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I do not think the connection to BD139 is correct; it is unlikely to work.
I do not understand what you mean?
Do you mean the connection method for BD139 is wrong? or do you mean i have to replace that transistor with another one?
You should modularize the circuit; even if it means more components. You will see that modifications will be easier and maintenance negligible.
1. Lamp driver should have some current control (brightness) and switched on /off with a transistor that works like a switch.
Do you mean change the 4,7 Ohms with a variable resistor?
2. Battery charger with over and under voltage cut-off. It should be permanently connected with trickle charge capability. The battery capacity, the load power consumption and the charger current should be properly evaluated.
True
But i don't know how can i make that?
3. Small LED indicators that show (i) battery charging (ii) replace battery (battery voltage lower than the under voltage cut-off value) (iii) mains power on (iv) battery down and mains power off (how the led will run then)
True
But i don't know how can i make that?
4. I suggest LEDs connected in series and the batteries too be connected in series
When LEDs connected in series the voltage on LEDs will rise and when the batteries connected in series the li-ion battery charger can't be used in this condition and i have to use BMS board?!
.
Let me assume that the power transistor pin labels are accurate. Try to see how the LED can get current at all; It will be easier to understand, for you and also for all of us, if you make a schematic diagram.
Commonly used, base acts as the current pin that controls the flow of current through the collector and emitter. In your circuit, the load (LEDs) is connected via the base pin which is wrong because the current at the base in input and not the other way.
If both transistors are conducting at the same time, the battery will be shorted.
The circuit is wrong.can you explain how the current flow through the transistors?
I believe the lights will still burn all the time according to your last schematic. If you notice the ground and positive from battery are directly connected to your led string. A suggestion would be to take the left side key/switch out and place it on ground on the opposite end of your led string. This could be used as a manual test switch for your leds. I'm also a newbie and I have a lot to learn yet and this thread probably helped me as much as it's helped you.