Series connection of two single-phase semi-converters (series converter) The input voltage is v = Vm sin(ωt). Two single-phase semi-converters are connected in series, with firing angles a1 and a2. If the first converter is fired at a1 = 0, determine the normalized average DC output voltage expression in terms of a2 (i.e., proportional to 1 + cos terms; ignore the common Vm/π factor).

Introduction / Context:
Series connection of semi-converters is a standard way to increase the controllable DC voltage. Each single-phase semi-converter produces an average voltage proportional to (1 + cos α), multiplied by Vm/π. When cascaded in series, the total average is the sum of individual averages (assuming identical input and continuous conduction).

Given Data / Assumptions:

• Two single-phase semi-converters in series.
• Input: v = Vm sin(ωt).
• Firing angles are a1 and a2; here a1 = 0.
• Resistive or continuous-current operation so standard average formulas apply.

Concept / Approach:

The average DC output of one semi-converter is Vdc1 = (Vm/π) * (1 + cos a1). For two in series, Vdc_total = (Vm/π) * [(1 + cos a1) + (1 + cos a2)]. Substitute a1 = 0 to obtain a compact expression in terms of a2 only.

Step-by-Step Solution:

Verification / Alternative check:

If a2 = 0 as well, total would be (Vm/π) * (4), consistent with both units firing at zero angle (maximum average).

Why Other Options Are Wrong:

• (1 + cos a2), (2 + cos a2), (4 + cos a2): do not correctly account for a1 = 0, which contributes +1 to the sum (becoming +2) and adds to the other converter’s 1 + cos a2, yielding 3 + cos a2.

Common Pitfalls:

Forgetting that each semi-converter yields (1 + cos α) and that series connection sums averages directly; omitting the constant from a1 = 0.

Final Answer:

(3 + cos a2)