Note: This post was hidden for the student only period
The worked solution...
Let us assume for now that the lamps have a fixed resistance and neglect the complex parts of their
impedance.
If we consider a 12V, 75W lamp, then by P = V^2/R, R = 1.92 ohms.
If we also consider a 12V, 35W lamp, then by the same maths, R = 4.11 ohms.
For a series connected string of 9 75W and 1 35W lamps, the total resistance = 9*1.92 + 4.11 = 21.39 ohms.
As they are connected in series, they must have the same
current flowing through them.
That
current can be given by
Ohm's law as V= IR.
So I = 5.609A
This is less than the case of 10 75W lamps, where I = 6.25A, so at first approach, the whole string will be
dimmer.
But that does not tell the full story. We need to examine what happens with the 35W lamp and what happens with each 75W lamp.
In the case of the 75W lamp, we multiply the 5.609A
current by the 1.92
ohm resistance to obtain the
voltage across the lamp. This gives us 10.77V. P=VI, so the
power being dissipated by each "75W" lamp will be 60W. This matches what we predicted about the string being
dimmer.
What is more interesting is to look at the case of the 35W lamp. Using
Ohm's law again, V= IR = 5.609*4.11 = 23.08V. P=VI gives us 129W being dissipated by the "35W" lamp.
So this means that the 35W lamp is running at almost double its rated
voltage. This is a problem...
If we consider the lamp formulas here:
http://www.controlbooth.com/wiki/Collaborative+Articles:Mathematical+Formulas+for+Lighting, and plugging the numbers in, we find that the
lumen output of the lamp will be 913% of the design, and the colour temperate increases by 31%.
But look at the lamp life. It's now a mere 0.02% of the design life.
So far we have only considered the steady state
circuit, but the reality is that all lamps take a little
bit of time to warm up, during which their resistance is much lower than when they warm up.
This thermal
shock will mean that it's a certain bet that the moment this string of lamps is powered up, the 35W lamp will blow. It may fail catastrophically and
throw glass everywhere. Or it may
short circuit and increase the
voltage through the remaining lamps. Or it may just be boring and blow boringly.
For the sake of completeness, the 75W lamps will now have 408% lamp life, 69% lumens output and lose 4.5% of their colour temperature.
While we see that practically you have a blown lamp and that's what matters, as an academic exercise, one should go back and reiterate the maths, because the resistance of a lamp changes depending on it's applied
voltage (mostly because of heat dissipation). So the resistance of the 75W lamps in this case would go down, 1.76 ohms would be the second iteration of the numbers and the 35W lamp resistance will go up, 6.94 ohms would be the second iteration value. At this
point, the
voltage across the 35w lamp is 36.5 volts... By the 3rd iteration it's up to 50V, and you can see how it would continue...
One should also consider the
inductance of the lamp
filament and this too will increase the
impedance, especially at the
switch on peak, but I think we've seen enough to know the results...
So a long explanation of a very simple blown lamp
