Some basic electronics and truth tables Some material on truth tables can be found in Chapter 1 of Computer Systems Organization & Architecture (Carpinelli)

Logic  Digital Electronics In Logic, one refers to Logical statements (propositions which can be true or false) What a computer scientist would call a Boolean variable In Electronics, one refers to inputs which will be high or low

Boole 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 Gate symbol NOT A A´ A AND B AB A OR B A+B A XOR B AB A NOR B (A+B) ´ A NAND B (AB) ´

A Truth Table A Truth table lists all possible inputs, that is, that is all possible values for the statements 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 It is also usual to interpret a high voltage as a true and a low voltage as a false

Generating Inputs Generating the inputs for a truth table is like counting in binary If there are two inputs, the combinations are 00, 01, 10 and 11 If there are three inputs, the combinations are 000, 001, 010, 011, 100, 101, 110, 111 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

Line by Line Input A Input B Expression A´B ´ 1 A´B AB ´ AB (Not A) AND (NOT B) A´B ´ 1 (Not A) AND B A´B A AND (NOT B) AB ´ A AND B AB

It’s true; it’s true 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

Example A B C Majority 1

Row Expressions A B C Row expressions A’B’C’ 1 A’B’C A’BC’ A’BC AB’C’ A’B’C’ 1 A’B’C A’BC’ A’BC AB’C’ AB’C ABC’ ABC

Majority rules A´BC + AB´C + ABC´ + ABC NOTs OR ANDs

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 false and B false B true, but A false A and B true A true, but 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)