After spending some time going over the numbers considering using a switching regulator to boost the super capacitor (SC) to 5V from 2.7 on down to 0.9V, it might work. Still may be a bad idea. You won't know until you put all the pieces in place.
Battery Capacity (actual)
Battery Discharge Percent
A typical phone comes with between 1500 and 2000mA hr. That capacity is when the battery is made. From that point on the battery capacity decreases.
Battery Degradation Factors
- Discharge Rate (talk vs. standby, etc)
- Number of Charge Cycles
- Depth of charge
- Battery chemistry (li-cobalt assumed for phones)
Acceptable battery life is 3 years and 500 charge cycles. A charge cycle is from 80% discharged to 100% charged.
After about 250 charge cycles battery capacity drops to around 80%. Less capacity also reduces charge time and SC energy requirements (joules).
The biggest performance factor is Depth of Discharge. Do you let the battery go dead or do you charge when 80% is left?
So the more times you charge and how much it needs to be charged is what governs the amount of energy needed. Reducing the charge percentage will also increase the number of charge cycles.
Energy capacity of SC: 3000F * (2.7V)2 / 2 = 10935 J
The boost regulator operated down to 0.0v which means not all the SC capacity can be used.
SC Unused: 3000F * (0.9V)2 / 2 = 1215 J
Amount SC energy available: 10935 - 1215 = 9720 J
So if you discharge a new 3.6v 2000mA hr battery you will need 25,920 Joules to fully charge battery.
The SC has 9720 / 25,920 = 37.5%
A slightly used battery has 90% capacity left raising SC capacity to 41.67%
If your phone uses a 1500mA hr battery we're up to 55.55%.
If you use a 3400 2.85v Farad SCp (new Maxwell Tech.), you are up to SC capacity 71% of phone charge.
You could discharge the phone down to 29% and the SC will bring it up to 100%.
The above would be if the boost regulator and battery charger were working at 100% efficiency.
The other problem is the charge time. A boost charger that has a 5v output, 0.9v minimum input voltage, generally has a lower output current than what is needed for a phone battery 1C charge rate.
A 1C charge is charging at the battery's 1 hour capacity , e.g. 2000mA hr = 2-3 hrs @ 2 Amp for the first charge stage (about 1 hr) which decrease to 3% after about 3 hrs.
Recommended charge rates are 0.5C to 0.8C.
If the boost regulator can provide 500mA, the charge rate of a new, with no compromises (dead back to 100%), 2000mA hr battery to 0.25C. Or an older 1500mA hr, with 75% capacity left, to about 0.45C.
Far from ideal but depending on your circumstance and how efficient you can convert the cap energy in to charging the phone, it may be doable.
End of Update
A 3000 Farad capacitor can store more than enough energy to charge the batteries. The if i,s can it be done efficiently enough. The rate can it be charged probably will be too long.
Let's say the phone battery is 1500mA Hr. We would need 1.5 Amp for the first hour. A boost regulator would have to operate from 2.7v down to 1V.
Bad, Bad, Bad Idea.
Supercap energy cells are not batteries. They do not hold a charge like a battery.
As an energy cell supercap is used to bridge power gaps lasting from a few seconds to a few minutes. Like to supply energy between power failure and backup generator starting.
I doubt you will want to use a supercap in place of a battery. The specific energy (capacity) of a supercap is 10-50 times less than a Li-ion battery. And they cost a lot more than a battery.
Why it will not work
The super cap will not hold its voltage like a battery. The voltage immediately will begin to drop. It will drop below the voltage necessary to continue charging very quickly. Like where the battery voltage (solid black line) crosses the Supercap's discharge voltage, about a half hour into charging time.
So you need a boost regulator to get the super cap up to the battery charger 4.2V output voltage. The efficiency of the boost will be 60-80%, then the charger will be 80-90% to what's left of the 60-80%. The supper cap will cease to work when hitting the boost converter's Minimum Input Voltage with a 1.5 Amp Output Current.
Charging a one energy cell with another energy cell? Where did you come up with that idea? How about plug your phone into the USB port?
If you need a portable battery for your phone, get a battery.