Question

You are given three circuit elements X, Y, and Z. They are connected one by one across a given ac source. It is found that \( V \) and \( I \) are in phase for element X. \( V \) leads \( I \) by \( \frac{\pi}{4} \) for element Y while \( I \) leads \( V \) by \( \frac{\pi}{4} \) for element Z. Identify elements X, Y, and Z.

Hint:

- In an AC circuit:
- For a resistor, voltage and current are in phase.
- For an inductor, current lags voltage, and at certain frequencies, voltage leads current.
- For a capacitor, current leads voltage.

Answer 1

Step 1: For element \( X \), voltage and current are in phase, which means that the impedance is purely resistive. Therefore, \( X \) must be a resistor.
Step 2: For element \( Y \), \( V \) leads \( I \) by \( \frac{\pi}{4} \). This suggests that element \( Y \) behaves like an inductor, because in an inductor, the current lags the voltage, and at high frequencies, the phase difference between voltage and current increases, but here we are given that voltage leads current, which is a characteristic of inductive reactance at a specific frequency.
Step 3: For element \( Z \), \( I \) leads \( V \) by \( \frac{\pi}{4} \), which indicates capacitive behavior, because in a capacitor, current leads voltage by \( \frac{\pi}{2} \) at high frequencies, and at lower frequencies, this phase difference decreases.
Conclusion:
- \( X \) is a resistor,
- \( Y \) is an inductor,
- \( Z \) is a capacitor.