Compare average dc output of half-wave versus full-wave (M–2) controlled rectifiers with a resistive load: relation between Vdc1 (half-wave) and Vdc2 (full-wave) for any firing angle α.

Introduction / Context:Average output voltage of phase-controlled rectifiers depends on firing angle and whether one or two half-cycles are used. Full-wave topologies deliver pulses twice per cycle, doubling the average (for the same Vm and α) relative to half-wave circuits.

Given Data / Assumptions:

• Resistive load.
• Half-wave controlled rectifier (single thyristor): Vdc1 formula.
• Full-wave controlled rectifier (M–2): Vdc2 formula.

Concept / Approach:

Standard results: For half-wave controlled rectifier, Vdc1 = (Vm / 2π) (1 + cos α). For the full-wave (M–2), Vdc2 = (Vm / π) (1 + cos α). Therefore Vdc1 = 0.5 * Vdc2 for any α in [0, π].

Step-by-Step Solution:

Verification / Alternative check:

At α = 0, Vdc1 = Vm/π, Vdc2 = 2Vm/π ⇒ ratio 0.5. At α = π/2, both reduce by the same factor, maintaining the 0.5 ratio.

Why Other Options Are Wrong:

• Equality or inequalities contradict the established factor-of-two difference due to pulse repetition rate.
• Dependence on α cancelling in the ratio is overlooked in the wrong choices.

Common Pitfalls:

• Using diode-rectifier (α = 0 only) values and generalizing incorrectly.
• Confusing Vm and Vrms when forming averages.

Final Answer:

Vdc1 = 0.5 Vdc2 for all firing angles