Series circuit reliability computation

[Electric-Light]

If all 100 were in parallel, you can expect to see 50 remaining after 1,000 hours.

From what I've been told, the time it takes for 10% to go out occurs at about 10% relative life sooner with two in series.


How does this calculation work?


How would I apply it in this example?

top is two independent parallel strings of

B10 of 100,000 hrs in series with two B50 of 25,000 in series. Any component failure stops one of the strings.

bottom is B10 of 60,000 hrs in series with four parallel sets of B50 of 36,000 hrs.
Alternatively, it could be set up up as two sets of ballast + 2 parallel lamps.

 

[Sahib]

The failures of ballast and lamp may be assumed to be independent events as a first approximation. The series and parallel circuits may change probabilities assigned to individual components: a filament bulb in parallel circuit with full applied voltage may have higher probability of failure than one in series circuit
with a lower applied voltage. So multiplication of probabilities of individual components of series and parallel circuits will result in two numbers, the lower of which belongs to the circuit with higher reliability. Now how to work out individual probabilities of ballasts and lamps from your data?

 

[wwhitney]

If all 100 were in parallel, you can expect to see 50 remaining after 1,000 hours.

From what I've been told, the time it takes for 10% to go out occurs at about 10% relative life sooner with two in series.

How does this calculation work?

The math is simple if you track the "stays lit" rate R, rather than the failure rate F. For individually wired fixtures, the two rates are related by R = 1 - F.

Call R_n the "stays lit" rate when the fixtures are wired in groups of size n, where the failure of any one light in the group will cause the whole group to go dark. Then with suitable independence assumptions, R_n = R₁ⁿ. This formula just says that a group stays lit only if all of the individual lights in the group are still working.

If R is considered a function of t, then the above equation holds at any given time t. If you are interested in questions like "at what point will 10% of the lights be out", then the answer for n > 1 depends on the shape of R₁(t).

Cheers, Wayne

 

[wwhitney]

If R is considered a function of t, then the above equation holds at any given time t. If you are interested in questions like "at what point will 10% of the lights be out", then the answer for n > 1 depends on the shape of R₁(t).

For example, with n = 2 and a 10% unlit threshold, note that sqrt(0.9) = 0.949. So if you have a graph of R₁, at the time t for which 5.1% of the individually fed lights would have failed, 10% of the groups of size 2 would be unlit.

Cheers, Wayne