ok, let me get this straight... you have 5 coils in parallel, and one of those shares a magnetic core with a transformer. the transformer drives a transistor which seems to be interrupting the DC applied across the 5 coils, i'm guessing to give the motor a kick in the butt to keep it moving. the back EMF from the motor is being siphoned off through a diode to charge the battery (which the OP wants to change to a supercap). the battery supplies 12V to the inverter, which generates 240VAC, some of which is the output, some of which drives the transformer, and we get 30V on the other side, which is rectified, and supposedly is going to supply 42 volts to the motor and the battery. so we go from 12V to 42V in the same circuit (i guess that's supposed to insure the supercap/battery stays charged and the motor keeps going).
looks like an overunity machine to me...
ok, lets look at this logically. we take an electric drill to get this thing in motion. we get the rotor moving at 100rpm, and we get up to a 12V charge on the battery/supercap. the inverter produces 240VAC, and to make everything play nicely, the AC is timed to the rotor. on the other side of the transformer we get 30VAC, and the rectifier pumps a pulse into the motor circuit at the right time to give the motor a kick , and then the pulse from the motor's transformer follows up by driving the transistor into conduction. the collapsing magnetic fields around the coils in the motor push against the magnets, so the motor keeps on moving. then the back EMF spikes, and is rectified, and puts more charge on the battery.... except, there's a voltage drop across the diode, and there's current flowing through the diode, so there's a little bit of heat emitted from the diode. working backwards, now, the transistor, when it went into conduction had a slight voltage drop across it and a large current flowing through it, and that generates some heat. the diode that rectifies the 30VAC has a voltage drop across it while there's current flowing through it, and so again theres some heat generated. the windings of the 240V:30V transformer have some resistance, probably 2 to 5 ohms on the 30V side, maybe 20-30 ohms on the primary, plus the iron core's electrons get shoved around a lot by the AC magnetic fields, so in the transformer we 3 sources of heat. the inverter circuit likely has a transformer in it, plus transistors, and other circuits, which all generate some heat. even the supercap/battery has ESR (effective series resistance) which makes for another source of heat. the motor windings also have some resistance, and so they produce some heat. the motor itself has shaft bearings, and the armature has air resistance, all of which create some heat. so, why all this focus on heat? because this machine isn't a completely closed system. when you generate heat, you radiate energy away from the system. once energy is converted to heat, you can't get it back. since heat is being generated while this machine is in motion, the system continues to lose energy, and the machine slows down and eventually stops (and we didn't even connect a load to the output terminals yet). even the wiring betwen components has some resistance, and will convert some of the energy to heat. once that energy is gone, you can't get it back.
when i was a teenager (and there are a lot of people on this board with the same experience), i cooked up a lot of perpetual motion ideas. i only came up with one that might actually keep in motion for a very long time and that would be a huge flywheel in the vacuum of space. eventually it would begin to slow down because space is not a total vacuum. and you could never extract energy from it, because any extraction would slow it down... actually i did once build a device with a very efficient motor, 2F supercap, and a fair sized flywheel. i could spin it up real fast, and it would take 20min to 1/2hr to slow down to a stop. wire resistance in the circuit, and in the motor, plus air resistance on the flywheel eventually drained the energy out of the system. that's why the laws of thermodynamics are so important in understanding how things work. in this internet age, the laws of thermodynamics can also help you keep your money in your wallet, because there's a ton of people out there selling all kinds of "free energy" stuff, and you see this stuff in operation on camera, and it looks so convincing. what you don't see are the things that have been done to make these gadgets work on camera. one guy electrolyzes water, and feeds the hydrogen oxygen mixture into the carburetor of a car, and shuts off the gasoline supply. then he shows an electrolysis cell connected to the alternator in the car, which is making more hydrogen/oxygen mixture. what you don't see is that yes the car is running off of hydrogen/oxygen mixture, but it's from the large electrolysis cell connected to 120V power, not from the little jar connected to the alternator. it takes more energy to dissociate water, than the energy you get back from burning it in the engine.