If you have one winding, it could push a battery to 15v as long as it is a fairly low capacity battery. If it is too big of a battery, you will need multiple power windings.

What battery?

You may be unhappy with the results based on the construction of the battery. It might help if you describe the battery is sufficient detail.

Ill make a few assumptions

your input voltage is similar to your charging battery
Your using a normal lead acid battery and not a gel type
you specific gravity of the acid is correct

Look at the Ah rating of your battery and divide that by 20, that would be your ideal input current draw.

If your input voltage or current is significantly lower than your charging battery you will need a higher current draw, eg more coils

If your Ah rating divided by 20 is significantly above your input current, raise the voltage of your input, thus raising the current too until you get closer to the C20 rate.

This sounds like you are using current to drive your battery but in reality you are creating the right conditions for your battery.

Consider that many battery manufacturers use a slightly lower SG of acid to improve life of the battery at the expense of Ah rating, hence a lower peak voltage. Increasing the SG above the recommended levels gives good results but shortens battery life.

After 6 charge/discharge cycles the battery is unlikely to improve much more.

Once you have the charge and discharge conditions correct for your battery, you often notice surprising results.