Single-phase dual converter (back-to-back converters) If α₁ and α₂ are the firing angles of the two converters operating in circulating-current mode, which relation must hold for equal and opposite average voltages?

Introduction / Context:
A single-phase dual converter consists of two line-commutated converters connected back-to-back to provide reversible DC output. In circulating-current mode, both converters can conduct simultaneously through an interconnecting reactor, and their average output voltages are controlled to be equal in magnitude and opposite in polarity.

Given Data / Assumptions:

• Two fully controlled single-phase converters: Converter 1 with firing angle α₁, Converter 2 with firing angle α₂.
• Circulating current reactor present; steady operation requires equal and opposite average DC voltages.

Concept / Approach:
For a line-commutated single-phase full converter, the average DC output voltage (neglecting overlap) is proportional to cos(α). Equal and opposite average voltages imply cos(α₁) = −cos(α₂), which yields α₁ + α₂ = π.

Step-by-Step Solution:
Let Vd1 ∝ cos(α₁) for Converter 1.Let Vd2 ∝ cos(α₂) for Converter 2.Condition for equal and opposite averages: Vd1 = −Vd2 ⇒ cos(α₁) = −cos(α₂).Hence α₁ + α₂ = π (principal solution for 0 ≤ α ≤ π).

Verification / Alternative check:
Phasor and waveform analysis of the dual converter confirm this relationship for zero net average across the circulating reactor.

Why Other Options Are Wrong:
α₁ − α₂ = 0: Would make average voltages equal, not opposite.
α₁ − α₂ = π / α₁ + α₂ = 2π: Not the standard balance condition in 0–π operating range.

Common Pitfalls:
Confusing inversion relations; remember the average voltage is proportional to cos(α).

Final Answer:
α₁ + α₂ = π