Learning Objectives
- Follow signal paths through a multi-gate circuit.
- Label and calculate intermediate gate outputs.
- Complete a truth table from a circuit with up to three inputs.
- Check that inversion bubbles are applied at the correct stage.
Key Terms
- Circuit tracing
- Following input values through every gate to determine the output.
- Signal path
- The route taken by a binary value through wires and gates.
- Stage
- One level of gate processing before the next gate.
- Fan-out or branch
- Use of one input signal by more than one gate.
- Final gate
- The last gate whose output is the circuit output.
- Dependency
- A relationship showing that one gate cannot be evaluated until its input gate results are known.
Start At The Inputs
A circuit should be traced in the direction of signal flow, usually from left to right. Identify the external inputs and locate the first gates they enter.
Do not begin with the final gate unless all of its input values are already known. A final gate often receives intermediate outputs rather than the original variables.
Label the output of each non-final gate as P, Q, R or another temporary variable. These labels become truth table columns.
An Example Circuit Structure
Suppose A and B enter an AND gate producing P. Input C enters a NOT gate producing Q. P and Q then enter an OR gate producing X.
The expression is X = (A AND B) OR (NOT C). The dependencies are P first, Q first, then X. P and Q can be calculated independently for each row because neither depends on the other.
A drawing may omit the expression, but the gate connections reveal the same structure.
Trace One Row Carefully
For A = 1, B = 1 and C = 1: P = A AND B = 1; Q = NOT C = 0; X = P OR Q = 1.
For A = 0, B = 1 and C = 1: P = 0; Q = 0; X = 0 OR 0 = 0.
Writing each stage prevents the common mistake of calculating A OR NOT C and ignoring B.
Parallel Paths
Some circuits split into two paths before combining at a final gate. Calculate each path separately. A branch of an input carries the same value on both paths.
For example, A may enter an AND gate with B and also a NOT gate. If A = 1, the AND path receives 1 while the NOT path outputs 0. The branch itself does not invert or alter A.
The final gate can be evaluated only after both path outputs are known.
Bubbles During Tracing
An output bubble means invert the result of that gate. For NAND, calculate AND and then invert. For NOR, calculate OR and then invert.
A NOT gate or output bubble at an intermediate stage affects every later gate that receives that signal. Do not wait until the end to invert unless the bubble is on the final gate.
Circle each bubble mentally or on rough working so that it is not overlooked.
A Full Trace Table
| A | B | C | P = A NAND B | Q = NOT C | X = P AND Q |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 |
Reverse Checking
After completing the table, select rows with distinctive values. In the example, Q = NOT C, so every row with C = 1 must have Q = 0. P = A NAND B, so only A = 1 and B = 1 gives P = 0.
These pattern checks can detect a column error before it reaches the final output. Then verify X using the final AND rule.
If one intermediate column is wrong, recalculate the final column for all affected rows.
Drawing A Circuit From A Trace
When a question presents gate outputs or an incomplete circuit, use dependencies to determine the missing connection. If X is P OR Q, the final gate must be OR and its inputs must be the lines P and Q.
Do not connect original variables directly if the table headings show that an intermediate result should be used.
Worked Examples
Tracing A Three-Gate Circuit
Question: A and B enter OR to give P. P and C enter NAND to give X. Find X for A = 0, B = 1, C = 1.
- P = 0 OR 1 = 1.
- X = 1 NAND 1.
- AND result is 1, then invert.
Answer: X = 0.
Finding A Missed Bubble
Question: A candidate calculates an AND-shaped gate with an output bubble as ordinary AND. What correction is needed?
- Recognise the symbol as NAND.
- Calculate the AND result.
- Invert that result before using it in later gates.
Answer: Every value in that gate output column must be changed to the opposite NAND value, then dependent columns recalculated.
Examination Guidance
- Label intermediate outputs on the circuit or beside the table.
- Work in dependency order from first-stage gates to the final gate.
- Treat parallel paths separately.
- Check every output bubble at the exact gate where it appears.
- Recalculate later columns after correcting an earlier gate result.
Common Mistakes
- Using the original inputs at the final gate instead of intermediate outputs.
- Applying a final inversion to a bubble that belongs to an earlier gate.
- Changing a signal value when a line branches.
- Skipping intermediate columns and making hidden errors.
- Tracing only selected rows rather than all required combinations.
Knowledge Check
1. Where should circuit tracing begin?
2. What should be labelled P or Q?
3. Does a branch change the signal?
4. When is a NAND inversion applied?
5. What happens after an intermediate error is corrected?
