Gates and Circuits

Nell Dale & John Lewis (adaptation by Erin Chambers and Michael Goldwasser)

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Computers and Electricity

A gate is a device that performs a basic operation on electrical signals

Gates are combined into circuits to perform more complicated tasks

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Gates

Let’s examine the processing of the following six types of gates

NOT

AND

OR

XOR

NAND

NOR

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Describing Gates and Circuits

There are three different, but equally powerful, notational methods
for describing the behavior of gates
and circuits

Boolean expressions

logic diagrams

truth tables

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Describing Gates and Circuits

Boolean algebra: expressions in this algebraic notation are an elegant and powerful way to demonstrate the activity of electrical circuits

Basic propositional statements are unambiguously either True or False

Operations such as AND or NOT are then performed on these values

A gate is simply a mechanical way to perform such a boolean operation

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Describing Gates and Circuits

Logic diagram: a graphical representation of a circuit

Each type of gate is represented by a specific graphical symbol

Truth table: defines the function of a gate by listing all possible input combinations that the gate could encounter, and the corresponding output

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NOT Gate

a NOT gate accepts one input value
and produces one output value

a NOT gate is sometimes referred to as an inverter because it inverts the input value

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AND Gate

An AND gate accepts two input signals

If the two input values for an AND gate are both 1, the output is 1; otherwise, the output is 0

Figure 4.2 Various representations of an AND gate

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OR Gate

If the two input values are both 0, the output value is 0; otherwise, the output is 1

Figure 4.3 Various representations of a OR gate

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XOR Gate

XOR, or exclusive OR, gate

An XOR gate produces 0 if its two inputs are the same, and a 1 otherwise

Note the difference between the XOR gate
and the OR gate; they differ only in one
input situation

When both input signals are 1, the OR gate produces a 1 and the XOR produces a 0

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XOR Gate

Figure 4.4 Various representations of an XOR gate

NAND and NOR Gates

The NAND and NOR gates are essentially the opposite of the AND and OR gates, respectively

Figure 4.5 Various representations of a NAND gate

Figure 4.6 Various representations of a NOR gate

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Gates with More Inputs

Gates can be designed to accept three or more input values

A three-input AND gate, for example, produces an output of 1 only if all input values are 1

Figure 4.7 Various representations of a three-input AND gate

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Constructing Gates

A transistor is a device that acts, depending on the voltage level of an input signal, either as a wire that conducts electricity or as a resistor that blocks the flow of electricity

A transistor has no moving parts, yet acts like
a switch

It is made of a semiconductor material, which is neither a particularly good conductor of electricity, such as copper, nor a particularly good insulator, such as rubber

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jasonm:

Redo 4.8 (crop)

Constructing Gates

A transistor has three terminals

A source

A base

An emitter, typically connected to a ground wire

If the electrical signal is grounded, it is allowed to flow through an alternative route to the ground (literally) where it can do no harm

Figure 4.8 The connections of a transistor

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Constructing Gates

It turns out that, because the way a transistor works, the easiest gates to create are the NOT, NAND, and NOR gates

Figure 4.9 Constructing gates using transistors

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Circuits

Two general categories

In a combinational circuit, the input values explicitly determine the output

In a sequential circuit, the output is a function of the input values as well as the existing state of the circuit

As with gates, we can describe the operations
of entire circuits using three notations

Boolean expressions

logic diagrams

truth tables

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Combinational Circuits

Gates are combined into circuits by using the output of one gate as the input for another

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Combinational Circuits

Because there are three inputs to this circuit, eight rows are required to describe all possible input combinations

This same circuit using Boolean algebra:

(AB + AC)

jasonm:

Redo to get white space around table (p100)

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Now let’s go the other way; let’s take a Boolean expression and draw

Consider the following Boolean expression: A(B + C)

jasonm:

Redo table to get white space (p101)

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Now compare the final result column in this truth table to the truth table for the previous example

They are identical

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Properties of Boolean Algebra

jasonm:

Redo table (p101)

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Adders

At the digital logic level, addition is performed in binary

Addition operations are carried out
by special circuits called, appropriately, adders

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Adders

The result of adding two binary digits could produce a carry value

Recall that 1 + 1 = 10
in base two

A circuit that computes the sum of two bits
and produces the correct carry bit is called a half adder

jasonm:

Redo table (p103)

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Adders

Circuit diagram representing
a half adder

Two Boolean expressions:

sum = A  B

carry = AB

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Adders

A circuit called a full adder takes the carry-in value into account

Figure 4.10 A full adder

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Multiplexers

Multiplexer is a general circuit that produces a single output signal

The output is equal to one of several input signals to the circuit

The multiplexer selects which input signal is used as an output signal based on the value represented by a few more input signals, called select signals or select control lines

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Multiplexers

The control lines S0, S1, and S2
determine which of eight other input lines
(D0 through D7) are routed to the output (F)

Figure 4.11 A block diagram of a multiplexer with three select control lines

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Circuits as Memory

Digital circuits can be used to store information

These circuits form a sequential circuit, because the output of the circuit is also used as input to the circuit

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Circuits as Memory

An S-R latch stores a single binary digit
(1 or 0)

There are several ways an S-R latch circuit could be designed using various kinds of gates

Figure 4.12 An S-R latch

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Circuits as Memory

The design of this circuit guarantees that the two outputs X and Y are always complements of each other

The value of X at any point in time is considered to be the current state of the circuit

Therefore, if X is 1, the circuit is storing a 1; if X is 0, the circuit is storing a 0

Figure 4.12 An S-R latch

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Integrated Circuits

An integrated circuit (also called a chip) is a piece of silicon on which multiple gates have been embedded

These silicon pieces are mounted on a plastic or ceramic package with pins along the edges that can be soldered onto circuit boards or inserted into appropriate sockets

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Integrated Circuits

Integrated circuits (IC) are classified by the number of gates contained in them

jasonm:

Redo table (p107)

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Integrated Circuits

Figure 4.13 An SSI chip contains independent NAND gates

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CPU Chips

The most important integrated circuit
in any computer is the Central Processing Unit, or CPU

Each CPU chip has a large number
of pins through which essentially all communication in a computer system occurs

Here's an approximate order I do in such a class:

* Have people assemble a simple OR gate, with two switches and one

output bulb. (this is to get used to the software)

* Have people assemble a 3-way OR gate, and play with the

combinations.

* Have people build verify some identities by using the input

switches to simultaneously feed both circuits, each with a

separate lightbulb.

-- A(B+C) == (AB) + (AC)

-- (AB)' = A' + B'

* Sometimes, I have them load my half-adder, and I discuss it a bit

while they play with it. Possibly even looking at the full-adder.

* Then I switch to sequential cirucits and feedback loops.

-- first looking at feedback.dat

-- then looking at feedback2.dat

-- then looking at SRlatch.dat

I have some other premade circuits available as well if you want

(e.g. a multiplexor, a latch). But I'm not sure whether to make time

for them.

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Jones & Bartlett Publishers

2002-06-03T22:19:42