Fundamental Hardware Elements of Computers: Building circuits

From Wikibooks, open books for an open world
Jump to: navigation, search

UNIT 2 - ⇑ Fundamental Hardware Elements of Computers ⇑

← Boolean gate combinations Building circuits Boolean algebra →

A common question in the exam is to be given some boolean algebra and be asked to express it as logic gates. Let's take a look at an addition and subtraction example that you should be familiar with:

9-(7+1)

First we are going to deal with the inner-most brackets

(7+1) = 8

Finally we combine this answer with the 9-

= 9-(8) = 1

It will work exactly in the same way for boolean algebra, but instead of using numbers to store our results, we'll use logic gates:

Example: Building circuits
C.(A+B)

As with any equation, we are going to deal with the inner-most brackets first(A+B), then combine this answer with the C.
CPT-logic-gate building circuit.svg

Exercise: Building circuits

Draw the circuit diagrams for the following (Remember: do we deal with AND or OR first?):

A.B+C

Answer :

CPT-logic-gate conversion (A.B)+C.svg

(\overline{A}+\overline{B}).C

Answer :

CPT-logic-gate conversion (-A+-B).C.svg

\overline{(A+B)}.C

Answer :

CPT-logic-gate conversion ¬(A.+B).C.svg

(A.\overline{B})\oplus(B.C)

Answer :

CPT-logic-gate conversion (A.¬B)(+)(B.C).svg

\overline{(A.B).(C+\overline{D})}

Answer :

CPT-logic-gate conversion ¬((A.B).(C+¬D)).svg

A common question in the exam is to give you a description of a system. You'll then be asked to create a boolean statement from this description, and finally build a logic gate circuit to show this system:

Example: Building circuits

Using boolean algebra describe the following scenario:

A car alarm is set off if a window is broken or if it senses something moving inside car, and the car is not being towed, or the engine is not on.

Where:

  • A = being towed,
  • B = window broken,
  • C = engine on,
  • D = senses movement

Before you rush into answering a question like this, let's try and break it down into its components. The questioner will often be trying to trick you. The two occasions that the alarm will sound are: B+D but there is a caveat, the alarm will sound if either of these are true AND two things are also true, namely the engine is NOT on, and the car is NOT being towed: \overline{A}+\overline{C}

Combining both we get (remember the brackets!): (B+D).(\overline{A}+\overline{C})

The next step is to create a diagram out of this: CPT-logic-gate conversion (B+D).(¬A+¬B).svg

Exercise: Building circuits

A security system allows people of two different clearance levels access to a building. Either they have low privileges and they have a card and they are not carrying a mobile. Alternatively they have a key and are allowed to carry a mobile.

The inputs available are:

  • A = carrying a card
  • B = carrying a mobile phone
  • C = carrying a key

Write down the boolean equation to express this:

Answer :

(A.\overline{B})+(C)

If you wrote: (A.\overline{B})+(C.B) You'd be wrong! The reason being the text says: Alternatively they have a key and are allowed to carry a mobile. This doesn't mean C.B, it means they can carry a mobile, or they can choose not to: C.(B+\overline{B}), which simplifies to: C.(B+\overline{B}) = C.(1) = C.

Draw the logic gate diagram to solve this:

Answer :

CPT-logic-gate conversion (A.¬B)+(C).svg