For a single-phase full-wave controlled rectifier (M-2 connection) feeding an R–L load with input v(t) = Vm sin(ωt), what is the expression for the average (dc) output voltage Vdc under continuous current?

Introduction / Context:
The average output voltage of a phase-controlled rectifier depends on the firing angle α and the rectifier topology. For a full-wave controlled rectifier using a center-tapped transformer (M-2), a standard closed-form expression exists when current is continuous through the R–L load.

Given Data / Assumptions:

• Single-phase full-wave controlled rectifier (M-2).
• Input v(t) = Vm sin(ωt).
• Load is R–L with continuous current (no extinction within the cycle).
• Firing angle α in the usual 0 to π range.

Concept / Approach:

For continuous current, each half-cycle contributes an identical controlled conduction segment. The average voltage over one electrical period can be integrated from α to π for each half, producing a well-known result. The total average over the full cycle yields a term proportional to (1 + cos α).

Step-by-Step Solution:

Verification / Alternative check:

At α = 0, Vdc = 2Vm/π which matches the diode full-wave average. As α increases to π, Vdc approaches 0, consistent with physical expectation.

Why Other Options Are Wrong:

• (Vm/2π)(1+cos α): half-wave result, not full-wave.
• (2Vm/π) cos α or (Vm/π) cos α: apply to certain bridge forms or different derivations; they do not match the M-2 continuous current case with the 1 + cos α term.
• (Vm/2)(1+cos α): wrong scaling and units.

Common Pitfalls:

• Using diode rectifier averages for controlled rectifiers without including α.
• Confusing M-2 with B-2 formulas; topology matters for prefactors.

Final Answer:

Vdc = (Vm / π) * (1 + cos α)