Taking apart a solar-powered lamp revealed a single-transistor voltage converter circuit that allowed an LED to be driven from a 1.2 V cell. The l/h diagram shows the circuit (with slight modifications). The circuit oscillates at about 500 kHz and, at a cell voltage of 1.4 V, draws 11 mA with a respectably bright LED. The circuit works down to a supply voltage of 0.8 V.
The oscilloscope shows 3 Vpp at the LED, as expected. The left-hand coil and the capacitor form a series resonant circuit, excited by the collector of the transistor which alternates periodically between conducting and blocking. When the transistor is off the upper coil dumps its stored energy so that the voltage on the collector rises to about double the cell voltage.
A sine wave voltage of 35 Vpp (!) was measured across the capacitor in the resonant circuit. Using a two-channel oscilloscope showed the phase relationships: the resonant circuit shifts the phase by about 90 degrees. The base resistor coupled with the base capacitance and the Miller capacitance (http://en.wikipedia.org/wiki/Miller_ effect) of the transistor add a further phase shift.
The voltage increase obtained using the series resonant circuit can be used to make a bipolar voltage converter, for example to power operational amplifiers (see r/h diagram). Two electrolytic capacitors and two diodes rectify the voltage. The circuit can deliver a voltage difference of 9 V at 0.2 mA, which is enough for a low-power opamp.