In order to step up the voltage output from four rechargeable AA batteries from around 5 Volts to the required 12 Volts the following circuit is used.
The key component is the LT1073-12 DC/DC Converter Chip from Linear Technology. Click here for the comprehensive LT1073 Specification Sheet which includes many example circuits using this chip.
This £2 device (the 5V output version LT1073-5 is pictured above) requires just a few off-the-shelf components and operates with supply voltages from 1V to 30V.
The only additional components required to use an LT1073-12 as a 5V-12V DC/DC converter are a 150uH inductor, two 100uF capacitors,and a 50 Ohm Resistor. Total cost of these components is around £1.