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?
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AVdc = (Vm / π) * (1 + cos α)
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BVdc = (Vm / 2π) * (1 + cos α)
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CVdc = (2Vm / π) * cos α
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DVdc = (Vm / π) * cos α
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EVdc = (Vm / 2) * (1 + cos α)
Answer
Correct Answer: Vdc = (Vm / π) * (1 + cos α)
Explanation
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:
For one half-cycle: v_o = Vm sin θ from θ = α to θ = π.Average over full cycle uses symmetry of two halves.Compute Vdc = (1/π) * ∫(α to π) Vm sin θ dθ + (1/π) * ∫(π+α to 2π) Vm sin θ dθ.Each integral gives Vm/π * (1 + cos α) / 2; combined over both halves yields Vdc = (Vm/π) * (1 + cos α).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)