Combining Logic Gates in Circuits

In real computer circuits, individual logic gates are combined to create more complex operations. OCR J277 Section 2.5 requires you to be able to trace through combined logic gate circuits, complete truth tables for multi-gate circuits, and write Boolean expressions from circuit diagrams.

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How Gates Are Combined

A logic circuit (also called a combinational circuit) connects the output of one gate to the input of another. By chaining gates together, we can create circuits that implement complex Boolean expressions.

For example, to implement the expression Q = (A AND B) OR C, we need:

1. An AND gate taking inputs A and B
2. An OR gate taking the output of the AND gate and input C

flowchart LR
    A((A)) --> AND["AND"]
    B((B)) --> AND
    AND --> OR["OR"]
    C((C)) --> OR
    OR --> Q((Q))

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Tracing Through a Circuit

To determine the output of a combined circuit for given inputs, follow these steps:

1. Label each gate's output with an intermediate variable (e.g., P, R)
2. Start from the inputs and work left to right through the circuit
3. Calculate each gate's output in order
4. The final gate's output is the circuit's output

Worked Example

Consider the circuit: Q = NOT (A OR B)

With A = 1, B = 0:

Step	Operation	Result
1	A OR B	1 OR 0 = 1
2	NOT (result of step 1)	NOT 1 = 0

So Q = 0.

OCR Exam Tip: When tracing through a circuit, work through one gate at a time, writing down intermediate results. Do not try to calculate the whole circuit in your head — this leads to errors.

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Building a Complete Truth Table for a Circuit

To build a truth table for a circuit with multiple inputs, you need columns for:

• All inputs (A, B, C, etc.)
• All intermediate results (outputs of gates that feed into other gates)
• The final output (Q)

Worked Example

Circuit: Q = (A AND B) OR (NOT A)

A	B	A AND B	NOT A	Q = (A AND B) OR (NOT A)
0	0	0	1	0 OR 1 = 1
0	1	0	1	0 OR 1 = 1
1	0	0	0	0 OR 0 = 0
1	1	1	0	1 OR 0 = 1

The intermediate columns (A AND B and NOT A) make it much easier to get the correct final answer.

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Three-Input Circuits

When a circuit has three inputs (A, B, C), the truth table has 2^3 = 8 rows:

Example: Q = (A AND B) OR C

A	B	C	A AND B	Q = (A AND B) OR C
0	0	0	0	0
0	0	1	0	1
0	1	0	0	0
0	1	1	0	1
1	0	0	0	0
1	0	1	0	1
1	1	0	1	1
1	1	1	1	1

OCR Exam Tip: For three-input truth tables, count in binary from 000 to 111 to list all eight input combinations systematically. Never try to list them randomly — you will miss one.

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Reading a Boolean Expression from a Circuit Diagram

When given a circuit diagram in the exam, convert it to a Boolean expression by:

1. Identify the final gate — its output is Q
2. Work backwards from Q to find what feeds into the final gate
3. Write the expression using brackets to show the order of operations

For example, if the final gate is OR, and its inputs come from an AND gate (with inputs A, B) and a NOT gate (with input C):

Q = (A AND B) OR (NOT C)

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Combinational Circuit Diagram: Q = (A AND B) OR (NOT C)

The diagram below shows how three inputs flow through three gates to produce Q. Trace from left to right: A and B feed into the AND gate; C feeds into the NOT gate; the AND output and the NOT output feed into the OR gate, which produces Q.

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

In an exam diagram the AND, NOT, and OR boxes would be drawn with their standard symbols (flat-back D, triangle-with-bubble, curved shield), but the wiring topology is exactly the same as shown here.

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Practical Circuit: Half Adder

The half adder adds two single-bit numbers and produces a sum and a carry:

flowchart LR
    A((A)) --> XOR["XOR"]
    B((B)) --> XOR
    A --> AND["AND"]
    B --> AND
    XOR --> Sum((Sum))
    AND --> Carry((Carry))

A	B	Sum (A XOR B)	Carry (A AND B)
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

This demonstrates how combining just two gates creates a useful arithmetic circuit.

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Common Exam Mistakes

1. Applying gates in the wrong order — always work from the inputs towards the output, not the other way round.
2. Forgetting intermediate steps — include columns for intermediate values to avoid errors.
3. Miscounting rows — for n inputs, there must be exactly 2^n rows.
4. Incorrect binary counting — always count 00, 01, 10, 11 (not 00, 10, 01, 11).

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Summary

• Logic gates are combined by connecting the output of one gate to the input of another
• To trace a circuit, work from inputs to output one gate at a time
• Always include intermediate columns in truth tables
• For n inputs, a truth table has 2^n rows
• List input combinations by counting in binary from all 0s to all 1s
• You must be able to read Boolean expressions from circuit diagrams and vice versa

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Extended Worked Example: Three-Gate Circuit

Consider the expression Q = (A . B) + ((NOT A) . C). This circuit uses four gates: one NOT, two ANDs, and one OR.

flowchart LR
    A((A)) --> NOT["NOT"]
    NOT --> AND1["AND 1"]
    B((B)) --> AND2["AND 2"]
    C((C)) --> AND2
    AND1 --> OR["OR"]
    AND2 --> OR
    OR --> Q((Q))

Wait — let me correct that. AND 1 takes (NOT A) and C, and AND 2 takes A and B. Redrawn clearly:

flowchart LR
    A((A)) --> NA["NOT A"]
    A --> AND2["AND 2: A . B"]
    B((B)) --> AND2
    NA --> AND1["AND 1: (NOT A) . C"]
    C((C)) --> AND1
    AND2 --> OR["OR"]
    AND1 --> OR
    OR --> Q((Q))

Truth table: