2. Multiple Sensor Alarm System
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A home security alarm system asserts its output if either sensor detects a trigger. Since standard OR gates are not available, the circuit is implemented using the Boolean identity:
Output = (Door Sensor AND Motion Sensor) XOR (Door Sensor XOR Motion Sensor)
To understand how this line is evaluated, we examine the truth table for all input combinations (A = Door Sensor, B = Motion Sensor):
| A | B | (A AND B) | (A XOR B) | Final Result: (AND) XOR (XOR) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 XOR 0 = 0 |
| 0 | 1 | 0 | 1 | 0 XOR 1 = 1 |
| 1 | 0 | 0 | 1 | 0 XOR 1 = 1 |
| 1 | 1 | 1 | 0 | 1 XOR 0 = 1 |
Calculation breakdown:
- Case (0,0): Both sensors inactive; result is 0.
- Cases (0,1) & (1,0): One sensor is active; the XOR returns 1. Since the AND returns 0, the final XOR (0,1) yields 1.
- Case (1,1): Both sensors are active; the AND returns 1, and the XOR returns 0. The final XOR (1,0) yields 1.
This demonstrates that the final output is high whenever at least one sensor is triggered, effectively synthesizing an OR operation without using a standard OR gate.