Thats a good idea ill look into it now. So basically If i break it into two segments of voltage being 1.25v-7.5V and 7.5V-15. I'll need to have transformer (C.T) that gives 2x9V? and togther 18v? to supply the 15V? Also what about the LEDS? do we need to account for their voltage drop?
Transformers with center tap will be listed in catalogues or online stores as total voltage, with a mention saying they're of center tap variety... for example you may see "18V CT" - that's 18 v AC rms, center tap , two secondary windings of 9v AC rms linked together in the middle.
So 24 AC CT will be two 12 v windings linked in the middle.
Center tap transformers only have three wires in the secondary side.. some allow you to separate the center wire in two to have two separate secondary windings, some don't. You have to visually see the terminals of the transformer to determine that.
Transformers with independent secondary windings have just that, two separate windings. They'll be listed as full voltage, or 2x half voltage.
You can link those in parallel to get more current, or in series to get more voltage, or you can link the middle two terminals to create a three terminal center tap transformer from it.
So again, most transformers are advertised as Voltage RMS - when you convert that to DC voltage, you multiply that value by 1.414.
So a 12v AC rms transformer will have a peak of 1.414 x 12 = 16.9 volts.
But when you use a bridge rectifier to convert to DC, there's a voltage drop on the diodes in the bridge rectifier. A diode drops about 0.4-1.1v depending on how big the bridge rectifier is and how much current goes through it, and in a bridge rectifier two diodes are always on, so you have to always keep in mind, you lose about 2 times the voltage drop on a diode inside the rectifier, about 0.8-2.2v.
So your 16.9 peak voltage becomes 16.9 - 2.2v = 14.7 volts. We're always using the worst case scenario values, you might have better bridge rectifier but we just have to be sure it would work in all cases.
Now, remember this is a PEAK voltage. The rectified voltage will not be a straight, smooth, DC line, it will vary between this peak of 14.7 volts and go down close to 0v for very short amounts of time. This is where the capacitor after the bridge rectifier comes in: it fills up when you get close to peak voltage and then when the voltage from transformer goes down, it fills the gap.
With an infinite size capacitor, you will get a perfect straight DC of about 14.7v but infinite capacitors are impossible. With more realistic values of something like 3300uF-4700uF, the DC voltage will fluctuate by about 1-2 volts, maybe a bit more depending on how much current your circuit uses.
So with this in mind, assuming in worst case scenario the voltage will vary by 2 volts in the capacitor, the linear regulator will at some points only receive about 14.7-2v = 12.7 volts.
The linear regulator itself needs about 2 volts above the output voltage to regulate the output properly, so that's what limits your output voltage ... 12.7v input - 2v drop on regulator = 10.7v ... so a range of 1.25-10v is adequate and safe.
Now with these values, 12.7v DC input , 1.25-10v @ max 1A your maximum wasted power will be (12.7v-1.25v )x1A = 11.45w , the lowest would be (12.7v-10v) x 1A = 2.7w.
Both are below the maximum of about 15 watts, so as long as you have a good heatsink on the chip, it's perfectly safe and good.
One more thing you have to keep in mind... transformers don't always output exactly 12v ac rms (if they're rated for that) ... at low amounts of current, they may output up to 10-15% more.
Let's say you only use 100mA ... in this case the transformer may output 10% more, or about 13v AC rms ... that converts into over 18 volts and even with the drops in the bridge rectifier you may have a voltage after the bridge rectifier of over 16v.
If you don't know this, you may say "well, 14.7v dc is less than 16v so i'll use 16v capacitors to smooth out the rectified voltage" but you really have to use capacitors rated for at least 25v, just for such cases.
LM317 is a good linear regulator. It's easy. Just use a transformer with smaller voltage like I've explained above.
SMPS are harder to make and not work right on prototyping boards so you'll have to solder them on pcb, get adequate inductors, low voltage drop diodes, it's much harder to get them right.
If you're a beginner like you seem to be, it's a good project and learning experience to make one of these linear power supplies first. You'll learn from everything a lot.
Regarding LEDs... leds are current devices, they don't care about voltage that much as long as it's over a certain value. You need to use a resistor to limit the current going into the led, and the resistor will also drop the voltage on it.
The formula is simple ... let's say your led is rated for 3v 10mA ... at 10mA it would be super bright, so use 5mA to make it light less bright and live longer.
You're going to power it from the input voltage as that won't vary... if your input is about 12v as I did the math above, you have the formula V = I x R so 12v = 0.005 x R so R = 12 / 0.005 = 2400 ohm. You round up to a standard value like 2700 ohm or you can put two 4700 ohm resistors in parallel to get 2350 ohm. It's not a critical value, if it varies by a few hundred ohms, you get 4 or 6 mA of current in the led, still below the maximum of 10mA.