# Average output voltage of single-phase thyristor-controlled bridge converter

This question is highly related to this one but I don't find the answers satisfying.

I'm a bit confused about the equation for the DC output voltage of phase-controlled bridge rectifiers.

$V_{o_{ave}} = \frac{V_{m}}{\pi}\int_\alpha^{\pi+\alpha}\sin\theta d\theta = \frac{2V_m}{\pi}\cos\alpha$

$vs.$

$V_{o_{ave}} = \frac{V_{m}}{\pi}\int_\alpha^{\pi}\sin\theta d\theta = \frac{V_m}{\pi}(1+\cos\alpha)$

I see both in books. Apparently they are both valid. So which one is used when? Does it depend on the kind of load? How can the bridge converter still conduct after 180$°$ as indicated in the first equation? I'm just looking for a clear distinction between the two in terms of usage. I'd also very much appreciate a thorough explanation.

• The answer I gave basically suggests that if $\alpha$ doesn't start until virtually 180 degrees then as soon as it triggers it switches off again hence the average voltage must be zero. So if you put $\pi$ into both those formulas, which one gives you the sensible result and, by a process of elimination the other must either be discarded or you must define precisely what $\alpha$ means. – Andy aka Aug 23 '17 at 16:26
• Yes thank you Andy. I think your answer was clear and helpful, but obviously there must be a reason why there are two different equations for the same circuit, and I think it points to the kind of load, as the other answer pointed out. So I need some clarification and elaboration to that as I am not very knowledgeable in waveform mathematics. – DorkOrc Aug 24 '17 at 5:07