Logic Gates (AQA A-Level Computer Science): Revision Notes

Logic Gates

Introduction

Logic gates are fundamental electronic components found in computer processors, memory chips, and registers. They work by evaluating Boolean expressions to produce digital outputs. Understanding logic gates is essential because there is a direct one-to-one relationship between Boolean expressions and the gates that implement them in hardware.

In electronic circuits, logic gates process electrical signals represented as voltages. A high voltage represents a binary 1 (true), whilst a low voltage represents a binary 0 (false). These gates form the building blocks of all digital circuits, from simple calculators to complex computer processors.

The six basic logic gates

There are six fundamental logic gates that you need to understand. Each gate performs a specific logical operation on its inputs to produce a single output.

Here is a summary of the six gates with their operators:

Gate	Operator	Description
AND	A·B	Output is true only when both inputs are true
OR	A+B	Output is true when at least one input is true
NOT	Ā	Output is the inverse of the input
NAND	A·B̄	Output is true when any input is false
NOR	Ā+B̄	Output is true only when both inputs are false
XOR	A⊕B	Output is true when inputs are different

AND gate

The AND gate produces a true output (1) only when both of its inputs are true. Think of it like a security system that requires two keys to open - both conditions must be met for the output to be true.

The Boolean expression for an AND gate is written as $A \cdot B$ or $A.B$, which means "A AND B".

Here is the truth table for a 2-input AND gate:

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

Notice that the output Q is only 1 in the final row, where both A and B are 1. In all other cases, the output is 0.

OR gate

The OR gate produces a true output (1) when either one or both of its inputs are true. Only when both inputs are false does the output become false. This is similar to a light that can be switched on from two different locations - either switch (or both) can turn it on.

The Boolean expression for an OR gate is $A+B$, which means "A OR B".

The truth table for a 2-input OR gate is:

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

The output is 1 whenever at least one input is 1. The output is only 0 when both inputs are 0.

NOT gate

The NOT gate is unique because it has only one input. It inverts whatever value is input - turning 1 into 0 and 0 into 1. This is also called an inverter.

The Boolean expression is $\bar{A}$ or NOT A, which means "NOT A" or "the inverse of A".

The truth table for a NOT gate is:

A	Q
0	1
1	0

NAND gate

The NAND gate is essentially an AND gate followed by a NOT gate. Its name means "NOT AND". The output is true (1) when any of the inputs are false - it's the opposite of the AND gate.

The Boolean expression is $\overline{A \cdot B}$, which means "NOT (A AND B)".

The truth table for a 2-input NAND gate is:

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

Notice this is the exact opposite of the AND gate - wherever the AND gate produces 0, the NAND gate produces 1, and vice versa. The NAND gate symbol looks like an AND gate with a small circle (bubble) at the output, indicating the inversion.

NOR gate

The NOR gate is an OR gate followed by a NOT gate. Its name means "NOT OR". The output is true (1) only when both inputs are false - it's the opposite of the OR gate.

The Boolean expression is $\overline{A + B}$, which means "NOT (A OR B)".

The truth table for a 2-input NOR gate is:

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

This is the inverse of the OR gate truth table. The NOR gate symbol looks like an OR gate with a small circle at the output.

XOR gate (exclusive OR)

The XOR gate stands for "exclusive OR". It produces a true output (1) when either input is true, but not when both inputs are true. In other words, the output is true when the inputs are different from each other.

The Boolean expression is $A \oplus B$, which means "A XOR B".

The truth table for a 2-input XOR gate is:

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

Combining logic gates

Logic circuits are created by connecting multiple logic gates together in sequence. These circuits can become very complex, with thousands of gates working together in modern processors. The output from one gate becomes the input to the next gate, creating a chain of logical operations.

When you combine gates, you build up more complex Boolean expressions. Each intermediate connection represents a partial result that feeds into the next stage of processing.

Building Boolean expressions from logic circuits

Just as you can create a logic circuit from a Boolean expression, you can also work backwards to create a Boolean expression from a circuit diagram. You simply follow the signal path through each gate, writing down the operation performed at each stage.

For more complex circuits, you trace through each gate systematically:

• Identify the inputs entering the first layer of gates
• Write the Boolean expression for each gate's output
• Use these intermediate results as inputs to the next layer of gates
• Continue until you reach the final output

Full and half adders

Adders are special logic circuits designed to perform binary addition. They are fundamental components of the Arithmetic Logic Unit (ALU) in a processor, which handles all mathematical and logical operations.

Half adder

A half adder is a circuit that adds two single bits together (A and B). It produces two outputs:

• S (Sum): the sum of A and B
• C (Carry): the carry bit generated when both inputs are 1

The half adder uses an XOR gate to calculate the sum (S) because the XOR gate outputs 1 when the inputs are different. It uses an AND gate to calculate the carry (C) because a carry is only generated when both inputs are 1.

The truth table for a half adder is:

A	B	C	S
0	0	0	0
1	0	0	1
0	1	0	1
1	1	1	0

Full adder

A half adder can only add two bits, but what if you need to add multiple bits or include a carry from a previous addition? This is where the full adder comes in.

A full adder adds three inputs:

• A: first bit
• B: second bit
• Cin: carry input from a previous addition

It produces two outputs:

• S (Sum): the sum result
• Cout: the carry output for the next addition

The truth table for a full adder shows all eight possible input combinations:

A	B	Cin	Cout	S
0	0	0	0	0
1	0	0	0	1
0	1	0	0	1
1	1	0	1	0
0	0	1	0	1
1	0	1	1	0
0	1	1	1	0
1	1	1	1	1

The full adder circuit works as follows:

• First XOR gate compares A and B to see if they're different
• Second XOR gate combines that result with Cin to produce the sum
• AND gates and OR gate work together to determine if a carry should be generated
• A carry is produced when either both original inputs are 1, or when the result of A and B produces a 1 that combines with Cin

Edge-triggered D-type flip-flop (A-level only)

Whilst logic gates show how binary values can be manipulated, they don't store data. Each time new inputs arrive, the previous results are lost. In real computer systems, we need some way to remember values temporarily - this is where flip-flops come in.

A flip-flop is a memory unit that can store a single bit of information. The edge-triggered D-type flip-flop is a specific type that changes its stored value only when triggered by the system clock.

The system diagram shows two types of inputs:

• Primary inputs: direct inputs to the logic circuit
• Secondary inputs: inputs that come from the memory unit (feedback from previous operations)

Similarly, there are two types of outputs:

• Primary outputs: the main results of the computation
• Secondary outputs: values sent back to the memory unit for storage

Clock signals

The clock is a device that generates regular pulses to synchronise all components in a computer. These pulses ensure that data moves through the system in an orderly, predictable manner.

A clock signal has several important characteristics:

Clock period: the time for one complete cycle (one pulse) of the clock

Clock width: the duration of the high portion of the pulse

Rising edge: the transition from low (0) to high (1) voltage - this is when the signal goes up

Falling edge: the transition from high (1) to low (0) voltage - this is when the signal goes down

How the D-type flip-flop works

An edge-triggered D-type flip-flop changes its state only on one edge of the clock pulse (usually the rising edge). This means:

• Data present at the input when the clock pulse arrives gets stored in the flip-flop
• This stored data remains at the output until the next clock pulse arrives
• Any changes to the input between clock pulses are ignored

Remember!