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The science
The six-pulse bridge
Six thyristors connect three AC phases to a positive and a negative DC rail. At any moment the upper valve of the highest phase and the lower valve of the lowest phase conduct, so the DC voltage is the envelope of the line-line voltages: ripple at six times the supply frequency and a mean Ud0 = (3√2/π) VLL = 1.35 VLL. A valve starts conducting only when it is fired; delaying the gate pulse by the firing angle α after the natural crossing gives Ud = Ud0 cos α, which falls to zero at 90 degrees and goes negative above it: the same bridge then inverts, moving power from DC to AC.
Commutation overlap
Transformer leakage reactance Xc stops the current from jumping between phases. During the overlap angle μ the outgoing and incoming valves conduct together and the rail voltage is their average. Solving the commutation gives cos α - cos(α + μ) = √2 Xc Id / VLL and the voltage drop Ud = Ud0 cos α - (3 Xc/π) Id. The bridge in this page is not evaluated from that formula: it is integrated in time, valve by valve, and the formulas are used only for the checks.
Power factor and reactive power
The fundamental AC current lags the voltage by about φ with cos φ ≈ (cos α + cos(α + μ))/2, so a converter always absorbs reactive power Q = P tan φ, both as rectifier and as inverter. Real stations install large AC filter and capacitor banks for it.
Harmonics and 12-pulse operation
The AC current of a six-pulse bridge contains harmonics of order 6k ± 1 (5, 7, 11, 13...). Two bridges fed by star-star and star-delta transformers are shifted by 30 degrees; connected in series on the DC side they cancel the 5th, 7th, 17th and 19th and leave 12k ± 1. The spectrum graph is computed by a discrete Fourier transform of the simulated current.
The link and its control
The rectifier controls the current: it picks α so that Id = (Udr - Udi) / Rdc equals the order. The inverter holds a constant extinction angle γ: Udi = Ud0 cos γ - (3 Xc/π) Id. Thyristors turn off only when the current falls to zero and then need a recovery time (modelled as a minimum angle γmin of reverse voltage). If a voltage dip or current rise lengthens the overlap so that γ falls below it, the outgoing valve turns on again: a commutation failure, which short-circuits the DC line for a moment and makes the current surge. The DC line obeys Ld dId/dt = Udr - Udi - Rdc Id.
Honest simplifications
Identical transformers and valves, ideal thyristors (no losses or snubbers), balanced sinusoidal AC supplies, a single pole, no AC filters or network impedance, no voltage-dependent current order, one valve model for both stations and a simple lag for the inverter control. Numbers are typical teaching values, not a design of any real project.