Question about battery series connection and battery life

[Jussi]

I have a simple question: My circuit has two 9 volt batteries in series. I have potentiometer to adjust the volts from 9 to 18 volts by resisting it. Will the batteries last as long as two 9 volt batteries in parallel would last when this potentiometer is adjusted to 9 volts?

The voltage adjust works like in this fuzz face circuit but I can change it if needed. http://beckyjc.files.wordpress.com/2010/10/fuzzfacelayout.gif

[R.G.]

Sadly, this seemingly simple question has a simple answer, but its not the one you want. Worse, the next question you will have (i.e. "Why is that?") has a much more complicated answer.

No, two batteries in series will not last as long as two 9V batteries in parallel in this setup.

Why they will not really gets back to two issues: (1) how that resistor adjusts "voltage", and (2) how loads and batteries work. Let's look at the second one first. Circuits pull current from their power supplies in various ways. They can pull a current proportional to the voltage, like a resistor would. They can also pull a constant current for a varying supply voltage, or even more current as the voltage goes up or down, but not proportionately. Which way this happens is dependent on what the circuit does, and is a property of the circuit's action. There is no substitute for knowing what the circuit acts like or how it works to figure this issue out.

That being said, most simple amplifier circuits either pull a constant current from their supply voltage, or act mostly resistive - they pull a current that goes up and down pretty much proportionately with the supply voltage, at least until something breaks or something cuts off.

Which leads us back to the first answer - a "voltage resistor" like shown in the link works by dropping a voltage proportionate to the current pulled by the circuit. So if the circuit current varies, so does the voltage dropped through the resistor. A simple series resistor also adds its resistance to the power supply line, and this can lead to either wanted or unwanted feedback and oscillation if the power at the circuit is not well decoupled. So it's not really a voltage dropping resistor so much as - well, a series resistance that drops a voltage dependent on what the circuit does.

And that leads us back to battery life. Battery life is measured in ampere-hours. Although battery voltage sags, it does not do this proportionate to the life of the battery. A "9V" battery may be 9.5V when fresh, and sag slowly to 7V when it's exhausted. A 9V battery puts out somewhere between 300ma-Hr and nearly 500ma-Hr depending on the internal construction and chemistry. So if you have a 300ma-H battery, and discharge it at 10ma, it will last 30Hr.
(Note: this is also a highly simplified view, but it's a first approximation for people who aren't battery-technical.) If you pull 50ma, it will last 6 hours.

So if your circuit pulls more current as its voltage increases, then having 18V supplied to it instead of 9V might make it pull twice the current. Probably not exactly twice, as the circuit does non-linear things with increasing voltage, but for simple understanding, assume it is about that much. So if you hook it up to 18V, it pulls twice as much current. With the batteries in series, they have identical currents coming out, so the same circuit on 18V, if it pulls twice the current, will make two 9V's in series last half as long as if it were running from one 9V, and using half the current in the circuit.

If the 9V batteries were in parallel, the circuit would be using its "9V" amount of current, but there would be twice as much ampere-hours available from two batteries in parallel, so the two in parallel would last twice as long as a single battery. If the circuit runs a certain time T from one battery, then it will run twice as long from two batteries in parallel as from one. And half as long from two batteries in series.  The "voltage resistor" changes this by reducing the amount of current the circuit uses, so it's not really as bad as a 4:1 change going from 2x9V to 18V, but still, you get shorter useful battery life. 

Whether the "voltage resistor" does useful things in terms of adjusting the sound is another issue.