1) Jerry..... the only way to convert your existing 120 volt Alternating Current supply voltage to use the 12 VDC LED's is with the use of a DC rectifier. I am surprised that the 120 volts AC did not immediately blow out or shatter the 12 VDC rated LED tubes?
2) AC and DC circuits do not mix well and if using one to convert to another type of power source, either a DC rectifier is required to convert from AC to DC or an Inverter is required to convert from DC to AC.
A plug-in DC type of transformer is also a form of a DC rectifier where it converts 120 volts AC to 12 volts DC (similar to a cell phone charger). If using one of these, you would need to purchase such a device with the exact supply current. This would also result in another issue since these types of transformers are plugged into a receptacle. Your ceiling box most likely does not have a receptacle. Even if it did, it would not look appropriate for aesthetic reasons.
3) A DC rectifier can be used to change the output 120 AC volts to 12 VDC but that would be somewhat of a costly solution. The DC rectifier would step-down the voltage to 12 volts and would simultaneously convert it to Direct Current on the output. I would not recommend using the 12VDC tubes, as you have already discovered, they don't work well with a 120 volt AC power source. You would require a DC rectifier to convert the voltage and step it down. This type of solution using 12VDC LED tubes is not economically feasible. Even if using a DC rectifier, these are typically a larger type of equipment and you would need to connect the rectifier to the 120 volt AC branch circuit in the ceiling. The DC rectifier would need to be located in the immediate vicinity such as on top of the cabinets or the countertop.
4) You can purchase 120 volt AC rated LED's from a local electrical supply that will work correctly in your application, thus no need to purchase a rectifier and/or re-design a circuit and do modifications. If converting from the traditional T12 or T8 or T5 to LED fluorescents, the ballast will need to be changed. 120VAC rated LED tubes are the most efficient as compared to even traditional T12, T8 or T5 fluorescent tubes. The only drawback is that 120VAC tubes are more costly.
5) The ballast is not the issue. The mixing of two different types of power inputs is the issue.
Hope this helps.........If you have any additional questions, let me know and I'll be glad to answer them for you.
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