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Some basic electronics and truth tables

Some material based on Chapters 3 through 5 of Digital Principles (Tokheim)

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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.

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Boola Boola! The expression

(Booleans) and the rules for combining them (Boolean algebra) are named after George Boole (1815-64), a British mathematician.

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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.

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Representations of Standard Boolean Operators

Boolean algebra

expressionGate symbol

NOT A A´

A AND B AB

A OR B A+B

A XOR B AB

A NOR B (A+B)´

A NAND B (AB)´

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Our Notation NOT is represented by a prime or an

apostrophe. A’ means NOT A

OR is represented by a plus sign. 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.

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Other Notations Ā means NOT A A means NOT A AB 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.

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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)

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(AB)’ A’B’

A B AB (AB)’

0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

A B A’ B’ A’B’

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 0 0 0

Note that the output is different

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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.

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The true one Traditionally we take a 1 to

represent true and a 0 to represent false. This is just a convention.

In addition, we will usually interpret a high voltage as a true and a low voltage as a false.

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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).

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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.

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A B A’B’

0 0 1

0 1 0

1 0 0

1 1 0

A B A’B

0 0 0

0 1 1

1 0 0

1 1 0

A B AB’

0 0 0

0 1 0

1 0 1

1 1 0

A B AB

0 0 0

0 1 0

1 0 0

1 1 1

Note that these expressions have the property that their truth table output has only one row with a 1.

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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

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It’s true; it’s true The following steps will allow you to generate

an expression for the output of any truth table. 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.

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Example: Majority Rules

A B C Majority

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

If two or more of the three inputs are high, then the output is high.

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Row Expressions

A B C Row expressions

0 0 0 A’B’C’

0 0 1 A’B’C

0 1 0 A’BC’

0 1 1 A’BC

1 0 0 AB’C’

1 0 1 AB’C

1 1 0 ABC’

1 1 1 ABC

The highlighted rows correspond to the high outputs.

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Sum of products Each row is represented by the ANDing of inputs

and/or inverses of inputs. E.g. A’BC Recall that ANDing is like Boolean multiplication

The overall expression for the truth table is then obtained by ORing the expressions for the individual rows. Recall that ORing is like Boolean addition E.g. A’BC + AB’C + ABC’ + ABC

This type of expression is known as a sum of products expression.

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Minterm The terms for the rows have a

particular form in which every input (or its inverse) is ANDed together.

Such a term is known an a minterm.

Minterms

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Majority rules A´BC + AB´C + ABC´ + ABC

ANDs

NOTs

OR

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Majority rules A´BC + AB´C + ABC´ + ABC

ANDs

NOTs

OR

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Another Example

A B C Out

0 0 0 1

0 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 1

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Another Example (Cont.) A’B’C’ + A’BC’ + AB’C + ABC The expression one arrives at in this way

is known as the sum of products. You take the product (the AND operation)

first to represent a given line. Then you sum (the OR operation) together

those expressions. It’s also called the minterm

expression.

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Yet Another Example

A B C Out

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

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Yet Another Example 2 (Cont.)

A’B’C + A’BC’ + A’BC + AB’C’ + AB’C + ABC’ + ABC

But isn’t that just the truth table for A+B+C?

There is another way to write the expression for truth tables.

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Another Example (Cont.)

A B C Out

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

In this approach, one looks at the 0’s instead of the 1’s.

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Another Example (Cont.) One writes expressions for the

lines which are 1 everywhere except the line one is focusing on.

Then one ANDs those expressions together.

The expression obtained this way is known as the product of sums.

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Expressions

A B C Expression

0 0 0 A + B + C

0 0 1 A + B + C’

0 1 0 A + B’ + C

0 1 1 A + B’ + C’

1 0 0 A’ + B + C

1 0 1 A’ + B + C’

1 1 0 A’ + B’ + C

1 1 1 A’ + B’ + C’

This is not yet a truth table. It has no outputs.

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Return to Example 1

A B C Out

0 0 0 1

0 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 1

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Return to Example 1 (Cont.)

The product of sums expression is (A+B+C’)(A+B’+C’)(A’+B+C)(A’+B’+C)

Each term has all of the inputs (or their inverses) ORed together.

Such terms are known as maxterms. Another name for the product of sums

expression is the maxterm expression.

Maxterm

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Comparing minterm and maxterm expressions

A B C Minterm Expression

Maxterm Expression

0 0 0 A’ B’ C’ A + B + C

0 0 1 A’ B’ C A + B + C’

0 1 0 A’ B C’ A + B’ + C

0 1 1 A’ B C A + B’ + C’

1 0 0 A B’ C’ A’ + B + C

1 0 1 A B’ C A’ + B + C’

1 1 0 A B C’ A’ + B’ + C

1 1 1 A B C A’ + B’ + C’

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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 it moves through a part of a circuit.

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Circuit A circuit is a closed path along which

charges flow. If there is not a closed path that allows

the charge to 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.

Open circuit, closed circuit

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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.

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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.

Resistors in series

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Series

Two resistors are in series if a charge passing through the first resistor must pass through the second resistor.

It has nowhere else to go.

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Resistors in series

Each resistor obeys Ohm’s law V1 = I1 R1 and V2 = I2 R2

The current through the resistors is the same I1 = I2 = I

R1 R2

I1 I2

V1 V2

a b

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Equivalent resistance (series) The equivalent resistance is the value of a

single resistor that can take the place of a combination Has same current and voltage drop as combo

Vab = V1 + V2 (the voltages add up to the total)

Vab = I1R1 + I2R2

Vab = I (R1 + R2) Vab = I Req

Req = R1 + R2

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Resistors in series Resistors in series add. The equivalent resistance is larger

than either individual resistance. If there are more things one has to

go through, it will be more difficult.

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Equivalent Resistance

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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.

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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.

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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).

Resistors in parallel

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Parallel

Two resistors are in parallel if the top ends of the two resistors are connected by wire and only wire and likewise for the bottom ends.

A charge will pass through one or the other but not both resistors.

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Resistors in parallel The voltage across the resistors is the

same V1 = V2 = Vab (you use up the same energy either way)

The current is split between the resistors I = I1 + I2

R1

R2

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Equivalent resistance (parallel)

I = I1 + I2

Vab=

V1+

V2

Req R1 R2

1=

1+

1

Req R1 R2

V’s are same, so they cancel

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Resistors in parallel Resistors in parallel add

reciprocally. The equivalent resistance will be

smaller than either individual resistance.

It is always easier if one has a choice of what one has to go through.

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Equivalent Resistance

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1=

1+

1

Req R1 R2

If R1 = 2.3 and R2 = 3.4, then do the following on the calculator: 2.3 → 1/x → + → 3.4 → 1/x → = → 1/x → 1.37

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Fire in a theater analogy If it bothers you that the resistance of two

resistors in parallel is lower than either resistor, consider the following. A fire starts in a packed theatre and there is one

door through which everyone must exit. It’s a difficult task to get everyone out. A second exit is found, the second exit is narrower and fewer people can use it. However, the theater can be emptied much faster using two exits than one – even if a given person can only use one of the exits.

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Series/Parallel Recap Series

Resistors in series have the same current. Their voltages add up to the total voltage. Rs = R1 + R2

Parallel Resistors in parallel have the same voltage. Their currents add up to the total current. 1/Rp = 1/R1 +1/R2

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Serial and parallel connections A connection is said to be serial if all

of the bits entering follow exactly the same path, bits then arrive one-by-one.

A connection is said to be parallel if there are a set of paths, bits can then take different paths and groups of bits can arrive simultaneously.

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Multi-meter A multi-meter can serve as a voltmeter,

ammeter or ohmmeter depending on its setting. To measure the voltage across a resistor, the

voltmeter is placed in parallel with the resistor.

To measure the current through a resistor, the ammeter is placed in series with the resistor.

To measure the resistance of a resistor, the resistor is removed from the circuit and each end is connected to an end of the ohmmeter.

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Voltmeter in parallel with 1-k Resistor

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Ammeter in series with 1-k Resistor

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Ohmmeter measuring resistance of 1-k and 2 -k resistors in series

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Checking continuity A wire or cable is metal (a conductor) on

the inside and thus has a low resistance. A broken cable has a high resistance. To check a cable,

remove the cable, set the multi-meter to ohmmeter Check each wire for “continuity” (should find a

low resistance).

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Heat A basic principle of physics is that

energy is conserved, that is, energy is never lost or gained but only rearranged and put in different forms.

When we have a simple resistor circuit, the potential energy that was in the battery becomes heat which is another form of energy.

References

Chapters 3 through 5 of Digital Principles (Tokheim)

http://en.wikipedia.org/wiki/Minterm http://www.physics.wisc.edu/

undergrads/courses/phys202fall96/?D=A

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