Transcript Some basic electronics and truth tables

Some basic electronics
and truth tables
Some material on truth
tables can be found in
Chapter 3 of Digital
Principles (Tokheim)
Logic  Digital Electronics

In Logic, one refers to Logical
statements (propositions which
can be true or false).


What a computer scientist would
represent by a Boolean variable.
In Electronics, one refers to inputs
which will be high or low.
Boola Boola!

The expression
(Booleans) and the
rules for combining
them (Boolean
algebra) are named
after George Boole
(1815-64), a British
mathematician.
Boolean operators



AND: when two or more Boolean
expressions are ANDed, both must be
true for the combination to be true.
OR: when two or more Boolean
expressions are ORed, if either one or the
other or both are true, then the
combination is true.
NOT: takes one Boolean expression and
yields the opposite of it, true  false and
vice versa.
Representations of
Standard Boolean Operators
Boolean
algebra
expression
NOT A
A´
A AND B
AB
A OR B
A NOR B
A+B
AB
(A+B)´
A NAND B
(AB)´
A XOR B
Gate symbol
Our Notation

NOT is represented by a prime or an
apostrophe.


OR is represented by a plus sign.


A’ means NOT A
A + B means A OR B
AND is represented by placing the two
variables next to one another.


AB means A AND B
The notation is like multiplication in regular
algebra since if A and B are 1’s or 0’s the only
product that gives 1 is when A and B are both
1.
Other Notations
means NOT A
 A means NOT A
 AB means A OR B
 A&B means A AND B
 Tokheim uses the overbar notation
for NOT, but we will use the prime
notation because it is easier to type.

Other vocabulary
We will tend to refer to A and B as
“inputs.” (Electronics)
 Another term for them is “Boolean
variables.” (Programming)
 Still another term for them is
“propositions.” (Logic)
 And yet another term for them is
“predicates.” (Logic and grammar)

(AB)’  A’B’
A
B
0
0
1
0
1
0
0
0
0
1
1
1
1
1
1
0
Note that the
output is different
AB (AB)’
A
B
A’
B’
A’B’
0
0
0
1
1
1
1
0
1
0
1
1
0
1
0
0
1
0
0
0
A Truth Table

A Truth table lists all possible
inputs, that is, all possible values
for the propositions.


For a given numbers of inputs, this is
always the same.
Then it lists the output for each
possible combination of inputs.

This varies from situation to situation.
The true one
Traditionally we take a 1 to
represent true and a 0 to
represent false
 In addition, we will usually
interpret a high voltage as a true
and a low voltage as a false

Generating Inputs


The truth-table inputs consist of all the
possible combinations of 0’s and 1’s for
that number of inputs.
One way to generate the inputs for is to
count in binary.



For two inputs, the combinations are 00, 01,
10 and 11 (binary for 0, 1, 2 and 3).
For three inputs, the combinations are 000,
001, 010, 011, 100, 101, 110 and 111 (binary
for 0, 1, 2, 3, 4, 5, 6 and 7).
For n inputs there are 2n combinations
(rows in the truth table).
Expressing truth tables



Every truth table can be expressed in
terms of the basic Boolean operators
AND, OR and NOT operators.
The circuits corresponding to those truth
tables can be build using AND, OR and
NOT gates.
The input in each line of a truth table can
be expressed in terms of AND’s and
NOT’s.
A B A’B’
0
0
1
1
0
1
0
1
1
0
0
0
A
0
0
1
1
A
0
0
1
1
B
0
1
0
1
AB’
0
0
1
0
A
0
0
1
1
B
0
1
0
1
A’B
0
1
0
0
B
0
1
0
1
AB
0
0
0
1
Note that
these
expressions
have the
property
that their
truth table
output has
only 1 row
with a 1.
In a sense, each line has an
expression
Input A Input B
Expression
0
0
(NOT A) AND (NOT B)
A´B´
0
1
(NOT A) AND B
A´B
1
0
A AND (NOT B)
AB´
1
1
A AND B
AB
It’s true; it’s true

The following steps will allow you to
generate an expression for the output of
any truth table.


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Take the true (1) outputs
Write the expressions for that input line (as
shown on the previous slide)
Then feed all of those expressions into an OR
gate
Sometimes we have multiple outputs
(e.g. bit addition had a sum output and a
carry output). Then each output is
treated separately.
Example: Majority Rules
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
Majority
0
0
0
1
0
1
1
1
If two or
more of the
three inputs
are high,
then the
output is
high.
Row Expressions
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
Row expressions
A’B’C’
A’B’C
A’BC’
A’BC
AB’C’
AB’C
ABC’
ABC
The
highlighted
rows
correspond
to the high
outputs.
Majority rules

A´BC + AB´C + ABC´ + ABC
NOTs
OR
ANDs
Electronics Workbench
Where the logic gates are
EWB
gates
chips
Be careful above the NAND chip is
the NOT gate.
NOT Gate and NOT Chip
Gates and Chips
EWB gates focus purely on the logic
(the inputs and outputs).
 EWB chips, while simulated, are
closer to a real-world device

Chips need power: VCC must be
connected to high (5 volts) and GND
must be connected to ground.
 Another complication is that the NOT
chip houses four NOT gates.

HELP! (Right click choose Help
from the menu)
Connectors: allow up to four wires
to meet
Resistors:
Component Properties: Right click
Value and units
Kilo-Ohm: Unit of resistance
Value of resistance
Switch
Battery and ground
Battery
Ground
Voltmeter, ammeter, 7-segment
display(s)
Venn Diagram
A Venn diagram is a pictorial
representation of a truth table.
 Venn diagrams come from set
theory.
 The correspondence between set
theory and logic is that either one
belongs to a set or one does not,
so set theory and logic go
together.

Venn (Cont.)
Does not
belong to
set  False
Belongs to
set  True
Overlapping sets
A
true,
but B
false
A
and
B
true
B
true,
but A
false
A false
and B
false
Ohm’s Law




V = I R, where
V is voltage: the amount of energy per
charge.
I is current: the rate at which charge
flows, e.g. how much charge goes by in a
second.
R is resistance: the “difficulty” a charge
encounters as moves through a part of a
circuit.
Circuit



A circuit is a closed path along which
charges flow.
If there is not a closed path that allows
that the charge can get back to where it
started (without retracing its steps), the
circuit is said to be “open” or “broken.”
The path doesn’t have to be unique;
there may be more than one path.
An analogy
A charge leaving a battery is like
you starting the day after a good
night’s rest; you are full of energy.
 Being the kind of person you are,
you will expend all of your energy
and collapse utterly exhausted into
bed at the end of the day; the
charge uses up all of its energy in
traversing a circuit.

Analogy (cont.)


You look ahead to the tasks of the day
and divide your energy accordingly, the
more difficult the task the more of your
energy it requires (resistors in series).
The tasks are resistors, so more energy
(voltage) is used up working through the
more difficult tasks (higher resistances).

The higher the resistance, the greater the
voltage drop (energy used up) across it.
One charge among many





You are just one charge among many.
If the task at hand is very difficult (the
resistance is high), not many will do it
(the current is low);
V=IR, if R is big, I must be small.
If the task is easy, everyone rushes to do
it.
V=IR, if R is small, I will be large.
More energetic
If we had more energy, more of us
would attempt a given task.
 V=IR, if V is bigger, I is bigger.
 If we are all tired out, few of us will
perform even the most basic task.
 V=IR, if V is small, I will be small.

Given the choice
Given the choice between a difficult
task and an easy task, most will
choose the easier task.
 If there is more than one path, most
take the “path of least resistance”
(resistors in parallel).
