Electronics Tutorials - Transistor Transistor Logic (TTL) (10)

7408 Chip Wikipedia Link :: Logic Goats (Google Video)

A mathematician called Boole invented a branch of maths for processing true and false values instead of numbers. This is called Boolean Algebra. Simple Boolean algebra is consistent with common sense but if you need to process decisions involving many values that might be true or false according to complex rules, you need this branch of mathematics.

Logic Gate Rules

There are several types of gate. Each follows a very simple set of rules. By combining many gates in suitable ways, processing devices can be produced. A computer CPU chip can have millions of gates fabricated onto it. The table below shows several gates with two inputs. Many of these gates are also available in three, four and eight input versions.

Rule	Symbol	Boolean	One Line Explanation
AND		Q = A . B	1 AND 1 gives 1. Any other input gives 0.
NAND		Q = A . B	(NOT AND) 1 AND 1 gives 0. Any other input gives 1.
OR		Q = A + B	0 OR 0 gives 0. Any other input gives 1.
NOR		Q = A + B	(NOT OR) 0 OR 0 gives 1. Any other input gives 0.
XOR		Q = A B	Equal inputs give 0. Non equal inputs give 1.
NOT		Q = A	Invert input bits. 0 becomes 1. 1 becomes 0.

Computers work using LOGIC. Displaying graphics such as the mouse cursor involves the XOR (Exclusive OR) opeartion. Addition makes use of AND and XOR. These and a few of the other uses of logic are described below.

Truth Tables

The one line descriptions of the rules above are clearer if shown in Truth Tables. These tables show the output for all possible input conditions. The inputs are always listed in the same order (counting in binary starting from zero).

Here is a summary. The yellow highlights match the rules above.

AND

NAND

NOR

XOR

NOT

A B   Q
0 0   0
0 1   0
1 0   0
1 1   1

A B   Q
0 0   1
0 1   1
1 0   1
1 1   0

A B   Q
0 0   0
0 1   1
1 0   1
1 1   1

A B   Q
0 0   1
0 1   0
1 0   0
1 1   0

A B   Q
0 0   0
0 1   1
1 0   1
1 1   0

A   Q
0   1
1   0

_B\^A	0	1
0	0	0
1	0	1

_B\^A	0	1
0	1	1
1	1	0

_B\^A	0	1
0	0	1
1	1	1

_B\^A	0	1
0	1	0
1	0	0

_B\^A	0	1
0	0	1
1	1	0

A	0	1
	1	0

Two ones
give a one.
Anything
else
gives 0.

Two ones
give a 0.
Anything
else
gives 1.

Two zeros
give a 0.
Anything
else gives 1.

Two zeros
give a 1.
Anything
else
gives 0.

Equal
inputs
give a 0.

Input
is
inverted.

Logic Gates - Remember to label the Inputs and Output!

Logic gates are the building blocks of all digital computers. Modern processors contain millions of gates. Each gate is built from a few transistors. The gates are used to store data, perform arithmetic and manipulate bits using the rules above. For example, the XOR rule can be used to test bits for equality.

AND

Both inputs must be true for the output to be true. AND is used for addition, decision making and bit masking.

The AND rule can be used in industrial safety. A sheet metal cutter might require two switches to be pressed before the cutter operates. One switch is is operated when the safety screen is closed. The other operates the cutter. This prevents the operator cutting off his/her arm.

 -----------
  A B Output
 -----------
  0 0    0
  0 1    0
  1 0    0
  1 1    1

OR

Both inputs must be false for the output to be false. OR is used in decision making and bit masking.

Fire alarm glasses (break to sound the alarm) obey the OR rule. One or more broken glasses cause the alarm to sound.

 -----------
  A B Output
 -----------
  0 0    0
  0 1    1
  1 0    1
  1 1    1

XOR

If the bits in a graphical image are XORed with other bits a new image appears. If the XORing is repeated the image disappears again. This is how the mouse and text cursors get moved around the screen. XOR is combined with AND for use in addition. XOR detects if the inputs are equal or not.

 -----------
  A B Output
 -----------
  0 0    0
  0 1    1
  1 0    1
  1 1    0

NAND

NAND is really AND followed by NOT. Electronic circuits are commonly built from NAND gates (circuits). Computer programming languages generally do not provide NAND. Use NOT AND instead. Gate arrays can be based on NAND gates.

 -----------
  A B Output
 -----------
  0 0    1
  0 1    1
  1 0    1
  1 1    0

NOR

NOR is really OR followed by NOT. Electronic circuits are commonly built from NOR gates (circuits). Computer programming languages generally do not provide NOR. Use NOT OR instead. Gate arrays can be based on NOR gates.

 -----------
  A B Output
 -----------
  0 0    1
  0 1    0
  1 0    0
  1 1    0

NOT

NOT is used to invert bits or True/False values. All the rules above had two inputs and one output. NOT has a single input and output.

 -----------
  A  Output
 -----------
  0     1
  1     0

The Half Adder Truth Table

The half adder does binary addition on two bits.
The AND gate computes the carry bit.
The XOR gate computes the sum bit.

0 + 0 = 0, carry 0
0 + 1 = 1, carry 0
1 + 0 = 1, carry 0
1 + 1 = 0, carry 1

 ------------------
  A B   SUM  CARRY
 ------------------
  0 0    0     0
  0 1    1     0
  1 0    1     0
  1 1    0     1