Logic Circuits

A logic circuit (also called a logic diagram or combinational logic circuit) is a diagram showing how logic gates are connected together to process inputs and produce outputs. In your GCSE exam, you must be able to read logic circuit diagrams, trace values through them, write their Boolean expressions and construct their truth tables.

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What Is a Logic Circuit?

A logic circuit is a visual representation of a Boolean expression using standard logic gate symbols. It shows:

• The inputs on the left (labelled with letters such as A, B, C).
• The logic gates in the middle, connected by wires.
• The output on the right (usually labelled Q or Z).

Logic circuits are read from left to right — inputs enter on the left, pass through one or more gates, and the final output exits on the right.

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Standard Gate Symbols Reference

Before working with circuits, make sure you can recognise each gate symbol:

Key identifiers:

• NOT: Triangle with bubble
• AND: Flat left, curved right, no bubble
• OR: Curved left, pointed right, no bubble
• XOR: Like OR but with extra curved line on left
• NAND: Like AND but with bubble on output
• NOR: Like OR but with bubble on output

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Tracing a Logic Circuit

To find the output of a logic circuit for given inputs, work through the circuit gate by gate, from left to right.

Example 1: Simple Two-Gate Circuit

Consider this circuit where inputs A and B pass through an AND gate, and the result passes through a NOT gate:

graph LR
  A(["A"]) --> AND["AND gate"]
  B(["B"]) --> AND
  AND --> NOT["NOT gate"]
  NOT --> Q(["Q"])

Boolean expression: Q = ¬(A ∧ B)

This is equivalent to a single NAND gate.

Truth table:

A	B	A ∧ B	Q = ¬(A ∧ B)
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

Example 2: Three-Gate Circuit

graph LR
  A(["A"]) --> AND["AND gate"]
  B(["B"]) --> AND
  C(["C"]) --> NOT["NOT gate"]
  AND --> OR["OR gate"]
  NOT --> OR
  OR --> Q(["Q"])

Boolean expression: Q = (A ∧ B) ∨ ¬C

The same circuit drawn as a flow diagram:

graph LR
    A((A)) --> AND["AND"]
    B((B)) --> AND
    C((C)) --> NOT["NOT"]
    AND --> OR["OR"]
    NOT --> OR
    OR --> Q((Q))

Truth table (8 rows for 3 inputs):

A	B	C	A ∧ B	¬C	Q = (A ∧ B) ∨ ¬C
0	0	0	0	1	1
0	0	1	0	0	0
0	1	0	0	1	1
0	1	1	0	0	0
1	0	0	0	1	1
1	0	1	0	0	0
1	1	0	1	1	1
1	1	1	1	0	1

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Drawing a Logic Circuit from a Boolean Expression

To draw a logic circuit from a Boolean expression:

1. Identify the final operation — this will be the last gate before the output.
2. Work backwards — determine what feeds into that final gate.
3. Draw the gates from left to right, connecting them with wires.
4. Label all inputs and the output.

Example: Draw the circuit for Q = (A ∨ B) ∧ ¬A

1. The final operation is AND (∧).
2. The AND gate receives two inputs: (A ∨ B) and ¬A.
3. (A ∨ B) comes from an OR gate with inputs A and B.
4. ¬A comes from a NOT gate with input A.

graph LR
  A(["A"]) --> OR["OR gate"]
  B(["B"]) --> OR
  A --> NOT["NOT gate"]
  OR --> AND["AND gate"]
  NOT --> AND
  AND --> Q(["Q"])

Note that input A is used twice — once going into the OR gate and once into the NOT gate. In circuit diagrams, a wire can split (branch) to feed multiple gates.

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Writing a Boolean Expression from a Logic Circuit

To write the expression from a circuit:

1. Start at the output and work backwards through the circuit.
2. At each gate, write the operation (AND, OR, NOT, etc.) with its inputs.
3. Use brackets to show the order of operations.

Systematic Approach

Alternatively, work forwards:

1. Label the output of each gate in the circuit (you can use letters like P, Q for intermediate values).
2. Write the expression for each gate in sequence.
3. Substitute to get the final expression in terms of the original inputs.

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Common Exam Circuit Patterns

Half Adder

A half adder adds two single-bit binary numbers:

graph LR
  A(["A"]) --> XOR["XOR gate"]
  B(["B"]) --> XOR
  A --> AND["AND gate"]
  B --> AND
  XOR --> SUM(["Sum (A ⊕ B)"])
  AND --> CARRY(["Carry (A ∧ B)"])

Alarm System

An alarm sounds when a sensor detects motion AND the system is armed:

graph LR
  SENSOR(["Sensor"]) --> AND["AND gate"]
  ARMED(["Armed"]) --> AND
  AND --> ALARM(["Alarm"])

Override Circuit

A device turns on when the normal switch is on OR when the override switch is on:

graph LR
  NORMAL(["Normal"]) --> OR["OR gate"]
  OVERRIDE(["Override"]) --> OR
  OR --> DEVICE(["Device"])

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Exam Technique for Logic Circuits

1. Label everything — mark the output of every gate, not just the final output.
2. Use intermediate columns in truth tables — one for each gate's output.
3. Check your work — verify at least two rows of your truth table by tracing through the circuit manually.
4. Count your rows — make sure you have 2^n rows for n inputs.
5. Read carefully — check whether the question asks for the expression, the truth table or both.

Exam Tip: When tracing a circuit, use a highlighter or pencil to mark the value on each wire as you work through it. This prevents errors caused by losing track of which wire carries which value.

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Summary

• A logic circuit shows how gates are connected to process inputs and produce outputs.
• Read circuits left to right, evaluating each gate in sequence.
• You must be able to trace values, write expressions and construct truth tables from circuits.
• You must also be able to draw circuits from given Boolean expressions.
• Use intermediate labels and intermediate columns to avoid errors.
• Common patterns include half adders, alarm systems and override circuits.

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Worked Examples: Tracing Logic Circuits

Worked Example 1: Two-Gate Circuit

A logic circuit takes inputs A and B. A passes through a NOT gate to give ¬A. ¬A and B feed an AND gate, producing Q.

Boolean expression: Q = ¬A ∧ B.

Truth table: