Single-phase Full-wave Midpoint Controlled Rectifier
This document explains the step-by-step derivation of average and RMS voltage for a single-
phase full-wave midpoint controlled rectifier (from Bimbra Power Electronics).
1. Transformer Secondary Voltages
From the transformer secondary winding (center tap = n):
v_an = V_m sin(ωt)
v_bn = -V_m sin(ωt)
Therefore, v_ab = v_an - v_bn = 2V_m sin(ωt).
2. Output Voltage Waveform
The output voltage across the load is given by:
v_o(ωt) = V_m sin(ωt), for α ≤ ωt < α+π
v_o(ωt) = 0, otherwise.
Thus conduction occurs over π radians, controlled by firing angle α.
3. Average DC Voltage
The average DC output voltage is:
V0 = (1/2π) ∫[α to α+π] V_m sinθ dθ
= (V_m/2π)[-cosθ] from α to α+π
= (2V_m/π) cosα
Hence, average DC voltage depends on firing angle α.
4. RMS Voltage
The RMS value is defined as:
Vrms = sqrt( (1/2π) ∫[α to α+π] (V_m sinθ)^2 dθ )
Expanding:
Vrms^2 = (V_m^2 / 2π) ∫[α to α+π] sin^2θ dθ
= (V_m^2 / 2π) * (π/2)
= V_m^2 / 4
Therefore:
Vrms = V_m / 2
Note: RMS is independent of firing angle α.
5. Physical Intuition
(i) RMS depends on the square of sine over an interval of π radians, which always integrates
to π/2 regardless of α.
(ii) Average value depends on the cosine term, hence it decreases as α increases.
πAverage DC voltage: V0 = (2V_m/α) cosα
✔ RMS voltage: Vrms = V_m / 2 (independent of α)
