I see your math but I disagree and believe there is an error with it, or best a missing variable...
Using your math your 9 LED flashlight looks like this...
1.25/9=0.139 approximate impedance of 9 LEDs
0.139 + 0.642 = 0.781 add in battery impedance
1.1 / 0.781 = 1.41 Amps
About 156mA an LED
Can you verify if said flashlight is drawing 1.41 Amps? My guess is closer to 800mA...
To verify my guess I just tested one of my cheap LED flashlights (mine had 6 LEDs) so by your math...
1.25/6= 0.208 approximate impedance of 6 LEDs
0.208 + 0.642 = 0.85 add in battery impedance
1.1 / 0.85 = 1.294 Amps
About 215mA an LED
BUT, when I measure my 6 LED flashlight (fresh batteries and all) it's only drawing 550mA, that is 43% of what your calculation said it would be... That is a HUGE discrepancy and IMO shows a clear fault in the calculation... It also indicates that 'something' else is limiting the current... As I have been stating that is the LED itself... These flashlights do not depend on the battery to limit the current, they simply exploit the fixed voltage and what the LED will pass at that voltage to limit current... It's still a poor way to manage current but it will work, in a calculated shortened life product...
BTW, yes I verified no resistors in the flash light...
My measurements are pretty much fully inline with what I would expect assuming that the LED is actually doing the limiting, not the battery... Look at this chart below of a typical super bright white LED... I measured the 3 AAA batteries in the flashlight they measure at 4.56 volts, I highlighted this point on the chart...
Based on this chart, I would expect each LED in my 6 LEDs flashlight to max out at about 85mA, or 510mA total... Since I actually measured 550mA (only a 7% error) I shows the LEDs to be the true source of the current limiting, in said flashlight...
And just for kicks I tested 1 single LED and 2 LEDs in parallel based on your math... They are 14000mcd bright white, no solid datasheet information...
1.25 + 0.642 = 1.892
1.1 / 1.892 = 581mA (1 LED)
1.25/2 + 0.642 = 1.267
1.1 / 1.267 = 868mA (2 LED)
Those values above will likely instantly pop the LEDs...
Real measurements, LEDs that were quite warm to the touch nearly instantly, so they were being pushed hard...
1 LED = 101mA ~ 17% of what was expected by your math
2 LED = 196mA ~ 23% "
Now as the number of LEDs increase X-fold there will certainly be a point where the batteries internal resistance comes into play and will supersede the existing limiting being done by the LEDs, but that is also the point at which you are drawing so much current the battery is in a rapid terminal overheat self destruct mode due to the high current drain, so there is no stability in such a circuit and it's actually a potential hazard...
Using your math your 9 LED flashlight looks like this...
1.25/9=0.139 approximate impedance of 9 LEDs
0.139 + 0.642 = 0.781 add in battery impedance
1.1 / 0.781 = 1.41 Amps
About 156mA an LED
Can you verify if said flashlight is drawing 1.41 Amps? My guess is closer to 800mA...
To verify my guess I just tested one of my cheap LED flashlights (mine had 6 LEDs) so by your math...
1.25/6= 0.208 approximate impedance of 6 LEDs
0.208 + 0.642 = 0.85 add in battery impedance
1.1 / 0.85 = 1.294 Amps
About 215mA an LED
BUT, when I measure my 6 LED flashlight (fresh batteries and all) it's only drawing 550mA, that is 43% of what your calculation said it would be... That is a HUGE discrepancy and IMO shows a clear fault in the calculation... It also indicates that 'something' else is limiting the current... As I have been stating that is the LED itself... These flashlights do not depend on the battery to limit the current, they simply exploit the fixed voltage and what the LED will pass at that voltage to limit current... It's still a poor way to manage current but it will work, in a calculated shortened life product...
BTW, yes I verified no resistors in the flash light...
My measurements are pretty much fully inline with what I would expect assuming that the LED is actually doing the limiting, not the battery... Look at this chart below of a typical super bright white LED... I measured the 3 AAA batteries in the flashlight they measure at 4.56 volts, I highlighted this point on the chart...
Based on this chart, I would expect each LED in my 6 LEDs flashlight to max out at about 85mA, or 510mA total... Since I actually measured 550mA (only a 7% error) I shows the LEDs to be the true source of the current limiting, in said flashlight...
And just for kicks I tested 1 single LED and 2 LEDs in parallel based on your math... They are 14000mcd bright white, no solid datasheet information...
1.25 + 0.642 = 1.892
1.1 / 1.892 = 581mA (1 LED)
1.25/2 + 0.642 = 1.267
1.1 / 1.267 = 868mA (2 LED)
Those values above will likely instantly pop the LEDs...
Real measurements, LEDs that were quite warm to the touch nearly instantly, so they were being pushed hard...
1 LED = 101mA ~ 17% of what was expected by your math
2 LED = 196mA ~ 23% "
Now as the number of LEDs increase X-fold there will certainly be a point where the batteries internal resistance comes into play and will supersede the existing limiting being done by the LEDs, but that is also the point at which you are drawing so much current the battery is in a rapid terminal overheat self destruct mode due to the high current drain, so there is no stability in such a circuit and it's actually a potential hazard...
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