Single-phase half-wave controlled rectifier with resistive load As the firing angle α increases, how do the DC output voltage and the RMS output voltage change?

Introduction / Context:
Phase control changes the conduction interval of a controlled rectifier. For a purely resistive load in a half-wave circuit, increasing the firing angle delays conduction, thereby reducing both the average (DC) and the RMS values of the output.

Given Data / Assumptions:

• Input: v(t) = Vm sin(ωt).
• Resistive load (no energy storage).
• Conduction over θ ∈ [α, π] each cycle.

Concept / Approach:

Average and RMS values are computed by integrating the output over the effective conduction interval. As α increases, the interval shortens. The RMS depends on the square of the waveform, so it does not scale linearly with the average.

Step-by-Step Solution:

Verification / Alternative check:

Check limiting cases: α = 0 gives maximum V_dc and V_rms; α → π gives both tending to zero.

Why Other Options Are Wrong:

“Increase” options contradict the physics of delayed conduction; “same proportion” is incorrect due to differing functional forms.

Common Pitfalls:

Confusing average with RMS or assuming a fixed proportionality between them; forgetting the squared nature of RMS calculation.

Final Answer:

Both decrease (not in the same proportion)