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Textbook Chapter 3

Digital Logic: From Transistors to Gates

Textbook Chapter 3

Transistor: building block of computers

Microprocessors contain tons of transistorsIntel Montecito (2005): 1 72 billion

The Transistor

Intel Montecito (2005): 1.72 billion

Intel Pentium 4 (2000): 48 million

IBM PowerPC 750FX (2002): 38 million

IBM/Apple PowerPC G5 (2003): 58 million

Intel 4004 (1971): 2500

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The Transistor: Past and Present

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Moore’s Law“The number of active components per chip will double every 18 months.”

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What Is a Transistor? Active device – signals are amplified, not

destroyed

A switch which can close between the source A switch, which can close between the source and the drain

Changing the voltage of the gate lets you change the current flow between the source and drain (closing or opening the switch)

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GPU Speed Compared to CPU

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Metal-Oxide-Semiconductor transistor

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How big is a transistor? If a CPU die were as big as this whole

classroom…

A transistor would be A transistor would be…

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Logically, each transistor is used as a switch

Combined to implement logic functions

AND OR NOT

What is a transistor?

AND, OR, NOT Combined to build higher-level structures

Adder, multiplexer, decoder, register, … Combined to build a processor

LC-3

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LC 3

n-type MOS transistorn-type MOS (nMOS)

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

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

Gate = 1

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Gate = 0Terminal #2 must be

connected to GND (0V).

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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)(switch open)

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

Gate = 1

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Gate = 0

Terminal #1 must beconnected to +2.9V in

this example.

Simple switch circuit

Switch open:No current through

circuitcircuitLight is offVout is +2.9V

Switch closed:Short circuit across

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switchCurrent flowsLight is onVout is 0V

Switch-based circuits can easily represent two states:

on/off, open/closed, voltage/no voltage.

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Digital Values for Analog Signals Use the switch behavior of MOS transistors to

implement logical functions: AND, OR, NOT

Digital symbols: Digital symbols:

We assign a range of analog voltages to each digital (logic) symbol

Assignment of voltage ranges depends on electrical properties of transistors being used

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CMOS circuit CMOS is Complementary Metal Oxide Semiconductor

Uses both n-type and p-type MOS transistors

p-type (pMOS) p type (pMOS)

Attached to + voltage

Pulls output voltage UP when input is zero

n-type (nMOS)

Attached to GND

Pulls output voltage DOWN when input is one

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Inverter (NOT gate)

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In Out0 V 2.9 V

2.9 V 0 V

In Out0 1

1 0

Truth Table The most basic

representation of a logic function

Lists the output for all possible input combinations

How many rows of the truth table needed? 2#inputs

X Y …A B …

OutputsInputs

X Y …A B …

OutputsInputs

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Truth Table: Inverter Inverted signals are

denoted with an overbar

Or with a prime symbol

Input Output

A Y = A’

A’

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Truth Table: AND Gate The result of an AND

operation is 1 if and only if all inputs are 1

Inputs Output

A B Y = A · B

Depict AND by the multiplication symbol

A·B

Or by lumping the signals together

AB

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AB

We don’t really build these gates…

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NAND gate (NOT-AND)

A B C

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0 0 1

0 1 1

1 0 1

1 1 0Note: Parallel structure on top, serial on bottom.

AND gate

A B C

0 0 0

0 1 0

1 0 0

1 1 1

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Add an inverter to a NAND.

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Truth Table: OR Gate The result of an OR

operation is 1 if and only if any inputs are 1

Inputs Output

A B Y = A + B

Depict OR by the addition symbol

A+B

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NOR Gate: NOT-OR

A B C

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0 0 1

0 1 0

1 0 0

1 1 0

Note: Serial structure on top, parallel on bottom.

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OR gate

A B C

0 0 0

0 1 1

1 0 1

1 1 1

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Add an inverter to a NOR gate.

About the Little Circle… The little circle is what inverts

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Sum of Products How do you get from a truth table to a logic

expression?

Sum of products is standard way of Sum of products is standard way of synthesizing simple circuits

Procedure:1. Find the rows with the ‘1’ output2. Write the product-form expression for the inputs

in that row (0=inverted, 1=normal)

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( , )3. Combine the products in step 2 into a sum (OR

the results of step 2)

Sum of Products1. Find the rows with the ‘1’

output2. Write the product-form

fA B Y

expression for the inputs in that row (0=inverted, 1=normal)

3. Combine the products in step 2 into a sum (OR the results of step 2)

0 0 0

0 1 1

1 0 1

1 1 0

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Logic MinimizationA B C Y

0 0 0 0

Example

0 0 1 0

0 1 0 1

0 1 1 1

1 0 0 1

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1 0 1 0

1 1 0 1

1 1 1 0

Synthesis of AOI Gates AOI means AND-OR-Invert

Truth table to a AOI gate (transistor-level)

R ll Recall: PMOS (with the bubbles) on top

NMOS (no bubbles) on bottom

Series structure makes AND

Parallel structure makes OR

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Synthesis of AOI Gates Method 1: sum of products

Start with ones

Build “pull-up” branch first Build pull up branch first

Method 2: sum of products

Start with zeros

Build “pull-down” branch first

Use inverters to complement inputs and output

Which method?

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Depends on the truth table

More ones: method 1

More zeros: method 2

Synthesis of AOI Gates Method 1: Sum of products for Y

Cover the ones

Build “pull-up” branch first using inverted inputs Build pull up branch first using inverted inputs

Derive “pull-down” branch as the dual of pull-up branch

Y=VDD

A B C Y

0 0 0 0

0 0 1 0

0 1 0 1

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Y

VSS

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

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Synthesis of AOI Gates Method 2: Sum of products for Y’

Cover the zeros

Build “pull-down” branch first, using asserted inputs Build pull down branch first, using asserted inputs

Derive “pull-up” branch as a dual of the pull-down branch

Y’=VDD

A B C Y

0 0 0 0

0 0 1 0

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Y

VSS

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

Recommended exercisesCombinational circuits

Ex 3.5, 3.6, 3.7, 3.8, 3.9

E 3 11 3 12 3 18 Ex 3.11, 3.12, 3.18

Ex 3.20, 3.22, 3.23, 3.24 with TA/Tut

Ex 3.30, 3.31, 3.35

Ex 3.44

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