Logic Circuits

A logic circuit is a physical implementation of a Boolean expression using interconnected logic gates. In OCR H446, you need to be able to draw circuits from Boolean expressions, trace circuits to produce truth tables, and understand multi-level circuits with several stages of gates.

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From Boolean Expression to Circuit

To convert a Boolean expression into a circuit:

1. Identify the main operation (the last operation to be evaluated) — this determines the output gate
2. Work backwards through the expression, adding gates for each sub-expression
3. Connect inputs to the appropriate gates
4. Label all wires clearly

Example: Draw the circuit for Q = (A AND B) OR (NOT C)

The main operation is OR, so the output gate is OR.

• Input 1 to the OR gate: A AND B (requires an AND gate with inputs A and B)
• Input 2 to the OR gate: NOT C (requires a NOT gate with input C)

Circuit structure:

A ---+
     |--- AND ---+
B ---+           |--- OR --- Q
                 |
C --- NOT -------+

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From Circuit to Boolean Expression

To determine the expression from a circuit:

1. Start at the inputs on the left
2. Trace through each gate, writing the operation it performs
3. Combine the intermediate expressions until you reach the output

Example: A circuit has:

• Gate 1: NOT gate, input A, output P
• Gate 2: AND gate, inputs P and B, output R
• Gate 3: OR gate, inputs R and C, output Q

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

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From Circuit to Truth Table

To produce a truth table from a circuit:

1. List all input variables
2. Trace through the circuit for every input combination
3. Calculate intermediate values at each gate
4. Record the final output

Worked Example:

Circuit: Q = (A XOR B) AND (B OR C)

A	B	C	A XOR B	B OR C	Q
0	0	0	0	0	0
0	0	1	0	1	0
0	1	0	1	1	1
0	1	1	1	1	1
1	0	0	1	0	0
1	0	1	1	1	1
1	1	0	0	1	0
1	1	1	0	1	0

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Multi-Level Circuits

A multi-level circuit has more than two stages of gates between inputs and output. Each stage introduces a gate delay — the time for a signal to propagate through a gate.

Example: A 3-level circuit:

Level 1:  P = A AND B
          R = C OR D
Level 2:  S = P XOR R
          T = NOT D
Level 3:  Q = S AND T

Expression: Q = ((A AND B) XOR (C OR D)) AND (NOT D)

Gate delay: The signal must pass through 3 gates sequentially to reach Q. If each gate has a delay of 10 nanoseconds, the total propagation delay is 30 nanoseconds.

Concept	Detail
Fan-in	The number of inputs to a gate
Fan-out	The number of gates an output connects to
Propagation delay	Time for a signal to pass through one gate
Circuit depth	Number of gate levels from input to output

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Designing Circuits from Truth Tables