Combinational logic (AQA A-Level Physics): Revision Notes

13.5.1 Combinational logic

• On ($1$) – Indicates the presence of a current.
• Off ($0$) – Indicates the absence of a current.

Combinational Logic

When several logic gates are combined to produce a specific output based on the inputs, this setup is called combinational logic. The output depends solely on the current input values, without any memory of previous inputs. Combinational logic is essential for analysing and creating digital circuits with desired functionalities.

Truth Tables

To understand how a combination of inputs affects the output, truth tables are used. A truth table lists all possible input values and the corresponding output for each input combination. This is particularly helpful for simplifying and analysing digital circuits. For a circuit with $n$ inputs, there are $2^n$ possible input combinations. For example, with 4 inputs, there are $2^4 = 16$ combinations.

Types of Basic Logic Gates

1. AND Gate

• Output is $1$ only if both inputs are $1$; otherwise, the output is $0$.
• Truth table:

A	B	Out
$0$	$0$	$0$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	1

2. OR Gate

• Output is $1$ if either input is $1$; otherwise, the output is $0$.
• Truth table:

A	B	Out
$0$	$0$	$0$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$1$

3. NOT Gate

• Inverts the input; output is $1$ if input is $0$, and vice versa.
• Truth table:

A	Out
$0$	1
$1$	0

4. NAND Gate

• Output is $0$ only if both inputs are $1$; otherwise, the output is $1$.
• This gate is essentially an AND gate followed by a NOT gate.
• Truth table: | A | B | Out | |---|---|---| | $0$ | $0$ | $1$ | | $0$ | $1$ | $1$ | | $1$ | $0$ | $1$ | | $1$ | $1$ | $0$ |

5. NOR Gate

• Output is $1$ only if both inputs are $0$; otherwise, the output is $0$.
• This gate is essentially an OR gate followed by a NOT gate.
• Truth table: | A | B | Out | |---|---|---| | $0$ | $0$ | $1$ | | $0$ | 1 | $0$ | | $1$ | 0 | $0$ | | $1$ | 1 | $0$ |

6. EOR (XOR) Gate

• Exclusive OR gate gives an output of $1$ if one input is $1$ and the other is $0$.
• Often used as a comparator.
• Truth table: | A | B | Out | |---|---|---| | $0$ | 0 | $0$ | | $0$ | 1 | $1$ | | $1$ | 0 | $1$ | | $1$ | 1 | $0$ |

Boolean Algebra and Logic Circuit Simplification

Boolean algebra is a mathematical tool used to describe and optimise logic circuits. Using Boolean algebra, complex circuits can be represented with expressions, and unnecessary gates can be eliminated.

Key identities in Boolean algebra:

• AND: $A \cdot B$
• OR: $A + B$
• NOT: $\overline{A}$

Key Boolean Laws

1. Associative Law

• ($A \cdot B) \cdot C = A \cdot (B \cdot C)$
• $(A + B) + C = A + (B + C)$

2. Commutative Law

• $A \cdot B = B \cdot A$
• $A + B = B + A$

3. Distributive Law

• $A \cdot (B + C)$ = $(A \cdot B)$ + ($A \cdot C)$
• $A + (B \cdot C) = (A + B) \cdot (A + C)$

Simplifications

• $A \cdot A = A$
• $A + \overline{A} = 1$
• $A \cdot 0 = 0$
• $A + 0 = A$ These simplifications allow circuits to be designed with fewer gates, saving space and power.