Bridge balance condition: In the given AC bridge circuit, Z1 = 200∠60° Ω, Z2 = 400∠−90° Ω, and Z3 = 300∠0° Ω. Determine the impedance Z4 that will balance the bridge.

Introduction / Context:
AC bridges are widely used for measuring impedances with accuracy. The condition of balance for a bridge is that the ratio of impedances in one arm must equal the ratio in the opposite arm. This problem tests understanding of phasor impedances and complex ratio arithmetic.

Given Data / Assumptions:

• Z1 = 200∠60° Ω
• Z2 = 400∠−90° Ω
• Z3 = 300∠0° Ω
• Z4 = ? (to be found for bridge balance)
• Standard balance condition: Z1 / Z2 = Z3 / Z4

Concept / Approach:

For a balanced bridge, the product of opposite arms must be equal: Z1 * Z4 = Z2 * Z3. Therefore, Z4 = (Z2 * Z3) / Z1. This requires multiplication and division of complex numbers in polar form, which involves multiplying/dividing magnitudes and adding/subtracting angles.

Step-by-Step Solution:

Verification / Alternative check:

Check ratio: Z1 / Z2 = 200∠60° / 400∠−90° = 0.5∠150°. Similarly, Z3 / Z4 = 300∠0° / 600∠−150° = 0.5∠150°. Both sides equal, confirming the balance condition is satisfied.

Why Other Options Are Wrong:

• 150∠30° Ω: incorrect magnitude and angle, does not satisfy balance condition.
• 400∠−90° Ω: simply repeats Z2, not derived from balance equation.
• 300∠90° Ω: wrong phase shift; fails ratio check.
• 600∠−150° Ω: correct value derived from balance condition (this is the answer).

Common Pitfalls:

• Mixing up balance condition (Z1/Z2 = Z3/Z4) with (Z1Z4 = Z2Z3). Both are equivalent but must be applied carefully.
• Forgetting to subtract angles properly when dividing in polar form.
• Neglecting to check with ratio verification after solving.

Final Answer:

600∠−150° Ω