Chapter 3Digital LogicStructures

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Transistor: Building Block of ComputersMicroprocessors contain millions of transistors

• Intel Pentium II: 7 million• Compaq Alpha 21264: 15 million• Intel Pentium III: 28 million• Intel Pentium4: 55 Million• Intel Core 2 Duo: 291 Million

Logically, each transistor acts as a switchCombined to implement logic functions

• AND, OR, NOTCombined to build higher-level structures

• Adder, multiplexor, decoder, register, …Combined to build processor

• LC-3

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Transistors

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First transistor: Bell Labs in 1947; developed byJ. Bardeen, W. Shockley & W. Brattain

A 2011 processor with 1.17 billion transistors positioned in 240 sq. millimeters

http://www.brew-wood.co.uk/computers/transistor.htm

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Simple Switch Circuit

Switch open:• No current through circuit• Light is off• Vout is +2.9V

Switch closed:• Short circuit across switch• Current flows• Light is on• Vout is 0V

Switch-based circuits can easily represent two states:on/off, open/closed, voltage/no voltage.

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N-type MOS TransistorMOS = Metal Oxide Semiconductor

• two types: N-type and P-typeN-type

• when Gate has positive voltage,short circuit between #1 and #2(switch closed)

• when Gate has zero voltage,open circuit between #1 and #2(switch open) Gate = 1

Gate = 0Terminal #2 must be

connected to GND (0V).

AnimationDrain

SourceGND

GND

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P-type MOS TransistorP-type is complementary to N-type

• when Gate has positive voltage,open circuit between #1 and #2(switch open)

• when Gate has zero voltage,short circuit between #1 and #2(switch closed)

Gate = 1

Gate = 0Terminal #1 must beconnected to +2.9V.

Source

Drain

GND

+2.9V

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CMOS CircuitComplementary MOSUses both N-type and P-type MOS transistors

• P-typeAttached to + voltagePulls output voltage UP when input is zero

• N-typeAttached to GNDPulls output voltage DOWN when input is one

For all inputs, make sure that output is either connected to GND or to +,but not both!

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Inverter (NOT Gate)

In Out0 V 2.9 V

2.9 V 0 V

In Out0 11 0

Truth table

High Voltage

Ground

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Logical Operations

In 1850, George Boole developed Boolean Algebra showing that all logical functions can be performed with just 3 operations (AND, OR & NOT). In 1937, Claude Shannon showed that Boolean Algebra could be applied to circuit design. 3-9

http://www.computerhistory.org/revolution/digital-logic/12/269

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NOR Gate (NOT OR)

A B C0 0 10 1 0

1 0 0

1 1 0

Note: Serial structure on top, parallel on bottom.

High Voltage

Ground

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OR Gate

Add inverter to NOR.

A B C0 0 00 1 1

1 0 1

1 1 1

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NAND Gate (NOT-AND)

A B C0 0 10 1 1

1 0 1

1 1 0

Note: Parallel structure on top, serial on bottom.

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AND Gate

Add inverter to NAND.

A B C0 0 00 1 0

1 0 0

1 1 1

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Basic Logic Gates

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More than 2 Inputs?AND/OR can take any number of inputs.

• AND = 1 if all inputs are 1.• OR = 1 if any input is 1.• Similar for NAND/NOR.• An AND gate with k inputs is called and ANDk gate (e.g., an AND2, AND3,

etc).Can implement AND3 with multiple AND2 gates,or with single transistor circuit.

• AND/OR are associative and commutative -- combine in any order.

ACB

C

AB

C

AB

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Logical CompletenessCan implement ANY truth table with AND, OR, NOT.

A B C D0 0 0 00 0 1 0

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 1

1 1 0 0

1 1 1 0

A B C

D

1.AND combinations

that yield a "1" in the truth table.

Put a “bubble” (inverter) for every 0, a straight-in for every 1 in a row

2. OR the resultsof the AND gates.

If there are N 1’s, there will be N and gates; the or gate will have N inputs

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DeMorgan's LawConverting AND to OR (with some help from NOT)Consider the following gate:

A B0 0 1 1 1 00 1 1 0 0 1

1 0 0 1 0 1

1 1 0 0 0 1

BA BA BA

Same as A OR B!

To convert AND to OR (or vice versa),

invert inputs and output.

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SummaryMOS transistors are used as switches to implementlogic functions.

• N-type: connect to GND, turn on (with 1) to pull down to 0• P-type: connect to +2.9V, turn on (with 0) to pull up to 1

Basic gates: NOT, NOR, NAND• Logic functions are usually expressed with AND, OR, and NOT

Properties of logic gates• Completeness

can implement any truth table with AND, OR, NOT• DeMorgan's Law

convert AND to OR by inverting inputs and output