I can hear the sounds of the frying pans clanking off in the distance.

I tried to repeatedly to explain to you what the base resistor was for you but you would have none of that.  I tried repeatedly to explain to you that your test that showed a limited amount of brightness change in the LED was just a distant secondary effect of changing the value of the base resistor but you wouldn't have any of that.  Just like the YouTube clip explaining how a Joule Thief works got you all confused, frustrated, and mad, you clearly have no understanding about how a Joule Thief works even though it must have been explained to you at least a dozen times.

Here is your big bamboozle moment:

The chances of TK changing the value of the base resistor in the second circuit to bring the Lux output to the same level as the first circuit while maintaining proper Joule Thief circuit operation are essentially nil.

Here is your big bamboozle moment II:

Okay!  So the supercapcap drops from 1.5 volts to say one volt.  You measure the output impedance of the battery when it also has dropped to one volt driving when the Joule Thief and say for illustrative purposes the output impedance of the battery is measured as being 10 ohms.

Here is where Brad's brain is on fire!

He takes his supercap which is outputting one volt, then adds the series resistor of 10 ohms, and then connects the Joule Thief load.  "We have the technology."

Then he sets the setup off to run, and WHOOPS!, he is not measuring one volt at the Joule Thief now.  He is only measuring 0.85 volts!

What's going on?  Brad says, "I know when my supercap is at one volt I must put a 10-ohm resistor in series.  But then the voltage at the Joule Thief is 0.85 volts."  "I am confused, because I know when my battery voltage is 0.85 volts, the output impedance is 12 ohms and I am supposed to put a 12-ohm resistor in series."

"But I just put a 10-ohm resistor in place but now I have to put a 12-ohm resistor in place??"

The steaming she is a starting, the sizzling sound she is a crackling.  Get your marshmallows out!

The fans from Bizarro World start up a chant, "More discombobulator!  More discombulator!" with the clanking sound of frying pans in the background.

The moral of the story:  Avoid the Logic Discombobulator and think first before you leap into the forum.

Isn't anyone going to analyze the graphs? Oh well....

Both Circuit 1 and Circuit 2 data were taken until the voltage dropped below 0.450 V, which was the predefined endpoint of the trials. Looking at the Lux - Seconds graph we can see something very interesting. While a rough integration using numerical methods shows that Circuit 1 is the _overall_ winner in terms of Lux-seconds, this is only due to the first three minutes of the data. Considering only the data after 210 seconds (corresponding to a voltage of somewhere around 0.9-1 volt), we see that Circuit 2 produces more total light.

In terms of (lux-seconds) per Joule, for the total data, we have :
Circuit 1 produces 8698.8 Lux-seconds of light and uses 11.16 Joules, for an energy efficiency of 779.15 LS/J.
Circuit 2 produces 7483.2 Lux-seconds of light and uses 11.77 Joules, for an energy efficiency of 635.60 LS/J.

But considering only the data from 210 seconds on, we have :
Circuit 1 produces 2035.8 Lux-seconds of light and uses 2.54 Joules, for an energy efficiency of 801.1 LS/J.
Circuit 2 produces 2698.2 Lux-seconds of light ....  but uses 3.82 Joules, for an energy efficiency of 706.9 LS/J.

Please check my math and my reasoning....

Yes, Brad, what you probably aren't realizing is that your model is wrong.

The proper model would be a fixed voltage source in series with a variable resistor, and of course the value of the resistor increases over time to model the battery getting discharged.

That simple model takes care of everything.  The voltage will drop over time under load with this simple model.  That's the model for a battery.

Your incorrect model is a dropping voltage source (the capacitor) in series with a variable resistor.  That model is no good because the lower voltage in the capacitor already represents the voltage drop associated with the impedance, but the actual output impedance is not correct.  Then when you tack on the variable resistor to match the impedance you cause another voltage drop that you don't want.  That new voltage drop represents another impedance and you end up chasing your tail around and around.  <<<  From below:  Or you can write a software control system if you wanted to torture yourself, perhaps some spaghetti code.  Or, you could do a table look-up and dumb it down.  >>>

All that you really need to do is write a simple litte microcontroller program, like in an Arduino.

The Arduino monitors the voltage and the current from the power supply which is set at 1.5 volts.  The program will monitor how much energy has been put into the load.  The microcontroller is connected to a little stepper motor that connects to a 10-turn pot. The 10-turn pot is used as the series output resistance for the 1.5 volt power supply.

So as the energy is delivered to the load the Arduino will adjust the 10-turn pot to emulate the increasing output impedance of the emulated battery.  So with not too much effort you can make a decent little battery emulator that also monitors energy delivered to the load.  You could even load in different battery profiles, regular, alkaline, etc.

Then there are some cat-calls from the Bizarro World supporters.  "Put a stepper motor on the voltage control also!!"  And indeed, if you had perhaps a bench power supply with a voltage control input, you could connect an Arduino D/A channel to the bench power supply.  Then you would hear the clanking from the rabid frying pan crowd.  You now have a Bizarro Whackadoo II self-monitoring power supply with variable voltage output and variable output resistance all under software control.  It can even play an mp3 song at the same time (Or perhaps maybe a software waveform generator perhaps??).

You see Brad, I can invent a project off the top of my head in five minutes that is probably more interesting than anything you have come up with over the past six years.  If I was so inclined I could build it too.

The pen is mightier than the bench.

https://www.youtube.com/watch?v=3zdcMXl3J0Q

And you can be such a complete bimbo sometimes, can't you Brad?  Forget about locking yourself up in a room for one month with four electronics books, how about much longer?

Even if the unloaded voltage of the battery decreases somewhat, this is pretty much junk data and it can be safely ignored.  The simple battery model works just fine.  Presumably you might indeed find differences between the absolute and the differential output impedance of a battery at a given operating point, I haven't really read in major depth about batteries.  However, I am figuring that if there were major differences in the absolute and differential impedance, I would have heard of it.  For sure there are much more complex battery models, but we aren't going there.

What do you think the "battery tester" function is there for on your multimeter?  It's because just measuring the open-circuit battery voltage with the voltmeter is no good.  You have to switch to battery tester mode to put a moderate load on the battery to get a better picture of the voltage drop/state of charge of the battery.

I know that you must know this, and yet you still stated what was quoted above.  It's like Neuron A can't talk to Neuron B sometimes.

  I've found the way to test AA cell is to set on the 10 amp setting, voltage doesn't
  tell you much.
            John.

And you can be such a complete bimbo sometimes, can't you Brad?

Forget about locking yourself up in a room for one month with four electronics books, how about much longer?

Even if the unloaded voltage of the battery decreases somewhat, this is pretty much junk data and it can be safely ignored.

The simple battery model works just fine.

Presumably you might indeed find differences between the absolute and the differential output impedance of a battery at a given operating point, I haven't really read in major depth about batteries.

  However, I am figuring that if there were major differences in the absolute and differential impedance, I would have heard of it.

For sure there are much more complex battery models, but we aren't going there.

What do you think the "battery tester" function is there for on your multimeter?  It's because just measuring the open-circuit battery voltage with the voltmeter is no good.  You have to switch to battery tester mode to put a moderate load on the battery to get a better picture of the voltage drop/state of charge of the battery.

I know that you must know this, and yet you still stated what was quoted above.  It's like Neuron A can't talk to Neuron B sometimes.

So, Circuit 1 ran at an average input power of 90 mW and produced 63.9 lux at the sensor, for an efficiency of 710 lux per Watt.
Circuit 2 ran at an average input power of 40 mW and produced 30.0 lux at the sensor, for an efficiency of 750 lux per Watt.
A second set of results at a lower input voltage of 1.52V:
Circuit 1 gave 49.3 Lux at an average input power of 54.6 mW for an efficiency of 903 Lux/Watt.
Circuit 2 gave 26.1 Lux at an average input power of 28 mW for an efficiency of 932 Lux/Watt.
Operating frequency is between 10 and 11 kHz.

One of the classic weaknesses on the forums is to use the term "efficiency" without even defining what it means.  Brad is someone that does this all the time.

Take a look at a Joule Thief.   Are we talking about electrical power in vs. electrical power out efficiency like Poynt just stated?  Or are we talking about electrical power in vs. light power out like TK just stated?

What about the LED itself?  Are you doing your "burn" at the optimum efficiency point for the LED where you get the most light out per milliwatt in?

How flat or sloped is the current discharge curve across the LED when you are doing a burn?  Does this have an impact on the power in to light out efficiency?

What about the flashing frequency and duty cycle and human perception of brightness?

What about the human perception of the light level?   How do you define an "adequate" level of light output from the Joule Thief?  Is it just bright enough to be a panel indicator light?  Or do you want a practical amount of light like a small night light?  Is there a sweet spot for human perception of the light output from a Joule Thief?

How you define efficiency for a Joule Thief is a big enough question for such a little circuit.  But it is what it is.

Just saying, "Wow, that looks like an efficient Joule Thief!" is essentially meaningless if you don't qualify it.