V; Output: 2.9V

- Input: 2.9V; Output: 0V

- Treat 0V, 2.9V as representing bits 0, 1 resp.

- Gate also needs to be connected to power

and ground.

Inverter Uses 2 Switches

• The bar at the top is

connected to voltage, the arrow at bottom is connected to ground.

• If input has voltage, we

want output to be connected to ground.

• If input is connected to

ground, we want output to be connected to voltage. 9

Inverting Input 0

• If the input is 0, then the

p-type switch on top is closed but the n-type switch below is open.

• Output is connected to

voltage and not to ground.

• (If output were

connected to both voltage and input, you'd have a short circuit.)

Other Gates

• Similar but more-complicated collections

of switches can be used to form devices

that take 2 inputs and produce 1 output

that is the AND, OR, NAND, or NOR of

the inputs

• On next slide, note use of + for OR,

juxtapositioning for AND, and overbar for

NOT. (A common notation.)

13

Basic Logic Gates

Logic Gate Symbols

15

• Note use of small circle to show negation

DeMorgan's Laws

• DeMorgan's laws show how^ AND/OR,

NAND/NOR, and NOT are interrelated:

- A^ B =^ A^ + B^ [NOT^ (A^ AND B) = NOT^ A^ OR NOT^ B]

- A^ + B =^ A^ B^ [NOT^ (A^ OR B) = NOT^ A^ AND NOT^ B]

- To negate an^ AND, negate the operands and

change the AND to an OR

- To negate an OR, negate the operands and

change the OR to an AND

Combining Gates to Form

Circuits

19

Two Kinds of Circuits

• We can identify two kinds of logic circuits

- Combinatorial logic circuits

+ Are pure functions

+ Outputs depend only on the inputs

+ Are stateless = have no internal state

- Sequential logic circuits

+ Do have an internal state, which can change

+ Output depends on state and inputs

Some Combinatorial Logic

Circuits

• Decoder (has n inputs, 2

n outputs)

- The n inputs correspond to a binary number

- Each output corresponds to one binary

number.

• Example:^ A^ 2-bit decoder has inputs^ A, B

and 4 outputs

- Think of^ A, B as forming a 2-bit number, then

output number (A followed by B) will be 1

21

2-Bit Decoder

Note circles

on inputs;

shorthand for

NOT gates

Half Adder

25

• A^ Half-Adder takes two

bits and produces two bits of output, a carry- out bit and a sum bit.

• If we want to add two

binary numbers, the rightmost column can use a half-adder.

• But the other columns

may have a carry-in. A B Cout Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0

Full Adder

• A^ full adder takes two

bits to add, plus a carry- in bit, and it produces a carry out bit and sum.

• A^ half adder can be

done as a full adder with a 0 for the carry in. A B Cin Cout Sum 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1

Four-bit Adder

• A!A"A#A$^ + B!B"B#B$^ = Cout^ S!S"S#S$.

27

Any Combinatorial Circuit

• A^ combinatorial circuit

takes some input bits and produces some output bits.

• Each row corresponds

to the AND of

- A^ or its negation

- B or its negation

- C or its negation

A B C Selector 0 0 0 A B C 0 0 1 A B C 0 1 0 A B C 0 1 1 A B C 1 0 0 A B C 1 0 1 A B C 1 1 0 A B C 1 1 1 A B C